tese_silo

KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT INGENIEURSWETENSCHAPPENDEPARTEMENT BURGERLĲKE BOUWKUNDEKasteelpark Arenberg 40, B-3001 Leuven

FLUID-STRUCTURE INTERACTION APPLIED TOFLEXIBLE SILO CONSTRUCTIONS

Promotoren: Proefschrift voorgedragen totProf. dr. ir. G. De Roeck het behalen van het doctoraatProf. dr. ir. G. Degrande in de ingenieurswetenschappen

door

David Dooms

Februari 2009

KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT INGENIEURSWETENSCHAPPENDEPARTEMENT BURGERLĲKE BOUWKUNDEKasteelpark Arenberg 40, B-3001 Leuven

FLUID-STRUCTURE INTERACTION APPLIED TOFLEXIBLE SILO CONSTRUCTIONS

Jury: Proefschrift voorgedragen totProf. dr. ir. D. Vandermeulen, voorzitter het behalen van het doctoraatProf. dr. ir. G. De Roeck, promotor in de ingenieurswetenschappenProf. dr. ir. G. Degrande, promotorProf. dr. ir. M. Baelmans doorProf. dr. ir. J. Vierendeels

(Universiteit Gent) David DoomsProf. dr. ir. B. Blocken

(Technische Universiteit Eindhoven)Prof. dr. rer. nat. M. Schafer

(Technische Universitat Darmstadt)

U.D.C. 519.6:551.55:624.042.4:624.954

Februari 2009

© Katholieke Universiteit Leuven – Faculteit IngenieurswetenschappenArenbergkasteel, B-3001 Leuven (Belgium)

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D/2009/7515/19ISBN 978-94-6018-036-1

Voorwoord

De afgelopen zeven jaren aan de afdeling bouwmechanica zĳn voor mĳ bĳzonderleerrĳk en verruimend geweest. Vandaag rolt het resultaat van al dit werk uit deprinter. Hoewel mĳn onderzoek vaak uit individueel werk bestond, droegen tochvele mensen een steentje bĳ. Ik wil hen dan ook graag bedanken voor hun bĳdrage.

Vooreerst gaat mĳn dank uit naar mĳn promotoren, Guido De Roeck en GeertDegrande. Zĳ boden mĳ de mogelĳkheid om dit doctoraat aan te vatten en hebbenme gedurende de afgelopen jaren opgevolgd en ondersteund. Ze gaven me dekansen en de vrĳheid om mĳn weg te zoeken binnen mĳn onderzoeksdomein enstonden steeds met nuttige tips en raad klaar. Teksten voor publicaties werdennauwgezet nagelezen. Met hun steun heb ik dit doctoraat tot een goed eindegebracht. Ook op menselĳk vlak kon ik steeds bĳ hen terecht.

Naast mĳn promotoren wil ik Tine Baelmans en Jan Vierendeels bedanken voorhun nuttige inbreng als lid van mĳn begeleidingscommissie. Verder dank ik ookBert Blocken voor de interessante discussies en om deel uit te maken van mĳnjury. I like to thank Michael Schafer for giving me the opportunity to stay duringa short period at the TU Darmstadt and for his willingness to be a member of myjury. Dirk Vandermeulen dank ik voor het vervullen van de voorzitterstaak.

Meer dan vĳf jaar heb ik de bureau gedeeld met Mattias Schevenels. De sfeer was ersteeds plezant en we hadden vele discussies over onderwĳs, onderzoek, architectuurof gewoonweg het leven zoals het is.

Ook de andere collega’s van de afdeling bouwmechanica wil ik bedanken voor degoede werksfeer: Danielle, Johan, Geert, Anne, Lincy, Serge, Ralf, Kathleen, Daan,Stĳn, Edwin, Hamid, Shashank, Jaime, Ozer, Ali, Kai, Eliz-Mari, Bram, Amin,Saartje, Suzhen, Hans en de vele buitenlandse bezoekers. Ik beleefde vele leukemomenten tĳdens het werk, op onze fietstochtjes aan de zee, op de verschillendebouwkunde-uitstappen en tĳdens de kerstfeestjes en etentjes met de afdeling. Metplezier denk ik terug aan de meetcampagne met de hoogtewerker op de silo teAntwerpen.

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ii VOORWOORD

Ik heb gedurende de afgelopen jaren mogen deelnemen aan een aantal interessantebuitenlandse congressen en workshops, waarvan spĳtig genoeg slechts enkele ingezelschap van collega’s. De spannende momenten van het te korte congres opStantorini zal ik - en waarschĳnlĳk ook Geert en Stĳn - niet snel vergeten. Meerontspannen ging het eraan toe bĳ onze verkenning van de steegjes van Venetie. Ookons bezoekje met de afdeling aan Geert in Parĳs behoort tot de beste herinneringendie ik overhoud.

Ik wil ook graag enkele mensen bedanken die vooral onrechtstreeks bĳgedragenhebben tot deze thesis. Mĳn ouders bedank ik voor de steun die ik altĳd hebmogen ervaren tĳdens mĳn studies en gedurende mĳn doctoraat. Stan en Lieseverwelkomden mĳ dagelĳks bĳ mĳn thuiskomst van het werk enthousiast met eenportie glimlachjes: alle zorgen waren er op slag door vergeten! Kristien stondsteeds voor mĳ klaar. Jouw bezorgdheid, steun en stimulans zĳn voor mĳ goudwaard.

David Dooms,Leuven, februari 2009

Fluıdum-structuur interactietoegepast op flexibelesiloconstructies

Inleiding

Het gedrag bĳ windbelasting van bouwkundige constructies die vrĳ flexibel zĳn,een complexe vorm hebben of in de nabĳheid van een ander gebouw liggen, ismoeilĳk te bepalen. Het gebrek aan kennis over wind-structuur interactie-effectenheeft in het verleden tot een aantal rampen geleid: in 1940 stortte de TacomaNarrows brug (Washington, US) in ten gevolge van flutter ; in 1965 stortten drievan een groep van acht koeltorens in te Ferrybridge (UK).

Tĳdens de storm op 27 oktober 2002 traden ovaliserende trillingen op bĳverschillende lege silo’s van een groep van veertig silo’s in de haven van Antwerpen(figuur 1). De verplaatsingen waren naar schatting van de grootte-orde vanmeerdere centimeters. Het ovaliseren is een aero-elastisch fenomeen waarbĳ denegatieve aerodynamische demping de structurele demping opheft. Gelĳkaardigevoorvallen in Duitsland en Schotland duiden erop dat de stormschade zich vooralop de hoeken van de groepen voordoet.

Om deze fenomenen te bestuderen is vandaag de dag de gebruikelĳke aanpak eenreeks windtunneltesten uit te voeren. Het doel van deze thesis is om gekoppeldenumerieke simulaties te gebruiken om dit gedrag te voorspellen. De verschillendenumerieke methodes die toegepast worden in de verschillende velden (fluıdumen structuur) en in de koppelingsprocedure worden verkend. Aangezien dezethesis binnen de onderzoeksgroep het eerste werk vormt dat gebruik maakt vanComputational Fluid Dynamics (CFD) en Fluid-Structure Interaction (FSI), is eenvan de objectieven om in ieder hoofdstuk een uitgebreid en coherent overzicht vande desbetreffende theorie te geven.

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Figuur 1: De silo groep in de haven van Antwerpen.

Dit werk focust op de simulatie van wind-structuur interactiefenomenen dieoptreden bĳ schaalconstructies. Het ovaliseren van de silo’s zal als voorbeelddoorheen de tekst gebruikt worden. Voor dit probleem worden nauwkeurigemodellen voor het fluıdum en de structuur opgesteld en ovaliserende trillingenvoor een silo gesimuleerd. In toekomstig werk kan de invloed van de smalletussenafstand tussen twee naburige silo’s op het optreden van ovaliserendetrillingen onderzocht worden en veranderingen aan het ontwerp voorgesteld wordenom het optreden van het fenomeen in de toekomst te vermĳden.

Methodologie

De methodes om wind-structuur interactiefenomenen die optreden bĳ schaal-constructies te simuleren worden geıntroduceerd aan de hand van het voorbeeldvan door de wind geınduceerde ovaliserende trillingen van silo’s. Ditdynamisch fluıdum-structuur interactieprobleem wordt voldoende representatiefbeschouwd voor de beoogde problemen. Hoewel het een eenvoudige cilindrischegeometrie betreft, komen alle specifieke aspecten van een wind-schaalstructuurinteractieprobleem aan bod: geometrisch niet-lineaire vervormingen van eenschaalstructuur, een turbulente windstroming met separatie en tenslotte het zelf-geexciteerd fenomeen dat slechts numeriek bestudeerd kan worden met methodesdie de interactie tussen het fluıdum en de structuur in rekening brengen. Degebruikte methodologie is algemeen en kan eveneens toegepast worden op deandere voorbeelden uit de inleiding.

Voor de numerieke simulatie van fluıdum-structuur interactie wordt eengepartitioneerde methode gebruikt: het fluıdum en de structuur worden

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beschouwd als geısoleerde velden die afzonderlĳk opgelost worden. Deinteractie-effecten worden aan de individuele velden opgelegd als uitwendigerandvoorwaarden die uitgewisseld worden tussen de velden. Het voornaamstevoordeel van deze gepartitioneerde aanpak is dat gevestigde discretisatietechniekenen oplossingsalgoritmes (en bĳgevolg bestaande software) ingezet kunnen wordenin de individuele velden. Deze technieken kunnen in hoge mate aangepast zĳnaan het karakteristieke gedrag van dit veld. Aangezien deze thesis binnen deonderzoeksgroep het eerste werk vormt dat gebruik maakt van CFD en FSI, iser vanaf het begin bewust voor gekozen om gebruik te maken van bestaandesoftware: Ansys (Ansys, 2005b) voor de structuurberekening en Flotran (Ansys,2005b) en CFX (Ansys, 2005a) voor de stromingsberekeningen. Bĳgevolg waren dekeuzes wat betreft algoritmes en modellen beperkt tot wat beschikbaar is in dezeprogramma’s (appendix A). Aangezien de stromingsvergelĳkingen (bv. de Navier-Stokes vergelĳkingen) niet-lineair zĳn en het niet-lineair gedrag van de structuurin rekening gebracht moet worden, wordt directe tĳdsintegratie gebruikt in beidevelden.

Alvorens de studie van het fluıdum-structuur interactiefenomeen aan te vatten, iseen grondige kennis van de individuele velden, nl. fluıdum en structuur, nodig.Nauwkeurige modellen zĳn vereist voor deze velden.

De structuur wordt gediscretiseerd aan de hand van de eindige-elementen methodeen geıntegreerd in de tĳd met het Newmark schema. De formulering brengtgeometrisch niet-lineair gedrag in rekening en laat grote rotaties en groteverplaatsingen toe. Vermits de reele waarde van de structurele modale dempingvan groot belang is voor het voorspellen van ovaliserende trillingen, is een insitu experiment uitgevoerd om deze te bepalen. De gemeten eigenfrequenties enmodevormen worden gebruikt om een drie-dimensionaal eindige-elementenmodelvan de silo te valideren.

Voor het fluıdum wordt een eindige-volumediscretisatie toegepast. De oplossingwordt geıntegreerd in de tĳd met de drie-punts-achterwaartse differentiemethode.De twee-dimensionale stroming rond een starre cilinder wordt berekend. Het effectvan de groepsopstelling op de stroming wordt bestudeerd voor een groep starrecilinders zoals in de haven van Antwerpen.

Tĳdens de FSI berekeningen wordt het rooster van het fluıdum gealigneerd met devervormende rand van de structuur. De stromingsberekeningen worden uitgevoerdop een vervormend rooster en de vergelĳkingen worden daarom geformuleerd in eenarbitraire Lagrangiaanse-Euleriaanse (ALE) beschrĳving. De vervorming van hetrooster wordt berekend aan de hand van een diffusievergelĳking met een variabelediffusiviteit en moet er bĳ voorkeur voor zorgen dat de kwaliteit en de verfijningenvan het rooster behouden blĳven.

Tussen Ansys en Flotran, en Ansys en CFX is een koppelingsalgoritme commercieelbeschikbaar. Twee gepartitioneerde algoritmes worden gebruikt: een zwak

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gekoppeld serieel staggered algoritme dat elk veld een maal oplost per tĳdstapen een sterk gekoppeld algoritme dat itereert tussen de twee velden binnen eentĳdstap. De roosters van het fluıdum en de structuur zĳn niet samenvallend aan deinterface. Een consistente interpolatie wordt gebruikt voor de overdracht van deverplaatsingen, terwĳl een knoop-naar-element conservatieve methode toegepastwordt bĳ de overdracht van de belastingen. De interactie tussen de windstromingen een silo wordt berekend.

Originele bĳdragen van de thesis

Een drie-dimensionaal eindige-elementenmodel van een cilindervormige silo isgevalideerd aan de hand van in situ metingen op een silo. De radiale versnellingenten gevolge van de heersende windbelasting zĳn gemeten in 10 punten op desilo. Aan de hand van de stochastische systeemidentificatietechniek zĳn demodale parameters bepaald op basis van uitgangssignalen. De invloed van derandvoorwaarden op de eigenfrequenties en modevormen is onderzocht.

Er wordt een overzicht gegeven van de huidige state-of-the-art methodes voor desimulatie van turbulente stromingen in functie van hun performantie en de vereisterekenkracht. De invloed van turbulentiemodellen, wandmodellen en niet-stationairgedrag is bestudeerd voor de twee-dimensionale stroming rond een cilinder in hetpostkritische regime. De resultaten worden vergeleken met experimentele data ennumerieke resultaten uit de literatuur. De twee-dimensionale stroming rond eengroep van 2-bĳ-2 en van 8-bĳ-5 cilinders toont dat er een grote invloed is van degroepsopstelling op de drukverdeling rond de cilinders.

Er wordt een coherent en uitgebreid overzicht gegeven van alle technieken vereistvoor de simulatie van dynamische fluıdum-structuur interactieproblemen. Devergelĳkingen voor de stroming in een Lagrangiaanse-Euleriaanse beschrĳvingzĳn afgeleid. De gevolgen van deze beschrĳving voor de nauwkeurigheid en destabiliteit van de tĳdsintegratie worden besproken. De verschillende methodes omde snelheid van de roosterpunten te berekenen worden toegelicht. Het gekoppeldeprobleem wordt opgelost via een gepartitioneerd algoritme. In het geval vanonsamendrukbare stromingen worden de nauwkeurigheid, stabiliteit en efficientievan zwak en sterk gekoppelde algoritmes besproken. De verschillende methodesvoor de overdracht van krachten en verplaatsingen tussen niet-samenvallenderoosters worden geevalueerd aan de hand van hun nauwkeurigheid en van de matewaarin de behoudswetten gerespecteerd worden.

Als praktische toepassing wordt de interactie tussen de windstroming en eencilinder berekend. Het drie-dimensionale eindige-elementenmodel van de silo isgekoppeld met de drie-dimensionale onsamendrukbare turbulente stroming omovaliserende trillingen te voorspellen.

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Structureel gedrag van de silo

Het structureel gedrag van de silo in de haven van Antwerpen wordt bestudeerd.De silo’s zĳn cilindrische schaalconstructies met een diameter van 5.5 m en eenhoogte van 25 m. Een cilinder bestaat uit 10 aluminium platen met een hoogtevan 2.5 m en een dikte die afneemt met de hoogte van 10.5 mm aan de onderkantnaar 6 mm aan de bovenkant.

Vermits de reele waarde van de structurele modale demping van groot belangis voor het voorspellen van ovaliserende trillingen, is in situ een experimentuitgevoerd om deze te bepalen. Om de meetopstelling af te stemmen op dete verwachten eigenmodes en eigenfrequenties werden deze vooraf berekend meteen harmonisch eindige-elementenmodel. De modevormen bestaan uit m halvesinussen over de hoogte en n sinussen over de omtrek. Terwĳl bĳ balken en platende complexiteit van de modevormen toeneemt bĳ stĳgende eigenfrequenties, is dithier niet het geval. De laagste eigenfrequentie wordt namelĳk teruggevonden bĳm = 1 en n = 4.

De radiale versnellingen ten gevolge van de heersende windbelasting werdengemeten in 10 punten op de silo. Aan de hand van de stochastische systeem-identificatietechniek zĳn de modale parameters (eigenfrequenties, modevormenen modale dempingsfactoren) bepaald. Er werden 60 eigenmodes met eeneigenfrequentie lager dan 20 Hz geıdentificeerd. De modale dempingsfactoren ξvarieren tussen 0.07 % en 1.32 %, wat realistische waarden zĳn voor een gelastealuminium structuur. De eigenmode met de laagste eigenfrequentie (3.94 Hz,m = 1, n = 3 of 4) heeft de grootste bĳdrage in de gemeten respons. Figuur 2toont een bovenaanzicht en een drie-dimensionaal zicht van drie geıdentificeerdeeigenmodes.

De laagste eigenfrequenties van de silo worden sterk overschat door het harmonischeindige-elementenmodel. De eigenfrequenties en modevormen van een cilindrischeschaalstructuur zĳn heel gevoelig voor de randvoorwaarden opgelegd aan de axialeverplaatsingen uz, terwĳl de invloed van de randvoorwaarden voor de rotaties ϕθbĳna verwaarloosbaar is (Forsberg, 1964; Koga, 1988; Leissa, 1993). Hetharmonisch eindige-elementenmodel belemmert de axiale verplaatsingen onderaanover de hele omtrek, terwĳl in realiteit de silo slechts in vier punten over deomtrek vastgebout is aan een achthoekige randbalk. Een drie-dimensionaaleindige-elementenmodel van de silo maakt het mogelĳk de verbindingen met dedraagstructuur correct in te rekenen. Figuren 3b tot 3f tonen een bovenaanzicht eneen drie-dimensionaal zicht van vĳf berekende modevormen. Het bovenaanzichttoont dat eigenmodes (1,5) en in mindere mate ook (1,3) een combinatie zĳnvan ovaliseren en van globale buiging van de silo. De eigenmodes met drie envier sinussen over de omtrek en een halve sinus over de hoogte hebben beide eeneigenfrequentie van 3.93 Hz.

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f = 4.00 Hz f = 4.01 Hz f = 17.95 Hz

ξ = 0.77 % ξ = 0.81 % ξ = 0.10 %

Figuur 2: Bovenaanzicht en drie-dimensionaal zicht van het reeel (volle lĳn) enhet imaginair deel (streeplĳn) van de geıdentificeerde eigenmodes 3, 4en 53 van de silo. De stippellĳn duidt de onvervormde silo aan.

De invloed van geometrisch niet-lineair gedrag op vervormingen van grootte-orde0.1 m (zoals gedurende de storm) is niet verwaarloosbaar: ten opzichte van eenlineaire berekening worden de verplaatsingen groter waar de kromming verminderten kleiner waar de kromming toeneemt. Het geometrisch niet-lineair gedrag wordtbĳ de verdere berekeningen in rekening gebracht.

Turbulente windstromingen rond cilinders

Windstromingen rond gebouwen worden door heel grote Reynoldsgetallengekenmerkt. Om de rekentĳd op een processor beperkt te houden, wordenin dit werk enkel niet-stationaire Reynolds gemiddelde Navier-Stokes (RANS)berekeningen toegepast. Om de invloed van de turbulentiemodellen, dewandmodellen en niet-stationair gedrag te verduidelĳken, wordt de twee-dimensionale stroming rond een cilinder in het postkritische regime bĳ een

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a.Model b.(1, 3) c.(1, 4) d.(1, 5) e.(1, 6) f.(1, 2)

3.93 Hz 3.93 Hz 5.25 Hz 7.37 Hz 7.75 Hz

Figuur 3: Drie-dimensionaal eindige-elementenmodel van de silo en een aantalgeselecteerde eigenmodes.

Reynoldsgetal 12.4 × 106 berekend. De stroming rond een cilinder is eenideale test omdat het verschillende stromingspatronen combineert zoals separatie,recirculatie, stagnatie en kromming van de stroomlĳnen. Bovendien is het eenvereenvoudiging van de stroming rond een silo. De resultaten worden vergelekenmet experimentele data en numerieke resultaten uit de literatuur. In Flotranbeınvloedt het al dan niet gebruiken van wandfuncties de resultaten in hogemate, terwĳl de resultaten bekomen met CFX hiervoor vrĳ ongevoelig zĳn.Resultaten berekend met verschillende turbulente-viscositeit-modellen vertonengrote verschillen, maar de minimale drukcoefficient is steeds onderschat en debasisdrukcoefficient steeds overschat. Voor de stroming rond een cilinder bĳ dezeReynoldsgetallen bekomt het Shih-Zhu-Lumley turbulentiemodel in Flotran debeste overeenkomst met de experimentele resultaten. In CFX levert het shearstress transport turbulentiemodel de beste resultaten op. Een niet-stationaireberekening met het shear stress transport turbulentiemodel laat toe de regelmatigewervelafscheiding in het postkritische regime te berekenen en wordt daaromverkozen boven een stationaire.

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Figuur 4 vergelĳkt de drukcoefficient van een stationaire berekening met detĳdsgemiddelde drukcoefficient Cp van een niet-stationaire berekening. Demaximale en minimale drukcoefficient gedurende de niet-stationaire berekeningzĳn eveneens aangeduid. Vergeleken met de stationaire resultaten, voorspelthet tĳdsgemiddelde van de niet-stationaire resultaten een lagere minimum

drukcoefficient Cmin

p en een lagere basisdrukcoefficient Cb

p. Vermits hettĳdsgemiddelde van de niet-stationaire berekening duidelĳk verschilt van destationaire oplossing, is de niet-stationaire berekening fysisch gezien meer correcten biedt als voordeel extra informatie over de tĳdsvariatie van de druk (Iaccarinoet al., 2003). Daarom worden enkel niet-stationaire berekeningen uitgevoerd voorde stroming rond een groep cilinders.

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Figuur 4: Drukcoefficient als functie van de hoek θ bekomen met een stationaireberekening (streeplĳn) en het tĳdsgemiddelde (volle lĳn), het minimum(streep-stippellĳn) en het maximum (stippellĳn) van de drukcoefficientbekomen met een niet-stationaire berekening voor (a) rooster A2 en (b)rooster B. De lichtgrĳze zone bevat alle beschikbare experimentele databĳ Reynoldsgetallen van 0.73× 107 tot 3.65× 107.

In de haven van Antwerpen zĳn de silo’s opgesteld in vĳf rĳen van acht silo’smet tussenruimtes van 30 cm tussen twee naburige silo’s. De twee-dimensionalestroming rond een groep van 2-bĳ-2 en van 8-bĳ-5 dicht bĳ elkaar staande cilinderstoont dat er een grote invloed is van de groepsopstelling op de drukverdeling rondde cilinders. De resultaten voor de groep van 8-bĳ-5 worden hier kort besproken.

Figuur 5 toont het tĳdsverloop en de frequentie-inhoud van de druk op hetoppervlak van een cilinder ter plekke van een smalle tussenruimte. Wervels wordenafgescheiden van de groep als geheel bĳ 0.165 Hz en van de individuele cilindersbĳ 2.85 Hz. De eerste frequentie komt overeen met een Strouhalgetal van 0.24, alsde geprojecteerde breedte 45.89 m van de groep gebruikt wordt als karakteristiekelengte, terwĳl de tweede frequentie een Strouhalgetal geeft van 0.49 als de cilinder

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diameter gebruikt wordt als karakteristieke lengte. Dit strookt met de hogereStrouhalgetallen die gemeten worden in compacte buizenbundels (Blevins, 1990).

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(b)

Figuur 5: (a) Tĳdsverloop en (b) frequentie-inhoud van de druk op het oppervlakvan een cilinder in de groep van 8-bĳ-5 cilinders.

Figuur 6 toont de tĳdsgemiddelde drukcoefficienten voor de groep. De streeplĳngeeft ter vergelĳking de drukcoefficient rond een alleenstaande cilinder weer. Degroepsopstelling verandert de drukverdeling rond de cilinders drastisch. Destuwpunten worden weg van de groep verschoven bĳ cilinders 2 tot 8. Dezone met positieve druk verkleint bĳ deze cilinders. Voor cilinders 9, 17, 25en 33 verplaatst het stuwpunt zich in de richting van de stroomopwaarts gelegencilinder. De zone met positieve druk is uitgebreid voor cilinders 1, 9 en 17. Deminimum drukcoefficient die optreedt vlak voordat de grenslaag loslaat, neemtvan cilinder 1 naar cilinder 8 toe. Cilinders 8 en 33 ondervinden heel grote zuigingvlak voordat de grenslaag loslaat. De hoge snelheden in smalle tussenruimtesveroorzaken uitgesproken minima in de drukverdeling op de plaats waar detussenruimte het smalst is. Alle cilinders die zich in het zog van andere cilindersbevinden, hebben twee maxima op de posities waar de stroming terug aanhechtaan de wand. Hoe meer stroomafwaarts de cilinders zich bevinden, hoe lager debasisdrukcoefficient. Sommige stroomopwaarts gelegen cilinders ondervinden eenpositieve basisdrukcoefficient.

De twee cilinders op de hoeken opzĳ van de groep ondervinden vrĳ grotehefcoefficienten. Eigenmodes met vier sinussen over de omtrek worden zwaarderbelast in de groepsopstelling dan in het geval van een alleenstaande cilinderomdat iedere cilinder omringd wordt door 4 cilinders. Vooral voor de cilindersop de hoeken opzĳ van de groep worden de eigenmodes met drie of vier sinussenover de omtrek sterk geexciteerd. Deze eigenmodes hebben vaak de laagsteeigenfrequenties. Dit verklaart waarom stormschade hoofdzakelĳk optreedt bĳhoeksilo’s.

xii SAMENVATTING

←

1 9 17 25 33

2 10 18 26 34

3 11 19 27 35

4 12 20 28 36

5 13 21 29 37

6 14 22 30 38

7 15 23 31 39

8 16 24 32 40

Figuur 6: Tĳdsgemiddelde drukcoefficienten Cp voor de stroming rond een groepvan 8-bĳ-5 cilinders (volle lĳn) en voor een alleenstaande cilinder(streeplĳn). De pĳl duidt de invalshoek van de wind aan.

SAMENVATTING xiii

Gekoppelde berekening van de windstroming rond eensilo

Het eindige-elementenmodel van de structuur wordt gekoppeld met de drie-dimensionale onsamendrukbare turbulente windstroming. Tussen de structuur eneen cilinder met als diameter twee maal de diameter van de silo wordt de stromingberekend op een vervormend rooster op basis van de arbitraire Lagrangiaanse-Euleriaanse beschrĳving (sectie 2.2.1). De verplaatsingen van de roosterpuntenworden berekend aan de hand van een diffusievergelĳking met een variabelediffusiviteit. Om de kwaliteit en de verfijningen van het rooster te behouden,wordt de diffusiviteit gelĳk gekozen aan het omgekeerde van het volume van deeindige volumes.

De roosters van het fluıdum en de structuur zĳn niet samenvallend aan deinterface. Een consistente interpolatie wordt gebruikt voor de overdracht van deverplaatsingen, terwĳl een knoop-naar-element conservatieve methode toegepastwordt bĳ de overdracht van de belastingen.

Als beginvoorwaarden worden de onvervormde structuur en de niet-stationairestroming rond een starre cilinder gekozen. De structurele demping wordt bepaaldaan de hand van de gemeten modale dempingsfactoren en toegevoegd onder devorm van Rayleigh demping. De structuur en het fluıdum worden sequentieelgekoppeld met twee van de beschikbare algoritmes: een zwak gekoppeld staggeredalgoritme (A) dat elk veld een maal oplost per tĳdstap en een sterk gekoppeldalgoritme (B) dat itereert tussen de twee velden binnen een tĳdstap. Dezeiteraties zorgen ervoor dat het evenwicht op de interface op ieder tĳdstip voldaanis. De overgedragen verplaatsingen en krachten worden niet gerelaxeerd. Er zĳnmaximaal vier iteraties vereist om de relatieve verandering van de overgedragengrootheden te beperken tot 0.001.

Figuur 7a vergelĳkt het tĳdsverloop van de radiale verplaatsingen berekend met dealgoritmes A en B in drie punten op halve hoogte. Binnen dit korte tĳdsintervalzĳn beide algoritmes stabiel. De verschillen tussen de resultaten van algoritmeA en B worden duidelĳk groter na verloop van tĳd. De nauwkeurigheid van hetzwak gekoppelde algoritme is lager dan die van het sterk gekoppelde algoritme.Figuur 7b toont de frequentie-inhoud van de radiale verplaatsingen berekend metalgoritme B. De resolutie ∆f = 0.4 in het frequentiedomein is vrĳ laag omwillevan het korte tĳdsinterval. De respons van de silo wordt gedomineerd door deeigenmodes (1,3) en (1,4) rond 4 Hz. De piek bĳ 2 Hz wĳst op de effecten vande wervelafscheiding op de structuur. De kleinere pieken boven 4 Hz horen bĳeigenmodes met hogere eigenfrequenties.

De vervormingen van de structuur beınvloeden de drukverdeling rond de structuur:bĳ de windstroming rond een starre silo waren er in het tĳdsverloop van de druk

xiv SAMENVATTING

5.5 6 6.5 7 7.5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time [s]

Dis

plac

emen

t [m

]

(a)0 2 4 6 8 10

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Frequency [Hz]

Dis

plac

emen

t [m

/Hz]

(b)

Figuur 7: (a) Tĳdsverloop van de radiale verplaatsingen op halve hoogte voorθ = 66 ( ), θ = 120 ( ) en θ = 180 ( )berekend met algoritme A(streeplĳn) en algoritme B (volle lĳn) en (b) frequentie-inhoud van deradiale verplaatsingen berekend met algoritme B. De hoek θ = 0 valtsamen met het stuwpunt.

voornamelĳk bĳdrages bĳ 2.16 en 4.31 Hz, terwĳl er bĳ de gekoppelde berekeningeveneens bĳdrages zĳn bĳ hogere frequenties. De belangrĳke bĳdrage rond 4 Hzkomt overeen met de eigenfrequenties van de eigenmodes (1,3) en (1,4). Door deinteractie is de grootte van de drukvariaties bĳ 2 Hz duidelĳk toegenomen, watduidt op de versterking van de wervelafscheiding.

Aangezien de structuur plots belast wordt en de structurele demping vrĳ laag is,moet de respons gedurende een langere periode berekend worden om te evaluerenof de ovaliserende trillingen geleidelĳk uitgedempt worden of integendeel versterktworden. De rekentĳd voor deze berekeningen op een computer met een processoris vrĳ lang.

Suggesties voor verder onderzoek

In deze tekst wordt de stroming op vervormende rekenroosters berekend methet commercieel softwarepakket CFX. De nauwkeurigheid en de stabiliteit vanstromingsberekeningen op vervormende roosters zou geevalueerd kunnen worden.Ten eerste kan gecontroleerd worden of de tweede-orde nauwkeurigheid in de tĳdvan het drie-punts-achterwaartse differentieschema behouden blĳft in de ALEbeschrĳving. Bĳvoorbeeld kan de twee-dimensionale stroming rond een cilinderdie vervormt bĳ een bepaalde frequentie met een bepaalde aantal sinussen over deomtrek gesimuleerd worden. Op basis van berekeningen met verschillende groottesvan tĳdstap kan de nauwkeurigheid in de tĳd bepaald worden.

SAMENVATTING xv

Vervolgens kan de implementatie van de geometrische behoudswet gecontroleerdworden door een uniforme stroming te berekenen terwĳl de interne knopen vanhet rooster op een arbitraire manier bewegen. De juiste oplossing moet bekomenworden onafhankelĳk van de vervormingen van het rooster en aan de massabalansmoet steeds voldaan zĳn.

In deze thesis wordt de twee-dimensionale turbulente luchtstroming bestudeerdrond een groep van 8-bĳ-5 cilinders met een tussenafstand van 30 cm voor eeninvalshoek van 30. Een pertinente vraag is hoe gevoelig de berekende resultatenzĳn voor veranderingen van de tussenafstand en van de invalshoek. Om dezeeffecten te begroten, moet een systematische studie van de stroming uitgevoerdworden voor een aantal invalshoeken en tussenafstanden.

De stroming rond een eindige cilinder met verhouding h/D = 4.55 is sterk drie-dimensionaal met hoefijzervormige wervels en boogvormige wervels. Het is teverwachten dat soortgelĳke drie-dimensionale wervelstructuren terug te vinden zĳnrond de groep silo’s als geheel. De numerieke simulatie van de drie-dimensionalestroming rond een groep van veertig dicht bĳ elkaar staande silo’s is een uitdagendprobleem wat betreft rekencapaciteit en kan slechts opgelost worden door deberekeningen te paralleliseren.

Als overgeschakeld wordt op drie-dimensionale berekeningen kunnen de RANSturbulentiemodellen vervangen worden door een hybride RANS/LES aanpak. Dezemethode is beter geschikt om de grootschalige turbulente structuren in het zog vanbouwkundige constructies te berekenen.

In het laatste hoofdstuk is de gekoppelde simulatie van de drie-dimensionaleonsamendrukbare turbulente windstroming rond een cilindervormige silo berekendom het optreden van ovaliserende trillingen te voorspellen. Aangezien de rekentĳdheel lang is, zĳn meer efficiente koppelingsalgoritmes vereist. Het gebruik vande commercieel beschikbare koppeling tussen Ansys en CFX heeft de keuze watbetreft koppelingsalgoritmes beperkt. Een eigen implementatie van de koppelingtussen de twee commerciele programma’s moet het gebruik van meer geavanceerdekoppelingsalgoritmes mogelĳk maken.

De nauwkeurigheid van het conventionele staggered algoritme kan verbeterdworden door geextrapoleerde structurele verplaatsingen op te leggen aan hetfluıdum en de overgedragen fluıdumkrachten overeenkomstig te corrigeren. Alsdit verbeterde algoritme stabiel zou zĳn voor zwak gekoppelde problemen metonsamendrukbare stromingen, kan het gebruik van iteraties binnen een tĳdstapvermeden worden en de rekentĳd substantieel verminderd worden.

Bĳ sterk gekoppelde problemen kan het algoritme met iteraties binnen eentĳdstap verbeterd worden door gebruik te maken van automatisch bepaalderelaxatiefactoren gedurende de iteraties tussen het fluıdum en de structuur. Dezetechniek verbetert de convergentie en reduceert het aantal benodigde iteraties.

xvi SAMENVATTING

Een interessante denkpiste is het aanmaken van gereduceerde orde modellen voorhet fluıdum en de structuur tĳdens de iteraties tussen het fluıdum en de structuur(Vierendeels, 2006). In elke iteratie wordt in het fluıdum op de fluıdum-structuurinterface de drukverdeling berekend die hoort bĳ de opgelegde verplaatsingen.Deze sets van opgelegde verplaatsingen en bĳhorende drukverdelingen kunnengebruikt worden om een gereduceerde orde model van het fluıdum op te stellen.Voor de structuur worden op een analoge manier in iedere iteratie de verplaatsingenop de interface berekend die horen bĳ een opgelegde drukverdeling. Deze setskunnen gebruikt worden om een gereduceerde orde model van de structuur op testellen. De resultaten van een gekoppelde berekening met de beide gereduceerdeorde modellen levert een startwaarde op voor de volgende iteratie. Bĳ elke iteratieworden de gereduceerde orde modellen verbeterd. Deze techniek laat toe om sterkgekoppelde problemen op te lossen met een klein aantal iteraties. De techniekkan eventueel verbeterd worden door de gereduceerde orde modellen opgesteldin de vorige tĳdstap te hergebruiken. Dit kan heel interessant zĳn bĳ aero-elastische problemen zoals flutter, galloping en het ovaliseren van silo’s, waarbĳveel berekeningen gemaakt worden voor bĳna identieke geometrieen.

Een globale multigrid techniek, die de gepartitioneerde koppeling tussen hetfluıdum en de structuur berekent op verschillende roosters, kan de stabiliteit, deefficientie en de nauwkeurigheid verhogen (Schafer et al., 2006; van Zuĳlen et al.,2007). De gekoppelde berekening wordt eerst uitgevoerd op een ruw rekenroostervoor zowel het fluıdum als de structuur. Deze oplossing kan gebruikt worden alsstartwaarde voor een gekoppelde berekening op fijnere roosters. De laagfrequentefouten in de oplossing op deze fijnere roosters kunnen gecorrigeerd worden dooreen nieuwe berekening op het ruwe rooster uit te voeren.

De efficientie, nauwkeurigheid en stabiliteit van de verschillende koppelings-algoritmes kan geverifieerd worden aan de hand van een reeks benchmarks bv. eenelastische balk achter een vierkant (Wall and Ramm, 1998) of achter een cilinder(Turek and Hron, 2006). Bathe and Ledezma (2007) geven een overzicht van eenreeks benchmarks om koppelingsalgoritmes en de overdracht van belastingen enverplaatsingen te evalueren.

De reductie van de dimensies van het probleem kan de rekentĳd eveneensverminderen. In het geval van de silo kan de eindige-strookmethode (appendix B)toegepast worden om een representatief twee-dimensionaal model van de structuurte maken. De structurele verplaatsingen worden verondersteld als een halve sinusof een halve cosinus te varieren over de hoogte. De eindige-strookmethode isgeımplementeerd als de combinatie van een twee-dimensionaal balkelement eneen door de gebruiker toegevoegd element. Een nadeel is dat het door degebruiker toegevoegd element onafhankelĳk is van de vervormingen, waardoorgeometrisch niet-lineair gedrag niet ingerekend wordt voor deze elementen.Dit eindige-strookmodel kan gekoppeld worden met een twee-dimensionalestromingsberekening. Bĳ de overdracht van de winddrukken naar de structuur

SAMENVATTING xvii

moet een aanname gemaakt worden voor de variatie van de drukken over dehoogte. Dit verloop is noch constant, noch sinusoıdaal. Als validatie kunnen deresultaten van een gekoppelde berekening met het eindige-strookmodel en de twee-dimensionale stroming vergeleken worden met de resultaten van de gekoppeldeberekening van de schaalconstructie en de drie-dimensionale stroming. De invloedvan de groepsopstelling op het optreden van ovaliserende trillingen kan vervolgensbestudeerd worden door het eindige-strookmodel van de silo’s te koppelen mettwee-dimensionale stroming rond de groep. Het aantal silo’s in de groep kanhierbĳ best geleidelĳk verhoogd worden, te beginnen met twee dicht bĳ elkaarstaande silo’s.

Hoewel in het voorliggende werk enkel het ovaliseren van silo’s bestudeerdwordt, kan dezelfde methodologie direct toegepast worden bĳ het berekenenvan de respons van zeilconstructies onder windbelasting, van flutter vanbruggen, van galloping van kabels en van trillingen van kabels geexciteerd doorwervelafscheiding.

Organisatie van de tekst

Alvorens de studie van het wind-structuur interactieprobleem aan te vatten, iseen grondige kennis van het gedrag van de structuur en van de stroming vereist.Voor beide deelproblemen zĳn nauwkeurige numerieke modellen nodig. Daarom isveel energie geınvesteerd in de oplossing van deze afzonderlĳke problemen. Dezeworden uiteindelĳk aangewend in hoofdstuk 5 om het ovaliseren te simuleren aande hand van een gekoppelde berekening van de windstroming rond de structuur.

Hoofdstuk 1 situeert het onderwerp en licht de objectieven en eigen bĳdragen toe.Het ovaliseren van cylindrische schaalconstructies wordt ingeleid. Dit dynamischfluıdum-structuur interactieprobleem wordt doorheen de hele thesis als voorbeeldgebruikt.

Hoofdstuk 2 behandelt de arbitraire Lagrangiaanse-Euleriaanse beschrĳving, dievereist is voor de berekening van stromingen in gebieden met veranderendebegrenzingen. De stabiliteit en de tĳdsnauwkeurigheid van deze berekeningenwordt besproken en gerelateerd aan de geometrische behoudswet. De verschillendemethodes voor de berekening van de vervormingen van het rekenrooster wordenbeschreven.

Hoofdstuk 3 beschrĳft de vergelĳkingen voor lineair en geometrisch niet-lineairgedrag van schaalconstructies. In situ is op een silo een experiment uitgevoerdom de eigenfrequenties, eigenmodes en modale dempingsfactoren te bepalen. Aande hand van deze experimentele modale parameters wordt een drie-dimensionaal

xviii SAMENVATTING

eindige-elementenmodel van de silo gevalideerd. Het belang van geometrisch niet-lineair gedrag tĳdens het ovaliseren van de silo’s wordt bestudeerd.

Hoofdstuk 4 behandelt de berekening van turbulente windstromingen rond starrebouwkundige constructies. Bestaande turbulentiemodellen worden besproken.Om een geschikt turbulentiemodel te selecteren, worden de resultaten van eenstationaire en een niet-stationaire Reynolds gemiddelde Navier-Stokes berekeningvan de stroming rond een cilinder vergeleken met experimentele data en metbeschikbare numerieke resultaten. De berekening van de stroming rond eengroep van 8-bĳ-5 dicht bĳ elkaar staande cilinders illustreert de invloed van degroepsopstelling op de wervelafscheiding en de drukverdelingen rond de cilinders.

Hoofdstuk 5 beschrĳft gepartitioneerde algoritmes die het gekoppelde fluıdum-structuur interactie probleem oplossen door de verschillende velden sequentieelof parallel te berekenen. De nauwkeurigheid, stabiliteit en efficientie van dezealgoritmes wordt behandeld. De overdracht van verplaatsingen en belastingentussen de niet-samenvallende rekenroosters van de structuur en het fluıdum opde interface wordt besproken. Het behoud van de totale belasting en van deenergie op de interface is van groot belang. Een gekoppelde berekening van dedrie-dimensionale windstroming rond een vervormende silo wordt uitgevoerd. Meerefficiente koppelingstechnieken moeten het mogelĳk maken de respons van de desilo gedurende veel langere periodes te simuleren.

Hoofdstuk 6 vat de belangrĳkste conclusies van deze thesis samen en geeftaanbevelingen voor verder onderzoek.

Summary

For modern slender structures, wind loading is often a critical load case. In thecase of quite flexible structures, the large structural displacements influence thewind pressure distribution around the structure. This may give rise to structuralinstabilities, such as flutter of bridges, galloping of cables and ovalling of silos. Thisthesis studies wind induced ovalling oscillations of a group of forty closely spacedsilos by means of coupled numerical simulations of the fluid and the structure.

First, a three-dimensional finite element model of the silo structure is validatedby means of modal parameters derived from an in situ experiment. Theboundary conditions strongly influence the eigenfrequencies and mode shapes. Theeigenmodes with the lowest eigenfrequencies (3.93 Hz) have half a wave along theheight and three or four waves around the circumference. All eigenmodes have alow damping ratio around 1%.

Next, the turbulent wind flow around a group of 8 by 5 silos at Reynolds number12.4 × 106 is studied using two-dimensional unsteady incompressible Reynoldsaveraged Navier-Stokes simulations. In order to clarify the influence of theturbulence models, the near-wall modelling and the unsteadiness, results for theflow around a single cylinder are first compared with experimental data andnumerical results reported in the literature. In the group configuration especiallythe cylinders on the side corners are heavily loaded.

Finally, the finite element computation of the structure is coupled with the three-dimensional computation of the flow around a single silo in order to predictovalling oscillations. At the fluid-structure interface boundary conditions areexchanged between both computations. Within each time step the method iteratesbetween the structure and the fluid until the coupling conditions are fulfilled. Theflow is computed on a deforming mesh, using the arbitrary Lagrangian Eulerianformulation. The grid point displacements of the fluid mesh are obtained bydiffusing the structural displacements at the interface through the fluid domain.Geometrically non-linear structural behaviour is taken into account through anupdated Lagrangian description. The computed response of the silo is dominatedby the eigenmodes with three or four waves around the circumference.

xix

List of Symbols

The following list provides an overview of symbols used throughout the text. Thephysical meaning of the symbols is explained in the text. Vectors, matrices andtensors are denoted by bold characters.

The general symbols, conventions and abbreviations are collected in the firsttwo sections. The remaining symbols are categorized in sections referring to thechapters or to the sections where they are first introduced.

General symbols and conventions

(x, y, z) Cartesian coordinates

(r, θ, z) cylindrical coordinates

n vector at time level n

(i) vector at iteration i

discretization of a vector

∂/∂ first order partial derivative with respect to the variable

∂2/∂2 second order partial derivative with respect to the variable

∂

∂t

∣∣

first order partial time derivative of the variable withvariable fixed

D

Dt first order material time derivative of the variable

first order material time derivative of the variable

second order material time derivative of the variable

∇ del operator

∇2 Laplace operator

∇ gradient of a vector field

∇ · divergence of a vector field

−1 inverse of a matrix

T transpose of a matrix

det determinant of a matrix

tr trace of a matrix

I identity matrix

xxi

xxii LIST OF SYMBOLS

δij Kronecker Delta

t time

f frequency

ω circular frequency

∆t time step

Acronyms

ALE Arbitrary Lagrangian-Eulerian

APR Adverse Pressure Recovery

CFD Computational Fluid Dynamics

DGCL Discrete Geometrical Conservation Law

DNS Direct Numerical Simulation

FEM Finite Element Method

FFT Fast Fourier Transform

FSI Fluid-Structure Interaction

GCL Geometrical Conservation Law

GLS Galerkin/Least Squares

LBB Ladyzhenskaya-Babuska-Brezzi

LES Large Eddy Simulation

PFEM Particle Finite Element Method

PSPG Pressure-Stabilizing/Petrov-Galerkin

RANS Reynolds averaged Navier-Stokes

RMS Root Mean Square

RSM Reynolds Stress Model

SST Shear Stress Transport

SUPG Streamline-Upwind Petrov-Galerkin

SZL Shih-Zhu-Lumley

TL Total Lagrangian

UL Updated Lagrangian

URANS Unsteady Reynolds averaged Navier-Stokes

X-FEM Extended Finite Element Method

Flow on a domain with deforming boundaries

x spatial coordinates

X material coordinates

χ referential coordinates

Ωx spatial domain

ΩX material domain

Ωχ referential domain

LIST OF SYMBOLS xxiii

x variable in the spatial domain

X variable in the material domain

χ variable in the referential domain

Γ area vector

n unit normal vector

variable concerning the mapping from the material to thespatial domain

variable concerning the mapping from the material to thereferential domain

variable concerning the mapping from the referential to thespatial domain

u displacement

v particle velocity (in the spatial domain)

v particle velocity in the referential domain

v velocity of the nodes of the mesh

c convective velocity

u displacement of the nodes of the mesh

x position of the nodes of the mesh

ϕ mapping

F deformation gradient

J Jacobian

M mass

M momentum

ρ density

ρb body forces

t traction vector

σ Cauchy stress tensor

ε strain rate tensor

ω vorticity tensor

p pressure

τ deviatoric stress tensor

µ dynamic viscosity

λ second viscosity constant

K bulk viscosity

ν kinematic viscosity

p? kinematic pressure

k diffusivity

xxiv LIST OF SYMBOLS

Shell structures

δu virtual displacement

δε virtual strain

D displacement gradient

C right Cauchy-Green deformation tensor

b left Cauchy-Green deformation tensor

E Green-Lagrange strain tensor

e Almansi strain tensor

R orthogonal rotation tensor

U right stretch tensor

V left stretch tensor

σ Cauchy stress tensor

τ Kirchhoff stress tensor

PX first Piola-Kirchhoff stress tensor

S second Piola-Kirchhoff stress tensor

α first Newmark parameter

δ second Newmark parameter

h height

L length

R radius

t thickness

D diameter

P center-to-center distance

λ first Lame constant

µ second Lame constant

E Young’s modulus

ν Poisson’s ratio

m axial half wave number

n circumferential wave number

Finite element method

h trial solution for the variable

R residual

w weighting functions for the momentum equations

q weighting function for the continuity equation

Nv

i finite element shape functions for the velocity

Npi finite element shape function for the pressure

M mass matrix

K stiffness matrix

LIST OF SYMBOLS xxv

KT tangent stiffness matrix

Kσ geometric stiffness matrix

C damping matrix

G discrete gradient operator

f load vector

wm modified weighting functions for the momentum equations

ν numerical diffusion

ν diffusivity tensor

τ stabilization parameter

ST stabilization term

Finite volume method

V conservative variables

Q source terms

Fd diffusive (viscous) fluxes

Fc convective (inviscid) fluxes

Fe convective fluxes corresponding to an Eulerian description

Fale additional convective fluxes corresponding to an ALE

descriptionFd numerical diffusive fluxes

Fc numerical convective fluxes

Fe numerical convective fluxes corresponding to an Euleriandescription

Fale additional numerical convective fluxes corresponding to anALE description

Computation of turbulent wind flows

mean part of the variable

′ fluctuating part of the variable

k turbulent kinetic energy

ω turbulent frequency

ε turbulence dissipation rate

P turbulence production

νt turbulent viscosity

y+ dimensionless wall distance

Re Reynolds number

St Strouhal number

Ma Mach number

Cd drag coefficient

xxvi LIST OF SYMBOLS

Cl lift coefficient

Cp pressure coefficient

Cminp minimum of the pressure coefficient

Cbp base pressure coefficient

θ circumferential angle

θs separation angle

α incidence angle of the flow

Coupled simulation of wind loading on structures

f variable in the fluid partitions variable in the structural partition

Ω variable in the interior of the domain

Γ variable on the fluid-structure interface

tf traction vector in the fluid on the fluid-structure interface

ts traction vector in the structure on the fluid-structureinterface

σf stress field in the structural partition

σs stress field in the fluid partition

uΓ

discretized displacements of the fluid mesh on the

fluid-structure interface

uΓ

discretized displacements in the structure on the

fluid-structure interface

ffΓ

nodal forces in the fluid on the fluid-structure interface

fsΓ

nodal forces in the structure on the fluid-structure interface

E energy

ω relaxation factor

F fluid solver

S structural solver

S Schur complement

H motion transfer matrix

L load transfer matrix

Contents

Voorwoord i

Samenvatting iii

Summary xix

List of Symbols xxi

Contents xxvii

List of Figures xxxi

List of Tables xxxix

1 Introduction 1

1.1 Problem outline and motivation . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Wind loading on civil engineering structures . . . . . . . . . 1

1.1.2 A wind-structure interaction problem: ovalling of silos . . . 2

1.2 State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Response of structures to wind loading . . . . . . . . . . . . 4

1.2.2 Ovalling oscillations . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Objectives and original contributions . . . . . . . . . . . . . . . . . 11

1.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 11

xxvii

xxviii CONTENTS

1.3.2 Original contributions . . . . . . . . . . . . . . . . . . . . . 13

1.4 Organization of the text . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Flow on a domain with deforming boundaries 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Flow on a domain with deforming boundaries . . . . . . . . . . . . 19

2.2.1 The arbitrary Lagrangian-Eulerian description . . . . . . . 19

2.2.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.3 Constitutive equations . . . . . . . . . . . . . . . . . . . . . 33

2.2.4 Governing equations . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.1 Finite Element Method . . . . . . . . . . . . . . . . . . . . 36

2.3.2 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . 46

2.4 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4.1 The Eulerian description . . . . . . . . . . . . . . . . . . . . 48

2.4.2 The ALE description . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Mesh deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Shell structures 65

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 Geometric non-linear behaviour . . . . . . . . . . . . . . . . . . . . 66

3.2.1 Virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.3 Conjugate stress and strain tensors . . . . . . . . . . . . . . 73

3.2.4 Incremental virtual work . . . . . . . . . . . . . . . . . . . . 76

3.2.5 Newton-Raphson procedure . . . . . . . . . . . . . . . . . . 80

3.2.6 Time integration . . . . . . . . . . . . . . . . . . . . . . . . 82

CONTENTS xxix

3.3 Structural behaviour of a silo . . . . . . . . . . . . . . . . . . . . . 84

3.3.1 Description of the silo structure . . . . . . . . . . . . . . . . 84

3.3.2 Harmonic finite element model . . . . . . . . . . . . . . . . 85

3.3.3 The experimental setup . . . . . . . . . . . . . . . . . . . . 88

3.3.4 Three-dimensional finite element model of the silo . . . . . 93

3.3.5 Non-linear behaviour of the silos . . . . . . . . . . . . . . . 99

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4 Computation of turbulent wind flows 105

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2 Computational approaches to turbulent flows . . . . . . . . . . . . 106

4.2.1 Direct numerical simulation . . . . . . . . . . . . . . . . . . 106

4.2.2 Reynolds averaged Navier-Stokes simulation . . . . . . . . . 107

4.2.3 Large eddy simulation . . . . . . . . . . . . . . . . . . . . . 109

4.3 RANS turbulence models . . . . . . . . . . . . . . . . . . . . . . . 111

4.3.1 Linear eddy viscosity models . . . . . . . . . . . . . . . . . 111

4.3.2 Non-linear eddy viscosity models . . . . . . . . . . . . . . . 117

4.3.3 Reynolds stress models . . . . . . . . . . . . . . . . . . . . . 118

4.4 Turbulent air flow around a single cylinder . . . . . . . . . . . . . . 119

4.4.1 Steady computation: problem domain, near-wallmodelling and eddy viscosity turbulence models . . . . . . . 123

4.4.2 Unsteady computation . . . . . . . . . . . . . . . . . . . . . 126

4.4.3 Mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . 128

4.4.4 Comparison with numerical results reported in the literature 132

4.5 Turbulent air flow around a cylinder group . . . . . . . . . . . . . 134

4.5.1 Group of 2 by 2 cylinders . . . . . . . . . . . . . . . . . . . 138

4.5.2 Group of 8 by 5 cylinders . . . . . . . . . . . . . . . . . . . 145

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

xxx CONTENTS

5 Coupled simulation of wind loading on structures 153

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.2 Coupling algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.2.1 The monolithic and the partitioned approach . . . . . . . . 155

5.2.2 Loosely coupled algorithms . . . . . . . . . . . . . . . . . . 158

5.2.3 Strongly coupled algorithms . . . . . . . . . . . . . . . . . . 169

5.3 Load and motion transfer . . . . . . . . . . . . . . . . . . . . . . . 174

5.3.1 Motion transfer or surface tracking . . . . . . . . . . . . . . 176

5.3.2 Load transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5.3.3 Conservation of energy . . . . . . . . . . . . . . . . . . . . . 184

5.4 Application: ovalling of silos . . . . . . . . . . . . . . . . . . . . . . 185

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6 Conclusions and recommendations for further research 195

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.2 Recommendations for further research . . . . . . . . . . . . . . . . 198

Bibliography 203

Curriculum vitae 221

A Possibilities of commercial software packages Flotran and CFX 225

B Finite strip model of the silo 229

List of Figures

1 De silo groep in de haven van Antwerpen. . . . . . . . . . . . . . . iv

2 Bovenaanzicht en drie-dimensionaal zicht van het reeel (volle lĳn)en het imaginair deel (streeplĳn) van de geıdentificeerde eigenmodes3, 4 en 53 van de silo. De stippellĳn duidt de onvervormde silo aan. viii

3 Drie-dimensionaal eindige-elementenmodel van de silo en een aantalgeselecteerde eigenmodes. . . . . . . . . . . . . . . . . . . . . . . . ix

4 Drukcoefficient als functie van de hoek θ bekomen met eenstationaire berekening (streeplĳn) en het tĳdsgemiddelde (volle lĳn),het minimum (streep-stippellĳn) en het maximum (stippellĳn) vande drukcoefficient bekomen met een niet-stationaire berekening voor(a) rooster A2 en (b) rooster B. De lichtgrĳze zone bevat allebeschikbare experimentele data bĳ Reynoldsgetallen van 0.73× 107

tot 3.65× 107. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

5 (a) Tĳdsverloop en (b) frequentie-inhoud van de druk op hetoppervlak van een cilinder in de groep van 8-bĳ-5 cilinders. . . . . xi

6 Tĳdsgemiddelde drukcoefficienten Cp voor de stroming rond eengroep van 8-bĳ-5 cilinders (volle lĳn) en voor een alleenstaandecilinder (streeplĳn). De pĳl duidt de invalshoek van de wind aan. . xii

7 (a) Tĳdsverloop van de radiale verplaatsingen op halve hoogte voorθ = 66 ( ), θ = 120 ( ) en θ = 180 ( )berekend met algoritmeA (streeplĳn) en algoritme B (volle lĳn) en (b) frequentie-inhoudvan de radiale verplaatsingen berekend met algoritme B. De hoekθ = 0 valt samen met het stuwpunt. . . . . . . . . . . . . . . . . . xiv

1.1 (a) Flutter of the Tacoma Narrows suspension bridge and (b) itsfinal collapse (Champneys, 2008). . . . . . . . . . . . . . . . . . . . 2

xxxi

xxxii LIST OF FIGURES

1.2 (a) Collapse of one of the cooling towers at Ferrybridge (UK) and(b) the final destruction of three cooling towers (Norfolk, 2008). . . 3

1.3 The silo group in the port of Antwerp. . . . . . . . . . . . . . . . . 3

1.4 Plan and lateral view of the silo group. . . . . . . . . . . . . . . . . 4

2.1 (a) Explicit and (b) implicit description of a deforming boundaryof the fluid domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Representations of (a) the Lagrangian, (b) the Eulerian and (c) thearbitrary Lagrangian-Eulerian descriptions in the spatial domain Ωx

at two different times t: continuum (dark gray), material particles(black dots) and mesh (black lines). Dashed lines show the meshmotion, while solid lines show the particle motion. . . . . . . . . . 20

2.3 Domains, meshes and mappings for an ALE description: fixedreferential domain Ωχ, changing material domain ΩX and changingspatial domain Ωx at time t (solid lines) and t+ ∆t (dashed lines). 21

2.4 Analytical solution for the velocity (black line), exact nodal solution(black dashed line) (2.128), Galerkin solution (dark grey line) (2.126)and full upwind solution (light grey line) (2.130) of a steady 1Dscalar convection-diffusion equation with a constant convection c =1 and a constant source term b = 1 using an element size h = 0.1for a Peclet number of (a) 0.25 and (b) 5. . . . . . . . . . . . . . . 40

3.1 Octagonal beam with four bolted connections at the bottom of thecylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.2 Harmonic finite element model of the silo and selected eigenmodes. 86

3.3 Eigenfrequencies of the harmonic finite element model of the silo asa function of m and n. . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.4 Dimensionless total strain energy (solid line) and dimensionlessstrain energy due to bending (dashed-dotted line) and stretching(dashed line) for m = 2 () and m = 3 (N) for a freely supportedcylinder with a height h = 25 m and a constant thickness t = 7 mm. 88

3.5 Location of the accelerometers on the silo wall. . . . . . . . . . . . 89

3.6 Power spectral density of the radial acceleration at the point HL09. 90

3.7 Top and three-dimensional view of the real (solid line) andimaginary (dashed line) part of the identified eigenmodes 3, 4 and53 of the silo. The dotted line shows the undeformed shape. . . . . 91

LIST OF FIGURES xxxiii

3.8 The radial displacements of mode 4 in the points HL6 to HL13 inthe complex plane. The displacements are scaled to unity in thepoint HL6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.9 Measured eigenfrequencies of the silo as a function of m and n. . . 92

3.10 Time history of the radial acceleration at the point VL15 (top), thenine modal contributions with the highest RMS value and the error(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.11 The approximated relative error on nine selected eigenfrequenciespredicted with the quarter three-dimensional finite element modelas a function of the mesh refinement factor. . . . . . . . . . . . . . 95

3.12 Top view of the eigenmode (1, 6) of the silo at a height of 15 mcomputed with (a,b) the quarter three-dimensional finite elementmodel with SS and AA boundary conditions and (c) the full three-dimensional finite element model. . . . . . . . . . . . . . . . . . . . 96

3.13 Top and lateral view of the eigenmode (1, 5) of the silo computedwith (a) the quarter three-dimensional finite element model withSA boundary conditions and (b) the full three-dimensional finiteelement model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.14 Three-dimensional finite element model of the silo and selectedeigenmodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.15 Eigenfrequencies of the three-dimensional finite element model ofthe silo as a function of m and n. . . . . . . . . . . . . . . . . . . . 98

3.16 Deformed shape of a cantilever beam of length L under a tip momentM = πEI/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.17 Imposed pressure distribution around the circumference of the silo. 101

3.18 (a) Finite element mesh and boundary conditions and (b)deformations magnified by factor of 3. . . . . . . . . . . . . . . . . 102

4.1 Pressure coefficient Cp for a single cylinder. . . . . . . . . . . . . . 120

4.2 Measured pressure coefficients at Reynolds numbers from 0.73×107

to 3.65× 107 (Zdravkovich, 1997). . . . . . . . . . . . . . . . . . . 121

4.3 Problem domain and mesh A2 for a single cylinder. . . . . . . . . . 124

xxxiv LIST OF FIGURES

4.4 Pressure coefficient as a function of the angle θ obtained withFlotran (solid lines) and CFX (dashed lines) turbulence modelsusing various eddy viscosity turbulence models ( k ε, SST,SZL) for (a) mesh A2 and (b) mesh B. The light grey zone containsall available experimental data at Reynolds numbers from 0.73×107

to 3.65× 107. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.5 (a) Turbulent kinetic energy and (b) wall shear stress as a functionof the angle θ at the cylinder’s surface using different eddy viscosityturbulence models ( k ε, SST, SZL) for mesh B: Flotran (solidlines) and CFX (dashed lines). . . . . . . . . . . . . . . . . . . . . 125

4.6 (a) Time history and (b) frequency content of the pressure at thecylinder’s surface at θ = 172 for a single cylinder using mesh B. . 126

4.7 (a) Time average and (b) standard deviation of the pressure p for asingle cylinder using mesh B. . . . . . . . . . . . . . . . . . . . . . 127

4.8 Pressure coefficient as a function of the angle θ obtained with thesteady state computation (dashed line) and the time average (solidline), minimum (dash-dotted line) and maximum (dotted line) ofthe pressure coefficient obtained with the transient computationfor (a) mesh A2 and (b) mesh B. The light grey zone contains allavailable experimental data at Reynolds numbers from 0.73 × 107

to 3.65× 107. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.9 (a) Pressure coefficient obtained on meshes A1 (black), A2 (blue),A3 (green) and A4 (red) and extrapolated solutions for meshrefinement study 1 (cyan) and 2 (magenta) and (b) local apparentorder of accuracy for mesh refinement study 1 (solid line) and 2(dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.10 (a) Approximated relative errors (dash dotted lines), extrapolatedrelative errors (dashed lines), GCI using the local (dotted lines)and the average (solid lines) apparent order of accuracy for meshrefinement study 1 (thin lines) and 2 (thick lines) and (b) timehistory of drag (thin) and lift (thick) coefficients obtained on meshesA1 (solid line), A2 (dash dotted line) , A3 (dashed line) and A4

(dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.11 Flow patterns around two closely spaced cylinders for (a) small, (b)medium and (c) large incidence angles (Sumner et al., 2000). . . . 135

4.12 Detail of the mesh for the group of 2 by 2 cylinders. . . . . . . . . 138

LIST OF FIGURES xxxv

4.13 Dimensionless distance y+ of the nodes next to the cylinder wallas a function of the angle θ for cylinder 1 (solid line), 2 (dashedline), 3 (dotted line) and 4 (dash-dotted line) of the group of 2 by2 cylinders in (a) mesh A and (b) mesh B. . . . . . . . . . . . . . . 139

4.14 Time average (thick line), maximum and minimum (thin lines) ofthe pressure coefficient Cp for mesh A (black) and mesh B (grey)as a function of the angle θ for (a) cylinder 1, (b) cylinder 2, (c)cylinder 3 and (d) cylinder 4 of the group of 2 by 2 cylinders. . . . 139

4.15 (a) Time history and (b) frequency content of the pressure at thecylinder’s surface in the point B of the group of 2 by 2 cylinders. . 140

4.16 (a) Time history and (b) frequency content of the pressure at thecylinder’s surface in the point E of the group of 2 by 2 cylinders. . 141

4.17 Streamlines at t = 14.075 s from a transient computation around agroup of 2 by 2 cylinders. . . . . . . . . . . . . . . . . . . . . . . . 141

4.18 Flow patterns around two closely spaced cylinders for (a) α = 25

and P/D = 1.3 and (b) α = 45 and P/D = 1.1 (Alam et al., 2005). 142

4.19 Time averaged pressure coefficients Cp for the flow around a groupof 2 by 2 cylinders (solid line) and for the flow around a singlecylinder (dashed line). The arrow indicates the incidence angle α. . 142

4.20 (a) Time averaged drag coefficient Cd, (b) fluctuating dragcoefficient C′d, (c) time averaged lift coefficient C l and (d)fluctuating lift coefficient C′l for the flow around the group of 2by 2 cylinders. The arrow indicates the incidence angle α. . . . . . 143

4.21 Decomposition of the time averaged pressure coefficient Cp into aseries of cosine functions with circumferential wavenumber n for theflow around the group of 2 by 2 cylinders (2 by 2 circles) and forthe flow around a single cylinder (single circle). The arrow indicatesthe incidence angle α. . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.22 Detail of the mesh for the group of 8 by 5 cylinders. . . . . . . . . 145

4.23 Dimensionless distance y+ of the nodes next to the cylinder wallas a function of the angle θ for all cylinders of the group of 8 by 5cylinders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.24 (a) Time history and (b) frequency content of the pressure at thecylinder’s surface in the point B of the group of 8 by 5 cylinders. . 146

4.25 (a) Time history and (b) frequency content of the pressure at thecylinder’s surface in the point E of the group of 8 by 5 cylinders. . 146

xxxvi LIST OF FIGURES

4.26 Streamlines at t = 50.94 s from a transient computation around agroup of 8 by 5 cylinders. . . . . . . . . . . . . . . . . . . . . . . . 147

4.27 Time averaged pressure coefficients Cp for the flow around a groupof 8 by 5 cylinders (solid line) and for the flow around a singlecylinder (dashed line). The arrow indicates the incidence angle α. . 148

4.28 (a) Time averaged drag coefficient Cd, (b) fluctuating dragcoefficient C′d, (c) time averaged lift coefficient C l and (d)fluctuating lift coefficient C′l for the flow around the group of 8by 5 cylinders. The arrow indicates the incidence angle α. . . . . . 149

4.29 Decomposition of the time averaged pressure coefficient Cp into aseries of cosine functions with circumferential wavenumber n for theflow around the group of 8 by 5 cylinders (8 by 5 circles) and forthe flow around a single cylinder (single circle). The arrow indicatesthe incidence angle α. . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.1 The serial staggered algorithm. . . . . . . . . . . . . . . . . . . . . 159

5.2 Non-collocated algorithm. . . . . . . . . . . . . . . . . . . . . . . . 163

5.3 The parallel algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.4 The enhanced parallel algorithm. . . . . . . . . . . . . . . . . . . . 165

5.5 Staggered algorithm with subcycling of the fluid. . . . . . . . . . . 166

5.6 The iteratively staggered algorithm. . . . . . . . . . . . . . . . . . 170

5.7 Meshes and displacement profiles at the fluid-structure interfacein the structure ( ) and the fluid partition ( ) obtained with theconsistent interpolation method when fluid mesh is (a) finer and (b)coarser than the solid mesh. . . . . . . . . . . . . . . . . . . . . . . 177

5.8 Meshes and displacement profiles at the fluid-structure interfacein the structure ( ) and the fluid partition ( ) obtained with theelement to element mapping when fluid mesh is (a) finer and (b)coarser than the solid mesh. . . . . . . . . . . . . . . . . . . . . . . 179

5.9 Meshes and nodal forces at the fluid-structure interface in thestructure ( ) and the fluid partition ( ) obtained with theconservative method when fluid mesh is (a) finer and (b) coarserthan the solid mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . 182

LIST OF FIGURES xxxvii

5.10 Meshes and nodal forces at the fluid-structure interface in thestructure ( ) and the fluid partition ( ) obtained with the elementto element conservative method when fluid mesh is (a) finer and (b)coarser than the solid mesh. . . . . . . . . . . . . . . . . . . . . . . 182

5.11 (a) Time history and (b) frequency content of the pressure at thecylinder’s surface at mid-height for θ = 112 ( ), θ = 174 ( ) andθ = 180( ). The angle θ = 0 coincides with the stagnation point. 187

5.12 Model for the coupled simulation of the three-dimensional wind flowaround a cylinder and the response of the silo structure. . . . . . . 188

5.13 (a) Time history of the radial displacements at mid-height for θ =66 ( ), θ = 120 ( ) and θ = 180 ( ) computed with algorithmA (dashed lines) and algorithm B (solid lines) and (b) frequencycontent of the radial displacements computed with algorithm B. Theangle θ = 0 coincides with the stagnation point. . . . . . . . . . . 189

5.14 Deformations (enlarged with a factor 5) of the structure between11.25 m and 13.75 m high at (a) t = 5.905 s, (b) t = 6.805 s and (c)t = 7.250 s. The wind flows from the left. . . . . . . . . . . . . . . 190

5.15 (a) Time history and (b) frequency content of the pressure at thecylinder’s surface at mid-height for θ = 112 ( ), θ = 174 ( ) andθ = 180( ) using algorithm B. The angle θ = 0 coincides withthe stagnation point. . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.16 Pressure field on the vertical plane through the cylinder axis parallelwith the inlet flow direction at t = 7.9 s. . . . . . . . . . . . . . . . 191

5.17 Pressure at the cylinder’s surface along the height for (a) θ = 0,(b) θ = 112 and (c) θ = 180 at t = 5.4 s ( ), t = 5.475 s ( ),t = 5.925 s ( ), t = 6.35 s ( ) and t = 6.605 s ( ). . . . . . . . . . . 191

5.18 (a) Time history and (b) frequency content of the first (solid line)and second principal component (dashed line) of the displacementsat mid-height with circumferential wavenumber n = 2 ( ), n = 3( ) and n = 4 ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

B.1 The relative error on the nine lowest eigenfrequencies predicted withthe finite strip model as a function of the number of elements in thecircumferential direction. . . . . . . . . . . . . . . . . . . . . . . . . 232

B.2 Eigenmodes of a silo with a height h = 25 m and a thickness t =7 mm, computed with a finite strip model. . . . . . . . . . . . . . . 233

List of Tables

3.1 Thickness of the aluminium plates of the silo as a function of theheight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2 Comparison of the eigenfrequencies computed with the harmonicfinite element model, the quarter three-dimensional finite elementmodel and the full three-dimensional finite element model with theexperimental eigenfrequencies. . . . . . . . . . . . . . . . . . . . . . 99

3.3 Horizontal and vertical tip displacement, number of load incrementsand equilibrium iterations as a function of the element type. . . . . 101

4.1 The time averaged drag coefficient Cd, the RMS of the dragcoefficient CRMS

d , the RMS of the lift coefficient CRMSl , the

fluctuation C′d of the drag coefficient, the fluctuation C′l of the liftcoefficient and the separation angle θs obtained on meshes A1, A2,A3 and A4 and the different approximated relative errors. . . . . . 131

4.2 Comparison of the Reynolds number Re, the Strouhal number

St, the separation angle θs, the base pressure coefficient Cb

p, the

minimum pressure coefficient Cmin

p , the drag coefficient Cd and

the adverse pressure recovery APR with numerical results from theliterature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.1 Density ratios ρs/ρf between some typical civil engineering materialsand air or water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.2 Eigenfrequencies (in Hz) computed with the coarser and thevalidated three-dimensional finite element model. . . . . . . . . . . 186

xxxix

xl LIST OF TABLES

A.1 Overview of the different possibilities for CFD computations withinFlotran 10.0 and CFX 10.0. . . . . . . . . . . . . . . . . . . . . . . 226

A.2 Overview of the different possibilities to couple Ansys 10.0 withFlotran 10.0 and CFX 10.0. . . . . . . . . . . . . . . . . . . . . . . 227

B.1 Comparison of the eigenfrequencies computed with the three-dimensional finite element model and the finite strip models forvarying n and m = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Chapter 1

Introduction

1.1 Problem outline and motivation

1.1.1 Wind loading on civil engineering structures

The behaviour of civil engineering structures under wind loads is an importantmatter of concern. Melchers (1987) mentions as prime cause of the failureof structures the inadequate estimation of loading conditions or the inaccurateassessment of structural behaviour. It is a trend in modern architecture toconstruct more slender and lightweight structures with often complex geometries.

The actual wind distribution around structures with a complex shape cannotbe derived from the literature or from the Eurocode (BIN, 1995). Moreover,the loadings specified in the Eurocode are generally based on wind tunnel testsperformed on free-standing structures in open surroundings. Wind loads onbuildings in realistic environments may be considerably different due to thepresence of neighbouring structures.

For lightweight structures, the ratio between the wind load and the dead loadcarried by the structure is substantially larger than for conventional structures.Therefore, wind loading has become an important design load case. The increasein span often results in more flexible structures with lower eigenfrequencies thatare more susceptible to dynamic vibrations. The shape of shell structures is oftendesigned to carry the static dead loads by normal forces, while the additional windloading causes considerable bending moments in the structure.

For relatively rigid structures, the wind load can be determined for the undeformedshape of the structure. This is no longer the case for flexible structures, where

1

2 INTRODUCTION

the wind pressures cause large structural displacements. The resulting change ofshape has an influence on the wind pressure distribution around the structure.This may give rise to structural instabilities, such as flutter of bridges, vortexinduced vibrations of tall buildings, galloping of cables and ovalling of silos (Simiuand Scanlan, 1986).

The lack of knowledge about wind-structure interaction effects has led to seriousdisasters in the past. The Tacoma Narrows bridge (Washington, US) collapsedin 1940 due to flutter (figures 1.1(a) and (b)). The H-shape of the section ofthe bridge resulted in a poor aerodynamic behaviour and made the bridge verysensitive to wind excitation.

(a) (b)

Figure 1.1: (a) Flutter of the Tacoma Narrows suspension bridge and (b) its finalcollapse (Champneys, 2008).

In 1965, three of a group of eight cooling towers at Ferrybridge (UK) collapsed,while the others were severely cracked (figures 1.2(a) and (b)). The cooling towershad been built closer together than usual and the design wind loads were basedon experiments for one isolated tower (Gunn and Malik, 1966). The towers thatcollapsed were not situated on the row facing the wind but in the row shelteredfrom the wind (the wind was blowing from the right in figure 1.2b).

These two examples illustrate that the behaviour of civil engineering structuresunder wind loading is difficult to predict.

1.1.2 A wind-structure interaction problem: ovalling of silos

During a storm on October 27, 2002, wind induced ovalling oscillations wereobserved on several empty silos of a group consisting of forty silos (figure 1.3),located in the port of Antwerp. These oscillations were estimated to have largeamplitudes of several centimeters. During the storm the hourly average wind speedin Deurne (about 7 kilometers to the east) ranged from 61 to 68 km/h with peakwind speeds up to 113 km/h. The wind direction was west-southwest.

PROBLEM OUTLINE AND MOTIVATION 3

(a) (b)

Figure 1.2: (a) Collapse of one of the cooling towers at Ferrybridge (UK) and (b)the final destruction of three cooling towers (Norfolk, 2008).

The silos are circular cylindrical shell structures with a diameter of 5.5 m and aheight of 25 m. One cylinder consists of 10 aluminium sheets with a height of 2.5 mand a thickness that decreases with the height from 10.5 mm at the bottom to 6 mmat the top. The height-to-radius ratio h/R = 9.1 and the radius-to-thickness ratioranges from R/t = 262 at the bottom to R/t = 458 at the top.

Figure 1.3: The silo group in the port of Antwerp.

4 INTRODUCTION

The forty silos of this group are placed in five rows of eight silos with gaps of 30 cmbetween two neighbouring silos (figure 1.4). The spacing ratio of the distance Pbetween the center of two cylinders to the cylinder diameter is P/D = 5.8/5.5 =1.05. The bottom of the silos is located at 16.66 m above ground level.

N

D

P

L

Wα

h = 25 m

16.66 m

Figure 1.4: Plan and lateral view of the silo group.

In ovalling, the cross section of a circular cylindrical structure deforms as a shellwithout bending deformation of the longitudinal axis of symmetry. The deformedshape consists of a number of waves in the circumferential direction and a numberof waves in the axial direction. Similar cases in Germany and Scotland indicatethat storm damage is mainly located on silos at the corners of the group.

The ovalling oscillations of this group of silos in the port of Antwerp will be usedas an example throughout the thesis. The ovalling phenomenon is described inthis introductory section in order to clarify the objectives of the experiments andsimulations in the following chapters.

1.2 State-of-the-art

1.2.1 Response of structures to wind loading

In this section an overview of possible methods to determine the response ofstructures to wind loading is given. First the experimental techniques for windtunnel testing are reviewed. Next, the numerical simulations are described.

Experimental techniques: wind tunnel tests

Rigid structures For relatively rigid structures, the behaviour under wind loadingis split in two separate steps: the determination of the pressure distribution aroundthe structure and the response of the structure to this loading.

STATE-OF-THE-ART 5

Usually, wind tunnel tests on rigid scale models are used to determine wind loadson buildings with complex geometries. The surroundings of the structure caneasily be included in this model. It is often impossible to correctly scale both themean wind profile and the turbulent Reynolds number Re = Dv/ν, where D isa characteristic length, v the fluid velocity and ν the kinematic viscosity. If theflow is dependent on the Reynolds number (e.g. for circular or rounded shapes),corrections should be made in the conversions to full scale results. To improve thelocal flow similarity, curved surfaces are usually roughened. The preparation ofthese tests is time-consuming and expensive. It is possible to cover a wide rangeof wind speeds, directions and turbulence intensities in a short time. Only globalpressure distributions are obtained. A change in the design requires a new modeland a complete re-run of the experiments.

The response of the structure is computed by applying the measured pressuredistributions to a numerical model of the structure. The numerical model yieldsstresses and forces in all structural members. Changes in the structural designthat do not influence the outer shape of the structure are easily evaluated.

Orlando (2001) used a linear finite element model to compute the quasi-static response of isolated and grouped cooling towers under measured pressuredistributions. Portela and Godoy (2007) performed an eigenvalue analysis and ageometric non-linear analysis on a finite element model of one of a group of steelstorage tanks to determine the buckling load. The applied pressure distributionswere obtained from a wind tunnel test on the group configuration. Lazzari et al.(2003) used pressure coefficients obtained by wind tunnel tests to define pressuretime histories that correspond to artificially generated space-varying velocity timehistories. A dynamic geometrically non-linear finite element analysis determinesthe response of a tensegrity stadium roof under these pressure time histories.

Flexible line-like structures For slender, line-like structures with a relativelyrigid cross section which is uniform along the length, as long-span bridges, thewind tunnel test can be performed on a model which represents a fraction ofthe total length. In the wind tunnel, a rigid section model of the structure ismoved harmonically with a controlled amplitude and frequency in a uniform flowin order to determine the resulting aerodynamic forces. From these measurementsaerodynamic or flutter derivatives are obtained for a number of dimensionlessfrequencies. The reduction of the model to a part of the structure enables the useof larger scale models at higher Reynolds numbers.

These aerodynamic derivatives are used to calculate the response of the structureby means of an analytical or a numerical model. The simplest structural modelconsists of a mass and a mass moment of inertia supported by a vertical springand a rotational spring. This model has only two eigenmodes and the responseis computed in the frequency domain (Simiu and Scanlan, 1986). Current state-

6 INTRODUCTION

of-the-art techniques use time domain methods in order to account for the non-linearity in the aerodynamic forces due to the change in angle of incidence (Chenand Kareem, 2001). Simple two degree-of-freedom models of the structure arereplaced by finite element models which include all contributing eigenmodes andare able to account for non-linear structural behaviour (Zahlten and Eusani, 2006).

Alternatively, if the rigid section model is mounted dynamically with a vertical anda rotational spring, the flutter wind speed can be determined from wind tunneltests.

Flexible 3D structures If the structural deformations are large enough tosubstantially change the pressure distribution around the structure, wind tunneltests on fully aeroelastic models are a possible approach. In the aeroelastic model,the mass, stiffness and damping of the structure should be scaled correctly. Thisis especially difficult if several eigenmodes contribute to the structural response.For shell structures, similarity requirements have to be fulfilled for the Poisson’sratio as well. Wind tunnel test results often have to be corrected to representthe full-scale behaviour. Vickery and Majowiecki (1992) studied the response ofa cable supported stadium roof on an aeroelastic model. Niemann and Kopper(1998) measured strains in a cooling tower by means of an aeroelastic wind tunneltest. The pressures and deformations can generally only be measured in a limitednumber of points.

Numerical simulations

CFD simulation of wind flow around rigid structures The fluid flow aroundrigid structures can be simulated numerically with Computational Fluid Dynamics(CFD). Wind flows around buildings are very high-Reynolds-number flows withunsteady separation and vortex shedding. This requires an appropriate turbulencemodel. Large eddy simulations (LES) are perfectly suited to capture these large-scale unsteadiness, but the computational cost is very high. Reynolds averagedNavier-Stokes (RANS) simulations are much cheaper and very often used tocompute these flows, but are, especially in the wake, less successful (Stathopoulos,2002). For a relatively simple shape, the averaged physical quantities, suchas the drag and the lift coefficients, can be computed with sufficient accuracy.The numerical prediction of peak pressures or spatial correlations of fluctuatingpressures with LES is still challenging. The application of hybrid RANS/LESmethods is very promising as they are able to capture the large scale turbulentstructures in the wake of bluff bodies at a more affordable computational cost(Spalart, 2000).

STATE-OF-THE-ART 7

These numerical simulations have the advantage of yielding much more detailedpressure distributions than in the case of wind tunnel testing. However, due to thehigh computational cost, quite coarse meshes and less expensive but less suitedturbulence models are often used. Therefore, the results are not always reliableand accurate.

Numerical simulation of fluid-structure interaction If the interaction betweenthe fluid and the structure is substantial, a CFD model of the flow can be coupledto a numerical model of the structure. Due to the increase in computing power,the numerical simulation of fluid-structure interaction (FSI) problems has gaineda lot of interest in the past decade. The main areas of interest are aeronauticalengineering, biomechanics and civil engineering. The aerospace applications(Dowell and Hall, 2001; Piperno and Farhat, 2001; Willcox and Peraire, 2002;Geuzaine et al., 2003) are mainly related to flutter of wings, while in biomechanics(Vierendeels et al., 2000; Baaĳens, 2001; Gerbeau and Vidrascu, 2003) blood flowsthrough the heart or the arteries or air flows in the lungs are studied. Withinthe context of civil engineering, challenging problems include the wind responseof tensile structures (Gluck et al., 2001, 2003; Haug et al., 2003; Hubner et al.,2004; Bletzinger et al., 2006), flutter of bridges (Hubner et al., 2002; Fouresteyand Piperno, 2004), vortex induced vibrations of marine riser pipes (Willden andGraham, 2004) or cables (Meynen et al., 2005) and ovalling of silos (Dooms et al.,2007).

Numerical methods used for fluid-structure interaction can be classified accordingto the complexity of the models and techniques employed in the structureand the fluid (Lohner and Cebral, 1996). For the structure, the followingmodels might be used (in increasing order of complexity): a system of springs,masses and dampers; linear finite element model; non-linear finite element model.For the fluid, typical options would be: potential flow; inviscid flow (Eulerequations); Reynolds averaged Navier-Stokes equations; large eddy simulations;direct numerical simulations. Nowadays, the available computational resourceslimit the computations at high Reynolds numbers to the coupled simulation ofReynolds averaged Navier-Stokes equations with a non-linear finite element model.

These numerical simulations have the advantage of yielding pressure distributionsand structural displacements and stresses for all elements, which is much moredetailed than in the case of wind tunnel testing. The main disadvantage isthe computational cost as the calculations do not run in real-time: they takemuch longer than the simulated time period. The determination of appropriateboundary conditions (e.g. velocity profiles, wall roughnesses) and turbulencemodel parameters is often difficult. The current accuracy and reliability of fluid-structure interaction computations implies that they are not competing with windtunnel testing, but are rather highly complementary: the use of these calculationsin an early design phase can reduce the required number of wind tunnel tests.

8 INTRODUCTION

1.2.2 Ovalling oscillations

As the ovalling oscillations of the group of silos in the port of Antwerp will be usedas an example throughout the thesis, the state-of-the-art with respect to ovallingoscillations of cylindrical shell structures is briefly described in this section.Ovalling oscillations were first reported in the fifties for steel chimney-stacks(Dockstader et al., 1956). The ovalling frequency was observed to be close to twicethe estimated vortex shedding frequency. Analogously to swaying oscillations ofbeam-like structures, ovalling oscillations were believed to be excited by periodicvortex shedding.

Wind tunnel tests

A lot of experimental research on ovalling has been carried out on scale modelsin wind tunnels. Johns and Sharma (1974) concluded from wind tunnel teststhat the ovalling frequency of the structure is equal to an integer multiple of thevortex shedding frequency. The vortex shedding frequency f was not measured butassumed to lie in the range corresponding to Strouhal numbers St = fD/v between0.166 and 0.2, where D is the diameter and v the flow velocity. The ovallingfrequency of the structure is very close or identical to one of its eigenfrequencies.Small discrepancies are probably due to the static deformation of the structureunder the wind pressures or the structural geometrically non-linear behaviour.The deformed shape has a node or an antinode facing the flow direction. Ovallingstarts at a threshold (or onset) flow velocity and its amplitude increases up toa peak flow velocity, which is substantially higher than the threshold velocity.Sometimes more than one eigenmode of the structure is excited simultaneouslyand the eigenmode that dominates the response changes with the flow velocity.

Paıdoussis and Helleur (1979) measured the ovalling frequency as well as thevortex shedding frequency in the wake. At the onset the ovalling frequency ofthe structure was equal to an integer multiple of the vortex shedding frequency.However, with increasing wind speeds, the ovalling frequency remained the same,while the vortex shedding frequency increased in accordance with a constantStrouhal number. The integer relationship between both frequencies ceased tohold, which meant that no ’lock-in’ occurred as in the case of swaying oscillationsof beam-like structures, where vortices continue being shed at the eigenfrequencyof the structure if the flow velocity is increased. In a second series of experiments,the periodic vortex shedding was suppressed by mounting a splitter plate behindthe cylinder. In this case, ovalling still occurred but at a slightly higher onsetvelocity. These two observations suggested that periodic vortex shedding is notthe cause of the ovalling. In a third series of experiments the flow pattern in thewake was altered by gluing flexible plastic tubes on the cylinder wall in its wake.Ovalling oscillations were totally suppressed.

STATE-OF-THE-ART 9

New experiments by Paıdoussis et al. (1982b) showed that, at the onset of ovalling,the ovalling frequency of the structure is not always equal to an integer multiple ofthe vortex shedding frequency, although this often occurs. It was suggested thatovalling is an aero-elastic flutter in a single eigenmode, which is associated withnegative aerodynamic damping (Paıdoussis et al., 1983). It should be distinguishedfrom classical flutter of wings and bridges where two eigenmodes of the structurecouple together during the oscillations. If the energy extracted from the flowexceeds the energy dissipated in the structure, the response of the structure to aninitial disturbance diverges. Ovalling especially occurs on thin welded structuresmade of high strength steel or aluminium which have a low internal damping.The often encountered integer relationship between the ovalling frequency andthe vortex shedding frequency at the onset of ovalling might be explained asfollows: ovalling starts at a slightly lower velocity because of the deformationsof the structure due to the periodic vortex shedding.

As also silos experience ovalling oscillations, Katsura (1985) performed windtunnel tests in a uniform flow at different turbulence intensities on cylinders withheight-to-radius and radius-to-thickness ratios typical for silos. The effective totaldamping of the structure was measured at different flow velocities and reducedto almost zero at the onset velocity where the vibration amplitudes increasesuddenly. At the lowest turbulence intensity, a negative phase lag between shelldisplacements and the wind pressures occurred above the onset velocity, whichcorresponds to a negative aerodynamic damping. At the peak flow velocity, thefrequency component in the pressure field close to the cylinder at the ovallingfrequency becomes dominant compared to the component at the vortex sheddingfrequency. At the highest turbulence intensity, the phase lags were always positiveand vibration amplitudes increase gradually in proportion with the pressure, whichis similar to turbulent buffeting. Uematsu and Uchiyama (1985) determinedfrom the cross-correlation of fluctuating pressures in two points in the wake ofa rigid cylinder that the pressure fluctuations in the wake are convected towardsthe separation point with a velocity that depends on their frequency and thefree stream velocity. This correlation might trigger the ovalling phenomenonfor eigenmodes with higher circumferential mode numbers. The eigenmode withlowest eigenfrequency is not always excited first.

Experiments by Panesar and Johns (1985) did not show any lock-in between theovalling frequency and the vortex shedding frequency. The presence of a splitterplate in the wake changed the ovalling onset velocity, but surprisingly did notsuppress the vortex shedding. Paıdoussis et al. (1991) showed that stiffening theupstream or the side part of the cylindrical shell changes the onset velocity, whilestiffening the leeward side totally eliminates the ovalling, which proves that thebehaviour in the wake is determining. Laneville and Mazouzi (1995) obtaineddifferent amplitudes for the ovalling vibrations at a certain flow velocity dependingon if the velocity is increasing or decreasing with time.

10 INTRODUCTION

Semi-analytical models

Simultaneously, semi-analytical models were developed to predict the onsetvelocity of ovalling (Paıdoussis et al., 1982a, 1991; Mazouzi et al., 1991; Lanevilleand Mazouzi, 1996). The linear behaviour of a cylindrical structure madeof an elastic, homogeneous and isotropic material is described by Donnell’sor Flugge’s theory (Kraus, 1967). The pressures acting on the structure areobtained from the superposition of a steady and an unsteady flow. The steadypart is represented as a potential flow which is tuned to experimental values.The separation angle is not affected by the oscillations. The unsteady part iscalculated as a potential flow caused by small amplitude vibrations. The effectsof periodic vortex shedding and turbulence are neglected. If the coupling betweendifferent eigenmodes during ovalling is neglected, the resulting eigenvalue problemyields the ovalling frequencies and the amount of aerodynamic damping. Theovalling frequencies are almost identical to the eigenfrequencies of the structure.Eigenmodes with an even circumferential wavenumber have negative aerodynamicdamping if a node faces the free stream direction and positive damping if anantinode faces the free stream direction. Eigenmodes with an odd circumferentialwavenumber have negative aerodynamic damping if an antinode faces the freestream direction and positive damping if a node faces the free stream direction.The aerodynamic damping increases with the free stream velocity. If the energygained from the flow equals the energy dissipated by the structure, ovalling setson. The accurate experimental determination of the structural modal dampingis therefore very important. Paıdoussis et al. (1991) included in the wake ofthe cylinder experimentally determined changes in the base pressure due to theshell deformations and the phase lag between the shell motion and the pressurein order to improve the predicted onset flow velocities. Mazouzi et al. (1991)used for the steady flow the potential flow model of Parkinson and Jandali (1970)which requires experimental values of the separation angle and the base pressurecoefficient. Through these values, the dependence of the pressure distributionon the Reynolds number is introduced. Laneville and Mazouzi (1996) simplifiedthese semi-analytical models and obtained an expression for the onset velocityas a function of the eigenfrequency and the Scruton number. As the simplifiedaerodynamic damping is independent of the circumferential wavenumber and theenergy dissipated by structural damping during ovalling is proportional to theproduct of the modal logarithmic decrement and the square of the eigenfrequency,the eigenmode with the lowest value for this product is excited first. Theseanalytical models generally require a substantial amount of experimental data likethe structural modal damping ratio, the separation angle and the base pressurecoefficient in order to obtain a good correspondence between measured andpredicted onset velocities. They do not provide physical insight in the occurringflow phenomena.

OBJECTIVES AND ORIGINAL CONTRIBUTIONS 11

Numerical simulations

Wuchner et al. (2005) and Bletzinger et al. (2006) numerically simulated ovallingoscillations by coupling the computation of the two-dimensional wind flow witha finite element model of a ring, which corresponds to an infinitely long circularcylinder. The three-dimensional behaviour in the structure, the flow and theinteraction are neglected. The eigenfrequencies and the sequence of the eigenmodesof an infinitely long cylinder differ considerably of cylinders of finite length.Any choice made for the boundary conditions of the ring structure changes theeigenfrequencies of some eigenmodes. The differences in dimensions, materialproperties and wind speed render the validation with the experimental resultsfrom Johns and Sharma (1974) impossible.

1.3 Objectives and original contributions

The behaviour under wind loading of civil engineering structures that are quiteflexible, have a complex geometry or are located in the presence of neighbouringstructures is very difficult to determine. Nowadays, the common approach isto perform a series of wind tunnel tests. The aim of the thesis is to studycoupled numerical simulations of the fluid and the structure to determine thisbehaviour. The different numerical methods applied in the individual fields (fluidand structure) and in the coupling procedure are explored. More specifically, thework focusses on the simulation of wind-structure interaction phenomena occurringon shell structures. As the present thesis is the first work dealing with CFD andFSI in the research group, one of the objectives is to give in each chapter anextensive and coherent overview of the related theory.

With regard to the example of wind induced ovalling oscillations of a silo, thecurrent work will focuss on the creation of accurate models for the fluid andstructural field and on the computation of the ovalling oscillations for a singlesilo. Future work should investigate the influence of the small distance betweentwo neighbouring silos on the occurrence of ovalling and propose design alterationsin order to avoid the occurrence of the oscillations.

1.3.1 Methodology

The methods to simulate wind-structure interaction phenomena occurring on shellstructures will be introduced by means of the example of wind induced ovallingoscillations of a silo. This dynamic fluid-structure interaction problem is estimatedrepresentative for the problems under consideration. Although it has a simplecircular cylindrical geometry, it covers all aspects specific for a general wind-shell

12 INTRODUCTION

interaction problem: the geometrical non-linear deformations of a shell structure,the turbulent wind flow with separation and finally the self-excited phenomenon,that can only be assessed numerically by methods which take the interactionbetween fluid and structure into account. The adapted methodology is generaland can also be applied to the problems mentioned in the previous sections.

For the numerical simulation of fluid-structure interaction a partitioned approachis used: the fluid and the structure are treated as isolated fields and solvedseparately. The interaction effects are applied on the individual fields as externalboundary conditions which are exchanged from one field to the other. Themain advantage of this partitioned approach is that well-established discretizationtechniques and solution algorithms (and by consequence existing software) whichare tailored to the characteristic behaviour of the individual fields can be usedin each field. As the present thesis is the first work dealing with CFD and FSIin the research group, it was a conscious choice at the beginning to make useof existing software: Ansys (Ansys, 2005b) for the structural computations andFlotran (Ansys, 2005b) and CFX (Ansys, 2005a) for the flow computations. Byconsequence, the possible choices regarding models and algorithms were restrictedto those available in these packages (appendix A). As the fluid flow equations(e.g. the Navier-Stokes equations) are non-linear and the non-linear behaviour ofthe structure should be taken into account, direct time integration is used in bothfields.

Prior to the study of fluid-structure interaction phenomena, a thorough knowledgeof the structural and the fluid field is required, for which accurate numerical modelsare needed.

The structure is discretized by means of the finite element method and integratedin time using the Newmark time integration scheme. The formulation takes intoaccount geometrically non-linear behaviour and allows for large rotations and largedisplacements. As for the prediction of ovalling oscillations the exact value of thestructural modal damping is very important, an in situ experiment is performedin order to determine the modal damping ratios of the silos. The measuredeigenfrequencies and eigenmodes are used to validate a three-dimensional finiteelement model of a silo.

For the fluid a finite volume discretization is applied and the solution is integratedin time using the three-point backward difference method. A two-dimensionalcomputation of the flow around one rigid cylinder is made. The effect of theclose spacing of cylinders on the flow pattern is studied for a group of rigidcylinders as in the port of Antwerp. For the FSI computations, the mesh in thefluid field is chosen to be aligned with the deforming boundary of the structure.The fluid computations are performed on a deforming mesh and the governingequations are formulated in an arbitrary Lagrangian-Eulerian description. Themesh deformation is computed by means of a diffusion equation with a variable

OBJECTIVES AND ORIGINAL CONTRIBUTIONS 13

diffusivity and should preferentially preserve the quality and the refinements ofthe mesh.

Between Ansys and Flotran and between Ansys and CFX, a coupling algorithmis commercially available. Two partitioned solution algorithms are employed: theloosely coupled serial staggered algorithm which solves each field one time in eachtime step and a strongly coupled staggered algorithm which iterates between thetwo fields in each time step. The meshes of the fluid and the structure are non-matching at the fluid-structure interface. A consistent interpolation method isused for the displacement transfer, while a node to element conservative methodis applied for the load transfer. The interaction between the wind flow and a singlesilo is computed.

1.3.2 Original contributions

A three-dimensional finite element model of a circular cylindrical silo has beenvalidated by means of in situ experiments performed on a single silo. Radialaccelerations at 10 points along the silo are measured under ambient wind loadingand modal parameters are extracted from the output-only data using the stochasticsubspace identification technique. The influence of the boundary conditions on theeigenfrequencies and mode shapes is investigated.

The current state-of-the-art methods for the simulation of turbulent flows arereviewed as a function of their performance and their computational requirements.The influence of turbulence models, near-wall mesh refinement and unsteadinessis studied for the two-dimensional flow around a single cylinder in the post-criticalregime. The results are compared with experimental data and numerical resultsavailable in the literature. The two-dimensional flow around a group of 2 by 2and of 8 by 5 cylinders is computed and shows a strong influence of the groupconfiguration on the pressure distribution around the cylinders.

A coherent and comprehensive overview of the whole set of numerical techniquesrequired for the simulation of dynamic fluid-structure interaction is given. Forthe fluid a boundary-fitted mesh is used together with the arbitrary Lagrangian-Eulerian description. The governing equations for the flow in an arbitraryLagrangian-Eulerian description are derived. The implication of this descriptionon the time-accuracy and the stability of the time integration is discussed. Thedifferent options to compute the fluid mesh velocity and deformation are reviewed.The coupled problem is solved by a partitioned algorithm, where the fluid and thestructure are separately integrated in time and the interaction effects are appliedas external boundary conditions. The accuracy, stability and efficiency of looselyand strongly coupled algorithms is discussed in the case of incompressible flows.Different methods for load and motion transfer between non-matching meshes aredescribed as a function of their accuracy and conservation properties.

14 INTRODUCTION

As a practical application, the interaction between the wind flow and a single silois computed. The three-dimensional finite element model of the silo is coupledwith the three-dimensional incompressible turbulent wind flow as to predictovalling oscillations. The results computed with the loosely coupled conventionalserial staggered algorithm show differences that increase in time with the resultscomputed with the strongly coupled algorithm using subiterations, which is moreaccurate.

1.4 Organization of the text

Prior to the study of wind-structure interaction phenomena, a thoroughunderstanding of the structural and the fluid behaviour is required, for whichaccurate numerical models are needed. Much effort has therefore been devoted tothe solution of the separate problems. These models are used in chapter 5 for acoupled numerical analysis of the wind flow around the structure that simulatesthe ovalling phenomenon.

Chapter 1 situates the subject of the thesis and highlights the objectives andoriginal contributions. The ovalling of cylindrical shell structures is introduced.This dynamic fluid-structure interaction problem is considered as an examplethroughout the thesis.

Chapter 2 describes the arbitrary Lagrangian-Eulerian description for flowcomputations on domains with deforming boundaries. The concepts ofstabilization for finite element discretizations and upwinding for finite volumediscretizations, which are needed for flow computations, are presented. Thestability and the time accuracy of computations on domains with deformingboundaries are discussed and related to the geometric conservation law. Themethods to compute the mesh deformation are reviewed.

Chapter 3 reviews the governing equations for linear and geometrically non-linear behaviour of shell structures. In situ experiments performed on a singlesilo in order to obtain the eigenfrequencies, eigenmodes and modal damping ratiosare described. A three-dimensional finite element model of a silo is validated bymeans of these experimental modal parameters. The importance of geometricallynon-linear behaviour during ovalling of the silos is evaluated.

ORGANIZATION OF THE TEXT 15

Chapter 4 treats the computation of turbulent wind flows around rigid civilengineering structures. Existing turbulence models are reviewed. The resultsof steady and unsteady Reynolds averaged Navier-Stokes simulations of the flowaround a single silo are compared with experimental data and few availablenumerical results in order to select a suitable turbulence model. The computationof the flow around a closely spaced group of 8 by 5 cylinders illustrates the influenceof the group configuration on the vortex shedding and the pressure distributionsaround the cylinders.

Chapter 5 describes the partitioned algorithms to solve the coupled dynamicfluid-structure interaction problem by computing the different fields sequentiallyor in parallel. The accuracy, stability and efficiency of these algorithms is discussed.The transfer of displacements and loads between non-matching discretizations ofthe fluid and the structure at the interface is treated. The conservation of the totalload and of the energy at the interface is of importance. A coupled simulation ofa three-dimensional wind flow around a single deforming silo is performed. Moreefficient coupling procedures should enable simulations during a much longer timeinterval.

Chapter 6 summarizes the conclusions of the thesis and presents recommenda-tions for further research.

Chapter 2

Flow on a domain withdeforming boundaries

2.1 Introduction

As the aim is to perform a coupled numerical simulation of the fluid and thestructure, the position of the structure determines at least partially the fluiddomain boundaries. If the structure undergoes large displacements, it is necessaryto perform the computations of the fluid flow on a domain with moving boundaries.

Many different approaches exist for the computations of flows on a domain withdeforming boundaries.

In a first approach, the deforming boundary is described explicitly. The mesh of thefluid is aligned with the deforming boundary (figure 2.1a). As the boundary moves,the mesh should follow this boundary. This can be achieved by deforming the meshwithout modifying its topology. If, however, the mesh becomes too distorted, thedomain has to be remeshed. In order to compute flow on a deforming mesh,the governing equations should be written in an arbitrary Lagrangian-Euleriandescription (Hughes et al., 1981; Donea et al., 1982). If this approach is used tocalculate free surface flows, it is called an interface tracking technique.

In a second approach, the deforming boundary is described implicitly. The flowcomputation is performed in an Eulerian description on a grid which is fixed inspace and covers the union of the fluid and the solid (or empty) domain. Thisextended domain has usually a simpler geometrical shape and does not change withtime. The solid domain is allowed to move with respect to the fixed grid (figure2.1b), so the fluid elements that contain the solid domain change continuously.

17

18 FLOW ON A DOMAIN WITH DEFORMING BOUNDARIES

The need for remeshing or mesh deformation is eliminated. As the meshes arenot boundary-fitted, structured refined boundary layer meshes next to the solidboundary are not possible. The governing equations of the fluid domain are alteredin order to reflect the presence of the solid domain. In the immersed boundarymethod (Peskin, 1972; Peskin and Mcqueen, 1980) a body force is added to the fluidequations. This force is equal to a Dirac delta function which differs from zero atthe fluid boundary. The fictitious domain method (Glowinski et al., 1994; Baaĳens,2001; De Hart et al., 2003) imposes the boundary conditions for the velocities atthe fluid-solid interface using Lagrange multipliers. In the extended finite elementmethod (X-FEM) (Gerstenberger and Wall, 2008), the approximation space isenriched with additional functions that might be discontinuous as well. In thisway the discontinuities in the pressure and the fluid field at the location wherea thin structure is present, can be included. Applied to free surface flows, thesetechniques are called interface capturing techniques.

(a) (b)

Figure 2.1: (a) Explicit and (b) implicit description of a deforming boundary ofthe fluid domain.

Combinations of the above approaches exist as well. For fluid flows around movingand rotating rigid bodies, the Chimera technique (Steger et al., 1983) is often used.The fluid domain consists of two meshes which partially overlap: a backgroundmesh which is fixed in space and a mesh which moves together with the rigidbody. This approach can be extended to a technique (Wall et al., 2006) where abackground mesh is fixed in space and the other mesh moves and deforms togetherwith the structure using an arbitrary Lagrangian-Eulerian description.

The particle finite element method (PFEM) (Idelsohn et al., 2004) uses thepositions of a number of particles to generate a new mesh at every time step.The particles correspond to the fluid nodes. If the distance between the particlesis too large, there is no element created between these particles. In this waythe domain may split in several parts or individual particles may leave the fluid

FLOW ON A DOMAIN WITH DEFORMING BOUNDARIES 19

and possibly join later again. As the fluid flow is computed using a Lagrangianformulation, convective terms are not present in the governing equations. Thismethod is especially suited to solve free surface problems, breaking waves andfluid particle separation.

For the applications in mind, no fluid particles are leaving the fluid domain nordoes the structural domain split due to fractures or explosions. The structuraldeformations are expected to be large, but no rotations of 360 degrees willoccur. Therefore, a boundary-fitted mesh can be used together with the arbitraryLagrangian-Eulerian description without frequent remeshing. Its ability to havestructured boundary layer meshes near the structure is very important. First,the governing equations for the fluid flow in an arbitrary Lagrangian-Euleriandescription are derived. The spatial discretization by means of the finite elementand the finite volume method is discussed. Using an Eulerian description, the timeintegration is straightforward. In the case of an arbitrary Lagrangian-Euleriandescription, specific choices have to be made for the time integration in order topreserve the time accuracy and the stability. Finally, different options to computethe fluid mesh deformations are reviewed.

2.2 Flow on a domain with deforming boundaries

2.2.1 The arbitrary Lagrangian-Eulerian description

Two classical kinematic descriptions are extensively used in continuum mechanics:the Lagrangian and the Eulerian description. The arbitrary Lagrangian-Euleriandescription (ALE) was developed to combine the advantages of both descriptions.

In a Lagrangian description (figure 2.2a) an observer is fixed to a material particleand follows its motion. This permits easy tracking of free surfaces and interfacesbetween different materials where boundary conditions are conveniently described.Physical quantities of a material particle are easily obtained. As the observerfollows the motion of material particles, the material time derivative reduces toa simple time derivative and convective terms do not occur in the governingequations. After discretization of these governing equations, the nodes of themesh function as observers. Each element of the mesh contains at each timethe same material particles, which enables to incorporate time dependent materialbehaviour. Conservation of mass is automatically satisfied. However, in non-linearcomputations large distortions of the continuum result in severe deterioration ofthe quality of the mesh and loss in accuracy. This description is most commonlyused in structural analysis.

In an Eulerian description (figure 2.2b) the observer is fixed in space. Physicalquantities are examined in a fixed point in space as material particles pass. The


material time derivative introduces convective terms in the governing equations.The material particles move with respect to the mesh and the continuum is allowedto undergo large distortions without loss of accuracy. Tracking of free surfaces andinterfaces is more difficult. For fluid analysis, an Eulerian description is often used,although a Lagrangian description might be used for contained fluids which onlyundergo small motions.

The ALE description (figure 2.2c) combines the advantages of the two descriptions:the observer moves arbitrarily through space. At an interface between differentmaterials or at a free surface, the observer may follow the material particles, whileat an inlet or outlet of the domain the observer may stay fixed in space. Inside thedomain, the movement of the nodes (or the observers) can be chosen arbitrarily butshould preferentially preserve the regularity of the mesh. The different possibilitiesto automatically specify suitable movement of the nodes will be treated in section2.5. In figure 2.2c the nodes of the left curved boundary follow the materialparticles, while the nodes at the right straight boundary are fixed in space. Thenodes in between these boundaries neither follow the material particles, neitherare fixed in space.

t

x1

x2

(a)t

x1

x2

(b)t

x1

x2

(c)

Figure 2.2: Representations of (a) the Lagrangian, (b) the Eulerian and (c) thearbitrary Lagrangian-Eulerian descriptions in the spatial domain Ωx

at two different times t: continuum (dark gray), material particles(black dots) and mesh (black lines). Dashed lines show the meshmotion, while solid lines show the particle motion.

The ALE description was first developed in the framework of the finite differencemethod by Hirt et al. (1974). Hughes et al. (1981) and Donea et al. (1982)described the first implementations for the finite element method. The ALEdescription was primarily used to compute flows on domains with movingboundaries like free surfaces or fluid-structure interfaces. Its use has been extendedto non-linear structural mechanics to deal with crack propagation, impacts,explosions, penetrations and forming processes (Liu et al., 1988), for which theALE description copes with the significant distortions of the material.

Computations using the ALE description are performed in the referential domainΩχ where the mesh is fixed at all time steps (figure 2.3): a node has the samereferential coordinates χ at each time t. The mapping ϕ defines for a node with


referential coordinates χ what the position x in space is of the material particlepresent at this node at time t:

ϕ : Ωχ × [t0,∞[→ Ωx : (χ, t) 7→ x = ϕ(χ, t) (2.1)

The spatial domain Ωx(t) which corresponds at time t + ∆t to the referentialdomain Ωχ is indicated by a dashed line. The mapping ϕ can be interpretedas the position of the nodes in the spatial domain. The time derivative of thismapping with the referential coordinate held fixed is denoted as v and describesthe velocity of the nodes of the mesh:

v =∂x(χ, t)

∂t

∣∣∣∣χ

(2.2)

The initial positions X where the material present in the mesh at time t waslocated at time t = 0 determine the material domain ΩX. The material domainΩX(t) at time t + ∆t is indicated by a dashed line. The mapping ϕ determinesfor a material particle with initial position X at time t = 0 what the referentialcoordinates χ in the mesh are at time t:

ϕ : ΩX × [t0,∞[→ Ωχ : (X, t) 7→ χ = ϕ(X, t) (2.3)

This mapping ϕ can be interpreted as the position of the material particles inthe referential domain. The time derivative of this mapping with the materialcoordinate held fixed is denoted as v and describes the particle velocity in thereferential domain:

v =∂χ(X, t)

∂t

∣∣∣∣X

(2.4)

Ωχ

Ωx

ΩX

ϕϕ

ϕ ϕ

Figure 2.3: Domains, meshes and mappings for an ALE description: fixedreferential domain Ωχ, changing material domain ΩX and changingspatial domain Ωx at time t (solid lines) and t+ ∆t (dashed lines).


The material particle motion ϕ, typically used in structural mechanics, relatesthe initial position of a material particle with its current position at time t and isobtained as a combination of the mappings ϕ and ϕ:

ϕ = ϕ ϕ : ΩX× [t0,∞[→ Ωx : (X, t) 7→ x = ϕ(X, t) = ϕ(ϕ(X, t), t) (2.5)

The displacements u of a particle are obtained by subtracting the initial positionX form the spatial position x:

u = x−X (2.6)

The time derivative of this mapping ϕ with the material coordinate held fixed isdenoted as v and describes the particle velocity in the spatial domain:

v =∂x(X, t)

∂t

∣∣∣∣X

(2.7)

In figure 2.3 the nodes of the left curved boundary follow the material particles andcoincide in material and referential domain, while the nodes at the right straightboundary are fixed in space and coincide in spatial and referential domain.

Both Lagrangian and Eulerian descriptions can be obtained as special cases ofthe ALE description: in a Lagrangian description the material and the referentialdomain coincide and ϕ = I. The mesh moves together with the material particles.In an Eulerian description the spatial and the referential domain coincide andϕ = I. The mesh is fixed in space.

The description of a scalar physical quantity f in the referential domain Ωχ involvesa complementary description g in the spatial domain:

g(x, t) = f(ϕ−1(x, t), t) = f(χ, t) (2.8)

g = f ϕ−1 (2.9)

and a complementary description h in the material domain:

h(X, t) = f(ϕ(X, t), t) = f(χ, t) (2.10)

h = f ϕ (2.11)

For simplicity, the complementary descriptions g and h will be denoted as f(x, t) =f(X, t) = f(χ, t) from here on.

The material time derivative (also known as the substantive derivative, thesubstantial derivative, the total time derivative, the convective derivative, theadvective derivative or the Lagrangian derivative) appears in the basic conservationlaws for mass, momentum and energy. An expression for the material time


derivative of a scalar quantity f is obtained in terms of the referential timederivative and a convective term taking into account the relative motion betweenthe material particle and the mesh:

f =Df

Dt=∂f(X, t)

∂t

∣∣∣∣X

=∂f(χ, t)

∂t

∣∣∣∣χ

+∂χ(X, t)

∂t

∣∣∣∣X

· ∇χf(χ, t) (2.12)

The first factor of the last term is equal to the particle velocity v in the referentialdomain (equation (2.4)):

Df

Dt=∂f(χ, t)

∂t

∣∣∣∣χ

+ v · ∇χf(χ, t) (2.13)

The application of this equation to the spatial coordinate x yields an expressionfor the particle velocity v:

v =Dx

Dt=∂x(χ, t)

∂t

∣∣∣∣χ

+ (v · ∇χ)x(χ, t) (2.14)

The first term on the right hand side is equal to the grid velocity v (equation(2.2)):

v = v + (v · ∇χ)x(χ, t) (2.15)

The second term on the right hand side of equation (2.15) is equal to the convectivevelocity c:

c = v− v = (v · ∇χ)x(χ, t) =∂x(X, t)

∂t

∣∣∣∣X

− ∂x(χ, t)

∂t

∣∣∣∣χ

(2.16)

The convective velocity c is the relative velocity between material particles andthe mesh in the spatial domain and differs from the particle velocity v in thereferential domain. The convective velocity c and the particle velocity v in thereferential domain are equal if ∇χx(χ, t) = I which is the case when the meshonly translates without any rotation or deformation.

As mentioned before, both Lagrangian and Eulerian descriptions can be obtainedas special cases of the ALE description. If the grid velocity v is equal to zero,the grid is fixed in space and an Eulerian description is recovered. The convectivevelocity c and the particle velocity v in the referential domain equal the materialparticle velocity v. If, on the other hand, the grid velocity v is equal to thematerial particle velocity v, the grid moves together with the material particlesand a Lagrangian description is retrieved. The convective velocity c and theparticle velocity v in the referential domain are zero.

In equation (2.12) the gradient of the quantity f in the referential domain hasto be computed. As constitutive relations are naturally expressed in the spatial


domain, equation (2.12) is rewritten as a function of the convective velocity c usingequation (2.16) and the chain rule:

f =Df

Dt=∂f(X, t)

∂t

∣∣∣∣X

=∂f(χ, t)

∂t

∣∣∣∣χ

+ v · ∇xf(x, t)∇χx(χ, t) (2.17)

=∂f(χ, t)

∂t

∣∣∣∣χ

+ c · ∇xf(x, t) (2.18)

This fundamental ALE equation shows that the material time derivative of ascalar quantity f is equal to its referential time derivative and a convective termconsisting of the convective velocity c and its spatial gradient. Application to thedifferent components of the velocity v yields an expression for the material timederivative of the particle velocity:

v =Dv

Dt=∂v(χ, t)

∂t

∣∣∣∣χ

+ (c · ∇x)v(x, t) (2.19)

The reader familiar with the derivation of the continuity and momentum equationsin the Eulerian, Lagrangian and ALE formulation may proceed immediately withsection 2.2.2.

Deformation gradients

The relation between an infinitesimal vector dX in the material domain and in thespatial domain dx is given by the deformation gradient F:

dx = ∇XxdX = FdX (2.20)

An analogous deformation gradient is defined between the referential and thespatial domain:

dx = ∇χxdχ = Fdχ (2.21)

and between the material and the referential domain:

dχ = ∇XχdX = FdX (2.22)

Volume and area change

The infinitesimal volume in the material domain with edges parallel to theCartesian axes is given as:

dΩX = dX1 · (dX2 × dX3) = dX1E1 · (dX2E2 × dX3E3) (2.23)

= dX1dX2dX3 (2.24)


where E1, E2 and E3 are orthogonal unit base vectors. The same infinitesimalvolume in the spatial domain is expressed as:

dΩx = dx1 · (dx2 × dx3) (2.25)

Using equation (2.20) the infinitesimal vectors dx1, dx2 and dx3 in the spatialdomain are obtained in terms of the infinitesimal vectors dX1, dX2 and dX3 inthe material domain:

dΩx = FdX1 · (FdX2 × FdX3) (2.26)

= (FE1 · (FE2 × FE3))dX1dX2dX3 (2.27)

= det FdΩX = JdΩX (2.28)

where the determinant of the deformation gradient F defines the Jacobian J , whichrelates the volume dΩX in the material domain to the volume dΩx in the spatialdomain.Analogously, the Jacobian J relates the volume dΩχ in the referential domain tothe volume dΩx in the spatial domain:

dΩx = det FdΩχ = JdΩχ (2.29)

and the Jacobian J relates the volume in the material domain dΩX to the volumedΩχ in the referential domain:

dΩχ = det FdΩX = JdΩX (2.30)

Substituting equation (2.30) in equation (2.29) gives:

dΩx = JdΩχ = J JdΩX (2.31)

Comparison with equation (2.28) yields the following relation:

J = J J (2.32)

The time rate of change of the Jacobians is given by:

DJ

Dt=∂J

∂t

∣∣∣∣X

= J∇x · v (2.33)

∂J

∂t

∣∣∣∣∣χ

= J∇x · v (2.34)

DJ

Dt=∂J

∂t

∣∣∣∣∣X

= J∇χ · v (2.35)


Expressing the infinitesimal volumes dΩx and dΩX as the dot product of a vectordl and an area dΓ, equation (2.28) for the volume change between the materialand the spatial domain yields:

dlx · dΓx = JdlX · dΓX (2.36)

Recalling equation (2.20), the vector dlx is expressed in terms of the vector dlX:

(FdlX) · dΓx = JdlX · dΓX (2.37)

As this expression is valid for any vector dlX, a relation between the infinitesimalareas in the material and the spatial domain is obtained:

FTdΓx = JdΓX (2.38)

or

dΓx = JF−TdΓX (2.39)

Analogously, a relation between the infinitesimal areas in the referential and thespatial domain is obtained:

dΓx = JF−TdΓχ (2.40)

The relation between the infinitesimal areas in the material and the referentialdomain is:

dΓχ = JF−TdΓX (2.41)

The material time derivative of an extensive property

An extensive property G(t) is defined as the volume integral of the mass densityρx times an intensive property g over the spatial domain:

G(t) =

∫

Ωx

ρxgdΩx (2.42)

The material time derivative of the extensive property G(t) is:

DG(t)

Dt=D

Dt

∫

Ωx

ρxgdΩx (2.43)

Recalling equation (2.28) for the change in volume between the spatial and thematerial domain, the material time derivative is rewritten:

DG(t)

Dt=D

Dt

∫

ΩX

ρxgJdΩX =

∫

ΩX

(Dρxg

DtJ + ρxg

DJ

Dt

)dΩX (2.44)


Using the fundamental ALE equation (2.19) for the material time derivative of thevector ρxg and equation (2.33) for the rate of change of the Jacobian J gives:

DG(t)

Dt=

∫

ΩX

((∂ρxg

∂t

∣∣∣∣χ

+ (c · ∇x)(ρxg)

)J + ρxgJ∇x · v

)dΩX (2.45)

or

DG(t)

Dt=

∫

ΩX

(∂ρxg

∂t

∣∣∣∣χ

+ (c · ∇x)(ρxg) + ρxg∇x · v)JdΩX (2.46)

Equation (2.28) enables the integral to be written again as an integral over thespatial domain:

DG(t)

Dt=

∫

Ωx

(∂ρxg

∂t

∣∣∣∣χ

+ (c · ∇x)(ρxg) + ρxg∇x · v)dΩx (2.47)

Adding and subtracting the term ρxg∇x · v to the integrand gives:

DG(t)

Dt=

∫

Ωx

(∂ρxg

∂t

∣∣∣∣χ

+ (c · ∇x)(ρxg) + ρxg∇x · v

− ρxg∇x · v + ρxg∇x · v)dΩx (2.48)

Using equation (2.16) the second, third and fourth term are combined and bymeans of equation (2.34) the last term is transformed:

DG(t)

Dt=

∫

Ωχ

∂ρxg

∂t

∣∣∣∣χ

+∇x · (ρxg⊗ c) + ρxg1

J

∂J

∂t

∣∣∣∣∣χ

dΩx (2.49)

Equation (2.29) enables the integral to be written as an integral over the referentialdomain:

DG(t)

Dt=

∫

Ωχ

J ∂ρxg

∂t

∣∣∣∣χ

+ J∇x · (ρxg⊗ c) + ρxg∂J

∂t

∣∣∣∣∣χ

dΩχ (2.50)

The first and third term of the integrand are combined:

DG(t)

Dt=

∫

Ωχ

∂Jρxg

∂t

∣∣∣∣∣χ

+ J∇x · (ρxg⊗ c)

dΩχ (2.51)


The referential time derivative in the first term can be brought outside the integral:

DG(t)

Dt=∂

∂t

(∫

Ωχ

JρxgdΩχ

)∣∣∣∣∣χ

+

∫

Ωx

J∇x · (ρxg⊗ c)dΩχ (2.52)

Equation (2.29) enables the integrals to be written again as an integral over thespatial domain:

DG(t)

Dt=∂

∂t

(∫

Ωx

ρxgdΩx

)∣∣∣∣χ

+

∫

Ωx

∇x · (ρxg⊗ c)dΩx (2.53)

Conservation of mass

If in equation (2.42) the intensive property g is equal to 1, the extensive propertyis the mass M(t) contained in the spatial domain Ωx:

M(t) =

∫

Ωx

ρxdΩx (2.54)

Conservation of mass states that the mass contained in a material volume isconserved or that the material time derivative of the the mass M(t) is equal tozero:

DM(t)

Dt=D

Dt

∫

Ωx

ρxdΩx = 0 (2.55)

Using equation (2.47) with g equal to 1, the expression for conservation of massis:

∫

Ωx

(∂ρx∂t

∣∣∣∣χ

+ c · ∇xρx + ρx∇x · v)dΩx = 0 (2.56)

Since the volume integral is zero for any arbitrarily chosen volume, the integrandmust be pointwise equal to zero in the spatial domain, which yields the differentialform of the law of conservation of mass, known as the continuity equation:

∂ρx∂t

∣∣∣∣χ

+ c · ∇xρx + ρx∇x · v = 0 (2.57)

The time derivative is taken holding the referential coordinate χ fixed, while allother derivatives are calculated in the spatial domain.

Second, equation (2.51) with g equal to 1 can be substituted in (2.55):

∫

Ωχ

∂Jρx∂t

∣∣∣∣∣χ

+ J∇x · (ρxc)

dΩχ = 0 (2.58)


As the integrand must be pointwise equal to zero in the referential domain, thecontinuity equation becomes:

∂Jρx∂t

∣∣∣∣∣χ

+ J∇x · (ρxc) = 0 (2.59)

Third, equation (2.53) with g equal to 1 can be substituted in (2.55):

∂

∂t

(∫

Ωx

ρxdΩx

)∣∣∣∣χ

+

∫

Ωx

∇x · (ρxc)dΩx = 0 (2.60)

The advantage of this equation is that the referential time derivative of a volumeintegral is taken and that both volume integrals are calculated in the spatialdomain.

From the continuity equations in the ALE description the continuity equation inEulerian description can be derived. In the Eulerian description the spatial andthe referential domain coincide and the convective velocity c and the materialparticle velocity v are equal. From equation (2.57) the differential form of thelaw of conservation of mass in an Eulerian description, as used in fluid mechanics,emerges:

∂ρx∂t

∣∣∣∣x

+ v · ∇xρx + ρx∇x · v = 0 (2.61)

As to formulate conservation of mass (2.55) in the material domain, the integrandcan be rewritten using equation (2.28):

ρxdΩx = ρxJdΩX = ρXdΩX (2.62)

where the density ρX is defined as ρX = Jρx. The expression for conservation ofmass in the material domain becomes:

DM(t)

Dt=D

Dt

∫

ΩX

ρXdΩX = 0 (2.63)

As the material time derivative is taken holding the material coordinate X fixed,the differential form of the law of conservation of mass in Lagrangian descriptionin the material domain, as used in structural mechanics, is easily obtained:

∂ρX∂t

∣∣∣∣X

= 0 (2.64)

This shows that the density ρX is constant in time in the material domain dΩX.


Conservation of momentum

If in equation (2.42) the intensive property g is equal to v, the extensive propertyis the momentum M(t) contained in the spatial domain Ωx:

M(t) =

∫

Ωx

ρxvdΩx (2.65)

Conservation of momentum states that the change in momentum of a materialvolume M(t) with respect to time is equal to the sum of all external forces on thisvolume:

DM(t)

Dt=D

Dt

∫

Ωx

ρxvdΩx =

∫

Γx

tdΓx +

∫

Ωx

ρxbdΩx (2.66)

where t is the traction that act on the boundary of the volume and ρxb is the bodyforce. The traction vector t on a surface with unit normal vector nx is obtainedas:

t = σnx (2.67)

where the second order tensor σ is called the Cauchy stress tensor. Using thisequation, Gauss’ theorem enables to write the integral over the boundary in thefirst term on the right hand side of equation (2.66) as a volume integral:

D

Dt

∫

Ωx

ρxvdΩx =

∫

Ωx

∇x · σdΩx +

∫

Ωx

ρxbdΩx (2.68)

Using equation (2.44) with g equal to v, the expression for conservation ofmomentum is:

∫

ΩX

(Dρxv

DtJ + ρxv

DJ

Dt

)dΩX =

∫

Ωx

∇x · σdΩx +

∫

Ωx

ρxbdΩx (2.69)

The left hand side is elaborated:∫

ΩX

(ρxDv

DtJ + v

DρxDtJ + ρxv

DJ

Dt

)dΩX (2.70)

=

∫

ΩX

(ρxDv

DtJ + v

[DρxDtJ + ρx

DJ

Dt

])dΩX (2.71)

The term in square brackets is equal to zero as this follows from the conservationof mass (equation (2.44) with g equal to 1). Using equation (2.19) for the materialtime derivative of the vector function v in the ALE description and equation (2.28)for the volume change, gives:

DM(t)

Dt=

∫

Ωx

ρx

(∂v

∂t

∣∣∣∣χ

+ (c · ∇x)v

)dΩx (2.72)


From the combination with equation (2.69), emerges the expression forconservation of momentum:

∫

Ωx

ρx

(∂v

∂t

∣∣∣∣χ

+ (c · ∇x)v

)dΩx =

∫

Ωx

∇x · σdΩx +

∫

Ωx

ρxbdΩx (2.73)

Since the volume integrals are equal for any arbitrarily chosen volume, theintegrand must be pointwise equal in the spatial domain, which yields the followingthe momentum equation:

ρx∂v

∂t

∣∣∣∣χ

+ ρx(c · ∇x)v = ∇x · σ + ρxb (2.74)

Second, equation (2.51) with g equal to v can be substituted in (2.68) and thevolume integrals are transformed to the referential domain (2.29):

∫

Ωχ

∂Jρxv

∂t

∣∣∣∣∣χ

+ J∇x · (ρxv⊗ c)

dΩχ =

∫

Ωx

J∇x · σdΩχ +

∫

Ωx

JρxbdΩχ

(2.75)

As the integrand must be pointwise equal to zero in the referential domain, themomentum equation becomes:

∂Jρxv

∂t

∣∣∣∣∣χ

+ J∇x · (ρxv⊗ c) = J∇x · σ + Jρxb (2.76)

Third, equation (2.53) with g equal to v can be substituted in (2.68):

∂

∂t

(∫

Ωx

ρxvdΩx

)∣∣∣∣χ

+

∫

Ωx

∇x·(ρxv⊗c)dΩx =

∫

Ωx

∇x·σdΩx+

∫

Ωx

ρxbdΩx (2.77)

The advantage of this equation is that the referential time derivative of a volumeintegral is taken and that both volume integrals are calculated in the spatialdomain.

From the momentum equations (2.74) in the ALE description the momentumequations in Eulerian and Lagrangian description can be derived. In the Euleriandescription the spatial and the referential domain coincide and the convectivevelocity c and the material particle velocity v are equal. From equation (2.74) thedifferential form of the law of conservation of momentum in Eulerian descriptionin the spatial domain, as used in fluid mechanics, emerges:

ρx∂v

∂t

∣∣∣∣x

+ ρx(v · ∇x)v = ∇x · σ + ρxb (2.78)


In the Lagrangian description the material and the referential domain coincideand the convective velocity c and the particle velocity v in the referential domainare zero. From equation (2.74) the differential form of the law of conservation ofmomentum in Lagrangian description in the material domain, as used in structuralmechanics, is easily obtained:

ρx∂v

∂t

∣∣∣∣X

= ∇x · σ + ρxb (2.79)

As to formulate conservation of momentum (2.66) in the material domain, theboundary integral in the first term on the right hand side should be transformedto the material domain. The integrand of this term is transformed using equation(2.67):

tdΓx = σnxdΓx = σdΓx (2.80)

where dΓx is the area vector. Recalling equation (2.39) for the relation betweenan area vector in the spatial and the material domain, yields:

tdΓx = σJF−TdΓX = JσF−TnXdΓX (2.81)

where nX denotes the unit normal vector to the same area in the material domain.In the material domain the traction tX = JσF−TnX is defined such that tdΓx =tXdΓX. The first Piola-Kirchhoff stress tensor PX = JσF−T is defined analogousto the Cauchy stress tensor (2.67) in the spatial domain:

tX = PXnX (2.82)

The first Piola-Kirchhoff stress tensor PX in the material domain relates the areavector dΓX in the material domain to the corresponding force vector tdΓx in thespatial domain.

The volume integrals on the left hand side and in the second term on the righthand side of equations (2.66) can be rewritten using equation (2.62). In the firstterm on the right hand side equations (2.81) and (2.82) are substituted.

D

Dt

∫

ΩX

ρXvdΩX =

∫

ΩX

∇X ·PXdΩX +

∫

ΩX

ρXbdΩX (2.83)

This is the expression for conservation of momentum in the material domain.The material time derivative is taken holding the material coordinate X fixed.Taking into account the conservation of mass (2.64), the momentum equation inLagrangian description in the material domain, which is useful for structures withnon-linear behaviour, is obtained:

ρX∂v

∂t

∣∣∣∣X

= ∇X ·PX + ρXb (2.84)


Conservation of moment of momentum

The conservation of moment of momentum is not elaborated here but implies thesymmetry of the Cauchy stress tensor:

σ = σT (2.85)

2.2.2 Kinematics

The velocity increment between two neighbouring particles can be expressed as:

dv = ∇xvdx (2.86)

The velocity gradient is decomposed into its symmetric and skew-symmetric parts:

∇xv =1

2(∇xv + (∇xv)T) +

1

2(∇xv− (∇xv)T) = ε + ω (2.87)

The symmetric tensor ε is called the strain rate tensor and describes thedeformations of the fluid volume, while the skew-symmetric tensor ω is calledthe vorticity tensor and describes rigid body rotations. The strain rate can besplit into its isotropic and deviatoric part:

∇xv =1

3(trε)I +

(ε− 1

3(trε)I

)+ ω (2.88)

2.2.3 Constitutive equations

The stress σ is split into its isotropic and its deviatoric part τ :

σ =1

3(trσ)I + τ (2.89)

The isotropic part is the sum of the negative pressure p and the product of thebulk viscosity K with the trace of the strain rate tensor:

1

3(trσ)I = −pI +K(trε)I (2.90)

In a fluid at rest, the shear stresses are zero. For a Newtonian fluid, a linearrelation between the deviatoric stress and the deviatoric strain rate is assumed:

τ = F(

ε− 1

3(trε)I

)(2.91)

If the fluid is isotropic, the relation becomes:

τ = 2µ

(ε− 1

3(trε)I

)(2.92)


where µ denotes the dynamic viscosity. Combining equations (2.89), (2.90) and(2.92) yields the expression for the Cauchy stress tensor as a function of the strainrate:

σ = −pI +K(trε)I + 2µ

(ε− 1

3(trε)I

)(2.93)

Reordering the terms, yields:

σ = −pI + λ(trε)I + 2µε (2.94)

where λ = K − 2/3µ is the second viscosity constant.

2.2.4 Governing equations

The continuity equation (2.57) and the momentum equation (2.74) in the spatialdomain using an ALE description, together with the constitutive equation (2.94)give:

∂ρx∂t

∣∣∣∣χ

+ c · ∇xρx + ρx∇x · v = 0 (2.95)

ρx∂v

∂t

∣∣∣∣χ

+ ρx(c · ∇x)v +∇xp = 2µ∇x · ε + λ∇x(trε) + ρxb (2.96)

For incompressible flows (Mach number Ma < 0.3), ρx may be assumed to beconstant. An equation for conservation of energy is not needed. The continuityequation (2.95) reduces to:

∇x · v = 0 (2.97)

From the continuity equation follows that the trace of the strain rate trε is equalto zero and the momentum equation (2.96) becomes:

ρx∂v

∂t

∣∣∣∣χ

+ ρx(c · ∇x)v +∇xp = 2µ∇x · ε + ρxb (2.98)

Dividing the momentum equations by ρx and defining the kinematic viscosityν = µ/ρx and the kinematic pressure p? = p/ρx, yields:

∂v

∂t

∣∣∣∣χ

+ (c · ∇x)v +∇xp? = 2ν∇x · ε + b (2.99)

Together with the boundary and initial conditions, equations (2.97) and (2.99)form the Navier-Stokes equations for an incompressible flow on a deforming


domain. This is a system of four non-linear second order differential equationsin four independent variables and is formulated here as a function of the primitivevariables v and p?. In the continuum equation (2.97), no time derivative occurs.This equation formulates a kinematical constraint on the velocity field. In themomentum equation (2.99), the first term on the left hand side is the timederivative of the velocity. The second term on the left hand side is a non-linearconvective (advective) term. Neglecting this term yields the Stokes equations forhighly viscous flow. The third term is the pressure gradient. Since only thegradient of the pressure appears in the equations, the pressure is determined onlyup to an arbitrary constant, which should be fixed by a boundary condition. Thefirst term on the right hand side is a linear viscous (diffusive) term. Dropping thisterm, the Euler equations for inviscid flow are obtained. Recalling the continuityequation and the definition of the strain rate, this term can be formulatedalternatively as ν∇2

xv.

Dirichlet (or essential, kinematic) boundary conditions impose the velocity v on apart ΓD of the boundary:

v = vD (2.100)

Neumann (or natural, mechanical) boundary conditions prescribe the traction t

on a part ΓN of the boundary:

t = σnx = −pnx + 2µεnx = tN (2.101)

After division by ρx, an alternative expression for the Neumann boundaryconditions emerges:

t? = σ?nx = −p?nx + 2νεnx = t?N (2.102)

Since no time derivative of the pressure appears in the governing equations, initialconditions are only needed for the velocity field at t = t0:

v = v0 (2.103)

and should fulfill the continuity equation:

∇x · v0 = 0 (2.104)

For the computation of the convective velocity c (equation (2.16)) the velocities v

of the nodes of the mesh are still needed. The different possibilities to obtain thevelocities of the nodes will be treated in section 2.5.


2.3 Spatial discretization

2.3.1 Finite Element Method

The standard finite element method (Hughes, 1987; Bathe, 1996; Zienkiewicz et al.,2005b; Zienkiewicz and Taylor, 2005; Zienkiewicz et al., 2005a) was originallydeveloped in structural mechanics. The typical applications (e.g. linear elasticity,heat conduction) were governed by diffusion-type differential equations. Thesolutions of these equations correspond to the minimum of an energy norm (e.g.total potential/thermal energy). Application of the standard finite element methodto convection-dominated problems in fluid mechanics caused several difficultiesand motivated the development of stabilization techniques. A detailed review ofthese methods is given by Donea and Huerta (2003), Gresho and Sani (2000a,b),Zienkiewicz et al. (2005a) and Lohner (2001). The CFD program Flotran (Ansys,2005b) is based on the finite element method. For ease of notation the subscriptsx referring to the spatial domain will be omitted in the following sections. First,the standard finite element method is applied to the incompressible Navier-Stokesequations (2.97) and (2.99):

∇ · v = 0 (2.105)

∂v

∂t

∣∣∣∣χ

+ (c · ∇)v− 1

ρ∇ · σ − b = 0 (2.106)

This is the strong form of the system of equations. If trial solutions for the velocityvh and for the pressure p?h which satisfy the Dirichlet boundary conditions (2.100)are introduced, the residuals are given by:

Rc = ∇ · vh (2.107)

Rm =∂vh

∂t

∣∣∣∣χ

+ (ch · ∇)vh − 1

ρ∇ · σh − bh (2.108)

A weighted residual formulation of the governing equations uses the weightingfunctions q and w for the continuity and momentum equations:

∫

Ω

q∇ · vhdΩ +

∫

Ω

w · ∂vh

∂t

∣∣∣∣χ

dΩ +

∫

Ω

w · (ch · ∇)vhdΩ

−∫

Ω

w ·(

1

ρ∇ · σh

)dΩ −

∫

Ω

w · bhdΩ = 0 (2.109)

SPATIAL DISCRETIZATION 37

The integral of the divergence of the stress tensor over the domain is integratedby parts:

∫

Ω

q∇ · vhdΩ +

∫

Ω

w · ∂vh

∂t

∣∣∣∣χ

dΩ +

∫

Ω


−∫

Ω

1

ρ∇ · (wσh)dΩ +

∫

Ω

1

ρσh : ∇wdΩ−

∫

Ω

w · bhdΩ = 0 (2.110)

Gauss’ theorem enables to write the volume integral in the fourth term on the lefthand side as a boundary integral:

∫

Ω

q∇ · vhdΩ +

∫

Ω

w · ∂vh

∂t

∣∣∣∣χ

dΩ +

∫

Ω


−∫

Γ

1

ρwσh · ndΓ +

∫

Ω

1

ρσh : ∇wdΩ−

∫

Ω

w · bhdΩ = 0 (2.111)

Introducing the Neumann boundary conditions (2.101) and recalling that theweighting functions w vanish on the part ΓD of the boundary, the weak formof the problem is obtained:

∫

Ω

q∇ · vhdΩ +

∫

Ω

w · ∂vh

∂t

∣∣∣∣χ

dΩ +

∫

Ω


−∫

ΓN

w · 1

ρtNdΓN +

∫

Ω

1

ρσh : ∇wdΩ−

∫

Ω

w · bhdΩ = 0 (2.112)

Recalling the constitutive equation (2.94) yields, after reorganizing of the terms:

∫

Ω

w · ∂vh

∂t

∣∣∣∣χ

dΩ +

∫

Ω

w · (ch · ∇)vhdΩ +

∫

Ω

2νεh : ∇wdΩ

−∫

Ω

p?h∇ ·wdΩ +

∫

Ω

q∇ · vhdΩ =

∫

Ω

w · bhdΩ +

∫

ΓN

w · t?NdΓN (2.113)

In the weak form, the continuity requirements for the trial solutions and weightingfunctions are modified. Whereas in the strong form the second order derivativeof the velocity trial solutions vh and the first order derivative of the pressuretrial solutions p?h appeared, now only the first order derivative of the velocitytrial solutions and the pressure trial solution must be square integrable. Whereasno requirement existed for the weighting functions w in the strong form, now


their first order derivative must also be square integrable. The trial solutions arediscretized using finite element approximations. The solutions for the velocityvh(x) are approximated as:

vh(x) =

nv∑

j

Nv

j (x)vj (2.114)

where Nv

j is the shape function (basis function) associated with node j and vjare the nodal unknowns, except on the Dirichlet boundaries where vj is equal tothe imposed value. nv nodes are used for the approximation of the velocity vector.The shape function associated with node i is a piecewise polynomial function whichonly differs from 0 in all connecting elements. Its value is equal to 1 for this nodeand equal to 0 for all other nodes:

Nv

i (xj) = δij (2.115)

The trial functions for the pressure p?h(x) are approximated as:

p?h(x) =

np∑

j

Npj (x)pj

(2.116)

The number np of nodes and the shape functions used for the approximation ofthe pressure may differ from those used for the approximation of the velocity.

In the Galerkin approach, every shape function is used once as a weightingfunction:

w(x) = Nv

i (x) (2.117)

q(x) = Npi (x) (2.118)

Substituting the discretizations of the velocity (2.114) and the pressure (2.116)and the weighting functions (2.118) in the weak form (2.113) yields:

nv∑

j

(∫

Ω

Nv

i ·Nv

j dΩ

)∂vj∂t

∣∣∣∣χ

+

nv∑

j

(∫

Ω

Nv

i · (ch · ∇)Nv

j dΩ

)vj

+

nv∑

j

(∫

Ω

ν(∇Nv

j + (∇Nv

j )T) : ∇Nv

i dΩ

)vj −

np∑

j

(∫

Ω

Npj∇ ·Nv

i dΩ

)pj

+

nv∑

j

(∫

Ω

Npi ∇ ·Nv

j dΩ

)vj =

∫

Ω

Nv

i · bhdΩ +

∫

ΓN

Nv

i · t?NdΓN (2.119)


This can be rewritten as a matrix system:

[M 0

0 0

] [v

p

]+

[K(v) G

−GT 0

] [v

p

]=

[f

0

](2.120)

where

Mij =

∫

Ω

Nv

i ·Nv

j dΩ (2.121)

Kij =

∫

Ω

Nv

i · (ch · ∇)Nv

j dΩ +

∫

Ω

ν(∇Nv

j + (∇Nv

j )T) : ∇Nv

i dΩ (2.122)

Gij = −∫

Ω

Npj∇ ·Nv

i dΩ (2.123)

fi

=

∫

Ω

Nv

i · bhdΩ +

∫

ΓN

Nv

i · t?NdΓN (2.124)

For convection-dominated problems, the stiffness matrix K is non-symmetric.

The advection-diffusion stability: Streamline-Upwind Petrov-Galerkin method(SUPG)

If the Galerkin approach is applied, the solutions for the velocities are sometimescorrupted by spurious node-to-node oscillations (also called wiggles) which pollutethe whole computational domain. In order to illustrate the deficiencies ofthe Galerkin approach for convection-dominated problems, equation (2.99) issimplified to a steady 1D scalar convection-diffusion equation with a constantconvection c and a constant source term b:

c∂v

∂x− ν ∂

2v

∂x2= b (2.125)

The combination of linear shape functions with the Galerkin approach yields ona uniform mesh with element length h a discretized equation for node i that isidentical to that obtained with second order central differences:

cvi+1 − vi−1

2h− ν vi+1 − 2vi + vi−1

h2= b (2.126)

The ratio of the convective to the diffusive transport is expressed by the meshPeclet number Pe that can be interpreted as a local Reynolds number:

Pe =ch

2ν(2.127)


When the Peclet number exceeds unity, the solution of equation (2.126) usingthe Galerkin approach is corrupted by spurious node-to-node oscillations (alsocalled wiggles) which pollute the whole computational domain. Figure 2.4a showsthe solution of a steady 1D scalar convection-diffusion equation with a constantconvection c = 1 and a constant source term b = 1 using an element size h = 0.1for a Peclet number smaller than one and figure 2.4b for a Peclet number largerthan one. The oscillations can be avoided by refining the mesh to obtain Pecletnumbers lower than one. The discretized equation which produces the exact nodalsolution with linear shape functions on a uniform mesh is (figure 2.4):

cvi+1 − vi−1

2h− (ν + ζ

ch

2)vi+1 − 2vi + vi−1

h2= b (2.128)

where ζ is defined as:

ζ = coth(Pe)− 1

Pe(2.129)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Distance [m]

Vel

ocity

[m/s

]

(a)0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Distance [m]

Vel

ocity

[m/s

]

(b)

Figure 2.4: Analytical solution for the velocity (black line), exact nodal solution(black dashed line) (2.128), Galerkin solution (dark grey line) (2.126)and full upwind solution (light grey line) (2.130) of a steady 1D scalarconvection-diffusion equation with a constant convection c = 1 and aconstant source term b = 1 using an element size h = 0.1 for a Pecletnumber of (a) 0.25 and (b) 5.

To remove these oscillations without any requirements on the mesh size, twopossibilities exist. First, comparison of equations (2.126) and (2.128) shows thatthe discretization error of the Galerkin finite element and the central differencediscretization methods is equal to a negative diffusion term with numericaldiffusion ν = ζ ch2 . This numerical diffusion ν can be added to the Galerkindiscretization of equation (2.126) to balance the negative diffusion inherent to


the Galerkin approach. The resulting discretization is identical to equation(2.128) where ζ may now differ from equation (2.129) and controls the amount ofnumerical diffusion.

Second, equation (2.128) can be rewritten as:

1− ζ2cvi+1 − vih

+1 + ζ

2cvi − vi−1

h− ν vi+1 − 2vi + vi−1

h2= 1 (2.130)

The convective flux is discretized by a weighted average of the fluxes from the leftand the right side, while the diffusive flux is still discretized with a second ordercentral difference scheme. ζ may differ from equation (2.129) and controls theweight given to the upstream and downstream element. The elements upstreamof a node can be more heavily weighted than those downstream for the convectiveterm. For a positive convective velocity c, a value of ζ equal to one correspondsto a full upwind discretization (figure 2.4), which is always stable but often toodiffusive (Donea and Huerta, 2003). For ζ equal to zero the equation reduces tothe standard Galerkin discretization. Using values between one and zero is calledhybrid differencing. The modified weighting functions wm differ from the Galerkinweighting functions w which are equal to the shape functions. A possible choiceis:

wm = w +ζh

2

∂w

∂x(2.131)

In both cases, the absolute value of ζ should be greater than a critical value inorder to avoid oscillations:

|ζ| ≥ 1− 1

|Pe| (2.132)

The extension of the convection-diffusion equation (2.125) to two dimensions gives:

c · ∇v − ν∇2v = b (2.133)

c1∂v

∂x1+ c2∂v

∂x2− ν

(∂2v

∂x21

+∂2v

∂x22

)= b (2.134)

Hybrid differencing can be applied separately to the terms c1∂v/∂x1 and c2∂v/∂x2,which corresponds to adding the following numerical dissipation:

ν11∂2v

∂x21

+ ν22∂2v

∂x22

(2.135)

This technique introduces numerical diffusion perpendicular to the convectiondirection (crosswind diffusion) which becomes large if the convective velocity is


not aligned with the mesh (Donea and Huerta, 2003). In order to add numericaldiffusion only in the convection direction and not in the transversal direction, adiffusivity tensor ν is constructed:

ν =ν

‖c‖2c⊗ c (2.136)

This approach does not introduce any crosswind diffusion. The magnitude of theconvective velocity is used to define the numerical viscosity:

ν = ζ‖c‖h

2(2.137)

Peclet numbers are computed using the magnitude of the convective velocity:

Pe =‖c‖h2ν

(2.138)

Adding the numerical diffusivity tensor corresponds to the use of a modifiedweighting function wm for the convective term:

wm = w +ζh

2‖c‖c · ∇w = w + τc · ∇w (2.139)

where τ is the stabilization parameter which is also called the intrinsic time.Several options exist for computing the element length h in equation (2.139). Thechoice largely influences the amount of numerical diffusion introduced. In the finiteelement code Flotran (Ansys, 2005b), the length of the projection of the elementon the direction of the convection velocity c is used. To reduce the computationalcosts the optimal blending function ζ (2.129) is replaced by its doubly asymptoticapproximation:

ζ =

Pe/3 if 0 ≤ Pe ≤ 3

1 if Pe > 3(2.140)

Other approximations exist, but the influence of the blending function is muchsmaller than the influence of the choice for the computation of the element length.

The approach where the modified weighting function of equation (2.139) is appliedto the convective term only, is called streamline upwind (SU). The exact solution ofthe differential equation is no longer a solution of the weak form. Generally, if thesame weighting function, different from the shape functions, is applied to all termsin the weak form of equation (2.113), a consistent formulation is obtained, which iscalled the Petrov-Galerkin approach. If the weighting function of equation (2.139)is used in all terms, the Streamline-Upwind Petrov-Galerkin (SUPG) method(Brooks and Hughes, 1982) is obtained, which can easily be implemented by adding


a stabilization term, emanating from the second term on the right hand side ofequation (2.139), to the Galerkin weak form of equation (2.113):

ST =

ne∑

i

∫

Ωi

τsupg(ch · ∇w) ·Rm(vh, p?h))dΩi (2.141)

This term should not be integrated by parts because then for linear elements thestabilization effects disappear. For linear elements the stabilization contributionin the diffusion term vanishes. As a result the spurious oscillations are localizedand do not pollute the whole domain.

The pressure stability: Pressure-Stabilizing/Petrov-Galerkin method (PSPG)

If the Galerkin approach is applied, the solution for the pressure sometimes exhibitsspurious pressure modes (e.g. checkerboard modes). This pressure instability canbe illustrated by means of the steady 2D incompressible Stokes equations for highlyviscous flow, which are obtained by neglecting the transient and the convectiveterm in the equations (2.97) and (2.99):

∇ · v = 0 (2.142)

∇p− ν∇2v = 0 (2.143)

By taking the divergence of the momentum equation (2.143) and inserting thecontinuity equation into the second term on the left hand side, a Laplace equationfor the pressure is obtained:

∇ · (∇p− ν∇2v) = ∇2p− ν∇2(∇ · v) = ∇2p = 0 (2.144)

If the velocity and the pressure are discretized in 2D using second order centraldifferences, the discrete Laplace equation becomes:

pi+2,j + pi−2,j + pi,j+2 + pi,j−2 − 4pi,j4h2

= 0 (2.145)

This equation is independent of the pressures at points i + 1, i − 1, j + 1 andj − 1. This is called the odd-even decoupling as the solution for odd pointsis independent of the solution for even points. This decoupling occurs if equalorder interpolations are used for the velocity and the pressure and results inspurious pressure oscillations. The Ladyzhenskaya-Babuska-Brezzi (LBB) orinf-sup condition (Babuska, 1971) specifies a relation between the velocity andpressure interpolation to guarantee the stability of a mixed method. Combinationsof trial solutions for velocity and pressure that satisfy the LBB condition arefor instance obtained using quadratic shape functions for the velocity and linearshape functions for the pressure. Stabilization methods enable to circumvent the


LBB condition (Hughes et al., 1986) and to use combinations of trial solutions forvelocity and pressure that are not stable in the Galerkin formulation e.g. equalorder interpolations for velocity and pressure. A stabilization term consistingof a stability parameter τpspg and the Laplacian of the pressure is added to thecontinuity equation (2.97):

∇ · v− τpspg∇2p = 0 (2.146)

If this stabilization term is integrated by parts in the weighted residual formulation,it becomes:

∫

Ω

τpspg∇q · ∇pdΩ (2.147)

This can be viewed as an additional term resulting from a modified weighting ofthe pressure gradient term in the momentum equation with the following weightingfunction:

wm = w + τpspg∇q (2.148)

A consistent approach, which is called the Pressure-Stabilizing/Petrov-Galerkin(PSPG) method (Hughes et al., 1986), is obtained if this weighting functionis applied to all terms of the momentum equation. This yields the followingstabilization term:

ST =

ne∑

i

∫

Ωi

τpspg∇q ·Rm(vh, p?h)dΩi (2.149)

The stability parameter τpspg is defined empirically as:

τpspg = αh2

2ν(2.150)

where α = 1/3 appears to be optimal for linear elements.

SUPG/PSPG method

The combination of the SUPG (2.139) and the PSPG method (2.148) leads to theSUPG/PSPG method which employs the following weighting function:

wm = w + τsupgch · ∇w + τpspg∇q (2.151)

This results in the corresponding stabilization term:

ST =

ne∑

i

∫

Ωi

(τsupgch · ∇w + τpspg∇q

)·Rm(vh, p?h)dΩi (2.152)


Galerkin/Least Squares method

The Galerkin/Least Squares (GLS) method (Hughes et al., 1989) is a generalizationof the SUPG and PSPG methods. It is a linear combination of the Galerkin andthe least squares method. The least-squares method minimizes the integral of thesquared residuals over the domain with respect to the nodal velocities vi:

∂

∂vi

∫

Ω

R2m(vh, p?h)dΩ = 2

∫

Ω

Rm(vh, p?h) · ∂Rm(vh, p?h)

∂vidΩ (2.153)

= 2

∫

Ω

Rm(vh, p?h) · (ch · ∇w− 2ν∇ · εh(w))dΩ

(2.154)

The integral of the squared residuals over the domain is minimized as well withrespect to the nodal pressures p

i:

∂

∂pi

∫

Ω

R2m(vh, p?h)dΩ = 2

∫

Ω

Rm(vh, p?h) · ∂Rm(vh, p?h)

∂pi

dΩ (2.155)

= 2

∫

Ω

Rm(vh, p?h) · (∇q)dΩ (2.156)

After multiplication with a stability parameter τgls, the sum of equations (2.154)and (2.156) yields the following stabilization term:

ST =

ne∑

i

∫

Ωi

(τgls(c

h · ∇w− 2ν∇ · εh(w) +∇q))·Rm(vh, p?h)dΩi (2.157)

The modified weighting function wm for the Galerkin/Least Squares (GLS) methodis:

wm = w + τgls(ch · ∇w− 2ν∇ · εh(w) +∇q) (2.158)

GLS and SUPG are identical for P1 elements, but differ for higher order elements.


2.3.2 Finite Volume Method

The finite volume method (Hirsch, 1995a,b; Anderson, 1995; Versteeg andMalalasekera, 1995; Ferziger and Peric, 2002) was originally developed in fluidmechanics. The CFD program CFX (Ansys, 2005a) is based on the finite volumemethod. In the finite volume method the conservation of mass (2.60) and theconservation of momentum (2.77) are expressed for a control volume Ωi:

∂

∂t

(∫

Ωi

ρdΩi

)∣∣∣∣χ

+

∫

Ωi

∇ · (ρc)dΩi = 0 (2.159)

∂

∂t

(∫

Ωi

ρvdΩi

)∣∣∣∣χ

+

∫

Ωi

∇ · (ρv ⊗ c)dΩi =

∫

Ωi

∇ · σdΩi +

∫

Ωi

ρbdΩi

(2.160)

By splitting the stress tensor in its hydrostatic and deviatoric part (2.89), theterms in equation (2.160) are reordered:

∂

∂t

(∫

Ωi

ρvdΩi

)∣∣∣∣χ

+

∫

Ωi

∇ · (ρv ⊗ c + pI)dΩi =

∫

Ωi

∇ · τdΩi +

∫

Ωi

ρbdΩi

(2.161)

Using Gauss’ theorem, the integrals of the divergence over the control volume inequations (2.159) and (2.161) can be rewritten as integrals on the boundary Γi ofthis control volume:

∂

∂t

(∫

Ωi

ρdΩi

)∣∣∣∣χ

+

∫

Γi

(ρc) · ndΓi = 0 (2.162)

∂

∂t

(∫

Ωi

ρvdΩi

)∣∣∣∣χ

+

∫

Γi

(ρv⊗ c + pI)ndΓi =

∫

Γi

τndΓi +

∫

Ωi

ρbdΩi

(2.163)

The vectors V and Q and the tensor Fd respectively represent the conservativevariables, the source terms and the diffusive (viscous) fluxes:

V =

[ρρv

]Q =

[0ρb

]F

d =

[0

τ

](2.164)

The tensor Fc contains the convective (inviscid) fluxes. Using the definition of theconvective velocity (2.16), the convective fluxes are split in a partFe correspondingto an Eulerian description and a contribution from the ALE description:

Fc =

[ρc

ρv⊗ c + pI

]=

[ρv

ρv⊗ v + pI

]−[ρvρv⊗ v

](2.165)

= Fe − V ⊗ v = Fe −Fale (2.166)


Using these definitions the equations are rewritten:

∂

∂t

(∫

Ωi

VdΩi

)∣∣∣∣χ

+

∫

Γi

FcndΓi =

∫

Γi

FdndΓi +

∫

Ωi

QdΩi (2.167)

In order to discretize these equations the computational domain is meshed. In thecell-centered approach the mesh cells are used as control volumes. The unknownsare located at the centroid of the mesh. In the vertex-centered approach theunknowns are located at the vertices of the mesh. The control volumes are definedby a dual mesh which connects the centroid of the mesh cells (centroid dual) or thecentroid of the mesh cells and the centroid of the faces of the mesh cells (mediandual). This latter approach is used in the finite volume code CFX (Ansys, 2005a).In the cell-vertex approach the unknowns are located at the vertices of the meshand the mesh cells are used as control volumes. In the sequel of this section itis supposed that the vertex-centered median dual approach is used, however thediscretization principles are similar for all approaches. One of the main differencesis the treatment of the boundary conditions.

As the control volume Ωi is closed by Ni faces, the boundary integrals aretransformed into a sum over these faces:

∂

∂t

(∫

Ωi

VdΩi

)∣∣∣∣χ

+

Ni∑

j

∫

Γij

FcndΓij =

Ni∑

j

∫

Γij

FdndΓij +

∫

Ωi

QdΩi

(2.168)

The volume integrals are replaced by the product of the volume and the averagevalues:

∂ΩiVi∂t

∣∣∣∣χ

+

Ni∑

j

∫

Γij

FcndΓij =

Ni∑

j

∫

Γij

FdndΓij + ΩiQi (2.169)

Vi contains the average values of the conservative variables over the control volumeand Qi the average values of the source terms.

The integrals of the fluxes Fc and Fd over the faces are numerically approximatedin two levels. First, the integration is performed numerically using a quadraturerule. The simplest approximation is obtained using the midpoint rule: the integralis approximated as the product of the integrand at the midpoint of the face and theface area. This approach is second order accurate. Higher-order approximationsneed the value of the integrand at the corners and centers of edges. Second, inorder to compute the fluxes at the midpoint (and eventually the corners and thecenters of the edges), the fluxes at these locations are interpolated from the fluxesat the vertices (flux averaging) or are computed from the conservative variablesat these locations, which are interpolated from the conservative variables at thevertices (variable averaging). In CFX, finite element shape functions are used


to interpolate the values of the conservative variables at these locations. As theintegral conservation applies to each control volume, the surface integrals should becomputed in a way that they cancel out for surfaces with opposite normals in orderto guaranty global conservation over the domain. The numerical approximationof the fluxes finally yields:

∂ΩiVi∂t

∣∣∣∣χ

+ Fci (V, x, v) = Fd

i (V, x) + ΩiQi (2.170)

where Fci and Fd

i are equal to the sum of all the numerical fluxes of the differentfaces of the control volume. Collecting all equations into a single system ofequations gives:

∂ΩV

∂t

∣∣∣∣χ

+ Fc(V , x, v) = Fd(V , x) + ΩQ (2.171)

where Ω is a diagonal matrix consisting of the volumes and V and Q are thevectors formed by the collection of respectively Vi and Qi.

2.4 Time integration

The system of equations which has already been discretized in space is nowintegrated in time. First, the different time integration methods for an Euleriandescription are reviewed. Next, the implication of the ALE description on theaccuracy and the stability of the time integration is discussed and related to thegeometric conservation law. The methods are described for the finite volumemethod, but analogous methods exist for the finite element method.

2.4.1 The Eulerian description

After spatial discretization a system of coupled first order ordinary differentialequations (2.171) is obtained. In the Eulerian description the control volumes arefixed in space and their volume is constant. The system of equations becomes:

Ω∂V

∂t+ Fc(V) = Fd(V) (2.172)

This is simplified to:

∂V

∂t= F(V) (2.173)

Direct time integration methods are used to integrate this initial value problem. Agroup of commonly used methods for integrating first order differential equations

TIME INTEGRATION 49

are the one-step θ methods. One-step methods only refer to the values at theprevious time t and the current time t + ∆t to determine the solution at timet + ∆t. In the one-step θ methods a weighted average of the right hand side ofequation (2.173) at time t and t+ ∆t is used:

Vn+1 − Vn

∆t= θF(Vn+1) + (1− θ)F(Vn) (2.174)

For θ = 0 the forward Euler method is obtained, which is an explicit method asVn+1 can be directly computed. All other values of θ yield implicit methods as a

system of equations for Vn+1 has to be solved.

The forward Euler method is conditionally stable. For convection dominatedproblems the stability is related to the Courant number C = v∆t

h where h isthe element size. As an example, for a one-dimensional convection equationthe combination of the forward Euler method for the time integration with thefirst order upwind difference scheme for the spatial discretization is stable if theCourant number C is smaller than one. This is known as the Courant-Friedrichs-Lewy (CFL) condition and limits the time step so that the distance travelled bya material particle during a the time step is smaller than element size. Methodswith θ ≥ 1/2 are unconditionally stable or A-stable. For θ = 1, the backwardEuler method is obtained and for θ = 1/2, the Crank-Nicolson method, which isalso called the trapezoidal method.

The forward and backward Euler are first order accurate in time while the Crank-Nicolson method is the only method which is second order accurate. A drawback ofthe Crank-Nicolson method is that it has no numerical dissipation and oscillatorysolutions might arise.

The one-step θ methods belong to a larger group of linear multistep (LMS)methods:

k∑

i=0

aiVn+1−i = ∆t

k∑

i=0

biF(Vn+1−i) (2.175)

These multistep methods refer to several previous time steps. A linear combinationof the values at the previous time steps is used. A-stable linear multistep methodsare at most second order accurate. Among these methods, the Crank-Nicolsonmethod has the smallest truncation error. If b0 = 0, the method is explicit, whileotherwise it is implicit. For a0 = 3/2, a1 = −2, a2 = 1/2 and b0 = 1 a second orderaccurate, implicit three-point backward difference method (BDF2) is obtained:

3

2Vn+1 − 2Vn +

1

2Vn−1 = ∆tF(Vn+1) (2.176)


This method is not self-starting as it requires the solutions at two previoustime steps. In Flotran and CFX this implicit backward difference method isimplemented.

2.4.2 The ALE description

In the ALE description the system of coupled first order ordinary differentialequations (2.171) obtained after spatial discretization has some importantdifferences in comparison with equation (2.173) for the Eulerian description:

∂ΩV

∂t

∣∣∣∣χ

= −Fc(V , x, v) + Fd(V, x) (2.177)

The diagonal matrix Ω with the volumes of the control volumes changes withtime. The convective fluxes Fc are integrated over a changing boundary anddepend on the mesh velocity. The diffusive fluxes Fd are as well integrated over achanging boundary. The possibilities to construct a time integration method forcomputations using the ALE description are studied is this section.

The geometrical conservation law

A first approach to construct a time integration procedure in an ALE descriptionis to require that a uniform flow can be computed exactly, independently of themesh deformations (Lesoinne and Farhat, 1996).

The time integration between tn and tn+1 of the finite volume equations (2.167)for a control volume Ωi yields:

(∫

Ωi

VdΩi

)n+1

−(∫

Ωi

VdΩi

)n+

∫ tn+1

tn

∫

Γi

FalendΓidt

=

∫ tn+1

tn

∫

Γi

FdndΓidt+

∫ tn+1

tn

∫

Ωi

QdΩidt

(2.178)

For a uniform flow the source term Q is equal to zero and the conservative variablesV are constant with time and denoted by V∗:

(∫

Ωi

V∗dΩi

)n+1

−(∫

Ωi

V∗dΩi

)n+

∫ tn+1

tn

∫

Γi

FalendΓidt

=

∫ tn+1

tn

∫

Γi

FdndΓidt (2.179)

TIME INTEGRATION 51

The convective fluxes Fc are split in two parts as in equation (2.166):

V∗(Ωn+1i − Ωni ) +

∫ tn+1

tn

∫

Γi

(Fe − V ⊗ v) ndΓidt =

∫ tn+1

tn

∫

Γi

FdndΓidt

(2.180)

The integrals on a closed boundary of the flux of a constant function, like Fe andF

d, are equal to zero:

V∗(Ωn+1i − Ωni )−

∫ tn+1

tn

∫

Γi

(V∗ ⊗ v)ndΓidt = 0 (2.181)

As V∗ is constant, the equation is rewritten as:

(Ωn+1i − Ωni ) =

∫ tn+1

tn

∫

Γi

v · ndΓidt (2.182)

This is the geometrical conservation law (GCL) or space conservation law(Demirdzic and Peric, 1988) for finite volume methods which is universal for alltime integration schemes: the change in volume of a control volume during a timestep should be equal to the volume swept by its boundaries during the same timestep. As the change in volume is computed exactly, any time integration schemeshould also compute the right hand of equation (2.182) side exactly. The GCL onlyincludes geometric quantities as the node positions x and the mesh velocities v.It only provides information for the time integration of the ALE convective fluxesas the viscous fluxes Fd automatically disappear from the equations. For a finiteelement discretization, an analogous GCL can be derived (Lesoinne and Farhat,1996; Formaggia and Nobile, 2004) if equation (2.76) is used as a starting point.The time derivative in this equation is applied to the product of the Jacobiandeterminant and the variables (density and velocity). If, however, equation (2.74),which only includes the time derivative of the variables, is chosen as a starting pointfor the finite element discretization, the geometric conservation law is satisfiedindependent of the time integration scheme (Formaggia and Nobile, 2004; Forsteret al., 2006). Space-time discretizations always satisfy the GCL.

The implications of the GCL will now be illustrated for two specific implicit timeintegration schemes: the backward Euler method and the three-point backwarddifference scheme. At first reading, the reader may wish to skip these twoparagraphs and continue with the paragraph on the importance of the GCL onpage 57.

If the semi-discretized equation (2.170) for one control volume is integrated in timebetween tn and tn+1, the following discrete equation is obtained:

(ΩiVi)n+1 − (ΩiV i)

n +

∫ tn+1

tnFci (V, x, v)dt =

∫ tn+1

tnFdi (V, x)dt (2.183)


The backward Euler method The backward Euler method (equation (2.174)for θ = 1) approximates these integrals as the fluxes Fi corresponding to Vn+1

multiplied with ∆t. The question arises on which mesh configuration x theseconvective fluxes and diffusive fluxes should be evaluated: at xn+1 , at xn or inbetween those two configurations. A similar question is which mesh velocities vshould be used and how they have to be computed from the node positions. If auniform flow is assumed, Vi = V∗i and the previous equation becomes:

Ωn+1i V

∗

i − Ωni V∗

i +

∫ tn+1

tnFci (V

∗, x, v)dt =

∫ tn+1

tnFdi (V

∗, x)dt (2.184)

For a uniform flow V∗i the diffusive flux Fdi (V

∗, x) is equal to zero. Theconvective flux is split in a part Fe

i corresponding to an Eulerian description anda contribution Fale

i from the ALE description:

Ωn+1i V

∗

i − Ωni V∗

i +

∫ tn+1

tnFei (V

∗, x)dt−∫ tn+1

tnFalei (V∗, x, v)dt = 0 (2.185)

For a uniform flow V∗i the flux Fei (V

∗, x) is equal to zero as well. From themathematical consistency of the numerical fluxes follows that the integral of theflux Fale

i (V∗, x, v) is given by:

∫ tn+1

tnFalei (V∗, x, v)dt = V∗i

∫ tn+1

tn

∫

Γi

v · ndΓidt (2.186)

By isolating the conservative variables V∗i from the convective fluxes Falei (V∗, x, v)

the function Gi(x, v) is defined which should satisfy:

V∗

i

∫ tn+1

tnGi(x, v)dt = V∗i

∫ tn+1

tn

∫

Γi

v · ndΓidt (2.187)

Substituting equation (2.187) in equation (2.184) yields:

Ωn+1i − Ωni =

∫ tn+1

tnGi(x, v)dt =

∫ tn+1

tn

∫

Γi

v · ndΓidt (2.188)

Generally the mesh positions x are only known at discrete time steps. The meshposition between tn and tn+1 is parameterized:

x(t) = ζn+1(t)xn+1 + ζn(t)xn (2.189)

where ζn(t) = 1 − ζn+1(t). The function ζn+1(t) satisfies ζn+1(tn+1) = 1 andζn+1(tn) = 0. The mesh velocity v becomes:

v(t) = ζn+1(t)xn+1 + ζn(t)xn (2.190)

TIME INTEGRATION 53

The sequel depends on the space dimensions of the problem. Using the aboveparameterizations, the integrand of the integral on the right hand side of equation(2.187) becomes a quadratic function of ζn+1 for three-dimensional computations.This integral can be integrated exactly if a Gaussian quadrature with twointegration points is used:

ζn+11 =

1

2

(1 +

1√3

)ζn1 =

1

2

(1− 1√

3

)wc

1 =1

2(2.191)

ζn+12 =

1

2

(1− 1√

3

)ζn2 =

1

2

(1 +

1√3

)wc

2 =1

2(2.192)

wc1 and wc

2 are the weight of the respective integration points. The Gaussianquadrature with two integration points corresponds to the assumption that thetime derivatives ζn+1 and ζn at the integration points are given by:

ζn+11 = ζn+1

2 =1

∆tζn1 = ζn2 = − 1

∆t(2.193)

Combining equations (2.189) and (2.192), the mesh position at the integrationspoints is:

x1c =

1

2

(1 +

1√3

)xn+1 +

1

2

(1− 1√

3

)xn (2.194)

x2c =

1

2

(1− 1√

3

)xn+1 +

1

2

(1 +

1√3

)xn (2.195)

Combining equations (2.190) and (2.193), the mesh velocities at the integrationspoints are computed as:

v1 = v2 =

xn+1 − xn

∆t(2.196)

Using the numerical integration equation (2.188) becomes:

Ωn+1i − Ωni = ∆t

Kc∑

k=1

wckGi(x

kc , vk) (2.197)

This is the discrete geometrical conservation law (DGCL) for the backward Euler

time integration scheme which is fulfilled for Kc = 2 and with xkc and vk

respectively given in equations (2.195) and (2.196). Therefore, the convectivefluxes should be integrated as:

∫ tn+1

tnFci (V , x, v)dt = ∆t

Kc∑

k=1

wckF

ci (V

n+1, xkc , vk) (2.198)


Demirdzic and Peric (1988) describe an alternative way to satisfy the DGCL forthe backward Euler scheme. Writing the right hand side of equation (2.188) as asum over all the faces of the control volume, gives:

Ωn+1i − Ωni =

Ni∑

j

∫ tn+1

tn

∫

Γij

v · ndΓijdt (2.199)

The integral on the right hand side should be computed so that:

∫ tn+1

tn

∫

Γij

v · ndΓijdt = ∆Ωn+1ij (2.200)

∆Ωn+1ij is the volume swept by the face Γij of the control volume Ωi between tn

and tn+1. The sum of all these volumes is equal to the change in volume of thecontrol volume. If these volumes ∆Ωn+1

ij are computed exactly based on the face

positions at tn and tn+1, they can be used to integrate the flux Falei in a way that

satisfies the DGCL:

∫ tn+1

tnFalei (V∗, x, v)dt =

Ni∑

j

∫ tn+1

tn

∫

Γij

V(v · n)dΓijdt (2.201)

=

Ni∑

j

Iij(Vn+1)∆Ωn+1

ij (2.202)

where Iij(Vn+1) interpolates the values of the conservative variables Vn+1 at the

cell centers to the midpoint of the face Γij . In this approach, the mesh velocitydoes not have to be calculated explicitly. It is not specified how to integrate thepart Fe

i corresponding to an Eulerian description of the convective fluxes.

The three-point backward difference scheme Koobus and Farhat (1999) derivedthe DGCL for the three-point backward difference scheme. If the semi-discretizedequation (2.170) for one control volume is integrated in time with the three-pointbackward difference scheme (2.176), the following discrete equation is obtained:

3

2(ΩiVi)

n+1 − 2(ΩiV i)n +

1

2(ΩiV i)

n−1 +

∫ tn+1

tnFci (V, x, v)dt (2.203)

=

∫ tn+1

tnFdi (V, x)dt (2.204)

TIME INTEGRATION 55

Assuming a uniform flow Vi = V∗i an analogous reasoning as for the backwardEuler scheme leads to:

3

2Ωn+1i − 2Ωni +

1

2Ωn−1i =

∫ tn+1

tnGi(V, x, v)dt (2.205)

The left hand side is reordered as:

3

2(Ωn+1i − Ωni )−

1

2(Ωni − Ωn−1

i ) =

∫ tn+1

tnGi(V, x, v)dt (2.206)

The terms in brackets on the left hand side are elaborated using the GCL (2.182).The mesh positions x are now parameterized between tn and tn+1 as:

x(t) = ζn+1(t)xn+1 + ζn(t)xn + ζn−1(t)xn−1 (2.207)

where ζn−1(t) = 1− ζn+1(t)− ζn(t). The function ζn+1(t) satisfies ζn+1(tn+1) = 1and ζn+1(tn) = 0 and the function ζn(t) satisfies ζn(tn+1) = 0 and ζn(tn) = 1.The mesh velocity v becomes:

v(t) = ζn+1(t)xn+1 + ζn(t)xn + ζn−1(t)xn−1 (2.208)

In order to satisfy equation (2.206) for three-dimensional computations, fourintegration points are used to compute the integral on the right hand side. Aninfinite number of choices exist for the integration points and the weights. Oneparticular choice is to use a two-point Gaussian quadrature between tn+1 and tn

and another two-point Gaussian quadrature between tn and tn−1:

ζn+11 =

1

2

(1 +

1√3

)ζn1 =

1

2

(1− 1√

3

)ζn−1

1 = 0 (2.209)

ζn+12 =

1

2

(1− 1√

3

)ζn2 =

1

2

(1 +

1√3

)ζn−1

2 = 0 (2.210)

ζn+13 = 0 ζn3 =

1

2

(1 +

1√3

)ζn−1

3 =1

2

(1− 1√

3

)(2.211)

ζn+14 = 0 ζn4 =

1

2

(1− 1√

3

)ζn−1

4 =1

2

(1 +

1√3

)(2.212)

The weights are the product of the weight of a two-point Gaussian quadrature withthe coefficients of the terms in brackets in the left hand side of equation (2.206):

wc1 =

1

2

3

2wc

2 =1

2

3

2wc

3 =1

2

(−1

2

)wc

4 =1

2

(−1

2

)(2.213)


The mesh position xkc at the kth-integration point is given by:

xkc = ζn+1

k xn+1 + ζnk x

n + ζn−1k x

n−1 (2.214)

The two Gaussian quadratures with two integration points correspond to theassumption that the mesh velocities at the integrations points are computed as:

v1 = v2 =

xn+1 − xn

∆tv

3 = v4 =xn − xn−1

∆t(2.215)

Using the numerical integration equation (2.205) becomes:

3

2Ωn+1i − 2Ωni +

1

2Ωn−1i = ∆t

Kc∑

k=1

wckGi(x

kc , vk) (2.216)

This is the DGCL for the three-point backward difference scheme which is fulfilledfor Kc = 4 and with xkc and vk respectively given in equations (2.214) and (2.215).Therefore, the convective fluxes should be integrated as:

∫ tn+1

tnFci (V , x, v)dt = ∆t

Kc∑

k=1

wckF

ci (V

n+1, xkc , vk) (2.217)

The approach of Demirdzic and Peric (1988) can be applied as well to the three-point backward difference scheme:

3

2Ωn+1i − 2Ωni +

1

2Ωn−1i =

Ni∑

j

∫ tn+1

tn

∫

Γij


The left hand side is reordered as:

3

2(Ωn+1i − Ωni )−

1

2(Ωni − Ωn−1

i ) =

Ni∑

j

∫ tn+1

tn

∫

Γij


The integral on the right hand side should be computed so that:

∫ tn+1

tn

∫

Γij

v · ndΓijdt =3

2∆Ωn+1ij − 1

2∆Ωnij (2.220)

If these volumes ∆Ωn+1ij and ∆Ωnij are computed exactly based on the face positions

at tn+1, tn and tn−1, they can be used to integrate the flux Falei in a way that it

satisfies the DGCL:

∫ tn+1

tnFalei (V∗, x, v)dt =

Ni∑

j

Iij(Vn+1)

(3

2∆Ωn+1ij − 1

2∆Ωnij

)(2.221)

This approach is implemented in CFX.

TIME INTEGRATION 57

The importance of the geometric conservation law The importance of fulfillingthe DGCL, which ensures that the particular case of a uniform flow is computedexactly whereas other flow solutions are still approximated, has been thoroughlystudied during the last years. Guillard and Farhat (2000) proved for the backwardEuler and the three-point backward difference scheme that the fulfillment of theDGCL is a sufficient condition to be at least first order time accurate in theALE description. In the case of a non-linear scalar hyperbolic conservation law,Farhat et al. (2001) show that, for the one-step θ schemes (2.174), the fulfillmentof the DGCL is a sufficient and a necessary condition to preserve in the ALEdescription the non-linear stability properties that these θ schemes exhibit inan Eulerian description. If the DGCL is violated, spurious oscillations developduring the computations, which may lead to instabilities in the flow solver orwrong predictions as for instance underestimated flutter speeds in aeroelasticcomputations. The magnitude of these oscillations is substantially larger for thethree-point backward difference scheme than for the backward Euler method whichmeans that it is more critical for time integration schemes which are second ordertime accurate in an Eulerian description to satisfy the DGCL than for first ordertime-accurate schemes. The magnitude of these oscillations increases with thecomputational time step. For sufficiently small time steps and smooth meshmotions the non-linear stability properties of time integration schemes are keptdespite the fact that they violate the DGCL. However, the estimation of thelargest time step which produces reasonable results is cumbersome and problemdependent. As both the three-point backward difference scheme and the backwardEuler method are implicit schemes, a limitation on the magnitude of the time stepis not desirable and might drastically increase the computational cost. Therefore, itis recommended to use time integration schemes that fulfill the DGCL. However,Boffi and Gastaldi (2004) and Formaggia and Nobile (2004) proved for a linearadvection-diffusion equation of a form similar to equation (2.76) discretized withfinite elements that, even if the DGCL is satisfied, the Crank-Nicolson and thethree-point backward difference scheme are only conditionally stable in an ALEdescription.

In equations (2.198) and (2.217), the numerical fluxes Fci have to be evaluated

Kc times on different mesh configurations. This raises the computational cost,although the cost for the evaluation of the fluxes is only small with respect tothe total cost. If, however, the convective fluxes Fc

i are computed using Roe’snumerical flux, the computational costs can be reduced. Roe’s numerical fluxconsists of a part that varies linearly with the mesh configuration and a part thatvaries non-linearly with it. As the non-linear part is equal to zero for a uniformflow, the DGCL can be satisfied as well by computing the flux once on a uniqueaveraged mesh configuration (Koobus and Farhat, 1999):

∫ tn+1

tnFci (V , x, v)dt = ∆tFc

i (Vn+1,

Kc∑

k=1

wckxkc ,

Kc∑

k=1

wckvk) (2.222)


The results computed with this equation generally differ from the results computedwith equation (2.217) as the flux is a non-linear function of the mesh configuration.

Time integration accuracy

A second approach to extend an Eulerian time integration procedure to an ALEdescription is to require that the order of time accuracy obtained in the Euleriandescription is preserved in the ALE description. For aeroelastic computations,the time accuracy is very important as it influences the energy exchange betweenstructure and fluid. As the fulfillment of the DGCL is a sufficient condition tobe at least first order time accurate in the ALE description, the conditions forthe the three-point backward difference scheme to be second order time accuratein the ALE description are derived (Geuzaine et al., 2003; Farhat and Geuzaine,2004). The DGCL only specifies rules (2.217) for the time integration of theconvective fluxes. For the time integration of the diffusive fluxes an analoguenumerical integration is assumed. The three-point backward difference scheme(2.204) becomes:

3

2(ΩiVi)

n+1 − 2(ΩiV i)n +

1

2(ΩiV i)

n−1

+ ∆t

Kc∑

k=1

wckF

ci(V

n+1, xkc , vk) = ∆t

Kd∑

k=1

wdkF

di (V

n+1, xkd) (2.223)

The integration points for the diffusive fluxes may differ from the integration pointsfor the convective fluxes as reflected in the possibly different mesh positions xkcand xkd. From the analysis of the truncation errors the conditions are derived forthe scheme to be second order time accurate. The conditions for the parametersassociated with the convective fluxes are independent of the conditions for theparameters associated with the diffusive fluxes.

For the integration of the convective fluxes the scheme with four integration pointsdefined by equations (2.212)-(2.215) satisfies these conditions for second order timeaccuracy as well as its DGCL. An alternative scheme which satisfies the conditionsfor second order time accuracy uses one integration point with a weight equal toone. As mesh position the position at tn+1 is taken:

ζn+11 = 1 ζn1 = 0 ζn−1

1 = 0 (2.224)

The coefficients for the computation of the mesh velocity correspond to thecoefficients of the first three terms in equation (2.223):

ζn+11 =

3

2

1

∆tζn1 = −2

1

∆tζn−1

1 =1

2

1

∆t(2.225)

TIME INTEGRATION 59

As this scheme is second order time accurate, but does not satisfy its DGCL, itshows that the fulfillment of the DGCL is not a necessary condition to preservethe order of time accuracy.

For the integration of the diffusive fluxes one of the possibilities is to use oneintegration point which uses the mesh position at tn+1 and has a weight equal toone.

By analogy with equation (2.222) a time integration scheme which computes thefluxes once on a unique averaged mesh configuration might be proposed:

3

2(ΩiVi)

n+1 − 2(ΩiV i)n +

1

2(ΩiV i)

n−1

+ ∆tFci (V

n+1,

Kc∑

k=1

wckxkc ,

Kc∑

k=1

wckvk) = ∆tFd

i (Vn+1,

Kd∑

k=1

wdkxkd) (2.226)

The conditions for second order time accuracy for this scheme are a little bitdifferent from those of the scheme of equation (2.223), but all schemes that satisfythe conditions related to equation (2.223) satisfy these conditions as well.

If the fluxes Falei are integrated as in equation (2.221) and the diffusive fluxes are

integrated using one mesh position at tn+1, the following time integration schemeis obtained:

3

2(ΩiVi)

n+1 − 2(ΩiV i)n +

1

2(ΩiV i)

n−1 + ∆tFei (V

n+1, xk)

−Ni∑

j

Iij(Vn+1)

(3

2∆Ωn+1ij − 1

2∆Ωnij

)= ∆tFd

i (Vn+1, xn+1) (2.227)

Only the mesh position xk which is used for the fluxes Fei has to be determined.

Geuzaine et al. (2003) proved that this results in a first order time-accurate schemefor k = n and in a second order time accurate scheme for k = n+ 1 which satisfiesas well its DGCL.

For the finite element discretizations that satisfy the GCL independent of thetime integration scheme, the order of time accuracy can be recovered if the meshvelocities are computed corresponding to the time integration scheme (Forsteret al., 2006). For the three-point backward difference scheme the mesh velocity isgiven by:

3

2xn+1 − 2xn +

1

2xn−1 = vn+1 (2.228)


2.5 Mesh deformation

In section 2.2.1 the governing equations (2.97)-(2.99) for the flow on a domain withdeforming boundaries are derived using the ALE description. In order to computethe convective velocity c (2.16) the velocities v of the nodes of the mesh still have tobe determined. The movement of these nodes can be chosen arbitrarily but shouldpreferentially preserve the quality and the refinements of the mesh. Computationson meshes consisting of distorted elements are less accurate and may require moreiterations and smaller time steps. Especially for the thin elements in the boundarylayer, the thin layer of fluid near the structure in which the velocity changesfrom the velocity of the structure to the free stream value, the limitation of thedistortions is very important and challenging. Ideally, these elements should movetogether with the neighbouring structure with the least amount of deformation.

If the quality of the mesh is still deteriorated too much, a new mesh should begenerated and the flow solution has to be projected from the old onto the new mesh.As this remeshing has a high computational cost and the projection introduceserrors in the flow solution, the challenge is to develop automatic mesh deformationtechniques that minimize the frequency of remeshing.

Meshes that deform to follow the change of the computational domain are calleddynamic meshes. A common method to obtain a suitable motion for the grid pointsis the spring analogy by Batina (1990). All the edges of the elements are replacedby fictitious linear springs. In order to prevent node collisions the spring stiffnessis proportional to the inverse of the edge length, which means that closely spacednodes exhibit stronger spring forces. This analogy works well if the fluid mesh isnot very fine and the mesh motion is relatively small. However, since the rotationof the edges does not induce any force in the linear springs, strongly distortedelements are easily obtained in the case of significant deformations. The distortedelements have very large or very small angles between adjacent edges. Ultimately,elements might be inverted, which is also called snap-through or cross-over (e.g.for two-dimensional triangular elements a vertex of the triangle passes through theopposite edge). For finite elements the sign of the Jacobian determinant shouldbe positive within the element domain.

Farhat et al. (1998a) added torsional springs between adjacent edges in the case oftwo-dimensional triangular unstructured meshes. The stiffness k of the torsionalsprings depends on the angle θ between the adjacent edges as:

k =1

1 + cos θ

1

1− cos θ=

1

sin2 θ(2.229)

Through the term sin θ the stiffness is related to the areas of triangles so that thearea of a triangle cannot become zero or negative. The addition of the torsionalsprings makes the method more robust. Degand and Farhat (2002) extended thismethod to three-dimensional unstructured meshes consisting of tetrahedra. By

MESH DEFORMATION 61

controlling the volume of the tetrahedra, all collapse mechanisms are prevented.The volume control is achieved indirectly by torsional springs which are added intwelve triangles constructed within the tetrahedron.

Analogously, Blom (2000) divided for two-dimensional triangular unstructuredmeshes the linear spring stiffness of an edge by the angle facing this edge.

As the spring analogy is essentially an elliptic problem, Saint-Venant’s principlewhich states that local perturbations of the solution have only a local impact, isapplicable. Therefore, Blom (2000) multiplied the spring stiffness locally with afactor of two in order to conserve the mesh quality in the boundary layers.

In the spring analogy a discrete pseudo-structural system is used to computethe mesh deformations. Johnson and Tezduyar (1994) used a continuous pseudo-structural system for the deformation of the mesh based on the equations of linearelasticity (2.79):

− ρx∂2u

∂t2

∣∣∣∣X

+ (λ+ µ)∇(∇ · u) + µ∇ · (∇u) = 0 (2.230)

The displacements of the structure on the fluid-structure interface are appliedas boundary conditions. The inertia term is usually taken equal to zero and aquasi-static equation is used. During the computation of the element stiffnessmatrices, the Jacobian determinant resulting from the parametric transformationbetween the natural coordinates and the physical coordinates, is dropped. Thismodification stiffens the smaller elements as compared to the larger elements. Thesmaller elements maintain their shape, while the larger elements, usually furtheraway from the structure, take a larger part of the deformations. In order tofurther reduce the distortions of the thin elements in the boundary layer near thestructure, the stiffness of these elements is increased by Stein et al. (2004). Forthe computation of the element stiffness matrices a modified equation is used:

Kji =

∫BeT

j DBeiJ

e

(J0

Je

)χdξdη (2.231)

where J0 is a constant. The case χ = 1 corresponds to dropping the Jacobiandeterminant. By changing χ = 2 for the elements close to the structure, theseelements are more stiffened and behave like an extension of the structural elements.Alternatively, Bar-Yoseph et al. (2001) made the Young’s modulus dependent onelement shape quality measure through a distortion measure.

A quite similar choice to compute the mesh deformation, is to solve the Laplaceequations for the mesh displacements u (Bathe et al., 1999; Robertson and Sherwin,1999):

∇2u = 0 (2.232)


In CFX, a diffusion equation with a variable diffusivity is applied to obtain themesh displacements u:

∇ · (k∇u) = 0 (2.233)

The diffusivity can be specified by the user. In order to maintain the quality ofthe smaller elements, the inverse of the volume of the finite volumes might be usedas diffusivity. As to conserve the mesh quality in the boundary layers the inverseof the wall distance is a valuable alternative.

Lohner and Yang (1996) applied a Laplacian smoother with a variable diffusivityto the mesh velocities v. The diffusivity k depends on the distance to the nearestdeforming boundary. Close to this boundary the diffusivity is large and leads toalmost constant mesh velocities, while far away from this boundary the diffusivityis equal to one which yields a uniform deformation of those elements.

Helenbrook (2003) proposed to use a fourth-order differential equation for the themesh displacements u which allows for the specification of two boundary conditionsalong each boundary: the mesh displacement u and the normal mesh spacing. Thebiharmonic equation is selected as a straightforward generalization of the Laplaceequation.

For all these approaches based on the connectivity between the nodes, a system ofequations has to be solved for all the nodes. Therefore, it is advisable to keep theALE part of the fluid domain as small as possible and use the Eulerian descriptionin the other parts of the domain. On the contrary in point-by-point schemes thedisplacements at the boundaries are interpolated to all nodes within the meshwithout solving a system of equations. Interpolation techniques as the transfiniteinterpolation (TFI) are only applicable to structured or block-structured meshes(Schafer et al., 2006). Kjellgren and Hyvarinen (1998) interpolated the meshvelocities v based on an analytical function of the distance to the closest point onthe moving boundary. In a layer around the moving boundary the mesh velocityis equal to the velocity at the deforming boundary. This method can be appliedto unstructured meshes and complex geometries. de Boer et al. (2007) used radialbasis functions to interpolate the displacements for unstructured meshes. Only asmall system of equations which consists of the nodes on the boundaries has to besolved.

CONCLUSION 63

2.6 Conclusion

In this chapter the techniques to perform fluid flow computations on a domain withmoving boundaries are described. The fluid mesh is aligned with the deformingboundary and should follow its deformations. Therefore, the governing equationsfor the fluid flow are derived in an ALE description. The convective velocity inthese equations is the relative velocity between fluid particles and the mesh. Themesh velocity is arbitrary but should preferentially preserve the quality and therefinements of the mesh.

The spatial discretization by means of the finite element (e.g. Flotran) and thefinite volume method (e.g. CFX) is discussed. The concepts of stabilizationfor finite element discretizations and upwind discretizations for the finite volumemethod, which are needed for flow computations, are presented.

The time integration techniques for an Eulerian description are described. Theimplication of the ALE on the time accuracy and the stability of the timeintegration is discussed and related to the geometric conservation law. The implicitthree-point backward difference method will be used.

Finally, the different options to compute the fluid mesh velocity and deformationare reviewed. In CFX the mesh deformation is computed by means of a diffusionequation with a variable diffusivity.

Chapter 3

Shell structures

3.1 Introduction

Shells are widely used in civil engineering structures as silos, chimneys, watertowers, cooling towers and tensile structures.

The present chapter focusses on the computation of the linear and the geometricalnon-linear response of shell structures. First, the virtual work formulation whichleads to the finite element method is briefly reviewed. The conjugate stresses andstrains used for the computation of geometrical non-linear structures are described.The Newton-Raphson technique solves the non-linear system of equations. In orderto compute the transient geometrical non-linear response of shell structures directtime integration is applied.

As an application, the cylindrical shell structure of the silos in the port of Antwerpis studied. As for the prediction of ovalling oscillations the exact value of thestructural modal damping is very important, an in situ experiment is performedin order to determine the modal damping ratios, eigenfrequencies and eigenmodesof the silos. These modal parameters allow for the validation of a three-dimensionalfinite element model of the silo. It is investigated if geometrical non-linearbehaviour should be taken into account during the ovalling of the silos.

65

66 SHELL STRUCTURES

3.2 Geometric non-linear behaviour

In structural mechanics, three types of non-linearity exist: material, geometric andcontact non-linearities. In the case of material non-linearities the relation betweenstresses and strains is non-linear: the material properties become dependent on thestress state (e.g. yield and creep). For geometric non-linearities, the deformationsare so large that the change in geometry influences the stiffness. The equilibriumshould be expressed with respect to the deformed configuration. Pressure loadsmay change direction during deformation, which are called follower forces. Inthe case of contact non-linearities, the boundary conditions change during thecomputation. The structural non-linear behaviour (Bonet and Wood, 1997) canbe computed using the finite element method (Crisfield, 1991, 1997; Bathe, 1996).

The current section focusses on geometrical non-linear behaviour. A furtherdistinction exists between cases where only the rotations and displacements arelarge and cases where the strains are large as well. The reader familiar with thetheoretical background to geometrical non-linear behaviour may skip this section.

3.2.1 Virtual work

In structural analysis a Lagrangian description is commonly used. As the materialparticles are followed in time, the representation in the material domain does notchange with time. The material configuration is therefore often called the initialconfiguration, while the representation in the spatial domain at time t is calledthe current configuration.

In the finite element method, a weighted residual formulation of the governingmomentum equation (2.79) in the spatial domain is used. In the context ofstructural mechanics, the weighting function might be interpreted as a virtualdisplacement δu and the weighted residual formulation expresses the virtual workdone by the virtual displacement:

δW =

∫

Ωx

(− ρx∂2u

∂t2

∣∣∣∣X

+∇x · σ + ρxb) · δudΩx = 0 (3.1)

Using integration by parts for the second term, equation (3.1) becomes:

∫

Ωx

ρx∂2u

∂t2

∣∣∣∣X

· δudΩx +

∫

Ωx

σ : ∇xδudΩx

=

∫

Ωx

∇x · (σTδu)dΩx +

∫

Ωx

ρxb · δudΩx (3.2)

GEOMETRIC NON-LINEAR BEHAVIOUR 67

Gauss’ theorem enables to write the volume integral in the first term on the righthand side as a boundary integral:

∫

Ωx

ρx∂2u

∂t2

∣∣∣∣X

· δudΩx +

∫

Ωx

σ : ∇xδudΩx

=

∫

Γx

σTδu · nxdΓx +

∫

Ωx


Writing the first term on the right hand as a function of the traction (2.67), thevirtual work becomes:

∫

Ωx

ρx∂2u

∂t2

∣∣∣∣X

· δudΩx +

∫

Ωx

σ : ∇xδudΩx

=

∫

Γx

t · δudΓx +

∫

Ωx


Using the symmetry of the Cauchy stress tensor (2.85), the integrand of the secondterm on the left hand side is written as:

σ : ∇xδu =1

2(σ + σT) : ∇xδu = σ :

1

2(∇xδu + (∇xδu)T) = σ : δε (3.5)

where δε is called the virtual strain. Substituting equation (3.5) in (3.4), theexpression for virtual work in the current configuration is obtained:

∫

Ωx

ρx∂2u

∂t2

∣∣∣∣X

· δudΩx +

∫

Ωx

σ : δεdΩx =

∫

Γx

δu · tdΓx +

∫

Ωx


The second term on the left hand side is the internal virtual work, while the righthand side describes the external virtual work.

When the displacements, rotations and strains are small, the difference betweenthe initial and the current configuration can be neglected. All integrations areperformed over the original volume and surface of the body and all derivatives aretaken with respect to the initial configuration:∫

ΩX

ρX∂2u

∂t2

∣∣∣∣X

·δudΩX+

∫

ΩX

σ : δεdΩX =

∫

ΓX

δu·tdΓX+

∫

ΩX

ρXb·δudΩX (3.7)

Hooke’s law assumes a linear relation between the stress tensor and the smallstrain tensor:

σ = C : ε (3.8)

In the case of an isotropic material, this relation can be described by twoindependent Lame constants λ and µ:

σ = λ(trε)I + 2µε (3.9)

68 SHELL STRUCTURES

The Lame constants are easily derived from the Young’s modulus E and thePoisson coefficient ν. After discretization of the displacements using locally definedshape functions, the finite element system of equations is obtained from equation(3.7):

Mu+ Ku = f (3.10)

In the case of geometric non-linear behaviour, the stiffness matrix and, possiblythe load vector, become dependent on the displacement vector:

Mu+ K(u)u = f (u) (3.11)

3.2.2 Kinematics

As the configuration of the structure changes continuously, appropriate strainmeasures should be defined. The deformation gradient (2.20) relates the relativeposition of two particles in the material domain to their relative position in thespatial domain. Substituting equation (2.6) for the spatial vector x into equation(2.20) for the deformation gradient F gives:

F = ∇Xx = ∇X(X + u) = I +∇Xu = I + D (3.12)

where D is the displacement gradient with respect to the material coordinates.The inverse deformation gradient F−1 relates an infinitesimal spatial vector dx tothe corresponding infinitesimal material vector dX:

dX = ∇xXdx = F−1dx (3.13)

Using equation (2.6), the inverse deformation gradient F−1 is expressed in termsof the displacement gradient with respect to the spatial coordinates:

F−1 = ∇xX = ∇x(x − u) = I−∇xu (3.14)

Strain tensors

The scalar product of two vectors includes the lengths of both vectors and theangle enclosed between the two vectors. Therefore, the change in the scalarproduct of two vectors between the material and the spatial domain is a measureof deformation. Using equation (2.20), the scalar product of two spatial vectors isexpressed in terms of the scalar product of the two material vectors:

dx1 · dx2 = dX1 · FTFdX2 = dX1 ·CdX2 (3.15)


where C = FTF denotes the right Cauchy-Green deformation tensor. The changein scalar product of two vectors from the material to the spatial domain is expressedwith reference to the material domain:

dx1 · dx2 − dX1 · dX2 = (dX1 ·CdX2)− dX1 · dX2 (3.16)

= dX1 · (C− I)dX2 (3.17)

= 2dX1 ·1

2(C− I)dX2 = 2dX1 · EdX2 (3.18)

where E is called the Green-Lagrange strain tensor. Recalling equation (3.12) theGreen-Lagrange strain tensor is obtained in terms of the displacements:

E =1

2(C− I) =

1

2(FTF− I) =

1

2((I +∇Xu)T(I +∇Xu)− I) (3.19)

=1

2(∇Xu + (∇Xu)T + (∇Xu)T∇Xu) (3.20)

=1

2(D + DT + DTD) (3.21)

The Green-Lagrange strain tensor is symmetric and objective, which means thatit is equal to zero if a rigid body rotation is applied to the body. The first andsecond terms are linear and only a quadratic higher-order term appears.

Alternatively, the scalar product of two material vectors is expressed in terms ofthe scalar product of the two spatial vectors:

dX1 · dX2 = dx1 ·F−TF−1dx2 = dx1 · b−1dx2 (3.22)

where b = FFT denotes the left Cauchy-Green deformation tensor. The change inscalar product of two vectors from the material to the spatial domain is expressedwith reference to the spatial domain:

dx1 · dx2 − dX1 · dX2 = dx1 · dx2 − dx1 · b−1dx2 (3.23)

= dx1 · (I− b−1)dx2 (3.24)

= 2dx1 ·1

2(I− b−1)dx2 = 2dx1 · edx2 (3.25)

where e is the Almansi strain tensor:

e =1

2(I− b−1) =

1

2(I− (F−1)TF−1) (3.26)

=1

2(I− (I−∇xu)T(I−∇xu)) (3.27)

=1

2(∇xu + (∇xu)T − (∇xu)T∇xu) (3.28)

70 SHELL STRUCTURES

The Almansi strain tensor is symmetric as well.If the quadratic term is dropped, the small strain tensor is recovered:

ε =1

2(∇xu + (∇xu)T) (3.29)

The change in squared length of a vector between the material and the spatialdomain is obtained from the change in the scalar product (3.18):

dx · dx− dX · dX = 2dXTEdX (3.30)

dl2 − dL2 = 2dXTEdX (3.31)

dl2 − dL2

2dL2=dX

dL· EdXdL

(3.32)

dl2 − dL2

2dL2= nX ·EnX (3.33)

Generally the strain is defined as:

1

a

dla − dLadLa

(3.34)

Commonly used strain measures correspond to different choices of the parametera:

a = 2 dl2−dL2

2dL2 = nX · EnX Green-Lagrange straina = 1 dl−dL

dL Biot (engineering) straina = 0 ln dl logarithmic strain

Polar decomposition

Applying the right polar decomposition, the deformation gradient F is decomposedas the product of an orthogonal tensor R and a symmetric tensor U:

F = RU (3.35)

The orthogonal tensor R corresponds to a rigid body rotation, while the symmetrictensor U corresponds to stretching of the material. The spatial vector dx isobtained by first applying the right stretch tensor U to the material vector dXand then the rotation tensor R. Substituting the right polar decomposition in thedefinition of the right Cauchy-Green deformation tensor C and recalling that R

is orthogonal and U is symmetric, gives:

C = FTF = UTRTRU = UTU = UU = U2 (3.36)


In order to obtain the stretch tensor U, the spectral decomposition of the rightCauchy-Green deformation tensor C is calculated:

C =

3∑

α=1

λ2αNα ⊗Nα (3.37)

The eigenvalues λ2α are equal to the square of the eigenvalues λα of U and the

eigenvectors Nα of the right Cauchy-Green deformation tensor C and the stretchtensor U are identical. The stretch tensor is computed as:

U =

3∑

α=1

λαNα ⊗Nα (3.38)

Using equation (3.35) the rotation tensor R is computed from U and F.

Alternatively, applying the left polar decomposition, the deformation gradient F isdecomposed as the product of a symmetric tensor V times the orthogonal rotationtensor R:

F = VR (3.39)

The spatial vector dx is now obtained by first applying the rotation tensor R tothe material vector dX and then the left stretch tensor V. The left stretch tensorV is expressed in terms of the right stretch tensor U:

V = RURT (3.40)

Substituting the left polar decomposition in the definition of the left Cauchy-Greendeformation tensor C and recalling that R is orthogonal and V is symmetric, gives:

b = FFT = VRRTVT = VVT = VV = V2 (3.41)

The spectral decomposition of the left stretch tensor V is similar to the spectraldecomposition of the right stretch tensor U in equation (3.38):

V =

3∑

α=1

λαnα ⊗ nα (3.42)

Combining equations (3.40) and (3.38), the spectral decomposition of the leftstretch tensor V reads as:

V = R(3∑

α=1

λαNα ⊗Nα)RT =3∑

α=1

λα(RNα)⊗ (RNα) (3.43)

72 SHELL STRUCTURES

The comparison of both expressions for the spectral decomposition of V yields:

nα = RNα

λα = λα (3.44)

The deformation gradient defined in equation (2.20) transforms a vector dX1 =dL1N1 aligned with N1 in the material domain into the vector dx1 in the spatialdomain. Using the right polar decomposition, the vector dx1 is expressed as:

dx1 = FdX1 = (RU)(dL1N1) = dL1(RUN1) = dL1(Rλ1N1)

= dL1λ1(RN1) = dL1λ1n1 = dl1n1 (3.45)

The eigenvalue λ1 gives the ratio between the length of a material vector in thedirection N1 and the length of the corresponding spatial vector, which is alignedwith n1:

λ1 =dl1dL1

(3.46)

Substituting equations (3.36) and (3.41) in respectively the definitions of theGreen-Lagrange strain tensor E (3.21) and the Almansi strain tensor e (3.28)and inserting the spectral decompositions gives:

E =1

2(C− I) =

1

2(U2 − I) =

1

2

3∑

α=1

(λ2α − 1)Nα ⊗Nα

e =1

2(I− b−1) =

1

2(I−V−2) =

1

2

3∑

α=1

(1− λ−2α )nα ⊗ nα (3.47)

The extension of the different strain measures defined in equation (3.34) yields thefollowing strain tensors with respect to the material or the spatial domain:

E(a) =1

a(Ua − I) =

1

a

3∑

α=1

(λaα − 1)Nα ⊗Nα

e(a) =1

a(I−V−a) =

1

a

3∑

α=1

(1− λ−aα )nα ⊗ nα (3.48)


3.2.3 Conjugate stress and strain tensors

In equation (3.6) the virtual work is expressed in the current configuration usingintegrals over the current volume Ωx and the current area Γx. For the computationof the virtual strain δε (3.5) derivatives with respect to the current configurationare needed. As the current configuration is unknown, an expression for virtualwork in the initial configuration is derived.

Kirchhoff stress tensor

Using the relation dΩx = JdΩX between the initial volume dΩX and the currentvolume dΩx and the relation dΓx = JF−TdΓX between the initial area dΓX andthe current area dΓx, an expression with integrals over the initial volume ΩX andthe initial area ΓX emerges from equation (3.6):

∫

ΩX

ρx∂2u

∂t2

∣∣∣∣X

·δuJdΩX+

∫

ΩX

σ : δεJdΩX =

∫

ΓX

t·δu dΓx

dΓX

dΓX+

∫

ΩX

ρxb·δuJdΩX

(3.49)

The Kirchhoff or nominal stress tensor τ is defined as:

τ = Jσ (3.50)

The Kirchhoff stress tensor τ is work conjugate to the strain ε with respect to theinitial volume. The Kirchhoff stress vector tX is defined as:

tX = tdΓx

dΓX

(3.51)

which is the traction vector per unit undeformed area. Both traction vectors tX

and t have the same direction. Finally the virtual work of equation (3.49) isrewritten inserting the above definitions and using equation (2.62):∫

ΩX

ρX∂2u

∂t2

∣∣∣∣X

·δudΩX+

∫

ΩX

τ : δεdΩX =

∫

ΓX

tX·δudΓX+

∫

ΩX

ρXb·δudΩX (3.52)

First Piola-Kirchhoff stress tensor

For the computation of the virtual strain δε (3.5) still derivatives with respectto the current configuration are needed. The relation between the gradient of avirtual displacement with respect to the current coordinates and the gradient withrespect to the initial coordinates is:

∇xδu = ∇XδuF−1 (3.53)

74 SHELL STRUCTURES

Substituting equations (3.5) and (3.53) into the second term on the left hand sideof equation (3.49) gives:

∫

ΩX

ρx∂2u

∂t2

∣∣∣∣X

· δuJdΩX +

∫

ΩX

σ : ∇XδuF−1JdΩX

=

∫

ΓX

t · δu dΓx

dΓX

dΓX +

∫

ΩX

ρxb · δuJdΩX (3.54)

Using the definition of the double product for the the second term on the left handside yields:

∫

ΩX

ρx∂2u

∂t2

∣∣∣∣X

· δuJdΩX +

∫

ΩX

JσF−T : ∇XδudΩX

=

∫

ΓX

t · δu dΓx

dΓX

dΓX +

∫

ΩX

ρxb · δuJdΩX (3.55)

The first factor in the double product is the first Piola-Kirchhoff stress tensorPX = JσF−T, which gives the current force per unit undeformed area and is notsymmetric. Its direction corresponds to the direction of the force in the currentconfiguration. The second factor is the virtual change of the displacement gradientD (3.12):

δD = ∇Xδu (3.56)

Combining equations (3.52), (3.55), (2.82) and (3.56), an expression for virtualwork in the initial configuration is obtained:

∫

ΩX

ρX∂2u

∂t2

∣∣∣∣X

·δudΩX +

∫

ΩX

PX : δDdΩX =

∫

ΓX

tX ·δudΓX +

∫

ΩX

ρXb ·δudΩX

(3.57)

This expression can also be obtained by using a weighted residual formulation ofthe governing momentum equation (2.84) in the material domain with an arbitraryvirtual displacement δu as weighting function.

Second Piola-Kirchhoff stress tensor

The Kirchhoff stress vector tX as defined in equation (2.82) still points in the samedirection as the Cauchy stress vector t. Using the deformation gradient F, it istransformed to the initial configuration:

F−1tX = F−1PXnX (3.58)


The second Piola-Kirchhoff stress tensor is defined as:

S = F−1PX = JF−1σF−T = F−1τF−T (3.59)

The definition shows that the second Piola-Kirchhoff stress tensor is symmetric.The stress vector t′

Xis similarly defined as:

t′X = F−1tX = SnX (3.60)

Introducing PX = FS in the second term on the left hand side of equation (3.57)gives:

∫

ΩX

ρX∂2u

∂t2

∣∣∣∣X

·δudΩX+

∫

ΩX

FS : ∇XδudΩX =

∫

ΓX

tX·δudΓX+

∫

ΩX

ρXb·δudΩX

(3.61)

Using the definition of the double product yields for the integrand of the secondterm on the left hand side:

FS : ∇Xδu = tr(STFT∇Xδu) (3.62)

Recalling that the second Piola-Kirchhoff stress tensor is symmetric, gives:

FS : ∇Xδu = tr(1

2(S + ST)TFT∇Xδu) (3.63)

= tr(ST 1

2(FT∇Xδu + (∇Xδu)TF)) (3.64)

Substituting the expression for the deformation gradient F (3.12), it becomes:

FS : ∇Xδu = tr(ST 1

2((I +∇Xu)T∇Xδu + (∇Xδu)T(I +∇Xu)) (3.65)

= tr(ST 1

2(∇Xδu + (∇Xδu)T + (∇Xu)T∇Xδu

+ (∇Xδu)T∇Xu)) (3.66)

where the second factor is the virtual Green-Lagrange strain:

δE =1

2(∇Xδu + (∇Xδu)T)︸︷︷︸

δEl

+1

2((∇Xu)T∇Xδu + (∇Xδu)T∇Xu)︸︷︷︸

δEnl

(3.67)

76 SHELL STRUCTURES

which finally yields:

FS : ∇Xδu = S : δE (3.68)

Combining equations (3.61) and (3.68), an expression for virtual work in the initialconfiguration is obtained:

∫

ΩX

ρX∂2u

∂t2

∣∣∣∣X

·δudΩX +

∫

ΩX

S : δEdΩX =

∫

ΓX

tX ·δudΓX +

∫

ΩX

ρXb ·δudΩX

(3.69)

3.2.4 Incremental virtual work

Suppose all the configurations up to time tn are known and the unknownconfiguration at time tn+1 has to be computed. For the reference configurationwith respect to which the virtual work (3.69) is expressed, any known configurationcan be used. In the total Lagrangian (TL) formulation all variables are referredto the initial configuration at time 0 for all time steps, while in the updatedLagrangian (UL) formulation the reference configuration is updated for every timestep and all variables are referred to the last calculated configuration at time tn.

Total Lagrangian formulation

In the total Lagrangian (TL) formulation, the displacements at time tn+1 areobtained by adding a displacement increment to the known displacements at timetn:

un+1 = un + ∆u (3.70)

The subscript indicates at which time the displacement occurs. The second Piola-Kirchhoff stress at time tn+1 is decomposed in the initial configuration as:

Sn+10 = Sn0 + ∆S0 (3.71)

The subscript indicates the configuration with respect to which the stress is defined.The stress increment ∆S0 is given by:

∆S0 = Cn+10 : ∆E0 (3.72)

where Cn+10 is the incremental stress-strain tensor at time tn+1 with respect to

the initial configuration. The Green-Lagrange strain at time tn+1 with respect tothe initial configuration is:

En+10 = En0 + ∆E0 (3.73)


The Green-Lagrange strain increment is expressed as a function of the displacementincrement:

∆E0 =1

2(∇X∆u + (∇X∆u)T)︸︷︷︸

∆El0

(3.74)

+1

2((∇Xun)T∇X∆u + (∇X∆u)T∇Xun

︸︷︷︸∆Enl

0

+ (∇X∆u)T∇X∆u︸︷︷︸neglected

) (3.75)

As the virtual displacements are not a function of time, from the virtual Green-Lagrange strain (3.67) the virtual strain increment is derived:

∆(δE0) =1

2((∇X∆u)T∇Xδu + (∇Xδu)T∇X∆u) (3.76)

Using equations (3.71- 3.76) the internal virtual work in equation (3.69) is rewrittenas a function of the displacement increment:

∫

ΩX

∆S0 : δE0dΩX +

∫

ΩX

S0 : ∆(δE0)dΩX (3.77)

Substituting equation (3.72) for the constitutive behaviour into the previousequation gives:

∫

ΩX

δE0 : Cn+10 : ∆E0dΩX +

∫

ΩX

S0 : ∆(δE0)dΩX (3.78)

Splitting the virtual Green Lagrange strain (3.67) and the strain increment (3.75)in their linear and non-linear parts yields:

∫

ΩX

δEl0 : Cn+1

0 : ∆El0dΩX +

∫

ΩX

δEl0 : Cn+1

0 : ∆Enl0 dΩX

+

∫

ΩX

δEnl0 : Cn+1

0 : ∆El0dΩX +

∫

ΩX

δEnl0 : Cn+1

0 : ∆Enl0 dΩX

+

∫

ΩX

S0 : ∆(δE0)dΩX (3.79)

After discretization the first term becomes the linear stiffness matrix K0. Thesecond, third and fourth term form the initial displacement (slope) stiffness matrixKL and the last term is the geometric stiffness (initial stress, stress stiffening)matrix Kσ.

78 SHELL STRUCTURES

Updated Lagrangian formulation

In the updated Lagrangian (UL) formulation, the reference configuration isupdated to the last calculated configuration. The updated material coordinatesbecome:

Xn+1 = Xn + un (3.80)

With respect to the updated reference configuration, the displacements at timetn+1 are:

un+1 = ∆u (3.81)

The second Piola-Kirchhoff stress tensor at time tn+1 with respect to the updatedreference configuration is:

Sn+1n = σn + ∆Sn (3.82)

where σn are the Cauchy stresses computed in the previous time step. The stressincrement ∆Sn is given by:

∆Sn = Cn+1n : ∆En (3.83)

where Cn+1n is the incremental stress-strain tensor at time tn+1 with respect to

the updated configuration. The Green-Lagrange strain at time tn+1 with respectto the updated configuration is:

En+1n = ∆En (3.84)

The Green-Lagrange strain increment is expressed as a function of the displacementincrement:

∆En =1

2(∇X∆u + (∇X∆u)T)︸︷︷︸

∆εn

+1

2((∇X∆u)T∇X∆u)︸︷︷︸

neglected

(3.85)

The first term is equal to the engineering strain increment computed at time tn.The virtual Green-Lagrange strain reduces to:

δEn =1

2(∇Xδu + (∇Xδu)T)︸︷︷︸

δεn

+1

2((∇Xδu)T∇Xδu)︸︷︷︸

neglected

(3.86)

The first term corresponds to the virtual engineering strain as computed at timetn. The expression for the virtual strain increment is identical to equation (3.76).Using equations (3.82-3.85) the internal virtual work in equation (3.69) is rewrittenas a function of the displacement increment:

∫

ΩX

∆Sn : δEndΩX +

∫

ΩX

σn : ∆(δEn)dΩX (3.87)


Substituting equation (3.83) for the constitutive behaviour into the previousequation gives:

∫

ΩX

δEn : Cn+1n : ∆EndΩX +

∫

ΩX

σn : ∆(δEn)dΩX (3.88)

Neglecting the higher order terms in the virtual Green Lagrange strain (3.86) andthe strain increment (3.85) yields:

∫

ΩX

δεn : Cn+1n : ∆εndΩX +

∫

ΩX

σn : ∆(δEn)dΩX (3.89)

After discretization the first term becomes the linear stiffness matrix K0. Thesecond term is the geometric stiffness matrix Kσ. In contrast to the totalLagrangian formulation, there is no initial displacement stiffness matrix.

Follower forces

In the external virtual work, body forces (e.g. gravity) and nodal forces actingin the global coordinate system are not deformation dependent. On the contrary,the traction applied on a surface changes if the area or the normal of this surfacechanges during deformation. Loads which are dependent on the deformations arecalled follower forces. In the case of a uniform normal pressure p applied to asurface, the external virtual work (3.6) in the current configuration becomes:

∫

Γx

(−p)nx · δudΓx (3.90)

If the surface is parameterized using a single isoparametric element, the normaland the infinitesimal area are expressed as a function of ∂x

∂ξ and ∂x

∂η :

∫

Γξ

(−p)(∂xn+1

∂ξ× ∂x

n+1

∂η

)· δudΓξ (3.91)

Rewriting this equation as a function of the displacement increment gives:

∫

Γξ

(−p)[∂xn

∂ξ·(∂∆u

∂η× δu

)− ∂x

n

∂η·(∂∆u

∂ξ× δu

)]dΓξ (3.92)

After discretization this becomes the follower load stiffness matrix Kp, which isnonsymmetric.

80 SHELL STRUCTURES

3.2.5 Newton-Raphson procedure

In order to solve the non-linear equations of the previous sections, the load isapplied in a number of load increments and an iterative procedure is used to obtainthe solution for each load increment. At each load level equilibrium equations areperformed in order to obtain convergence:

1. Set i = 1

2. Predict the displacements u(0). Usually the converged solution of theprevious load increment is used for the displacement u(0) and the restoringforce f (0)

r. If a different prediction is used, compute the values corresponding

to this prediction. Compute the out-of-balance load vector:

r(0) = f (0)

e− f (0)

r(3.93)

3. Compute the tangent stiffness matrix:

K(i−1)T =

∂f (i−1)

r

∂u(i−1)(3.94)

4. Compute the displacement increment ∆u(i) from:

K(i−1)T ∆u(i) = r(i−1) (3.95)

and a new total displacement from:

u(i) = u(i−1) + ∆u(i) (3.96)

5. Compute the restoring force f (i)

rcorresponding to the element stresses.

For material non-linear behaviour a strain path has to be chosen and theconstitutive equation has to be integrated along this path.

6. Compute the external applied force f (i)

eand the out-of-balance load vector

r(i):

r(i) = f (i)

e− f (i)

r(3.97)

Only in the case of follower forces (as pressures) the external applied forcemight change each iteration.

7. If the maximum number of equilibrium iterations is reached: i = imax, stop.


8. Check the convergence for the displacements:

‖∆u(i)‖L2< εu‖u(i)‖L2

(3.98)

and for the out-of-balance load vector:

‖r(i)‖L2< εr‖f (i)

e‖L2

(3.99)

The L2-norm consists of the square root of the sum of squares (SRSS) ofthe nodal values. Quantities with different units should not be summedin the convergence criteria, which are therefore checked separately fordisplacements and rotations and for forces and moments. If the convergencecriteria are fulfilled, stop. Otherwise set i = i+ 1 and go to step 3.

A number of modifications can be made to this scheme.

In the modified Newton-Raphson method, the tangent stiffness matrix K(0)T of the

first iteration is used for all equilibrium iterations during a load increment. Thetangent stiffness matrix only has to be factorized once per load increment. In

the initial-stiffness Newton-Raphson method, the tangent stiffness matrix K(0)T is

evaluated only once in the first iteration of the first load increment and is used inall iterations of all load increments.

Adaptive descent uses a weighted average of the secant and the tangent stiffnessmatrix to compute the displacement increment (3.95):

K(i−1) = ξK(i−2)T + (1− ξ)K(i−1)

T (3.100)

The descent parameter ξ is adapted according to the convergence behaviour.

The line search technique alternatively computes a new total displacement from:

u(i) = u(i−1) + µ∆u(i) (3.101)

where the underrelaxation factor µ is determined by minimizing the total potentialenergy of the structure.

The magnitude of a new load increment is predicted based on the number ofequilibrium iterations needed at the previous load increment, the maximumallowable strain increment and the response frequency for dynamic computations.If the convergence criteria have not been satisfied within the maximum number ofequilibrium iterations, the load increment is bisected and the iterative procedureis repeated.

If the behaviour of a structure beyond a (local) maximum in the load-deflectioncurve has to be studied, methods which allow for an increase in displacement withdecrease in load are needed. A possibility is to apply displacement increments

82 SHELL STRUCTURES

instead of load increments. A more general approach is provided by the arc-lengthmethod, which uses a variable load increment during the equilibrium iterations. Aconstraint equation relates the load increment to the displacement increment:

(∆f (i)

e)2 +

(∑i∆u

(i))T(∑i∆u

(i))

β2= (∆l)2 (3.102)

where ∆l is the arc-length.

3.2.6 Time integration

The direct time integration is performed using the Newmark method. Theequilibrium equations are formulated at time tn+1:

Mun+1 + Cun+1 + Kun+1 = fn+1 (3.103)

The acceleration un+1 and the velocity un+1 at time tn+1 are expressed as afunction of the displacement un+1 at time tn+1 and known quantities at time tn

(Bathe, 1996):

un+1 =δ

α∆t

(un+1 − un

)− δ − ααun − δ − 2α

2α∆tun (3.104)

un+1 =1

α∆t2(un+1 − un

)− 1

α∆tun − 1− 2α

2αun (3.105)

Substituting expressions (3.104) and (3.105) in the equilibrium equations (3.103)at time tn+1 yields:

(M

α∆t2+δC

α∆t+ K

)un+1 = fn+1 + M

(un

α∆t2+un

α∆t+

1− 2α

2αun)

+C

(δ

α∆tun +

δ − ααun +

δ − 2α

2α∆tun

)

(3.106)

This is the general equation for all Newmark time integration schemes, which areimplicit schemes. For α = 1

4 and δ = 12 the constant-average-acceleration method

or trapezoidal rule is obtained which is unconditionally stable and second-ordertime accurate. In order to reduce the period elongation and obtain sufficientlyaccurate results, the time step should be a fraction (e.g. 1/20) of the eigenperiodsthat contribute to the response.

An alternative time integration method is the midpoint rule. The equilibriumequations are formulated at time tn+ 1

2 :

Mun+1/2 + Cun+1/2 + Kun+1/2 = fn+1/2 (3.107)


The acceleration un+1/2, velocity un+1/2 and displacement un+1/2 at the midpointare approximated as:

un+1/2 =1

2

(un+1 + un

)(3.108)

un+1/2 =1

2

(un+1 + un

)(3.109)

un+1/2 =1

2

(un+1 + un

)(3.110)

The combination of these equations with the equations (3.104) and (3.105) yieldsthe midpoint time integration scheme. In the linear case this method is identicalto the trapezoidal rule.

84 SHELL STRUCTURES

3.3 Structural behaviour of a silo

In this section the structural behaviour of the silos in the port of Antwerp isanalysed. As for the prediction of ovalling oscillations the exact value of thestructural modal damping is very important, an in situ experiment is performedin order to determine the modal damping ratios, eigenfrequencies and eigenmodesof the silos. The mode shapes and frequencies of a single silo are first computedwith a harmonic finite element model, in order to support the design of the in situexperiment. Radial accelerations are measured in situ under ambient wind loadingand modal parameters are extracted from the output-only data in order to validatea finite element model of the silo (Dooms et al., 2006b). A three-dimensional finiteelement model of the silo accounts for the influence of the boundary conditionson the eigenfrequencies and mode shapes. Finally, the importance of geometricalnon-linear behaviour is studied for deformations of order of magnitude of 0.1 m.

3.3.1 Description of the silo structure

The silos are circular cylindrical shell structures with a diameter D = 5.5 m anda height h = 25 m (figure 1.3). One cylinder consists of 10 aluminium sheets witha height of 2.5 m and a thickness that decreases with the height (table 3.1). Theheight-to-radius ratio h/R = 9.1 and the radius-to-thickness ratio ranges fromR/t = 262 at the bottom to R/t = 458 at the top.

At the top and the bottom, a cone is welded to the cylinder at an angle of 15

and 60 with the horizontal plane, respectively. An octagonal beam supports thebottom of the cylinder and is bolted to the silo at 4 points around the circumference(figure 3.1). The silos are made of aluminium with a Young’s modulus E = 67600×106 N/m

2, a Poisson’s ratio ν = 0.35 and a density ρ = 2700 kg/m

3.

Height Thickness[m] [mm]

0.0 - 2.5 10.52.5 - 5.0 9.05.0 - 7.5 8.57.5 - 10.0 7.510.0 - 12.5 7.012.5 - 15.0 6.515.0 - 25.0 6.0

Table 3.1: Thickness of the aluminium plates of the silo as a function of the height.

STRUCTURAL BEHAVIOUR OF A SILO 85

Figure 3.1: Octagonal beam with four bolted connections at the bottom of thecylinder.

3.3.2 Harmonic finite element model

In order to design a proper experimental setup, the eigenfrequencies and modeshapes are computed with a harmonic finite element model of a single silo. In acylindrical frame of reference, the displacements of an axisymmetric structure aredecomposed as (Zienkiewicz and Taylor, 2005):

ur(r, θ, z, t)uθ(r, θ, z, t)uz(r, θ, z, t)

=

∞∑

n=0

cos(nθ) 0 0

0 sin(nθ) 00 0 cos(nθ)

usrn(r, z, t)usθn(r, z, t)uszn(r, z, t)

(3.111)

+

∞∑

n=0

sin(nθ) 0 0

0 cos(nθ) 00 0 sin(nθ)

uarn(r, z, t)uaθn(r, z, t)uazn(r, z, t)

(3.112)

where n indicates the circumferential wave number and the superscripts s and arefer to the symmetric and antisymmetric terms with respect to θ. For each valueof n, the eigenfrequencies and mode shapes for the symmetric case are calculatedusing the harmonic 2-node SHELL61 element of the finite element program Ansys(Ansys, 2005b). Each mode shape is referred to by a couple (m,n), where mdenotes the half wave number in the axial direction (m/2 is the number of axialwaves) and n is the number of circumferential waves. The antisymmetric caseyields identical eigenfrequencies and similar mode shapes.

86 SHELL STRUCTURES

Figure 3.2a shows the finite element model and the applied boundary conditions.On the axis of symmetry, the radial displacement ur and the rotation ϕθ areconstrained. The vertical displacements uz are restricted around the circumferenceat the bottom of the cylinder.

a. Model b. (1, 4) c. (1, 5) d. (1, 3) e. (1, 6) f. (2, 5)

4.65 Hz 5.52 Hz 5.99 Hz 7.38 Hz 8.78 Hz

Figure 3.2: Harmonic finite element model of the silo and selected eigenmodes.

A finite element length le = 0.83 m is used, except near the connections betweenthe cylinder and the cones, where local bending waves occur. At the top of thecylinder, the axial wavelength is estimated as (Billington, 1965):

λz =2π√Rt

4√

3(1− ν2)(3.113)

resulting into λz = 0.634 m for a cylinder with a radius R = 2.75 m and a shellthickness t = 0.006 m. The number of elements per wavelength is equal to 12, sothat the element length is equal to le = 0.053 m near the connections.


Figures 3.2b up to 3.2f show five mode shapes referred to by the couple (m,n).Mode (1,4) at 4.65 Hz has the lowest frequency. The degree of clamping of thebottom of the cylinder increases with increasing circumferential wave number n.Figure 3.3 summarizes the eigenfrequencies as a function of m and n. For modescorresponding to m = 1, the eigenfrequency is minimal for n = 4. Whereas forbeam and plate structures, the complexity of the modes shapes generally increasesfor increasing eigenfrequencies, this appears not to be the case here.

Figure 3.3: Eigenfrequencies of the harmonic finite element model of the silo as afunction of m and n.

The dimensionless strain energy S of a freely supported cylinder is defined as(Arnold and Warburton, 1949):

S =4S(1− ν2)

Eπh|ur|2(3.114)

Figure 3.4 shows the dimensionless strain energy for a cylinder with a heighth = 25 m and a constant shell thickness t = 7 mm as a function of n for m = 2and m = 3. The total strain energy associated with a mode shape is equal tothe sum of the bending and the stretching energy. The bending energy is almostindependent on m and strongly increases with n. The stretching energy increaseswith m and strongly decreases with n. Consequently, the total strain energy hasa minimum as a function of n, which shifts to higher values of n for increasingm. The order of the eigenfrequencies of the silo (figure 3.3) is related to the total

88 SHELL STRUCTURES

strain energy associated with a mode shape. The dotted line in figure 3.3 shows theeigenfrequencies for such a tube, representing the silo without the cones. As onlythe bending energy is important for a tube, this curve is a good approximationfor the eigenfrequencies of the silo for high values of n, for which the stretchingenergy is small. This approximation is best at low values of m since the stretchingstrain energy increases with m.

Figure 3.4: Dimensionless total strain energy (solid line) and dimensionless strainenergy due to bending (dashed-dotted line) and stretching (dashedline) for m = 2 () and m = 3 (N) for a freely supported cylinderwith a height h = 25 m and a constant thickness t = 7 mm.

3.3.3 The experimental setup

Measurements (Dooms et al., 2003) have been performed on the silo on the cornerof the group where the largest amplitudes were observed during the October 2002storm. Two filling pipes are attached to this silo in discrete points along the height(figure 1.3).

In the case of wind excitation, the eigenmodes below 15 Hz have the highestcontribution to the structural response. The results of the harmonic finite elementcalculation (figure 3.3) indicate that these eigenmodes incorporate maximum sevencircumferential waves (n = 8) and a single axial wave (m = 3).


Ten accelerometers are glued to the silo wall to measure the radial acceleration.Eight accelerometers are placed along a horizontal arc of 7π/24 radians at a heightof 15 m (figure 3.5); the angle between two accelerometers is equal to π/24 radians,corresponding to 6 accelerometers per wavelength for a circumferential wave withn = 8. Two additional accelerometers are placed on a vertical line at a height of2 m and 10 m (figure 3.5) in order to identify the number m.

Figure 3.5: Location of the accelerometers on the silo wall.

The accelerations are sampled at fmax = 100 Hz and 18432 time steps are recorded,resulting in a measurement time of about three minutes.

90 SHELL STRUCTURES

Figure 3.6 presents the power spectral density of the radial acceleration measuredat the point HL09. The spectrum clearly shows peaks around 4 Hz and 5.3 Hz.

0 5 10 15 200

0.05

0.1

0.15

0.2

Frequency [Hz]

Pow

er s

pect

rum

[(m

/s²/

Hz)

²]

Figure 3.6: Power spectral density of the radial acceleration at the point HL09.

The stochastic subspace identification method (Van Overschee and De Moor, 1996;Peeters and De Roeck, 1999) is used to estimate the modal parameters of astructure based on output-only time data. The unknown excitation is assumedto be a realization of a stochastic process (white noise). From the measuredaccelerations, stochastic state space models are identified of different order N ,ranging from 2 to 350 in steps of 2. N/2 modal parameters (eigenfrequencies,complex mode shape vectors and modal damping ratios) are extracted from amodel of order N . If similar modal parameters are obtained with increasing modelorder, a physical eigenmode is identified. Sixty eigenmodes are identified below20 Hz. The modal damping ratios ξ vary between 0.07 % and 1.32 %, which arelow values as expected for a welded aluminium structure. Figure 3.7 shows a topview and an isometric view of three identified eigenmodes. Whereas the modeshapes 3 and 53 are almost entirely real and correspond to standing waves alongthe circumferential direction, mode 4 has an imaginary part with a comparablemagnitude as the real part, resulting in a propagating wave. The latter is expectedonly for axisymmetric structures, while the axisymmetry is slightly disturbed hereby the filling pipes along the height of the silo. A perfect axisymmetric structurehas pairs of eigenmodes corresponding to identical eigenfrequencies, reflecting thatthese mode shapes are indefinite and propagate in the circumferential direction.The radial modal displacement urmn(r, θ, z) at the eigenfrequency ωmn is equalto urmn(r, z)[cos(nθ) + i sin(nθ)], which is represented by a circle with radiusurmn(r, z) and center at the origin of the complex plane. Figure 3.8 depicts thedisplacements of mode 4 in the points HL6 to HL13 (corresponding to differentvalues of θ) in the complex plane, resulting in an almost circular shape.


8

157

9

16

6

10

1112

13

6

157

16

8

910

1112

13

7

6

16

8

15

9

11

10

12

13

a. Mode 3 b. Mode 4 c. Mode 53

f = 4.00 Hz f = 4.01 Hz f = 17.95 Hz

ξ = 0.77 % ξ = 0.81 % ξ = 0.10 %

Figure 3.7: Top and three-dimensional view of the real (solid line) and imaginary(dashed line) part of the identified eigenmodes 3, 4 and 53 of the silo.The dotted line shows the undeformed shape.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real part

Imag

inar

y pa

rt

Figure 3.8: The radial displacements of mode 4 in the points HL6 to HL13 in thecomplex plane. The displacements are scaled to unity in the pointHL6.

92 SHELL STRUCTURES

Radial modal displacements urmn(r, θ, z) at a height z are written as a linearcombination of the radial displacements corresponding to the symmetric andantisymmetric eigenmodes at this frequency:

urmn(r, θ, z) = αusrmn(r, z) cos(nθ) + βua

rmn(r, z) sin(nθ) (3.115)

A least squares analysis is performed to determine the values of the parameters αand β and the circumferential wave number n that give the best correspondencewith the measured mode shapes at a height z = 15 m. As no analytical expressionis available for the mode shape as a function of the height z, the number of halfwavelengths m is estimated from the experimental results. Figure 3.9 shows theexperimental eigenfrequencies of the silo as a function of m and n. Comparisonof these results with the predicted eigenfrequencies (figure 3.3) reveals that theharmonic finite element model overestimates the lowest eigenfrequencies. Thisdifference cannot be explained by a variation of the Young’s modulus withinreasonable bounds. The eigenfrequencies and mode shapes of a circular cylindricalshell structure are very sensitive to the boundary conditions imposed on the axialdisplacements uz, while the influence of the boundary condition for the rotation ϕθis almost negligible (Forsberg, 1964; Koga, 1988; Leissa, 1993). The harmonic finiteelement model imposes zero axial displacements around the whole circumferenceat the bottom of the silo. In reality, each silo is only bolted to the octagonal beamin four points around the circumference.

Figure 3.9: Measured eigenfrequencies of the silo as a function of m and n.


The time history of the radial acceleration in each point is decomposed into 68modal contributions by a stochastic state space model of order N = 136 (Peetersand De Roeck, 1999). Most of these modes are physical, although spurious modesmay exist, which are identified by a high damping ratio. The modal contributionsare ordered according to decreasing Root Mean Square (RMS) values of their timehistory. Figure 3.10 shows the time history of the radial acceleration at the pointVL15, the nine modal contributions with the highest RMS value and the error,which is defined as the difference between the measured response and the sum of68 modal contributions. In order to determine the importance of an eigenmodein the global response of the silo, the average of the RMS values of the modalcontributions in all channels is calculated. The dominant eigenfrequencies areidentified as 3.94 Hz, 4.00 Hz and 4.49 Hz. These eigenfrequencies may slightlydiffer from the previously identified eigenfrequencies, as the latter are selected atdifferent orders N , while the former are all selected at order N = 136.

3.3.4 Three-dimensional finite element model of the silo

In order to reduce the difference between the measured and computedeigenfrequencies and mode shapes, a realistic modelling of the four boltedconnections is required, necessitating a three-dimensional finite element modelof the silo.

A first model exploits the symmetric geometry to reduce the model to one fourthof the silo, and will be referred to as the quarter three-dimensional finite elementmodel. The silo is modelled with the 8-node quadrilateral SHELL93 element(Ansys, 2005b). Four symmetry planes are defined by the vertical axis and theposition of the bolted connections, where all degrees of freedom are constrained.The finite element mesh is similar to the three-dimensional finite element mesh ofthe complete silo that will be introduced later in figure 3.14a. For the eight centralaluminium sheets (with a height of 2.5 m each), 15 shell elements are used alongan arc in the circumferential direction with an angle π/2 and 16 elements are usedalong the vertical direction. In the zones near the lower and upper edges of the siloand on both cones, smaller finite elements are used as to fulfill the requirementdefined in equation (3.113). The total number of shell elements in the quarterthree-dimensional finite element model is equal to 4747.

Convergence is checked by computing the eigenfrequencies of alternative finiteelement models where the mesh has been uniformly refined with a factor of

√2, 2

and 2√

2. The approximated relative error εj on an eigenfrequency ωj is definedas:

εj =| ωj − ωres

j |ωresj

(3.116)

94 SHELL STRUCTURES

Figure 3.10: Time history of the radial acceleration at the point VL15 (top), thenine modal contributions with the highest RMS value and the error(bottom).


where the reference frequency ωresj is computed with the finest finite element mesh

(mesh refinement factor 2√

2).

Figure 3.11 shows the approximated relative error on the predicted eigenfrequen-cies as a function of the mesh refinement factor. For the original mesh (meshrefinement factor 1), all relative errors are below 10−2 and most of the values areeven below 10−3. This is sufficiently accurate for the subsequent comparison withthe experimental eigenfrequencies, as the latter are also subject to measurementand identification errors.

1 1.4 210

−5

10−4

10−3

10−2

Refinement factor [−]

Rel

ativ

e er

ror

[−]

(1,4)(2,6)(3,10)(1,3)(1,5)(4,9)(1,4)(1,2)(3,10)

Figure 3.11: The approximated relative error on nine selected eigenfrequenciespredicted with the quarter three-dimensional finite element modelas a function of the mesh refinement factor.

Four different models with symmetric and antisymmetric boundary conditionson the symmetry planes are analysed. The two models with symmetrical andantisymmetrical boundary conditions (SA and AS) result in similar mode shapeswith identical eigenfrequencies, corresponding to odd values of the circumferentialwave number n. The eigenmodes of the models with symmetrical-symmetrical(SS) and antisymmetrical-antisymmetrical (AA) boundary conditions correspondto even values of n and have different mode shapes and eigenfrequencies due to thedifferent relative position of the bolted connections and the ovalling mode shapes(figures 3.12a and 3.12b). Although in these models the four bolted connections atthe bottom of the silo violate axisymmetry, still two eigenmodes can be associatedwith every couple (m,n), as for the harmonic finite element model.

The disadvantage of the quarter three-dimensional finite element model is that itfixes the relative position of the four bolted connections and the ovalling modeshapes. The eigenmodes (1,6), computed with the quarter three-dimensionalfinite element models with SS (figures 3.12a) and AA boundary conditions (figure3.12b), have fixed relative positions between the ovalling mode shapes and thebolted connections (mode shapes are shown of the part of the cylindrical shell at

96 SHELL STRUCTURES

a height between 15 and 17.5 m above the bottom cone). A three-dimensionalfinite element model of the complete silo with constrained degrees of freedom atthe four bolted connections enables to realize other relative positions between theovalling mode shapes and the bolted connections (figure 3.12c). As in the quarterthree-dimensional model, two eigenmodes are associated with every couple (m,n).

a. SS model b. AA model c. 3D model

Figure 3.12: Top view of the eigenmode (1, 6) of the silo at a height of 15 mcomputed with (a,b) the quarter three-dimensional finite elementmodel with SS and AA boundary conditions and (c) the full three-dimensional finite element model.

Figure 3.13a shows a top and a lateral view of the eigenmode (1,5) computed withthe quarter three-dimensional finite element model with AS boundary conditions,fixing the relative position of the global bending modes and the ovalling modes.

a. SA model b. 3D model

Figure 3.13: Top and lateral view of the eigenmode (1, 5) of the silo computedwith (a) the quarter three-dimensional finite element model with SAboundary conditions and (b) the full three-dimensional finite elementmodel.


The three-dimensional finite element model of the complete silo allows for anycombination of bending and ovalling (figure 3.13b) and is therefore preferred; itis obtained by mirroring the quarter model on two symmetry planes through thebolted connections (figure 3.14a).

Figures 3.14b up to 3.14f show a top view and a three-dimensional view of fivecalculated mode shapes of the silo. The top view corresponds to the displacementsof the part of the cylindrical shell at a height between 15 and 17.5 m above thebottom cone and reveals that mode shape (1,5) and, to a lesser extent, mode shape(1,3) are a combination of global bending of the silo and ovalling.

a.Model b.(1, 3) c.(1, 4) d.(1, 5) e.(1, 6) f.(1, 2)

3.93 Hz 3.93 Hz 5.25 Hz 7.37 Hz 7.75 Hz

Figure 3.14: Three-dimensional finite element model of the silo and selectedeigenmodes.

98 SHELL STRUCTURES

Figure 3.15 summarizes the eigenfrequencies calculated with the full three-dimensional finite element model. Two mode shapes and eigenfrequencies areassociated with every couple (m,n). Table 3.2 compares the eigenfrequenciescomputed with the harmonic finite element model, the quarter three-dimensionalfinite element model and the full three-dimensional finite element modelwith the experimental eigenfrequencies. The eigenfrequencies of the quartermodel are computed with the different sets of symmetric and antisymmetricboundary conditions. For the full three-dimensional finite element model, theeigenfrequencies of the two different eigenmodes that can be associated with everycouple (m,n) are given. The correct representation of the bolted connectionsstrongly influences the eigenfrequencies of eigenmodes with a low circumferentialwave number n, more specifically those eigenmodes where the contribution ofthe stretching energy to the total strain energy is large. Differences between thequarter model and the full three-dimensional finite element model are apparentfor the eigenfrequencies of the modes (1,3), (1,5) and (2,5), whereas all othereigenmodes have the same eigenfrequencies. The eigenfrequencies of the fullthree-dimensional model agree better with the experimental eigenfrequencies. Theeigenfrequencies of modes (1,4) and (1,3) are almost identical. Discrepancies canbe explained by the presence of the filling pipes, the assumed boundary conditions,and the uncertainty on the modal parameters derived from the dynamic systemidentification procedure.

Figure 3.15: Eigenfrequencies of the three-dimensional finite element model of thesilo as a function of m and n.


(m,n) Harm. Quarter 3D Full 3D ExperimentalAA SS SA AS

(1,2) 10.8 7.75 8.48 7.75 8.48 7.87(1,3) 5.99 4.19 4.19 3.93 3.93 3.91 4.49(1,4) 4.65 3.93 4.04 3.93 4.04 3.91 4.00 4.08 4.49(1,5) 5.52 5.42 5.42 5.25 5.25 5.31 5.34(1,6) 7.38 7.37 7.37 7.37 7.37 7.07 7.23 7.49(1,7) 9.73 9.72 9.72 9.72 9.72 9.84 10.7(1,8) 12.5 12.5 12.5 12.5 12.5 12.8(1,9) 15.8 15.7 15.7 15.7 15.7(1,10) 19.4 19.4 19.4 19.4 19.4 19.8(2,3) 16.5 13.9 13.9 13.9 13.9(2,4) 10.7 8.71 8.94 8.71 8.94(2,5) 8.78 8.08 8.08 8.01 8.01(2,6) 9.47 9.29 9.39 9.29 9.39 9.05 9.25(2,7) 11.5 11.5 11.5 11.5 11.5 11.1 11.4(2,8) 14.1 14.1 14.1 14.1 14.1 14.2(2,9) 17.1 17.1 17.1 17.1 17.1(2,10) 20.5 20.5 20.5 20.5 20.5(3,4) 20.1 17.5 17.7 17.5 17.7(3,5) 14.7 13.0 13.0 13.0 13.0(3,6) 12.9 12.0 12.2 12.0 12.2 11.9 12.0(3,7) 13.8 13.6 13.6 13.6 13.6 13.6(3,8) 16.1 16.1 16.1 16.1 16.1 15.9(3,9) 19.0 19.0 19.0 19.0 19.0 19.0 19.1(4,5) 22.8 20.8 20.8 20.8 20.8(4,6) 18.2 16.8 17.0 16.8 17.0(4,7) 17.0 16.3 16.3 16.3 16.3 16.3 16.5(4,8) 18.4 18.3 18.3 18.3 18.3

Table 3.2: Comparison of the eigenfrequencies computed with the harmonic finiteelement model, the quarter three-dimensional finite element model andthe full three-dimensional finite element model with the experimentaleigenfrequencies.

3.3.5 Non-linear behaviour of the silos

In this section, it is investigated if geometrical non-linear behaviour should betaken into account during the ovalling of the silos. Based on a movie of the ovallingof the silos taken during the storm in October 2002, the amplitude of the ovallingoscillations is estimated to be maximally of an order of 0.1 m. Therefore, theresults of a static linear computation under a pressure load which yields a maximaldisplacement of 0.1 m, is compared to the results of a non-linear computation withthe same loading.

In order to check the parameters and element types employed during thisgeometrical non-linear computation, first the well-known deformation of acantilever beam of length L under a tip momentM (Cook et al., 2002) is computed.

100 SHELL STRUCTURES

The analytical solution for the vertical tip displacement uy is:

uy =EI

M

(1− cos

ML

EI

)(3.117)

If the tip momentM = πEIL , the deformed shape of the beam is half a circle (figure

3.16) and the horizontal and vertical tip displacement are respectively:

ux = −L uy =2L

π(3.118)

The cantilever beam is modelled using 1 element along the width and 15 shellelements along the length. Two different shell elements (Ansys, 2005b) were usedto compute the deformations.

The 8-node quadrilateral SHELL93 and the 4-node quadrilateral SHELL181element are based on the Mindlin-Reissner plate theory, which includes transverseshear deformations. An updated Lagrangian formulation which uses logarithmicstrains and Cauchy true stresses, is adopted. The formulation is suited to accountfor large strains. In the case of the SHELL181 element, the follower load stiffnessmatrix is included in the tangent stiffness matrix, while it is not included for theSHELL93 element.

The geometric stiffness matrix Kσ was not included in the non-linear computations.The tolerance for the convergence check is set to εr = 0.5% for forces and momentsand to εu = 5% for the displacements.

Figure 3.16 shows the deformed shape. Table 3.3 lists the horizontal and verticaltip displacement, number of load increments and equilibrium iterations as afunction of the element type. The displacements obtained with SHELL181 arecloser to the analytical solution, but a few more equilibrium iterations wereexecuted. For the geometrical non-linear computation of the deformation of thesilo, the SHELL181 element will be used.

Figure 3.16: Deformed shape of a cantilever beam of length L under a tip momentM = πEI/L.


ux uy increments iterationsSHELL93 −0.9968L 0.6412L 10 78SHELL181 −1.0L 0.6378L 10 85

Table 3.3: Horizontal and vertical tip displacement, number of load incrementsand equilibrium iterations as a function of the element type.

The comparison between a linear and a non-linear computation focusses on ovallingoscillations in eigenmode (1,4). Therefore, a pressure distribution is imposed whichvaries as p sin(4θ) around the circumference (figure 3.17). Within each element onlythe linear part of the variation is taken into account. The silos are simplified to acylindrical shell with a diameter of 5.5 m, a height of 25 m and a constant thicknessof 7 mm. The cylinder is made of aluminium. The magnitude p of the loading isequal to 1004.8 N/m2 which yields in the case of a static linear computation amaximal radial displacement of 0.1 m.

Figure 3.17: Imposed pressure distribution around the circumference of the silo.

The finite element model exploits the symmetry of the geometry and the loading toreduce the model to one eighth of the cylinder in the circumferential direction andhalf a cylinder along the height. Using the antisymmetry of the loading as well,the model could be reduced to one sixteenth of the cylinder in the circumferentialdirection for the linear computation. However for the geometrical non-linearcomputation, antisymmetric boundary conditions should not be applied as largestructural deformations out of the plane of antisymmetry might occur and disturbthe validity of the symmetry of the geometry. At the bottom of the cylinder asupport is supposed which constrains around the circumference the radial andcircumferential displacements and the rotations around the z-axis. Figure 3.18ashows the finite element mesh and the boundary conditions of the model. 8 shellelements are used along an arc with an angle π/4 in the circumferential directionand 10 elements are used along the vertical direction.

102 SHELL STRUCTURES

(a) (b)

Figure 3.18: (a) Finite element mesh and boundary conditions and (b)deformations magnified by factor of 3.

The geometric stiffness matrix Kσ was not included in the non-linear computations.The loading was applied in 12 increments and totally 24 equilibrium iterations wereperformed.

Figure 3.18b shows the deformations of the cylinder magnified by factor of 3. Inthe left symmetry plane the structure displaces outward. The curvature increasesand the structure stiffens. The maximal radial displacement decreases from 0.1 to0.093 m by including the non-linear behaviour. In the right symmetry plane thestructure displaces inward. The curvature decreases and the structure softens. Themagnitude of the radial displacement increases from 0.1 to 0.125 m by includingthe non-linear behaviour. The influence of geometrical non-linear behaviour is notvery large, nor negligible. The geometrical non-linear behaviour will be includedin further computations.

CONCLUSION 103

3.4 Conclusion

This chapter treats the computation of the linear and the geometrical non-linearresponse of shell structures. First, the virtual work formulation which leads tothe finite element method is briefly reviewed. The conjugate stresses and strainsused for the computation of geometrical non-linear structures are described. TheNewton-Raphson technique solves the non-linear system of equations. In order tocompute the transient geometrical non-linear response of shell structures directtime integration is applied.

As an example, the cylindrical shell structure of the silos in the port of Antwerphas been studied. The eigenmodes and eigenfrequencies of a single silo havebeen first computed with a harmonic finite element model in order to enable thedesign of a proper experimental set-up. Measurements of the radial accelerationunder ambient wind loading have been carried out at 10 points on the silo.Modal parameters (eigenfrequencies, mode shapes and modal damping ratios) wereextracted from the output-only data using the stochastic subspace identificationtechnique. The eigenmode with the lowest eigenfrequency at 3.94 Hz (m = 1, n = 3or 4) has the largest contribution to the measured response. All modes have a lowdamping ratio around 1%. A three-dimensional finite element model of the silousing shell elements enables to model correctly the connections between the siloand the supporting structure. The eigenmodes with three and four waves alongthe circumference (n = 4) and half a wave along the height (m = 1) are bothsituated at 3.93 Hz. Finally, the influence of geometrical non-linear behaviouron deformations of the silos of order of magnitude of 0.1 m is not negligible:the magnitude of the displacements increases where the curvature decreases anddecreases where the curvature increases. The geometrical non-linear behaviourwill be included in further computations.

Chapter 4

Computation of turbulent windflows

4.1 Introduction

In this chapter, some aspects of the computation of turbulent wind flows aroundcivil engineering structures which are relevant to the work in this thesis, will betreated. First current state-of-the-art methods for the simulation of turbulentflows are reviewed as a function of their performance and their computationalrequirements.

In order to evaluate the feasibility of numerical flow simulations and to clarify theinfluence of turbulence models, wall treatment, mesh refinement and unsteadiness,first an example for which the solution is known is computed. The flow arounda single cylinder is an ideal test case because it combines several types offlow structures as separation, recirculation, stagnation and streamline curvature.Moreover it is a simplification of the wind flow around a single silo. The results arecompared with the pressure coefficients from Eurocode 1 (BIN, 1995), experimentaldata (Zdravkovich, 1997) and the few numerical results reported in the literature.

In the port of Antwerp the group of 8 by 5 silos (figure 1.3) has very small gapsof 30 cm between two neighbouring silos. As this configuration is expected tohave a large influence on the wind pressure distribution around the silos andsecondarily on the occurrence of ovalling oscillations, a numerical study of thewind flow should provide a more realistic estimation of the pressure coefficientsand forces. The influence of surrounding structures on the flow pattern arounda structure is called interference effect. The flow around the group of silos is

105

106 COMPUTATION OF TURBULENT WIND FLOWS

simplified to the two-dimensional flow around a group of 8 by 5 cylinders. Beforethe extensive computation for the group of 8 by 5 cylinders, the less comprehensivecase of a group of 2 by 2 cylinders is modelled.

4.2 Computational approaches to turbulent flows

The simulation of turbulent flows (Versteeg and Malalasekera, 1995; Wilcox, 1998;Pope, 2000) is still a challenging task. Turbulent flows are characterized by avelocity field that varies randomly both in space and time. They are inherentlythree-dimensional and time-dependent. A large range of time and length scales arepresent in a turbulent flow. The incompressible Navier-Stokes equations (2.97)-(2.99) are applicable for isothermal turbulent flows of Newtonian fluids and arerecalled here in an Eulerian description:

∇ · v = 0 (4.1)

∂v

∂t+∇ · (v ⊗ v) +

1

ρ∇p = ν∇2v + b (4.2)

The challenge is to accurately compute the relevant statistics of such flows.

Turbulent kinetic energy is produced at the largest scales of motion, which are aslarge as a characteristic dimension of the flow. These largest turbulent motions(eddies) are directly affected by the geometry and the boundary conditions andtherefore anisotropic. As direct effects of viscosity are very small for the largeeddies, they are unstable and break up transferring their energy to successivelysmaller eddies. This energy cascade continues up to scales that are small enoughto dissipate the energy by viscous forces. These smallest eddies are statisticallyisotropic. The rate at which energy is transferred from larger to smaller scales isdetermined at the largest scales and nearly equal to the dissipation rate ε. Thesmallest length, velocity and time scales, which are called the Kolmogorov scales,are determined by the kinematic viscosity ν and the dissipation rate ε. The ratio

of the smallest to the largest length scales is equal to Re−34 and for the time scales

to Re−12 (Pope, 2000). The turbulence energy is predominantly contained in the

larger anisotropic eddies.

4.2.1 Direct numerical simulation

In order to numerically simulate turbulent flows, a number of approaches exists.Conceptually, direct numerical simulation (DNS) is the simplest approach. TheNavier-Stokes equations (4.1)-(4.2) are solved, resolving all length and time scales.For a three-dimensional time-dependent computation, the number of grid points

COMPUTATIONAL APPROACHES TO TURBULENT FLOWS 107

has generally to be proportional to Re94 and the number of time steps proportional

to Re34 . As the computational cost increases drastically with the Reynolds number,

DNS is limited to flows of low and moderate Reynolds numbers. Therefore,DNS is mainly a research tool to perform numerical experiments that enables theimprovement of turbulence models and increases insight into turbulence physics(Moin and Mahesh, 1998).

4.2.2 Reynolds averaged Navier-Stokes simulation

In Reynolds averaged Navier-Stokes (RANS) simulation the velocity v isdecomposed in the mean velocity v and the fluctuating velocity v′:

v = v + v′ (4.3)

Analogously, the pressure p is decomposed the mean pressure p and the fluctuatingpressure p′:

p = p+ p′ (4.4)

For statistically stationary flows the mean can be thought of as the timeaverage over a long period, while for statistically non-stationary flows the meancorresponds to an ensemble average: the unsteady Reynolds averaged Navier-Stokes (URANS) simulation produces the average over a large number of identicalexperiments. Taking the mean of the Navier-Stokes equations (4.1)-(4.2) andsubstituting (4.3) in the convective term, the Reynolds equations for the meanvelocity are obtained:

∇ · v = 0 (4.5)

∂v

∂t+∇ · (v ⊗ v) +∇ · v′ ⊗ v′ +

1

ρ∇p = ν∇2v + b (4.6)

Because of the non-linear convective term in the Navier-Stokes equations, thedivergence of the velocity covariances v′ ⊗ v′ appears in the momentum equations.The velocity covariances are called the Reynolds stresses 1, although they stemfrom the convective term. These four equations contain, in addition to the threecomponents of the mean velocity and the mean pressure, the six Reynolds stresses.Without additional equations for these Reynolds stresses, this set of equationscannot be solved. The transport equations for the Reynolds stresses can be derived,but a lot of additional unknowns are present in those equations. This is theclosure problem: the number of equations never balances the number of unknowns.Therefore, some of the unknowns have to be modelled. In turbulent viscosity

1Although the apparent stress is equal to −ρv′ ⊗ v′, v

′ ⊗ v′ are conventionally called the

Reynolds stresses.


models the Reynolds stresses in equation (4.6) are determined by a turbulencemodel. In Reynolds stress models (RSM) the unknown terms in the transportequations for the Reynolds stresses are modelled. Using RANS all the turbulentfluctuations are modelled and consequently all the turbulent kinetic energy as well.

A huge number of turbulence models exist. In these models many constants appear,which are calibrated by means of experimental results for simple shear flows.

As all turbulent wind flows are bounded by walls, the grid requirements near wallsin RANS are generally very important. The properties of a straight fully developedboundary layer flow, as obtained from experiments and DNS, are very illustrative.The total shear stress is the sum of the Reynolds and the viscous stresses. At thewall the Reynolds shear stress is zero and the wall shear stress τw is entirely due tothe viscous contribution. In the region close to the wall the viscous contributiondominates, while further away from the wall the Reynolds stress is dominant. Inthe former region, a viscous velocity scale and length scale are defined from thewall shear stress τw and the viscosity ν: the friction velocity vτ =

√τw/ρ and the

viscous length scale δv = ν√ρ/τw = ν/vτ .

Dividing the distance from the wall y by the viscous length scale δv yields adimensionless expression for the distance y+ = y/δv = vτy/ν. This dimensionlessdistance y+ can be considered as a local Reynolds number, which gives the relativeimportance of viscous and turbulent processes. Analogously, a dimensionlessvelocity v+ = v/vτ is obtained.

Close to the wall, an inner layer exists where the dimensionless mean velocityprofile v+ only depends on the dimensionless wall distance y+. The viscouscontribution to the shear stress decreases from 100 % at the wall to 50% at y+ ≈ 12and less than 10% at y+ = 50 (Pope, 2000). In the viscous sublayer y+ < 5, whereviscosity effects dominate, v+ equals y+. In the outer layer y+ > 50, the effectsof the viscosity on the mean flow are negligible. At high Reynolds numbers, alayer exists where the mean velocity v+ is not influenced by the viscosity and onlydepends on the dimensionless wall distance y+:

v+ =1

κln y+ +B =

1

κlnvτy

ν+B (4.7)

This is the logarithmic law of the wall (Pope, 2000) for smooth surfaces. The vonKarman constant κ is equal to 0.41 and B is equal to 5.2. The region of validity ofthe logarithmic law can be extended from y+ > 50 to y+ > 30. In the logarithmiclaw region, production and dissipation of turbulence are almost in equilibrium.

COMPUTATIONAL APPROACHES TO TURBULENT FLOWS 109

For rough surfaces the logarithmic law can be modified in order to account for theeffects of sand-grain roughness ks:

v+ =1

κln

vτy

ν(1 + 0.3ks)+B (4.8)

This roughness modification is inconsistent with the requirements for anatmospheric boundary layer flow and might compromise the accuracy of thesesimulations (Blocken et al., 2007).

For the computations there are two possibilities. In a low Reynolds formulation,the RANS equations can be solved up to the wall. Ideally, five nodes are located inthe viscous sublayer (y+ ≈ 1 for the node next to the wall). The turbulence modelsfor the Reynolds stress have to be modified in order to be valid in the region whereviscosity has direct effects. In a high Reynolds formulation, the nodes next to thewall are placed in the logarithmic law region. So-called wall functions apply thelogarithmic law as a boundary condition on the nodes next to the wall. The near-wall mesh is much coarser in comparison with the low Reynolds formulation. Thesewall functions are not suitable if the logarithmic layer does not exist, for instancefor separated or impinging flows, or in case of curved streamlines. In these cases,there is no equilibrium between production and dissipation of turbulence and thesolution might be dependent on the location of the node next to the wall.

4.2.3 Large eddy simulation

In large eddy simulation (LES) only the larger scale turbulent motions, whichdepend on the flow geometry and are anisotropic, are directly computed. Theinfluence of the smaller scale motions, which are isotropic and universal, can betaken into account by a simple turbulence model. Therefore, a velocity field v

where smaller scale motions are filtered out, is defined:

v(x) =

∫

Ω

G(r,x)v(x − r)dr (4.9)

where G(r,x) is a filter with∫G(r,x)dr = 1 and the integration is performed

over the entire fluid domain. The velocity field v is decomposed into the filteredcomponent v and a residual component v′. If the specified filter width is equalto the size of the smallest energy containing eddies, about 80 % of the turbulentkinetic energy is resolved. The number of grid points has to be just large enoughto compute accurately this filtered velocity field. The computational cost is muchsmaller than for a DNS. Filtering the Navier-Stokes equations (4.1)-(4.2) with a


spatially uniform filter G(r) yields:

∇ · v = 0 (4.10)

∂v

∂t+∇ ·

(˜v⊗ v + ˜v⊗ v′ + v′ ⊗ v + v′ ⊗ v′

)+

1

ρ∇p = ν∇2v + b (4.11)

To close the set of equations, a residual stress tensor τR is defined as the difference

between the term in brackets, which is equal to v⊗ v, and the product of thefiltered velocities v⊗ v. The momentum equation becomes:

∂v

∂t+∇ · (v ⊗ v) +∇ · τR +

1

ρ∇p = ν∇2v + b (4.12)

This residual stress tensor τR has to be modelled taking into account the filterwidth and filter type. The filtered Navier-Stokes equations (4.10)-(4.12) are solvedfor the filtered velocity v. The filter characteristics only enter indirectly in theseequations through the model for the residual stress tensor.

LES is especially suited for flows with large scale unsteadiness, like flows over bluffbodies with unsteady separation and vortex shedding. Unfortunately, close to asolid wall, all eddies, including the energy containing eddies, are very small andtheir size is dependent on the Reynolds number. As these small energy containingeddies should be resolved, very fine grids and small time steps are required nearthe wall. To reduce the computational cost two options exist. As in the wallfunctions approach for RANS, appropriate boundary conditions can be applied atthe first grid point away from the wall. This option is not satisfactory if there isno equilibrium between production and dissipation, as in the case of separatingflows. The second possibility is to solve RANS boundary layer equations on anembedded grid, which produce boundary conditions for the LES computation.

Due to the poor performance of RANS for massively separated flows and thehigh computational cost of wall-resolved LES, hybrid RANS/LES models arebeing developed, which are a generalization of the second approach for LES nearwalls. These models switch to RANS or LES behaviour depending on the gridresolution. In the RANS regions the turbulence model controls the solution, whilein the LES region the larger eddies are resolved. In very large eddy simulations(VLES) (Speziale, 1998), the filter width (or grid spacing) is compared with aturbulent length scale. If the filter width is larger than the size of the smallestenergy containing eddies, the computation switches to RANS behaviour, whileLES behaviour is activated when the filter width is smaller. In detached eddysimulations (DES) (Spalart et al., 1997; Travin et al., 2000), the ratio of thewall distance and the grid spacing determines if the RANS or LES mode is used.The model operates in RANS mode near the wall if the grid spacing normal orparallel to the wall is large compared with the boundary-layer thickness. Thedetached eddies further from the wall are simulated in LES mode. The design of

RANS TURBULENCE MODELS 111

the grid controls the behaviour of these hybrid models. The computational costsare considerably lower than for a wall-resolved LES.

The application of hybrid RANS/LES methods is very promising as they are able tocapture the large scale turbulent structures in the wake of bluff bodies at a moreaffordable computational cost. However, for the use in coupled fluid-structureproblems, this computational cost today is still very high and in this thesis onlyRANS models will be used. The next sections treats those models in more detail.

4.3 RANS turbulence models

4.3.1 Linear eddy viscosity models

The Reynolds stress tensor v′ ⊗ v′ can be split in its isotropic and deviatoric part:

v′ ⊗ v′ =1

3v′ · v′ I +

(v′ ⊗ v′ − 1

3v′ · v′ I

)(4.13)

The turbulent kinetic energy k is defined as half the trace of the Reynolds stresstensor v′ ⊗ v′:

k =1

2v′ · v′ (4.14)

The isotropic part of the Reynolds stress tensor is equal to 2/3kI. The turbulentviscosity hypothesis by Boussinesq (Pope, 2000) supposes that the deviatoric partof the Reynolds stress tensor is proportional to the mean strain rate ε = 1/2(∇v+(∇v)T):

v′ ⊗ v′ =2

3kI− 2νtε =

2

3kI− νt(∇v + (∇v)T) (4.15)

Equation (4.15) is analogous to the expression for the stress in a Newtonian fluid(2.94). The main assumption is that the mean velocity gradients determine thedeviatoric part of the Reynolds stress tensor. As the molecular motion, responsiblefor the viscous stresses, rapidly adjusts to the mean straining, equation (2.94)is appropriate and the viscous stresses are mainly isotropic. Turbulent motionoften does not rapidly adapt to the mean straining. The history of strainingis important and the Reynolds stresses are largely anisotropic. The turbulentviscosity hypothesis is not generally applicable. As for simple turbulent shearflows the mean velocity gradients change slowly and are a good measure for the


straining history, the hypothesis is in these cases reasonable. The production anddissipation of turbulence are approximately equal.

More specifically, in equation (4.15) a linear relation is assumed between thedeviatoric part and the mean strain rate. The scalar νt is the turbulent viscosity.In reality the principal axes of the deviatoric part of the Reynolds stress tensorand the mean strain rate tensor are not aligned. For simple shear flows, onlyone component of the Reynolds stress tensor is of interest and the hypothesis israther a definition of the turbulent viscosity νt. For swirling flows or flows withsubstantial streamline curvature, the linear relation is not valid.

Substituting the turbulent viscosity hypothesis (4.15) in the momentum equation(4.6) yields:

∂v

∂t+∇ · (v ⊗ v) +

1

ρ∇(p+

2

3ρk) = (ν + νt)∇2v + b (4.16)

This equation is similar to the Navier-Stokes momentum equation (2.99), but thevelocity, viscosity and pressure are replaced by the mean velocity, the effectiveviscosity and the modified mean pressure, respectively.

The turbulent viscosity νt still has to be determined. At high Reynolds numbersand far away from the wall, it scales with a length scale and a velocity scale of theflow. For algebraic models as the mixing-length model, no additional transportequations need to be solved, but they are incomplete as an appropriate mixinglength should be specified which depends on the flow geometry. One- and two-equation models require the solution of respectively one or two additional transportequations to determine the turbulent viscosity νt. The Spalart-Allmaras model(Spalart and Allmaras, 1994) is a complete one-equation model which performswell for aerodynamic flows with separation.

The standard k-ε model

The standard k-ε model (Jones and Launder, 1972) is a two-equation model. Theturbulent viscosity νt is expressed as a function of the turbulent kinetic energy kand the turbulent energy dissipation rate ε:

νt = Cµk2

ε(4.17)


Two additional transport equations are solved for the turbulent kinetic energy kand the turbulent energy dissipation rate ε:

∂k

∂t+∇ · (kv) = ∇ ·

((ν +νtσk

)∇k)

+ P − ε (4.18)

∂ε

∂t+∇ · (εv) = ∇ ·

((ν +νtσε

)∇ε)

+ Cε1ε

kP − Cε2

ε2

k(4.19)

where the turbulence production P is, using the turbulent viscosity hypothesis(4.15), equal to:

P = νt∇v : (∇v + (∇v)T) (4.20)

The time derivative, the convective term and the production term of the transportequation (4.18) stem from the exact equation for the turbulent kinetic energy k,while the first term on the right hand side models the diffusion terms with thegradient-diffusion hypothesis. In the transport equation (4.19) for the turbulentenergy dissipation rate ε only the left hand side corresponds to the exact equation.The right hand side is entirely modelled and mainly follows from dimensionalanalysis. The standard k-ε model includes five constants Cµ, σk, σε, Cε1 and Cε2which are calibrated from a number of turbulent flows e.g. decaying homogeneousturbulence, a homogeneous shear flow and a turbulent boundary layer. For aparticular flow the accuracy of the model can be improved by tuning the constants.

The standard k-ε model yields acceptable results for simple turbulent shearflows, but has considerable shortcomings for more complex flows with streamlinecurvature and boundary layer separation under adverse pressure gradients.

The k-ω model

In many two-equation models, the turbulent kinetic energy k is taken as one ofthe variables. In the k-ω model (Wilcox, 1988) the turbulent frequency ω is thesecond variable. The turbulent viscosity νt is expressed as:

νt =k

ω(4.21)

Comparison of equations (4.17) and (4.21) yields the relation between the turbulentfrequency ω and the turbulent energy dissipation rate ε:

ω =1

Cµ

ε

k(4.22)


The transport equations for the turbulent kinetic energy k and the turbulentfrequency ω are:

∂k

∂t+∇ · (kv) = ∇ ·

((ν +νtσk1

)∇k)

+ P − β∗kω (4.23)

∂ω

∂t+∇ · (ωv) = ∇ ·

((ν +νtσω1

)∇ω)

+γ1νtP − β1ω

2 (4.24)

The transport equation for the turbulent kinetic energy k is, except for the values ofthe constants, identical to one of the standard k-ε model. The transport equation(4.19) for the turbulent energy dissipation rate ε can be transformed in a transportequation for the turbulent frequency ω:

∂ω

∂t+∇·(ωv) = ∇·

((ν +νtσω2

)∇ω)

+γ2νtP−β2ω

2+2

k

(ν +νtσω2

)∇k·∇ω (4.25)

Except for values of the constants, the only difference between equations (4.24) and(4.25) is the appearance of the cross diffusion term on the right hand side (Wilcox,1998). This cross diffusion term deteriorates the quality of the computationof turbulent boundary layers under adverse pressure gradients. The k-ω modeltherefore predicts the onset and the amount of separation under adverse pressuregradients much more accurate than the standard k-ε model. In contrast to thestandard k-ε model, it does not need any damping function to be used withinthe viscous sublayer. However, in free shear flows the cross diffusion increasesthe production of ω, which results in an increased dissipation of kinetic energy k.The absence of this term in the k-ω model makes the model very sensitive to freestream turbulence levels, while the standard k-ε model is independent from thefree stream levels.

The shear stress transport model

The shear stress transport (SST) model (Menter, 1994) combines the independenceof the free stream turbulence levels of the k-ε model with the robust and accuratecomputation of boundary layers under adverse pressure gradients of the k-ω model.Therefore, the standard k-ε model is transformed into a k-ω formulation. Theequation for the turbulent kinetic energy k is, except for the values of the constants,identical to equation (4.23). The transformation of the transport equation for theturbulent energy dissipation rate ε is given in equation (4.25).

A blending function F1 is defined which is equal to one in the near-wall regionand zero far away from the wall. The transport equations of the k-ω model aremultiplied with F1 while the transport equations of the standard k-ε model aremultiplied with 1 − F1. The corresponding transport equations are added. The


resulting model uses the k-ω model in the boundary layer and the standard k-εmodel far away from the walls.

In turbulent boundary layers under adverse pressure gradients the production canbe much larger than the dissipation. In this case the turbulent viscosity νt isoverestimated by equation (4.21). This definition is slightly modified in order toaccount for the transport of turbulent shear stresses:

νt =a1k

max(a1ω, F2Ω)(4.26)

where Ω is the absolute value of the mean vorticity. F2 is, similar to F1, a blendingfunction which is equal to one in the near-wall region and zero far away fromthe wall. Far away from the walls, the original definition (4.21) of the turbulentviscosity νt is recovered. Close to the walls, the original definition is as wellrecovered if a1ω ≥ F2Ω. For a1ω < F2Ω the deviatoric part of the Reynoldsstresses becomes:

v′ ⊗ v′ − 2

3kI = − a1k

F2Ω(∇v + (∇v)T) (4.27)

In boundary layers, where mainly one velocity gradient is of importance, thiscorresponds to the assumption of Bradshaw (Menter, 1994) that in a turbulentboundary layer the Reynolds shear stress is proportional to the turbulent kineticenergy. This modification makes the SST model very appropriate for the predictionof the onset and the amount of separation.

Finally, in order to avoid excessive build-up of turbulence in stagnation regions, alimiter is applied to the productions terms:

P = min(P , climε) (4.28)

where the limiter clim defaults to 10.

The realizable k-ε model

In the Reynolds stress tensor v′ ⊗ v′ the normal Reynolds stresses are alwayspositive:

v′iv′

i ≥ 0 (4.29)

and the correlations between fluctuating velocities fulfill the Cauchy-Schwartzinequality:

v′iv′

j

2

v′2i v′2j

≤ 1 (4.30)


By the turbulent viscosity hypothesis (4.15) these properties of the Reynolds stresstensor are not automatically guarantied and for instance violated in the case oflarge mean strains ε. Equations (4.29) and (4.30) are called the realizabilityconditions. In order to fulfill these conditions, Cµ (equation (4.17)) cannot bea constant and should be related to the mean strain rate (Shih et al., 1995a):

Cµ =1

A0 +Askε

√ε : ε + ω : ω

(4.31)

where ω is the mean rotation rate:

ω =1

2(∇v − (∇v)T)− CrIΩ (4.32)

The vector Ω defines the angular velocity of the frame of reference. The parametersCr and A0 are respectively equal to 3 and 4 and As is given by:

As =√

6 cosφ, φ =1

3arccos(

√6W ), W =

(ε ε) : εT

(√ε : ε

)3 (4.33)

In the realizable k-ε model also a new transport equation for the turbulent energydissipation rate ε is obtained from the dynamic equation for the mean-square ofthe vorticity fluctuation:

∂ε

∂t+∇ · (εv) = ∇ ·

((ν +νtσε

)∇ε)

+ Cε1εε− Cε2ε2

k +√νε

(4.34)

where ε is:

ε =√

2ε : ε (4.35)

If in the last term on the right hand side of equation (4.34) k >√νε, as is the case

at high Reynolds numbers, the production term is the only difference with theequation (4.19) of the standard k-ε model. The values of the constants σε and Cε2in equation (4.34) differ from those of the standard k-ε model. Cε1 is determinedby:

Cε1 = max

0.43,

η

η + 5

(4.36)

where η is the ratio of a time scale of the turbulence and the mean strain:

η =k

εε (4.37)


4.3.2 Non-linear eddy viscosity models

The turbulent viscosity hypothesis (4.15) supposes that a linear relation existsbetween the deviatoric part of the Reynolds stress tensor and the mean strain ratetensor. The deviatoric part of the Reynolds stress tensor could more generally berelated to the mean velocity gradient, which consists of the mean strain rate tensorand the mean rotation rate tensor (4.32). After multiplication with a turbulencetime scale k/ε the normalized mean strain rate ε and the normalized mean rotationrate ω are obtained:

ε =k

εε (4.38)

ω =k

εω (4.39)

The Reynolds stress tensor is now modelled as:

v′ ⊗ v′ =2

3kI + 2kB(ε, ω) (4.40)

where B is a general function of the normalized mean rate of strain ε andnormalized the mean rate of rotation ω. An integrity basis for all symmetric anddeviatoric tensors formed from the normalized mean strain rate ε and rotationrate ω consists of ten combinations. Craft et al. (1996) introduced a model whereall combinations of the strain rate and the rotation rate tensor up to the thirdorder are included, which corresponds to six combinations:

v′ ⊗ v′ =2

3kI + 2kc1ε + 2kc2(εω − ωε) + 2kc3(ε2 − 1

3tr(ε2)I) (4.41)

+ 2kc4(ω2 − 1

3tr(ω2)I) + 2kc5(ωε2 − ε2ω) (4.42)

+ 2kc6(ω2ε + εω2 − 2

3tr(εω2)I) (4.43)

Generally the coefficients c1 till c6 might depend on the five independent tensorinvariants composed of the normalized mean strain rate ε and rotation rate ω.The cubic terms enable to capture the streamline curvature and swirl effects.

Shih et al. (1993, 1995b) only retained combinations up to the second order (c5 =c6 = 0). The realizability conditions (4.29) and (4.30) were used to determine thecoefficients, which finally only depend on two tensor invariants:

η =√

2ε : ε (4.44)

ζ =√

2ω : ω (4.45)


where η was already defined in equation (4.37). In the earliest version (Shih et al.,1993) the coefficient c1 for the linear term is given by:

c1 =As1

As2 + η +As3ζ(4.46)

where As1 = 2/3, As2 = 1.25 and As3 = 0.9.

The Shih-Zhu-Lumley model (SZL) in Flotran uses this coefficient, but drops allhigher order terms.

4.3.3 Reynolds stress models

In the previous sections the Reynolds stresses v′ ⊗ v′ are modelled as a function ofthe mean velocity gradient. As to avoid this assumption, Reynolds stress models(RSM) start from the exact transport equations for the Reynolds stresses v′iv

′

j :

∂v′iv′

j

∂t+ vk∂v′iv

′

j

∂xk= Rij +

∂

∂xk

(ν∂v′iv

′

j

∂xk+ Tkij

)+ Pij − εij (4.47)

The production tensor Pij is in closed form:

Pij = −v′iv′k(∂vj∂xk

)− v′jv′k

(∂vi∂xk

)(4.48)

The pressure-rate-of-strain tensor Rij is equal to:

Rij =p′

ρ

(∂v′i∂xj

+∂v′j∂xi

)(4.49)

This expression introduces new unknowns and is the most important term to bemodelled. The part Tkij of the turbulent transport is given by:

Tkij = −v′iv′jv′k −1

ρv′ip′δjk −

1

ρv′jp′δik (4.50)

This is usually modelled using a gradient-diffusion model. The dissipation tensorεij is equal to:

εij = 2ν∂v′i∂xk

∂v′j∂xk

(4.51)

Because dissipation occurs at the smallest scales, often isotropy is assumed for thedissipation tensor:

εij =2

3εδij (4.52)

TURBULENT AIR FLOW AROUND A SINGLE CYLINDER 119

where the turbulent energy dissipation rate ε = 1/2εii is the same as appearsin the standard k-ε models. An additional transport equation is solved for theturbulent energy dissipation rate ε:

∂ε

∂t+ vi∂ε

∂xi=∂

∂xi

(Cεk

εv′iv′

j

∂ε

∂xj

)+ Cε1

ε

kP − Cε2

ε2

k(4.53)

Globally seven transport equations (for the six Reynolds stresses v′iv′

j and theturbulent energy dissipation rate ε) have to be solved instead of two equations forthe two-equation eddy viscosity models. The hypothesis that the Reynolds stresstensor is determined by the velocity gradients is removed which enables the modelsto capture the streamline curvature and swirl effects. The transport equations forthe Reynolds stresses make it possible to take the effects of the flow history intoaccount.

4.4 Turbulent air flow around a single cylinder

Circular cylinders are widely used in civil engineering constructions as silos,chimneys, water towers, power transmission lines, offshore structures andsuspension bridge cables.

The turbulent air flow around a single cylinder is computed in order to clarifythe influence of turbulence models, near-wall mesh refinement and unsteadiness(Dooms et al., 2006a).

The cylinder represents a silo in the port of Antwerp with a diameter D = 5.5 m.For the air, a density ρ = 1.25 kg/m3 and a dynamic viscosity µ = 1.76×10−5 Pa sare used. The bottom of the silos is located at 16.66 m above ground level. Due tothe proximity of the river Scheldt, the surroundings are quite flat and vast, which isreflected in terrain category II (BIN, 1995) with an aerodynamic roughness lengthof 0.05 m. The mean wind velocity at half the height of the silo or z = 30 m aboveground level is equal to v = 31.8 m/s. The Reynolds number Re = Dv/ν is equalto 12.4× 106.

The flow around a finite circular cylinder of height-to-diameter ratio 4.55 (as thesilos in the port of Antwerp) is strongly three-dimensional. The critical ratio belowwhich the free end effect extends to the base of the cylinder and Karman vortexshedding is suppressed, varies between the different studies in the literature from 2to 6 (Pattenden et al., 2005). The three-dimensional flow around one finite cylinderof small aspect ratio embedded in an atmospheric boundary layer has been oftenstudied experimentally. However, the numerical simulation of this flow (usingLES) is still challenging from a computational point of view at moderate Reynoldsnumbers (Re = 20000 − 200000) (Frohlich and Rodi, 2004; Afgan et al., 2007;Pattenden et al., 2007). Computations of flows around more complex geometries


in atmospheric boundary layers at high Reynolds number have been performed,but with quite coarse near-wall meshes (Nozu et al., 2008).

Using a single processor, the numerical simulation of the three-dimensional flowaround forty closely-spaced finite circular cylinders at Re = 12.4 × 106 is out ofreach. Therefore, all flow computations around cylinders will be limited to twodimensions, corresponding to an infinite cylinder.

For the two-dimensional flow around a cylinder, the fluid particles reach thecylinder first at the stagnation point, where the velocity is equal to zero andthe streamline is perpendicular to the wall. The angle θ = 0 coincides withthe upstream stagnation point. The pressure coefficient Cp = 2(p − pf)/(ρv2f )(figure 4.1) is a dimensionless expression for the pressure at the cylinder’s surface,where pf and vf are the free stream pressure and velocity, respectively. It ismaximal and equal to 1 at the stagnation point and gradually decreases aroundthe circumference. The boundary layers stay attached in this favourable pressureregion. The pressure coefficient attains a minimum Cmin

p and starts to increase.The boundary layers now experience an adverse pressure gradient and separatesfrom the wall at θs. A recirculation zone is formed behind the cylinder. Thepressure is approximately constant in the recirculation zone. The base pressurecoefficient Cb

p is the pressure coefficient at θ = 180. The drag coefficient

Cd =∫ 2π

0 Cp cos θdθ is a dimensionless expression for the force experienced bythe cylinder in the flow direction. The flow around a cylinder is a an ideal testcase for the turbulence models because it combines several types of flow structuresas separation, recirculation, stagnation and streamline curvature.

Cminp

Cbp1

θsv

Figure 4.1: Pressure coefficient Cp for a single cylinder.

As the Reynolds number Re is larger than 3.5 − 6 × 106, the regime of the flowaround the cylinder is post-critical (Zdravkovich, 1997): the wake and the shearlayers are fully turbulent and the boundary layers become fully turbulent prior


to separation. In the boundary layers, the transition from laminar to turbulentflow takes place between the stagnation and the separation point. Regular vortexshedding reappears, while it is absent at lower Reynolds numbers.

Eurocode 1 (BIN, 1995) describes the pressure coefficient as a function of thecircumferential angle for a Reynolds number Re = 107. The flow around a circularcylinder has been widely studied, mainly focussing on the regular vortex sheddingin the subcritical regime and the drag crisis in the critical regime. Zdravkovich(1997) gives an overview of experimental data. There is a lack of detailedexperimental data at post-critical Reynolds numbers, while available data showconsiderable scatter, which may be explained by the high sensitivity of the flowto perturbations due to surface roughness and free-stream turbulence. Numericalresults in the present section are always compared with 12 experimentally obtainedpressure coefficients at Reynolds numbers from 0.73 × 107 to 3.65 × 107 (figure4.2). Because of the amount of scatter and the number of experimental curves, itis reasonable to consider numerical results as accurate if they fall within the zonedefined by these curves.

0 45 90 135 180−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

Figure 4.2: Measured pressure coefficients at Reynolds numbers from 0.73 × 107

to 3.65× 107 (Zdravkovich, 1997).

Few computations of the flow in the post-critical regime have been performed.A number of two-dimensional URANS simulations are reported in the literature.Celik and Shaffer (1995) used URANS with an empirically fixed transition pointto compute the flow for Reynolds numbers up to 3.6 × 106. The predictions arestrongly influenced by grid refinement and especially by the grid distribution inthe boundary layer. The best results are obtained with the first grid point locatedin the viscous sublayer. Holloway et al. (2004) applied an URANS techniquecapable of resolving boundary layer transition for Reynolds numbers up to 107. AtReynolds number 107, similar results are obtained for the fully turbulent model andthe transition model. The magnitude of the time averaged drag coefficient doesnot agree with experimental values due to a rather small overestimation in the


separation angle. Saghafian et al. (2003) compared the flow computed with a non-linear eddy viscosity model and with the standard k−ε model. The cubic terms inthe non-linear model account for effects due to streamline curvature. The pressurecoefficient around the circumference at Reynolds number 8.4×106 has a significantdecrease between 165 and 180, which is explained by the assumption that theflow is two-dimensional, causing vortices to roll up close to the cylinder. Younisand Przulj (2006) modified the k− ε model to account for the direct energy inputat the vortex shedding frequency from the mean flow into the random turbulencemotions. This effect is introduced by an additional source term in the dissipationrate equation. The results are compared with the RNG k− ε model. The pressurecoefficient around the circumference at Reynolds number 3.5 × 106 significantlydecreases as well between 150 and 180.

Travin et al. (2000) applied three-dimensional DES for Reynolds numbers up to 3×106. For the turbulent separation cases, the results obtained with two-dimensionalURANS computations are very close to those obtained with three-dimensionalDES. Adding a curvature correction term to the turbulence model improves theestimation of the separation angle, the base pressure and the drag coefficient.

Catalano et al. (2003) used three-dimensional LES with the dynamic Smagorinskymodel to compute the flow for Reynolds numbers up to 2 × 106 and comparedit with three-dimensional RANS and URANS results. The solutions showrelative insensitivity to the Reynolds number and inaccurate predictions at higherReynolds numbers (2 × 106) which are probably due to poor grid resolution.Tamura et al. (1990) computed the flow in two and three dimensions without anyturbulence model, which corresponds to a LES using the discretization schemeas a filter with numerical dissipation. Using this technique, grid refinement inthe circumferential direction strongly influences the results obtained with a two-dimensional computation. At Re = 106, the time averaged drag coefficients Cd ofthe two-dimensional and three-dimensional computations are the same. For super-critical and post-critical flow, the effects of unsteadiness and three-dimensionalvortex structures in the flow are less important and mainly located in the wake.Singh and Mittal (2005) performed a similar analysis without any turbulence modelin two dimensions for Reynolds numbers up to 107. At Reynolds number 106, thepressure coefficient around the circumference has a significant decrease between140 and 180.

Some results of these numerical simulations reported in the literature are listed intable 4.2.


A two-dimensional RANS and URANS simulation will be performed with theFlotran finite element code (Ansys, 2005b) and the CFX finite volume code (Ansys,2005a) using a number of eddy viscosity turbulence models on two different meshes.In Flotran the second order accurate Collocated Galerkin (COLG) approach isused for the spatial discretization. To handle the coupling between the pressureand momentum equations, an enhanced segregated solution algorithm (SIMPLEN)(Wang, 2001) is employed. In CFX the High Resolution spatial discretization isused, which is an automatically determined blend of a first and a second orderaccurate scheme. The coupled algorithm solves the momentum and continuityequation as a single system. These options are used for all computations in thisthesis.

4.4.1 Steady computation: problem domain, near-wallmodelling and eddy viscosity turbulence models

At the inlet a uniform velocity v = 31.8 m/s, a turbulent kinetic energy

k = 0.1521 m2/s2 and a turbulent energy dissipation rate ε =Cµk

3/2

0.99 =0.0053929 m2/s3 are imposed. This corresponds to a turbulence intensity I =√

23k/v = 1.00%. In atmospheric boundary layers, the turbulence intensity is

higher, but this low turbulence intensity enables to compare with the results fromwind tunnel tests. At the lateral boundaries symmetry is imposed. The walls areconsidered smooth and no-slip boundary conditions are applied. In Flotran zeropressure is imposed at the outlet, while in CFX the average of the pressure overthe outlet should be zero. For the steady computations, symmetry permits theflow to be computed on one side of the cylinder, with reflective symmetry imposedalong the centerline.

The boundaries of the fluid domain should be sufficiently far from the region closeto the cylinder where the accuracy of solution is important. Behr et al. (1991,1995) suggest a distance of at least 8D for the inlet and the lateral boundariesand a distance of 22.5D for the outlet, with D the diameter of the cylinder. Forthe present computations, a distance of 9D is adopted for the inlet and the lateralboundaries and 30D for the outlet (figure 4.3). Results on a larger problem domain,where the inlet and the lateral boundaries are located at 12D and the outlet at40D, are comparable.

All simulations are performed on two different meshes that differ only in the near-wall region. In mesh A2, a high Reynolds formulation is used and the nodes nextto the cylinder wall are predominantly placed in the logarithmic law region. Closeto the cylinder wall, the mesh is structured and consists of 138 quadrilateralsaround the circumference and 70 elements in the radial direction. Far away fromthe cylinder wall, an unstructured mesh consisting of triangles is used. The mesh


9D 9D 21D

9D

Figure 4.3: Problem domain and mesh A2 for a single cylinder.

consists of 26765 elements and 22899 nodes. The dimensionless wall distance y+

of the nodes next to the cylinder wall varies from 0 at the stagnation points to 130.The choice of the wall functions (table A.1) has no important influence on thepressure coefficient, both within Flotran (equilibrium, Van Driest and Spaldingwall functions) and within CFX (standard and scalable wall functions).

In mesh B, a low-Reynolds formulation is used and the nodes next to the cylinderwall are located in the viscous sublayer at y+ ≈ 1. Mesh B consists of 54694elements and 51009 nodes. The structured mesh close to the cylinder wall consistsof 409 quadrilaterals around the circumference and 80 elements in the radialdirection. The dimensionless wall distance y+ of the nodes next to the cylinderwall is smaller than 1.6.

0 45 90 135 180−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

(a)0 45 90 135 180

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

(b)

Figure 4.4: Pressure coefficient as a function of the angle θ obtained with Flotran(solid lines) and CFX (dashed lines) turbulence models using variouseddy viscosity turbulence models ( k ε, SST, SZL) for (a)mesh A2 and (b) mesh B. The light grey zone contains all availableexperimental data at Reynolds numbers from 0.73×107 to 3.65×107.

Figure 4.4 shows the pressure coefficient computed for various turbulence models


with a steady simulation. Near-wall modelling strongly influences the pressurecoefficient computed with Flotran, while the results obtained with CFX are quitesimilar for a high- and a low-Reynolds formulation. A light grey zone containingall available experimental data at Reynolds numbers from 0.73× 107 to 3.65× 107

is superimposed on this and the following figures. In Flotran, changing from ahigh- to a low-Reynolds formulation the correspondence between measurementsand results improves for the standard k − ε and the SZL model, but not for theSST model.

Figure 4.4 as well shows that the influence of the different eddy viscosity turbulencemodels (table A.1) on the pressure coefficient is large. The results obtainedwith the SZL turbulence model of Flotran have the best correspondence with theexperimental results. Within CFX, the SST turbulence model produces reasonableresults.

The standard k-ε model overestimates the turbulence production in front of thecylinder (figure 4.5a), does not account for streamline curvature and predicts theonset of separation too late (figure 4.5b). Different results are obtained withFlotran and CFX using the same turbulence models with the same parameters.These are probably caused by differences in the near-wall modelling employed inFlotran and CFX.

0 45 90 135 180

0

5

10

15

20

Angle[°]

Tur

bule

nt k

inet

ic e

nerg

y [m

2 /s2 ]

(a)0 45 90 135 180

0

2

4

6

8

10

12

14

Angle[°]

Wal

l she

ar s

tres

s [P

a]

(b)

Figure 4.5: (a) Turbulent kinetic energy and (b) wall shear stress as a functionof the angle θ at the cylinder’s surface using different eddy viscosityturbulence models ( k ε, SST, SZL) for mesh B: Flotran (solidlines) and CFX (dashed lines).

The SZL turbulence model of Flotran produces the best overall correspondencewith the experimental results. Since computations converge faster in CFX due tothe use of a multigrid solver, the SST turbulence model of CFX, whose resultscorrespond, of all the turbulence models in CFX, best with the experimentalresults, is used for all subsequent calculations.


4.4.2 Unsteady computation

In the previous section, steady state solutions have been computed, althoughregular vortex shedding is present in the post-critical regime, which demands anunsteady RANS computation (Iaccarino et al., 2003). In this section, the resultsfrom a transient computation of the flow are compared with the steady stateresults.

The vortex shedding frequency fvs is described by the dimensionless Strouhalnumber St = fvsD/v. For flows with Re ≈ 107, experimental values for theStrouhal number range from 0.27 to 0.32 (Zdravkovich, 1997). Eurocode 1 suggestsa constant value of 0.2, independent of the Reynolds number.

The transient solution is integrated by the three-point backward differencescheme with a dimensionless time step ∆t v/D = 0.029, which corresponds toapproximately 100 time steps per vortex shedding period. Within every time step,maximum 5 iterations are performed in order to reduce the RMS of the normalizedresiduals to 10−6. The computed time window corresponds to ten vortex sheddingperiods. For mesh A2, 937 time steps are computed, which results in a timewindow of 4.68 s. With a shedding frequency of 2.13 Hz, a Strouhal number of 0.37is calculated. For mesh B, 954 time steps are computed, which results in a timewindow of 4.765 s. With a shedding frequency of 2.10 Hz, a Strouhal number of0.36 is calculated.

Figure 4.6 shows the time history and the frequency content of the pressure atθ = 172 at the cylinder’s surface for mesh B. The vortex shedding frequency andsome higher harmonics are clearly visible in the frequency content.

0 1 2 3 4

−350

−300

−250

−200

Time [s]

Pre

ssur

e [P

a]

(a)0 5 10 15 20

0

5

10

15

20

25

30

35

40

Frequency [Hz]

Pre

ssur

e [P

a/H

z]

(b)

Figure 4.6: (a) Time history and (b) frequency content of the pressure at thecylinder’s surface at θ = 172 for a single cylinder using mesh B.


Figure 4.7 shows the time average and the standard deviation of the pressure p.The stagnation pressure at the windward side and the suction in the wake areclearly visible. The largest time variations of the pressure occur in the wake.

(a) (b)

Figure 4.7: (a) Time average and (b) standard deviation of the pressure p for asingle cylinder using mesh B.

Figure 4.8 compares the pressure coefficient of the steady state computation withthe time averaged pressure coefficient Cp of the transient computation. Themaxima and minima of the pressure coefficient during the transient computationare depicted as well. Compared to the steady state results, the time average of the

transient computation predicts a lower minimum pressure coefficient Cmin

p and

a lower base pressure coefficient Cb

p, which is a deterioration for the minimum

pressure coefficient Cminp and an improvement for the base pressure coefficient Cb

p

with regard to the experimental data.

0 45 90 135 180−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

(a)0 45 90 135 180

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

(b)

Figure 4.8: Pressure coefficient as a function of the angle θ obtained with thesteady state computation (dashed line) and the time average (solidline), minimum (dash-dotted line) and maximum (dotted line) of thepressure coefficient obtained with the transient computation for (a)mesh A2 and (b) mesh B. The light grey zone contains all availableexperimental data at Reynolds numbers from 0.73×107 to 3.65×107.


As the time average of the transient solution differs clearly from the steady statesolution, a transient approach is more physically sound and offers additionalinformation about the time variation of the pressure (Iaccarino et al., 2003).Therefore, only transient computations will be performed for the flow around acylinder group.

4.4.3 Mesh refinement

In order to study the influence of mesh refinement, the unsteady flow is computedon four different meshes, with a mesh refinement factor r equal to

√2. The finest

mesh A1 is a factor√

2 finer than mesh A2. The coarsest mesh A4 is a factor 2coarser than mesh A2. Mesh A3 is a factor

√2 coarser than mesh A2. For each

set of three meshes a mesh refinement study is made: study 1 starting from thesolutions on meshes A2, A3 and A4 and study 2 with the meshes A1, A2 and A3.

For all meshes the time averaged pressure coefficient at the cylinder wall iscomputed (figure 4.9a). At the stagnation point, the lowest pressure coefficient1.015 is obtained using the finest mesh, while the coarsest mesh gives a value 1.038.The sequence of the solutions changes between 55 and 67. Near the minimum,the finest mesh yields the highest value -2.444 and the coarsest mesh the lowestvalue -2.468. The sequence of the solutions changes again around the separationpoint, between 115 and 121. In the recirculation zone the largest differencesoccur and the finest mesh produces a value that is about 0.05 lower than the valueof the coarsest mesh. The pressure coefficient is symmetrical, so the conclusionsbetween 181 and 360 are similar to those for 0 to 180.

0 90 180 270 360−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

(a)0 90 180 270 360

0

1

2

3

4

5

Angle [°]

Ord

er o

f acc

urac

y [−

]

(b)

Figure 4.9: (a) Pressure coefficient obtained on meshes A1 (black), A2 (blue), A3

(green) and A4 (red) and extrapolated solutions for mesh refinementstudy 1 (cyan) and 2 (magenta) and (b) local apparent order ofaccuracy for mesh refinement study 1 (solid line) and 2 (dashed line).


The solutions on the three meshes should be known at a common location.Therefore, the solutions on the fine and the intermediate mesh are interpolated tothe locations of the nodes of the coarse grid using piecewise cubic splines. Based onthree solutions φk computed on meshes Ak, the convergence ratio s is calculated:

s =φ2 − φ1

φ3 − φ2(4.54)

If s is between 0 and 1, the convergence is monotonic. For s smaller than 0,the convergence is oscillatory. If s is larger than 1, the solution diverges. Inthe zones where the sequence of the solutions changes (between 55 and 67 andbetween 115 and 121), the convergence is oscillatory or divergent. Near thestagnation point (between 357 and 360) the solution diverges as well. For meshrefinement study 1 and 2 respectively 3.6 % and 6.7 % of the points exhibitedoscillatory convergence or divergence. At the points with monotonic convergence,the apparent order of accuracy p is computed:

p =ln(φ3−φ2

φ2−φ1

)

ln r(4.55)

Figure 4.9b shows the apparent order of accuracy. For mesh refinement study 1the local order of accuracy p ranges from 0.04 to 8.52 with an average of 1.27, whilefor study 2 the local order of accuracy p ranges from 0.05 to 13.00 with an averageof 1.70. The variability of the apparent order is large, which is explained by thefact that it is poorly estimated except in the asymptotic range (Stern et al., 2001).For the High Resolution discretization, in the asymptotic range theoretically anorder of accuracy between 1 and 2 is expected. Ferziger and Peric (2002) state thatfor turbulent flows the definition of order is often difficult. In the neighbourhoodof the separation points, very large and very small apparent orders occur. Nearthe stagnation points the apparent order becomes very small. In the recirculationzone an almost constant, but very low value of 0.5 is obtained. Around 49 and311 the apparent order becomes very large.

Starting from the solutions φ1 and φ2 and the apparent order p, an extrapolatedsolution φ21

e is computed:

φ21e =

rpφ1 − φ2

r − 1(4.56)

If the apparent order is quite unrealistic, this leads to unrealistic extrapolatedvalues (Celik and Karatekin, 1997). Therefore, points where the apparent order issmaller than 0.5 or larger than 5 are disregarded here. For mesh refinement study1 and 2 respectively 12.3 % and 7.2 % of the points are ignored. The extrapolated


solutions for both mesh refinement studies are shown as well in figure 4.9a. Thedifferences between the solution on the finest mesh and the extrapolated solutionsis small except in the recirculation zone, where the difference increases to 0.05 dueto the small apparent order. The time averaged drag coefficient can be estimatedas 0.40 using the extrapolated pressure coefficient of mesh refinement study 2.

As to quantify the errors, the approximated relative error ε21a is computed:

ε21a =

∣∣∣∣φ1 − φ2

φ1

∣∣∣∣ (4.57)

Based on the extrapolated solution φ21e the extrapolated relative error ε21

e iscalculated:

ε21e =

∣∣∣∣φ21

e − φ1

φ21e

∣∣∣∣ (4.58)

The grid convergence index GCI21 is defined as (Celik and Karatekin, 1997):

GCI21 =1.25ε21

a

rp − 1(4.59)

The grid convergence index (GCI) is computed twice: once using the local apparentorder of accuracy and once with the average apparent order of accuracy.

Figure 4.10a shows the different errors. The relative errors become very largebetween 28 and 35 and between 326 and 332, as the pressure coefficient isthere around zero. The largest errors are present in the recirculation zone. There,the GCI using the local apparent order of accuracy predicts errors of about 20 %and 14 % for respectively mesh A2 and A1. The extrapolated relative errors arerespectively around 14 % and 10 %. Using the average apparent order of accuracy,the GCI yields 8 % and 4 %. The approximated relative errors are about 3.5 %and 2.5 %.

Figure 4.10b shows the time history of the drag and the lift coefficients as obtainedon the different meshes. The time averaged drag coefficient Cd and the fluctuation

C′d =√∑

(Cd − Cd)2/N of the drag coefficient increase when the mesh is refined.

The fluctuation C′l =√∑

(Cl − C l)2/N of the lift coefficient increases as well when

the mesh is refined. Table 4.1 gives the values obtained on the different meshesand the approximated relative errors for the time averaged drag coefficient Cd, theRMS of the drag coefficient CRMS

d =√∑C2

d/N , the RMS of the lift coefficient

CRMSl =

√∑C2

l /N , the fluctuation C′d of the drag coefficient, the fluctuation C′lof the lift coefficient and the separation angle θs. None of the integral variables


0 90 180 270 3600

10

20

30

40

50

Angle [°]

Err

or [%

]

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time [s]

Dra

g/lif

t coe

ffici

ent [

−]

(b)

Figure 4.10: (a) Approximated relative errors (dash dotted lines), extrapolatedrelative errors (dashed lines), GCI using the local (dotted lines)and the average (solid lines) apparent order of accuracy for meshrefinement study 1 (thin lines) and 2 (thick lines) and (b) time historyof drag (thin) and lift (thick) coefficients obtained on meshes A1

(solid line), A2 (dash dotted line) , A3 (dashed line) and A4 (dottedline).

(Cd, CRMSd , CRMS

l , C′d and C′l ) is converging (s > 1). The order of convergence forintegral quantities is clearly not the same as for field values (Ferziger and Peric,2002). The errors for the fluctuating parts are still quite large. For the three finestmeshes the separation angle θs converges to a value of 116.

φ4 φ3 φ2 φ1 ε43a ε32

a ε21a

Cd 0.341 0.342 0.349 0.358 0.5 % 2.1 % 2.5 %CRMS

d 0.341 0.342 0.349 0.358 0.5 % 2.1 % 2.5 %CRMS

l 0.0753 0.0834 0.0923 0.1023 9.7 % 9.6 % 9.8 %C′d 0.00155 0.00203 0.00260 0.00334 23.7 % 22.0 % 22.2 %C′l 0.0753 0.0834 0.0923 0.1023 9.7 % 9.6 % 9.8 %θs 119 118 118 117 0.4 % 0.4 % 0.3 %

Table 4.1: The time averaged drag coefficient Cd, the RMS of the drag coefficientCRMS

d , the RMS of the lift coefficient CRMSl , the fluctuation C′d of

the drag coefficient, the fluctuation C′l of the lift coefficient and theseparation angle θs obtained on meshes A1, A2, A3 and A4 and thedifferent approximated relative errors.


4.4.4 Comparison with numerical results reported in theliterature

Table 4.2 compares the Reynolds number Re, the Strouhal number St, the

separation angle θs, the base pressure coefficient Cb

p, the minimum pressure

coefficient Cmin

p , the drag coefficient Cd and the adverse pressure recovery APR =

Cb

p−Cmin

p from the present computations with numerical results from the literature.The Reynolds numbers of the numerical results published in the literature areslightly lower than the Reynolds number of the present calculations.

In the different experiments (Zdravkovich, 1997), Strouhal numbers lie within therange of 0.27 ≤ St ≤ 0.32, and the boundary layer separates between 100 and110. After separation, the base pressure coefficient Cb

p is constant and lies withinthe range −0.5 to −0.8. The adverse pressure recovery varies from 1.0 to 1.5. Dragcoefficients coefficients vary from 0.40 to 0.80.

According to Eurocode 1 (BIN, 1995), the minimum value of the pressure coefficientCmin

p equals −1.5 at an angle of 75. The boundary layer separates at 105. After

separation, the base pressure coefficient Cbp is constant and equal to −0.8. This

corresponds to an extremely low adverse pressure recovery of 0.7. Eurocode 1mentions the value 0.2 for the Strouhal number, which corresponds to laminarvortex shedding. The drag coefficient is equal to 0.72 for a smooth surface (k/b =10−5).

Simulations with the standard k-εmodel predict separation much too late, yieldinga smaller and shorter wake with vortices shed at a higher frequency. The suctionin the wake is too low, which results in a lower drag coefficient.

If the present computations are compared with the 12 experimentally obtainedpressure coefficients (figure 4.2), they overestimate the base pressure andconsequently underestimate the drag coefficient, but some of the more advancedmodels in the literature do as well. The Shih-Zhu-Lumley model yields relativelygood results on mesh B, though the base pressure coefficient is quite high. In theunsteady computations with the SST model, the boundary layer separates slightlytoo late, and therefore the drag coefficient is underestimated and the Strouhalnumber is overestimated. The time averaged base pressure coefficient is lowerthan in steady computations.


Re St θs -Cb

p -Cmin

p Cd APR×106 []

2-D RANS k − ε Flotran mesh A2 12.4 158 -0.08 2.6 0.30 2.72-D RANS k − ε Flotran mesh B 12.4 143 0.08 2.5 0.31 2.42-D RANS SST Flotran mesh A2 12.4 113 0.37 2.1 0.36 1.72-D RANS SST Flotran mesh B 12.4 118 0.31 2.3 0.30 2.02-D RANS SZL Flotran mesh A2 12.4 116 0.35 2.0 0.35 1.72-D RANS SZL Flotran mesh B 12.4 108 0.41 1.8 0.42 1.42-D RANS k − ε CFX mesh A2 12.4 147 0.10 2.4 0.33 2.32-D RANS k − ε CFX mesh B 12.4 149 0.10 2.4 0.34 2.32-D RANS SST CFX mesh A2 12.4 114 0.38 2.2 0.29 1.92-D RANS SST CFX mesh B 12.4 113 0.39 2.2 0.30 1.82-D URANS SST CFX mesh A2 12.4 0.37 117 0.45 2.4 0.35 2.02-D URANS SST CFX mesh B 12.4 0.36 116 0.46 2.4 0.36 2.0Eurocode (BIN, 1995) 10.0 0.2 105 0.80 1.5 0.72 0.72-D URANS k − ε transition 3.6 118 0.35 2.3 2.0(Celik and Shaffer, 1995)2-D URANS realizable k − ε 10.0 120 0.26(Holloway et al., 2004)2-D URANS transition 10.0 119 0.25(Holloway et al., 2004)2-D URANS RNG k − ε 3.5 0.28 122 0.8 2.5 0.56 1.7(Younis and Przulj, 2006)2-D URANS modified k − ε 3.5 0.28 120 1.25 2.5 0.72 1.3(Younis and Przulj, 2006)2-D URANS k − ε 8.4 0.25 104 0.72 1.8 0.66 1.1(Saghafian et al., 2003)2-D URANS non-linear 8.4 0.33 125 1.15 2.6 0.61 1.5(Saghafian et al., 2003)3-D DES 3.0 0.35 111 0.53 2.2 0.41 1.7(Travin et al., 2000)3-D DES + curvature 3.0 0.33 106 0.64 2.1 0.51 1.5(Travin et al., 2000)3-D URANS k − ε 1.0 0.31 0.41 2.3 0.40 1.9(Catalano et al., 2003)3-D LES Smagorinsky 1.0 0.35 0.32 2.4 0.31 2.1(Catalano et al., 2003)2-D LES discretization 10.0 1.25 0.85(Singh and Mittal, 2005)

Table 4.2: Comparison of the Reynolds number Re, the Strouhal number St, the

separation angle θs, the base pressure coefficient Cb

p, the minimum

pressure coefficient Cmin

p , the drag coefficient Cd and the adverse

pressure recovery APR with numerical results from the literature.


4.5 Turbulent air flow around a cylinder group

As cylinders are often placed in groups and the configuration of these groupslargely influences the pressure distribution around the cylinders, an experimentalor numerical study of the wind flow should provide a more realistic estimation ofthe pressure coefficients and forces.

For the silo group in the port of Antwerp (figure 1.3), the forty silos are placed infive rows of eight silos with gaps of 30 cm between two neighbouring silos (figure1.4). The spacing ratio of the distance P between the center of two cylinders to thecylinder diameter is P/D = 5.8/5.5 = 1.05. In this section, the turbulent air flowaround cylinder groups with spacing ratio P/D = 1.05 is studied. No experimentalor numerical data are available for the flow around a group of cylinders with sucha small spacing ratio and at the present high Reynolds number. Therefore, theincompressible turbulent wind flow around a group of 8 by 5 cylinders is simulated.Before, the less comprehensive case of a group of 2 by 2 cylinders is modelled.

The flow around two cylinders in a staggered configuration has been thoroughlystudied experimentally, mostly at high subcritical Reynolds numbers. Gu and Sun(1999) visualized the flow at Reynolds number 5.6× 103, measured time averagedpressure distributions at Reynolds number 2.2 × 105, and derived lift and dragforces. Extensive flow visualization studies by Sumner et al. (2000) at Reynoldsnumber 850 ≤ Re ≤ 1900 identified nine different flow patterns. Alam et al.(2005) measured time averaged and fluctuating pressure distributions at Reynoldsnumber 5.5 × 104 for spacing ratios P/D from 1.1 to 6 and for incidence anglesα between 10 to 75. Closely spaced cylinder configurations behave similarly asa single bluff body: Karman vortex shedding occurs from the group as a whole,and one Karman vortex street exists in the combined wake. The time averagedaerodynamic forces undergo large changes with the incidence angle α between thewind direction and the longitudinal group axis (Sumner et al., 2005). Upstreamcylinders experience smaller fluctuating forces than a single cylinder.

• For small incidence angles α < 10−20 (nearly tandem configuration - figure4.11a) the inner separated shear layer from the upstream cylinder reattachesonto the outer side of the downstream cylinder, preventing flow through thegap (shear layer reattachment (Sumner et al., 2000) or pattern IB (Gu andSun, 1999)). The pressure distribution of the downstream cylinder attainsa maximum where reattachment occurs and a minimum where the outerboundary layer separates. The downstream cylinder experiences a negativedrag. Only one Strouhal number is measured in the wake, which is slightlyhigher than the value for a single cylinder.

TURBULENT AIR FLOW AROUND A CYLINDER GROUP 135

• At incidence angles 20 ≤ α ≤ 45 (figure 4.11b) the inner shear layer fromthe upstream cylinder is directed through the gap and reattaches with itshigh-speed side onto the inner side of the downstream cylinder. It inducesseparation on the inner side of the downstream cylinder (induced separation(Sumner et al., 2000) or pattern IIB (Gu and Sun, 1999)). A large area ofsuction is developed on the downstream cylinder in the gap region, resultingin a peak in the lift coefficient of the downstream cylinder. This correspondswith a local maximum drag for the upstream and a local minimum drag forthe downstream cylinder. Vortex shedding occurs at different frequenciesfrom the outer shear layers of the upstream and downstream cylinder.

• For large angles α between 45 and 90 (figure 4.11c), fluid enters throughthe gap in the base region, causing a lengthening of the near-wake region(base bleed (Sumner et al., 2000) or pattern IIIB (Gu and Sun, 1999)). Thisgap flow between the shear layers, separated from the inner side of bothcylinders, is typically deflected towards the upstream cylinder, although itmight also be instantaneously deflected towards the downstream cylinder.For large angles, the lift forces pull the cylinders away from each other. Thesame low-frequency Strouhal number is measured behind both cylinders. Itdecreases with increasing α to 0.10 at α = 90.

(a) (b) (c)

Figure 4.11: Flow patterns around two closely spaced cylinders for (a) small, (b)medium and (c) large incidence angles (Sumner et al., 2000).

Gu (1996) measured pressure distributions at supercritical Reynolds numbers 4.5×105, which differ from those at subcritical Reynolds number. The downstreamcylinder experiences higher drag coefficients than a single cylinder for angles α >45 due to an increased base pressure and an extended stagnation area.

Numerical simulations are mainly available at low Reynolds numbers (Re ≤ 1000).Flow patterns computed using the vortex method for the laminar two-dimensionalflow at Reynolds number 800 (Akbari and Price, 2005) correspond well withexperimental data for a large number of configurations. Jester and Kallinderis(2003) used the finite element method to compute time averaged drag coefficientsfor spacing ratios P/D ≥ 1.5.


Studies of the flow around arrays with more than two cylinders are quite rare andmainly treat side-by-side or tandem configurations. Experiments on groups of fourcylinders in a square configuration were performed by Lam and Fang (1995). Thetime averaged pressure distributions were measured at Reynolds number 12.8×103

for spacing ratios P/D from 1.26 to 5.80 in 15 steps and for angles α between 0 to45 in steps of 15. Lift and drag forces were obtained by integrating the pressuredistributions. Flow patterns change rapidly as the spacing ratio is decreased andhighly depend on the incidence angle. Consequently, time averaged drag and liftcoefficients of closely spaced arrays are very sensitive to changes in spacing ratio.Unique characteristics were observed at the minimum spacing ratio. Lam et al.(2003) measured the time averaged and fluctuating drag and lift forces at thesubcritical Reynolds number 4.1 × 104 for spacing ratios P/D from 1.69 to 3.83and for angles α from 0 to 45 in steps of 15. Strouhal numbers were derivedfrom the power spectral density of the forces. Different cylinders may have distinctStrouhal numbers that change with the incidence angle, due to the existence ofwider and narrower wakes behind the cylinders. Farrant et al. (2000) computedthe laminar flow around an array of four cylinders using the cell boundary elementmethod at Reynolds number 200 for large spacing ratios P/D = 3 and 5 and forangles α = 0 and 45.

A lot of experimental research focuses on the flow in tube bundles in heatexchangers (Zdravkovich, 2003). Weaver et al. (1993) studied the periodicexcitation mechanism in a rotated square (α = 45) arrangement with spacingratios 1.21 ≤ P/D ≤ 2.83. At the smallest spacing ratio, only one Strouhalnumber was found, while at higher spacing ratios, two distinct numbers wereobserved, which correspond to vortex shedding from the first and the secondtube rows. A flow visualization in a rotated square arrangement with spacingratio P/D = 1.50 at Reynolds numbers 80 ≤ Re ≤ 1300 was carried out byPrice et al. (1995) and only revealed one Strouhal number. Using periodicboundary conditions, the flow computation in rotated square arrangements formsan attractive test for turbulence models due to the contractions and expansions ofthe flow, the high turbulence intensities (35%) and streamline curvatures. RANSmodels generally predict the velocity profiles reasonably well, but Reynoldsstress profiles are poor because the size of the larger eddies is comparable tothe cylinder radius and fluctuating velocities are larger than the mean velocitiesin some regions (Rollet-Miet et al., 1999). In the stagnation region in front ofthe cylinders, the turbulent kinetic energy is decreasing which is not predictedby RANS models. Benhamadouche and Laurence (2003) obtained satisfactoryresults for both velocity and Reynolds stress profiles with a LES and a Reynoldsstress model on a fine and a coarse three-dimensional mesh for a rotated squarearray with spacing ratio P/D = 1.40 at Reynolds number 9000. The Reynoldsnumber is based on the bulk velocity in the gaps. Results computed with aRSM on a two-dimensional mesh show strong vortex shedding and severelyoverestimate the normal Reynolds stresses, because the flow structures are highly


three-dimensional. Beale and Spalding (1999) imposed spanwise perturbationsto compute a two-dimensional transient periodic laminar flow in in-line squareand rotated square arrays with spacing ratio P/D = 2 at Reynolds numbers30 ≤ Re ≤ 3000. Derived Strouhal numbers are in agreement with experimentaldata. Hassan and Barsamian (2004) performed a LES of the flow around a fullthree-dimensional non-symmetric bundle consisting of five rows of 2 cylinderswith a spacing ratio P/D = 2.07 and an angle α = 45 at Reynolds number 21700.Cylinders of the last row have larger wakes and highest pressures occur at theside of the bundle. No distinct peaks are observed in the power spectral densityof the lift and drag force. Large scale three-dimensional effects are absent at themidplane.

In this section, the two-dimensional flow around groups with a spacing ratio P/D =1.05 are computed in the post-critical regime (Reynolds number 12.4× 106) usingthe same turbulence model as in the previous section. This spacing ratio is smallerthan the ratios studied in the literature. Experiments and simulations mainly treatthe laminar and the subcritical regime. The RANS model is not ideal, but predictsthe velocity profiles quite well and has a reasonable computational cost.

The flow around groups of cylinders is strongly three-dimensional with probablyhorseshoe vortices, trailing vortices and arch vortices around the group as a whole.In a three-dimensional computation a lot of the fluid will pass left, right andabove the group, while in a two-dimensional computation the fluid is only allowedto pass left or right and a larger amount of fluid is forced to pass through thesmall gaps. Therefore, in the two-dimensional computations higher pressures willprobably be found in the gaps than in the case of a three-dimensional computation.The velocity gradients of an atmospheric boundary layer cannot be considered ina two-dimensional computation. The turbulent kinetic energy and the turbulentenergy dissipation rate at the inlet are equal to the low values used for the case ofa single cylinder. All these important assumptions complicate the extrapolation ofthe results of the two-dimensional computations to groups of cylindrical structureslocated in an atmospheric boundary layer.

Two configurations are studied: first the less comprehensive group of 2 by 2cylinders and then a group of 8 by 5 cylinders, as in the port of Antwerp.The incidence angle α (figure 4.12) between the wind flow direction and thecylinder group is equal to 30 for both configurations, corresponding to thepredominant wind direction at the site in Antwerp. During the storm on October27, 2002, the wind direction (west-southwest) corresponded to this incidence angle.Computations for other incidence angles should be performed as well, but areskipped as they are completely analogous and just a matter of computation time.The lateral boundaries in the model are located at a distance of 9 times the widthL sin(30) +W cos(30) of the group, the inlet at a distance of 9 times the depthL cos(30) +W sin(30) of the group and the outlet at a distance of 30 times the


depth of the group (figure 1.4). A smaller problem domain would be unlikely toinfluence the results near the cylinders, but nevertheless the number of elementsin the far field is anyhow negligible with respect to the number of elements nextto the group.

4.5.1 Group of 2 by 2 cylinders

Figure 4.12 shows a detail of the mesh A near the group of 2 by 2 cylinders.A structured mesh consisting of 280 elements around the circumference and 30elements in the radial direction is used around each cylinder. Mesh A consists of201184 elements and 243348 nodes. The four cylinders are numbered 1 to 4 asshown in the figure. The angle θ is defined independently of the wind direction.

qA

E

B

D

F

C

1

3

2

4

a

vf

Figure 4.12: Detail of the mesh for the group of 2 by 2 cylinders.

The transient solution is integrated with a dimensionless time step v∆t/D = 0.029.Maximum 10 iterations are performed within every time step in order to reducethe RMS of the normalized residuals to 10−6. 2965 time steps are computed, whichresults in a time window of 14.820 s.

Figure 4.13a shows the dimensionless distance y+ of the nodes next to the cylinderwall as a function of the angle θ for the four cylinders at time t = 14.820 s . Mostpoints lie within the region where the logarithmic law is valid.

The flow is also computed on a finer mesh B, where the nodes next to the cylinderwall lie within the viscous sublayer (figure 4.13b). The structured mesh aroundeach cylinder consists of 800 elements around the circumference and 90 elementsin the radial direction. Mesh B is made up of 511444 elements and 810088 nodes.

1933 time steps are computed, which results in a time window of 9.665 s. Figure4.14 compares the time averaged pressure coefficients on meshes A and B as a


0 90 180 270 3600

50

100

150

200

Angle [°]

Dim

ensi

onle

ss d

ista

nce

[−]

(a)0 90 180 270 360

0

0.5

1

1.5

2

2.5

Angle [°]

Dim

ensi

onle

ss d

ista

nce

[−]

(b)

Figure 4.13: Dimensionless distance y+ of the nodes next to the cylinder wall asa function of the angle θ for cylinder 1 (solid line), 2 (dashed line), 3(dotted line) and 4 (dash-dotted line) of the group of 2 by 2 cylindersin (a) mesh A and (b) mesh B.

0 90 180 270 360

−3

−2

−1

0

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

(a)0 90 180 270 360

−3

−2

−1

0

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

(c)

0 90 180 270 360

−3

−2

−1

0

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

(b)0 90 180 270 360

−3

−2

−1

0

1

Angle [°]

Pre

ssur

e co

effic

ient

[−]

(d)

Figure 4.14: Time average (thick line), maximum and minimum (thin lines) ofthe pressure coefficient Cp for mesh A (black) and mesh B (grey)as a function of the angle θ for (a) cylinder 1, (b) cylinder 2, (c)cylinder 3 and (d) cylinder 4 of the group of 2 by 2 cylinders.


function of the angle θ for the four cylinders. The minima and maxima arealso indicated. For the mesh B, the time average is computed from t = 5.170 sto t = 9.665 s. As only small differences between both meshes are noticeable,most pronounced at the small gaps and around the separation points, the resultsobtained with mesh A will be discussed further.

Figures 4.15 and 4.16 show the time history and the frequency content of thepressure between t = 7.385 s and t = 14.820 s at a small gap (B) and in the middleof the group (E), as indicated in figure 4.12. The largest variation of the pressureoccurs at the small gaps and at the separation points. The dominant peak in thefrequency content at 0.67 Hz (figures 4.15b and 4.16b) corresponds to a Strouhalnumber of 0.28, using the projected width 13.42 m of the group as a characteristiclength. In the middle of the group and in the wake, higher harmonics are clearlypresent in the pressure time history. From the third peak in the frequency contentat 2.03 Hz (figure 4.16b), a Strouhal number of 0.35 is derived using the cylinderdiameter as a characteristic length. Time variations of the pressure are generallylarger at the leeward side of the cylinder group (figure 4.14). At the stagnationpoints, the pressure is constant in time.

8 9 10 11 12 13 14

−1900

−1800

−1700

−1600

−1500

Time [s]

Pre

ssur

e [P

a]

(a)0 1 2 3 4 5

0

50

100

150

200

Frequency [Hz]

Pre

ssur

e [P

a/H

z]

(b)

Figure 4.15: (a) Time history and (b) frequency content of the pressure at thecylinder’s surface in the point B of the group of 2 by 2 cylinders.

Figure 4.17 shows the streamlines at t = 14.075 s during the last vortex sheddingperiod. The outer boundary layer F1 of cylinder 1 separates at 320 ≤ θ ≤ 322,but reattaches at 333 ≤ θ ≤ 334. At θ = 12 the boundary layer separatesagain and reattaches to cylinder 4. The gap flow F1 separates from cylinder 2 atθ = 78 and is deflected towards the downstream cylinder 4. It finally separatesfrom cylinder 4 at 54 ≤ θ ≤ 66. This yields a very wide recirculation zone W1behind cylinder 2, where vortices shed at a lower frequency alternately from thisdeflected gap flow F1 and the outer shear layer separated at 324 ≤ θ ≤ 326 fromcylinder 2. This flow pattern upstream and between cylinders 1 and 2 is similar to


8 9 10 11 12 13 14

−400

−300

−200

−100

0

Time [s]

Pre

ssur

e [P

a]

(a)0 1 2 3 4 5

0

50

100

150

200

Frequency [Hz]

Pre

ssur

e [P

a/H

z]

(b)

Figure 4.16: (a) Time history and (b) frequency content of the pressure at thecylinder’s surface in the point E of the group of 2 by 2 cylinders.

← F2

W1

W3

W4

F1

W2

1

2

4

3

q

Figure 4.17: Streamlines at t = 14.075 s from a transient computation around agroup of 2 by 2 cylinders.

the patterns described by Alam et al. (2005) for α = 25 and P/D = 1.3 (figure4.18a) and for α = 45 and P/D = 1.1 (figure 4.18b). The wake behaviour isclearly different due to the presence of the two other cylinders.

The inner boundary layer F2 of cylinder 1 separates at θ = 78 and reattachesto cylinder 4. The gap flow F2 separates from cylinder 3 at θ = 12 and isdeflected towards the downstream cylinder 4. It finally separates from cylinder 4at 95 ≤ θ ≤ 107. The recirculation zone W2 behind cylinder 3 is constrainedbetween this gap flow F2 and the outer shear layer separated at 100 ≤ θ ≤ 102

from cylinder 3.

Both deflected gap flows F1 and F2 constrain the spatial development of therecirculation zone W3 behind cylinder 4, yielding a very small recirculation zone.


(a) (b)

Figure 4.18: Flow patterns around two closely spaced cylinders for (a) α = 25

and P/D = 1.3 and (b) α = 45 and P/D = 1.1 (Alam et al., 2005).

The recirculation zone W4 behind cylinder 1 is highly constrained between thegap flows F1 and F2 attached to cylinders 2 and 3 and is biased away from theflow axis. The flow pattern upstream and between cylinders 1 and 3 is similar tothe base bleed pattern (Sumner et al., 2000) and pattern IIIB (Gu and Sun, 1999).

←

1 3

2 4

Figure 4.19: Time averaged pressure coefficients Cp for the flow around a group of2 by 2 cylinders (solid line) and for the flow around a single cylinder(dashed line). The arrow indicates the incidence angle α.

Figure 4.19 shows the time averaged pressure coefficients for the flow around thegroup compared with the pressure coefficient for the flow around a single cylinder.The time average is computed from t = 7.385 s to t = 14.820 s. The groupconfiguration drastically changes the pressure distribution around the cylinders.The stagnation points of cylinders 1 and 3 are shifted towards each other, whilethe stagnation point of cylinder 2 is shifted away from cylinder 1. Similar shifts are


measured by Lam and Fang (1995) at Reynolds number 12.8×103 for P/D = 1.26and angle α = 30. The positive pressure area on cylinder 1 is largely extended.Cylinder 4 is submerged in the wake of the three other cylinders, yielding suctionover the whole circumference. The reattachment of the gap flows to cylinder 4produces the maxima in its pressure distribution. The largest suction occurs wherethe outer boundary layers separate from cylinders 2 and 3. The large velocitiesin the gaps produce distinct minima in the pressure distributions where the gaparea is minimum. The results by Lam and Fang (1995) for P/D = 1.26 also showdistinct minima at some gaps, but not at all gaps. The larger spacing ratio and thelower Reynolds number of the experiments might explain these differences. Thebase pressure of cylinder 2, 3 and 4 is decreased, compared with a single cylinder.

Figure 4.20 shows the time averaged and fluctuating drag and lift coefficients forthe flow around the group. Cylinder 1, at the head of the group, experiences ahigher time averaged drag coefficient Cd in comparison with the flow around asingle cylinder (0.35 ≤ Cd ≤ 0.36) due to the extended area of positive pressureand the large suction in the gaps. Cylinder 4 is completely immersed in the wakeof the three other cylinders and is pulled forward towards these cylinders by thelarge suction in the gaps, resulting in a negative drag coefficient. Negative dragwas measured for a group of 4 cylinders by Lam and Fang (1995) for P/D = 1.26and angle α = 0. In two-cylinder configurations, Sumner et al. (2005) observednegative drag coefficients for P/D = 1.125 and angle 0 ≤ α ≤ 15 and Gu (1996)did at supercritical Reynolds number for P/D = 1.2 and angle 10 ≤ α ≤ 45.

← ← ← ←

0.98

0.18

0.51

−0.18

0.01

0.08

0.06

0.02

−0.61

−0.56

0.68

−0.27

0.04

0.07

0.10

0.19

(a) (b) (c) (d)

Figure 4.20: (a) Time averaged drag coefficient Cd, (b) fluctuating drag coefficientC′d, (c) time averaged lift coefficient C l and (d) fluctuating liftcoefficient C′l for the flow around the group of 2 by 2 cylinders. Thearrow indicates the incidence angle α.

Time averaged lift coefficients C l are quite high (figure 4.20c). As the inner part ofcylinders 2 and 3 are immersed in the wake of cylinder 1 and large suction occurswhere the outer boundary layer separates from these cylinders, they are pulledaway from the group. For cylinder 1, the large suction around θ = 280 and theextended area of positive pressure yield a negative lift coefficient. At P/D = 1.26,Lam and Fang (1995) report attractive forces between cylinders 2 and 3 for anangle α = 30, but repulsive forces at α = 15. Sumner et al. (2005) measuredattractive forces between two staggered cylinders with P/D = 1.125 in the range


0 ≤ α ≤ 25 and repulsive forces in the range 60 ≤ α ≤ 90. Fluctuations oncylinder 1 are small, while cylinder 4 in the wake experiences a larger fluctuatinglift coefficient.

The time averaged pressure coefficient is decomposed into a series of cosinefunctions with circumferential wavenumber n, corresponding to the mode shapesof an axisymmetric structure (figure 3.14):

Cp =∞∑

n=0

Cn

p cos(nθ + ϕn) (4.60)

The eigenmodes with n = 3 or n = 4 have the highest contribution to the responseof the silos under wind loading (Dooms et al., 2003).

Figure 4.21 shows the amplitudes Cn

p for the flow around the group for n =0− 10, which represent the major part of the variation of the pressure coefficient,compared with the amplitudes for the flow around a single cylinder. For all valuesof n, except n = 2, larger amplitudes C

n

p than for a single cylinder occur for oneor more cylinders in the group. For the cylinders on the transverse corners of thegroup, the amplitudes C

n

p for n = 3 and n = 4 are larger than the amplitudes fora single cylinder.

←

0

0.5

1

1.5

n = 0

n = 1 n = 2 n = 3 n = 4 n = 5

n = 6 n = 7 n = 8 n = 9 n = 10

Figure 4.21: Decomposition of the time averaged pressure coefficient Cp into aseries of cosine functions with circumferential wavenumber n for theflow around the group of 2 by 2 cylinders (2 by 2 circles) and for theflow around a single cylinder (single circle). The arrow indicates theincidence angle α.


4.5.2 Group of 8 by 5 cylinders

Figure 4.22 shows a detail of the mesh near the group of 8 by 5 cylinders. Thesame structured mesh consisting of 280 elements around the circumference and 30elements in the radial direction is used around each cylinder. The mesh consistsof 566866 elements and 921438 nodes. The cylinders are numbered as shown inthe figure.

12

34

56

78

1615

1413

1211

109

2423

2221

2019

1817

3231

3029

2827

2625

3334

3536

3738

3940

vf

a

E

F

D

A

C

B

Figure 4.22: Detail of the mesh for the group of 8 by 5 cylinders.

The transient solution is integrated with a dimensionless time step v∆t/D = 0.058.A maximum of 10 iterations are performed within every time step in order to reducethe RMS of the normalized residuals to 10−6. 5170 time steps are computed, whichresults in a time window of 51.70 s.

0 90 180 270 3600

50

100

150

200

Angle[°]

Dim

ensi

onle

ss d

ista

nce

[−]

Figure 4.23: Dimensionless distance y+ of the nodes next to the cylinder wall asa function of the angle θ for all cylinders of the group of 8 by 5cylinders.

Figure 4.23 shows for the forty cylinders the dimensionless distance y+ of the nodesnext to the cylinder wall as a function of the angle θ at time t = 51.70 s. Mostpoints lie within the region where the logarithmic law is valid.


Figures 4.24 and 4.25 show the time history and the frequency content between t =33.48 s and t = 51.70 s of the pressure at the cylinder’s surface at a separation point(B) and at a small gap (E), as indicated in figure 4.22. At the stagnation points,the pressure is constant in time. Generally, the pressure variation on cylindersat the windward side primarily has components at approximately 0.165 Hz (figure4.24b), which corresponds to a Strouhal number of 0.24, using the projected width45.89 m of the group as a characteristic length. In the middle of the group andin the wake, the pressure spectrum has also a significant peak at 2.85 Hz (figure4.25b). A vortex shedding frequency of 2.85 Hz corresponds to a Strouhal numberof 0.49 using the cylinder diameter as a characteristic length. This is consistentwith higher Strouhal numbers measured in closely spaced tube arrays (Blevins,1990). Generally, time variations of the pressure are larger at the leeward side ofthe cylinder group.

35 40 45 50

−2350

−2300

−2250

−2200

−2150

−2100

Time [s]

Pre

ssur

e [P

a]

(a)0 1 2 3 4 5

0

5

10

15

20

25

30

35

40

Frequency [Hz]

Pre

ssur

e [P

a/H

z]

(b)

Figure 4.24: (a) Time history and (b) frequency content of the pressure at thecylinder’s surface in the point B of the group of 8 by 5 cylinders.

35 40 45 50

−850

−800

−750

−700

−650

−600

Time [s]

Pre

ssur

e [P

a]

(a)0 1 2 3 4 5

0

5

10

15

20

25

30

35

40

Frequency [Hz]

Pre

ssur

e [P

a/H

z]

(b)

Figure 4.25: (a) Time history and (b) frequency content of the pressure at thecylinder’s surface in the point E of the group of 8 by 5 cylinders.


Figure 4.26 shows the streamlines at t = 50.94 s during the last vortex sheddingperiod. The outer boundary layers of cylinders 1-7 separate around θ = 313

and reattach around θ = 341 before entering the gaps. The gap flows betweenthe cylinders have a sinusoidal pattern which makes an angle of 15 with the flowdirection. When leaving the group, most of the gap flows are deflected towards thedownstream cylinders and join up with other deflected gap flows. These unifiedgap flows finally separate from cylinder 40. The recirculation zone behind most ofthe cylinders on the leeward borders of the group are strongly constrained betweenthe different gap flows and biased away from the flow axis. The outer shear layer

←

7

8

65

4

32

1

16

1314

15

12

11

2423

2221

20

1918

17

10

28

2526

27

30

299

33

3435

3738

32

31

3940

36

W2

W3

W1

Figure 4.26: Streamlines at t = 50.94 s from a transient computation around agroup of 8 by 5 cylinders.

separated from cylinder 8 yields a very wide recirculation zone W1 behind thiscylinder, where vortices shed at a low frequency alternately from this separatedboundary layer and the unified gap flow separated from cylinder 40. The gap flowbetween cylinders 34 and 35 is deflected towards the upstream cylinder, yielding asecond very large recirculation zone W2 behind cylinder 35. The outer shear layerseparated from cylinder 33 and the gap flow between cylinders 33 and 34 form athird recirculation zone W3 behind cylinder 33. The recirculation zones behindcylinders within the group are highly constrained between the gap flows attachedto neighbouring cylinders and extend up to the downstream cylinder.

Figure 4.27 shows the time averaged pressure coefficients for the flow aroundthe group compared with the pressure coefficient for the flow around a singlecylinder. The time average is computed from t = 33.48 s to t = 51.70 s. The groupconfiguration drastically changes the pressure distribution around the cylinders.The stagnation points are shifted towards the outer side for cylinders 2 to 8.The positive pressure area on these cylinders is reduced. For cylinders 9, 17,25 and 33, the stagnation points are shifted towards the upstream cylinder. Thepositive pressure area on cylinder 1, 9 and 17 is largely extended. The minimum


←1 9 17 25 33

2 10 18 26 34

3 11 19 27 35

4 12 20 28 36

5 13 21 29 37

6 14 22 30 38

7 15 23 31 39

8 16 24 32 40

Figure 4.27: Time averaged pressure coefficients Cp for the flow around a group of8 by 5 cylinders (solid line) and for the flow around a single cylinder(dashed line). The arrow indicates the incidence angle α.

pressure coefficient upstream of the boundary layer separation increases fromcylinder 1 to cylinder 8. On cylinders 2 to 7, after separation, a zone of constantpressure is formed up to the reattachment. On cylinder 33, large suction isproduced upstream of the boundary layer separation. The large velocities in thegaps produce distinct minima in the pressure distributions where the gap areais minimum. All cylinders submerged in the wake of other cylinders have twomaxima in the pressure coefficient at the locations where the gap flows reattach.


The base pressure generally decreases for cylinders located further downstream.For some of the upstream cylinders, positive base pressures were computed.

← ←

0.27

0.05

0.05

0.03

0.03

0.06

0.16

0.20

0.61

0.16

0.16

0.13

0.13

0.13

0.13

0.27

0.67

0.19

0.16

0.14

0.15

0.15

0.15

0.34

0.74

0.21

0.15

0.09

0.14

0.14

0.12

0.22

0.13

0.16

0.15

0.01

0.19

0.17

0.14

0.02

0.01

0.01

0.00

0.00

0.00

0.00

0.01

0.01

0.00

0.00

0.00

0.00

0.00

0.01

0.01

0.01

0.01

0.00

0.00

0.01

0.00

0.01

0.01

0.02

0.01

0.00

0.01

0.02

0.03

0.03

0.01

0.03

0.01

0.00

0.00

0.00

0.02

0.03

0.03

0.02

(a) (b)

← ←

−0.50

−0.06

−0.09

−0.14

−0.20

−0.27

−0.36

−0.84

−0.22

0.14

0.08

0.08

0.05

−0.02

−0.16

−0.69

−0.13

0.23

0.21

0.16

0.14

0.10

0.04

−0.19

0.01

0.37

0.31

0.25

0.16

0.13

0.09

−0.09

1.01

0.22

0.22

0.52

0.37

0.30

0.23

0.00

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.00

0.00

0.01

0.01

0.01

0.01

0.01

0.01

0.00

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.00

0.01

0.01

0.02

0.02

0.03

0.04

0.02

0.01

0.01

0.01

0.01

0.02

0.03

0.03

0.05

(c) (d)

Figure 4.28: (a) Time averaged drag coefficient Cd, (b) fluctuating drag coefficientC′d, (c) time averaged lift coefficient C l and (d) fluctuating liftcoefficient C′l for the flow around the group of 8 by 5 cylinders. Thearrow indicates the incidence angle α.

Figure 4.28 shows the time averaged and fluctuating drag and lift coefficients forthe flow around the group. At the windward side of the group, cylinders 9, 17


and 25 experience a higher time averaged drag coefficient Cd in comparison withthe flow around a single cylinder (0.35 ≤ Cd ≤ 0.36), while in the middle of thegroup the drag coefficients are considerably lower. At the two transverse corners(cylinders 8 and 33) the drag coefficient is also low. Figure 4.28c shows the timeaveraged lift coefficients C l. The cylinders on the borders of the group are pulledaway from the group, which results in relatively high lift coefficients for thesecylinders, especially for the two on the transverse corners (cylinders 8 and 33).

Fluctuating drag and lift coefficients (figures 4.28b and 4.28d) are generally small.The largest fluctuations occur at the leeward side of the group.

As for the 2 by 2 group, the time averaged pressure coefficient is decomposed intoa series of cosine functions with circumferential wavenumber n. Figure 4.29 showsthe amplitude C

n

p for the flow around the group for 0 ≤ n ≤ 10, compared withthe amplitudes for the flow around a single cylinder. For all values of n, exceptn = 2, larger amplitudes C

n

p than for the flow around a single cylinder occur forone or more cylinders in the group. The highest values are always situated onthe borders of the group. Eigenmodes with circumferential wavenumber n equalto a multiple of 4 are heavily loaded in the group configuration due to the foursmall gaps surrounding most of the cylinders. Especially for the cylinders on the

←

0

0.5

1

1.5

n = 0

n = 1 n = 2 n = 3 n = 4 n = 5

n = 6 n = 7 n = 8 n = 9 n = 10

Figure 4.29: Decomposition of the time averaged pressure coefficient Cp into aseries of cosine functions with circumferential wavenumber n for theflow around the group of 8 by 5 cylinders (8 by 5 circles) and for theflow around a single cylinder (single circle). The arrow indicates theincidence angle α.

CONCLUSION 151

transverse corners of the group, eigenmodes with n = 3 or n = 4, which for silostypically have the lowest eigenfrequencies, are strongly excited.

The large values for the time averaged lift coefficients and large amplitudes Cn

p

for n = 3 and n = 4 for the cylinders on the transverse corners of the group areconsistent with the observations that that storm damage is mainly located on silosat the corners of the groups.

4.6 Conclusion

In this chapter, some aspects of the computation of turbulent wind flows aroundcivil engineering structures which are relevant to the work in this thesis, havebeen addressed. First the current state-of-the-art methods for the simulationof turbulent flows are reviewed as a function of their performance and theircomputational requirements. For wind flows around buildings, which have a veryhigh Reynolds number, DNS and LES are too demanding from a computationalpoint of view. The application of hybrid RANS/LES methods is very promisingas they are able to capture the large scale turbulent structures in the wake of bluffbodies at a more affordable computational cost. However, for the use in coupledfluid-structure problems, this computational cost today is still very high and forthe coupled problem in the next chapter URANS simulations will be performed.

In order to clarify the influence of the RANS turbulence models, the near-wall meshrefinement and the unsteadiness, first the steady state two-dimensional turbulentair flow around a single cylinder in the post-critical regime at a Reynolds number12.4 × 106 is computed. Within Flotran, near-wall modelling strongly influencesthe results, while the results obtained with CFX are quite insensitive. Resultscomputed with various eddy viscosity turbulence models strongly differ, but theminimum pressure coefficient is always underestimated, while the base pressurecoefficient is overestimated. For the flow around a circular cylinder at theseReynolds numbers, the Shih-Zhu-Lumley turbulence model in Flotran has thebest correspondence with experimental results. Within CFX, the shear stresstransport turbulence model produces the best correspondence. Different resultsare obtained with Flotran and CFX using the same turbulence models. A transientsimulation using the SST turbulence model captures the regular vortex sheddingin the post-critical regime and is therefore preferred.

As an application, the interference effects are studied for a group of 8 by 5cylinders, which is a simplification of the silo group in the port of Antwerp. Atwo-dimensional computation is performed which neglects the three-dimensionalcharacter of the flow around the group and the velocity gradients of theatmospheric boundary layer. Vortices shed at 0.165 Hz from the group as a wholeand at 2.85 Hz from the individual cylinders. These vortex shedding frequencies


are lower than the lowest eigenfrequency of the structure (3.93 Hz). The largevelocities in the gaps produce distinct minima in the pressure distributions. Thetwo cylinders on the transverse corners of the group experience quite high liftcoefficients. Eigenmodes with circumferential wavenumber n equal to a multipleof 4 are more heavily loaded in the group configuration, where four small gapssurround most of the cylinders, compared with the single cylinder configuration.Especially for the cylinders on the transverse corners of the group, eigenmodeswith n = 3 or n = 4, which for silos typically have the lowest eigenfrequencies, arestrongly excited. This explains why storm damage is mainly located on silos atthe corners of the group.

Chapter 5

Coupled simulation of windloading on structures

5.1 Introduction

The behaviour under wind loading of civil engineering structures that are quiteflexible, have a complex geometry or are located in the presence of neighbouringstructures is very difficult to determine. The aim of this chapter is to study couplednumerical simulations of the fluid and the structure to determine this behaviour.First an extensive and coherent overview of the state-of-the-art coupling algorithmsand load and motion transfer methods is given. As a practical application, theinteraction between the wind flow and a single silo as in the port of Antwerp iscomputed.

In the case of large deformations of structures due to fluid loads, the structuralresponse is primarily of interest. The most straightforward way would be toeliminate the fluid field and only retain the structural degrees of freedom. Becausethe Navier-Stokes equations which represent the fluid field are non-linear thisis impossible. If the structure undergoes large deformations and this non-linearbehaviour of the structure has to be incorporated, direct time integration methodswill be needed for both fields.

This non-linear transient aeroelastic problem is formulated as a three field problem:the fluid, the structure and the deforming mesh. The governing equations of thedifferent fields were derived in the previous chapters. In the fluid field, the Navier-Stokes equations (2.59- 2.76) are formulated in an Arbitrary Lagrangian Eulerian

153

154 COUPLED SIMULATION OF WIND LOADING ON STRUCTURES

description:

∂Jρf∂t

∣∣∣∣∣χ

+ J∇ · (ρfc) = 0 (5.1)

∂Jρfv

∂t

∣∣∣∣∣χ

+ J∇ · (ρfv⊗ c) = J∇ · σf + Jρfbf (5.2)

The governing equations of the structure are formulated in a Lagrangiandescription (2.79):

ρs∂2u

∂t2

∣∣∣∣X

= ∇ · σs + ρsbs (5.3)

For the deformation of the fluid mesh a diffusion equation with a variable diffusivityis used (2.233):

∇ · (k∇u) = 0 (5.4)

The computation of the fluid flow around a deforming structure is a surface coupledproblem: the fluid and the structure only interact at the fluid-structure interfaceΓ. At the interface the following boundary conditions should be satisfied: thetraction on the surface of the structure should be in equilibrium with the tractionon the surface of the fluid:

ts + tf = 0 (5.5)

σsns + σfnf = 0 (5.6)

σsns = pfnf − τfnf (5.7)

The velocity of the structure u should be equal to the fluid velocity v. At thefluid-structure interface a Lagrangian description is used in the fluid domain, sothe fluid velocity v is equal to the mesh velocity v:

u = v = v (5.8)

The displacements of the structure u are equal to the mesh displacements u:

u = u (5.9)

First the partitioned algorithms to solve the coupled problem by computing thedifferent fields sequentially or in parallel are discussed. Section 5.3 treats thetransfer of displacements and loads between non-matching discretizations of thefluid and the structure field. In the last section the coupled computation of ovallingoscillations of silos is studied.

COUPLING ALGORITHMS 155

5.2 Coupling algorithms

A number of techniques exist to solve coupled field problems with direct timeintegration. A short review with the advantages and disadvantages of all theapproaches will be given.

5.2.1 The monolithic and the partitioned approach

Generally two approaches exist to solve the coupled problem: the monolithic andthe partitioned approach. In the monolithic approach both fields are joined intoone fully coupled system and are simultaneously advanced in time. In partitionedapproaches (Felippa and Park, 1980; Felippa et al., 2001) the coupled system isspatially decomposed into partitions which coincide with the physical fields (e.g.fluid and structure). The partitions are treated as isolated entities and separatelyintegrated in time. The interaction effects are applied on the individual partitionsas external boundary conditions which are exchanged from one partition to theother.

The monolithic approach

As in the monolithic approach both fields are solved simultaneously, the couplingconditions at the fluid-structure interface are exactly satisfied. This improvesgreatly the accuracy and stability of the computations in strongly coupledproblems, where the behaviour of the coupled system differs largely from thebehaviour of the individual fields.

A first drawback is that the number of equations to be solved at the same timeis equal to the sum of the fluid and the structural degrees of freedom. Moreovera completely new implementation is required to assemble the global matrix andto solve the resulting system. The global matrix is composed of submatrices witha different topology and different numerical properties (e.g. for finite elementdiscretizations the stiffness matrix of the structure is symmetric while the stiffnessmatrix of the fluid is nonsymmetric) and might be ill-conditioned. An appropriateand optimal iterative solver is difficult to find. Although different time scalesmight be present in the individual fields, the monolithic system is advanced witha single time step which should be small enough to compute accurately and stablyboth fields.

An inherent difficulty is that the structural equations involve second order time-derivatives, whereas the fluid equations involve only first order derivatives. Takingboth the structural velocities and displacements as degrees of freedom (Hron andTurek, 2006), the second order differential equations can be recast in first order


form, but this doubles the number of degrees of freedom. Additional equationsimpose the relation between the structural velocities and displacements:

∂u

∂t= u (5.10)

The resulting coupled system of first order differential equations is integrated intime with the same method and the same time step for both fields. Hron andTurek (2006) used the Crank-Nicolson time integration method to compute thecoupled behaviour of an incompressible fluid and an incompressible solid. Theincompressible solid is described with a mixed u− p formulation. Both fields havedisplacement, velocity and pressure degrees of freedom and are discretized withQ2P1 finite elements.

Hubner et al. (2004) used for the structure a mixed formulation with the structuralvelocities and second Piola-Kirchhoff stresses as variables. This yields twopartial differential equations with first order time-derivatives. After the staticcondensation on element level of these stresses, only the structural velocitiesand the boundary tractions remain as degrees of freedom. The coupled systemis subsequently discretized with time-discontinuous stabilized space-time finiteelements for both fields.

Rugonyi and Bathe (2001) and Zhang et al. (2003) simply use the structuraldisplacements and the fluid velocities and pressures as degrees of freedom. Afterspace and time discretization, the degrees of freedom of the structure aresubdivided into displacements u

Γat the fluid-structure interface and the remaining

displacements uΩ

in the interior of the domain. Similarly, the velocity degrees offreedom of the fluid are divided into v

Γat the interface and v

Ωin the interior of

the domain. At the fluid-structure interface the velocities vΓ

are expressed as afunction of the displacements u

Γ. If the movement of the interior nodes of the fluid

is omitted for ease of notation, the discretized coupled system might be expressedas:

KfΩΩ

KfΩp

KfΩΓ

0

KfpΩ

Kfpp

KfpΓ

0

KfΓΩ

KfΓp

KfΓΓ

+ KsΓΓ

KsΓΩ

0 0 KsΩΓ

KsΩΩ

vΩ

p

uΓ

uΩ

=

ffΩ

ffp

ffΓ

+ fsΓ

fsΩ

(5.11)

This monolithic approach is implemented as well in the finite element programAdina.


The partitioned approach

In partitioned approaches (Felippa and Park, 1980; Felippa et al., 2001) the coupledsystem is spatially decomposed into partitions which coincide with the physicalfields (e.g. fluid and structure). The partitions are treated as isolated entities andseparately integrated in time. The interaction effects are applied on the individualpartitions as external boundary conditions which are exchanged from one partitionto the other.

As an example the most straightforward partitioned algorithm for fluid-structureinteraction is described. Suppose the solution of the coupled problem is knownat tn. First, the structural displacements un

Γof the fluid-structure interface are

imposed as boundary conditions on the fluid field and the fluid field is advancedto time level tn+1:

[Kf

ΩΩKf

Ωp

KfpΩ

Kfpp

] [vn+1

Ω

pn+1

]=

[f

fΩ−Kf

ΩΓun

Γ

ffp −Kf

pΓun

Γ

](5.12)

This yields the new solution of the fluid field vn+1Ω

and pn+1. The forces exertedby the fluid on the fluid-structure interface are applied to the structure and thestructure is advanced to time level tn+1:

[Ks

ΓΓKs

ΓΩ

KsΩΓ

KsΩΩ

] [un+1

Γ

un+1Ω

]=

[f

sΓ−Kf

ΓΩvn+1

Ω−Kf

Γppn+1 −Kf

ΓΓun

Γ

fsΩ

](5.13)

The structural displacements un+1Γ

of the fluid-structure interface and un+1Ω

of therest of the structure are obtained. The systems (5.12) and (5.13) computed in thisexample can be compared to the coupled system (5.11).

A first advantage of partitioned approaches is that for each field well-establisheddiscretization techniques and solution algorithms can be used which are tailored tothe different behaviour of the individual fields. Different time integration schemesaccount for the specific requirements of the different fields: an implicit or explicitmethod might be chosen; possibly different time steps are used to advance the fluidand structure, what is called subcycling. Finally different solvers can be appliedwhich take the characteristics and the conditioning of the resulting matrices intoaccount.

Already available software for the individual fields can be reused. This softwareis validated and highly adapted to the specific applications. A modularimplementation of the partitioned procedure should enable that the software usedfor one of the fields is easily replaced by another program with different capabilities.Adaptations to the models and techniques applied in one of the partitions can beimplemented and validated in the respective software independent of the couplingprocedure and the other fields.


The main disadvantage of partitioned approaches is that the coupling conditions(5.8) might not exactly be fulfilled. In the example the structural displacements un

Γ

of the fluid-structure interface applied to obtain the fluid solution at the new timelevel tn+1 differ from the the structural displacements un+1

Γof the fluid-structure

interface at this time level. Generally, the violation of the coupling conditionsmight deteriorate the stability and the accuracy of the coupled computation. Thechallenge is to design efficient partitioned procedures which do not degrade thestability and the accuracy of the methods applied in the individual partitions.

The computation of the solutions in the individual partitions, which are of smallersize than the coupled problem, requires less computational time and memory.The efficiency of the coupling procedure will determine if overall the partitionedprocedure is more efficient than the monolithic approach.

In the remainder of this section the partitioned algorithms are further elaborated.First loosely coupled and then strongly coupled algorithms are treated.

5.2.2 Loosely coupled algorithms

In loosely coupled algorithms the coupling conditions (5.8) at the fluid-structureinterface are not exactly fulfilled. Each field is solved one or a few times in eachtime step.

Method 1: conventional serial staggered algorithm

The conventional serial staggered algorithm (figure 5.1) was described in theprevious section. It consists of the following steps:

1. Transfer the structural displacements unΓ

of the fluid-structure interface tothe fluid mesh

2. Compute the mesh deformation at time tn+1

3. Transfer the mesh position xn+1 to the fluid

4. Advance the fluid field to time level tn+1

5. Transfer the fluid forces fn+1

fΓon the fluid-structure interface to the structure

6. Advance the structure to time level tn+1

In this algorithm each field is solved once per time step step. The boundaryconditions are exchanged once per time step. Because the two fields are solvedsequentially, this is called a staggered method.


un un+1

vn vn+1

xn

xn+1

Mesh

Fluid

Structure

∆t

1

2

3

4

5

6

Figure 5.1: The serial staggered algorithm.

The displacements at the fluid-structure interface at tn are used to computethe fluid flow at tn+1. This introduces explicitness in the coupling procedure,irrespective of the fact that the individual fields are integrated in time with anexplicit or implicit scheme. Consequently, a restriction on the time step is expected,even if both individual fields are integrated in time with an unconditionally stablemethod.

As the displacements imposed on the fluid field at tn+1 differ from thedisplacements computed subsequently in the structure, the kinematical continuityis not fulfilled. Consequently the conservation of mass, impulse and energy isnot guaranteed at the interface. This lack of conservation at the fluid-structureinterface causes a loss in time-accuracy and numerical stability. Even if second-order accurate time integration schemes are used for the fluid and the structure,the coupled computation is only first-order time-accurate. As the objective is tocompute physical instabilities as flutter, galloping and ovalling, the stability ofthe coupled procedure is very important as weak numerical instabilities might beconfused with physical instabilities.

The above described approach is called Dirichlet-Neumann coupling: on onepartition Dirichlet boundary conditions are imposed, while on the other partitionNeumann boundary conditions are applied. Depending on the order in which thepartitions are solved and on which boundary condition is imposed to which field,four different coupling procedures exist.

Commonly, the fluid field is chosen as Dirichlet partition and has prescribedvelocities at the fluid-structure interface, while the structure is the Neumannpartition with imposed traction on the interface. The fluid is solved first andthe structure secondly.


Method 2: enhanced serial staggered algorithms

A first way to improve the conventional serial staggered algorithm is to transfer aprediction un+1

Γp of the structural displacements at tn+1 to the fluid mesh in thefirst step of the algorithm:

un+1Γp = un

Γ+ α0∆tun

Γ+ α1∆t(un

Γ− un−1

Γ) (5.14)

The trivial prediction, which is equal to the value computed at tn, is recoveredwith α0 = α1 = 0. A first order accurate prediction is obtained with α0 = 1 andα1 = 0 and a second order accurate with α0 = 1 and α1 = 1

2 .

Next, a corrected fluid force fn+1

fΓcis transferred to the structure in the fifth step

of the algorithm. This corrected value is for instance chosen so as to minimize theamount of momentum and energy that is artificially created at the fluid-structureinterface (Piperno and Farhat, 2001) or as to obtain a global order of time-accuracy(Farhat et al., 2006).

If the trapezoidal rule is used for the structure, the momentum artificially createdby the staggering at the interface is (Piperno, 1997):

∆Ms + ∆Mf =

(fn

fΓc+ fn+1

fΓc

2− fn+1

fΓ

)∆t (5.15)

fn+1

fΓdepends on the time integrator that is used for the fluid field. For the

forward Euler, the backward Euler and second order time-accurate methods fn+1

fΓ

is respectively equal to:

fn+1

fΓ= fn

fΓ(5.16)

fn+1

fΓ= fn+1

fΓ(5.17)

fn+1

fΓ≈fn

fΓ+ fn+1

fΓ

2(5.18)

From equation (5.15) follows that conservation of momentum is satisfied at theinterface if:

fn+1

fΓc= 2f

n+1

fΓ− fn

fΓc(5.19)

If a second order time-accurate method (5.18) is used in the fluid field, this reducesto:

fn+1

fΓc= fn+1

fΓ(5.20)


Another choice for the corrected fluid force fn+1

fΓcis to transfer the forces computed

at tn:

fn+1

fΓc= fn

fΓ(5.21)

This enables to compute the fluid and the structure in parallel. A last possibilityis to transfer the average of the forces computed at tn and tn+1:

fn+1

fΓc=fn+1

fΓ+ fn

fΓ

2(5.22)

The loss of numerical stability and accuracy of loosely coupled methods is relatedto the conservation of energy at the fluid-structure interface. If the trapezoidal ruleis used for the structure, the energy that is artificially created by the staggeringat the fluid-structure interface is (Piperno and Farhat, 2001):

∆Es + ∆Ef = (un+1Γ− un

Γ)Tfn

sΓc+ fn+1

sΓc

2− (un+1

Γp − unΓp)Tf

n+1

fΓ(5.23)

where un+1Γp are the predicted displacements after transfer from the structure to the

possibly non-matching fluid mesh and fnsΓc

are the corrected forces after transferfrom the fluid to the possibly non-matching structure mesh. As the predicteddisplacements un+1

Γp differ from the computed displacements un+1Γ

which are not

yet known at the moment the corrected force fn+1

fΓcis transferred, generally the

energy cannot be conserved at the fluid-structure interface. The predictor andcorrector can be adjusted in order to reduce the imbalance as much as possible.

Therefore Piperno and Farhat (2001) studied the accuracy of the conservationof energy at the fluid-structure interface for the simplified case of a structurevibrating at a certain frequency with a constant amplitude. For the conventionalserial staggered algorithm (un+1

Γp = unΓ

and fn+1

fΓc= fn+1

fΓ), the conservation of

energy is first order accurate independent of the time integration scheme which isused in the fluid field. With the first order predictor the conservation of energyof several algorithms becomes second order accurate, among others all algorithmswhere the corrected fluid force is given by (5.19) and which thus conserve themomentum at the fluid-structure interface. With the second order predictor theconservation of energy of these momentum-conserving algorithms becomes third-order accurate.

Farhat et al. (2006) developed a loosely-coupled globally second order time-accurate algorithm. Therefore, they combined the second order time-accuratethree-point backward difference scheme (2.226) which satisfies the DGCL, withthe second order predictor (5.14) and the latest computed fluid force (5.20) ascorrector. This approach conserves the momentum at the fluid-structure interfaceand the conservation of energy at the interface is third-order accurate. Finally the


mesh deformation algorithm should be adapted: the common approach which usesthe stiffness matrix corresponding to the predicted displacements un

Γp to compute

the predicted displacements un+1Γp , can be viewed as forward Euler time integration

of the mesh motion equations (2.230). In order to obtain a globally second ordertime-accurate algorithm, the stiffness matrices of the mesh at both the time levelstn and tn−1 are needed to compute the predicted displacements un+1

Γp .

Method 3: non-collocated algorithms

At the fluid-structure interface the displacements (5.9) and the velocities (5.8) ofthe fluid and the structure should be equal. The above partitioned procedurestry to enforce displacement continuity by transferring the predicted structuraldisplacements to the fluid mesh. Suppose that the displacement continuity isfulfilled. In the fluid the mesh velocity is computed from the mesh positions basedon equation (2.215) in order to satisfy the geometric conservation law (section2.4.2). If the structure is advanced with a second order accurate time integrationprocedure, the structural velocity usΓ

differs from the mesh velocity and velocitycontinuity will be violated. This violation results in an error in the exchange ofkinetic energy between the fluid and the structure.

Therefore Lesoinne and Farhat (1996) and Farhat and Lesoinne (2000) proposed anon-collocated algorithm (figure 5.2) which satisfies both displacement and velocitycontinuity and the GCL. In the above algorithms the solutions for the fluid andthe structure are computed at the same time levels. In the non-collocated (orasynchronous) algorithm, the fluid field is computed at half time levels tn+ 1

2 ,while the structure is computed at full time levels tn+1: the fluid and structurecomputations are offset by half a time step. The structure is advanced using themidpoint rule.

1. Predict the structural displacements un+ 1

2Γp of the fluid-structure interface

based on the displacements unΓ

and transfer them to the fluid mesh

2. Compute the mesh deformation at time tn+ 12

3. Transfer the mesh position xn+ 12 and mesh velocities vn to the fluid

4. Advance the fluid field to time level tn+ 12

5. Transfer the fluid forces fn+ 12

fΓon the fluid-structure interface to the structure


The first order accurate predicted structural displacements un+ 1

2Γp are given by:

un+ 1

2Γp = un

Γ+

∆t

2un

Γ(5.24)


un un+1

vn−1/2 vn+1/2

xn−1/2 x

n+1/2

Mesh

Fluid

Structure

∆t

1

2

3

4

5

6

Figure 5.2: Non-collocated algorithm.

If the structure is advanced using the midpoint rule, this predictor automaticallyimplies the velocity of the fluid nodes vn

Γat the fluid-structure interface

corresponds to the velocity of the structure unΓ. If the trapezoidal rule is used

for the structure, the same result is obtained by transferring a corrected fluidforce fn+1

fΓcto the structure in step 5:

fn+1

fΓc= 2fn+ 1

2

fΓ− fn

fΓc(5.25)

For the trapezoidal rule the energy that is artificially created by the staggering atthe fluid-structure interface is (Piperno and Farhat, 2001):

∆Es + ∆Ef = (un+1Γ− un

Γ)Tfn

sΓc+ fn+1

sΓc

2(5.26)

− 1

2(un+ 1

2sΓp − un−

12

sΓp )Tfn+ 1

2

fΓ− 1

2(un+ 3

2sΓp − un+ 1

2sΓp )Tf

n+ 32

fΓ(5.27)

where fn+ 1

2

fΓdepends on the time integrator that is used for the fluid field and is

analogous to equations (5.17) to (5.18).

As for the enhanced serial staggered algorithms, the accuracy of the conservation ofenergy at the fluid-structure interface was studied by Piperno and Farhat (2001) forthe simplified case of a structure vibrating at a certain frequency with a constantamplitude. For the corrected fluid force (5.25) alternatives are analysed, whichare given by the following expressions:

fn+1

fΓc= f

n+ 12

fΓ(5.28)

fn+1

fΓc= 2f

n+ 12

fΓ− fn

fΓc(5.29)


For most algorithms the conservation of energy is first order accurate, althoughfor some it is second order accurate. The combination of the corrected fluid force(5.25) with a second order accurate time integration in the fluid field yields theonly algorithm where the conservation of energy is third-order accurate.

Based on this algorithm Farhat et al. (2006) developed a loosely-coupled globallysecond order time-accurate algorithm. It consists of the combination of the secondorder time-accurate three-point backward difference scheme (2.226) which satisfiesthe DGCL for the fluid, the midpoint rule for the structure, the latest computed

fluid force fn+ 12

fΓas corrector and a second order accurate predictor:

un+ 1

2Γp = un

Γ+

∆t

2un

Γ+

∆t

8

(un

Γ− un−1

Γ

)(5.30)

The stiffness matrix of the mesh at time level tn should be used to compute the

predicted displacements un+ 12

p . As the mesh is unknown at this time level, themesh deformation problem becomes non-linear. Farhat et al. (2006) linearizedthese equations around tn−

12 .

For aeroelastic problems in compressible flows, these improved algorithms(enhanced serial staggered and non-collocated algorithms) enable accurate andstable computations with time steps that are 5 to 10 times larger than the time stepused in the conventional staggered procedure. These time steps are comparable toones used in monolithic algorithms with implicit time integration schemes in bothfluid and structure.

Method 4: parallel loosely-coupled algorithms

The basic parallel algorithm (figure 5.3) sends the displacements unΓ

to the fluidand the fluid forces fn

fΓto the structure and then advances both to tn+1 in parallel.

Only at the beginning of the time step the coupling conditions are exchanged. Thisalgorithm is first order time accurate, but half as accurate as the conventional serialstaggered algorithm. It requires small time steps to be stable and sufficientlyaccurate.

An enhanced parallel algorithm (figure 5.4) which has an increased time-accuracyis proposed by (Piperno et al., 1995; Farhat and Lesoinne, 2000). In this algorithmthe boundary conditions are exchanged twice per time step: first the displacementsun

Γare sent to the fluid and the fluid forces fn

fΓto the structure. The fluid is

advanced to tn+ 12 and the structure to tn+1. The structural displacement un+1

Γ

is used to advance the fluid from tn+ 12 to tn+1. The fluid force fn+ 1

2

fΓis used to

advance the structure again from tn to to tn+1. Each field is solved twice per timestep.


un un+1

vn vn+1

xn

xn+1

Mesh

Fluid

Structure∆t

11

2

3

4

4

Figure 5.3: The parallel algorithm.

un un+1

vn vn+1/2 vn+1

xn

xn+1/2 x

n+1

Mesh

Fluid

Structure∆t

1

1

2

3

4

4

55

6

7

8

8

Figure 5.4: The enhanced parallel algorithm.

Method 5: subcycling

The fluid and structure are often characterized by different time scales. Usuallythe fluid requires smaller time steps than the structure. This is especially the caseif an explicit time integration procedure is used for the fluid, while an implicitmethod is applied to the structure. The subcycling factor m is defined as the ratioof the time step ∆ts in the structure and the time step ∆tf in the fluid:

m =∆ts∆tf

(5.31)

Piperno et al. (1995) and Piperno (1997) presented loosely coupled algorithmswith subcycling of the fluid (figure 5.5). Starting from the second order accuratepredicted structural displacements at tn+1 (5.14) the grid deformation xn+1 iscomputed. During all the fluid time steps within one coupling time step, the mesh


velocity is assumed constant and equal to:

v =xn+1 − xn

∆ts(5.32)

For each fluid time step the grid positions are linearly interpolated between xn andxn+1. Analogously to equation (5.19) conservation of momentum at the interface

is satisfied if the corrected fluid force is given by:

fn+1

fΓc=

2

∆ts

∫ tn+1

tnf

fΓdt− fn

fΓc(5.33)

Piperno and Farhat (2001) showed by means of a simplified case that theconservation of energy of this algorithm is third-order accurate.

un un+1

vk vk+5

xn

xn+1

Mesh

Fluid

Structure

∆ts

∆tf

1

2

3

4

5

6

Figure 5.5: Staggered algorithm with subcycling of the fluid.

Added-mass effect

Farhat et al. (2006) proved the second-order time accuracy of two loosely coupledalgorithms. The numerical stability these algorithms is not formally proved, but inthe case of aeroelastic computations with compressible flows all numerical resultssuggest a very stable behaviour.

However when dealing with incompressible flows, different authors report stabilityproblems. The more time accurate the coupling algorithm is, the sooner theinstability manifests itself. Surprisingly decreasing the time step does not suppressor postpone the instability, on the contrary the instability occurs earlier.

These instabilities are related to the added-mass effect and were studied byLe Tallec and Mouro (2001), Causin et al. (2005) and Forster et al. (2007). As the


instability usually occurs in the first time steps, the non-linear (convective) termin the fluid equations can be neglected. Because the instability occurs even whenvery small time steps are used, the fluid stiffness (viscosity) can be neglected aswell. The resulting fluid equations only consist of an acceleration and a pressure-gradient term. If these equations are condensed on the fluid-structure interface,the following interface equation is obtained:

fn+1

fΓ= mfMAvΓ

(5.34)

where mf is a characteristic fluid mass, for instance the nodal mass of a lumpedmass matrix. The added mass operatorMA is a purely geometrical quantity. Forstabilized finite element discretizations the stabilization makes the added massoperatorMA dependent on the time step: its largest eigenvalue max µi increaseswith decreasing time steps (Forster et al., 2007).

As the instability occurs even when very small time steps are used, the structuralstiffness can be neglected. The fluid force fn+1

fΓis imposed on the structure. The

fluid acceleration vΓ

at the interface is expressed as a function of the predictedstructural displacements. After the eigenvalue decomposition of the added massoperator, an stability condition is derived from the structural equations:

mf

msmaxµi < Cstab (5.35)

where ms is the structural nodal mass of a lumped mass matrix. max µi is thelargest eigenvalue of the added mass operatorMA. The stability constant Cstab

depends on the accuracy of the coupling algorithm: the accuracy of the fluid timeintegration and the accuracy of the structural predictor. The constant decreaseswith increasing time-accuracy. As the largest eigenvalue max µi increases withdecreasing time steps, the instabilities cannot be suppressed by decreasing thetime steps.

If the fluid acceleration vn+1Γ

depends on the fluid acceleration vnΓ

at the previoustime level, as for one-step θ methods (2.174) with θ 6= 1 (e.g. the Crank-Nicolsonscheme), the stability condition is even more restrictive:

mf

msmaxµi <

Cstab

n(5.36)

where n is the number of the time step. The condition becomes every time stepmore restrictive and instabilities will occur from a certain time step irrespectivelyof the values of the other parameters.

The factor mf

msshows that the mass density ratio ρf/ρs is a determining factor

in the stability of loosely coupled partitioned procedures. Every loosely coupledpartitioned procedure has a limiting mass density ratio beyond which it becomesunstable.


Table 5.1 gives the density ratio ρs/ρf between some typical civil engineeringmaterials and air or water. These density ratios indicate that stable loosely coupledsimulations for civil engineering structures in contact with water are not possible.The interaction between structures and air might be computed with a looselycoupled algorithm for the materials with the highest density, although it dependson the time accuracy of the simulations and it is not possible to precisely predictthe limiting density ratio.

air water1.25 1000

steel 7850 6280 7.85aluminium 2700 2160 2.70concrete 2500 2000 2.50

wood 290-900 232-720 0.29-0.90tensile 80 64 0.08

Table 5.1: Density ratios ρs/ρf between some typical civil engineering materialsand air or water.

For larger time steps, the fluid viscosity and the structural stiffness cannot beneglected. Generally, increasing the viscosity destabilizes the computations, whileincreasing the structural stiffness slightly improves the stability. For very stiffstructures, the density ratio does not influence the stability anymore.

The above conclusions are valid in the case the fluid partition is solved first withimposed Dirichlet boundary conditions and the structure is computed secondlywith Neumann boundary conditions. Matthies et al. (2006) studied the three othercombinations of boundary conditions and order in which the fields are solved. Ifthe boundary conditions are kept the same, but the structure is solved first andthe fluid secondly, still for stability the density ratio ρf/ρs should be small enough.However if the structure is computed with imposed displacements at the fluid-structure interface and the fluid with imposed stresses, the density ratio ρs/ρfshould be small enough, no matter in which sequence they are solved.

The stability problems are related with the equations used for incompressible flows.Farhat (2006) computed the numerical examples of Causin et al. (2005) using thediscretized equations for compressible flows and did not encounter any stabilityproblems in this case.


5.2.3 Strongly coupled algorithms

In strongly coupled algorithms each field is solved multiple times per time step inorder to fulfill within a certain tolerance the coupling conditions (5.8) at the fluid-structure interface. This improves the accuracy and the numerical stability of thecoupled computations. The solution obtained with a strongly coupled algorithmcorresponds to the solution obtained with a monolithic algorithm.

Method 1: Iteratively staggered algorithms

A first approach is to introduce iterations between the two partitions within onetime step. The first solution for the fluid and the structure at time level tn+1

is obtained by means of a loosely-coupled algorithm. The newly computed valuefor the displacement is subsequently transferred to the fluid domain in order tocompute the solution for the same time step again. The obtained fluid forcesare transferred to the structure and the structure is computed again. Thisprocedure is repeated until the change of the displacements or the forces atthe fluid-structure interface is smaller than a certain tolerance. These iterationsare called subiterations, (Dirichlet-Neumann) inner iterations, predictor-correctoriterations or interfield iterations. If these subiterations converge, they converge tothe solution obtained with the monolithic approach. At convergence, the kinematicand dynamic continuity at the fluid-structure interface are almost exactly fulfilled.Consequently the conservation of mass, momentum and energy at the interface issatisfied.

This method is sometimes called the block Gauss-Seidel approach in analogy tothe Gauss-Seidel method used to solve iteratively systems of equations. It usesnew interface results as soon as they are available and is a serial algorithm. Theconvergence depends on the order in which the fields are solved.

By analogy to iterative Dirichlet-Neumann substructure methods or non-stationary Richardson iteration methods for solving systems of equations, arelaxation parameter ωn+1(i) can be introduced for the interface displacements:

un+1(i+1)Γ = ωn+1(i)u

n+1(i+1)Γ

+ (1− ωn+1(i))un+1(i)Γ (5.37)

= un+1(i)Γ + ωn+1(i)(un+1(i+1)

Γ− un+1(i)

Γ ) (5.38)

The interface displacements un+1(i+1)Γ that will be transferred in the next

iteration to the fluid, are a linear combination of the newly computed interface

displacements un+1(i+1)Γ

in the structure and the interface displacements un+1(i)Γ

transferred to the fluid at the beginning of the iteration. The relaxation of interfacedisplacements should guaranty and accelerate the convergence of the subiterations.The iteratively staggered algorithm consists of the following steps:


1. Predict the structural displacements un+1(0)Γ of the fluid-structure interface

at time level tn+1

2. Set i = 0

3. Transfer the structural displacements un+1(i)Γ of the fluid-structure interface

to the fluid mesh

4. Compute the mesh deformation at time tn+1

5. Transfer the mesh position xn+1(i) to the fluid

6. Advance the fluid field to time level tn+1

7. Transfer the fluid forces fn+1(i+1)

fΓon the fluid-structure interface to the

structure


9. Relax ωn+1(i) the interface displacements un+1(i+1)Γ

10. Check the convergence for the displacements and the forces. If theconvergence criteria are fulfilled, stop. Otherwise set i = i + 1 and go tostep 3.

Figure 5.6 shows these steps for an iteration i.

un un+1(i)

vn vn+1(i)

xn

xn+1(i)

Mesh

Fluid

Structure

∆t

7

8

3

4

5

6

Figure 5.6: The iteratively staggered algorithm.

The convergence of the subiterations is reached if:

‖φn+1(i+1)Γ − φ

n+1(i)Γ ‖L2

‖φn+1(i+1)Γ ‖L2

< εφ (5.39)

where φΓ are the interface displacements or forces.


This iteratively staggered approach with a fixed relaxation factor is available inmost of the commercial packages e.g. Flotran, CFX and Adina.

The choice of the value of the relaxation factor is crucial but problematic: it isoften obtained with trial and error or based on empirical formulas. An optimalrelaxation factor results in a minimal number of subiterations.

The added mass effect, which causes the stability problems of loosely coupledalgorithms, also influences the convergence of the subiterations for incompressibleflows. Generally, problems that are more likely to become unstable using looselycoupled algorithms require a lower relaxation factor which results in higher numberof subiterations (Mok, 2001; Causin et al., 2005; Heck et al., 2006): in order toobtain convergence the relaxation factor ωn+1(i) should decrease if the densityratio ρf/ρs is increased, the time step is decreased or the stiffness of the structureis reduced. In addition the problems that are more likely to become unstable usingloosely coupled algorithms, are more sensitive to the value of the relaxation factorωn+1(i): a small variation away from the optimal value of the relaxation factorincreases the number of subiterations drastically or even leads to divergence of theinterfield iterations.

Furthermore the relaxation factor which is optimal for the first subiterations andtime steps, is not necessarily optimal during the following subiterations and timesteps due to the non-linear behaviour of the coupled problem.

A solution is to automatically calculate the relaxation factors. Le Tallec and Mouro(2001) used the steepest descent method to determine the relaxation factors. Thestructure and fluid field have to be computed once with as only loading the residualinterface displacements in order to determine the relaxation parameter. As thecomputational cost is quite high, it is recommended to update the relaxationparameters only after a few iterations or time steps.

Mok et al. (2001) and Mok (2001) applied the Aitkin method, originally employedto accelerate modified Newton-Raphson methods for non-linear computations, tothe iteratively coupled FSI problem in order to obtain an optimal relaxation factorωn+1(i) for each iteration i:

ωn+1(i) = ωn+1(i−1)

1−

(∆un+1(i)Γ −∆u

n+1(i+1)Γ

)T

·∆un+1(i+1)Γ

(∆un+1(i)Γ −∆u

n+1(i+1)Γ

)2

(5.40)

where

∆un+1(i)Γ = ∆u

n+1(i−1)Γ −∆un+1(i)

Γ(5.41)

For the first iteration of the first time step the relaxation factor ω1(0) has to besupplied based on experience. The relaxation factors for the first iteration of thefollowing time steps ωn+1(0) is equal to the last relaxation factor of the previous


time step ωn(imax). The extra computational cost of the Aitkin method is almostnegligible.

As mentioned before, the convergence depends on the order in which the fieldsare solved and on which boundary condition is imposed to which field. Hubneret al. (2007) computed the structure first with imposed forces and achieved thebest convergence by relaxing the transferred forces instead of the displacements.Causin et al. (2005) showed that if the structure is computed first with imposeddisplacements at the fluid-structure interface and the fluid next with imposedstresses, the relaxation factor should go to zero if the mesh is refined, which limitsits practical use.

Often this Aitkin method accelerates the convergence very well and lesssubiterations are needed. No trial and error or based on empirical formulas todetermine a relaxation factor which converges and needs a minimal number ofsubiterations.

However, for strongly coupled problems, even with the Aitkin convergenceacceleration the required number of subiterations to obtain convergence maybecome very high and make the computations inefficient (Vierendeels, 2006).Therefore alternative strongly coupled partitioned methods were developed.

Method 2: Newton’s method

At time level tn+1 the iterative solvers F and S compute the new value of thevariables for respectively the fluid and the solid subproblems:

v(i+1) = F(v(i),u) (5.42)

u(i+1) = S(u(i),v) (5.43)

These solvers are in fixed-point form. The residuals associated with each solverare:

Rf(v(i),u(i)) = v(i) −F(v(i),u(i)) (5.44)

Rs(u(i),v(i)) = u(i) − S(u(i),v(i)) (5.45)

Upon convergence these residuals are equal to 0. The block Newton method triesto find the root of the residual:

[DvRf(v

(i),u(i)) DuRf(v(i),u(i))

DvRs(u(i),v(i)) DuRs(u

(i),v(i))

] [∆v(i)

∆u(i)

]= −

[Rf(v

(i),u(i))Rs(u

(i),v(i))

](5.46)

where ∆v(i) = v(i+1) − v(i) and ∆u(i) = u(i+1) − u(i). The first matrix on theleft hand side consists of the Jacobians of the residuals with respect to the fluid


and the structural degrees of freedom. The cross Jacobians DuRf and DvRs

are a measure of the sensitivity of respectively the fluid solution with respect tostructural motions and the structural solution with respect to changes in the fluiddomain. Matthies and Steindorf (2002, 2003) eliminated ∆v(i) from the systemof equations (5.46) and solved the resulting system with an iterative method(e.g. GMRES, Bi-CGStab). The cross Jacobians are approximated by finitedifferences. Fernandez and Moubachir (2005) derived the exact expressions forthe cross Jacobians of the linearized subproblems.

If for simplicity the pressure degrees of freedom p are included in vΩ, the discretized

coupled system (5.11) can be simplified to:

KfΩΩ

KfΩΓ

0

KfΓΩ

KfΓΓ

+ KsΓΓ

KsΓΩ

0 KsΩΓ

KsΩΩ

v

Ω

uΓ

uΩ

=

f

fΩ

ffΓ

+ fsΓ

fsΩ

(5.47)

If this system is condensed on the fluid-structure interface, the following interfaceequation is obtained:

(Sf + Ss)uΓ= f

fΓ+ f

sΓ−Kf

ΓΩKf−1

ΩΩf

fΩ−Ks

ΓΩKs−1

ΩΩf

sΩ(5.48)

where Sf and Ss are respectively the Schur complement for the fluid and thestructure field:

Sf = KfΓΓ−Kf

ΓΩKf−1

ΩΩKf

ΩΓSs = Ks

ΓΓ−Ks

ΓΩKs−1

ΩΩKs

ΩΓ(5.49)

Starting from the interface equation (5.48), the fluid-structure interaction problemcan be formulated as a fixed-point problem (or Picard iteration) for the interfacedisplacement u

Γ:

u(i+1)Γ = S−1

s (−Sfu(i)Γ + f

fΓ+ f

sΓ−Kf

ΓΩKf−1

ΩΩf

fΩ−Ks

ΓΩKs−1

ΩΩf

sΩ) (5.50)

As an alternative to the above described Newton’s method, the residual of thestructural degrees of freedom at the interface can be used:

RΓ(u(i)Γ ) = u

(i)Γ −S−1

s (−Sf(u(i)Γ )+f

fΓ+f

sΓ−Kf

ΓΩKf−1

ΩΩf

fΩ−Ks

ΓΩKs−1

ΩΩf

sΩ) (5.51)

The application of Newton’s method yields:

DuΓRΓ(u

(i)Γ )∆u

(i)Γ = −RΓ(u

(i)Γ ) (5.52)

where ∆u(i)Γ = u

(i+1)Γ − u(i)

Γ . Gerbeau and Vidrascu (2003) solved this systemwith the iterative GMRES method. In order to approximate the Jacobian asimplified linear and inviscid model for the fluid is used. Deparis et al. (2006)applied domain decomposition techniques to the interface equation (5.48) whichenables the parallel computation of both fields.


Generally, these Newton’s methods need less iterations per time step than thefixed-point methods, but the iterations are computationally more expensive. Onlyfor strongly coupled problems, which require lots of iterations even with Aitkinacceleration technique, Newton’s method will outperform the fixed-point methods.

5.3 Load and motion transfer

When the fluid and structure domain are discretized, two options exist. If thediscretizations of the fluid and the structure are the same at the fluid-structureinterface, the meshes are called matching or conforming. In this case the transferof displacements and forces becomes straightforward as the nodes of both meshescoincide.

However, very often the mesh requirements at the fluid-structure interface differfor the fluid and the structure. Typically the fluid mesh has to be finer than thestructure mesh. The use of non-matching meshes offers the flexibility to designand refine independently the meshes of the fluid and the structure. The elementsof the structure mesh might differ from those of the fluid mesh in shape (e.g.hexahedrons versus tetrahedrons) or in order.

The drawback is that two different discrete representations of the fluid-structureinterface exist. If the fluid-structure interface is not piecewise planar, the nodesof one mesh might lay inside or outside the second mesh or accidentally on itssurface. This complicates the load and motion transfer, which will be treated inthe following sections.

If no-slip boundary conditions (5.7)-(5.9) are applied at the fluid-structureinterface and the connectivity between the nodes and the elements of both themeshes of the fluid and the structure is kept during the computations, the matricesfor load and displacement transfer have to be computed only once at the beginningof the calculation. When the fields are remeshed or when the meshes are adaptivelyrefined during the computations, the load and displacement transfer has to berecalculated.

First the methods for associating points and surfaces of one mesh with points orsurfaces of the other mesh are reviewed.

Point to point mapping

A first method to transfer data between two non-matching meshes, searches fora point of the first mesh the point of the second mesh closest to this point. Thepoints might be nodes or a Gauss integration points.

LOAD AND MOTION TRANSFER 175

A brute force search loops for each point of the first mesh over all the points on theinterface of the other mesh in order to find the best point to be mapped to. As abrute force search is computationally inefficient for problems with a large numberof points, more efficient search algorithms have been developed.

The bucket search method (Asano et al., 1985) divides the space around the fluid-structure interface into a number of buckets (Cartesian boxes). First all points ofthe second mesh are located in one or more buckets. Next the point of the firstmesh is located in a bucket. The loop over the candidate elements is now restrictedto the elements within that bucket. While the bucket search method uses equalsized Cartesian boxes, the octree search method (Knuth, 1998) uses adaptivelyrefined Cartesian boxes and is able to account for large differences in element sizewithin one of the meshes.

Point to element mapping

The second approach tries to find for a point of the first mesh the element of thesecond mesh closest to this point. The best element should fulfill two matchingcriteria.

First, the projection of the point onto the element should lay within the element.If the elements of the second mesh form a concave ridge, a point can be sometimesmapped onto more than one element. The element which has the smallest normaldistance is selected. If the elements of the second mesh form a convex ridge orif the edges of the fluid-structure interface are misaligned, the projection of thepoint may not lay within any element. In these cases the point should be mappedonto the closest element edge.

Next, the normal distance between the point and the selected element shouldbe smaller than a specified tolerance. In Ansys (Ansys, 2005c) this tolerance isa fraction (e.g. 10−6) of the largest dimension of the Cartesian bounding boxaround the specific fluid-structure interface. Lohner et al. (1995) uses a tolerancethat varies for the different elements on the interface and is equal to five percentof the square root of the area of the element.

In this case, a brute force search loops for each point of the first mesh over all theelements on the interface of the other mesh in order to find the best element to bemapped onto. The more efficient bucket search and octree search methods can beused as well.

Lohner (1995) developed for this specific case the vectorized advancing frontneighbour-to-neighbour algorithm. If for a point an element in the vicinity of thispoint is already known and the neighbouring elements of this element are known,it is possible to go, based on the matching criteria, from element to element untilthe best element is found. If the best element is known for a point, it is very easy


to find, using this technique, the best element for the neighbouring points . Thedrawback is that a detailed knowledge of the topologies of both meshes is required.

By default the more efficient search algorithms should be used. If these fail, therecan be switched to more robust algorithms like the brute force search.

Element to element mapping

The third method projects all elements of the second mesh onto one element ofthe first mesh. The ratio of the overlapped area of the element of the first meshand an element of the second mesh to the area of the element of the first meshdetermines the weight used during the transfer.

In Ansys the element of the first mesh is first divided into as many faces as theelement has nodes. The overlapped areas are not computed exactly, but obtainedby converting all areas in pixel images which have a default resolution of 100 by100 pixels.

5.3.1 Motion transfer or surface tracking

On the fluid-structure interface Γ equation (5.9) should be fulfilled. Thecompatibility between the discretized displacement fields of fluid and structureon Γ can generally be expressed as:

uΓ

= HuΓ

(5.53)

where uΓ

is the vector with the displacements of the nodes of the fluid meshat the fluid-structure interface and u

Γis the vector with the displacements of

the nodes of the structure mesh at the fluid-structure interface. The matrix H

depends on the method used for transferring the displacements. Alternatively, thefluid displacements of the node j can be expressed as a function of the structuraldisplacements of the node i:

uΓj

=

is∑

i=1

HjiuΓi(5.54)

Nearest neighbour interpolation

The nearest neighbour interpolation uses the point to point mapping. For eachnode of the fluid mesh the closest node of the structure mesh is determined. Thecomputed structural displacements at this node are transferred to the fluid node.This method results in a matrix H with a single one on each row. It only workswell if the nodes of both meshes are almost coincident.


Consistent interpolation method

The consistent interpolation method is based on a point to element mapping: eachnode of the fluid mesh is mapped onto one element of the mesh of the structure.The value of the displacement at the projection of the fluid node onto the structuralelement is interpolated using the shape functions Ns

i of the structure mesh.

uΓj

= u(xj) =

is∑

i=1

Nsi(xj)uΓi

(5.55)

where is is the number of nodes of the structure at the fluid-structure interface andxj the coordinates of the projection of the fluid node onto the structural element.Comparison of equation 5.55 and 5.53 gives:

Hji = Nsi(xj) (5.56)

In this case the fluid nodes, which receive the data, are projected onto the elementsof the structure which send the data.

Figure 5.7 shows the displacement profiles in the fluid that correspond to anarbitrarily created displacement profile in the structure using the consistentinterpolation. If the mesh of the fluid is finer than the mesh of the structure(figure 5.7a), this method preserves the displacement profile. If the mesh of thefluid is coarser than the mesh of the structure (figure 5.7b), the number of nodesof the fluid mesh is insufficient to accurately capture the displacement profile.

(a) (b)

Figure 5.7: Meshes and displacement profiles at the fluid-structure interface in thestructure ( ) and the fluid partition ( ) obtained with the consistentinterpolation method when fluid mesh is (a) finer and (b) coarser thanthe solid mesh.


Quadratic interpolation

If the fluid mesh is finer than the structure mesh and linear elements are usedfor the structure, a consistent interpolation would yield a piecewise planar fluiddomain. Using the displacements of the nodes as well as the normals defined inthe nodes of one element of the structure, a quadratic interpolation yields thedisplacements in the fluid nodes (Lohner et al., 1995). The normal defined in anode is the average of the normals of the surrounding elements.

Incompatible dimensionality of the fluid and the structure mesh

In several cases the fluid mesh and the structure mesh have a differentdimensionality: usually the fluid mesh has a higher dimensionality than thestructure mesh e.g. a plate model of a bridge, a beam model of a chimney or a shellmodel of thick wall are coupled to a three-dimensional mesh for the flow aroundit. In these cases the discretizations of the fluid and the structure are separated inspace. The initial distances between the fluid nodes and the structure mesh have tobe computed at the beginning of the computation. During the computation theseinitial distance vectors should translated and rotated together with the structureelements. The easiest approach is to translate and rotate the initial distancevectors together with the element normals of the structure. A more advancedapproach uses for each fluid grid point a normal that is interpolated betweenthe normals defined in the nodes of the element of the structure. This yields asmoother deformed mesh (Cebral and Lohner, 1997b).

Element to element mapping

If an element to element mapping is used to transfer the displacements, the ratioof the overlapped area of the fluid element and an structural element to the areaof the fluid element of the first mesh determines the weight used to compute thefluid displacements.

Figure 5.8 shows the displacement profiles in the fluid that correspond to thearbitrarily created displacement profile in the structure using the element toelement mapping.

If the mesh of the fluid is finer than the mesh of the structure (figure 5.8a),the method preserves the displacement profile. If the fluid mesh is much finerthan the mesh of the structure, several neighbouring fluid nodes may receive thedisplacement from the same node of the structure which yields constant pieces inthe displacement profile of the fluid.


If the mesh of the fluid is coarser than the mesh of the structure (figure 5.8b),the number of nodes of the fluid mesh is insufficient to accurately capture thedisplacement profile. Any local oscillations present in the structural displacementprofile are smoothed by taking an area-weighted average of the displacements inseveral nodes of the structure.

(a) (b)

Figure 5.8: Meshes and displacement profiles at the fluid-structure interface in thestructure ( ) and the fluid partition ( ) obtained with the elementto element mapping when fluid mesh is (a) finer and (b) coarser thanthe solid mesh.

5.3.2 Load transfer

At fluid structure interface Γ the traction on the surface of the structure shouldbe in equilibrium with the traction on the surface of the fluid (equation 5.7).

The total force exerted by the fluid over the fluid-structure interface should beequal to the total force imposed on the structure at this interface:

∫

Γ

(σsn)dΓ =

∫

Γ

(−pfn + τfn)dΓ (5.57)

A conservative load transfer method fulfills exactly the above equation. Not allload transfer methods are conservative.

Consistent interpolation method

If the structure is discretized by the finite element method, the energy-equivalentnodal forces f

sΓiin the element coordinate system can be computed from the


traction transferred from the fluid to the structure:

fsΓi

=

∫

Γs

NsT

i df =

∫

Γs

NsT

i (−pfn + τfn)dΓs (5.58)

This integral can be approximated using a Gaussian quadrature:

fsΓi

=

ng∑

g=1

wgNsT

i (xg)(−pf(xg)n(xg) + τf(xg)n(xg)) det Fs (5.59)

where wg is weight of the Gauss point xg, ng is the number of Gauss points usedfor approximating the integral.

In order to determine the values of the fluid pressure pf(xg) and stress τf(xg)at the Gauss points, the consistent interpolation method uses a point to elementmapping: each Gauss point (equation (5.59)) of the structure mesh is mappedonto one element of the mesh of the fluid. The values of pf(xg) and τf(xg) at theprojection of the Gauss point onto the fluid element are interpolated using thediscretization technique intrinsic to the flow solver.

The consistent interpolation transfers the traction from the fluid to the structurepartition and integrates this traction over the surface of the structure. This meansthat the method does not conserve exactly the total force (equation (5.57)): e.g. ifthe fluid mesh is finer than the structure mesh, the projections of the Gauss pointsof the structure mesh are located in only a part of the fluid elements. The tractionin the other fluid elements are not used to compute the nodal forces. Howeverthe number of Gauss points ng used for evaluating the nodal forces f

sΓidoes not

have to be equal to the number of Gauss points selected for computing the elementstiffness matrices (e.g. when a high pressure gradient is expected, a larger numberof Gauss points can be used).

Farhat et al. (1998b) shows that for aeroelastic simulations the consistentinterpolation method produces practically conservative results although it is notexactly conservative.

Node to element conservative method

A conservative method is obtained if the traction is first integrated over the surfaceof the fluid independently of the structure. The energy-equivalent nodal forces f

fΓj

in the element coordinate system can be computed from the work performed inone element at the fluid-structure interface Γf :

δW f =

∫

Γf

σfn · udΓf =

jf∑

j=1

∫

Γf

(σfn)T

NfjuΓjdΓf =

jf∑

j=1

fT

fΓju

Γj(5.60)


where the energy-equivalent nodal force ffΓj

is equal to:

ffΓj

=

∫

Γf

NfT

j σfndΓf (5.61)

If the finite volume method is used to discretize the fluid domain, the functionsNfj are step functions. After transformation to the global coordinate system and

addition of the contributions of the neighbouring elements, the resulting nodalforce is transferred to the structure using a point to element mapping: each nodeof the fluid mesh is mapped onto one element of the mesh of the structure. Thetransferred force is distributed over the nodes of this element according to thevalue of the shape functions:

fsΓi

=

jf∑

j=1

NsT

i (xj)f fΓj(5.62)

As the sum of the shape functions is equal to one the total force is conserved:

is∑

i=1

fsΓi

=

is∑

i=1

jf∑

j=1

NsT

i (xj)f fΓj=

jf∑

j=1

ffΓj

(5.63)

In this case the fluid nodes, which send the data, are projected onto the elementsof the structure which receive the data. The relations between the nodal forces offluid and structure on Γ can generally be expressed as:

fsΓ

= LffΓ

(5.64)

Figure 5.9 shows the nodal forces on the structure that correspond to an arbitrarilycreated pressure profile in the fluid using the node to element conservative method.If the mesh of the fluid is finer than the mesh of the structure (figure 5.9a), theload distribution is adequately captured. If the mesh of the fluid is coarser thanthe mesh of the structure (figure 5.9b), some nodes of the structure mesh may notreceive any loads. The total force is conserved, but the load distribution is notaccurate.

Element to element conservative method

In order to remove the disadvantage of the above conservative method that somenodes of the structure mesh may not receive any loads if the mesh of the fluid iscoarser than the mesh of the structure, an element to element mapping can beused. The traction is first integrated over the surface of the fluid independently ofthe structure. The fluid element is divided into as many faces as the element hasnodes. All neighbouring elements of the structure are projected onto each of these


(a) (b)

Figure 5.9: Meshes and nodal forces at the fluid-structure interface in thestructure ( ) and the fluid partition ( ) obtained with theconservative method when fluid mesh is (a) finer and (b) coarser thanthe solid mesh.

faces. The ratio of the overlapped area of a face and an element of the structureto the area of this face determines the weight used during the transfer.

Figure 5.10 shows the nodal forces on the structure that correspond to anarbitrarily created pressure profile in the fluid using the element to elementconservative method. The total force is conserved and the load distribution isnow adequately captured independent of the fact that the mesh of the fluid is finer(figure 5.10a) or coarser (figure 5.10b) than the mesh of the structure.

(a) (b)

Figure 5.10: Meshes and nodal forces at the fluid-structure interface in thestructure ( ) and the fluid partition ( ) obtained with the elementto element conservative method when fluid mesh is (a) finer and (b)coarser than the solid mesh.


Weighted residual formulation

At fluid structure interface Γ the traction on the surface of the structure should bein equilibrium with this on the surface of the fluid (equation (5.7)). To ensure thisequilibrium, a weighted residual formulation can be used with the shape functionsN si of the structure as weight functions (Cebral and Lohner, 1997a):

∫

Γ

NsT

i (σsn)dΓ =

∫

Γ

NsT

i (−pfn + τfn)dΓ (5.65)

If the shape functions of the structure are used to impose the traction on thefluid-structure interface and the shape functions Nf

j are used to approximate thefluid stress, the weighted residual formulation becomes:

∫

Γs

NsT

i

is∑

i=1

Nsi(σsin)dΓ =

∫

Γf

NsT

i

jf∑

j=1

Nfj(−pfjn + τ fjn)dΓ (5.66)

The integral at the right hand side will be approximated using a Gaussianquadrature. A first possibility is to perform a loop over the Gauss points of themesh of the structure:

∫

Γs

NsT

i

is∑

i=1

Nsi(σsin)dΓ =

nes∑

es=1

ngs∑

gs=1

wgsNsT

i (xgs)Nfj(xgs

)(−pfj

n + τ fjn) det Fs

(5.67)

The right hand side corresponds to the nodal forces obtained with the consistentinterpolation method. As mentioned before, if the fluid mesh is finer than thestructure mesh, the projections of the Gauss points of the structure mesh arelocated in only a part of the fluid elements. The traction in the other fluid elementsare not used to compute the nodal forces. The second possibility is to perform aloop over the Gauss points of the mesh of the fluid:

∫

Γs

NsT

i

is∑

i=1

Nsi(σsin)dΓ =

nef∑

ef=1

ngf∑

gf=1

wgfNsT

i (xgf)Nfj(xgf

)(−pfj

n + τ fjn) det Ff

(5.68)

The right hand side corresponds to the nodal forces obtained with the node toelement conservative method. The advantage is that the fluid pressure of allelements is taken into account. By first integrating over the surface of the fluidthe total force is conserved. If the mesh of the fluid is coarser than the mesh ofthe structure, the load distribution is not accurate. Therefore, Cebral and Lohner


(1997b) suggested an adaptive Gaussian quadrature: if the structure elements aresmaller than the fluid elements, the number of Gauss points is increased by dividingthe fluid elements into smaller elements. The ratio of the area of the fluid elementto the areas of the structure elements that host its Gauss points determines thenumber of subdivisions.

5.3.3 Conservation of energy

As the fluid and the structure are closed system, the energy released or absorbedby the structure (except for structural damping) should be equal to the energygained or released by the fluid. Conservation of energy at the fluid-structureinterface yields:

W s =

∫

Γs

σsn · udΓs =W f =

∫

Γf

σfn · udΓf (5.69)

After discretization of the displacement fields, the energy is expressed as a functionof the nodal displacements and the nodal forces (equations (5.58) and (5.61)):

uTΓf

sΓ= uT

Γf

fΓ(5.70)

The combination of any motion transfer method (5.53) with the expression forconservation of energy (5.70) determines the corresponding load transfer whichconserves the energy:

uTΓf

sΓ= uT

ΓHTf

fΓ(5.71)

fsΓ

= HTffΓ

(5.72)

If the consistent interpolation method (5.56) is used for the displacement transfer,the corresponding load transfer which conserves the energy becomes (Farhat et al.,1998b):

fsΓi

=

jf∑

j=1

NsT

i (xj)f fΓj(5.73)

This is the node to element load transfer method as described in equation (5.62)and which conserves the total force.

As mentioned in section 5.2.2, the violation of the velocity continuity due to theuse of different time integration methods in the fluid and the structure still resultsin an error in the exchange of kinetic energy.

Similarly, the combination of any load transfer method (5.64) with the expressionfor conservation of energy (5.70) determines the corresponding motion transfer

APPLICATION: OVALLING OF SILOS 185

which conserves the energy:

fT

fΓu

Γ= fT

fΓLTu

Γ(5.74)

uΓ

= LTuΓ

(5.75)

For the load transfer a method which conserves the total force can be selected.According to Lohner (2006), this might lead to non-smooth deformations whichare not locally accurate. If you are dealing with transient aeroelastic problems, amethod which conserves the energy might be preferred, while in the case of shocks,it is important to conserve the total force.

5.4 Application: ovalling of silos

In this section the wind induced ovalling oscillations of the silos located in theport of Antwerp, are studied using fluid-structure interaction. In order to limitthe computational cost, the computation is performed for one silo in a three-dimensional flow.

In chapter 3 a three-dimensional finite element model with 18988 elements wasvalidated using experimental modal analysis. For the fluid-structure interactioncomputation, a coarser finite element model without the local refinements nearthe boundaries is used. The cylindrical part of the silo is modelled with 60 8-node quadrilateral shell elements in the circumferential direction and 20 in theaxial direction. The total number of shell elements in this model is equal to 1756.The boundary conditions are the same as in chapter 3. Table 5.2 compares theeigenfrequencies of the coarser model with eigenfrequencies of the experimentallyvalidated model (table 3.2). For the lowest eigenfrequencies around 4 Hz, thedifferences between eigenfrequencies computed with the two models are smallerthan 2%.

To advance the finite element solution of the structure in time, the Newmarkmethod (3.106) with α = 0.25 and δ = 0.5 is used. The time step ∆t is equalto 0.005 s, which is small enough to compute accurately the contributions ofeigenmodes up to 10 Hz. For the structure Rayleigh damping is added. Thedamping matrix C is constructed as a linear combination of the mass and thestiffness matrix:

C = αM + βK (5.76)

The modal damping ratios ξk at two different frequencies ωk determine themultipliers α and β for respectively the mass and the stiffness matrix by means ofthe following system of equations:

2ωkξk = α+ βω2k (5.77)


(m,n) Coarser mesh Validated mesh(1,2) 7.90 8.80 7.75 8.48(1,3) 4.00 4.00 3.93 3.93(1,4) 3.93 4.05 3.93 4.04(1,5) 5.37 5.37 5.25 5.25(1,6) 7.37 7.37 7.37 7.37(1,7) 9.72 9.72 9.72 9.72(2,4) 8.71 8.97 8.71 8.94(2,5) 5.93 5.93 5.56 5.56(2,5) 8.08 8.08 8.01 8.01(2,6) 9.29 9.49 9.29 9.39

Table 5.2: Eigenfrequencies (in Hz) computed with the coarser and the validatedthree-dimensional finite element model.

A modal damping ratio ξ1 = 0.25% at f1 = 3.93 Hz and ξ2 = 0.50% at f2 = 20 Hzcorresponds to α = 0.078 s−1 and β = 0.75 · 10−4 s. The modal damping ratios areestimated from the measured modal damping ratios for all eigenmodes between3.93 Hz and 20 Hz (Dooms et al., 2003).

In chapter 4 the two-dimensional turbulent wind flow around a single silo andaround a group of 8 by 5 silos was computed. For the fluid-structure interactioncomputation, the three-dimensional wind flow around a single cylinder is modelledwith symmetric boundary conditions on top and bottom surfaces. The velocityprofile at the inlet is uniform. The turbulent kinetic energy and the turbulentenergy dissipation rate at the inlet are equal to the low values used for the caseof the two-dimensional flow. This simulation is not meant to resemble to anatmospheric boundary layer flow around a free-standing cylinder as there are novelocity gradients in the inlet profile and the turbulence intensity is low. Anunsteady RANS simulation using the Shear Stress Transport turbulence model isperformed.

The three-dimensional mesh is obtained by copying the mesh A2 used for theunsteady flow around a single silo (chapter 4) 12 times in the axial direction. Thenumber of elements in the axial direction is sufficient to compute the flow arounda silo which deforms according to an eigenmode with m = 1. This number ishowever too low to generate any variation in the vortex shedding pattern alongthe axial direction. The total number of elements is equal to 642360.

The transient solution is integrated by the three-point backward difference scheme(2.176) with a time step ∆t = 0.005 s, which is equal to the time step used for thestructure and small enough to take the vortex shedding into account. Within everytime step, maximum 5 iterations are performed to obtain a converged solution.


In order to obtain an initial solution for the fluid-structure computation, the fluidfield is first computed independently during 1080 time steps, which results in atime window of 5.4 s. Figure 5.11 shows the time history and the frequency contentof the pressure between t = 3.55 s and t = 5.4 s at the cylinder’s surface at mid-height for θ = 112, θ = 174 and θ = 180. The frequency resolution ∆f = 0.54 isquite low due to the short analysed time interval. The vortex shedding frequencyat 2.16 Hz and a higher harmonic at 4.31 Hz are clearly visible in the frequencycontent. This vortex shedding frequency is slightly higher than the frequencyobtained in chapter 4 with a two-dimensional computation (2.13 Hz).

4 4.5 5−900

−800

−700

−600

−500

−400

−300

−200

−100

Time [s]

Pre

ssur

e [P

a]

(a)0 2 4 6 8 10

0

20

40

60

80

100

120

140

160

180

Frequency [Hz]

Pre

ssur

e [P

a/H

z]

(b)

Figure 5.11: (a) Time history and (b) frequency content of the pressure at thecylinder’s surface at mid-height for θ = 112 ( ), θ = 174 ( ) andθ = 180( ). The angle θ = 0 coincides with the stagnation point.

Next, the shell model of the structure is coupled with the three-dimensionalincompressible turbulent wind flow (figure 5.12). Between the structure and acylindrical surface with a diameter equal to twice the silo diameter, the fluidflow is computed on a deforming mesh, using the Arbitrary Lagrangian Eulerianformulation (section 2.2.1). The grid point displacements of the fluid mesh areobtained by diffusing the displacements of the structure through this domain(equation (2.233)). As to preserve the quality of the mesh in refined regions, thediffusivity of a finite volume is equal to the inverse of its volume. The grid pointdisplacements are equal to zero at the inlet and the outlet. On the symmetryplanes, the grid point displacements are not specified.

A node to element conservative method (equations (5.61)-(5.62)) is used for theload transfer between the non-matching grids, while the consistent interpolationmethod (equation (5.55)) is used for the displacement transfer.

As initial conditions the undeformed structure and the transient solution of thefluid flow without interaction at t = 5.4 s are used. 500 time steps are computed,which results in a time window of 2.5 s. The structure and the fluid field are


Figure 5.12: Model for the coupled simulation of the three-dimensional wind flowaround a cylinder and the response of the silo structure.

sequentially coupled using some of the available algorithms (table A.2). Firstthe conventional serial staggered algorithm (algorithm A - figure 5.1) is applied,in which every field is computed once at each time level tn. In a secondcomputation an iteratively staggered algorithm (algorithm B - figure 5.6) ensuresthe equilibrium between the two fields at each time level tn. The transferredinterface displacements and forces are not relaxed. Maximum four subiterationsare needed to obtain a relative change of the transferred quantities smaller thanεφ = 0.001 (equation (5.39)).

Figure 5.13a compares the time history of the radial displacements in three pointsat mid-height of the silo computed with algorithm A and B. Within this shorttime window both computations are stable. For a more rigourous evaluation ofthe stability more time steps should be computed. Clearly, the results computedwith algorithm A show differences that increase in time with the results ofalgorithm B because the accuracy of the conventional serial staggered algorithmis lower than the accuracy of the iteratively staggered algorithm. The accuracyof the conventional serial staggered algorithm could be improved by the use of aprediction for the structural displacements and a corrected fluid force, but thisoption is not available in the coupling between Ansys and CFX (table A.2). As


the staggered coupling algorithm is stable for this example, alternatively the timestep could be reduced in order to improve the accuracy. This might be cheaperthan the use of subiterations.

5.5 6 6.5 7 7.5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time [s]

Dis

plac

emen

t [m

]

(a)0 2 4 6 8 10

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Frequency [Hz]

Dis

plac

emen

t [m

/Hz]

(b)

Figure 5.13: (a) Time history of the radial displacements at mid-height for θ =66 ( ), θ = 120 ( ) and θ = 180 ( ) computed with algorithm A(dashed lines) and algorithm B (solid lines) and (b) frequency contentof the radial displacements computed with algorithm B. The angleθ = 0 coincides with the stagnation point.

Figure 5.13b shows the frequency content of the radial displacements in the samepoints computed with algorithm B. The frequency resolution ∆f = 0.4 is quite lowdue to the short time interval. The response of the silo is dominated by eigenmodes(1,3) and (1,4) around 4 Hz. The peak around 2 Hz indicates the effects of vortexshedding on the silo structure. The smaller peaks above 4 Hz are related to theeigenmodes with higher frequencies.

Figure 5.14 shows the deformations (enlarged with a factor 5) of the structurebetween a height of 11.25 m and 13.75 m at three different times. At all times anantinode faces the free stream direction. At t = 5.905 s and t = 6.805 s the responseis dominated by eigenmodes (1,4) with respectively a negative and a positive radialdisplacement at θ = 0. At t = 7.250 s eigenmode (1,3) is dominant. The maximalradial displacement is 0.105 m at t = 1.015 s and occurs at a height of 15 m andθ = 0.

Figure 5.15 shows the time history and frequency content of the pressure betweent = 5.4 s and t = 7.9 s at the cylinder’s surface at mid-height for θ = 112, θ = 174

and θ = 180. The comparison with figure 5.11 indicates that the structuraldeformations influence the pressure field near the wall. While the pressure timehistories of the flow simulation around a rigid silo mainly showed contributionsat 2.16 and 4.31 Hz, in the coupled computation contributions are present as wellat higher frequencies. An important contribution is present around 4 Hz whichcorresponds to the eigenfrequencies of eigenmodes (1,3) and (1,4). Due to the


(a) (b) (c)

Figure 5.14: Deformations (enlarged with a factor 5) of the structure between11.25 m and 13.75 m high at (a) t = 5.905 s, (b) t = 6.805 s and (c)t = 7.250 s. The wind flows from the left.

5.5 6 6.5 7 7.5−900

−800

−700

−600

−500

−400

−300

−200

−100

Time [s]

Pre

ssur

e [P

a]

(a)0 2 4 6 8 10

0

20

40

60

80

100

120

140

160

180

Frequency [Hz]

Pre

ssur

e [P

a/H

z]

(b)

Figure 5.15: (a) Time history and (b) frequency content of the pressure at thecylinder’s surface at mid-height for θ = 112 ( ), θ = 174 ( ) andθ = 180( ) using algorithm B. The angle θ = 0 coincides with thestagnation point.

interaction the magnitude of the pressure fluctuations around 2 Hz has clearlyincreased, which indicates an amplification of the vortex shedding.

The pressure field on the vertical plane through the cylinder axis parallel with theinlet flow direction at t = 7.9 s is shown in figure 5.16. The pressure field behindthe cylinder is clearly three-dimensional. Figure 5.17 shows the pressure at thecylinder’s surface along the height at five different times for three circumferentialangles. At the beginning of the simulation (at t = 5.4 s), the pressure is constantalong the height in the stagnation point (θ = 0) and for θ = 180. At θ = 112

the pressure varies slightly with the height. During the coupled simulation thelargest variations along the height occur at θ = 112, but also at θ = 0 and


θ = 180 considerable variations take place.

Figure 5.16: Pressure field on the vertical plane through the cylinder axis parallelwith the inlet flow direction at t = 7.9 s.

400 600 800 10000

5

10

15

20

25

Pressure [Pa]

Hei

ght [

m]

(a)−800 −600 −400 −200

0

5

10

15

20

25

Pressure [Pa]

Hei

ght [

m]

(b)−600 −400 −200 0

0

5

10

15

20

25

Pressure [Pa]

Hei

ght [

m]

(c)

Figure 5.17: Pressure at the cylinder’s surface along the height for (a) θ = 0,(b) θ = 112 and (c) θ = 180 at t = 5.4 s ( ), t = 5.475 s ( ),t = 5.925 s ( ), t = 6.35 s ( ) and t = 6.605 s ( ).

At every time the radial displacements along the circumference at mid-height aredecomposed into a Fourier series of modes with circumferential wavenumbers n.For each mode, principle component analysis (Pearson, 1901) of the time seriesyields the position of the first and the second principle component with respectto the silo. For all circumferential wavenumbers n the first principle componentis positioned roughly with an antinode facing the flow and the second with anode facing the flow. Figure 5.18 shows the time history and frequency contentof the first and the second principle component corresponding to circumferentialwavenumbers n = 2, 3 or 4. The response of the silo mainly consists of modeswith circumferential wavenumber n = 3 and 4. Their contribution varies stronglywith time.

In order to evaluate the occurrence of ovalling oscillations, the response of thestructure should be computed during a much longer time interval (e.g. 40 s),


5.5 6 6.5 7 7.5−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time [s]

Dis

plac

emen

t [m

]

(a)0 2 4 6 8 10

0

0.005

0.01

0.015

0.02

0.025

0.03

Frequency [Hz]

Dis

plac

emen

t [m

/Hz]

(b)

Figure 5.18: (a) Time history and (b) frequency content of the first (solid line)and second principal component (dashed line) of the displacementsat mid-height with circumferential wavenumber n = 2 ( ), n = 3( ) and n = 4 ( ).

as the structure is suddenly loaded and the modal damping ratios are very low.Nowadays, the computation times using a single processor are very high (312hours for the coupled simulation of the wind flow around one silo during 2.5 s).The major part (306 hours) is spent on the fluid partition. In order to makecoupled simulations of ovalling feasible, more efficient coupling procedures arerequired or the computation should be parallelized. A reduction of the dimensionsof the problem could decrease the computation times as well. The finite stripmethod (appendix B) enables to build an approximate model of the structure intwo dimensions. This finite strip model of the silo could be coupled with a twodimensional flow.

5.5 Conclusion

This chapter is concerned with some aspects of the coupled computation of fluidflows around deforming structures. First an overview is given of the existingpartitioned methods where the fluid and the structure are separately integratedin time and the interaction effects are applied as external boundary conditions.Loosely coupled algorithms solve each field one or a few times in each time stepand do not exactly fulfill the coupling conditions at the fluid-structure interface.The accuracy of these algorithms can be increased by using a suitable predictorfor the transferred structural displacements at the interface and the correspondingcorrector for the transferred fluid forces at the interface. The conservation of theenergy at the fluid-structure interface is an interesting tool to evaluate the differentalgorithms. For incompressible flows, loosely coupled algorithms are only stable for

CONCLUSION 193

loosely coupled problems, where the density of the structure is much higher thanthe density of the fluid. Strongly coupled algorithms solve each field multiple timesper time step in order to fulfill the coupling conditions within a certain tolerance.Automatically determined relaxation factors for the subiterations between the fluidand the structure partition improve the convergence and reduce the number ofiterations for strongly-coupled problems.

Next, methods for load and motion transfer between non-matching meshes aredescribed. The ratio of the number of elements in the fluid at the interfaceto the number of elements in the structure at the interface strongly influencesthe results of the different methods. For the transfer of displacements mainlythe accuracy of the transferred displacement profile is of concern. During thetransfer of the fluid forces to the structure the total load should be conserved. Fortransient aeroelastic computations the conservation of energy at the interface isvery important. The combination of a consistent interpolation for motion transferwith a node to element conservative method for load transfer conserves the energyat the fluid-structure interface and performs well for problems where the fluidmesh is finer than the structure mesh.

As an application, the shell model of the circular cylindrical silo is coupled withthe three-dimensional incompressible turbulent flow around a cylinder as to predictovalling oscillations. The inlet velocity is uniform along the height. The resultscomputed with the conventional serial staggered algorithm show differences thatincrease in time with the results computed using subiterations, which are moreaccurate. The accuracy of the staggered algorithm could be improved using apredictor for the structural displacements and the corresponding corrector for thefluid forces, but this option is not available in the coupling between Ansys andCFX. The response of the silo is dominated by the lowest eigenmodes (1,3) and(1,4) around 4 Hz. In order to evaluate the occurrence of ovalling oscillations, theresponse of the structure should be computed during a much longer time interval.Therefore, more efficient coupling procedures are required or the computationshould be parallelized. A reduction of the dimensions of the problem (e.g. thefinite strip method) could decrease the computation times as well.

Chapter 6

Conclusions andrecommendations for furtherresearch

6.1 Conclusions

The behaviour of civil engineering structures under wind loads is an importantmatter of concern. It is a trend in modern architecture to construct more slenderand lightweight structures with often complex geometries. The actual winddistribution around these complex structures cannot be derived from the literatureand may be strongly influenced by the presence of neighbouring structures. Forrelatively rigid structures, the wind load can be determined for the undeformedshape of the structure. This is no longer the case for flexible structures, wherethe pressures cause large structural displacements. The resulting change of shapehas an influence on the wind pressure distribution around the structure. This maygive rise to structural instabilities, such as flutter of bridges, galloping of cables,ovalling of silos and vortex induced vibrations of tall buildings.

This thesis studies coupled numerical simulations of the fluid and the structure todetermine the behaviour of civil engineering structures under wind loads. As anexample throughout the text, wind induced ovalling oscillations of a group of fortysilos are studied.

195

196 CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH

For the numerical simulation of fluid-structure interaction a partitioned approachis used: the fluid and the structure are treated as isolated fields and solvedseparately. The interaction effects are applied on the individual fields as externalboundary conditions which are exchanged from one field to the other. Themain advantage of this partitioned approach is that well-established discretizationtechniques and solution algorithms (and by consequence existing software) can beused in the individual fields which are tailored to the characteristic behaviourof this field. Due to the use of existing software (Ansys for the structuralcomputations and Flotran and CFX for the flow computations), the possiblechoices regarding models and algorithms were restricted to those available in thesepackages.

As the aim is to perform a coupled numerical simulation of the fluid and thestructure, the position of the structure determines at least partially the fluiddomain boundaries. If the structure undergoes large displacements, it is necessaryto perform the computations of the fluid flow on a domain with deformingboundaries. The fluid mesh is aligned with this deforming boundary and shouldfollow its deformations. Therefore, the governing equations for the fluid flow inan arbitrary Lagrangian-Eulerian description are derived. The convective velocityin these equations is the relative velocity between fluid particles and the mesh.The mesh velocity is arbitrary but should preferentially preserve the quality andthe refinements of the mesh. The mesh deformation is computed by means ofa diffusion equation with a variable diffusivity. The fluid domain is discretizedin space by means of the finite volume method. In the case of an arbitraryLagrangian-Eulerian description specific choices have to be made in order topreserve the accuracy and the stability of the time integration. The implicit three-point backward difference method which respects the geometric conservation law,is used.

The structure is discretized by means of the finite element method. The cylindricalshell of the silos in the port of Antwerp is studied with Ansys. As for the predictionof ovalling oscillations the exact value of the structural modal damping is veryimportant, an in situ experiment is performed in order to determine the modaldamping ratios, eigenfrequencies and eigenmodes of the silos. In order to supportthe design of the in situ experiment, the mode shapes and eigenfrequencies of asingle silo are first computed with a harmonic finite element model. The modeshapes consist of a number of waves in the circumferential direction and a numberof waves in the axial direction. Radial accelerations at 10 points along the silo aremeasured under ambient wind loading and modal parameters are extracted fromthe output-only data using the stochastic subspace identification technique. Theeigenmode with the lowest eigenfrequency at 3.94 Hz has the largest contribution tothe measured response. All modes have a low damping ratio around 1%. A three-dimensional finite element model of the silo is validated by means of the modalparameters. The boundary conditions strongly influence the eigenfrequencies and

CONCLUSIONS 197

mode shapes. The eigenmodes with three and four waves along the circumference(n = 3 or 4) and half a wave along the height (m = 1) are both situated at 3.93 Hz.

Geometrically non-linear behaviour is taken into account through an updatedLagrangian description and allows for large rotations and large displacements.The Newton-Raphson technique solves the non-linear system of equations. Theinfluence of geometrical non-linear behaviour on deformations of the silos of orderof magnitude of 0.1 m is not negligible: the magnitude of the displacementsincreases where the curvature decreases and decreases where the curvatureincreases.

Turbulent wind flows around civil engineering structures have a very high Reynoldsnumber. LES and hybrid RANS/LES approaches are well suited to capture thelarge scale turbulent structures in the wake of bluff bodies. However, as theircomputational cost is today still too high for the use in coupled fluid-structureproblems, URANS simulations are performed.

In order to clarify the influence of the RANS turbulence models, the near-wall meshrefinement and the unsteadiness, first the steady state two-dimensional turbulentair flow around a single cylinder in the post-critical regime at Reynolds number12.4 × 106 is computed with Flotran and CFX. Within Flotran, the near-wallmodelling strongly influences the results, while the results obtained with CFXare quite insensitive. Results computed with various eddy viscosity turbulencemodels strongly differ, but with respect to experimental results reported in theliterature the minimum pressure coefficient is always underestimated, while thebase pressure coefficient is overestimated. For the flow around a circular cylinderat these Reynolds numbers, the Shih-Zhu-Lumley turbulence model of Flotranhas the best correspondence with experimental results. Within CFX, the shearstress transport turbulence model produces the best results. Different results areobtained with Flotran and CFX using the same turbulence models. A transientsimulation model captures the regular vortex shedding in the post-critical regimeand is therefore preferred.

As an application the interference effects for the group of 8 by 5 silos in the portof Antwerp with spacing ratio P/D = 1.05 are studied at an incidence angle of30. Vortices shed at 0.165 Hz from the group as a whole and at 2.85 Hz from theindividual cylinders. The large velocities in the gaps produce distinct minima inthe pressure distributions where the gap area is minimum. The two cylinders onthe side corners of the group experience quite high lift coefficients. Eigenmodeswith circumferential wavenumber n equal to a multiple of 4 are more heavily loadedin the group configuration, where four small gaps surround most of the cylinders,compared with the single cylinder configuration. Especially for the cylinders onthe side corners of the group, eigenmodes with n = 3 or n = 4, which have thelowest eigenfrequencies for these silos, are strongly excited. This explains whystorm damage is mainly located on silos on the corners of the group.


The coupled numerical simulation of the response of a structure under fluidloads is computed using a partitioned approach. As the fluid and the structuralfield are non-linear, direct time integration methods are applied in both fields.Two partitioned solution algorithms which are available in Ansys and CFX, areemployed: the first, a loosely coupled serial staggered algorithm, solves each fieldonce in each time step and transfers the last computed solution of one field tothe other field. It does not exactly fulfill the coupling conditions at the fluid-structure interface, what may deteriorate the stability and the accuracy of thecoupled computation. For incompressible flows, loosely coupled algorithms areonly stable for loosely coupled problems, where the density of the structure ismuch higher than the density of the fluid. The second, a strongly coupled staggeredalgorithm, iterates between the two fields in each time step and fulfills the couplingconditions within a certain tolerance.

The meshes of the fluid and the structure are non-matching at the fluid-structureinterface. During the displacement transfer, the consistent interpolation methodguarantees the accuracy of the transferred displacement profile, while for the loadtransfer a node to element method conserves the total load. This combinationconserves the energy at the fluid-structure interface and performs well for problemswhere the fluid mesh is finer than the structure mesh.

The interaction between the wind flow and a single silo is computed. A shell modelof a circular cylindrical silo is coupled with the three-dimensional incompressibleturbulent wind flow as to predict ovalling oscillations. The results computed withthe conventional serial staggered algorithm show differences that increase in timewith the results computed using subiterations, which are more accurate. Theresponse of the silo is dominated by the lowest eigenmodes (1,3) and (1,4) around4 Hz. In order to evaluate the occurrence of ovalling oscillations, the response ofthe structure should be computed during a much longer time interval.

6.2 Recommendations for further research

In this text, the flow is computed on deforming meshes with the commercialsoftware CFX. The accuracy and the stability of flow computations on deformingmeshes could be evaluated. First it should be checked if the second order timeaccuracy of the three-point backward difference scheme is preserved in the ALEdescription. For instance the two-dimensional flow around a cylinder with aprescribed deformation with a given frequency, circumferential wavenumber andamplitude could be computed. Based on computations with different time stepsthe order of time accuracy can be derived.

Next, the implementation of the geometric conservation law could be checkedby computing a uniform flow and moving all the inner nodes of the mesh in

RECOMMENDATIONS FOR FURTHER RESEARCH 199

an arbitrary way. It is required that this solution can be computed exactlyindependently of the mesh deformations and that the mass balance is fulfilledfor the computational domain.

In this thesis, the two-dimensional turbulent air flow around a group of 8 by 5cylinders with spacing ratio P/D = 1.05 is studied at an incidence angle of 30.A significant question that remains is how sensitive these computed results are tothe incidence angle and the spacing ratio. In order to quantify these effects, asystematic study of the flow for a number of incidence angles and spacing ratioscan be performed.

The flow around one finite circular cylinder of aspect ratio h/D = 4.55 is stronglythree-dimensional with horseshoe vortices, trailing vortices and arch vortices. It isexpected that comparable three-dimensional vortex structures are found aroundthe group as a whole. The numerical simulation of the three-dimensional flowaround one finite cylinder could be calculated but is still challenging from acomputational point of view. Using multiple processors, the numerical simulationof the three-dimensional flow around forty closely-spaced finite circular cylindersmight become feasible.

If the three-dimensional flow around one finite cylinder is computed the RANSturbulence models could be replaced by a hybrid RANS/LES approach which isable to capture the large scale turbulent structures in the wake of bluff bodies.

In the last chapter the coupled simulation of three-dimensional incompressibleturbulent wind flow around a circular cylindrical silo is computed in order topredict ovalling oscillations. As the computational cost is very high, more efficientcoupling procedures are required. The use of the commercially available couplingbetween Ansys and CFX, limited the choice of employed coupling algorithms. Anew implementation of the coupling between the two commercial packages shouldenable the use of more advanced coupling algorithms.

The accuracy of the conventional staggered algorithm could be improved using apredictor for the structural displacements and the corresponding corrector for thefluid forces. If this enhanced staggered algorithm is still stable for loosely coupledproblems with incompressible flows, the use of subiterations could be avoided,which substantially decreases the computational cost.

For strongly coupled problems, the algorithm with subiterations can be enhancedusing automatically determined relaxation factors for the subiterations betweenthe fluid and the structure partition in order to improve the convergence and toreduce the number of subiterations.


An interesting alternative is to construct reduced order models for the fluid andthe structure during the subiterations (Vierendeels, 2006). For the fluid in eachsubiteration the pressure distribution at the fluid-structure interface due to theimposed displacements at this interface is computed. These sets of imposeddisplacements and corresponding pressure distributions can be used to constructa reduced order model for the fluid. Analogously, for the structure in eachsubiteration the displacements at the fluid-structure interface which correspondto the imposed pressures at this interface are computed. These sets can be usedto construct a reduced order model for the structure. The result of the coupledcomputation using the two reduced order models can be used as an initial guessfor a subiteration. Each subiteration the reduced order models are improved.This technique allows for the solution of strongly coupled problems with a smallnumber of subiterations. It can possibly be improved by reusing in the currenttime step the reduced order models constructed during the previous time steps.This might be very interesting for aeroelastic problems as flutter, galloping andovalling because a lot of computations of the fluid domain are required for nearlyidentical geometries.

A global multigrid technique, which computes partitioned coupling between fluidand structure at different grid levels, might as well enhance the stability, efficiencyand accuracy (Schafer et al., 2006; van Zuĳlen et al., 2007). The computation ofthe coupled problem is first performed on a coarse fluid and a coarse structure mesh.This solution can be used as a coarse grid prediction for a coupled computationon a finer grid. The low frequency error in the solution obtained on a fine grid canbe corrected by computing a new solution on the coarse meshes.

The efficiency, accuracy and stability of the different coupling algorithms can beverified on a set of benchmark examples e.g. an elastic beam behind a square(Wall and Ramm, 1998) or behind a circle (Turek and Hron, 2006). Bathe andLedezma (2007) give an overview of a series of benchmarks in order to evaluatethe coupling algorithms and load and motion transfer algorithms.

A reduction of the dimensions of the problem could decrease the computationalcost as well. For the example of the silo, the finite strip method (appendix B) maybe applied to make a representative two-dimensional model of the structure. Thestructural displacements are supposed to vary as half a sine or cosine function inthe axial direction. The finite strip element is implemented as a combination of atwo-dimensional beam element and a user defined element. A drawback is that thestiffness matrix of the user defined elements is independent of the deformation, sothat geometrical non-linear effects are not included in these additional terms. Thisfinite strip model could be coupled to a two-dimensional flow computation. Forthe transfer of the fluid forces to the structure, an assumption for the variation ofthe pressure along the height of the silo has to be made. As figure 5.17 shows, thevariations are neither constant, neither sinusoidal. As a validation, the results of acoupled simulation with the finite strip model and the two-dimensional flow should

RECOMMENDATIONS FOR FURTHER RESEARCH 201

be compared to the results of a coupled simulation of the three-dimensional flowaround the shell model of the structure. The influence of the group configurationon the occurrence of ovalling oscillations can be subsequently studied by couplingfinite strip models for the silos with a two-dimensional flow around the group.Obviously, the number of silos should be gradually increased, starting with twoclosely-spaced silos.

While only the ovalling of silos is considered in the present work, the methodologycan also be applied to wind response of tensile structures, flutter of bridges,galloping of cables and vortex induced vibrations of marine riser pipes or cables.

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Curriculum vitae

David Dooms18 September 1978, Sint-Niklaas (Belgium)

Education

2001-2008PhD in Engineering, Department of Civil Engineering, K.U.Leuven

1996-2001MSc in Engineering: Architecture , K.U.Leuven

1992-1998Secondary education, Latin–Mathematics, Sint-Jozef-Klein-Seminarie, Sint-Niklaas

Work

2001-2007Assistant, Department of Civil Engineering, K.U.Leuven

2007-2009Research assistant, OI-project ’An interactive and adaptive application for thestatic and dynamic analysis of structures’, Department of Civil Engineering,K.U.Leuven

221

222 CURRICULUM VITAE

Publications

International journal papers

D. Dooms, G. Degrande, G. De Roeck, and E. Reynders. Finite element modellingof a silo based on experimental modal analysis. Engineering Structures, 28(4):532–542, 2006.

L. Karl, W. Haegeman, G. Degrande, and D. Dooms. Determination of thematerial damping ratio with the bender element test. Journal of Geotechnicaland Geoenvironmental Engineering, Proceedings of the ASCE, 134(12):1743–1756,2008.

International conference papers

D. Dooms, G. De Roeck, and G. Degrande. Fluid-structure interaction applied toovalling oscillations of a silo. In 8th World Congress on Computational Mechanicsand 5th European Congress on Computational Methods in Applied Sciences andEngineering, Venice, Italy, July 2008.

D. Dooms, G. De Roeck, and G. Degrande. Wind induced ovalling oscillations ofthin-walled cylindrical structures. In International Workshop on Fluid-StructureInteraction: Theory, Numerics and Applications, Herrsching am Ammersee,Germany, September 2008.

D. Dooms, G. De Roeck, and G. Degrande. Fluid-structure interaction appliedto ovalling oscillations of a silo. In Fourth International Conference on AdvancedComputational Methods in Engineering, Liege, Belgium, May 2008.

S. Arnout, D. Dooms, and G. De Roeck. Shape and size optimization ofshell structures with variable thickness. In 6th International Conference on thecomputation of shell & spatial structures, Ithaca, New York, USA, May 2008.

W. Figeys, S. Ignoul, D. Dooms, L. Schueremans, D. Van Gemert, and G. De Roeck.Strengthening of an industrial cylindrical shell damaged by a collision. InD. D’Ayala, editor, Proceedings of the Structural Analysis of Historic Construction,volume 2, pages 1087–1094, Bath, United Kingdom, July 2008.

CURRICULUM VITAE 223

D. Dooms, G. De Roeck, and G. Degrande. Fluid-structure interaction appliedto ovalling oscillations of silos. In E. Onate, M. Papadrakakis, and B. Schrefler,editors, Computational Methods for Coupled Problems in Science and EngineeringII, pages 586–589, Santa Eulalia, Ibiza, Spain, May 2007.

W. Figeys, S. Ignoul, D. Dooms, L. Schueremans, D. Van Gemert, and G. De Roeck.Repair of impact damage in an industrial cylindrical shell. In Proceedings of 3ndInternational fib Congress, page 305, Dubrovnik, Croatia, May 2007.

D. Dooms, G. De Roeck, and G. Degrande. Influence of the group positioningof cylinders on the wind pressure distribution in the post-critical regime. InP. Wesseling, E. Onate, and J. Periaux, editors, Proceedings of EuropeanConference on Computational Fluid Dynamics ECCOMAS CFD 2006, Egmondaan Zee, the Netherlands, September 2006.

D. Dooms, G. De Roeck, and G. Degrande. Fluid-structure interaction applied toovalling oscillations of silos. In P. Sas and M. De Munck, editors, Proceedings ofISMA2006 International Conference on Noise and Vibration Engineering, pages4557–4571, Leuven, Belgium, September 2006.

D. Dooms, G. Degrande, G. De Roeck, and E. Reynders. Wind inducedoscillations of thin-walled silos. In M. Papadrakakis, E. Onate, and B. Schrefler,editors, Computational Methods for Coupled Problems in Science and Engineering,Santorini, Greece, May 2005. CD-ROM.

D. Dooms, G. Degrande, G. De Roeck, and E. Reynders. Wind induced vibrationsof thin-walled cylindrical structures. In ISMA2004 International Conference onNoise and Vibration Engineering, pages 781–796, Leuven, Belgium, September2004.

National conference papers

D. Dooms, G. De Roeck, and G. Degrande. Reynolds Averaged Navier Stokessimulation of the post-critical flow around a single circular cylinder and groups ofcylinders. In Proceedings of the 7th National Congress on Theoretical and AppliedMechanics, Mons, Belgium, May 2006. National Committee for Theoretical andApplied Mechanics.

224 CURRICULUM VITAE

Internal reports

D. Dooms, G. Degrande, and G. De Roeck. Dynamic analysis of a group of silosunder dynamic wind loading: modelling of the wind flow around a group of silos.Report to Ellimetal BWM-2006-06, Department of Civil Engineering, K.U.Leuven,April 2006.

M. Schevenels, G. Degrande, G. Lombaert, and D. Dooms. Trillingsmetingenin een woning aan de Loverstraat 37 te Sint-Baafs-Vĳve. Report to F. DoomsBWM-2004-05, Department of Civil Engineering, K.U.Leuven, May 2004.

D. Dooms, G. Degrande, and G. De Roeck. Dynamische analyse van een groepsilo’s onder dynamische windbelasting: Eindige elementenmodellering van eenlege silo. Report to Ellimetal BWM-2003-08, Department of Civil Engineering,K.U.Leuven, June 2003.

D. Dooms, S. Jacobs, G. Degrande, and G. De Roeck. Dynamische analyse vaneen groep silo’s onder dynamische windbelasting: In situ metingen op een legehoeksilo. Report to Ellimetal BWM-2003-09, Department of Civil Engineering,K.U.Leuven, June 2003.

D. Dooms, G. Degrande, G. De Roeck, and S. Jacobs. Dynamische analyse vaneen dansvloer in de Bowling Factory te Braine l’Alleud. Report to J. De LaereBWM-2002-06, Department of Civil Engineering, K.U.Leuven, May 2002.

Appendix A

Possibilities of commercialsoftware packages Flotran andCFX

225

226 POSSIBILITIES OF COMMERCIAL SOFTWARE PACKAGES FLOTRAN AND CFX

Flo

tran

CF

XT

echniq

ue

Fin

ite

elem

ent

met

hod

Fin

ite

volu

me

met

hod

Spati

al

dis

cret

izati

on

Monoto

ne

Str

eam

line

Upw

ind

(MSU

)U

pw

ind

Diff

eren

ceSch

eme

(UD

S)

Str

eam

line

Upw

ind

Pet

rov-G

ale

rkin

(SU

PG

)Sp

ecifi

edble

nd

Colloca

ted

Gale

rkin

(CO

LG

)H

igh

Res

olu

tion

(HiR

es)

Tim

edis

cret

izati

on

back

ward

Eule

rback

ward

Eule

rN

ewm

ark

thre

e-p

oin

tback

ward

diff

eren

ce(B

DF

2)

Alg

ori

thm

sSem

i-Im

plici

tM

ethod

for

Pre

ssure

-L

inked

Equati

ons

(SIM

PL

E)

couple

dso

lver

Turb

ule

nce

Zer

oE

quati

on

Zer

oE

quati

on

k−ε

k−ε

RN

Gk−ε

RN

Gk−ε

k−ω

k−ω

Shea

rStr

ess

Tra

nsp

ort

(SST

)Shea

rStr

ess

Tra

nsp

ort

(SST

)Shih

-Zhu-L

um

ley

(SZ

L)

Rea

liza

blek−ε

(NK

E)

Nonlinea

rM

odel

of

Gir

imaji

(GIR

)R

eynold

sStr

ess

Model

(RSM

)D

etach

edE

ddy

Sim

ula

tion

(DE

S)

Larg

eE

ddy

Sim

ula

tion

(LE

S)

Wall

funct

ions

equilib

rium

standard

Van

Dri

est

scala

ble

/auto

mati

cSpald

ing

Table A.1: Overview of the different possibilities for CFD computations withinFlotran 10.0 and CFX 10.0.

POSSIBILITIES OF COMMERCIAL SOFTWARE PACKAGES FLOTRAN AND CFX 227

Flo

tran

CF

XD

escr

ipti

on

arb

itra

ryL

agra

ngia

n-E

ule

rian

(AL

E)

arb

itra

ryL

agra

ngia

n-E

ule

rian

(AL

E)

Mes

hdef

orm

ati

on

pse

udo-s

truct

ure

diff

usi

on

equati

on

Coupling

alg

ori

thm

spart

itio

ned

part

itio

ned

iter

ati

vel

yst

agger

edit

erati

vel

yst

agger

edfixed

rela

xati

on

fact

or

fixed

rela

xati

on

fact

or

sub

cycl

ing

sub

cycl

ing

seri

al

seri

al

or

para

llel

Load

transf

erco

nsi

sten

tin

terp

ola

tion

consi

sten

tin

terp

ola

tion

node

toel

emen

tco

nse

rvati

ve

node

toel

emen

tco

nse

rvati

ve

Moti

on

transf

erco

nsi

sten

tin

terp

ola

tion

consi

sten

tin

terp

ola

tion

Table A.2: Overview of the different possibilities to couple Ansys 10.0 withFlotran 10.0 and CFX 10.0.

Appendix B

Finite strip model of the silo

A fluid-structure interaction analysis aims to predict the ovalling onset flow velocityand to investigate the influence of the distance between the silos. As the coupledthree-dimensional calculation of turbulent wind flow around a group of silos is toodemanding from the computational point of view, an approximate analysis will beperformed by reducing the structure to two dimensions and by coupling it with atwo dimensional flow.

Using a finite strip formulation (Cheung, 1976), the displacements of the three-dimensional structure are decomposed into a series of orthogonal functions thatsatisfy a priori the Dirichlet boundary conditions in the axial z-direction and a two-dimensional displacement field in the (r, θ)-plane. The use of orthogonal functionsresults in a decoupled system of equations for every term in the series. The sinefunctions reflect that the radial and circumferential displacements are assumed tobe zero at both ends of the cylinder, while the cosine function allows for free axialdisplacements at both ends:

ur(r, θ, z, t)uθ(r, θ, z, t)uz(r, θ, z, t)

=

∞∑

m=0

sin(mπzh ) 0 0

0 sin(mπzh ) 00 0 cos(mπzh )

urm(r, θ, t)uθm(r, θ, t)uzm(r, θ, t)

(B.1)

In reality, however, the cylinder is welded to the cones at both ends and bolted tothe supporting structure at the bottom in four points along the circumference.

As the experimental results indicate that the eigenmodes with the highest modalcontributions consists of half a wavelength along the height (m = 1), the series inequation (B.1) is limited to the term m = 1.

229

230 FINITE STRIP MODEL OF THE SILO

Using equation (B.1), the in-plane stiffness matrix for a finite strip membraneelement for m = 1 is obtained (Cheung, 1976):

KIP = t

Eph/2L +GLπ

2

6h symmetric

νEpπ4 −Gπ4 Ep

Lπ2

6h +G h2L−Ep

h/2L +GLπ

2

12h −νEpπ4 −Gπ4 Ep

h/2L +GLπ

2

6hνE

1−ν2π4 +Gπ4 Ep

Lπ2

12h −G h2L −νEpπ4 +Gπ4 Ep

Lπ2

6h +G h2L

(B.2)

where Ep and G are:

Ep =E

1− ν2(B.3)

G =E

2(1 + ν)(B.4)

The out-of-plane stiffness matrix for a finite strip plate-bending element for m = 1consists of:

[KOOP]11 = [KOOP]33 =13Lπ4D

70h3+

12π2Dxy

5Lh+

6π2νD

5Lh+

12h2D

L3(B.5)

[KOOP]21 = −[KOOP]43 =3π2νD

5h+π2Dxy

5h+

11L2π4D

420h3+

6h2D

L2(B.6)

[KOOP]22 = [KOOP]44 =L3π4D

210h3+

4Lπ2Dxy

15h+

2Lπ2νD

15h+

4h2D

L(B.7)

[KOOP]31 =9Lπ4D

140h3− 12π2Dxy

5Lh− 6π2νD

5Lh−12h2D

L3(B.8)

[KOOP]32 = −[KOOP]41 =13L2π4D

840h3− π

2Dxy

5h− π

2νD

10h−6h2D

L2(B.9)

[KOOP]42 = −3L3π4D

840h3− Lπ

2Dxy

15h− Lπ

2νD

30h+

2h2D

L(B.10)

where the bending stiffness D and Dxy are equal to:

D =Et3

12(1− ν2)(B.11)

Dxy =Gt3

12(B.12)

FINITE STRIP MODEL OF THE SILO 231

The in-plane mass matrix for a finite strip membrane element for m = 1 is:

MIP = ρth

2

L3

0 L3

L6 0 L

3

0 L6 0 L

3

(B.13)

The out-of-plane mass matrix for a finite strip plate-bending element for m = 1 is:

MOOP = ρth

2

13L35

11L2

210L3

1059L70

13L2

42013L35

− 13L2

420 − 3L1

140 − 11L2

210L3

105

(B.14)

The stiffness and mass matrix for a finite strip shell element are obtained bycombining the matrices for in-plane and out-of-plane behaviour. The resulting2-node element has four degrees of freedom per node: the displacements in theradial, circumferential and axial directions and the rotation around the z-axis.

All terms marked in grey in the stiffness and mass matrices of equations (B.2)-(B.14) are similar to terms of a two-dimensional beam element. The finite stripelement is implemented in the finite element program Ansys as a combination of atwo-dimensional BEAM3 (Ansys, 2005b) element and a user defined MATRIX27element, which adds all the black terms. The terms marked in grey becomeidentical to those of a two-dimensional beam element if the area and the momentof inertia are computed based on the thickness and half the height of the finitestrip:

A =(h/2)t

1− ν2(B.15)

I =(h/2)t3

12(1− ν2)(B.16)

The area and moment of inertia are divided by (1−ν2) to incorporate the increasein stiffness caused by the Poisson effect. The density is multiplied by a factor(1 − ν2) to obtain the correct mass:

ρ′ = ρ(1− ν2) (B.17)

The stiffness matrix of the user defined elements is independent of the deformation,so that geometrical non-linear effects are not included in these additional terms.

Figure B.1 shows the relative error on the predicted eigenfrequencies as a functionof the number of elements along the circumference. A reference solution is obtainedbased on a very fine mesh with 1152 elements in the circumferential direction. The

232 FINITE STRIP MODEL OF THE SILO

72 90 144 28810

−6

10−5

10−4

10−3

10−2

Number of elements [−]

Rel

ativ

e er

ror

[−]

(1,4)(1,3)(1,5)(1,2)(1,6)(1,7)(1,8)(1,9)(1,10)

Figure B.1: The relative error on the nine lowest eigenfrequencies predicted withthe finite strip model as a function of the number of elements in thecircumferential direction.

relative errors on the nine lowest eigenfrequencies, predicted with a finite stripmodel with 90 elements along the circumference of the silo are lower than 10−3,so that this model can be used with confidence.

In the finite strip method, a constant thickness of the aluminium shell along theheight of the silo is assumed, whereas, in reality, this thickness reduces with theheight (table 3.1). Furthermore, the radial modal displacement varies as a sinefunction along the height. As the degree of clamping at the top and the bottomof the silo increases with increasing circumferential wave number n, an equivalentheight can be derived that depends on n (Arnold and Warburton, 1953). A betterprediction accuracy could be pursued by formulating an optimization problemwhere the objective function is defined as the weighted squared difference betweenthe predicted and experimental eigenfrequencies and minimized with respect tothe (uniform) shell thickness t and the height h in the Fourier expansion (B.1) ofthe displacements. Alternatively, a sensitivity analysis is performed here wherethe eigenfrequencies predicted with three finite strip models with different shellthickness and height are compared with the results of the three-dimensional finiteelement model (table B.1). Every couple (m,n) corresponds to two eigenmodes,that have the same eigenfrequencies in case of the finite strip model and may haveslightly different eigenfrequencies for the three-dimensional finite element model.For a model with a height of 25 m and a thickness of 7 mm (figure B.2), theeigenfrequencies for n = 3 and n = 4 correspond well with the results of the three-dimensional calculation, while the differences between both models increase forhigher values of n. The correspondence for n = 3 and n = 4 improves by changingthe height and the shell thickness to 25.34 m and 7.215 mm, respectively (tableB.1).

FINITE STRIP MODEL OF THE SILO 233

n 3D model Finite strip modelh = 24.70 m h = 25.00 m h = 25.34 mt = 6.600 mm t = 7.000 mm t = 7.215 mm

2 7.75 8.48 7.67 7.50 7.313 3.93 3.93 4.03 4.00 3.934 3.93 4.04 3.80 3.93 3.995 5.25 5.25 5.25 5.54 5.696 7.37 7.37 7.49 7.93 8.177 9.72 9.72 10.2 10.8 11.28 12.5 12.5 13.4 14.2 14.79 15.7 15.7 17.1 18.1 18.710 19.4 19.4 21.1 21.2 23.1

Table B.1: Comparison of the eigenfrequencies computed with the three-dimensional finite element model and the finite strip models for varyingn and m = 1.

a. (1, 4) b. (1, 3) c. (1, 5) d. (1, 2)

3.93 Hz 4.00 Hz 5.54 Hz 7.50 Hz

Figure B.2: Eigenmodes of a silo with a height h = 25 m and a thickness t = 7 mm,computed with a finite strip model.

As the degree of clamping increases mainly at the bottom of the cylinder withincreasing circumferential wave number n and as the shell thickness decreaseswith the height, the eigenfrequencies of the eigenmodes with high circumferentialwave number n are better approximated by finite strip models with a lower heightand a smaller thickness. For n = 5, for example, results obtained with a finitestrip model with a height h = 24.7 m and a thickness t = 6.6 mm agree well withthe results of the three-dimensional model (table B.1).

The influence of the height h of the finite strip model decreases for increasingvalues of the circumferential wave number n, while the shell thickness t stronglyaffects the eigenfrequencies of the modes with a high circumferential wave number,but has a minor influence on the modes corresponding to low values of n.

tese_silo

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