no title

Aerosp. Sci. Technol.4 (2000) 249–262 2000 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS1270-9638(00)00137-1/FLA

On the spatial structure of global linear instabilities and their experimentalidentification *

Vassilios Theofilisa,1

a DLR Institute of Fluid Mechanics, Division of Transition and Turbulence, Bunsenstraße 10, 37073 Göttingen, Germany

Received 5 August 1999; revised and accepted 8 March 2000

Abstract The purpose of the present paper is to discuss the spatial structure of global instabilities, solutionsof the partial derivative eigenvalue problem resulting from a nonparallel linear instability analysis of theincompressible Navier–Stokes and continuity equations, as developing upon four prototype essentially two-dimensional steady laminar flows. Theoretical knowledge of these eigendisturbances is instrumental todevising measurement techniques appropriate for their experimental recovery and ultimate control of laminar-turbulent transition mechanisms. Raising the awareness of the global linear flow eigenmodes contributes toredefining the boundaries between experimental observations which may be attributed to linear global asopposed to nonlinear mechanisms. 2000 Éditions scientifiques et médicales Elsevier SAS

laminar-turbulent flow transition / nonparallel linear instability / partial-derivative eigenvalue problem

Zusammenfassung Über die räumliche Struktur globaler linearer Instabilitäten und deren experimenteller Erkennung.Zweck dieses Artikels ist eine Diskussion der räumlichen Struktur globaler Instabilitäten, die von einer nicht-parallelen linearen Instabilitätsanalyse der inkompressiblen Navier–Stokeschen und Kontinuitätsgleichungenmittels Lösungen eines partiellen Differentialgleichungs-Eigenwertproblems gewonnen werden. Dazu wurdenvier prototype Konfigurationen gewählt, die essentielen zweidimensionalen stationären laminaren Strömungenentsprechen. Theoretische Kentnisse über diese Eigenstörungen ist eine Grundvoraussetzung für die Konstruk-tion experimenteller Methoden, um die globalen Instabilitäten messen zu können und zu steuern. Eine erhöhteWahrnehmung der globalen linearen Strömungsmoden trägt dazu bei, die Grenzen zwischen experimentellenBeobachtungen, die als lineare globale bzw. nichtlineare Phenomäne anerkannt werden, neu zu definieren. 2000 Éditions scientifiques et médicales Elsevier SAS

laminare-turbulente Transition von Strömungen / nichtparallele lineare Instabilität / partielles Differen-tialgleichungs-Eigenwertproblem

Nomenclature

Latin SymbolsA aspect ratioDx ∂/∂x

* This paper is based on the DLR Zentrumskolloquium entitled “On the origins of transition and turbulence in three-dimensional flows” delivered on15 April 1999 at the Institute of Fluid Mechanics of the DLR Göttingen

1 E-mail: [email protected]

D2x ∂2/∂x2

Dy ∂/∂y

D2y ∂2/∂y2

i imaginary unit

250 V. Theofilis / Aerosp. Sci. Technol. 4 (2000) 249–262

p disturbance pressureq a three-dimensional unsteady solution of the

incompressible equations of motionq a steady-state two-dimensional solution of the

equations of motionq small-amplitude three-dimensional unsteady dis-

turbances superimposed uponqq amplitude functions pertaining toqRe Reynolds numbert time(u, v, w) basic flow velocity components along(x, y, z)(u, v, w) disturbance velocity components along(x, y, z)(u, v, w, p) amplitude functions pertaining to the eigen-

mode(u, v, w, p)(x, y, z) Cartesian spatial coordinates

Greek Symbolsβ wavenumber in the homogeneous directionzε amplitude parameter,ε 1ω complex eigenvalueωr <ωωi =ωΩ nonperiodic two-dimensional spatial domain on

which q andq are defined

SuperscriptsT transpose

1. Introduction

The quest for identification of deterministic routes inthe phenomenon of laminar-turbulent fluid flow transi-tion has been pursued, for the best part of last century,through numerical solution of the linear eigenvalue prob-lem proposed by Tollmien [26]. The premises on whichTollmien’s theory is based is the decomposition of anyflow quantity into a steady one-dimensional profile andan unsteady three-dimensional part, superimposed uponthe steady so-called basic flow at a linearly small am-plitudeε. The resulting system of equations at O(1) de-scribes the basic state (e.g. the Blasius profile in the in-compressible flat-plate boundary layer or the parabolicprofile of plane Poiseuille flow), the system at O(ε) isthe one solved while that at O(ε2) is neglected. Separ-ability of the linearised system of equations is achievedif, firstly, two out of the three spatial directions are con-sidered as homogeneous and treated as periodic and, sec-ondly, the basic flow velocity component in the spatialdirection normal to the wall in a boundary-layer typeof flow is neglected. The latter so-called ‘parallel-flow’assumption severely constrains the type and number offlows on which Tollmien’s classic linear instability theorymay be applied and, even within the realm of its applica-bility, has led to a considerable amount of controversy inthe past. The disturbance components along the homo-geneous spatial directions are decomposed in multiplesof fundamental wavenumbers in each spatial direction.The third spatial direction, the same as that on which the

basic flow depends, is resolved. The system at O(ε) inincompressible flow is none other than that which givesrise to the Orr-Sommerfeld and Squire equations [6]. Thelinear instabilities in physical space may then be inter-preted as plane or oblique wave-like disturbances, de-pending on whether a non-zero fundamental wavenum-ber is considered in one or two spatial directions, respec-tively. Following the celebrated experiment of Schubauerand Skramstadt [21], who demonstrated the existence andlinear amplification of wave-like disturbances (what laterbecame known as the Tollmien–Schlichting instabilitywaves) prior to laminar-turbulent breakdown in the in-compressible flat-plate boundary-layer flow, intense ex-ploitation of Tollmien’s linear theory and its extensionsto compressible flow [13] has created a wealth of infor-mation regarding the transition phenomenon on accountof this type of linear instability mechanism [16].

A widespread misconception in the community isthat linear instability is associated with wave-like dis-turbances alone. Consequently, experimental work aim-ing at the identification of linear instability mechanismsis currently almost exclusively utilising single hot-wireprobes which are transversed along a single spatial direc-tion (e.g. the wall-normal direction in a boundary-layertype of flow) in order for information on the spatial struc-ture of the disturbances to be collected. Such an approachimplicitly assumes that the only linear disturbances inthe flow are of wave-like nature and the aim of the ex-perimentation is to identify the spatial structure of theamplitude functions of different instability modes. Clearlythe success of such an approach is limited by its inher-ent assumptions; one either has no means of identifyingstructures in the flow other than the ones aimed at, or ifindeed identified, global linear instabilities which exist inthe flow may well be confused with nonlinear flow be-haviour.

On account of the property of resolution of a singlespatial direction we term Tollmien’s linear instability the-ory a ‘one-dimensional’ theory. Potential of confusionis immediately generated with both theoretical and ex-perimental work in the literature, which terms the caseof existence of a nonzero wavenumber along one of thetwo aforementioned homogeneous spatial directions as a‘two-dimensional’ and that in which nonzero wavenum-bers along both periodic spatial directions exist as a‘three-dimensional’ linear instability (e.g. [2,20]); in ourterminology both are approaches aiming at the quantifica-tion of one-dimensional disturbances. Within the frame-work of the parabolised stability equations (PSE), on theother hand, one resolves one spatial direction and con-siders only mild dependence of the basic flow and theinstabilities on a second spatial direction (see [10] andreferences therein); the third spatial direction is againtaken to be periodic. Such a treatment of the instabilityproblem could be termed a quasi-two-dimensional ap-proach.

V. Theofilis / Aerosp. Sci. Technol. 4 (2000) 249–262 251

Here we are concerned with essentially two-dimensio-nal linear instabilities. In a manner analogous to Toll-mien’s theory, the two-dimensional (also herein referredto as global) linear instability analysis is concerned withthe decomposition of a flow quantity into steady basicand unsteady disturbance fields, the latter superimposedat a small amplitudeε upon the former. However, both thebasic and the disturbance quantities are two-dimensionalfunctions of two resolved spatial directions. The basicflow is again obtained at O(1) and the O(ε2) terms are ne-glected; the O(ε) system of linear equations, on the otherhand, has coefficients which are independent of one spa-tial direction and time and, as such, may be written in theform of a complex nonsymmetric linear partial deriva-tive eigenvalue problem. In incompressible flow the onlyfree parameter of the problem, besides the Reynolds num-ber, is the wavenumber of the disturbances in the sin-gle periodic spatial direction. The solution of a systemof partial differential equations in two-dimensional lin-ear instability analysis, as opposed to the system of ordi-nary differential equations that the one-dimensional the-ory addresses, offers a great deal of more freedom inthe choice of boundary conditions associated with theproblem of global linear instability and, hence, has ob-vious consequences for the character of the solutions ob-tained.

The purpose of the present paper is to discuss somesuch structures with the aid of four examples of flow inboth closed and open systems, under a variety of bound-ary conditions imposed by the flow physics. We dis-cuss the eigenvectors delivered by the partial derivativeeigenvalue problem analysis of global linear instabilityin the rectangular duct, the infinite-aspect-ratio limit ofwhich is plane Poiseuille flow, the lid-driven cavity forwhich no rational approximation exists which may re-duce the linear instability problem to one for which theone-dimensional theory may be used (except, perhaps,in the large aspect-ratio limit of the flow), boundary-layer flow incorporating a laminar separation bubblefor which plenty of work exists using one-dimensionalanalysis but virtually none using two-dimensional lin-ear theory and, finally, the infinite swept attachment-line boundary layer formed on the windward face ofswept cylinders and wings. The latter problem is onein which experimentation on the basis of the assump-tions of the one-dimensional theory, in conjunction witha particular wall-normal structure of the global linearinstabilities has resulted in the latter having remainedconcealed in the results of continuous experimentationover four decades. Both the numerical aspects and de-tailed presentation of the results obtained from a phys-ical point of view are discussed elsewhere. In the lim-ited space available here the focus is placed on expo-sure of the spatial structure of the eigenvectors in anattempt to raise the awareness of the community, espe-cially of experimentalists, with respect to the expectedlinear disturbances potentially to be found in a given

flow problem. In section 2 we briefly discuss the par-tial derivative eigenvalue problem theory before present-ing in section 3 results for the most significant, froma linear instability analysis point of view, eigenmodesin the aforementioned examples as well as their spa-tial structure. Closing remarks are presented in sec-tion 4.

2. The partial-derivative eigenvalue problem theory

Alternative formulations exist which address thepartial-derivative eigenvalue problem governing global(two-dimensional) linear flow instability. Here we re-strict ourselves to the presentation of the most general,primitive-variables formulation, for the sake of intro-duction of the necessary notation. Taking incompress-ible flow, a solution of the equations of motion,q =(u, v,w,p)T, may be considered as composed of small-amplitude three-dimensional perturbationsq = (u, v, w,p)T superimposed upon a steady two-dimensional basicflow q= (u, v, w, p)T according to

q(x, y, z, t)= q(x, y)+ ε q(x, y, z, t), (1)

with ε 1. Here the three spatial directions are denotedby x, y, andz, respectively andt denotes time. Linearisa-tion aboutq is permissible on account of the smallness ofperturbations compared with the steady-stateq and theresulting system for the determination ofq is separablein both t andz on account of the steadiness and the two-dimensionality of the basic flowq. Eigenmodes are intro-duced in these directions such that

q(x, y, z, t)= q(x, y)ei[βz−ωt ] + c.c. (2)

with q = (u, v, w, p)T containing the perturbation velo-city components and disturbance pressure amplitudefunctions and i= √−1. Complex conjugation is intro-duced in (2) sinceq is real while all three ofq, β andωmay be complex. In the framework of the temporal linearnonparallel instability analysis used presently we writethe linearised system in the form of an eigenvalue prob-lem forω, while β is taken to be a real wavenumber pa-rameter describing an eigenmode in thez-direction. Thereal part ofω is related with the frequency of the instabil-ity mode while its imaginary part is the growth/dampingrate; a positive value ofωi ≡ =ω indicates exponentialgrowth of q in time t while ωi < 0 denotes decay ofqin time. In the present framework the three-dimensionalspace comprises an arbitrary two-dimensional domainΩextended periodically inz and characterised by a wave-lengthLz in this direction, which is in turn associatedwith the wavenumber of each eigenmode,β , throughLz = 2π/β . The system for the determination ofω andq takes the form of a complex nonsymmetric generalisedeigenvalue problem


[L− (Dx u)

]u− (Dy u)v−Dxp=−iωu, (3)

−(Dxv)u+[L− (Dyv)

]v −Dyp=−iωv, (4)

−(Dxw)u− (Dyw)v+Lw− iβp=−iωw, (5)

Dx u+Dyv + iβw= 0, (6)

subject to appropriate boundary conditions on∂Ω . Thelinear operatorL = (1/Re) (D2

x + D2y − β2) − uDx −

vDy − iβw. Some details on the accurate and efficientnumerical solution of the partial-derivative eigenvalueproblem (3)–(6) have been presented by Theofilis [24].The reader’s attention is drawn to the boundary condi-tions associated with this system. This issue is straight-forwardly resolved in the one-dimensional limit of (3)–(6) where one imposes homogeneous Dirichlet bound-ary conditions on the disturbance velocity componentson both solid boundaries and the free-stream. The reso-

Figure 1. Upper: the Orr-Sommerfeld and Squire spectra obtained by numerical solution of the partial-derivative eigenvalue problemin the rectangular duct atRe= 104, β = 1,A= 100. In both cases P and S correspond to the continuous spectrum; the Orr-Sommerfelddiscrete eigenspectrum is denoted by A [15] and that of the Squire modes falls into three families, denoted by K, L and M. Lower: thedisturbance eigenfunctions corresponding to the unstable mode; solid lines denote the real and dashed-dotted the imaginary parts of theeigenfunctions.


lution of two spatial directions,x andy in (3)–(6), on theother hand results in an elliptic problem for which bound-ary conditions on∂Ω are required. Here intuition offeredby the flow physics alone may provide plausible answers.We discuss in what follows both closed and open prob-lems, as examples of successful resolution of the issue ofboundary conditions associated with the partial derivativeeigenvalue problem.

Regarding the numerical methods for the solution (3)–(6), briefly, application of the classic QZ algorithm for therecovery of the full spectrum of eigenvalues/eigenvectorsof the partial derivative eigenvalue problem is not advis-able on account of both the memory and runtime require-ments of this algorithm which scales with the square andthe cube, respectively, of the product of the number ofpoints resolving each of the spatial directionsx andy.Krylov subspace iteration, on the other hand, offers anefficient means of recovering the most interesting, froman instability analysis point of view, part of the eigen-

spectrum at a fraction of the cost of the QZ algorithm.This permits probing into flow instability at substantiallyhigher Reynolds numbers at a given level of computingcost, compared with the QZ.

3. Results

3.1. Flow in a rectangular duct

A minimal test to which a new extension of an estab-lished theory must be subjected is the reproduction bythe former of the results of the latter theory. The rec-tangular duct flow is well suited for this exercise, sinceits large aspect ratio may be described by the classicplane Poiseuille flow (PPF), the linear instability resultsof which are well-known [18]. Tatsumi and Yoshimura[22], who were the first to address the problem of non-parallel linear instability in the rectangular duct albeitusing different numerical methods and more restrictive

Figure 2.The spatial structure of the neutral wall mode in rectangular duct flow atRe = 7500, β = 1, A= 6: normalised magnitude ofthe disturbance (top left) velocity componentu; (top right) velocity componentv; (bottom left) velocity componentw; (bottom right)pressurep.


Figure 3. The spatial structure of a decaying centre mode in rectangular duct flow atRe = 4000, β = 1: normalised magnitude ofthe disturbance (top left) velocity componentu; (top right) velocity componentv; (bottom left) velocity componentw; (bottom right)pressurep.

boundary conditions compared with those used presently,have mentioned that they also subjected their algorithm tothe same test by explicitly neglecting the dependence ofthe problem on one spatial dimension. By contrast herewe letA→∞, in practice permitting the aspect ratio totake large finite values. In a series of careful experimentsNishioka and co-workers (e.g. [17]) have established thatan aspect ratio in excess ofA = 25 is necessary in or-der for the PPF instability results to be recovered exper-imentally; a valueA = 100 is taken presently. Calcula-tions have been performed at the parameters of the testcase of Orszag [18],Re = 104, β = 1 usingNy = 48and a variable number ofNx collocation points resol-ving they ∈ [−1,1] andx ∈ [−A,A] directions respec-tively. As the aspect ratio increases, the eigenvalue spec-trum is found to be composed of clusters ofNx copies ofthe PPF Orr-Sommerfeld and Squire spectra, recoveredto within six significant digits. The spectra are presentedin figure 1clustered into families of eigenmodes [15]; the

eigenfunctions corresponding to the most unstable modeare also shown in this figure.

The aspect ratio is subsequently decreased to O(1)values and the results of Tatsumi and Yoshimura [22]are fully recovered [24]. From the point of view of thetheme of the present paper, the spatial structure of thedisturbance velocity components pertaining to the criti-cal wall mode atA = 6 are shown infigure 2. Parame-ters used for this calculation areRe = 7500, β = 1; thecalculated eigenvalue isω = (0.2542,0.0). Several as-pects of these results deserve discussion. The symmetriespresent in these results were imposed by Tatsumi andYoshimura [22] for reasons of numerical efficiency; bycontrast we recover these symmetries as a result of ourcalculations. Taking cuts through the line of symmetryx = 0 delivers ay-distribution of the eigenmodes whichis very similar to that found in the one-dimensional limitof the analysis discussed infigure 1; this fact is partic-ularly evident in the results for thev, w and p distur-


Figure 4. Isosurface of the disturbance vorticity magnitude ofthe neutral wall mode pertaining to flow in anA = 6 duct atRe = 7500, β = 1.

Figure 5. Isosurface of the disturbance vorticity magnitude ofa decaying centre mode pertaining to flow in a square duct atRe = 4000, β = 1.

bances. However, the most striking feature of the resultsis the pronounced wall-effects which are inaccessible tothe one-dimensional theory. Elaborating on the last point,we mention that the scope of application of the tradi-tional approach of transversing hot-wire probes throughthe flowfield in order for global two-dimensional lineareigenfunctions of the form presented in the results offig-ure 2 to be recovered is very limited indeed. In order forsuch results to be recovered reliably the experimental ap-proach used must be based on global field measurements,capable of recording the flowfield simultaneously at sev-eral different times; the decomposition (1)–(2) may then

be used in order to calculate the amplitude functionsq.An analogous conclusion may be drawn by inspection ofthe centre modes pertaining to flow in the square duct, atypical example of which atRe = 4000, β = 1 having aneigenvalueω = (0.959,−0.041) is presented infigure 3.A further point worthy of discussion here is that knowl-edge of the spatial structure of the two-dimensional lin-ear eigenmodes limits the possibility of confusing suchglobal coherent structures with nonlinear flow behaviour.This point may be strengthened by viewing isosurfacesof the magnitude of the disturbance vorticity pertainingto the results presented, which are presented infigures 4and 5; arbitrary scales have been used for the presentlinear results. Needless to say, such structures manifestthemselves as global flowfield oscillations without anyevidence of travelling wave-like behaviour.

3.2. The lid-driven cavity

Another closed-system flow problem in which linearinstability analysis may only be performed in the presentnonparallel linear analysis framework is that of a flow de-veloping in the square lid-driven cavity. Here no simplifi-cations exist which may reduce the instability analysis toa (one-dimensional) ordinary-differential-equation-basedeigenvalue problem and one has to resort to the two-dimensional partial-derivative eigenvalue problem or, al-ternatively, to direct numerical simulation [12]. Shortof following the latter path, which is not well-suitedfor parametric studies and in which different instabilitymechanisms might be confused, we solve the eigenvalueproblem (3)–(6) at given real parametersβ in order to re-cover the frequencyωr and the growth/damping rateωiof the respective eigenmode. Alongside, we recover thespatial structure of the eigenfunctions pertaining to eachmode. In line with the spirit of the present contributionwe refrain from discussing either numerical aspects orthe open issue of the critical Reynolds number in this flow[4,19] and concentrate on the qualitative discussion of thespatial structure of the eigenfunctions and of the flowfieldwhich is set up by linear superposition of the two mostunstable three-dimensional (β 6= 0) eigenmodes upon thesteady basic flow. The aim here is to demonstrate thata) linear mechanisms are indeed at play in the nonpar-allel laminar steady basic flow which b) have nothing todo with the classic wave-like structures encompassed inTollmien’s theory.

Figure 6 (left) shows a perspective view of the spa-tial distribution of the least-stable stationary eigenmodeat a low Reynolds number,Re = 100 and a wavenumberβ = 1. In figure 6 (right) the next in significance trav-elling eigenmode is presented. In all plots the lower leftcorner of the domain corresponds to that of the cavityand the origin of the axes(0,0), with the lid defined atx ∈ [0,1], y = 1; the upper row shows the real parts ofthe eigendisturbances, the lower row the correspondingimaginary parts, and the first to fourth column contains


Figure 6. Contour lines of the least-stable (damped) stationary mode (left), and travelling mode (right) in a square lid-driven cavityatRe = 100, β = 1. Shown is the two-dimensional eigenvector(u, v, w, p)T as a function of the cavity coordinates(x ∈ [0,1] × y ∈[0,1]).

the eigenvectors(u, v, w, p)T. Arbitrary scales have beenused for the present linear results. Qualitative analogiesbetween the results of the two figures exist with respect tothe following characteristics. First, the (damped at theseparameters) linear eigenmode is shown to inherit the gen-eral features of the basic flow to which it corresponds;the flowfield set up by thex-wise velocity component isin the direction of the lid motion in the upper half andin the opposite direction in the lower half of the cavity.Analogously, the disturbance velocity component in they-direction is such that fluid flows in the positive sensealong this axis iny < 0.5 and in the opposite sense iny > 0.5. The disturbance velocity component in the thirdspatial directionz provides a clear view of the vorticalmotion set up in the cavity. Finally, the disturbance pres-sure has been found to be the hardest eigenfunction toresolve; its correct capturing is key to reliability of thepartial-derivative eigenvalue problem results. In a man-ner analogous to the one-dimensional linear instabilityanalysis theory, the higher the eigenmode the more com-plicated its structure, as can clearly be seen by comparingthe results of thefigure 6.

At a much higher Reynolds numberRe = 1000,predicted by Ding and Kawahara [4] to lie within theneutral loop of this flow, the first travelling eigenmodegenerates a rather complicated flowfield, which has beenvisualised by means of isocontours of the spanwisedisturbance velocity componentw and shown at anarbitrarily-defined amplitude infigure 7. For simplicityonly one of the complex conjugate pair of eigenmodesis presented; in the flow a linear superposition of both,forming a standing-wave pattern, is to be expected.The edges(0,0) and (1,1) of the cavity correspond tothe lower left and upper right corners in this figure,respectively, and one periodicity length is shown in the

third (spanwise) spatial direction. One may appreciatein these results the complexity of the flowfield set upby the linear mechanism discussed here; well-definedstructures exist adjacent to all three stationary walls,x = 0, y = 0 andx = 1 while disconnected structures,all pertaining to the same linear eigenmode are to befound in the centre of the cavity. The point to be madewith the aid of these results is, again, twofold. Firstly,any attempt to recover such structures by single-probetransversing is doomed to failure; global simultaneousfield measurements are necessary for structures of thekind shown infigure 7 to be identified. Second, powerspectral analysis of the flowfield at conditions favouringgrowth of several eigenmodes makes identification ofthe latter in three-dimensional physical space a ratherlaborious task. In such cases it is advisable to focus onslightly supercritical conditions in the neighbourhood ofthe neutral loop in order to isolate and identify individualeigenmodes. In this respect, prior theoretical informationbased on the numerical solution of the partial-derivativeeigenvalue problem (3)–(6) is indispensable.

3.3. The laminar separation bubble

A flow whose linear instability still entails many openquestions, despite the large amount of knowledge havingbeen amassed over decades of research [5], is that in aboundary layer which encompasses a closed recirculationzone generated by an adverse pressure gradient, such asthat depicted infigure 8. A detailed discussion of the dif-ferent instability mechanisms operative in this flow maybe found in [25]; an analogous global linear instabilityproblem has been recently studied in separated-flow setup by the backward-facing step geometry [1]. The ma-jor difference, from a numerical point of view, with both


Figure 7. Isosurface of the spanwise disturbance velocity component of the most unstable mode atRe = 1000, β = 7.5. Flow is in thedirection of the arrow.

flows of sections 3.1 and 3.2 concerns the boundary con-ditions for the elliptic partial-derivative eigenvalue prob-lem (3)–(6). Flow in closed domains may be tackled us-ing the usual viscous boundary conditions for the distur-bance velocity components alongside compatibility con-ditions for the disturbance pressure to close the systemof equations. In the case of an open flow system, on theother hand, the issue of boundary conditions must be con-sidered carefully with respect to the plausible physicalsituations that one wishes to model. We have addressedthis problem by imposing homogeneous Dirichlet condi-tions on all perturbations at inflow, thus prohibiting anydisturbances to enter the calculation domain and allow-ing only those generated by the presence of the steadylaminar separation bubble. Extrapolation of informationfrom the interior of the integration domain was used atoutflow. On the wall and at the far-field the usual viscousboundary conditions are employed.

The implications on unsteadiness of the steady lami-nar flow deriving from linear amplification of the globaleigenmode discovered theoretically by [25] to grow lin-early upon the two-dimensional laminar basic flow of [3]atRe = 106/48 have been discussed extensively in [25].There we also discussed one scenario potentially explain-ing the three-dimensionalisation of the reattachment-line as observed in experiment. Here, we present infigure 9 in perspective view colour-coded contours of

the wall-normal disturbance velocity componentv, aswell as isosurfaces of the (actual reason for three-dimensionalisation) disturbance velocity componentw.From a numerical point of view, the effects of the im-posed boundary conditions are clearly visible in thisfigure, namely that activity is mainly concentrated inthe neighbourhood of the basic flow laminar separationbubble. Interestingly, large-scale structures analogous tothose which have been identified in three-dimensionalspatial DNS are present in these global linear instabil-ity results. From the point of view of identification of theglobal modes it is clear that the mode shown infigure 9is distinct from those of one-dimensional theory whoseessential element is theirx-wise periodicity. In our viewexperimentation on linear instability in the laminar sep-aration bubble open flow system should divert its tradi-tional attention from the well-known scenario of exist-ing x-wise periodic Tollmien–Schlichting wave-like dis-turbances, which experience explosive amplification af-ter they enter the separation bubble, and consider dis-turbances generated within the laminar separation bub-ble such as that presented here. The existence and linearamplification of the latter yields inoperative any meansof flow control which focusses exclusively on the for-mer type of linear instability. Again, the first step towardsidentification of the global disturbances experimentally is


Figure 8. The two-dimensional laminar separation bubble basic flow solved by Briley [3], presented as colour-coded contour lines ofthe basic flow vorticity.

Figure 9. Perspective view of the most unstable linear global eigenmode in Briley’s laminar separation bubble flow [25] presented ascolour-coded contour lines of the disturbance velocity componentv and isosurfaces of the disturbance velocity componentw at an(arbitrary) positive level+0.02 (light-coloured surface) and its opposite (dark-coloured surface). Flow is in the direction of the arrowand the boundaries of the steady laminar separation bubble are outlined by dashed lines.


Figure 10.Streamwise disturbance velocity component of the GH mode in the incompressible infinite swept attachment-line boundarylayer flow.Re= 800, β = 0.25.

obtaining information from a numerical solution of thepartial-derivative eigenvalue problem.

3.4. The swept attachment-line boundary layer

This flow is one in which a number of factors haveworked together in the direction of concealing unstableglobal linear eigenmodes despite over four decades ofcontinuous experimentation. The new modes were dis-covered by solution of the partial derivative eigenvalueproblem by Lin and Malik [14], their existence was con-firmed by Joslin using spatial DNS [11] and their spatialstructure was analysed by Theofilis [23], who also de-rived a one-dimensional model which extends the 1955Görtler–Hämmerlin (GH) Ansatz to three spatial dimen-sions; here we briefly expose the spatial structure of theglobal eigenmodes. The latter are present in the flow ata certain Reynolds number and wavenumber parametersas families in which the mode postulated by Görtler andHämmerlin and recovered numerically by Hall et al. [8]is the most unstable. The simple spatial structure of theamplitude functions of the GH mode, whose disturbanceeigenvector satisfies [7,9]

u(x, y)= u′(y)x, (7)

v(x, y)= v(y), (8)

w(x, y)= w(y), (9)

wherex andy respectively denote the spatial coordinatesalong the chordwise and wall-normal directions, permitsrecovery of the GH mode by a one-dimensional lineareigenvalue problem, that solved by Hall et al. [8]. Inthree-dimensional space the GH mode is taken to be peri-odic along the spanwisez direction along the attachmentline, such that

u(x, y, z, t)= u(x, y)E, (10)

v(x, y, z, t)= v(x, y)E, (11)

w(x, y, z, t)= w(x, y)E, (12)

with E ≡ eiβz−ωt. However, the work of Lin and Malik[14] has revealed that global linear instabilities with am-plitude functions other than the simple GH structure existin this flow. Solution of the partial-derivative eigenvalueproblem [23,24] reveals the spatial structure of the ampli-tude eigenfunctions. The chordwise disturbance velocitycomponents pertaining to the three most unstable globallinear disturbances atRe = 800, β = 0.25 is shown infigures 10to 12 as function of the chordwise and wall-normal spatial coordinates; an algebraic (polynomial)dependence ofu(x, y) on x is clearly visible in theseresults.

The work of Lin and Malik [14] revealed that theadditional modes have frequencies which are different


Figure 11. Real part (left) and imaginary part (right) of the streamwise disturbance velocity component of the A1 mode in theincompressible infinite swept attachment-line boundary layer flow.Re= 800, β = 0.25.

Figure 12. Real part (left) and imaginary part (right) of the streamwise disturbance velocity component of the S2 mode in theincompressible infinite swept attachment-line boundary layer flow.Re= 800, β = 0.25.

in relative terms to that of the GH mode by a fewpercent, making their identification impossible by powerspectral analysis alone. The analysis of Theofilis [23] hasrevealed that on one hand they-dependence of the neweigenmodes is identical with that of the GH mode and onthe other hand the global modes may be classified intotwo classes of either symmetric

u(x, y)=M∑m=1

x2m−1u2m−1(y),

v(x, y)=M∑m=1

x2m−2v2m−2(y),

w(x, y)=M∑m=1

x2m−2w2m−2(y), (13)

or antisymmetric

u(x, y)=M∑m=1

x2mu2m(y),

v(x, y)=M∑m=1

x2m−1v2m−1(y),

w(x, y)=M∑m=1

x2m−1w2m−1(y). (14)

linear disturbances. The GH mode is a symmetric moderecovered from (13) by settingM = 1, the first antisym-metric mode A1 is that havingM = 1 in (14), followedby the second symmetric mode S2 which corresponds toM = 2 in (13), the second antisymmetric mode A2 which


corresponds toM = 2 in (14), etc. This classification per-mits recovery of the partial-derivative eigenvalue prob-lem results from an ordinary-differential-equation-basedeigenvalue problem which may be solved at a negligi-ble fraction of the cost of solving (3)–(6); details may befound in [23].

From the point of view of the theme of the present pa-per, the attachment-line boundary layer is a classic ex-ample of physical information being hidden or misinter-preted on account of experimental techniques which areinherently based on the assumptions of Tollmien’s theory.The most common means of identification of linear in-stabilities, that based on power spectral density informa-tion, cannot deliver the additional to the GH eigenmodesof the flow on account of their near-identical frequencieswith that of the GH mode. The more sophisticated exper-imental approach based on a single hot-wire transversingin the wall-normal direction will also deliver incompleteresults on account of the identicaly-dependence of theamplitude functions. One must resort to the only remain-ing discriminating characteristic of the eigenmodes de-livered by the global linear instability analysis, namelytheir x-dependence. Use of multiple hot-wire probes (D.I. A. Poll, personal communication) offers a straightfor-ward means for identification of the global linear insta-bilities (13)–(14).

4. Conclusions

Our objective here has been to present the spatial struc-ture of global flow instabilities which may grow lin-early upon four essentially two-dimensional incompress-ible laminar basic flows. In all examples presented thepoint was stressed that traditional means of identifica-tion of linear instabilities, deriving from the results ofTollmien’s classic theory on wave-like disturbances de-veloping upon a flat-plate boundary layer, are hardly ad-equate for the identification of the structures pertainingto the global eigenmodes discussed. Yet, these modesare significant on (at least) two counts. Firstly, homo-geneity of physical space in the present nonparallel lin-ear instability theory is assumed in one spatial direc-tion alone. Consequently, the global eigenmodes forma much wider basis on which an arbitrary initial con-dition may be decomposed, compared with that formedby the eigenmodes of the one-dimensional Tollmien the-ory. Secondly, in a manner analogous to that experi-enced in the one-dimensional theory, once generatedwithin the flow these global linear instabilities will growexponentially at appropriate parameter ranges and leadlaminar flow into transition and breakdown to turbu-lence.

Another analogy of the two-dimensional nonparallelextension of Tollmien’s theory with the latter is that afeature of the global eigenmodes discovered is that theirspatial structure is conditioned by the respective basic

flow. Examination of the full spectrum reveals that in-creasingly higher modes possess increasingly more com-plicated structure compared with both the lower eigen-modes and the basic flow upon which they develop.This feature is reminiscent of the multi-peak structureof the Tollmien–Schlichting instabilities developing upona Blasius boundary-layer or those in the plane channelflow. In the Tollmien theory the one-dimensional natureof the amplitude functions leaves room for (careful) sin-gle hot-wire probe measurements to be performed. Thisis no longer possible in the case of global linear in-stabilities; alternative means, based on modern globalmeasurement techniques or developments thereof arecalled for. While frequency information is certainly theeasiest to acquire, the critical measurement to be per-formed is, in our view, that of two-dimensional eigen-functions.

The present theoretical extension of both Tollmien’stheory and PSE to two resolved directions has only beenmade possible on account of an ever-increasing com-puting capacity. In the light of the significance of thenew results obtained it would be highly desirable thatthis development be accompanied by renewed experi-mental efforts to identify and quantify some of the lin-ear instability phenomena which up to the present timewere erroneously classified as an unquantifiable nonlin-earity.

References

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[8] Hall P., Malik M.R., Poll D.I.A., On the stability of aninfinite swept attachment-line boundary layer, Proc. Roy.Soc. Lond. A 395 (1984) 229–245.

[9] Hämmerlin G., Zur Instabilitätstheorie der ebenenStaupunktströmung, in: Görtler H., Tollmien W. (eds), 50


Jahre Grenzschichtforschung, Vieweg und Sohn, 1955,pp. 315–327.

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[23] Theofilis V., On the verification and extension of theGörtler–Hämmerlin assumption in three-dimensional in-compressible swept attachment-line boundary layer flow,Technical Report IB 223-97 A 44, DLR, 1997.

[24] Theofilis V., Linear instability in two spatial dimensionsin: Papailiou K.D. (ed.), Fourth European ComputationalFluid Dynamics Conference ECCOMAS’98, Chichester,N. York, J. Wiley and Sons, 1998, pp. 547–552.

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