101098rsta20000706

On the origins of unsteadinessand three-dimensionality in a

laminar separation bubble

By Vassilios Theofilis Stefan Hein a n d U w e Dallmann

Deutsches Zentrum fur Luft- und Raumfahrt eV (DLR)Institute of Fluid Mechanics Transition and Turbulence

Bunsenstrafrac14 e 10 D-37073 Gottingen Germany

We analyse the three-dimensional non-parallel instability mechanisms responsible fortransition to turbulence in regions of recirculating steady laminar two-dimensionalincompressible separation bubble regow in a twofold manner First we revisit theproblem of TollmienSchlichting (TS)-like disturbances and we demonstrate for theshy rst time for this type of regow excellent agreement between the parabolized stabil-ity equation results and those of independently performed direct numerical simula-tions Second we perform a partial-derivative eigenvalue problem stability analysisby discretizing the two spatial directions on which the basic regow depends precludingTS-like waves from entering the calculation domain A new two-dimensional set ofglobal amplishy ed instability modes is thus discovered In order to prove earlier topo-logical conjectures about the regow structural changes occurring prior to the onset ofbubble unsteadiness we reconstruct the total regowshy eld by linear superposition of thesteady two-dimensional basic regow and the new most-amplishy ed global eigenmodes Inthe parameter range investigated the result is a bifurcation into a three-dimensionalregowshy eld in which the separation line remains unanotected while the primary reattach-ment line becomes three dimensional in line with the analogous result of a multitudeof experimental observations

Keywords linear non-local instability topological deg ow changeslaminar separation bubble global instability structural instability

1 Introduction

The subjects of laminar regow separation its linear and nonlinear instability andtransition to turbulence have attracted experimental and theoretical interest for thebest part of the last century The understanding of the physics governing thesephenomena especially with respect to vortex shedding and its relation to instabilitymechanisms is still far from being satisfactory as witnessed by the plethora anddiversity of contributions to this issue Motivation for continuous study from anaerodynamic point of view is primarily onotered by the formation of laminar separationbubbles in the neighbourhood of the mid-chord region of an aerofoil at moderateangles of attack and low Reynolds numbers Under the inreguence of environmentalexcitation the laminar separated regow has been demonstrated to undergo transitionand reattach as turbulent regow in a multitude of experiments (see for example Dovgal

Phil Trans R Soc Lond A (2000) 358 32293246

3229

creg 2000 The Royal Society

3230 V Theomacrlis S Hein and U Dallmann

et al 1994) and numerical simulations (see for example Bestek et al 1989) As inattached boundary-layer regows essential features of the transition process are theregow unsteadiness and three dimensionality

It should be noted that several parameters|such as Reynolds number means ofintroduction extent of application and strength of an adverse pressure gradient|should be taken into account in order for a laminar separation bubble to be char-acterized Rather than relying on parametric studies however our concern is withthe identishy cation of the physical mechanisms underlying the onset of unsteadinessand three dimensionality of the regow Specishy cally we have the following questions inmind

(i) Does the bubble only act as an amplishy er of environmental disturbances enteringthe separated regow region or does it also have the potential to generate unstablemodes in the absence of incoming disturbances

(ii) In either case what are the structural changes experienced by the laminarseparation bubble on account of linear instability mechanisms

(iii) Further how do these structural changes relate to the topological regow changeswhich have been conjectured in the literature as being responsible for the onsetof vortex shedding from separation bubbles

Abandoning any intention of onotering a complete discussion here of the large body ofwork concerning instability and transition of regow encompassing a laminar separationbubble we conshy ne ourselves to drawing the demarcation line between the approachesused in the past and those presently utilized in order to address the above questionsWe also stress that rather than performing a study at Reynolds numbers relevantto industrial applications (as done for example by Spalart amp Coleman (1997)) weconshy ne ourselves to low transitional Reynolds numbers

We address the questions posed within the frame of linear instability analysesin which disturbances are treated as non-periodic two-dimensional functions of thechordwise and wall-normal spatial coordinates Two complementary approaches havebeen followed First we consider slow development of the basic regow in the streamwisedirection and apply our non-local analysis tools based on the parabolized stabilityequations (PSE) developed by Hein et al (1994) Second we perform a partial-derivative eigenvalue problem stability (PEPS) analysis (Theoshy lis 1998) in whichboth the chordwise and wall-normal directions are fully resolved In both approachesthe linear disturbances are taken to be periodic in the spanwise direction It shouldbe noted that we study solely instabilities of a convective nature when employingthe PSE since evidence exists (Allen amp Riley 1995) that breakdown to turbulence inbasic regows with small separation bubbles in terms of extent of the bubble comparedwith a typical length and in terms of magnitude of the steady recirculating regowcompared with the freestream velocity is of a convective nature

In x 2 we present the two basic regows utilized for our numerical linear instabilityexperiments Elements of the theories governing the non-local and global instabilityanalyses based on the PSE and the PEPS respectively are presented in x 3 Inx 4 we present a comparison of our PSE results with those of the spatial directnumerical simulations (DNS) of Rist amp Maucher (1994) and classic OrrSommerfeldtheory New global linear instability modes discovered by the PEPS analysis are alsopresented in x 4 Section 5 is devoted to discussing our shy ndings from a topological

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3231

analysis point of view Concluding remarks relating our results to earlier work areonotered in x 6

2 The basic deg ows

The instability properties and characteristics of two dinoterent incompressible two-dimensional regat-plate laminar boundary-layer regows with separation bubbles encom-passed are analysed In both cases laminar separation is caused by a deceleratedfreestream velocity Ue(x) prescribed at some shy xed distance from the wall The insta-bilities developing upon the shy rst basic regow (BF1) were studied in a DNS by Ristamp Maucher (1994) They prescribed a smooth deceleration of centUe = 009 betweenx1 = 071 and x2 = 243 where x and y denote streamwise and wall-normal direc-tion respectively and (U V )T denotes the corresponding velocity vector Upstreamof x1 and downstream of x2 the freestream velocity is kept constant Flow quantitiesare made non-dimensional with the freestream velocity U curren

1 = 30 m siexcl1 upstreamof x1 and a global reference length L curren = 005 m The global Reynolds numberRe = U curren

1 L curren =cedil curren was set to Re = 105 The steady basic regow was calculated byRist amp Maucher (1994) using their DNS code with a Blasius solution as the inregowcondition prescribed at x0 = 037 The regow separates at x S ordm 173 and reattaches atxR ordm 221 Downstream of the bubble the regow asymptotically recovers to a Blasiusboundary layer as indicated by the shape factor H12 (Hein et al 1998)

Linear instability mechanisms supported by a second basic regow (BF2) that de-scribed by Briley (1971) have also been studied This regow is calculated in twostages First the non-similar boundary-layer equations are solved subject to the clas-sic linearly decelerating streamwise velocity component distribution due to Howarth(1938) Second a two-dimensional incompressible DNS is performed using an emacr -cient and spectrally accurate algorithm based on matrix diagonalization The inregowboundary data for the DNS are provided by the boundary-layer solution taken ata location well before separation occurs The imposition in the DNS of the samefreestream velocity distribution as in Howarthrsquos (1938) case up to a given down-stream position beyond which the freestream velocity is kept constant results inthe formation of a separated regow region which compares very well with those pre-sented by both Briley (1971) and Cebeci amp Stewartson (1983) The parameters heldconstant were the Reynolds number Re = 106=48 and the inregow location sup1 in f = 005(corresponding to xin f = (shy 0=shy 1) sup1 in f = 005=3) while those varied were the down-stream extent of the calculation domain taking values from sup1 = 05 (Briley 1971)to sup1 = 15 (Cebeci amp Stewartson 1983) and the (scaled) location in the wall-normaldirection where the pressure gradient is imposed varying from sup2 = 54 in the formerreference to sup2 = 75 in the latter reference

One essential dinoterence between BF1 and BF2 is that in the former basic regowthe Reynolds number based on the displacement thickness macr 1 at x1 = 071 is Re macr 1

ordm460 According to linear local instability theory the critical Reynolds number ofBlasius regow is Re macr 1

= 420 (see for example Schlichting 1979) Hence for this basicregow TollmienSchlichting instability sets in upstream of the position x1 where regowdeceleration starts In BF2 on the other hand the Reynolds number at the outregowboundary sup1 ou t = 15 where a Blasius boundary layer is recovered is Re macr 1

= 290well upstream of the Blasius critical Re macr 1

Phil Trans R Soc Lond A (2000)

3232 V Theomacrlis S Hein and U Dallmann

3 Linear instability theories

The non-localglobal nature of the instability analyses used derives from the decom-position of a perturbed regowshy eld into a steady two-dimensional basic and an unsteadythree-dimensional disturbance component according to

Q(x y z t) = Q b (x y) + Q p (x y) exp i pound + cc (31)

with Q b = (U V 0 P )T indicating the real basic regow and Qp = (u v w p)T denotingcomplex-valued perturbations to it Here x denotes the streamwise direction y thewall-normal direction and z the spanwise spatial direction the imaginary unit isi =

piexcl 1 and cc denotes complex conjugation in order for the total shy eld to remain

real Linearization about Q b follows based on the argument of smallness of and thebasic regow terms themselves satisfying the NavierStokes and continuity equationsare subtracted out The dinoterence between the two approaches stems from the ansatzmade for the complex function pound as follows

(a) Non-local analysis based on the PSE

The linear non-local instability analyses were performed with the NOnLOcal Tran-sition analysis (NOLOT)PSE code developed in cooperation between DLR and Fly-gtekniska Forsokanstalten (Hein et al 1994)y In the NOLOTPSE analysis bothQ b and Q p are functions of y slowly varying in x we call this approach a non-localanalysis as opposed to the classic local theory in which both Q b and Q p are functionsof y alone The wave function

pound =x

x0

not ( sup1 ) d sup1 + shy z iexcl laquo t (32)

with not a slowly varying streamwise wavenumber is intended to capture practicallyall oscillatory-type streamwise variations of the disturbance that is the variations onthe fast scale The disturbances are assumed to grow in the streamwise direction xonly and spatially evolving wave-like instabilities are considered Hence the stream-wise wavenumber not is represented by a complex quantity whereas the spanwisewavenumber shy and the circular frequency laquo are real parameters

Since both the amplitude function and the wave function depend on the x-direc-tion a normalization condition is required which removes this ambiguity as intro-duced by Bertolotti et al (1992)

ye

0

Q yp

Q p

xdy = 0 (33)

The superscript y refers to the complex conjugate and ye stands for the upper bound-ary of the discretized domain The normalization condition is used as an iterationcondition in order to ensure that as much as possible of the streamwise variationof the disturbance is transferred into the wave function The remaining part of thestreamwise variations is described by the amplitude functions and is small conse-quently second derivatives of the amplitude functions in x are discarded This leads

y For the purposes of the present study the code was run in the incompressible limit at which excellentagreement with results obtained by incompressible codes has been shown by Hein et al (1994)

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3233

to a system of partial dinoterential equations of the form

AQ p + BQ p

y+ C

2Q p

y2+ D

Q p

x= 0 (34)

Details of the consistent multiple-scales approach based on a Reynolds numberexpansion used to derive the non-local stability equations which are closely relatedto the PSE (Bertolotti et al 1992 Herbert 1997) and the elements of the matrices AB C D can be found in Hein et al (1994) They represent an initial-boundary-valueproblem that can be solved emacr ciently by a marching procedure in the x-directionwith upstream enotects neglected an approach valid for convectively unstable regowsonly The partial dinoterential equations system (34) is discretized by a shy rst-orderbackward Euler scheme in the x spatial direction and fourth-order compact shy nitedinoterences in the y spatial direction On the amplitude functions (u v w) we imposehomogeneous Dirichlet boundary conditions at the wall (y = 0) and asymptoticboundary conditions enforcing exponential decay at the upper boundary ye (Hein etal 1994) Initial conditions are provided by a local parallel instability analysis usingthe same code with the matrix D all streamwise derivatives in matrices A B andC and the wallnormal velocity component V of the basic regow set to zero

The spatial disturbance growth rate frac14 in local instability theory is described bythe negative imaginary part of the complex wavenumber not ie frac14 = iexcl not i whereasin non-local theory there is an additional contribution from the amplitude functionsand various reasonable alternative deshy nitions are possible here we use either

frac14 E = iexcl not i +

xln(

pE) with E =

ye

0

(kuk2 + kvk2 + kwk2) dy (35)

or

frac14 u = iexcl not i + Re1

um

um

x (36)

where um denotes the value of u at the wall-normal coordinate where kuk reaches itsmaximum

(b) The partial derivative eigenvalue problem

The assumptions of slow growth of Q b and Q p inherent in the PSE may berelaxed by fully resolving both the streamwise and the wall-normal spatial directionson account of this property we term the PEPS a global instability analysis In thetemporal framework as considered here the complex function pound becomes simply

pound = shy z iexcl laquo t (37)

where shy is a real wavenumber parameter and laquo is the sought complex eigenvaluewhose real part indicates frequency and whose positive imaginary part is the growthrate in time t of global disturbances This deshy nition of pound suggests the second essentialdinoterence between the PSE and the PEPS approaches namely that any structurethat an eigenmode may have in the streamwise direction x including that of a wave-like disturbance considered by the PSE will be captured by Q p in (31)

Numerical aspects of the solution of the two-dimensional partial derivative eigen-value problem are discussed by Theoshy lis (1998) Compared with that work we have

Phil Trans R Soc Lond A (2000)

3234 V Theomacrlis S Hein and U Dallmann

implemented here a novel version of the algorithm that applies to regow problems inwhich the wavenumber vector is perpendicular to the plane on which the basic regowis deshy ned (Theoshy lis 2000) Specishy cally straightforward redeshy nitions of the spanwisedisturbance velocity component w and the eigenvalue laquo by iw and i laquo respectivelyresult in a disturbance regow system with real coemacr cients that compared with thefull problem requires approximately half the storage and runtime to solve The realnon-symmetric generalized eigenvalue problem for the determination of laquo and Q p

which results is as follows

[ L iexcl (DxU )]u iexcl (DyU )v iexcl Dxp = laquo u (38)

iexcl (DxV )u + [ L iexcl (DyV )]v iexcl Dy p = laquo v (39)

L w iexcl shy p = laquo w (310)

Dxu + Dy v iexcl shy w = 0 (311)

where

L =1

Re(D2

x + D2y iexcl shy 2) iexcl UDx iexcl V Dy Dx =

xand Dy =

y

For its solution we use spectral collocation on a rectangular Cartesian grid Theboundary conditions imposed are the straightforward viscous boundary conditionson all disturbance velocity components at the wall and compatibility conditions forthe pressure in the freestream the disturbance velocity components and disturbancepressure are required to decay In an attempt to preclude disturbances from enteringthe recirculating regow area from upstream homogeneous Dirichlet boundary condi-tions are imposed at the inregow At the outregow boundary linear extrapolation of alldisturbance quantities from the interior of the integration domain is performed

4 Results

(a) Short chordwise-wavelength instabilities studied by the PSE

A classic means of linear analysis of short-wavelength instabilities in laminar separa-tion bubbles is based on local theory in which the parallel regow assumption is madeand the OrrSommerfeld equation (OSE) is solved (Allen amp Riley 1995) There is norational approach leading to the latter equation that ignores both upstream inreguenceand downstream development of the basic regow nevertheless both the OSE and spa-tial DNS predict in close agreement an explosive growth of convective instabilitiesentering the laminar separated regow (Bestek et al 1989) On the other hand an insta-bility analysis approach which takes into account downstream development of theregow is based on the PSE Due to the backregow within a separation bubble one mightsuspect that there is a signishy cant amount of information travelling upstream anda marching procedure as used in non-local instability analyses is not appropriateHowever there is strong indication from both experiment and DNS that thin lam-inar separation bubbles are convectively unstable to short-wavelength disturbancesof small amplitude initiated upstream of the bubble (Dovgal et al 1994 Rist et al 1996) This encourages us to embark upon application of PSE for this type of regow

As a shy rst basic regow we take BF1 of Rist amp Maucher (1994) They comparedthe results of linear local theory with those of their DNS the latter run at low

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3235

15 20 25x

0

100

200

300

400

grow

th r

ate

(m- 1

)

DNS = 0 m - 1

DNS = 400 m - 1

DNS = 800 m - 1

PSEOSE

S R

bbb

Figure 1 Comparison of non-local growth rates frac14 u for BF1 obtained by NOLOTPSE withthe DNS data of Rist amp Maucher (1994) and our OSE solutions Separation and reattachmentare denoted by Srsquo and Rrsquo respectively

disturbance amplitudes we use their growth rate data as a reference Figure 1 showsthe disturbance growth rates for f ordm 1719 Hz (reduced frequency F = 18 pound 10iexcl4)and three dinoterent (dimensional) spanwise wavenumbers shy = 0 400 and 800 miexcl1The corresponding wave angles iquest at separation point x S are iquest ordm 0 22 and 41macr Thelinear local (OSE) and the non-local (PSE) results are compared with those obtainedin the DNS of Rist amp Maucher (1994) The non-local growth rate is deshy ned as in theDNS ie on the maximum streamwise disturbance velocity (36)

The result of major signishy cance in this shy gure is the agreement between the linearPSE and DNS results which is found to be inreguenced by the wave angle For two-dimensional waves PSE delivers instability results that are in excellent agreementwith those obtained by DNS at all x positions with the obvious advantage that PSEresults are obtained at orders-of-magnitude lower computing enotort compared withDNS The agreement is still very convincing for three-dimensional waves althougha discrepancy between PSE and DNS results that increases with wave angle is tobe found upstream of the bubble There are at least three possible explanations forthis wave-angle dependent discrepancy First it might be thought that the upstreaminreguence neglected in PSE might cause PSE results to deviate from those of theDNS Second violation of the PSE assumption of a weakly non-parallel regow in theneighbourhood of the separation point may be considered to be responsible for thediscrepancy Third transients in the DNS data may cause the small dinoterences Theshy rst reason cannot explain the discrepancies because these are present upstreamof separation in a region of attached boundary-layer regow where upstream enotectsare indeed negligible but disappear in the second half of the bubble despite theexisting backregow In order for the second reason to be verishy ed one should refer to

Phil Trans R Soc Lond A (2000)

3236 V Theomacrlis S Hein and U Dallmann

0 04 08

0

001

002

003

004

005

006

UV 100

0 04 08 ||Qp||

||u||||v||||p||

yL

QbU yen

Figure 2 Velocity promacrles of the laminar basic deg ow BF1 and the amplitude functions of atwo-dimensional wave of f = 1719 Hz at x = 1984 within the recirculating deg ow

the magnitude of the non-parallel terms in the attached decelerating part of the regowdomain compared with that of the non-parallel terms in the recirculating zone

We believe that the third reason plays an essential role since disturbances aregenerated in the DNS by periodic blowing and suction at the wall immediatelyupstream of the region where regow deceleration is enforced It therefore takes somedownstream distance until these artishy cially introduced disturbances are convertedinto TS-like instabilities whose growth rate is no longer anotected by the actual meansof their generation On the other hand memory enotects are shorter in our PSE anal-yses which were started with TS waves (solutions of the OSE equation) imposedat the inregow boundary the distance from the end of the suction strip to x = 15corresponds to less than six disturbance wavefronts It is well known that both theenotects of regow non-parallelism and the enotects of disturbance history increase withdisturbance wave angle If as we think the dinoterences are indeed caused by remain-ing transients they should be more pronounced at smaller values of x However thiscould not be verishy ed because DNS growth-rate data for x lt 15 are not availablefrom Rist amp Maucher (1994) These authors also performed an OSE analysis (notpresented here) the results of which compare very well with our linear local anal-ysis (also shown in shy gure 1) The agreement between our OSE growth rate resultsand DNS is quite close in line with the shy ndings of Bestek et al (1989) and othersHowever the agreement between OSE and DNS can be seen to be much inferior tothat between PSE and DNS results

Note that the largest growth rates are of the same order of magnitude as the dis-turbance wavenumbers which means that the disturbances grow much more rapidly

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3237

1 2 3 4 50

1000

2000

3000

4000

5000

S Rx

f (H

z)

05 10 150

500

1000

1500

2000

2500

3000

S Rx

(a) (a)

Figure 3 Isolines of constant non-local growth rate frac14 E for BF1 (a) and BF2 (b) Frequenciesf of unstable waves against scaled downstream distance are plotted The outermost curvescorrespond to neutral stability and the increments in contour lines are macr frac14 E = 25 and 10 for BF1and BF2 respectively

than those in Blasius regow The non-local amplitude functions of the two-dimensionalwave of f ordm 1719 Hz are plotted in shy gure 2 for the downstream position x = 1984alongside those of the corresponding velocity proshy les of the basic regow u has maximainside the backregow region close to the wall at the inregection point of the stream-wise velocity proshy le of the basic regow and at the edge of the separated boundarylayer while v is rather large Since the two-dimensional wave is the most amplishy edat almost all downstream positions (as seen in shy gure 1) we focus all subsequentNOLOTPSE studies on two-dimensional waves The instability diagram for two-dimensional short-wavelength disturbances is shown in shy gure 3 Isolines of constantnon-local growth rate (deshy ned by 35) are plotted the outermost isoline representsthe curve of neutral stability This growth-rate deshy nition was chosen because in theseparated regow region the disturbance amplitude functions have several maxima inthe wall-normal direction (as seen in shy gure 2) The instability diagram for BF2 isalso shown in shy gure 3 Its general shape appears dinoterent from that of BF1 thelatter having an instability region of the decelerating and separated regow embeddedinto the region of TS instability unlike BF2 where the two are clearly separated (seealso comments in x 2)

(b) Global disturbances addressed by the PEPS

In the previous section we focused on instabilities of an oscillatory nature thestreamwise length-scale of which is small compared with that of the basic regowthis structure is exploited by the ansatz (32) and emacr cient marching proceduresmake a NOLOTPSE approach the method of choice for accurate results to bedelivered within reasonable computing time for this type of instability On the otherhand an analysis of disturbances developing over length-scales comparable withthat of the basic regow using NOLOTPSE may deliver inaccurate results Short ofresorting to a spatial DNS in order to take both upstream inreguence and downstream

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

3230 V Theomacrlis S Hein and U Dallmann

et al 1994) and numerical simulations (see for example Bestek et al 1989) As inattached boundary-layer regows essential features of the transition process are theregow unsteadiness and three dimensionality

It should be noted that several parameters|such as Reynolds number means ofintroduction extent of application and strength of an adverse pressure gradient|should be taken into account in order for a laminar separation bubble to be char-acterized Rather than relying on parametric studies however our concern is withthe identishy cation of the physical mechanisms underlying the onset of unsteadinessand three dimensionality of the regow Specishy cally we have the following questions inmind

(i) Does the bubble only act as an amplishy er of environmental disturbances enteringthe separated regow region or does it also have the potential to generate unstablemodes in the absence of incoming disturbances

(ii) In either case what are the structural changes experienced by the laminarseparation bubble on account of linear instability mechanisms

(iii) Further how do these structural changes relate to the topological regow changeswhich have been conjectured in the literature as being responsible for the onsetof vortex shedding from separation bubbles

Abandoning any intention of onotering a complete discussion here of the large body ofwork concerning instability and transition of regow encompassing a laminar separationbubble we conshy ne ourselves to drawing the demarcation line between the approachesused in the past and those presently utilized in order to address the above questionsWe also stress that rather than performing a study at Reynolds numbers relevantto industrial applications (as done for example by Spalart amp Coleman (1997)) weconshy ne ourselves to low transitional Reynolds numbers

We address the questions posed within the frame of linear instability analysesin which disturbances are treated as non-periodic two-dimensional functions of thechordwise and wall-normal spatial coordinates Two complementary approaches havebeen followed First we consider slow development of the basic regow in the streamwisedirection and apply our non-local analysis tools based on the parabolized stabilityequations (PSE) developed by Hein et al (1994) Second we perform a partial-derivative eigenvalue problem stability (PEPS) analysis (Theoshy lis 1998) in whichboth the chordwise and wall-normal directions are fully resolved In both approachesthe linear disturbances are taken to be periodic in the spanwise direction It shouldbe noted that we study solely instabilities of a convective nature when employingthe PSE since evidence exists (Allen amp Riley 1995) that breakdown to turbulence inbasic regows with small separation bubbles in terms of extent of the bubble comparedwith a typical length and in terms of magnitude of the steady recirculating regowcompared with the freestream velocity is of a convective nature

In x 2 we present the two basic regows utilized for our numerical linear instabilityexperiments Elements of the theories governing the non-local and global instabilityanalyses based on the PSE and the PEPS respectively are presented in x 3 Inx 4 we present a comparison of our PSE results with those of the spatial directnumerical simulations (DNS) of Rist amp Maucher (1994) and classic OrrSommerfeldtheory New global linear instability modes discovered by the PEPS analysis are alsopresented in x 4 Section 5 is devoted to discussing our shy ndings from a topological

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3231

analysis point of view Concluding remarks relating our results to earlier work areonotered in x 6

2 The basic deg ows

The instability properties and characteristics of two dinoterent incompressible two-dimensional regat-plate laminar boundary-layer regows with separation bubbles encom-passed are analysed In both cases laminar separation is caused by a deceleratedfreestream velocity Ue(x) prescribed at some shy xed distance from the wall The insta-bilities developing upon the shy rst basic regow (BF1) were studied in a DNS by Ristamp Maucher (1994) They prescribed a smooth deceleration of centUe = 009 betweenx1 = 071 and x2 = 243 where x and y denote streamwise and wall-normal direc-tion respectively and (U V )T denotes the corresponding velocity vector Upstreamof x1 and downstream of x2 the freestream velocity is kept constant Flow quantitiesare made non-dimensional with the freestream velocity U curren

1 = 30 m siexcl1 upstreamof x1 and a global reference length L curren = 005 m The global Reynolds numberRe = U curren

1 L curren =cedil curren was set to Re = 105 The steady basic regow was calculated byRist amp Maucher (1994) using their DNS code with a Blasius solution as the inregowcondition prescribed at x0 = 037 The regow separates at x S ordm 173 and reattaches atxR ordm 221 Downstream of the bubble the regow asymptotically recovers to a Blasiusboundary layer as indicated by the shape factor H12 (Hein et al 1998)

Linear instability mechanisms supported by a second basic regow (BF2) that de-scribed by Briley (1971) have also been studied This regow is calculated in twostages First the non-similar boundary-layer equations are solved subject to the clas-sic linearly decelerating streamwise velocity component distribution due to Howarth(1938) Second a two-dimensional incompressible DNS is performed using an emacr -cient and spectrally accurate algorithm based on matrix diagonalization The inregowboundary data for the DNS are provided by the boundary-layer solution taken ata location well before separation occurs The imposition in the DNS of the samefreestream velocity distribution as in Howarthrsquos (1938) case up to a given down-stream position beyond which the freestream velocity is kept constant results inthe formation of a separated regow region which compares very well with those pre-sented by both Briley (1971) and Cebeci amp Stewartson (1983) The parameters heldconstant were the Reynolds number Re = 106=48 and the inregow location sup1 in f = 005(corresponding to xin f = (shy 0=shy 1) sup1 in f = 005=3) while those varied were the down-stream extent of the calculation domain taking values from sup1 = 05 (Briley 1971)to sup1 = 15 (Cebeci amp Stewartson 1983) and the (scaled) location in the wall-normaldirection where the pressure gradient is imposed varying from sup2 = 54 in the formerreference to sup2 = 75 in the latter reference

One essential dinoterence between BF1 and BF2 is that in the former basic regowthe Reynolds number based on the displacement thickness macr 1 at x1 = 071 is Re macr 1

ordm460 According to linear local instability theory the critical Reynolds number ofBlasius regow is Re macr 1

= 420 (see for example Schlichting 1979) Hence for this basicregow TollmienSchlichting instability sets in upstream of the position x1 where regowdeceleration starts In BF2 on the other hand the Reynolds number at the outregowboundary sup1 ou t = 15 where a Blasius boundary layer is recovered is Re macr 1

= 290well upstream of the Blasius critical Re macr 1

Phil Trans R Soc Lond A (2000)

3232 V Theomacrlis S Hein and U Dallmann

3 Linear instability theories

The non-localglobal nature of the instability analyses used derives from the decom-position of a perturbed regowshy eld into a steady two-dimensional basic and an unsteadythree-dimensional disturbance component according to

Q(x y z t) = Q b (x y) + Q p (x y) exp i pound + cc (31)

with Q b = (U V 0 P )T indicating the real basic regow and Qp = (u v w p)T denotingcomplex-valued perturbations to it Here x denotes the streamwise direction y thewall-normal direction and z the spanwise spatial direction the imaginary unit isi =

piexcl 1 and cc denotes complex conjugation in order for the total shy eld to remain

real Linearization about Q b follows based on the argument of smallness of and thebasic regow terms themselves satisfying the NavierStokes and continuity equationsare subtracted out The dinoterence between the two approaches stems from the ansatzmade for the complex function pound as follows

(a) Non-local analysis based on the PSE

The linear non-local instability analyses were performed with the NOnLOcal Tran-sition analysis (NOLOT)PSE code developed in cooperation between DLR and Fly-gtekniska Forsokanstalten (Hein et al 1994)y In the NOLOTPSE analysis bothQ b and Q p are functions of y slowly varying in x we call this approach a non-localanalysis as opposed to the classic local theory in which both Q b and Q p are functionsof y alone The wave function

pound =x

x0

not ( sup1 ) d sup1 + shy z iexcl laquo t (32)

with not a slowly varying streamwise wavenumber is intended to capture practicallyall oscillatory-type streamwise variations of the disturbance that is the variations onthe fast scale The disturbances are assumed to grow in the streamwise direction xonly and spatially evolving wave-like instabilities are considered Hence the stream-wise wavenumber not is represented by a complex quantity whereas the spanwisewavenumber shy and the circular frequency laquo are real parameters

Since both the amplitude function and the wave function depend on the x-direc-tion a normalization condition is required which removes this ambiguity as intro-duced by Bertolotti et al (1992)

ye

0

Q yp

Q p

xdy = 0 (33)

The superscript y refers to the complex conjugate and ye stands for the upper bound-ary of the discretized domain The normalization condition is used as an iterationcondition in order to ensure that as much as possible of the streamwise variationof the disturbance is transferred into the wave function The remaining part of thestreamwise variations is described by the amplitude functions and is small conse-quently second derivatives of the amplitude functions in x are discarded This leads

y For the purposes of the present study the code was run in the incompressible limit at which excellentagreement with results obtained by incompressible codes has been shown by Hein et al (1994)

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3233

to a system of partial dinoterential equations of the form

AQ p + BQ p

y+ C

2Q p

y2+ D

Q p

x= 0 (34)

Details of the consistent multiple-scales approach based on a Reynolds numberexpansion used to derive the non-local stability equations which are closely relatedto the PSE (Bertolotti et al 1992 Herbert 1997) and the elements of the matrices AB C D can be found in Hein et al (1994) They represent an initial-boundary-valueproblem that can be solved emacr ciently by a marching procedure in the x-directionwith upstream enotects neglected an approach valid for convectively unstable regowsonly The partial dinoterential equations system (34) is discretized by a shy rst-orderbackward Euler scheme in the x spatial direction and fourth-order compact shy nitedinoterences in the y spatial direction On the amplitude functions (u v w) we imposehomogeneous Dirichlet boundary conditions at the wall (y = 0) and asymptoticboundary conditions enforcing exponential decay at the upper boundary ye (Hein etal 1994) Initial conditions are provided by a local parallel instability analysis usingthe same code with the matrix D all streamwise derivatives in matrices A B andC and the wallnormal velocity component V of the basic regow set to zero

The spatial disturbance growth rate frac14 in local instability theory is described bythe negative imaginary part of the complex wavenumber not ie frac14 = iexcl not i whereasin non-local theory there is an additional contribution from the amplitude functionsand various reasonable alternative deshy nitions are possible here we use either

frac14 E = iexcl not i +

xln(

pE) with E =

ye

0

(kuk2 + kvk2 + kwk2) dy (35)

or

frac14 u = iexcl not i + Re1

um

um

x (36)

where um denotes the value of u at the wall-normal coordinate where kuk reaches itsmaximum

(b) The partial derivative eigenvalue problem

The assumptions of slow growth of Q b and Q p inherent in the PSE may berelaxed by fully resolving both the streamwise and the wall-normal spatial directionson account of this property we term the PEPS a global instability analysis In thetemporal framework as considered here the complex function pound becomes simply

pound = shy z iexcl laquo t (37)

where shy is a real wavenumber parameter and laquo is the sought complex eigenvaluewhose real part indicates frequency and whose positive imaginary part is the growthrate in time t of global disturbances This deshy nition of pound suggests the second essentialdinoterence between the PSE and the PEPS approaches namely that any structurethat an eigenmode may have in the streamwise direction x including that of a wave-like disturbance considered by the PSE will be captured by Q p in (31)

Numerical aspects of the solution of the two-dimensional partial derivative eigen-value problem are discussed by Theoshy lis (1998) Compared with that work we have

Phil Trans R Soc Lond A (2000)

3234 V Theomacrlis S Hein and U Dallmann

implemented here a novel version of the algorithm that applies to regow problems inwhich the wavenumber vector is perpendicular to the plane on which the basic regowis deshy ned (Theoshy lis 2000) Specishy cally straightforward redeshy nitions of the spanwisedisturbance velocity component w and the eigenvalue laquo by iw and i laquo respectivelyresult in a disturbance regow system with real coemacr cients that compared with thefull problem requires approximately half the storage and runtime to solve The realnon-symmetric generalized eigenvalue problem for the determination of laquo and Q p

which results is as follows

[ L iexcl (DxU )]u iexcl (DyU )v iexcl Dxp = laquo u (38)

iexcl (DxV )u + [ L iexcl (DyV )]v iexcl Dy p = laquo v (39)

L w iexcl shy p = laquo w (310)

Dxu + Dy v iexcl shy w = 0 (311)

where

L =1

Re(D2

x + D2y iexcl shy 2) iexcl UDx iexcl V Dy Dx =

xand Dy =

y

For its solution we use spectral collocation on a rectangular Cartesian grid Theboundary conditions imposed are the straightforward viscous boundary conditionson all disturbance velocity components at the wall and compatibility conditions forthe pressure in the freestream the disturbance velocity components and disturbancepressure are required to decay In an attempt to preclude disturbances from enteringthe recirculating regow area from upstream homogeneous Dirichlet boundary condi-tions are imposed at the inregow At the outregow boundary linear extrapolation of alldisturbance quantities from the interior of the integration domain is performed

4 Results

(a) Short chordwise-wavelength instabilities studied by the PSE

A classic means of linear analysis of short-wavelength instabilities in laminar separa-tion bubbles is based on local theory in which the parallel regow assumption is madeand the OrrSommerfeld equation (OSE) is solved (Allen amp Riley 1995) There is norational approach leading to the latter equation that ignores both upstream inreguenceand downstream development of the basic regow nevertheless both the OSE and spa-tial DNS predict in close agreement an explosive growth of convective instabilitiesentering the laminar separated regow (Bestek et al 1989) On the other hand an insta-bility analysis approach which takes into account downstream development of theregow is based on the PSE Due to the backregow within a separation bubble one mightsuspect that there is a signishy cant amount of information travelling upstream anda marching procedure as used in non-local instability analyses is not appropriateHowever there is strong indication from both experiment and DNS that thin lam-inar separation bubbles are convectively unstable to short-wavelength disturbancesof small amplitude initiated upstream of the bubble (Dovgal et al 1994 Rist et al 1996) This encourages us to embark upon application of PSE for this type of regow

As a shy rst basic regow we take BF1 of Rist amp Maucher (1994) They comparedthe results of linear local theory with those of their DNS the latter run at low

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3235

15 20 25x

0

100

200

300

400

grow

th r

ate

(m- 1

)

DNS = 0 m - 1

DNS = 400 m - 1

DNS = 800 m - 1

PSEOSE

S R

bbb

Figure 1 Comparison of non-local growth rates frac14 u for BF1 obtained by NOLOTPSE withthe DNS data of Rist amp Maucher (1994) and our OSE solutions Separation and reattachmentare denoted by Srsquo and Rrsquo respectively

disturbance amplitudes we use their growth rate data as a reference Figure 1 showsthe disturbance growth rates for f ordm 1719 Hz (reduced frequency F = 18 pound 10iexcl4)and three dinoterent (dimensional) spanwise wavenumbers shy = 0 400 and 800 miexcl1The corresponding wave angles iquest at separation point x S are iquest ordm 0 22 and 41macr Thelinear local (OSE) and the non-local (PSE) results are compared with those obtainedin the DNS of Rist amp Maucher (1994) The non-local growth rate is deshy ned as in theDNS ie on the maximum streamwise disturbance velocity (36)

The result of major signishy cance in this shy gure is the agreement between the linearPSE and DNS results which is found to be inreguenced by the wave angle For two-dimensional waves PSE delivers instability results that are in excellent agreementwith those obtained by DNS at all x positions with the obvious advantage that PSEresults are obtained at orders-of-magnitude lower computing enotort compared withDNS The agreement is still very convincing for three-dimensional waves althougha discrepancy between PSE and DNS results that increases with wave angle is tobe found upstream of the bubble There are at least three possible explanations forthis wave-angle dependent discrepancy First it might be thought that the upstreaminreguence neglected in PSE might cause PSE results to deviate from those of theDNS Second violation of the PSE assumption of a weakly non-parallel regow in theneighbourhood of the separation point may be considered to be responsible for thediscrepancy Third transients in the DNS data may cause the small dinoterences Theshy rst reason cannot explain the discrepancies because these are present upstreamof separation in a region of attached boundary-layer regow where upstream enotectsare indeed negligible but disappear in the second half of the bubble despite theexisting backregow In order for the second reason to be verishy ed one should refer to

Phil Trans R Soc Lond A (2000)

3236 V Theomacrlis S Hein and U Dallmann

0 04 08

0

001

002

003

004

005

006

UV 100

0 04 08 ||Qp||

||u||||v||||p||

yL

QbU yen

Figure 2 Velocity promacrles of the laminar basic deg ow BF1 and the amplitude functions of atwo-dimensional wave of f = 1719 Hz at x = 1984 within the recirculating deg ow

the magnitude of the non-parallel terms in the attached decelerating part of the regowdomain compared with that of the non-parallel terms in the recirculating zone

We believe that the third reason plays an essential role since disturbances aregenerated in the DNS by periodic blowing and suction at the wall immediatelyupstream of the region where regow deceleration is enforced It therefore takes somedownstream distance until these artishy cially introduced disturbances are convertedinto TS-like instabilities whose growth rate is no longer anotected by the actual meansof their generation On the other hand memory enotects are shorter in our PSE anal-yses which were started with TS waves (solutions of the OSE equation) imposedat the inregow boundary the distance from the end of the suction strip to x = 15corresponds to less than six disturbance wavefronts It is well known that both theenotects of regow non-parallelism and the enotects of disturbance history increase withdisturbance wave angle If as we think the dinoterences are indeed caused by remain-ing transients they should be more pronounced at smaller values of x However thiscould not be verishy ed because DNS growth-rate data for x lt 15 are not availablefrom Rist amp Maucher (1994) These authors also performed an OSE analysis (notpresented here) the results of which compare very well with our linear local anal-ysis (also shown in shy gure 1) The agreement between our OSE growth rate resultsand DNS is quite close in line with the shy ndings of Bestek et al (1989) and othersHowever the agreement between OSE and DNS can be seen to be much inferior tothat between PSE and DNS results

Note that the largest growth rates are of the same order of magnitude as the dis-turbance wavenumbers which means that the disturbances grow much more rapidly

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3237

1 2 3 4 50

1000

2000

3000

4000

5000

S Rx

f (H

z)

05 10 150

500

1000

1500

2000

2500

3000

S Rx

(a) (a)

Figure 3 Isolines of constant non-local growth rate frac14 E for BF1 (a) and BF2 (b) Frequenciesf of unstable waves against scaled downstream distance are plotted The outermost curvescorrespond to neutral stability and the increments in contour lines are macr frac14 E = 25 and 10 for BF1and BF2 respectively

than those in Blasius regow The non-local amplitude functions of the two-dimensionalwave of f ordm 1719 Hz are plotted in shy gure 2 for the downstream position x = 1984alongside those of the corresponding velocity proshy les of the basic regow u has maximainside the backregow region close to the wall at the inregection point of the stream-wise velocity proshy le of the basic regow and at the edge of the separated boundarylayer while v is rather large Since the two-dimensional wave is the most amplishy edat almost all downstream positions (as seen in shy gure 1) we focus all subsequentNOLOTPSE studies on two-dimensional waves The instability diagram for two-dimensional short-wavelength disturbances is shown in shy gure 3 Isolines of constantnon-local growth rate (deshy ned by 35) are plotted the outermost isoline representsthe curve of neutral stability This growth-rate deshy nition was chosen because in theseparated regow region the disturbance amplitude functions have several maxima inthe wall-normal direction (as seen in shy gure 2) The instability diagram for BF2 isalso shown in shy gure 3 Its general shape appears dinoterent from that of BF1 thelatter having an instability region of the decelerating and separated regow embeddedinto the region of TS instability unlike BF2 where the two are clearly separated (seealso comments in x 2)

(b) Global disturbances addressed by the PEPS

In the previous section we focused on instabilities of an oscillatory nature thestreamwise length-scale of which is small compared with that of the basic regowthis structure is exploited by the ansatz (32) and emacr cient marching proceduresmake a NOLOTPSE approach the method of choice for accurate results to bedelivered within reasonable computing time for this type of instability On the otherhand an analysis of disturbances developing over length-scales comparable withthat of the basic regow using NOLOTPSE may deliver inaccurate results Short ofresorting to a spatial DNS in order to take both upstream inreguence and downstream

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3231

analysis point of view Concluding remarks relating our results to earlier work areonotered in x 6

2 The basic deg ows

The instability properties and characteristics of two dinoterent incompressible two-dimensional regat-plate laminar boundary-layer regows with separation bubbles encom-passed are analysed In both cases laminar separation is caused by a deceleratedfreestream velocity Ue(x) prescribed at some shy xed distance from the wall The insta-bilities developing upon the shy rst basic regow (BF1) were studied in a DNS by Ristamp Maucher (1994) They prescribed a smooth deceleration of centUe = 009 betweenx1 = 071 and x2 = 243 where x and y denote streamwise and wall-normal direc-tion respectively and (U V )T denotes the corresponding velocity vector Upstreamof x1 and downstream of x2 the freestream velocity is kept constant Flow quantitiesare made non-dimensional with the freestream velocity U curren

1 = 30 m siexcl1 upstreamof x1 and a global reference length L curren = 005 m The global Reynolds numberRe = U curren

1 L curren =cedil curren was set to Re = 105 The steady basic regow was calculated byRist amp Maucher (1994) using their DNS code with a Blasius solution as the inregowcondition prescribed at x0 = 037 The regow separates at x S ordm 173 and reattaches atxR ordm 221 Downstream of the bubble the regow asymptotically recovers to a Blasiusboundary layer as indicated by the shape factor H12 (Hein et al 1998)

Linear instability mechanisms supported by a second basic regow (BF2) that de-scribed by Briley (1971) have also been studied This regow is calculated in twostages First the non-similar boundary-layer equations are solved subject to the clas-sic linearly decelerating streamwise velocity component distribution due to Howarth(1938) Second a two-dimensional incompressible DNS is performed using an emacr -cient and spectrally accurate algorithm based on matrix diagonalization The inregowboundary data for the DNS are provided by the boundary-layer solution taken ata location well before separation occurs The imposition in the DNS of the samefreestream velocity distribution as in Howarthrsquos (1938) case up to a given down-stream position beyond which the freestream velocity is kept constant results inthe formation of a separated regow region which compares very well with those pre-sented by both Briley (1971) and Cebeci amp Stewartson (1983) The parameters heldconstant were the Reynolds number Re = 106=48 and the inregow location sup1 in f = 005(corresponding to xin f = (shy 0=shy 1) sup1 in f = 005=3) while those varied were the down-stream extent of the calculation domain taking values from sup1 = 05 (Briley 1971)to sup1 = 15 (Cebeci amp Stewartson 1983) and the (scaled) location in the wall-normaldirection where the pressure gradient is imposed varying from sup2 = 54 in the formerreference to sup2 = 75 in the latter reference

One essential dinoterence between BF1 and BF2 is that in the former basic regowthe Reynolds number based on the displacement thickness macr 1 at x1 = 071 is Re macr 1

ordm460 According to linear local instability theory the critical Reynolds number ofBlasius regow is Re macr 1

= 420 (see for example Schlichting 1979) Hence for this basicregow TollmienSchlichting instability sets in upstream of the position x1 where regowdeceleration starts In BF2 on the other hand the Reynolds number at the outregowboundary sup1 ou t = 15 where a Blasius boundary layer is recovered is Re macr 1

= 290well upstream of the Blasius critical Re macr 1

Phil Trans R Soc Lond A (2000)

3232 V Theomacrlis S Hein and U Dallmann

3 Linear instability theories

The non-localglobal nature of the instability analyses used derives from the decom-position of a perturbed regowshy eld into a steady two-dimensional basic and an unsteadythree-dimensional disturbance component according to

Q(x y z t) = Q b (x y) + Q p (x y) exp i pound + cc (31)

with Q b = (U V 0 P )T indicating the real basic regow and Qp = (u v w p)T denotingcomplex-valued perturbations to it Here x denotes the streamwise direction y thewall-normal direction and z the spanwise spatial direction the imaginary unit isi =

piexcl 1 and cc denotes complex conjugation in order for the total shy eld to remain

real Linearization about Q b follows based on the argument of smallness of and thebasic regow terms themselves satisfying the NavierStokes and continuity equationsare subtracted out The dinoterence between the two approaches stems from the ansatzmade for the complex function pound as follows

(a) Non-local analysis based on the PSE

The linear non-local instability analyses were performed with the NOnLOcal Tran-sition analysis (NOLOT)PSE code developed in cooperation between DLR and Fly-gtekniska Forsokanstalten (Hein et al 1994)y In the NOLOTPSE analysis bothQ b and Q p are functions of y slowly varying in x we call this approach a non-localanalysis as opposed to the classic local theory in which both Q b and Q p are functionsof y alone The wave function

pound =x

x0

not ( sup1 ) d sup1 + shy z iexcl laquo t (32)

with not a slowly varying streamwise wavenumber is intended to capture practicallyall oscillatory-type streamwise variations of the disturbance that is the variations onthe fast scale The disturbances are assumed to grow in the streamwise direction xonly and spatially evolving wave-like instabilities are considered Hence the stream-wise wavenumber not is represented by a complex quantity whereas the spanwisewavenumber shy and the circular frequency laquo are real parameters

Since both the amplitude function and the wave function depend on the x-direc-tion a normalization condition is required which removes this ambiguity as intro-duced by Bertolotti et al (1992)

ye

0

Q yp

Q p

xdy = 0 (33)

The superscript y refers to the complex conjugate and ye stands for the upper bound-ary of the discretized domain The normalization condition is used as an iterationcondition in order to ensure that as much as possible of the streamwise variationof the disturbance is transferred into the wave function The remaining part of thestreamwise variations is described by the amplitude functions and is small conse-quently second derivatives of the amplitude functions in x are discarded This leads

y For the purposes of the present study the code was run in the incompressible limit at which excellentagreement with results obtained by incompressible codes has been shown by Hein et al (1994)

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3233

to a system of partial dinoterential equations of the form

AQ p + BQ p

y+ C

2Q p

y2+ D

Q p

x= 0 (34)

Details of the consistent multiple-scales approach based on a Reynolds numberexpansion used to derive the non-local stability equations which are closely relatedto the PSE (Bertolotti et al 1992 Herbert 1997) and the elements of the matrices AB C D can be found in Hein et al (1994) They represent an initial-boundary-valueproblem that can be solved emacr ciently by a marching procedure in the x-directionwith upstream enotects neglected an approach valid for convectively unstable regowsonly The partial dinoterential equations system (34) is discretized by a shy rst-orderbackward Euler scheme in the x spatial direction and fourth-order compact shy nitedinoterences in the y spatial direction On the amplitude functions (u v w) we imposehomogeneous Dirichlet boundary conditions at the wall (y = 0) and asymptoticboundary conditions enforcing exponential decay at the upper boundary ye (Hein etal 1994) Initial conditions are provided by a local parallel instability analysis usingthe same code with the matrix D all streamwise derivatives in matrices A B andC and the wallnormal velocity component V of the basic regow set to zero

The spatial disturbance growth rate frac14 in local instability theory is described bythe negative imaginary part of the complex wavenumber not ie frac14 = iexcl not i whereasin non-local theory there is an additional contribution from the amplitude functionsand various reasonable alternative deshy nitions are possible here we use either

frac14 E = iexcl not i +

xln(

pE) with E =

ye

0

(kuk2 + kvk2 + kwk2) dy (35)

or

frac14 u = iexcl not i + Re1

um

um

x (36)

where um denotes the value of u at the wall-normal coordinate where kuk reaches itsmaximum

(b) The partial derivative eigenvalue problem

The assumptions of slow growth of Q b and Q p inherent in the PSE may berelaxed by fully resolving both the streamwise and the wall-normal spatial directionson account of this property we term the PEPS a global instability analysis In thetemporal framework as considered here the complex function pound becomes simply

pound = shy z iexcl laquo t (37)

where shy is a real wavenumber parameter and laquo is the sought complex eigenvaluewhose real part indicates frequency and whose positive imaginary part is the growthrate in time t of global disturbances This deshy nition of pound suggests the second essentialdinoterence between the PSE and the PEPS approaches namely that any structurethat an eigenmode may have in the streamwise direction x including that of a wave-like disturbance considered by the PSE will be captured by Q p in (31)

Numerical aspects of the solution of the two-dimensional partial derivative eigen-value problem are discussed by Theoshy lis (1998) Compared with that work we have

Phil Trans R Soc Lond A (2000)

3234 V Theomacrlis S Hein and U Dallmann

implemented here a novel version of the algorithm that applies to regow problems inwhich the wavenumber vector is perpendicular to the plane on which the basic regowis deshy ned (Theoshy lis 2000) Specishy cally straightforward redeshy nitions of the spanwisedisturbance velocity component w and the eigenvalue laquo by iw and i laquo respectivelyresult in a disturbance regow system with real coemacr cients that compared with thefull problem requires approximately half the storage and runtime to solve The realnon-symmetric generalized eigenvalue problem for the determination of laquo and Q p

which results is as follows

[ L iexcl (DxU )]u iexcl (DyU )v iexcl Dxp = laquo u (38)

iexcl (DxV )u + [ L iexcl (DyV )]v iexcl Dy p = laquo v (39)

L w iexcl shy p = laquo w (310)

Dxu + Dy v iexcl shy w = 0 (311)

where

L =1

Re(D2

x + D2y iexcl shy 2) iexcl UDx iexcl V Dy Dx =

xand Dy =

y

For its solution we use spectral collocation on a rectangular Cartesian grid Theboundary conditions imposed are the straightforward viscous boundary conditionson all disturbance velocity components at the wall and compatibility conditions forthe pressure in the freestream the disturbance velocity components and disturbancepressure are required to decay In an attempt to preclude disturbances from enteringthe recirculating regow area from upstream homogeneous Dirichlet boundary condi-tions are imposed at the inregow At the outregow boundary linear extrapolation of alldisturbance quantities from the interior of the integration domain is performed

4 Results

(a) Short chordwise-wavelength instabilities studied by the PSE

A classic means of linear analysis of short-wavelength instabilities in laminar separa-tion bubbles is based on local theory in which the parallel regow assumption is madeand the OrrSommerfeld equation (OSE) is solved (Allen amp Riley 1995) There is norational approach leading to the latter equation that ignores both upstream inreguenceand downstream development of the basic regow nevertheless both the OSE and spa-tial DNS predict in close agreement an explosive growth of convective instabilitiesentering the laminar separated regow (Bestek et al 1989) On the other hand an insta-bility analysis approach which takes into account downstream development of theregow is based on the PSE Due to the backregow within a separation bubble one mightsuspect that there is a signishy cant amount of information travelling upstream anda marching procedure as used in non-local instability analyses is not appropriateHowever there is strong indication from both experiment and DNS that thin lam-inar separation bubbles are convectively unstable to short-wavelength disturbancesof small amplitude initiated upstream of the bubble (Dovgal et al 1994 Rist et al 1996) This encourages us to embark upon application of PSE for this type of regow

As a shy rst basic regow we take BF1 of Rist amp Maucher (1994) They comparedthe results of linear local theory with those of their DNS the latter run at low

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3235

15 20 25x

0

100

200

300

400

grow

th r

ate

(m- 1

)

DNS = 0 m - 1

DNS = 400 m - 1

DNS = 800 m - 1

PSEOSE

S R

bbb

Figure 1 Comparison of non-local growth rates frac14 u for BF1 obtained by NOLOTPSE withthe DNS data of Rist amp Maucher (1994) and our OSE solutions Separation and reattachmentare denoted by Srsquo and Rrsquo respectively

disturbance amplitudes we use their growth rate data as a reference Figure 1 showsthe disturbance growth rates for f ordm 1719 Hz (reduced frequency F = 18 pound 10iexcl4)and three dinoterent (dimensional) spanwise wavenumbers shy = 0 400 and 800 miexcl1The corresponding wave angles iquest at separation point x S are iquest ordm 0 22 and 41macr Thelinear local (OSE) and the non-local (PSE) results are compared with those obtainedin the DNS of Rist amp Maucher (1994) The non-local growth rate is deshy ned as in theDNS ie on the maximum streamwise disturbance velocity (36)

The result of major signishy cance in this shy gure is the agreement between the linearPSE and DNS results which is found to be inreguenced by the wave angle For two-dimensional waves PSE delivers instability results that are in excellent agreementwith those obtained by DNS at all x positions with the obvious advantage that PSEresults are obtained at orders-of-magnitude lower computing enotort compared withDNS The agreement is still very convincing for three-dimensional waves althougha discrepancy between PSE and DNS results that increases with wave angle is tobe found upstream of the bubble There are at least three possible explanations forthis wave-angle dependent discrepancy First it might be thought that the upstreaminreguence neglected in PSE might cause PSE results to deviate from those of theDNS Second violation of the PSE assumption of a weakly non-parallel regow in theneighbourhood of the separation point may be considered to be responsible for thediscrepancy Third transients in the DNS data may cause the small dinoterences Theshy rst reason cannot explain the discrepancies because these are present upstreamof separation in a region of attached boundary-layer regow where upstream enotectsare indeed negligible but disappear in the second half of the bubble despite theexisting backregow In order for the second reason to be verishy ed one should refer to

Phil Trans R Soc Lond A (2000)

3236 V Theomacrlis S Hein and U Dallmann

0 04 08

0

001

002

003

004

005

006

UV 100

0 04 08 ||Qp||

||u||||v||||p||

yL

QbU yen

Figure 2 Velocity promacrles of the laminar basic deg ow BF1 and the amplitude functions of atwo-dimensional wave of f = 1719 Hz at x = 1984 within the recirculating deg ow

the magnitude of the non-parallel terms in the attached decelerating part of the regowdomain compared with that of the non-parallel terms in the recirculating zone

We believe that the third reason plays an essential role since disturbances aregenerated in the DNS by periodic blowing and suction at the wall immediatelyupstream of the region where regow deceleration is enforced It therefore takes somedownstream distance until these artishy cially introduced disturbances are convertedinto TS-like instabilities whose growth rate is no longer anotected by the actual meansof their generation On the other hand memory enotects are shorter in our PSE anal-yses which were started with TS waves (solutions of the OSE equation) imposedat the inregow boundary the distance from the end of the suction strip to x = 15corresponds to less than six disturbance wavefronts It is well known that both theenotects of regow non-parallelism and the enotects of disturbance history increase withdisturbance wave angle If as we think the dinoterences are indeed caused by remain-ing transients they should be more pronounced at smaller values of x However thiscould not be verishy ed because DNS growth-rate data for x lt 15 are not availablefrom Rist amp Maucher (1994) These authors also performed an OSE analysis (notpresented here) the results of which compare very well with our linear local anal-ysis (also shown in shy gure 1) The agreement between our OSE growth rate resultsand DNS is quite close in line with the shy ndings of Bestek et al (1989) and othersHowever the agreement between OSE and DNS can be seen to be much inferior tothat between PSE and DNS results

Note that the largest growth rates are of the same order of magnitude as the dis-turbance wavenumbers which means that the disturbances grow much more rapidly

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3237

1 2 3 4 50

1000

2000

3000

4000

5000

S Rx

f (H

z)

05 10 150

500

1000

1500

2000

2500

3000

S Rx

(a) (a)

Figure 3 Isolines of constant non-local growth rate frac14 E for BF1 (a) and BF2 (b) Frequenciesf of unstable waves against scaled downstream distance are plotted The outermost curvescorrespond to neutral stability and the increments in contour lines are macr frac14 E = 25 and 10 for BF1and BF2 respectively

than those in Blasius regow The non-local amplitude functions of the two-dimensionalwave of f ordm 1719 Hz are plotted in shy gure 2 for the downstream position x = 1984alongside those of the corresponding velocity proshy les of the basic regow u has maximainside the backregow region close to the wall at the inregection point of the stream-wise velocity proshy le of the basic regow and at the edge of the separated boundarylayer while v is rather large Since the two-dimensional wave is the most amplishy edat almost all downstream positions (as seen in shy gure 1) we focus all subsequentNOLOTPSE studies on two-dimensional waves The instability diagram for two-dimensional short-wavelength disturbances is shown in shy gure 3 Isolines of constantnon-local growth rate (deshy ned by 35) are plotted the outermost isoline representsthe curve of neutral stability This growth-rate deshy nition was chosen because in theseparated regow region the disturbance amplitude functions have several maxima inthe wall-normal direction (as seen in shy gure 2) The instability diagram for BF2 isalso shown in shy gure 3 Its general shape appears dinoterent from that of BF1 thelatter having an instability region of the decelerating and separated regow embeddedinto the region of TS instability unlike BF2 where the two are clearly separated (seealso comments in x 2)

(b) Global disturbances addressed by the PEPS

In the previous section we focused on instabilities of an oscillatory nature thestreamwise length-scale of which is small compared with that of the basic regowthis structure is exploited by the ansatz (32) and emacr cient marching proceduresmake a NOLOTPSE approach the method of choice for accurate results to bedelivered within reasonable computing time for this type of instability On the otherhand an analysis of disturbances developing over length-scales comparable withthat of the basic regow using NOLOTPSE may deliver inaccurate results Short ofresorting to a spatial DNS in order to take both upstream inreguence and downstream

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

3232 V Theomacrlis S Hein and U Dallmann

3 Linear instability theories

The non-localglobal nature of the instability analyses used derives from the decom-position of a perturbed regowshy eld into a steady two-dimensional basic and an unsteadythree-dimensional disturbance component according to

Q(x y z t) = Q b (x y) + Q p (x y) exp i pound + cc (31)

with Q b = (U V 0 P )T indicating the real basic regow and Qp = (u v w p)T denotingcomplex-valued perturbations to it Here x denotes the streamwise direction y thewall-normal direction and z the spanwise spatial direction the imaginary unit isi =

piexcl 1 and cc denotes complex conjugation in order for the total shy eld to remain

real Linearization about Q b follows based on the argument of smallness of and thebasic regow terms themselves satisfying the NavierStokes and continuity equationsare subtracted out The dinoterence between the two approaches stems from the ansatzmade for the complex function pound as follows

(a) Non-local analysis based on the PSE

The linear non-local instability analyses were performed with the NOnLOcal Tran-sition analysis (NOLOT)PSE code developed in cooperation between DLR and Fly-gtekniska Forsokanstalten (Hein et al 1994)y In the NOLOTPSE analysis bothQ b and Q p are functions of y slowly varying in x we call this approach a non-localanalysis as opposed to the classic local theory in which both Q b and Q p are functionsof y alone The wave function

pound =x

x0

not ( sup1 ) d sup1 + shy z iexcl laquo t (32)

with not a slowly varying streamwise wavenumber is intended to capture practicallyall oscillatory-type streamwise variations of the disturbance that is the variations onthe fast scale The disturbances are assumed to grow in the streamwise direction xonly and spatially evolving wave-like instabilities are considered Hence the stream-wise wavenumber not is represented by a complex quantity whereas the spanwisewavenumber shy and the circular frequency laquo are real parameters

Since both the amplitude function and the wave function depend on the x-direc-tion a normalization condition is required which removes this ambiguity as intro-duced by Bertolotti et al (1992)

ye

0

Q yp

Q p

xdy = 0 (33)

The superscript y refers to the complex conjugate and ye stands for the upper bound-ary of the discretized domain The normalization condition is used as an iterationcondition in order to ensure that as much as possible of the streamwise variationof the disturbance is transferred into the wave function The remaining part of thestreamwise variations is described by the amplitude functions and is small conse-quently second derivatives of the amplitude functions in x are discarded This leads

y For the purposes of the present study the code was run in the incompressible limit at which excellentagreement with results obtained by incompressible codes has been shown by Hein et al (1994)

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3233

to a system of partial dinoterential equations of the form

AQ p + BQ p

y+ C

2Q p

y2+ D

Q p

x= 0 (34)

Details of the consistent multiple-scales approach based on a Reynolds numberexpansion used to derive the non-local stability equations which are closely relatedto the PSE (Bertolotti et al 1992 Herbert 1997) and the elements of the matrices AB C D can be found in Hein et al (1994) They represent an initial-boundary-valueproblem that can be solved emacr ciently by a marching procedure in the x-directionwith upstream enotects neglected an approach valid for convectively unstable regowsonly The partial dinoterential equations system (34) is discretized by a shy rst-orderbackward Euler scheme in the x spatial direction and fourth-order compact shy nitedinoterences in the y spatial direction On the amplitude functions (u v w) we imposehomogeneous Dirichlet boundary conditions at the wall (y = 0) and asymptoticboundary conditions enforcing exponential decay at the upper boundary ye (Hein etal 1994) Initial conditions are provided by a local parallel instability analysis usingthe same code with the matrix D all streamwise derivatives in matrices A B andC and the wallnormal velocity component V of the basic regow set to zero

The spatial disturbance growth rate frac14 in local instability theory is described bythe negative imaginary part of the complex wavenumber not ie frac14 = iexcl not i whereasin non-local theory there is an additional contribution from the amplitude functionsand various reasonable alternative deshy nitions are possible here we use either

frac14 E = iexcl not i +

xln(

pE) with E =

ye

0

(kuk2 + kvk2 + kwk2) dy (35)

or

frac14 u = iexcl not i + Re1

um

um

x (36)

where um denotes the value of u at the wall-normal coordinate where kuk reaches itsmaximum

(b) The partial derivative eigenvalue problem

The assumptions of slow growth of Q b and Q p inherent in the PSE may berelaxed by fully resolving both the streamwise and the wall-normal spatial directionson account of this property we term the PEPS a global instability analysis In thetemporal framework as considered here the complex function pound becomes simply

pound = shy z iexcl laquo t (37)

where shy is a real wavenumber parameter and laquo is the sought complex eigenvaluewhose real part indicates frequency and whose positive imaginary part is the growthrate in time t of global disturbances This deshy nition of pound suggests the second essentialdinoterence between the PSE and the PEPS approaches namely that any structurethat an eigenmode may have in the streamwise direction x including that of a wave-like disturbance considered by the PSE will be captured by Q p in (31)

Numerical aspects of the solution of the two-dimensional partial derivative eigen-value problem are discussed by Theoshy lis (1998) Compared with that work we have

Phil Trans R Soc Lond A (2000)

3234 V Theomacrlis S Hein and U Dallmann

implemented here a novel version of the algorithm that applies to regow problems inwhich the wavenumber vector is perpendicular to the plane on which the basic regowis deshy ned (Theoshy lis 2000) Specishy cally straightforward redeshy nitions of the spanwisedisturbance velocity component w and the eigenvalue laquo by iw and i laquo respectivelyresult in a disturbance regow system with real coemacr cients that compared with thefull problem requires approximately half the storage and runtime to solve The realnon-symmetric generalized eigenvalue problem for the determination of laquo and Q p

which results is as follows

[ L iexcl (DxU )]u iexcl (DyU )v iexcl Dxp = laquo u (38)

iexcl (DxV )u + [ L iexcl (DyV )]v iexcl Dy p = laquo v (39)

L w iexcl shy p = laquo w (310)

Dxu + Dy v iexcl shy w = 0 (311)

where

L =1

Re(D2

x + D2y iexcl shy 2) iexcl UDx iexcl V Dy Dx =

xand Dy =

y

For its solution we use spectral collocation on a rectangular Cartesian grid Theboundary conditions imposed are the straightforward viscous boundary conditionson all disturbance velocity components at the wall and compatibility conditions forthe pressure in the freestream the disturbance velocity components and disturbancepressure are required to decay In an attempt to preclude disturbances from enteringthe recirculating regow area from upstream homogeneous Dirichlet boundary condi-tions are imposed at the inregow At the outregow boundary linear extrapolation of alldisturbance quantities from the interior of the integration domain is performed

4 Results

(a) Short chordwise-wavelength instabilities studied by the PSE

A classic means of linear analysis of short-wavelength instabilities in laminar separa-tion bubbles is based on local theory in which the parallel regow assumption is madeand the OrrSommerfeld equation (OSE) is solved (Allen amp Riley 1995) There is norational approach leading to the latter equation that ignores both upstream inreguenceand downstream development of the basic regow nevertheless both the OSE and spa-tial DNS predict in close agreement an explosive growth of convective instabilitiesentering the laminar separated regow (Bestek et al 1989) On the other hand an insta-bility analysis approach which takes into account downstream development of theregow is based on the PSE Due to the backregow within a separation bubble one mightsuspect that there is a signishy cant amount of information travelling upstream anda marching procedure as used in non-local instability analyses is not appropriateHowever there is strong indication from both experiment and DNS that thin lam-inar separation bubbles are convectively unstable to short-wavelength disturbancesof small amplitude initiated upstream of the bubble (Dovgal et al 1994 Rist et al 1996) This encourages us to embark upon application of PSE for this type of regow

As a shy rst basic regow we take BF1 of Rist amp Maucher (1994) They comparedthe results of linear local theory with those of their DNS the latter run at low

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3235

15 20 25x

0

100

200

300

400

grow

th r

ate

(m- 1

)

DNS = 0 m - 1

DNS = 400 m - 1

DNS = 800 m - 1

PSEOSE

S R

bbb

Figure 1 Comparison of non-local growth rates frac14 u for BF1 obtained by NOLOTPSE withthe DNS data of Rist amp Maucher (1994) and our OSE solutions Separation and reattachmentare denoted by Srsquo and Rrsquo respectively

disturbance amplitudes we use their growth rate data as a reference Figure 1 showsthe disturbance growth rates for f ordm 1719 Hz (reduced frequency F = 18 pound 10iexcl4)and three dinoterent (dimensional) spanwise wavenumbers shy = 0 400 and 800 miexcl1The corresponding wave angles iquest at separation point x S are iquest ordm 0 22 and 41macr Thelinear local (OSE) and the non-local (PSE) results are compared with those obtainedin the DNS of Rist amp Maucher (1994) The non-local growth rate is deshy ned as in theDNS ie on the maximum streamwise disturbance velocity (36)

The result of major signishy cance in this shy gure is the agreement between the linearPSE and DNS results which is found to be inreguenced by the wave angle For two-dimensional waves PSE delivers instability results that are in excellent agreementwith those obtained by DNS at all x positions with the obvious advantage that PSEresults are obtained at orders-of-magnitude lower computing enotort compared withDNS The agreement is still very convincing for three-dimensional waves althougha discrepancy between PSE and DNS results that increases with wave angle is tobe found upstream of the bubble There are at least three possible explanations forthis wave-angle dependent discrepancy First it might be thought that the upstreaminreguence neglected in PSE might cause PSE results to deviate from those of theDNS Second violation of the PSE assumption of a weakly non-parallel regow in theneighbourhood of the separation point may be considered to be responsible for thediscrepancy Third transients in the DNS data may cause the small dinoterences Theshy rst reason cannot explain the discrepancies because these are present upstreamof separation in a region of attached boundary-layer regow where upstream enotectsare indeed negligible but disappear in the second half of the bubble despite theexisting backregow In order for the second reason to be verishy ed one should refer to

Phil Trans R Soc Lond A (2000)

3236 V Theomacrlis S Hein and U Dallmann

0 04 08

0

001

002

003

004

005

006

UV 100

0 04 08 ||Qp||

||u||||v||||p||

yL

QbU yen

Figure 2 Velocity promacrles of the laminar basic deg ow BF1 and the amplitude functions of atwo-dimensional wave of f = 1719 Hz at x = 1984 within the recirculating deg ow

the magnitude of the non-parallel terms in the attached decelerating part of the regowdomain compared with that of the non-parallel terms in the recirculating zone

We believe that the third reason plays an essential role since disturbances aregenerated in the DNS by periodic blowing and suction at the wall immediatelyupstream of the region where regow deceleration is enforced It therefore takes somedownstream distance until these artishy cially introduced disturbances are convertedinto TS-like instabilities whose growth rate is no longer anotected by the actual meansof their generation On the other hand memory enotects are shorter in our PSE anal-yses which were started with TS waves (solutions of the OSE equation) imposedat the inregow boundary the distance from the end of the suction strip to x = 15corresponds to less than six disturbance wavefronts It is well known that both theenotects of regow non-parallelism and the enotects of disturbance history increase withdisturbance wave angle If as we think the dinoterences are indeed caused by remain-ing transients they should be more pronounced at smaller values of x However thiscould not be verishy ed because DNS growth-rate data for x lt 15 are not availablefrom Rist amp Maucher (1994) These authors also performed an OSE analysis (notpresented here) the results of which compare very well with our linear local anal-ysis (also shown in shy gure 1) The agreement between our OSE growth rate resultsand DNS is quite close in line with the shy ndings of Bestek et al (1989) and othersHowever the agreement between OSE and DNS can be seen to be much inferior tothat between PSE and DNS results

Note that the largest growth rates are of the same order of magnitude as the dis-turbance wavenumbers which means that the disturbances grow much more rapidly

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3237

1 2 3 4 50

1000

2000

3000

4000

5000

S Rx

f (H

z)

05 10 150

500

1000

1500

2000

2500

3000

S Rx

(a) (a)

Figure 3 Isolines of constant non-local growth rate frac14 E for BF1 (a) and BF2 (b) Frequenciesf of unstable waves against scaled downstream distance are plotted The outermost curvescorrespond to neutral stability and the increments in contour lines are macr frac14 E = 25 and 10 for BF1and BF2 respectively

than those in Blasius regow The non-local amplitude functions of the two-dimensionalwave of f ordm 1719 Hz are plotted in shy gure 2 for the downstream position x = 1984alongside those of the corresponding velocity proshy les of the basic regow u has maximainside the backregow region close to the wall at the inregection point of the stream-wise velocity proshy le of the basic regow and at the edge of the separated boundarylayer while v is rather large Since the two-dimensional wave is the most amplishy edat almost all downstream positions (as seen in shy gure 1) we focus all subsequentNOLOTPSE studies on two-dimensional waves The instability diagram for two-dimensional short-wavelength disturbances is shown in shy gure 3 Isolines of constantnon-local growth rate (deshy ned by 35) are plotted the outermost isoline representsthe curve of neutral stability This growth-rate deshy nition was chosen because in theseparated regow region the disturbance amplitude functions have several maxima inthe wall-normal direction (as seen in shy gure 2) The instability diagram for BF2 isalso shown in shy gure 3 Its general shape appears dinoterent from that of BF1 thelatter having an instability region of the decelerating and separated regow embeddedinto the region of TS instability unlike BF2 where the two are clearly separated (seealso comments in x 2)

(b) Global disturbances addressed by the PEPS

In the previous section we focused on instabilities of an oscillatory nature thestreamwise length-scale of which is small compared with that of the basic regowthis structure is exploited by the ansatz (32) and emacr cient marching proceduresmake a NOLOTPSE approach the method of choice for accurate results to bedelivered within reasonable computing time for this type of instability On the otherhand an analysis of disturbances developing over length-scales comparable withthat of the basic regow using NOLOTPSE may deliver inaccurate results Short ofresorting to a spatial DNS in order to take both upstream inreguence and downstream

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3233

to a system of partial dinoterential equations of the form

AQ p + BQ p

y+ C

2Q p

y2+ D

Q p

x= 0 (34)

Details of the consistent multiple-scales approach based on a Reynolds numberexpansion used to derive the non-local stability equations which are closely relatedto the PSE (Bertolotti et al 1992 Herbert 1997) and the elements of the matrices AB C D can be found in Hein et al (1994) They represent an initial-boundary-valueproblem that can be solved emacr ciently by a marching procedure in the x-directionwith upstream enotects neglected an approach valid for convectively unstable regowsonly The partial dinoterential equations system (34) is discretized by a shy rst-orderbackward Euler scheme in the x spatial direction and fourth-order compact shy nitedinoterences in the y spatial direction On the amplitude functions (u v w) we imposehomogeneous Dirichlet boundary conditions at the wall (y = 0) and asymptoticboundary conditions enforcing exponential decay at the upper boundary ye (Hein etal 1994) Initial conditions are provided by a local parallel instability analysis usingthe same code with the matrix D all streamwise derivatives in matrices A B andC and the wallnormal velocity component V of the basic regow set to zero

The spatial disturbance growth rate frac14 in local instability theory is described bythe negative imaginary part of the complex wavenumber not ie frac14 = iexcl not i whereasin non-local theory there is an additional contribution from the amplitude functionsand various reasonable alternative deshy nitions are possible here we use either

frac14 E = iexcl not i +

xln(

pE) with E =

ye

0

(kuk2 + kvk2 + kwk2) dy (35)

or

frac14 u = iexcl not i + Re1

um

um

x (36)

where um denotes the value of u at the wall-normal coordinate where kuk reaches itsmaximum

(b) The partial derivative eigenvalue problem

The assumptions of slow growth of Q b and Q p inherent in the PSE may berelaxed by fully resolving both the streamwise and the wall-normal spatial directionson account of this property we term the PEPS a global instability analysis In thetemporal framework as considered here the complex function pound becomes simply

pound = shy z iexcl laquo t (37)

where shy is a real wavenumber parameter and laquo is the sought complex eigenvaluewhose real part indicates frequency and whose positive imaginary part is the growthrate in time t of global disturbances This deshy nition of pound suggests the second essentialdinoterence between the PSE and the PEPS approaches namely that any structurethat an eigenmode may have in the streamwise direction x including that of a wave-like disturbance considered by the PSE will be captured by Q p in (31)

Numerical aspects of the solution of the two-dimensional partial derivative eigen-value problem are discussed by Theoshy lis (1998) Compared with that work we have

Phil Trans R Soc Lond A (2000)

3234 V Theomacrlis S Hein and U Dallmann

implemented here a novel version of the algorithm that applies to regow problems inwhich the wavenumber vector is perpendicular to the plane on which the basic regowis deshy ned (Theoshy lis 2000) Specishy cally straightforward redeshy nitions of the spanwisedisturbance velocity component w and the eigenvalue laquo by iw and i laquo respectivelyresult in a disturbance regow system with real coemacr cients that compared with thefull problem requires approximately half the storage and runtime to solve The realnon-symmetric generalized eigenvalue problem for the determination of laquo and Q p

which results is as follows

[ L iexcl (DxU )]u iexcl (DyU )v iexcl Dxp = laquo u (38)

iexcl (DxV )u + [ L iexcl (DyV )]v iexcl Dy p = laquo v (39)

L w iexcl shy p = laquo w (310)

Dxu + Dy v iexcl shy w = 0 (311)

where

L =1

Re(D2

x + D2y iexcl shy 2) iexcl UDx iexcl V Dy Dx =

xand Dy =

y

For its solution we use spectral collocation on a rectangular Cartesian grid Theboundary conditions imposed are the straightforward viscous boundary conditionson all disturbance velocity components at the wall and compatibility conditions forthe pressure in the freestream the disturbance velocity components and disturbancepressure are required to decay In an attempt to preclude disturbances from enteringthe recirculating regow area from upstream homogeneous Dirichlet boundary condi-tions are imposed at the inregow At the outregow boundary linear extrapolation of alldisturbance quantities from the interior of the integration domain is performed

4 Results

(a) Short chordwise-wavelength instabilities studied by the PSE

A classic means of linear analysis of short-wavelength instabilities in laminar separa-tion bubbles is based on local theory in which the parallel regow assumption is madeand the OrrSommerfeld equation (OSE) is solved (Allen amp Riley 1995) There is norational approach leading to the latter equation that ignores both upstream inreguenceand downstream development of the basic regow nevertheless both the OSE and spa-tial DNS predict in close agreement an explosive growth of convective instabilitiesentering the laminar separated regow (Bestek et al 1989) On the other hand an insta-bility analysis approach which takes into account downstream development of theregow is based on the PSE Due to the backregow within a separation bubble one mightsuspect that there is a signishy cant amount of information travelling upstream anda marching procedure as used in non-local instability analyses is not appropriateHowever there is strong indication from both experiment and DNS that thin lam-inar separation bubbles are convectively unstable to short-wavelength disturbancesof small amplitude initiated upstream of the bubble (Dovgal et al 1994 Rist et al 1996) This encourages us to embark upon application of PSE for this type of regow

As a shy rst basic regow we take BF1 of Rist amp Maucher (1994) They comparedthe results of linear local theory with those of their DNS the latter run at low

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3235

15 20 25x

0

100

200

300

400

grow

th r

ate

(m- 1

)

DNS = 0 m - 1

DNS = 400 m - 1

DNS = 800 m - 1

PSEOSE

S R

bbb

Figure 1 Comparison of non-local growth rates frac14 u for BF1 obtained by NOLOTPSE withthe DNS data of Rist amp Maucher (1994) and our OSE solutions Separation and reattachmentare denoted by Srsquo and Rrsquo respectively

disturbance amplitudes we use their growth rate data as a reference Figure 1 showsthe disturbance growth rates for f ordm 1719 Hz (reduced frequency F = 18 pound 10iexcl4)and three dinoterent (dimensional) spanwise wavenumbers shy = 0 400 and 800 miexcl1The corresponding wave angles iquest at separation point x S are iquest ordm 0 22 and 41macr Thelinear local (OSE) and the non-local (PSE) results are compared with those obtainedin the DNS of Rist amp Maucher (1994) The non-local growth rate is deshy ned as in theDNS ie on the maximum streamwise disturbance velocity (36)

The result of major signishy cance in this shy gure is the agreement between the linearPSE and DNS results which is found to be inreguenced by the wave angle For two-dimensional waves PSE delivers instability results that are in excellent agreementwith those obtained by DNS at all x positions with the obvious advantage that PSEresults are obtained at orders-of-magnitude lower computing enotort compared withDNS The agreement is still very convincing for three-dimensional waves althougha discrepancy between PSE and DNS results that increases with wave angle is tobe found upstream of the bubble There are at least three possible explanations forthis wave-angle dependent discrepancy First it might be thought that the upstreaminreguence neglected in PSE might cause PSE results to deviate from those of theDNS Second violation of the PSE assumption of a weakly non-parallel regow in theneighbourhood of the separation point may be considered to be responsible for thediscrepancy Third transients in the DNS data may cause the small dinoterences Theshy rst reason cannot explain the discrepancies because these are present upstreamof separation in a region of attached boundary-layer regow where upstream enotectsare indeed negligible but disappear in the second half of the bubble despite theexisting backregow In order for the second reason to be verishy ed one should refer to

Phil Trans R Soc Lond A (2000)

3236 V Theomacrlis S Hein and U Dallmann

0 04 08

0

001

002

003

004

005

006

UV 100

0 04 08 ||Qp||

||u||||v||||p||

yL

QbU yen

Figure 2 Velocity promacrles of the laminar basic deg ow BF1 and the amplitude functions of atwo-dimensional wave of f = 1719 Hz at x = 1984 within the recirculating deg ow

the magnitude of the non-parallel terms in the attached decelerating part of the regowdomain compared with that of the non-parallel terms in the recirculating zone

We believe that the third reason plays an essential role since disturbances aregenerated in the DNS by periodic blowing and suction at the wall immediatelyupstream of the region where regow deceleration is enforced It therefore takes somedownstream distance until these artishy cially introduced disturbances are convertedinto TS-like instabilities whose growth rate is no longer anotected by the actual meansof their generation On the other hand memory enotects are shorter in our PSE anal-yses which were started with TS waves (solutions of the OSE equation) imposedat the inregow boundary the distance from the end of the suction strip to x = 15corresponds to less than six disturbance wavefronts It is well known that both theenotects of regow non-parallelism and the enotects of disturbance history increase withdisturbance wave angle If as we think the dinoterences are indeed caused by remain-ing transients they should be more pronounced at smaller values of x However thiscould not be verishy ed because DNS growth-rate data for x lt 15 are not availablefrom Rist amp Maucher (1994) These authors also performed an OSE analysis (notpresented here) the results of which compare very well with our linear local anal-ysis (also shown in shy gure 1) The agreement between our OSE growth rate resultsand DNS is quite close in line with the shy ndings of Bestek et al (1989) and othersHowever the agreement between OSE and DNS can be seen to be much inferior tothat between PSE and DNS results

Note that the largest growth rates are of the same order of magnitude as the dis-turbance wavenumbers which means that the disturbances grow much more rapidly

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3237

1 2 3 4 50

1000

2000

3000

4000

5000

S Rx

f (H

z)

05 10 150

500

1000

1500

2000

2500

3000

S Rx

(a) (a)

Figure 3 Isolines of constant non-local growth rate frac14 E for BF1 (a) and BF2 (b) Frequenciesf of unstable waves against scaled downstream distance are plotted The outermost curvescorrespond to neutral stability and the increments in contour lines are macr frac14 E = 25 and 10 for BF1and BF2 respectively

than those in Blasius regow The non-local amplitude functions of the two-dimensionalwave of f ordm 1719 Hz are plotted in shy gure 2 for the downstream position x = 1984alongside those of the corresponding velocity proshy les of the basic regow u has maximainside the backregow region close to the wall at the inregection point of the stream-wise velocity proshy le of the basic regow and at the edge of the separated boundarylayer while v is rather large Since the two-dimensional wave is the most amplishy edat almost all downstream positions (as seen in shy gure 1) we focus all subsequentNOLOTPSE studies on two-dimensional waves The instability diagram for two-dimensional short-wavelength disturbances is shown in shy gure 3 Isolines of constantnon-local growth rate (deshy ned by 35) are plotted the outermost isoline representsthe curve of neutral stability This growth-rate deshy nition was chosen because in theseparated regow region the disturbance amplitude functions have several maxima inthe wall-normal direction (as seen in shy gure 2) The instability diagram for BF2 isalso shown in shy gure 3 Its general shape appears dinoterent from that of BF1 thelatter having an instability region of the decelerating and separated regow embeddedinto the region of TS instability unlike BF2 where the two are clearly separated (seealso comments in x 2)

(b) Global disturbances addressed by the PEPS

In the previous section we focused on instabilities of an oscillatory nature thestreamwise length-scale of which is small compared with that of the basic regowthis structure is exploited by the ansatz (32) and emacr cient marching proceduresmake a NOLOTPSE approach the method of choice for accurate results to bedelivered within reasonable computing time for this type of instability On the otherhand an analysis of disturbances developing over length-scales comparable withthat of the basic regow using NOLOTPSE may deliver inaccurate results Short ofresorting to a spatial DNS in order to take both upstream inreguence and downstream

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

3234 V Theomacrlis S Hein and U Dallmann

implemented here a novel version of the algorithm that applies to regow problems inwhich the wavenumber vector is perpendicular to the plane on which the basic regowis deshy ned (Theoshy lis 2000) Specishy cally straightforward redeshy nitions of the spanwisedisturbance velocity component w and the eigenvalue laquo by iw and i laquo respectivelyresult in a disturbance regow system with real coemacr cients that compared with thefull problem requires approximately half the storage and runtime to solve The realnon-symmetric generalized eigenvalue problem for the determination of laquo and Q p

which results is as follows

[ L iexcl (DxU )]u iexcl (DyU )v iexcl Dxp = laquo u (38)

iexcl (DxV )u + [ L iexcl (DyV )]v iexcl Dy p = laquo v (39)

L w iexcl shy p = laquo w (310)

Dxu + Dy v iexcl shy w = 0 (311)

where

L =1

Re(D2

x + D2y iexcl shy 2) iexcl UDx iexcl V Dy Dx =

xand Dy =

y

For its solution we use spectral collocation on a rectangular Cartesian grid Theboundary conditions imposed are the straightforward viscous boundary conditionson all disturbance velocity components at the wall and compatibility conditions forthe pressure in the freestream the disturbance velocity components and disturbancepressure are required to decay In an attempt to preclude disturbances from enteringthe recirculating regow area from upstream homogeneous Dirichlet boundary condi-tions are imposed at the inregow At the outregow boundary linear extrapolation of alldisturbance quantities from the interior of the integration domain is performed

4 Results

(a) Short chordwise-wavelength instabilities studied by the PSE

A classic means of linear analysis of short-wavelength instabilities in laminar separa-tion bubbles is based on local theory in which the parallel regow assumption is madeand the OrrSommerfeld equation (OSE) is solved (Allen amp Riley 1995) There is norational approach leading to the latter equation that ignores both upstream inreguenceand downstream development of the basic regow nevertheless both the OSE and spa-tial DNS predict in close agreement an explosive growth of convective instabilitiesentering the laminar separated regow (Bestek et al 1989) On the other hand an insta-bility analysis approach which takes into account downstream development of theregow is based on the PSE Due to the backregow within a separation bubble one mightsuspect that there is a signishy cant amount of information travelling upstream anda marching procedure as used in non-local instability analyses is not appropriateHowever there is strong indication from both experiment and DNS that thin lam-inar separation bubbles are convectively unstable to short-wavelength disturbancesof small amplitude initiated upstream of the bubble (Dovgal et al 1994 Rist et al 1996) This encourages us to embark upon application of PSE for this type of regow

As a shy rst basic regow we take BF1 of Rist amp Maucher (1994) They comparedthe results of linear local theory with those of their DNS the latter run at low

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3235

15 20 25x

0

100

200

300

400

grow

th r

ate

(m- 1

)

DNS = 0 m - 1

DNS = 400 m - 1

DNS = 800 m - 1

PSEOSE

S R

bbb

Figure 1 Comparison of non-local growth rates frac14 u for BF1 obtained by NOLOTPSE withthe DNS data of Rist amp Maucher (1994) and our OSE solutions Separation and reattachmentare denoted by Srsquo and Rrsquo respectively

disturbance amplitudes we use their growth rate data as a reference Figure 1 showsthe disturbance growth rates for f ordm 1719 Hz (reduced frequency F = 18 pound 10iexcl4)and three dinoterent (dimensional) spanwise wavenumbers shy = 0 400 and 800 miexcl1The corresponding wave angles iquest at separation point x S are iquest ordm 0 22 and 41macr Thelinear local (OSE) and the non-local (PSE) results are compared with those obtainedin the DNS of Rist amp Maucher (1994) The non-local growth rate is deshy ned as in theDNS ie on the maximum streamwise disturbance velocity (36)

The result of major signishy cance in this shy gure is the agreement between the linearPSE and DNS results which is found to be inreguenced by the wave angle For two-dimensional waves PSE delivers instability results that are in excellent agreementwith those obtained by DNS at all x positions with the obvious advantage that PSEresults are obtained at orders-of-magnitude lower computing enotort compared withDNS The agreement is still very convincing for three-dimensional waves althougha discrepancy between PSE and DNS results that increases with wave angle is tobe found upstream of the bubble There are at least three possible explanations forthis wave-angle dependent discrepancy First it might be thought that the upstreaminreguence neglected in PSE might cause PSE results to deviate from those of theDNS Second violation of the PSE assumption of a weakly non-parallel regow in theneighbourhood of the separation point may be considered to be responsible for thediscrepancy Third transients in the DNS data may cause the small dinoterences Theshy rst reason cannot explain the discrepancies because these are present upstreamof separation in a region of attached boundary-layer regow where upstream enotectsare indeed negligible but disappear in the second half of the bubble despite theexisting backregow In order for the second reason to be verishy ed one should refer to

Phil Trans R Soc Lond A (2000)

3236 V Theomacrlis S Hein and U Dallmann

0 04 08

0

001

002

003

004

005

006

UV 100

0 04 08 ||Qp||

||u||||v||||p||

yL

QbU yen

Figure 2 Velocity promacrles of the laminar basic deg ow BF1 and the amplitude functions of atwo-dimensional wave of f = 1719 Hz at x = 1984 within the recirculating deg ow

the magnitude of the non-parallel terms in the attached decelerating part of the regowdomain compared with that of the non-parallel terms in the recirculating zone

We believe that the third reason plays an essential role since disturbances aregenerated in the DNS by periodic blowing and suction at the wall immediatelyupstream of the region where regow deceleration is enforced It therefore takes somedownstream distance until these artishy cially introduced disturbances are convertedinto TS-like instabilities whose growth rate is no longer anotected by the actual meansof their generation On the other hand memory enotects are shorter in our PSE anal-yses which were started with TS waves (solutions of the OSE equation) imposedat the inregow boundary the distance from the end of the suction strip to x = 15corresponds to less than six disturbance wavefronts It is well known that both theenotects of regow non-parallelism and the enotects of disturbance history increase withdisturbance wave angle If as we think the dinoterences are indeed caused by remain-ing transients they should be more pronounced at smaller values of x However thiscould not be verishy ed because DNS growth-rate data for x lt 15 are not availablefrom Rist amp Maucher (1994) These authors also performed an OSE analysis (notpresented here) the results of which compare very well with our linear local anal-ysis (also shown in shy gure 1) The agreement between our OSE growth rate resultsand DNS is quite close in line with the shy ndings of Bestek et al (1989) and othersHowever the agreement between OSE and DNS can be seen to be much inferior tothat between PSE and DNS results

Note that the largest growth rates are of the same order of magnitude as the dis-turbance wavenumbers which means that the disturbances grow much more rapidly

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3237

1 2 3 4 50

1000

2000

3000

4000

5000

S Rx

f (H

z)

05 10 150

500

1000

1500

2000

2500

3000

S Rx

(a) (a)

Figure 3 Isolines of constant non-local growth rate frac14 E for BF1 (a) and BF2 (b) Frequenciesf of unstable waves against scaled downstream distance are plotted The outermost curvescorrespond to neutral stability and the increments in contour lines are macr frac14 E = 25 and 10 for BF1and BF2 respectively

than those in Blasius regow The non-local amplitude functions of the two-dimensionalwave of f ordm 1719 Hz are plotted in shy gure 2 for the downstream position x = 1984alongside those of the corresponding velocity proshy les of the basic regow u has maximainside the backregow region close to the wall at the inregection point of the stream-wise velocity proshy le of the basic regow and at the edge of the separated boundarylayer while v is rather large Since the two-dimensional wave is the most amplishy edat almost all downstream positions (as seen in shy gure 1) we focus all subsequentNOLOTPSE studies on two-dimensional waves The instability diagram for two-dimensional short-wavelength disturbances is shown in shy gure 3 Isolines of constantnon-local growth rate (deshy ned by 35) are plotted the outermost isoline representsthe curve of neutral stability This growth-rate deshy nition was chosen because in theseparated regow region the disturbance amplitude functions have several maxima inthe wall-normal direction (as seen in shy gure 2) The instability diagram for BF2 isalso shown in shy gure 3 Its general shape appears dinoterent from that of BF1 thelatter having an instability region of the decelerating and separated regow embeddedinto the region of TS instability unlike BF2 where the two are clearly separated (seealso comments in x 2)

(b) Global disturbances addressed by the PEPS

In the previous section we focused on instabilities of an oscillatory nature thestreamwise length-scale of which is small compared with that of the basic regowthis structure is exploited by the ansatz (32) and emacr cient marching proceduresmake a NOLOTPSE approach the method of choice for accurate results to bedelivered within reasonable computing time for this type of instability On the otherhand an analysis of disturbances developing over length-scales comparable withthat of the basic regow using NOLOTPSE may deliver inaccurate results Short ofresorting to a spatial DNS in order to take both upstream inreguence and downstream

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3235

15 20 25x

0

100

200

300

400

grow

th r

ate

(m- 1

)

DNS = 0 m - 1

DNS = 400 m - 1

DNS = 800 m - 1

PSEOSE

S R

bbb

Figure 1 Comparison of non-local growth rates frac14 u for BF1 obtained by NOLOTPSE withthe DNS data of Rist amp Maucher (1994) and our OSE solutions Separation and reattachmentare denoted by Srsquo and Rrsquo respectively

disturbance amplitudes we use their growth rate data as a reference Figure 1 showsthe disturbance growth rates for f ordm 1719 Hz (reduced frequency F = 18 pound 10iexcl4)and three dinoterent (dimensional) spanwise wavenumbers shy = 0 400 and 800 miexcl1The corresponding wave angles iquest at separation point x S are iquest ordm 0 22 and 41macr Thelinear local (OSE) and the non-local (PSE) results are compared with those obtainedin the DNS of Rist amp Maucher (1994) The non-local growth rate is deshy ned as in theDNS ie on the maximum streamwise disturbance velocity (36)

The result of major signishy cance in this shy gure is the agreement between the linearPSE and DNS results which is found to be inreguenced by the wave angle For two-dimensional waves PSE delivers instability results that are in excellent agreementwith those obtained by DNS at all x positions with the obvious advantage that PSEresults are obtained at orders-of-magnitude lower computing enotort compared withDNS The agreement is still very convincing for three-dimensional waves althougha discrepancy between PSE and DNS results that increases with wave angle is tobe found upstream of the bubble There are at least three possible explanations forthis wave-angle dependent discrepancy First it might be thought that the upstreaminreguence neglected in PSE might cause PSE results to deviate from those of theDNS Second violation of the PSE assumption of a weakly non-parallel regow in theneighbourhood of the separation point may be considered to be responsible for thediscrepancy Third transients in the DNS data may cause the small dinoterences Theshy rst reason cannot explain the discrepancies because these are present upstreamof separation in a region of attached boundary-layer regow where upstream enotectsare indeed negligible but disappear in the second half of the bubble despite theexisting backregow In order for the second reason to be verishy ed one should refer to

Phil Trans R Soc Lond A (2000)

3236 V Theomacrlis S Hein and U Dallmann

0 04 08

0

001

002

003

004

005

006

UV 100

0 04 08 ||Qp||

||u||||v||||p||

yL

QbU yen

Figure 2 Velocity promacrles of the laminar basic deg ow BF1 and the amplitude functions of atwo-dimensional wave of f = 1719 Hz at x = 1984 within the recirculating deg ow

the magnitude of the non-parallel terms in the attached decelerating part of the regowdomain compared with that of the non-parallel terms in the recirculating zone

We believe that the third reason plays an essential role since disturbances aregenerated in the DNS by periodic blowing and suction at the wall immediatelyupstream of the region where regow deceleration is enforced It therefore takes somedownstream distance until these artishy cially introduced disturbances are convertedinto TS-like instabilities whose growth rate is no longer anotected by the actual meansof their generation On the other hand memory enotects are shorter in our PSE anal-yses which were started with TS waves (solutions of the OSE equation) imposedat the inregow boundary the distance from the end of the suction strip to x = 15corresponds to less than six disturbance wavefronts It is well known that both theenotects of regow non-parallelism and the enotects of disturbance history increase withdisturbance wave angle If as we think the dinoterences are indeed caused by remain-ing transients they should be more pronounced at smaller values of x However thiscould not be verishy ed because DNS growth-rate data for x lt 15 are not availablefrom Rist amp Maucher (1994) These authors also performed an OSE analysis (notpresented here) the results of which compare very well with our linear local anal-ysis (also shown in shy gure 1) The agreement between our OSE growth rate resultsand DNS is quite close in line with the shy ndings of Bestek et al (1989) and othersHowever the agreement between OSE and DNS can be seen to be much inferior tothat between PSE and DNS results

Note that the largest growth rates are of the same order of magnitude as the dis-turbance wavenumbers which means that the disturbances grow much more rapidly

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3237

1 2 3 4 50

1000

2000

3000

4000

5000

S Rx

f (H

z)

05 10 150

500

1000

1500

2000

2500

3000

S Rx

(a) (a)

Figure 3 Isolines of constant non-local growth rate frac14 E for BF1 (a) and BF2 (b) Frequenciesf of unstable waves against scaled downstream distance are plotted The outermost curvescorrespond to neutral stability and the increments in contour lines are macr frac14 E = 25 and 10 for BF1and BF2 respectively

than those in Blasius regow The non-local amplitude functions of the two-dimensionalwave of f ordm 1719 Hz are plotted in shy gure 2 for the downstream position x = 1984alongside those of the corresponding velocity proshy les of the basic regow u has maximainside the backregow region close to the wall at the inregection point of the stream-wise velocity proshy le of the basic regow and at the edge of the separated boundarylayer while v is rather large Since the two-dimensional wave is the most amplishy edat almost all downstream positions (as seen in shy gure 1) we focus all subsequentNOLOTPSE studies on two-dimensional waves The instability diagram for two-dimensional short-wavelength disturbances is shown in shy gure 3 Isolines of constantnon-local growth rate (deshy ned by 35) are plotted the outermost isoline representsthe curve of neutral stability This growth-rate deshy nition was chosen because in theseparated regow region the disturbance amplitude functions have several maxima inthe wall-normal direction (as seen in shy gure 2) The instability diagram for BF2 isalso shown in shy gure 3 Its general shape appears dinoterent from that of BF1 thelatter having an instability region of the decelerating and separated regow embeddedinto the region of TS instability unlike BF2 where the two are clearly separated (seealso comments in x 2)

(b) Global disturbances addressed by the PEPS

In the previous section we focused on instabilities of an oscillatory nature thestreamwise length-scale of which is small compared with that of the basic regowthis structure is exploited by the ansatz (32) and emacr cient marching proceduresmake a NOLOTPSE approach the method of choice for accurate results to bedelivered within reasonable computing time for this type of instability On the otherhand an analysis of disturbances developing over length-scales comparable withthat of the basic regow using NOLOTPSE may deliver inaccurate results Short ofresorting to a spatial DNS in order to take both upstream inreguence and downstream

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

3236 V Theomacrlis S Hein and U Dallmann

0 04 08

0

001

002

003

004

005

006

UV 100

0 04 08 ||Qp||

||u||||v||||p||

yL

QbU yen

Figure 2 Velocity promacrles of the laminar basic deg ow BF1 and the amplitude functions of atwo-dimensional wave of f = 1719 Hz at x = 1984 within the recirculating deg ow

the magnitude of the non-parallel terms in the attached decelerating part of the regowdomain compared with that of the non-parallel terms in the recirculating zone

We believe that the third reason plays an essential role since disturbances aregenerated in the DNS by periodic blowing and suction at the wall immediatelyupstream of the region where regow deceleration is enforced It therefore takes somedownstream distance until these artishy cially introduced disturbances are convertedinto TS-like instabilities whose growth rate is no longer anotected by the actual meansof their generation On the other hand memory enotects are shorter in our PSE anal-yses which were started with TS waves (solutions of the OSE equation) imposedat the inregow boundary the distance from the end of the suction strip to x = 15corresponds to less than six disturbance wavefronts It is well known that both theenotects of regow non-parallelism and the enotects of disturbance history increase withdisturbance wave angle If as we think the dinoterences are indeed caused by remain-ing transients they should be more pronounced at smaller values of x However thiscould not be verishy ed because DNS growth-rate data for x lt 15 are not availablefrom Rist amp Maucher (1994) These authors also performed an OSE analysis (notpresented here) the results of which compare very well with our linear local anal-ysis (also shown in shy gure 1) The agreement between our OSE growth rate resultsand DNS is quite close in line with the shy ndings of Bestek et al (1989) and othersHowever the agreement between OSE and DNS can be seen to be much inferior tothat between PSE and DNS results

Note that the largest growth rates are of the same order of magnitude as the dis-turbance wavenumbers which means that the disturbances grow much more rapidly

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3237

1 2 3 4 50

1000

2000

3000

4000

5000

S Rx

f (H

z)

05 10 150

500

1000

1500

2000

2500

3000

S Rx

(a) (a)

Figure 3 Isolines of constant non-local growth rate frac14 E for BF1 (a) and BF2 (b) Frequenciesf of unstable waves against scaled downstream distance are plotted The outermost curvescorrespond to neutral stability and the increments in contour lines are macr frac14 E = 25 and 10 for BF1and BF2 respectively

than those in Blasius regow The non-local amplitude functions of the two-dimensionalwave of f ordm 1719 Hz are plotted in shy gure 2 for the downstream position x = 1984alongside those of the corresponding velocity proshy les of the basic regow u has maximainside the backregow region close to the wall at the inregection point of the stream-wise velocity proshy le of the basic regow and at the edge of the separated boundarylayer while v is rather large Since the two-dimensional wave is the most amplishy edat almost all downstream positions (as seen in shy gure 1) we focus all subsequentNOLOTPSE studies on two-dimensional waves The instability diagram for two-dimensional short-wavelength disturbances is shown in shy gure 3 Isolines of constantnon-local growth rate (deshy ned by 35) are plotted the outermost isoline representsthe curve of neutral stability This growth-rate deshy nition was chosen because in theseparated regow region the disturbance amplitude functions have several maxima inthe wall-normal direction (as seen in shy gure 2) The instability diagram for BF2 isalso shown in shy gure 3 Its general shape appears dinoterent from that of BF1 thelatter having an instability region of the decelerating and separated regow embeddedinto the region of TS instability unlike BF2 where the two are clearly separated (seealso comments in x 2)

(b) Global disturbances addressed by the PEPS

In the previous section we focused on instabilities of an oscillatory nature thestreamwise length-scale of which is small compared with that of the basic regowthis structure is exploited by the ansatz (32) and emacr cient marching proceduresmake a NOLOTPSE approach the method of choice for accurate results to bedelivered within reasonable computing time for this type of instability On the otherhand an analysis of disturbances developing over length-scales comparable withthat of the basic regow using NOLOTPSE may deliver inaccurate results Short ofresorting to a spatial DNS in order to take both upstream inreguence and downstream

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3237

1 2 3 4 50

1000

2000

3000

4000

5000

S Rx

f (H

z)

05 10 150

500

1000

1500

2000

2500

3000

S Rx

(a) (a)

Figure 3 Isolines of constant non-local growth rate frac14 E for BF1 (a) and BF2 (b) Frequenciesf of unstable waves against scaled downstream distance are plotted The outermost curvescorrespond to neutral stability and the increments in contour lines are macr frac14 E = 25 and 10 for BF1and BF2 respectively

than those in Blasius regow The non-local amplitude functions of the two-dimensionalwave of f ordm 1719 Hz are plotted in shy gure 2 for the downstream position x = 1984alongside those of the corresponding velocity proshy les of the basic regow u has maximainside the backregow region close to the wall at the inregection point of the stream-wise velocity proshy le of the basic regow and at the edge of the separated boundarylayer while v is rather large Since the two-dimensional wave is the most amplishy edat almost all downstream positions (as seen in shy gure 1) we focus all subsequentNOLOTPSE studies on two-dimensional waves The instability diagram for two-dimensional short-wavelength disturbances is shown in shy gure 3 Isolines of constantnon-local growth rate (deshy ned by 35) are plotted the outermost isoline representsthe curve of neutral stability This growth-rate deshy nition was chosen because in theseparated regow region the disturbance amplitude functions have several maxima inthe wall-normal direction (as seen in shy gure 2) The instability diagram for BF2 isalso shown in shy gure 3 Its general shape appears dinoterent from that of BF1 thelatter having an instability region of the decelerating and separated regow embeddedinto the region of TS instability unlike BF2 where the two are clearly separated (seealso comments in x 2)

(b) Global disturbances addressed by the PEPS

In the previous section we focused on instabilities of an oscillatory nature thestreamwise length-scale of which is small compared with that of the basic regowthis structure is exploited by the ansatz (32) and emacr cient marching proceduresmake a NOLOTPSE approach the method of choice for accurate results to bedelivered within reasonable computing time for this type of instability On the otherhand an analysis of disturbances developing over length-scales comparable withthat of the basic regow using NOLOTPSE may deliver inaccurate results Short ofresorting to a spatial DNS in order to take both upstream inreguence and downstream

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

3238 V Theomacrlis S Hein and U Dallmann

- 2 - 1 0 1 2- 5

- 4

- 3

- 2

- 1

0

1

W i

W r

Figure 4 The window of the spectrum of global PEPS disturbancesin the neighbourhood of laquo = 0

development into account disturbances having length-scales comparable with thatof the basic regow are best analysed using numerical solutions of the partial derivativeeigenvalue problem in which the available computing resources are devoted to thesimultaneous resolution of both the x and y spatial directions We concentrate hereon presenting results for the BF2 basic regow obtained using the pressure-gradientparameters shy 0 = 100 and shy 1 = 300 at Re = 106=48 The eigenvalue spectrum inthe neighbourhood of laquo = 0 and for a spanwise wavenumber shy = 20 is shown inshy gure 4 Stationary (Reflaquo g = 0) as well as travelling (Reflaquo g 6= 0) modes are to befound in this window of the spectrum calculated by the Arnoldi algorithm (Theoshy lis1998) The travelling modes appear in symmetric pairs indicating that there is nopreferential direction in z

At this set of parameters the most unstable mode is a stationary disturbancey theeigenvector Q p of this mode is presented in shy gure 5 In order to aid the discussionon this shy gure the position of the laminar basic regow separation bubble is indicatedby a dotted line From a numerical point of view we stress that our results showthat no enotect of the downstream boundary conditions can be found in these PEPSsolutions a result which we attribute to a novel treatment of the pressure bound-ary conditions which will be presented elsewhere It sumacr ces to mention here thatthe same boundary conditions were shown to perform equally well in another open-system problem solved that in the inshy nite swept attachment-line boundary layer(Lin amp Malik 1996 Theoshy lis 1997) Most of the activity in all disturbance eigenfunc-tions is conshy ned within the boundary layer with the neighbourhood of the inregowregion being innocuous as imposed by the inregow boundary condition Signishy cantly

y A standing-wave pattern results from the linear superposition of the symmetric pairs of travellingmodes

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3239

0

0002

0004

0006

0008

0010

y

0

001

002

003

004 008 012 0160

0002

0004

0006

0008

0010

x

y

004 008 012 0160

0004

0008

0012

0016

0020

x

(a) (b)

(c) (d)

Figure 5 Normalized disturbance velocity components and pressure distribution of the unstablestationary global mode in macrgure 4 Contour levels to be read as outer-to-inner values whereavailable (a) Refug 01(02)09 (solid) iexcl01(iexcl01) iexcl 03 (dashed) (b) Refvg iexcl02(iexcl02) iexcl 08(solid) 02(02)08 (dashed) (c) Refwg 02(02)08 (solid) iexcl02(iexcl02)iexcl08 (dashed) (d) Refpg02(01)06 (solid) iexcl02 iexcl03 iexcl04 (dashed) The phase of this eigenvector sup3 (x y) is approxi-mately constant sup3 ordm ordm In all results the location of the laminar separation bubble is indicatedby the dotted line Note the direg erent y-scales in the direg erent eigenvectors

the neighbourhood of the basic laminar regow separation point is also unanotected as isclearly demonstrated by the level of activity of all disturbance velocity componentsand pressure in that region The peak of both the chordwise and the wall-normal lin-ear disturbance velocity components is to be found just downstream of the maximumextent of the primary recirculation region x ordm 01 with lower-level linear activitycontinuing past the point of primary reattachment Interestingly w (which is theonly source of three dimensionality in this linear framework) is mostly distributedwithin the primary separation bubble and has a tendency to split the latter intotwo regions of reguid moving in opposite directions Pressure also has an interestingsignature with almost purely sinusoidal disturbances generated within the primarylaminar bubble and persisting well after the laminar basic regow has reattached

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

3240 V Theomacrlis S Hein and U Dallmann

(a)

(b)

Figure 6 Local topological changes of deg ow structures associated with the formation of separationbubbles and global topological change associated with the onset of vortex shedding behind ablunt body

(a) (b)

Figure 7 Two direg erent conjectures ((a) and (b)) for topological changes of a separationbubblersquo s structure associated with the onset of vortex shedding

5 Topological conjectures for structural separation-bubble instability

A separation bubble appears as a region of recirculating regow between a separationpoint and a point of regow attachment located at a wall (shy gure 6a)y These two pointsform half-saddles in a cross-sectional plane that is parallel to the streamlines Atthe wall the locus of these points creates separation and reattachment lines respec-tively where a direction reversal of the near-wall regow occurs since the wall-shearstress changes its sign The single streamline (streamsurface) that connects these

y Figures 6 8 10 and 11 are also available at httpwwwsmgodlrdesm-sm infoTRTinfogalleryhtm

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3241

Figure 8 Generic wall-shear stress distributions associated with the formation of multi-ple-structured separation bubbles The asterisks indicate degenerate critical points for localtopological changes

two `criticalrsquo points (lines) and the wall itself deshy ne a boundary of the region ofseparated recirculating reguid which is characterized additionally by a centre pointof a `vortexrsquo (Dallmann et al 1997) Hence at incipient separation (not coveredby our present investigation) the two half-saddles and the centre point are createdsimultaneously at the wall and a separation bubble and a `vortexrsquo are created vialocal structural (topological) changes of a regow In the structurally stable regime thesimple bubble changes shape and grows in size but does not change its streamlinetopology However the topological structure of the pressure shy eld will in generalchange at a dinoterent set of critical parameters and this in turn can initiate a changein the bubble structure A more complicated but nevertheless local structural regowchange occurs at the onset of regow separation behind a blunt body (shy gure 6b) In two-dimensional regows around blunt bodies it is well known that the onset of vortex shed-ding is associated with a break-up of saddle connections How can a simple bubblebecome unsteady One conjecture might be that due to a hydroaerodynamic insta-bility a region of recirculating reguid might be instantaneously swept downstream in away sketched in shy gure 7a such that the points of separation and attachment mergeand release a patch of circulating reguid to convect downstream while a new bubbleforms at the wall Only local structural regow changes are associated with this type ofbubble unsteadiness and vortex shedding Dallmann et al (1995) provide numericalevidence for the correctness of an earlier topological conjecture namely that priorto unsteadiness andor vortex shedding multiple recirculation zones appear withina separation bubble and shy nally lead to a global structural regow change with mul-tiple structurally unstable saddle-to-saddle connections The separation line couldthen stay almost stationary despite the vortex shedding present as sketched in thesequence of instantaneous patterns of shy gure 7b

There are several routes leading from a simple generic bubble to a multiple-structured one before unsteadiness and vortex shedding sets in In the most simplemultiple-structured bubble case with several recirculating regow regions the generic

Phil Trans R Soc Lond A (2000)

3242 V Theomacrlis S Hein and U Dallmann

0 004 008 012 016 020- 2

- 1

0

1

2

3

4

5

6

7

x

| e |

0

5

10

15

S

S

S

RR

R

paraUpara

y +

parau

paray

acute 10

4e

^

Figure 9 Wall-shear distribution due to the basic and direg erent linear amounts of the disturbancestreamwise velocity component Shown are the separation (S) and reattachment (R) points asgenerated by the presence of the two-dimensional linear disturbance

structure of the associated wall-shear stress will be of the kind sketched in shy gure 8The so-called singularity or degeneracy in the wall-shear stress graph (which appearswhere its magnitude together with shy rst and possibly higher streamwise derivativesvanish) indicates a local degeneracy where a secondary bubble may form In shy gure 9the wall-shear pattern set up by the linear superposition of the basic regow and alinearly small amount of the streamwise disturbance velocity component u shown inshy gure 5 is shown the latter amount is indicated on the shy gure in percentage termsThe enotect of the presence of the stationary unstable mode on the primary separationand reattachment points may be seen in this shy gure While the separation line (inthree dimensions) remains unanotected in the presence of a linearly unstable globalmode the reattachment point of the basic regow at sumacr ciently high amplitudes of thelinearly unstable mode becomes a secondary separation point while both upstreamand downstream of this point secondary reattachment is generated linearly

Let us now consider the onset of three dimensionality in a simple two-dimensionalseparation bubble as sketched in shy gure 10y The separation and attachment lines ofthe two-dimensional regow (upper and middle parts of shy gure 10) will always be struc-turally unstable against any three-dimensional (steady or unsteady) perturbation

y The mathematical framework of a description of the structural changes associated with local and(instantaneous) global topological regow changes and the set of `elementary (three-dimensional) topologicalregow structuresrsquo which can appear at a rigid wall has been given by Dallmann (1988)

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3243

Figure 10 Streamline topology of a simple generic separation bubble (upper) Wall-deg owpattern for a two-dimensional simple separation bubble (middle) Wall-deg ow pattern for athree-dimensional separation bubblersquo (lower)

Figure 11 Streamline topology of a multiple-structured separation bubble (upper) Wall-deg owpattern for the three-dimensional multiple-structured separation bubble (lower)

Phil Trans R Soc Lond A (2000)

3244 V Theomacrlis S Hein and U Dallmann

The structurally stable conshy guration will be formed via a sequence of saddle andnodal points (either nodes or foci) along the separation as well as the attachmentline(s) The resulting wall-streamline pattern (lower part of shy gure 10) does not nec-essarily dinoter much from the wall-regow pattern for a two-dimensional regow however`three dimensionalityrsquo will always be observed close to the critical points ie close topoints of separation andor attachment What will be the enotect of three-dimensionaldisturbances on the above-discussed incipient structural change from a simple two-dimensional bubble into a multiple-structured one (which may appear and exist asa steady regow structure prior to regow unsteadiness and vortex shedding) For sim-plicity let us assume that the disturbance regow is periodic in the spanwise direction(as assumed in the PEPS and NOLOTPSE analyses) The additional amplishy ca-tion of a wall-shear stress of the disturbance regow will increase and reduce the localmean-regow wall-shear stress in a spanwise-periodic way The result may be foundin shy gure 8 where the singularities in the wall-shear stress distribution (along x)will be structurally stabilized with additional critical points of attachment and sep-aration occurring at one spanwise location while the simple bubble structure willbe preserved at other spanwise locations Hence the x spatial oscillatory wall-shearstress behaviour (shy gure 8) of a spanwise-periodic disturbance regow of sumacr cient ampli-tude must immediately lead to a pronounced three-dimensional wall-regow pattern ifa multiple-structured bubble occurs (upper part of shy gure 11) The above-presentedresults of the PEPS instability analysis clearly support these topological conjecturesnamely the growth of a stationary disturbance mode leads to a multiple-structuredseparation `bubblersquo prior to regow unsteadiness and vortex shedding However such a`bubblersquo breaks up into a structure with complex three-dimensional topology due tothe inherent three-dimensional character of the stationary spanwise-periodic distur-bance mode Three dimensionality will leave its footprints in the wall-regow patternespecially within the `reattachment zonersquo ie where secondary `bubblesrsquo form

6 Discussion and outlook

In this paper we subject a steady laminar two-dimensional boundary-layer regow thatincorporates a recirculation zone to two distinct linear instability analyses and reporttwo sets of signishy cant shy ndings First we apply PSE in order to study instability ofa wall-bounded regow that encompasses a separation bubble The agreement betweenPSE and spatial DNS results ranges from very good for three-dimensional distur-bances to excellent for plane instability waves Pauley et al (1990) performed two-dimensional DNS and reported that the Strouhal number St = f macr 2S =Ue based onthe vortex-shedding frequency f the momentum thickness macr 2 and the boundary-layer edge velocity Ue at separation is St = 0006 86 sect 06 and is independent ofReynolds number and pressure gradient this result corresponds to dimensional fre-quencies of f = 1180 Hz and f = 986 Hz for BF1 and BF2 respectively Both matchthe most unstable frequencies found in our non-local instability analyses very well ascan be seen in shy gure 3 This shy nding is in line with those of Bestek et al (1989) whoreported that the mechanism of unsteadiness of separation bubbles is closely linkedto the growth of disturbances travelling through the bubble The agreement betweenPSE and DNS indicates that the upstream inreguence within a laminar separationbubble is indeed negligible for the short-wavelength disturbances considered as longas the prevailing instabilities are of a convective nature However the extremely

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3245

large disturbance growth rates in regows with laminar separation renders nonlinearPSE analyses much more challenging than in attached boundary layers

The second signishy cant shy nding presented in this paper concerns the existence ofglobal instability modes inaccessible to local analyses Solutions of the partial deriva-tive eigenvalue problem have been discovered that correspond to both stationary andtravelling global instabilities periodic in the spanwise z-direction While we presentsolutions at conditions supporting linear growth of one stationary mode alone resultsnot presented here indicate that both travelling and other stationary modes maybecome linearly unstable at dinoterent parameters From the point of view of mainte-nance of laminar regow by instability control the existence of the new modes suggeststhat engineering methods of control of small-amplitude disturbances that aim at thefrequencies of TS-like disturbances|as delivered by DNS or PSE and well approxi-mated by an OSE solution|will have no impact on the global modes the frequenciesof the latter also have to be identishy ed However a control method based on frequencyinformation alone is bound to fail altogether if the unstable global mode is a sta-tionary disturbance A further signishy cant result related to the new global modesconcerns past conjectures that used topological arguments in an attempt to explainthe origins of unsteadiness and three dimensionality of laminar separated regow Alinear mechanism is found and presented which on account of the instability of theglobal mode and its spanwise periodicity respectively may lead to unsteadiness andthree dimensionality in line with a multitude of experimental observations (see forexample Dovgal et al 1994) Being solutions of the equations of motion the globalmodes should be reproducible by spatial DNS on the other hand the wave-likeansatz made in the PSE may question the applicability of the latter approach inidentifying the PEPS modes both questions are currently under investigation

This paper is devoted to the memory of our colleague and friend Horst Bestek Discussions withUlrich Rist as well as his DNS data are appreciated

References

Allen T amp Riley N 1995 Absolute and convective instabilities in separation bubbles Aero J99 439448

Bertolotti F P Herbert Th amp Spalart P R 1992 Linear and nonlinear stability of the Blasiusboundary layer J Fluid Mech 242 441474

Bestek H Gruber K amp Fasel H 1989 Self-excited unsteadiness of laminar separation bubblescaused by natural transition In Proc Conf on the Prediction and Exploitation of SeparatedFlows 1820 October 1989 The Royal Aeronautical Society

Briley W R 1971 A numerical study of laminar separation bubbles using the NavierStokesequations J Fluid Mech 47 713736

Cebeci T amp Stewartson K 1983 On the calculation of separation bubbles J Fluid Mech 133287296

Dallmann U 1988 Three-dimensional vortex structures and vorticity topology In Proc IUTAMSymp on Fundamental Aspects of Vortex Motion Tokyo Japan pp 183189

Dallmann U Herberg Th Gebing H Su W-H amp Zhang H-Q 1995 Flow macreld diagnosticstopological deg ow changes and spatio-temporal deg ow structures AIAA 33rd Aerospace SciencesMeeting and Exhibit AIAA-95-0791

Dallmann U Vollmers H Su W-H amp Zhang H-Q 1997 Flow topology and tomographyfor vortex identimacrcation in unsteady and in three-dimensional deg ows In Proc IUTAM Sympon Simulation and Identimacrcation of Organized Structures in Flows Lyngby Denmark 2529May 1997

Phil Trans R Soc Lond A (2000)

3242 V Theomacrlis S Hein and U Dallmann

0 004 008 012 016 020- 2

- 1

0

1

2

3

4

5

6

7

x

| e |

0

5

10

15

S

S

S

RR

R

paraUpara

y +

parau

paray

acute 10

4e

^

Figure 9 Wall-shear distribution due to the basic and direg erent linear amounts of the disturbancestreamwise velocity component Shown are the separation (S) and reattachment (R) points asgenerated by the presence of the two-dimensional linear disturbance

structure of the associated wall-shear stress will be of the kind sketched in shy gure 8The so-called singularity or degeneracy in the wall-shear stress graph (which appearswhere its magnitude together with shy rst and possibly higher streamwise derivativesvanish) indicates a local degeneracy where a secondary bubble may form In shy gure 9the wall-shear pattern set up by the linear superposition of the basic regow and alinearly small amount of the streamwise disturbance velocity component u shown inshy gure 5 is shown the latter amount is indicated on the shy gure in percentage termsThe enotect of the presence of the stationary unstable mode on the primary separationand reattachment points may be seen in this shy gure While the separation line (inthree dimensions) remains unanotected in the presence of a linearly unstable globalmode the reattachment point of the basic regow at sumacr ciently high amplitudes of thelinearly unstable mode becomes a secondary separation point while both upstreamand downstream of this point secondary reattachment is generated linearly

Let us now consider the onset of three dimensionality in a simple two-dimensionalseparation bubble as sketched in shy gure 10y The separation and attachment lines ofthe two-dimensional regow (upper and middle parts of shy gure 10) will always be struc-turally unstable against any three-dimensional (steady or unsteady) perturbation

y The mathematical framework of a description of the structural changes associated with local and(instantaneous) global topological regow changes and the set of `elementary (three-dimensional) topologicalregow structuresrsquo which can appear at a rigid wall has been given by Dallmann (1988)

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3243

Figure 10 Streamline topology of a simple generic separation bubble (upper) Wall-deg owpattern for a two-dimensional simple separation bubble (middle) Wall-deg ow pattern for athree-dimensional separation bubblersquo (lower)

Figure 11 Streamline topology of a multiple-structured separation bubble (upper) Wall-deg owpattern for the three-dimensional multiple-structured separation bubble (lower)

Phil Trans R Soc Lond A (2000)

3244 V Theomacrlis S Hein and U Dallmann

The structurally stable conshy guration will be formed via a sequence of saddle andnodal points (either nodes or foci) along the separation as well as the attachmentline(s) The resulting wall-streamline pattern (lower part of shy gure 10) does not nec-essarily dinoter much from the wall-regow pattern for a two-dimensional regow however`three dimensionalityrsquo will always be observed close to the critical points ie close topoints of separation andor attachment What will be the enotect of three-dimensionaldisturbances on the above-discussed incipient structural change from a simple two-dimensional bubble into a multiple-structured one (which may appear and exist asa steady regow structure prior to regow unsteadiness and vortex shedding) For sim-plicity let us assume that the disturbance regow is periodic in the spanwise direction(as assumed in the PEPS and NOLOTPSE analyses) The additional amplishy ca-tion of a wall-shear stress of the disturbance regow will increase and reduce the localmean-regow wall-shear stress in a spanwise-periodic way The result may be foundin shy gure 8 where the singularities in the wall-shear stress distribution (along x)will be structurally stabilized with additional critical points of attachment and sep-aration occurring at one spanwise location while the simple bubble structure willbe preserved at other spanwise locations Hence the x spatial oscillatory wall-shearstress behaviour (shy gure 8) of a spanwise-periodic disturbance regow of sumacr cient ampli-tude must immediately lead to a pronounced three-dimensional wall-regow pattern ifa multiple-structured bubble occurs (upper part of shy gure 11) The above-presentedresults of the PEPS instability analysis clearly support these topological conjecturesnamely the growth of a stationary disturbance mode leads to a multiple-structuredseparation `bubblersquo prior to regow unsteadiness and vortex shedding However such a`bubblersquo breaks up into a structure with complex three-dimensional topology due tothe inherent three-dimensional character of the stationary spanwise-periodic distur-bance mode Three dimensionality will leave its footprints in the wall-regow patternespecially within the `reattachment zonersquo ie where secondary `bubblesrsquo form

6 Discussion and outlook

In this paper we subject a steady laminar two-dimensional boundary-layer regow thatincorporates a recirculation zone to two distinct linear instability analyses and reporttwo sets of signishy cant shy ndings First we apply PSE in order to study instability ofa wall-bounded regow that encompasses a separation bubble The agreement betweenPSE and spatial DNS results ranges from very good for three-dimensional distur-bances to excellent for plane instability waves Pauley et al (1990) performed two-dimensional DNS and reported that the Strouhal number St = f macr 2S =Ue based onthe vortex-shedding frequency f the momentum thickness macr 2 and the boundary-layer edge velocity Ue at separation is St = 0006 86 sect 06 and is independent ofReynolds number and pressure gradient this result corresponds to dimensional fre-quencies of f = 1180 Hz and f = 986 Hz for BF1 and BF2 respectively Both matchthe most unstable frequencies found in our non-local instability analyses very well ascan be seen in shy gure 3 This shy nding is in line with those of Bestek et al (1989) whoreported that the mechanism of unsteadiness of separation bubbles is closely linkedto the growth of disturbances travelling through the bubble The agreement betweenPSE and DNS indicates that the upstream inreguence within a laminar separationbubble is indeed negligible for the short-wavelength disturbances considered as longas the prevailing instabilities are of a convective nature However the extremely

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3245

large disturbance growth rates in regows with laminar separation renders nonlinearPSE analyses much more challenging than in attached boundary layers

The second signishy cant shy nding presented in this paper concerns the existence ofglobal instability modes inaccessible to local analyses Solutions of the partial deriva-tive eigenvalue problem have been discovered that correspond to both stationary andtravelling global instabilities periodic in the spanwise z-direction While we presentsolutions at conditions supporting linear growth of one stationary mode alone resultsnot presented here indicate that both travelling and other stationary modes maybecome linearly unstable at dinoterent parameters From the point of view of mainte-nance of laminar regow by instability control the existence of the new modes suggeststhat engineering methods of control of small-amplitude disturbances that aim at thefrequencies of TS-like disturbances|as delivered by DNS or PSE and well approxi-mated by an OSE solution|will have no impact on the global modes the frequenciesof the latter also have to be identishy ed However a control method based on frequencyinformation alone is bound to fail altogether if the unstable global mode is a sta-tionary disturbance A further signishy cant result related to the new global modesconcerns past conjectures that used topological arguments in an attempt to explainthe origins of unsteadiness and three dimensionality of laminar separated regow Alinear mechanism is found and presented which on account of the instability of theglobal mode and its spanwise periodicity respectively may lead to unsteadiness andthree dimensionality in line with a multitude of experimental observations (see forexample Dovgal et al 1994) Being solutions of the equations of motion the globalmodes should be reproducible by spatial DNS on the other hand the wave-likeansatz made in the PSE may question the applicability of the latter approach inidentifying the PEPS modes both questions are currently under investigation

This paper is devoted to the memory of our colleague and friend Horst Bestek Discussions withUlrich Rist as well as his DNS data are appreciated

References

Allen T amp Riley N 1995 Absolute and convective instabilities in separation bubbles Aero J99 439448

Bertolotti F P Herbert Th amp Spalart P R 1992 Linear and nonlinear stability of the Blasiusboundary layer J Fluid Mech 242 441474

Bestek H Gruber K amp Fasel H 1989 Self-excited unsteadiness of laminar separation bubblescaused by natural transition In Proc Conf on the Prediction and Exploitation of SeparatedFlows 1820 October 1989 The Royal Aeronautical Society

Briley W R 1971 A numerical study of laminar separation bubbles using the NavierStokesequations J Fluid Mech 47 713736

Cebeci T amp Stewartson K 1983 On the calculation of separation bubbles J Fluid Mech 133287296

Dallmann U 1988 Three-dimensional vortex structures and vorticity topology In Proc IUTAMSymp on Fundamental Aspects of Vortex Motion Tokyo Japan pp 183189

Dallmann U Herberg Th Gebing H Su W-H amp Zhang H-Q 1995 Flow macreld diagnosticstopological deg ow changes and spatio-temporal deg ow structures AIAA 33rd Aerospace SciencesMeeting and Exhibit AIAA-95-0791

Dallmann U Vollmers H Su W-H amp Zhang H-Q 1997 Flow topology and tomographyfor vortex identimacrcation in unsteady and in three-dimensional deg ows In Proc IUTAM Sympon Simulation and Identimacrcation of Organized Structures in Flows Lyngby Denmark 2529May 1997

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3243

Figure 10 Streamline topology of a simple generic separation bubble (upper) Wall-deg owpattern for a two-dimensional simple separation bubble (middle) Wall-deg ow pattern for athree-dimensional separation bubblersquo (lower)

Figure 11 Streamline topology of a multiple-structured separation bubble (upper) Wall-deg owpattern for the three-dimensional multiple-structured separation bubble (lower)

Phil Trans R Soc Lond A (2000)

3244 V Theomacrlis S Hein and U Dallmann

The structurally stable conshy guration will be formed via a sequence of saddle andnodal points (either nodes or foci) along the separation as well as the attachmentline(s) The resulting wall-streamline pattern (lower part of shy gure 10) does not nec-essarily dinoter much from the wall-regow pattern for a two-dimensional regow however`three dimensionalityrsquo will always be observed close to the critical points ie close topoints of separation andor attachment What will be the enotect of three-dimensionaldisturbances on the above-discussed incipient structural change from a simple two-dimensional bubble into a multiple-structured one (which may appear and exist asa steady regow structure prior to regow unsteadiness and vortex shedding) For sim-plicity let us assume that the disturbance regow is periodic in the spanwise direction(as assumed in the PEPS and NOLOTPSE analyses) The additional amplishy ca-tion of a wall-shear stress of the disturbance regow will increase and reduce the localmean-regow wall-shear stress in a spanwise-periodic way The result may be foundin shy gure 8 where the singularities in the wall-shear stress distribution (along x)will be structurally stabilized with additional critical points of attachment and sep-aration occurring at one spanwise location while the simple bubble structure willbe preserved at other spanwise locations Hence the x spatial oscillatory wall-shearstress behaviour (shy gure 8) of a spanwise-periodic disturbance regow of sumacr cient ampli-tude must immediately lead to a pronounced three-dimensional wall-regow pattern ifa multiple-structured bubble occurs (upper part of shy gure 11) The above-presentedresults of the PEPS instability analysis clearly support these topological conjecturesnamely the growth of a stationary disturbance mode leads to a multiple-structuredseparation `bubblersquo prior to regow unsteadiness and vortex shedding However such a`bubblersquo breaks up into a structure with complex three-dimensional topology due tothe inherent three-dimensional character of the stationary spanwise-periodic distur-bance mode Three dimensionality will leave its footprints in the wall-regow patternespecially within the `reattachment zonersquo ie where secondary `bubblesrsquo form

6 Discussion and outlook

In this paper we subject a steady laminar two-dimensional boundary-layer regow thatincorporates a recirculation zone to two distinct linear instability analyses and reporttwo sets of signishy cant shy ndings First we apply PSE in order to study instability ofa wall-bounded regow that encompasses a separation bubble The agreement betweenPSE and spatial DNS results ranges from very good for three-dimensional distur-bances to excellent for plane instability waves Pauley et al (1990) performed two-dimensional DNS and reported that the Strouhal number St = f macr 2S =Ue based onthe vortex-shedding frequency f the momentum thickness macr 2 and the boundary-layer edge velocity Ue at separation is St = 0006 86 sect 06 and is independent ofReynolds number and pressure gradient this result corresponds to dimensional fre-quencies of f = 1180 Hz and f = 986 Hz for BF1 and BF2 respectively Both matchthe most unstable frequencies found in our non-local instability analyses very well ascan be seen in shy gure 3 This shy nding is in line with those of Bestek et al (1989) whoreported that the mechanism of unsteadiness of separation bubbles is closely linkedto the growth of disturbances travelling through the bubble The agreement betweenPSE and DNS indicates that the upstream inreguence within a laminar separationbubble is indeed negligible for the short-wavelength disturbances considered as longas the prevailing instabilities are of a convective nature However the extremely

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3245

large disturbance growth rates in regows with laminar separation renders nonlinearPSE analyses much more challenging than in attached boundary layers

The second signishy cant shy nding presented in this paper concerns the existence ofglobal instability modes inaccessible to local analyses Solutions of the partial deriva-tive eigenvalue problem have been discovered that correspond to both stationary andtravelling global instabilities periodic in the spanwise z-direction While we presentsolutions at conditions supporting linear growth of one stationary mode alone resultsnot presented here indicate that both travelling and other stationary modes maybecome linearly unstable at dinoterent parameters From the point of view of mainte-nance of laminar regow by instability control the existence of the new modes suggeststhat engineering methods of control of small-amplitude disturbances that aim at thefrequencies of TS-like disturbances|as delivered by DNS or PSE and well approxi-mated by an OSE solution|will have no impact on the global modes the frequenciesof the latter also have to be identishy ed However a control method based on frequencyinformation alone is bound to fail altogether if the unstable global mode is a sta-tionary disturbance A further signishy cant result related to the new global modesconcerns past conjectures that used topological arguments in an attempt to explainthe origins of unsteadiness and three dimensionality of laminar separated regow Alinear mechanism is found and presented which on account of the instability of theglobal mode and its spanwise periodicity respectively may lead to unsteadiness andthree dimensionality in line with a multitude of experimental observations (see forexample Dovgal et al 1994) Being solutions of the equations of motion the globalmodes should be reproducible by spatial DNS on the other hand the wave-likeansatz made in the PSE may question the applicability of the latter approach inidentifying the PEPS modes both questions are currently under investigation

This paper is devoted to the memory of our colleague and friend Horst Bestek Discussions withUlrich Rist as well as his DNS data are appreciated

References

Allen T amp Riley N 1995 Absolute and convective instabilities in separation bubbles Aero J99 439448

Bertolotti F P Herbert Th amp Spalart P R 1992 Linear and nonlinear stability of the Blasiusboundary layer J Fluid Mech 242 441474

Bestek H Gruber K amp Fasel H 1989 Self-excited unsteadiness of laminar separation bubblescaused by natural transition In Proc Conf on the Prediction and Exploitation of SeparatedFlows 1820 October 1989 The Royal Aeronautical Society

Briley W R 1971 A numerical study of laminar separation bubbles using the NavierStokesequations J Fluid Mech 47 713736

Cebeci T amp Stewartson K 1983 On the calculation of separation bubbles J Fluid Mech 133287296

Dallmann U 1988 Three-dimensional vortex structures and vorticity topology In Proc IUTAMSymp on Fundamental Aspects of Vortex Motion Tokyo Japan pp 183189

Dallmann U Herberg Th Gebing H Su W-H amp Zhang H-Q 1995 Flow macreld diagnosticstopological deg ow changes and spatio-temporal deg ow structures AIAA 33rd Aerospace SciencesMeeting and Exhibit AIAA-95-0791

Dallmann U Vollmers H Su W-H amp Zhang H-Q 1997 Flow topology and tomographyfor vortex identimacrcation in unsteady and in three-dimensional deg ows In Proc IUTAM Sympon Simulation and Identimacrcation of Organized Structures in Flows Lyngby Denmark 2529May 1997

Phil Trans R Soc Lond A (2000)

3244 V Theomacrlis S Hein and U Dallmann

The structurally stable conshy guration will be formed via a sequence of saddle andnodal points (either nodes or foci) along the separation as well as the attachmentline(s) The resulting wall-streamline pattern (lower part of shy gure 10) does not nec-essarily dinoter much from the wall-regow pattern for a two-dimensional regow however`three dimensionalityrsquo will always be observed close to the critical points ie close topoints of separation andor attachment What will be the enotect of three-dimensionaldisturbances on the above-discussed incipient structural change from a simple two-dimensional bubble into a multiple-structured one (which may appear and exist asa steady regow structure prior to regow unsteadiness and vortex shedding) For sim-plicity let us assume that the disturbance regow is periodic in the spanwise direction(as assumed in the PEPS and NOLOTPSE analyses) The additional amplishy ca-tion of a wall-shear stress of the disturbance regow will increase and reduce the localmean-regow wall-shear stress in a spanwise-periodic way The result may be foundin shy gure 8 where the singularities in the wall-shear stress distribution (along x)will be structurally stabilized with additional critical points of attachment and sep-aration occurring at one spanwise location while the simple bubble structure willbe preserved at other spanwise locations Hence the x spatial oscillatory wall-shearstress behaviour (shy gure 8) of a spanwise-periodic disturbance regow of sumacr cient ampli-tude must immediately lead to a pronounced three-dimensional wall-regow pattern ifa multiple-structured bubble occurs (upper part of shy gure 11) The above-presentedresults of the PEPS instability analysis clearly support these topological conjecturesnamely the growth of a stationary disturbance mode leads to a multiple-structuredseparation `bubblersquo prior to regow unsteadiness and vortex shedding However such a`bubblersquo breaks up into a structure with complex three-dimensional topology due tothe inherent three-dimensional character of the stationary spanwise-periodic distur-bance mode Three dimensionality will leave its footprints in the wall-regow patternespecially within the `reattachment zonersquo ie where secondary `bubblesrsquo form

6 Discussion and outlook

In this paper we subject a steady laminar two-dimensional boundary-layer regow thatincorporates a recirculation zone to two distinct linear instability analyses and reporttwo sets of signishy cant shy ndings First we apply PSE in order to study instability ofa wall-bounded regow that encompasses a separation bubble The agreement betweenPSE and spatial DNS results ranges from very good for three-dimensional distur-bances to excellent for plane instability waves Pauley et al (1990) performed two-dimensional DNS and reported that the Strouhal number St = f macr 2S =Ue based onthe vortex-shedding frequency f the momentum thickness macr 2 and the boundary-layer edge velocity Ue at separation is St = 0006 86 sect 06 and is independent ofReynolds number and pressure gradient this result corresponds to dimensional fre-quencies of f = 1180 Hz and f = 986 Hz for BF1 and BF2 respectively Both matchthe most unstable frequencies found in our non-local instability analyses very well ascan be seen in shy gure 3 This shy nding is in line with those of Bestek et al (1989) whoreported that the mechanism of unsteadiness of separation bubbles is closely linkedto the growth of disturbances travelling through the bubble The agreement betweenPSE and DNS indicates that the upstream inreguence within a laminar separationbubble is indeed negligible for the short-wavelength disturbances considered as longas the prevailing instabilities are of a convective nature However the extremely

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3245

large disturbance growth rates in regows with laminar separation renders nonlinearPSE analyses much more challenging than in attached boundary layers

The second signishy cant shy nding presented in this paper concerns the existence ofglobal instability modes inaccessible to local analyses Solutions of the partial deriva-tive eigenvalue problem have been discovered that correspond to both stationary andtravelling global instabilities periodic in the spanwise z-direction While we presentsolutions at conditions supporting linear growth of one stationary mode alone resultsnot presented here indicate that both travelling and other stationary modes maybecome linearly unstable at dinoterent parameters From the point of view of mainte-nance of laminar regow by instability control the existence of the new modes suggeststhat engineering methods of control of small-amplitude disturbances that aim at thefrequencies of TS-like disturbances|as delivered by DNS or PSE and well approxi-mated by an OSE solution|will have no impact on the global modes the frequenciesof the latter also have to be identishy ed However a control method based on frequencyinformation alone is bound to fail altogether if the unstable global mode is a sta-tionary disturbance A further signishy cant result related to the new global modesconcerns past conjectures that used topological arguments in an attempt to explainthe origins of unsteadiness and three dimensionality of laminar separated regow Alinear mechanism is found and presented which on account of the instability of theglobal mode and its spanwise periodicity respectively may lead to unsteadiness andthree dimensionality in line with a multitude of experimental observations (see forexample Dovgal et al 1994) Being solutions of the equations of motion the globalmodes should be reproducible by spatial DNS on the other hand the wave-likeansatz made in the PSE may question the applicability of the latter approach inidentifying the PEPS modes both questions are currently under investigation

This paper is devoted to the memory of our colleague and friend Horst Bestek Discussions withUlrich Rist as well as his DNS data are appreciated

References

Allen T amp Riley N 1995 Absolute and convective instabilities in separation bubbles Aero J99 439448

Bertolotti F P Herbert Th amp Spalart P R 1992 Linear and nonlinear stability of the Blasiusboundary layer J Fluid Mech 242 441474

Bestek H Gruber K amp Fasel H 1989 Self-excited unsteadiness of laminar separation bubblescaused by natural transition In Proc Conf on the Prediction and Exploitation of SeparatedFlows 1820 October 1989 The Royal Aeronautical Society

Briley W R 1971 A numerical study of laminar separation bubbles using the NavierStokesequations J Fluid Mech 47 713736

Cebeci T amp Stewartson K 1983 On the calculation of separation bubbles J Fluid Mech 133287296

Dallmann U 1988 Three-dimensional vortex structures and vorticity topology In Proc IUTAMSymp on Fundamental Aspects of Vortex Motion Tokyo Japan pp 183189

Dallmann U Herberg Th Gebing H Su W-H amp Zhang H-Q 1995 Flow macreld diagnosticstopological deg ow changes and spatio-temporal deg ow structures AIAA 33rd Aerospace SciencesMeeting and Exhibit AIAA-95-0791

Dallmann U Vollmers H Su W-H amp Zhang H-Q 1997 Flow topology and tomographyfor vortex identimacrcation in unsteady and in three-dimensional deg ows In Proc IUTAM Sympon Simulation and Identimacrcation of Organized Structures in Flows Lyngby Denmark 2529May 1997

Phil Trans R Soc Lond A (2000)

Unsteadiness and three-dimensionality in separation bubbles 3245

large disturbance growth rates in regows with laminar separation renders nonlinearPSE analyses much more challenging than in attached boundary layers

The second signishy cant shy nding presented in this paper concerns the existence ofglobal instability modes inaccessible to local analyses Solutions of the partial deriva-tive eigenvalue problem have been discovered that correspond to both stationary andtravelling global instabilities periodic in the spanwise z-direction While we presentsolutions at conditions supporting linear growth of one stationary mode alone resultsnot presented here indicate that both travelling and other stationary modes maybecome linearly unstable at dinoterent parameters From the point of view of mainte-nance of laminar regow by instability control the existence of the new modes suggeststhat engineering methods of control of small-amplitude disturbances that aim at thefrequencies of TS-like disturbances|as delivered by DNS or PSE and well approxi-mated by an OSE solution|will have no impact on the global modes the frequenciesof the latter also have to be identishy ed However a control method based on frequencyinformation alone is bound to fail altogether if the unstable global mode is a sta-tionary disturbance A further signishy cant result related to the new global modesconcerns past conjectures that used topological arguments in an attempt to explainthe origins of unsteadiness and three dimensionality of laminar separated regow Alinear mechanism is found and presented which on account of the instability of theglobal mode and its spanwise periodicity respectively may lead to unsteadiness andthree dimensionality in line with a multitude of experimental observations (see forexample Dovgal et al 1994) Being solutions of the equations of motion the globalmodes should be reproducible by spatial DNS on the other hand the wave-likeansatz made in the PSE may question the applicability of the latter approach inidentifying the PEPS modes both questions are currently under investigation

This paper is devoted to the memory of our colleague and friend Horst Bestek Discussions withUlrich Rist as well as his DNS data are appreciated

References

Allen T amp Riley N 1995 Absolute and convective instabilities in separation bubbles Aero J99 439448

Bertolotti F P Herbert Th amp Spalart P R 1992 Linear and nonlinear stability of the Blasiusboundary layer J Fluid Mech 242 441474

Bestek H Gruber K amp Fasel H 1989 Self-excited unsteadiness of laminar separation bubblescaused by natural transition In Proc Conf on the Prediction and Exploitation of SeparatedFlows 1820 October 1989 The Royal Aeronautical Society

Briley W R 1971 A numerical study of laminar separation bubbles using the NavierStokesequations J Fluid Mech 47 713736

Cebeci T amp Stewartson K 1983 On the calculation of separation bubbles J Fluid Mech 133287296

Dallmann U 1988 Three-dimensional vortex structures and vorticity topology In Proc IUTAMSymp on Fundamental Aspects of Vortex Motion Tokyo Japan pp 183189

Dallmann U Herberg Th Gebing H Su W-H amp Zhang H-Q 1995 Flow macreld diagnosticstopological deg ow changes and spatio-temporal deg ow structures AIAA 33rd Aerospace SciencesMeeting and Exhibit AIAA-95-0791

Dallmann U Vollmers H Su W-H amp Zhang H-Q 1997 Flow topology and tomographyfor vortex identimacrcation in unsteady and in three-dimensional deg ows In Proc IUTAM Sympon Simulation and Identimacrcation of Organized Structures in Flows Lyngby Denmark 2529May 1997

Phil Trans R Soc Lond A (2000)

3246 V Theomacrlis S Hein and U Dallmann

Dovgal A V Kozlov V V amp Michalke A 1994 Laminar boundary layer separation instabilityand associated phenomena Prog Aerospace Sci 30 6194

Hein S Bertolotti F P Simen M Hanimacr A amp Henningson D 1994 Linear nonlocal insta-bility analysis|the linear NOLOT code Internal report DLR IB 223-94 A56

Hein S Theomacrlis V amp Dallmann U 1998 Unsteadiness and three-dimensionality of steady two-dimensional laminar separation bubbles as result of linear instability mechanisms Internalreport DLR IB 223-98 A39

Herbert Th 1997 Parabolized stability equations A Rev Fluid Mech 29 245283

Howarth L 1938 On the solution of the laminar boundary layer equations Proc R Soc LondA 164 547579

Lin R-S amp Malik M R 1996 On the stability of the attachment-line boundary layers Part 1The incompressible swept Hiemenz deg ow J Fluid Mech 311 239255

Pauley L L Moin P amp Reynolds W C 1990 The structure of two-dimensional separationJ Fluid Mech 220 397411

Rist U amp Maucher U 1994 Direct numerical simulation of 2-D and 3-D instability wavesin a laminar separation bubble In AGARD-CP-551 Application of Direct and Large EddySimulation to Transition and Turbulence pp 34-134-7

Rist U Maucher U amp Wagner S 1996 Direct numerical simulation of some fundamentalproblems related to transition in laminar separation bubbles In Proc Computational FluidDynamics Conf ECCOMAS rsquo96 pp 319325

Schlichting H 1979 Boundary-layer theory McGraw-Hill

Spalart P R amp Coleman G N 1997 Numerical study of a separation bubble with heat transferEur J Mech B 16 169189

Theomacrlis V 1997 On the verimacrcation and extension of the GortlerHammerlin assumption inthree-dimensional incompressible swept attachment-line boundary layer deg ow Internal reportDLR IB 223-97 A44

Theomacrlis V 1998 Linear instability in two spatial dimensions In Proc Computational FluidDynamics Conf ECCOMAS rsquo98 Athens Greece 711 September 1998 pp 547552

Theomacrlis V 2000 Globally unstable deg ows in open cavities AIAA paper no 2000-1965

Phil Trans R Soc Lond A (2000)