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Experiments in Fluids 4:97-106 (1986) Expemnents n Flui Springer-Verlag 1986

Th e structure of the turbulent boundary layer over a mobi leand deformable boundary *Y. Papadimitrakis**, En Hsu , and R. StreetEnvironmental Fluid Mechanics Lab, Dept. of Civil Engineering, Stanford University, Stanford, CA 94305, USA

Abstract. The structure of the velocity field above a propagatingwater wave of fixed frequency was investigated in order toevaluate the transport of wind momentum to water waves and theinfluence of a mobile and deformable boundary on the burstingcycle. The vertical and horizontal velocities were measured in atransformed Eulerian wave-following frame of reference with theaid of a cross hot film, in a wind-wave research facility atStanford University.The mean velocity profiles have a log-linear form with a wakefree-stream characteristic. The wave-coherent motion in the free-stream is irrotational; in the boundary layer, it has a strong shearbehavior related to the wave-associated stress. The wave-inducedvelocity field and the wave-perturbed turbulence depend stronglyon the ratio of the wave-speed to the mean free-stream velocity,c /U &. 'The presence of the propagating waves affects the burstingcycle, making the contribution of sweeps and ejections almostequal and dependent on the ratio c/Uao. The magnitudes of thecontribution of the bursting events are generally enhanced by thepresence of water waves. The time interval between ejections orsweeps does not scale with either the inner and/or outer flowvariables.

1 IntroductionThe knowledge of the detailed structure of the velocityfield above water waves is valuable for understanding avariety of different problems such as: 1. the generationand growth of water waves; 2. the gas and heat exchangeprocesses at the air-water interface; and 3. the transport ofpollutants in the atmospheric boundary layer and waterenclosures. Although there exist several experimental andtheoretical investigations of both the pressure and velocityfield in the air boundary layer above water waves, theirresults show considerable disagreement. Among them,only a few have considered t he Reynol ds stress pr oducti onmechanism in the water proximity. Considerable theoreti-cal and experimental work, however, using both flow

* This paper was presented at the Ninth Symposium on Turbu-lence, University of Missouri-Rolla, October 1-3, 1984** Now at the College of Marine Studies, University of Delaware,Lewes, Delaware 19958

visualization and either velocity or pressure sensors, hasbeen done for wall flows. Experimental studies of thebursting phenomenon, a process responsible for most ofthe Reynolds stress production in these flows, generallyconcentrate on determining the average spatial and tem-poral scales of ejection and sweep events. Although thespatial configuration and scales of the wall structure aregenerally agreed upon by different investigators, there isconsiderable disagreement concerning the frequency ofoccurrence and the scaling of the bursting events inbounded shear flows. These differences have been at-tributed to the different detection schemes; Bogard andTiederman (1981) have suggested that the probe measure-ments probably detect something different from the burstsidentified using flow-visualization techniques.

Previous field and laboratory measurements of thevelocity and pressure fields over water waves, conductedby Kendall (1970), Takeuchi etal. (1977), Hsu eta!.(1981) and Hsu and Hsu (1983), have shown considerablealteration of the wind turbulence due to the presence ofwaves. Therefore, it will be of great importance to knowthe characteristics of the bursting cycle under the presenceof a mobile and deformable boundary. In this study, thevelocity field was measured with an "X" hot film mount-ed on a wave-following device operating in a transformedcoordinate system similar to that used by Hsu et ai.(1981), and Hsu and Hsu (1983). This paper, being part ofthe doctoral work of the first author (Papadimitrak!s1982), describes in detail the structures of the mean, wave-induced, and turbulent velocity fields in the air, and theinfluence of water waves on the turbulent Reynolds stressproduction mechanism and on the bursting-related phe-nomenon.

2 Experimental apparatus and data analysi sThe experiments were conducted in the Stanford wind-wave research facility (Hsu 1965). The depth of the airflow H (= 260) measured from the mean water level


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98 Experiments in Fluids 4 (1986)( MW L) to the channe l r oo f was 1 .07 m. The wa te r dep th dwas 0 .83 m. The 1 Hz mec hanica l ly - gen er a ted wave was indeep wa te r , wi th a wave length L = 1 .56m, a wavenu mb er k= 4 .03 m - I and a phase speed c = 1 .56 ms - l .T h e a i r f l o w a b o v e t h e p r o p a g a t i n g w a v e i s c o n s i d e r e d t obe two- d imens iona l .

I n the Car tes ian coor d ina te sys tem used , x i s measur edi n t h e w a v e - p r o p a g a t i o n o r m e a n a i r - f lo w d i r e c t i o n a n d yi s t h e v e r t i c a l c o o r d i n a t e m e a s u r e d u p w a r d f r o m t h eM W L . T h e c o o r d i n a t e t r a n s f o r m a t i o n u s e d c o n t a i n s o n l yver t ica l t r ans la t ion ; namely ,t = t * , x = x * , Y = Y * + f ( Y * ) O ( 2 .1a , b , c )

s inh ( k H - k y* )f ( y * ) = (2.2)s inh ( k H )and has been desc r ibed by Hsu e t a l. ( 1981) ; F / r epresen tsthe s inuso ida l wa te r sur f ace d i sp lacement f r om the MW L.

W a v e - h e i g h t g a u g e s w e r e u s e d t o o b t a i n t h e w a v echar ac te r i s t i c s and dr ive the wave f o l lower . They wer e ofc a p a c i t a n c e t y p e w i t h a n a c c u r a c y i n s t a t i c a n d d y n a m i ccalib rat ion tes ts of __ 0.25 ram. Th e ve loci ty f ie ld in the airwas meas ur ed wi th a ho t f i lm . I t was a TSI Mod e l 1248-10end- f low "X " p r obe , 0 .0254 mm in d iamete r , 0 . 508 m mlong, wi th a f l a t f r equency r esponse up to 40 ,000 Hz ; i tw a s o p e r a t e d i n a c o n s t a n t - t e m p e r a t u r e m o d e . F o r t h ewind speeds examined in th i s inves t iga t ion , i t s v i scouslength 1 was in the r ange 1 .6 < / + < 5 .0 . Th e uncer -ta in ty f or the mean ve loc i ty was 3%. However , f o r them e a s u r e d t u r b u l e n t q u a n t i t i e s s u c h as t u r b u l e n t R e y n o l d ss t r es ses , i t was h igher and was f ound to be appr oximate ly10%.

The hot - f i lm s igna ls wer e ze r o- suppr es sed , ampl i f ied ,and low- pass f i l t e r ed a t 250 Hz to f u l f i l l the N yqu is tc r i t e rion , a s s amples wer e taken ever y 0 .002sec f or3 minutes and r e cor ded on a t ape f or l a te r ana lys i s . Theda ta t aken cor r espond to s even d i f f e r en t f r ee - s t r eam meanwind ve loc i t ie s in the r ange 14 0- 4 00 cm/sec , wi th I Hz2 .5 4 c m n o m i n a l a m p l i t u d e , m e c h a n i c a l l y - g e n e r a t e dwaves . They wer e co l lec ted a t 20 or 21 e leva t ions , r angingf r om 0 .75 and 53 .3 cm above th e wa te r sur f ace .

S ince the ins tan taneous ve loc i ty f i e ld above the wavescons i st s o f a mean , a wave- in duce d pe r tur ba t ion , and at u r b u l e n t c o m p o n e n t , v i z .ui = Ui + t2g + u} (i = 1, 2 ), (2.3 )f ami l ia r t ime and phase aver ages wer e used to ex t r ac t thewave- induced f luc tua t ions f r om the to ta l ve loc i ty s igna ls .The ve r t ica l wave- in duce d ve loc i ty was cor rec ted to ac -c o u n t f o r t h e s p u r i o u s c o m p o n e n t i n t r o d u c e d b y t h e w a v e-fol lower motion. Cross- and auto-spectral analys is for ~i ,us ing ~/ a s a r e fe r ence , wer e pe r f o r med by f as t - Fo ur ie r

1 Defined as l + = l" u, /v where l, u, and v are probe length, airfriction velocity and kinem atic viscosity o f the air, respectively

t r a n s f o r m a t i o n ( F F T ) t o d e t e r m i n e t h e a m p l i t u d e a n dphase of each ha r mon ic conta ine d in f fi ( Hsu e t a l . 1981) .

The bur s t ing events wer e iden t i f i ed by c las s i f ica t ion ofthe me asur ed Reyn olds s t re s s u v accor d ing to the s igns ofu (= ~+ u ' ) and v (= ~r+ v ' ) . Br odk ey et a l . ' s (1974) ter -minology was used in th i s s tudy to char ac te r ize thevar ious events o f the cyc le . For com par i so n o f our r e su l tswith others , the s tress u v was f ur the r c la s s i f ied accor d ingt o t h e " h o l e " m e t h o d i n t r o d u c e d b y L u a n d W i l l m a r t h(1973).

3 Re s u l t s

3.1 M ean s ta t i st i csThe exper imenta l da ta f or the mean ve loc i t i e s wer e cur ve-f i t ted, in a leas t- squares sense, to the wake log- l inear ex-pr es s ion :U~0- U 1 In + 1 + cos (3.1)H, )4

to de term ine 5, u, , and ~(x ) ; )4, 5, and ~(x) are theV o n K a r m a n c o n s ta n t , b o u n d a r y - l a y e r t h i c kn e s s, a n d t h ewake pa r amete r , r e spec tive ly . I n wa l l coor d ina tes , theabove expr es s ion y ie lds :u+ = 1 ln y + + C + 1 - cos (3.2))4 )4 \- U- j/withu _ U . , 6 + = c Su , ," y + _ y * u ,b/, v V

+ _ l l n 6 - 2 n ( x)C = u~0 - - ( 3 .3 a , b , c , d )Table 1 l is ts the de te r min ed pa r ame te r s u , , n ( x), C , c5+,and F ig . 1 shows the me an w ind pr of i le s in wa l l coor d i -na tes ( u + vs. I n y+) . The lower pa r t r epr esen ts the log ar i th -m i c r e g i o n o f a ty p i c a l t u r b u l e n t - b o u n d a r y - l a y e r v e l o c i typr of i le , whi le the upper pa r t shows the wake char ac -te r i s t i c . For the h igher wind speeds , the wake becomesmor e pr onounced , the e leva t ion wher e i t beg ins to be f e l t ,lower , and the lower por t ion of the pr of i l e c lose to the a i r-wa te r in te r f ace devia tes f r om the logar i thmic law. Ther e ,Table 1. Mean velocity pro file characteristicsU~o u , 7~ x) C ~+(cm/sec) (cm/sec)141 4.25 0.53 12.76 1,453179 5.52 0.34 12.35 1,886200 6.20 0.30 11.87 1,922231 7.24 0.43 10.77 2,425346 11.91 0.31 7.48 2,725402 15.57 0.26 4.12 3,556


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Y . Papadimitrakis et al.: Turbulen t bounda ry over a mobile and deformab le bound ary 9935 I I I I I I I

30

u + 26

20

15101

Fig. 1.

I I I I I I I I I IO = 141 cm s "1 = 47,500U~o ReSo = 179 60,300[] = 200 67,600%, = 231 77,900O = 346 = 402 = 115,700

= 135,700R e = U 6 ~o/t~so oooA~ G ~ ~

v %,

I I I I I I I II IlO 2

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F i g . 1 . Mean velocityprofiles

70

60

50

403020

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f) O ) I I I 1

R e ~ o = 1 3 5 , 7 0 0 "U~ = 2.5 8cU ' U ' ,

oQ

oQ

I I I e l ~0 0 . 2 0 . 4 0 . 6 0 . 8Y / 8 0

1 4

12I 0

86

l I I I

o o I o~- - - . - - - f X

QO

QQ

I I I O /0 0 . 2 0 . 4 0 . 6 v 8 -Y / 8 o

14 I i I I

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e Uso8

6

4 oQ2

oO 0 I I I I ~0 . 2 0 . 4 0 . 6 0 .8

Y/8o1. 0

F i g. 2 . T y p i ca l m e a n t u r b u l e n t R e y n o l d ss t r e s s d i s t r i b u t i o n

the ve loc i ty magni tude s a re l a rger than those predic ted b ythe log l aw. Thi s devia t ion may be a t t r ibuted to thegenera t ion o f the sur face dr i f t cur rent , whose m agni tude i spropor t iona l to the wind speed (Wu 1975) . The wakecharacteris t ic of the mean veloci ty profi les was alsoobserv ed b y Hsu et a l . (1981), and Hsu a nd H su (1983).The var i a t ion of the in te rcept s of these prof i l e s wi thincreas ing wind speed or Reynolds number, as also indi-cated by the decreas ing values of constant C, can beat t r ibuted to the changing surface roughness condi t ion(see also sect ion 3.6). For low wind speeds the f lowbecomes super smooth due to the re la t ive sur face s t res sdecrease caused by the dri f t current , resul t ing in values ofC grea te r than the i r f ixed-sur face counte rpar t s . For h ighwind speeds (c/U~o< 0 .45) , r ipples becom e p romin entsur face roughness on the wa te r waves caus ing a shears t ress increase which is greater than the correspondingstress reduct ion produced by the dri f t current , resul t ing insmaller values of C.

3.2 Mean turbulent and mean-wave associatedReynolds stressesTypica l mea n turbu lent Reynolds s t res s prof i l e s a re shownin F ig. 2 . The magni tudes of u ' u ' a re a pproxim ate ly one

order l a rger than those o f - u ' v ' and v ' v ' . An a lm os tconstant s t ress layer near the wave surface is observed ine a c h p r o f i l e o f - u ' v'. The decrease of shear s t ress atthe lowest port ion of the constant s t ress layer can bea t t r ibuted to the d amp ing of turbulence by v i scos i ty . Thenon-normal ized mean turbulent Reynolds s t res ses t end toincrease with wind speed.

F igure 3 shows a typica l d i s t r ibut ion of the m eanwave-a ssociated Reyn olds s t ress ui uj ( - 10%) as a func-t ion of y*/6o. As this f igure shows, the s ign of the s t resschanges throughout the boundary l ayer , in agreement wi ththe ob servat ions o f Hsu et a l. (1981), and Hsu an d H su(1983). For c~U~0 = 0.45 and 0.39, ~7 v- rem ain s surp risi ng lyenough posi t ive. This picture suggests that the energydrains ei ther from the mean f ield to the waves or viceversa , depending on c~ U~o and the specif ic locat ion withinthe boundary layer. Kendal l (1970) and, with some excep-t ions , Takeuchi etal . (1977) found the wave-associateds t res s to be pos i t ive and nega t ive be low and above thecri t ical height (where U = c) . This contradicts the resul tsof La i and Shemdin (1971) , who found l a rge nega t ivecontribut ions to the u, v cospectra (a downward f lux) atthe wave f requency when c/U~o< 1. Our da ta suppor tTakeuchi e t a l . (1977) measurements and , wi th someexcept ions , those o f Kend al l (1970).


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1003 5

3 0

2 52 01 5

1 050

I i ! IR%=7 7,90,U 0

~o= 1.48cd U x [ O4

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o oI ? ? io0 0.2 0.4 0.6 0.8

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Experiments in Fluids 4 (1986)

Fig. 3. Typical mea n wave-associatedReynolds stress distribution

2OOI0 0

< ~ ' 7 ) o( C 2/sec2_) 0 0

-200 ~--1 I I0 0.2 0.4 0.6

t (see)Fig. 4. Typical Ev" pha se-ave raged distribution

eo Uso/c = 2.58o o ooo oo

cpe e oo~

I0.8 1.0

20 (I0(( u' v') 0(cm2/sec z)

- I 0 0-20C

c/U8 ,= 0.39 Re6o= 135,700

0.0 0.2 0 . 4 0.6 0.8t ( s e c )

F i g . 5 . T y p i c a l u ' v ' p h a s e - a v e r a g e d d i s t r i b u t i o n

GOOJ~Q OOo O~O G (9OO O o ~ O O OO OO O O@ O -. . . . . . . . . o - - o - . . . . . . . . . . . . . . . . Q __e__Q GO~ o20 Q ~ O ~ O O O Q

I I I 1 1. 0

3.3 Phase-averaged wave-associated and wave-inducedReynolds stressesT h e m a i n c o m p o n e n t ( f u n d a m e n t a l m o d e ) o f t h e p ha s e -averaged wave-as soc ia ted s t ress (~ i u j ) appears a t tw icethe fundamenta l wa te r wave f requency, a s i t can readi lybe seen f rom the re la t ionship .

1(~e~j)=~lc~,ll~jlcos(o~t-o~,)cos(~ot-o~,). (3.4)There fore , the ha rmo nics of the ( ff i i f/ ) s tress ap pea r a tfrequenc ies 4, 6, 8 Hz, e tc ., for a 1 Hz mec hanica l ly-genera ted wate r wave . Thi s behavior i s apparent ly shownin Fig. 4, represe nt ing a typical pha se avera ged ff~

s tress dis t r ibut ion for the various wind speeds consideredin this study.

Typica l pha se -averag ed resu l ts for the u' v' stress at thelowes t poin t o f measurements , shown in F ig . 5 , dem on-s trate that this s t ress is relat ively high on the windwards ide of the wa te r wave , wi th the except ion of lowes t windspeed run, where c /U6o> 1. This behavior is consis tentwith the m easu rem ents o f Okud a et a l . (1977), H su et a l .(1981) , Okuda (1982) and the predic t ion of Gent andTai lor (1976), and can be at t r ibuted to the response ofturbulence to a va ry ing mean f low.3.4 Wave-indu ced Reyn olds stresses 2Figure 6 shows a typica l amp l i tude (+ 8%) and pha se(_ 10 ) dis t r ibut ion of these s t resses . The amp li tude sIi / I are large near the interface and decrease as y*/5oincreases . This decrease, however, is not monotonic, butshows an osci l la tory behavior. A dip is occas ional lyobserved in the i~ i/ I prof i l e s with a cor responding phasejump, but no cons i s t en t pa t t e rn can be d i sce rned . Theosci l la tory beh avio r of the I /z i; I amp li tude s was also fou ndby Gent and T aylor (1976) in ca lcula tions of the in te r facef low in a curv i l inea r coordina te sys tem under f in i t e -amp l i tude wave condi t ions . A cons i st en t inc rease of thepea k value of ]~i / ] with increas ing wind speed was alsofound. The decrease in ]By] as 9 / 6 o ~ 0 is again due tothe v i scosi ty damping.3.5 Classification of Reyn olds stressesFigure 7 shows the fract ional contribut ions u v i /u v to thetotal m ean Reyn olds s t ress u v at the lowest point ofmea surem ents , as a funct ion of wind speed ; u vi representsthe contrib ut ion of the i th quad rant . I t is seen that thesecond quadrant (eject ions) is the larges t contr ibutor tot h e ~ - s t r e s s , w i t h t h e f o u r t h q u a d r a n t ( s w e e p s ) t h esecond l a rges t . The cont r ibut ions f rom outward and wal l -ward interact ions (f i rs t and third qu adrants) are also

2 De fine as: ?) /=


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Y. Papadimitrakis et al. : Turbulent boundary ov er a mobile and deformable boundary2520

AI r l l _~ l I 5Us o2x 10410

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i i i iR else= 77 19 000g o = 1 A 8

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2 701 8 0

900 , 3 6 0

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O?2a 900~360

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0 0.2 0.4 0.6 0.8-N-Y 18o

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Fig. 6. Typical amplitu de and ph ase distributionso f Vii , F12 an d F22

2.01.61.20.80.4UV-- 0UV

-0,4-0.8

"-r~/

--//-

-1.2-1.6-2.0 L.L//0.2

"B' t ,~aB,

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014 016 018 1.10 1.2cU8o

Fig. 7. Fractio nal contributions uvi/~ at the lowest point ofmeasurements: y*/&0= 0.014; 0.015; 0.015; 0.015; 0.018; 0.019;0.019 in o rder o f increasing wind speed. ~ 1st quadra nt; "~ 2ndquadra nt; j~ 3rd quadran t; R 4th quadran t

s i gn i fi c an t , b u t s m a l l e r . F o r c/U6o>0.68, e ject ions con-t r ibu te on the aver age 90% to the total mean s tress . Thisr esu l t agr ees wi th the smooth- wal l tu r bu len t boundar ylayer behavior obse r ved by Cor ino and Br odkey ( 1969) ,Kim et a l . (1971) , Lu and W illm ar th (1973) , and others .I ndeed , th i s f r ac t iona l cont r ibu t ion i s somewhat g r ea te rthan the value 77% or 80% repo r ted by Ki m et a l . (1971) ,and Lu and W i l lmar th ( 1973) , r e spec tive ly . I n the s amer ange of wind speeds , sweeps cont r ibu te abou t 77% to theto ta l m ean s t res s, l eav ing - 67% to the o the r two nega t ivecont r ibu tor s . As can be seen f r om F ig . 7 bo th f i rs t andth i r d quadr an ts have iden t ica l cont r ibu t ions ( 34% and33%, respectively) . Lu and Willmar th (1973) have repor tedva lues of 55% and - 32% f or the co nt r ibu t ions of sweeps,wa l lwar d , and ou twar d in te r ac t ions . The inc r eased sweepc o n t r i b u ti o n s w e r e a l so o b s e r v e d b y N a k a g a w a a n d N e z u( 1977) in the i r r ough- bed , open- channe l f low measur e -ments . They a t t r ibu ted th i s inc r ease to the bed r oughnessand conc luded tha t , " . . . a s r oughness inc r eases and the


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102 Experim ents in Fluids 4 (1986)di s t ance f rom the wa l l dec reases ; sweeps appear to bemore impor tant than e jec t ions . " For c/U~o < 0.68, a signif-i cant change in the beh avior of f rac tiona l cont r ibut ionsu v i / ~ i s obse rved . The cont r ibut ions of a l l quadrant sprogres s ive ly inc rease , but the ra t ios u-vS '2 /~ and ~ /~rem ain alm ost unal tered. Values as high as 192%, 174%,- 146% and - 120% were found at c/U~o = 0.39.Whi le the turbulent cor re la t ion coef fi c i en t R '= - u ' v' /[RMS (u ' ) ' RMS (v ' )] was found to rem ain cons tant wi thwind speed, the correla t ion coefficient R = - u T /[RMS (u) RMS (v)] decreases progress ively from about 0.3at c/U~o~_0.68 to 0.1 at c/U~o~_0.39. This decrease iscaused by the m odi f i ca t ion of the a i r f low f i e ld due to thewave- induced per turba t ions , a s can be seen by expres s ingR in t e rms of the wave-coherent and turbulence s t res sesviz.

12Y+ u'v 'R (3.5)As the wind speed increases , the s t ress t r decreasestoward ze ro due to the rap id decrease of Y per turba t ions ,provided the s t ress u 'v ' doubles only a t the h ighes t windspeed. For c/U6o>0.45, the wave-associated s t ress re-mains pos i t ive c lose to the wa te r sur face , a l though smal l ,and therefo re lu T] < ] u ' v ' [. In contras t , tg2-~~ , a n d v~2increase wi th wind speed (but not monotonica l ly) , andthus the coefficient R decreases . Since the fract ional con-t r ibut ions u vi/uv increase with decreas ing R (Lu andWil lmarth 1973), i t becomes evident that the wave-coherent p e r turba t ions a f fec t the d i s t r ibut ion o f thesecont r ibut ions for d i f fe rent wind condi t ions through thevariat io n of the wave-asso ciated s t ress , which in turnreflects the cri t ical and viscous (Stokes) layer dynamics , asHsu and Hsu (1983) ha ve shown. A de ta il ed desc r ip t ion ofboth the turbulent and w ave-coherent ve loc i ty f i e lds inthe air can be found in Papadimitrakis (1982). Variat ionsin sur face roughness , due to the wave growth , probablyaffect the skewness and diffus ion factors , as a lso discussedby Nakagawa and Nezu (1977) for open-channe l f lows ,and the re fore a l t e r the probabi l i ty dens i ty d i s t r ibut ion ofthe Reyno lds s t ress u v, and ul t ima tely the fract iona l con-t r ib u t i on s ~ / ~ , .

3. 6 Distributions of the contributions of the various eventsthroughout the boundary layerFigures 8 show typical dis t r ibu t ions of the fract ional co n-t r ibut ions of the va r ious events to the mea n ( ~ ) Reynoldss tress as a funct ion of y*/6o. The d i s t r ibut ions of pos i t iveand nega t ive f rac t iona l cont r ibut ions to u 'v ' t h r o u g h o u tthe boundary l ayer a re a l so inc luded in these f igures(Papadimitrakis 1982). As can be seen, the intens i t ies ofthe various events sat is fy the relat ion: e ject ion > sweep> outward in te rac t ion > inward in te rac t ion wi th in theobserved range of y*/6o. The d i f fe rence be tween the twointeract ions , however, is negl igibly small . This behavior

was a l so observed by Lu and Wi l lmar th (1973) , Nakagawaand Nezu (1977), and others . Close to the water surface andup to the middle of the equi l ibr ium region (y*/6o < 0.3),the de ta i l ed d i s t r ibut ion of the va r ious events depends onthe rat io c/U~o. Except for the highest wind speed, theintens i t ies of a l l the events increase with increas ing y*/6o,reach a maximum, and then e i the r decrease unt i l theyreach a minimum or remain fa i r ly cons tant . The widthand loca t ion of the two ext rem a depend c lea r ly on c/U~o.At the highest wind speed, the intens i t ies decrease withincreas ing y*/3o, in agreement wi th the resu l t s ofTakeuchi etal . (1977). Lu and Wil lmarth (1973) andBrodk ey et a l . (1974) reporte d an inc rease of the variousintensities as y*/6o ~ O. N a k a g a w a a n d N e z u ( 1 9 7 7 )observed a s imi la r .beh avio r for the i r smooth-bed , openchannel f lows, but they found a decrease in the intens i tyof e j ec t ions and sweeps in the i r rough-bed case . However ,the i r rough bed measurements do not ex tend deeplyenough in the wa l l reg ion (down to about y*/6o ~- 0.085),and thus i t i s unknown whether these t rends cont inue there.

In the equi l ibrium region (0.1 < y*/6o < 0.6), the inten-s i ty of each event is nearly constant , i rrespect ive of y*/6o.The ra tes of in tens ity cont r ibuted by e jec t ions and sweepsare in excess of 100%, respect ively, and the excess st ressba lances the sum of the cont r ibut ions of inward andoutward in terac tions . In the f ree -s t ream (y*/6o > 0.6), theintens i ty of each event rapidly increases with y*/3o. T h esame behavior , obse rved in the equi l ibr ium and f ree -s t ream regions , has a l so been repor ted by a l l the above-ment ioned inves t iga tors . The re la t ion be tween the in ten-s i t ies of each event shown in the previous f igures confirmsthe observa t ion made by Gras s (1971) tha t both e jec t ionsand sweeps exis t i rrespec t ive o f the rough nesscondi t ion .

The d i f fe rences found in the wa te r proximi ty a rethought to be the phys ica l consequence of the changingsur face- roughness . The sur face roughness condi t ion char -acterized by the roughness Reynolds number Re , = u, yo/vwas found to va ry f rom aerodynamica l ly smooth toroug h in this investigation; the rough ness h eight, y0 = I/v/ '2,where ~ ' represents the ins tantaneo us r ipp le height , wasca lcula ted f rom the wave-he ight records . Other mecha-ni sms, depending on the dynamica l beh avior of the wave-coherent ve loci ty pe r turba t ions and Reynolds num bereffects , are also responsible for the observed behavior.The fac t tha t both e jec t ions and sweeps , which a re thepredominant event s in the burs t ing phenomenon, a rea f fec ted by the roughness condi t ion i s ve ry impor tan t andwas d i scussed by N akag awa and Nez u (1977).

I t i s s een tha t the av erage va lue o f the ra t io ~ / u v 4 , atthe lowes t poin t of measurements , i s appro xima te ly 1 .15,muc h smal le r than 1 .85 repor ted b y Lu and Wi l lmar th(1973) close to the wal l . A sharp r ise near the interface isob ser ve d for low wind speeds , whi le for mos t of theboundary layer this rat io is nearly constant , with valuesbe tween 1 .05-1 .20 . Lu and Wi l lmar th (1973) repor ted thesame beh avior , bu t they found a va lue of 1 .35 . As the


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Y. Papadimitrakis et al.: Turbulent boundary over a mobile and deformable boundary

U V i ( U ' V ' ) jUV uV

5"t2

I

0

- I

-2- 3-4 -

- 5

I I I I

c / = 0 . 4 5 Re s o 1 1 6 , 7 0 0UBo"n

I "D [3"

,o" ~ ,o" ,o" ..o

4~

I I I t.0 02 0.4 0.6 0.8 1.0

Y/goI I I I

CA =0 . 78 Re8~ 67 , 600- UB o

5

4.

3

2I

0

-I

-2- 3-4

- 5

EL n. n.

UVi ( U'V'>i ~ I JU~Ja"_ - - , . . . . ..... . . . . . . . J % p'UV uV

t I I I I0 0.2 0.4 0.6 0.8b Y/Bo

J:

1.0

103

Fig. 8a and b. Distributions of ~Y~,/~ and (u~?)j/u' v' with H= O. ~ 1st quadrant; 'El 2rid quadrant;3rd quadrant; ~, 4th quadrant; ~ negative part;e positive part

wind speed increases, this ratio increases in the equilib-rium and most of the free-stream regions, with a tendencyto decrease in the centerline channel region. This de-creasing behavior towards the water surface for high windspeeds agrees well with the results of Wallace et al. (1972),Brodkey et al. (1974) and Nakagawa and Nezu (1977) andmay be attributed to the effect of surface roughness. Rey-nolds number effects may also be responsible. Away fromthe interface, where surface effects decrease, the ratiouv2/ gF 4 increases toward a value of approx imately 1.4, in

agreement with smooth, flat-wall observations. It is seenfrom these characteristics, as well as from the contribu-tions of the various events to u v, that the burst ing processnear the free stream may consist of smoother and moreisotropic events.3. 7 Class i f icat ion o f Reyn olds s t ress accordingt o t he "ho l e' " me t hodTypical contributions of the various events of the cycle tothe mean Reynolds stress u v as a function of the "hole"


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104 Experiments in Fluids 4 (1986)

iUViU V

2.0

1. 6

i 2

0 . 80.4.0 . 0

- 0 . 4

- 0 . 8- I .2

- I . 6

- 2 . 00

1 I I

C /us o= 0 .8 7 R e 8 o - - 6 0 ,5 0 0

X X X X X X X0 0"IS] x0

~ o

~ ~" ~ ~' ~

I I lZ 3 ,4

H OL E S IZE , H

Fig, 9. Measurements of the contributions to fly fromdifferent events. ~" 1st quadra nt; "~ 2nd qua drant;U 3rd quadrant; ~ 4th quadrant; o hole; x fraction oftime spent in the "h ole"

s ize H , a t the lowes t poin t of measu remen ts a re shown inF ig. 9 . The f rac t ion of t ime spent in the "hole" reg ion i salso included in these f igures . As can be seen from thecontrib ut ion curve related to the "ho le" , u v has an inter-mi t t ency fac tor of the ord er of 0 .53. L u and Wi l lmar th(1973) and Nakagawa and Nezu (1977) a l so observed thesame behavior. Eject ions are always the larges t contr ibutorto ~ , wi th sweeps the s econd l a rges t , i rrespec t ive of windspeed and "hole" s i ze , whi le the cont r ibut ions f rom u v land u v3 are nega t ive and re la t ive ly smal l . However , forc/U~


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~om

Y. Papadimitrakis et al.: Turbulent boundary o ver a mobile and deform able boundary

10 7

I0 1

10 0

10"1

0 1 2 3 4I / ] I I

o

c/U,5 : 1. 11 o: 0.87 A= 0.78 X= 0 .68 [ ]= 0.55 v= 0.4 5 (b) ~ ~ N: 0 .39 A~ o

| x ov! ~ o

v ~ o|v ~ oo

o II v

~ o

o

I I I I1 2 3 4

Ho l e S i ze , H

103

!10 2

o

- - l O ~

II10 0v

- - 1 f f l

I5

T S U6

Fig. 10. Mean tim e interval between ejections an d/o r sweeps atthe lowest point of measu remen t

Table 2. Ejection and sweep tim e intervalsUao (cm /sec) 141 179 200 231 285 346 402Reo 4,180 4,620 5,100 6,460 6,050 7,500 9,150T~ 51 66 80 109 339 611 1,065T s 125 201 286 354 928 1,0 53 1,814iPe(H=4)(sec) 0.443 0.341 0.326 0.325 0.511 0.673 0.685

there fore , m ax ]u v I ~ - 1 0 1 ~ I . For the h igher windspeeds, max luv [ wi ll be even grea te r due to the de -creas ing correlat ion coefficient R. These eject ions arecer t a in ly v io lent and mus t come f rom la rge sp ikes in theu v s igna l. T he n ondimen s iona l mean t ime in te rva l s be -tween e jec t ions and/or sweeps T& s= TB, s" u 2 /v, scaledwith the inner variables and corresponding to H = 4 werenot found to remain cons tant , bu t inc rease wi th momen-tum th ickness Reynolds number Reo as shown in Table 2,in agreement aga in wi th Bogard and Tiederman ( t981) .The same holds t rue for any o ther v a lue of the thresholdand/or loca t ion y*/6o th roughout the boundary l ayer .Since the viscous lengths of the film w ere < 20, we hav e noreason to be l i eve tha t our resu l t s a re contamina ted bywire-length effects , as discussed by Blackwelder and Hari-tonidis (1983) an d Alfredsson and Joh ansson (1983). T he

lack of e jec t ion and /or sweep t ime sca ling with the inner

105and/or oute r f low var iab les , however , should not be sur -pris ing, because Jackson (1976) has pointed out thatnone of the da ta f rom geophys ica l boundary f lows suppor tthis t ime scal ing, a t leas t with the outer f low paramete rs .

4. Discussion4.1 Influence o f the mobile boundary on the product(onof R eynold stresses and the bursting-related pheno meno nFo r H = 4, the d ime nsio nal ejection t im e interv als 2PB (sec)show a som ewhat d i f fe rent behavior , a s s een in Tab le 2 ,inasmuch as they s l ight ly decrease with wind speed in therange of U&= 141-179 cm/sec, ins tead of increas ing. I t i sremarkable, however, that these eject ion t ime intervalsclosely correspond to the t ime between the larges t con-secut ive nega t ive or pos i t ive peaks of the phase -averagedReynolds s t resses , as can be seen by combining Figs . 4and 5. The fol lowing s imple analys is a lso demonstrates theva l id ity of the abov e a rgument . T o a f i r st approxim at ion:( u L ,) ~_ ( ~ ) + ( u ' ~ , ' ) = 1~ I I~1 cos(Zcot-Oa-O,~)+ l,~121 c o s( ~ o t- % ) +1 al le l co s( 0~ - 0,~) + u 'L, ' . (4.1)There fore , the main co mpo nent of the wave-as soc ia teds tress ffzr appe ars at 2 Hz, with a t im e interval betweensuccess ive negat ive or posi t ive peaks of about 0.5 sec. Thewave-coherent turbulent s t res s then modula tes th i s t imeinterval according to the phase lags of Oa, Oe an d 0e1~. Th us,the t ime period between success ive eject ions in interfacef lows i s roughly de te rmined by both the wind condi t ionsand the moving boundary characteris t ics . This conclus ionsensibly agrees with Kendal l ' s (1970) observat ion that" . . . an in t r ins ic t ime cons tant i s an impo r tant fea ture ofthe f low" , provided the turbulence s t ruc ture l ags behindthe wave.

Since only a s ingle-frequency external wave osci l la t ionwas examined in this inves t igat ion, no defini te conclus ion:wi l l be drawn on the re la t ion be tween the mean e jec t ionper iod and the t ime in te rva l be tween the two consecut ivela rgest peaks of the ph ase -average d shear s tress d i s t r ibu-t ion. Further research is being conducted on this aspect .

5 Summary and conclusionsThe resul ts and discuss ions presented in the previous sec-t ions suggest the fol lowing co nclus ions:

(1) The constant C characterizing the me an v eloci typrofi les decreases with increas ing wind speed as a resul tof the va r i a t ion of the wa te r sur face roughness condi t ionbe tween sm ooth and fu l ly rough regimes ; accordingly , thef r i c t ion ve loc i ty u , becomes smal le r and/or grea te r thani ts counterpart value over a f ixed surface, depending onthe local wind speed.


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106( 2) T h e l o g - l i n e a r f o r m o f m e a n v e l o c i t y p r o f i l e s , t h e i r

w a k e f r e e - s t r e a m c h a r a c t e r i s t i c , a n d t h e s i m i l a r i t y b e -t w e e n t h e m e a n t u r b u l e n t R e y n o l d s s tr e ss p r o f i le s a n dt h o s e o b s e r v e d o v e r a f l a t su r f a c e c l e a r l y i n d i c a t e t h a t t h et r a n s f o r m e d c o o r d i n a t e sy s t e m g i v e n b y ( 2 . 1 a - c ) d e -s c r i b e s b e s t t h e a i r f l o w a b o v e w a t e r w a v e s i n t h e p r e s e n c eof a swel l .( 3 ) T h e d a t a p r e s e n t e d s h o w t h a t w a v e s i n f l u e n c e t h ea i r f l o w a b o v e t h e m . B o t h t h e w a v e - i n d u c e d a n d t u r b u -l e n t v e l o c i t y - f i e l d c h a r a c t e r i s t i c s d e p e n d o n c/U6o. T h ep h a s e - a v e r a g e d w a v e - a s s o c i a t e d s tr e ss e s w i t h t h e i r m a i nc o m p o n e n t a t t w i ce t h e fu n d a m e n t a l w a v e f r e q u e n c y a r es i gn i f i can t .

( 4 ) T h e p r e s e n c e o f w a t e r w a v e s e f f e c ts t h e m e c h a n i s mo f R e y n o l d s s tr e ss p r o d u c t i o n t h r o u g h t h e v a r i a t i o n o fs u r fa c e r o u g h n e s s c o n d i t i o n a n d t h e d y n a m i c s o f t h ew a v e - c o h e r e n t v e l o c i t y f i e l d in t h e a i r . T h e b u r s t i n ge v e nt s a r e g e n e r a l l y e n h a n c e d b y t h e p r e s e n c e o f w a t e rw a v e s . T h r o u g h o u t t h e b o u n d a r y l a y e r , e j e c t i o n s a r e t h el a r g e s t c o n t r i b u t o r s t o u - ~ , w i t h s w e e p s t h e s e c o n d l a r g e s t .F o r c/U6o> 0 . 6 8 , e j e c t i o n s a n d s w e e p s c o n t r i b u t e a b o u t90% a n d 7 7% t o t h e m e a n R e y n o l d s s t r e ss , w h i l e t h eo u t w a r d a n d i n w a r d i n t e r a c t io n s c o n t r i b u t e - 3 4 % a n d- 3 3 % , r e s p ec t iv e l y . F o r c/U6o

fully developed trubulencet

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