Optimal disturbances and bypass transition in boundary layers

Optimal disturbances and bypass transition in boundary layersPaul Andersson, Martin Berggren, and Dan S. Henningson Citation: Physics of Fluids (1994-present) 11, 134 (1999); doi: 10.1063/1.869908 View online: http://dx.doi.org/10.1063/1.869908 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/11/1?ver=pdfcov Published by the AIP Publishing

This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.143.23.241 On: Mon, 11 Nov 2013 14:01:57

http://scitation.aip.org/content/aip/journal/pof2?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/2127325285/x01/AIP-PT/PoF_CoverPg_101613/aipToCAlerts_Large.png/5532386d4f314a53757a6b4144615953?x

http://scitation.aip.org/search?value1=Paul+Andersson&option1=author

http://scitation.aip.org/search?value1=Martin+Berggren&option1=author

http://scitation.aip.org/search?value1=Dan+S.+Henningson&option1=author

http://scitation.aip.org/content/aip/journal/pof2?ver=pdfcov

http://dx.doi.org/10.1063/1.869908

http://scitation.aip.org/content/aip/journal/pof2/11/1?ver=pdfcov

http://scitation.aip.org/content/aip?ver=pdfcov

PHYSICS OF FLUIDS VOLUME 11, NUMBER 1 JANUARY 1999

This ar

Optimal disturbances and bypass transition in boundary layersPaul Andersson,a) Martin Berggren, and Dan S. Henningsona)

FFA, the Aeronautical Research Institute of Sweden, Computational Aerodynamics Department,P.O. Box 11021, S-161 11 Bromma, Sweden

~Received 6 March 1998; accepted 29 September 1998!

Streamwise streaks are ubiquitous in transitional boundary layers, particularly when subjected tohigh levels of free-stream turbulence. Using the steady boundary-layer approximation, the upstreamdisturbances experiencing maximum spatial energy growth are numerically calculated. Thecalculations use techniques commonly employed when solving optimal-control problems fordistributed parameter systems. The calculated optimal disturbances consist of streamwise vorticesdeveloping into streamwise streaks. The maximum spatial energy growth was found to scale linearlywith the distance from the leading edge. Based on these results, a simple model for prediction oftransition location is proposed. Available experiments have been used to correlate the singleconstant appearing in the model. ©1999 American Institute of Physics.@S1070-6631~99!01601-3#

inpason

arp

cuhencrere

t ool-

chnad-ee

tru

eswi

itth

he

cend-enec-sticsperdis-r toowed

-ndase-he

er-g a

om

p-

ex-s-

s of

se.ur-rr–

gy

I. INTRODUCTION

A. Free-stream turbulence and streaky structures

The streamwise location on an airplane wing or a turbblade where the flow turns turbulent is a critical designrameter. The standard transition prediction tool is thecalledeN method which empirically correlates the transitiolocation with the amplitude of exponentially growing linemodes. However, given a large enough disturbance amtude, transition in the boundary-layer flow may even ocbelow the onset of exponential instability. This is often tcase at moderate and high levels of free-stream turbuleThis problem with linear stability analysis has recentlyceived renewed attention, and transition scenarios diffefrom those involving unstable eigenmodes~Tollmien–Schlichting waves! have in particular been studied.

Numerous experiments have verified the strong effecfree-stream turbulence on transition. Experiments since thof Klebanoff et al.1 have displayed features, in particular aternating longitudinal high and low velocity streaks, whiare independent of the linear theory of two-dimensioTollmien–Schlichting waves. Flow visualizations of bounary layers subjected to moderate and high levels of frstream turbulence have been made by Kendall,2 Gulyaevet al.,3 and Alfredsson and Matsubara.4 Evident in these vi-sualizations is the occurrence of elongated streamwise stures with narrow spanwise scales~in the following denotedstreaky structures!. Kendall2 denoted these structuresKle-banoff modes.5 These boundary layer disturbances manifthemselves as a periodic spanwise modulation of streamvelocity and are believed to give rise to the so-calledbypasstransition, a term coined by Morkovin,6,7 describing transi-tion emanating from disturbance growth not associated wexponential instabilities. Experimental evidence reveals

a!Also at: Department of Mechanics, Royal Institute of TechnoloS-100 44 Stockholm, Sweden.

1341070-6631/99/11(1)/134/17/$15.00

ticle is copyrighted as indicated in the abstract. Reuse of AIP content is sub

128.143.23.241 On: Mon,

e--

li-r

e.-nt

fse

l

-

c-

tse

hat

the Klebanoff mode is strongly linked to the strength of tfree-stream turbulence.

Bradshaw8 measured the spanwise variations of surfashear stress, produced in nominally two-dimensional bouary layers by a spatial instability of the flow through thwind-tunnel damping screens. He also discussed the contion between the spanwise variations and the characteriof wind-tunnel damping screens. Moreover, the same papresented the result from an approximate analysis thatplayed the strong sensitivity of the Blasius boundary layesteady spanwise variations in the free stream. Later Cr9

derived this result more rigorously by using matchasymptotic expansions.

B. Algebraic instability and transient growth

Ellingsen and Palm10 introduced an infinitesimal disturbance without streamwise variation in a shear layer, ashowed that the streamwise velocity component can increlinearly with time, within the inviscid approximation, producing alternating low and high velocity streaks in tstreamwise velocity component.

Landahl11,12 argued that a fluid element in a shear laywould initially retain its horizontal momentum when displaced a distance in the wall-normal direction, thus causinperturbation in the streamwise velocity component. Frthis physical interpretation he coined the termlift-up effect.This inviscid phenomenon together with the viscous daming constitutes what became known astransient growth~Hultgren and Gustavsson,13 Boberg and Brosa,14

Gustavsson,15 Farrell,16 Butler and Farrell,17 Reddy andHenningson,18 Henningsonet al.,19 Trefethen et al.20!. Inthese papers, parallel flow is assumed which allows anplicit computation of the maximum transient growth and asociated disturbances by optimizing over the eigenmodethe Orr–Sommerfeld operator.

This technique is not applicable to the nonparallel caBelow, we propose a method to obtain the optimal distbance without determining the eigenmodes of the O

,

© 1999 American Institute of Physics

ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

11 Nov 2013 14:01:57

ze

cieteana

d--

laopiov

n

e

nth

arinamk

tivgesice

eisiob

n-

s0

-eric-het,lothmlrme

ure-ws

arever,er-

ssedmsic

theion.e

hethe

untpro-msu-oftinort-ge

ua-al-r-

thec.u-la-

theeed,eri-

heA,

s-at

atce.am-

esetial

135Phys. Fluids, Vol. 11, No. 1, January 1999 Andersson, Berggren, and Henningson

This ar

Sommerfeld operator. Starting from the steady, lineariboundary layer equations, but otherwise without anya prioriassumptions of the shape or scaling of the different velocomponents, we use an optimization procedure to numcally compute the optimal spatial growth rate and associadisturbances as a function of streamwise coordinatespanwise wave number. This is a particular example ofoptimal-control problem for a parabolic, distributeparameter system,21 and we use techniques commonly employed in the numerical solution of such problems,22–26suchas adjoint equations and least-squares techniques. Fromresults of the computation, we propose a simple scalingfor the transitional Reynolds number associated with thetimal disturbances. This is a first step toward a transitprediction tool in environments with moderate and high leels of free-stream turbulence.

A similar optimization procedure is proposed, indepedent of ours, in a paper by Luchini.27 At the high-Reynolds-number limit, our optimal growth rates agree with the onfound by Luchini. Previously, Luchini28 studied the sameequations in the limit of small spanwise wave number afound a set of solutions exhibiting unbounded spatial grow

Using the parabolized stability equations~PSEs!,Bertolotti29 studies the response of the Blasius boundlayer to two- and three-dimensional vortical modes satisfythe linearized Navier–Stokes equations in the free streThese vortical modes are used to represent some of thefeatures of low-level turbulence, thus constituting a recepity mechanism for the flat plate, excluding the leading edWhen steady and low-frequency vortical modes are conered, the analysis yields results that successfully reprodunumber of the experimental measurements of Kendall30,2 onstreaky structures.

C. Transition prediction methods

Transition prediction with theeN method is based on threlative amplification of the specific discrete-frequency dturbance which first reaches a preset ad hoc ‘‘transitlevel’’ of eN. This method was developed independentlySmith and Gamberoni31 and van Ingen.32 Mack33 used amodified eN method and suggested the empirical relatioship, N528.43– 2.4 ln~Tu!, between the free-stream turbulence level Tu and theN factor at the transition location. Thimodel gives reasonable transition locations in the range,Tu,2%.

Other empirical correlations for transition criteria involving the combined effects of Tu and the streamwise prsure gradient have been developed. For example, van Dand Blumer34 arrive at a semiempirical model, by introduing a critical vorticity Reynolds number, that correlates tpressure gradient and free-stream turbulence level withReynolds number at transition. In the model of Dunham35

the value of the Reynolds number based on momentum-thickness at the transition point is given as a function ofPohlhausen~pressure gradient! parameter and the free-streaturbulence level. Abu-Ghannam and Shaw36 suggest a modethat gives the start and end of the transition region in teof the Reynolds number based on momentum-loss thickn


128.143.23.241 On: Mon,

d

tyri-ddn

thew-

n-

-

s

d.

yg

.ey-.

d-a

-ny

-

.1

s-est

he

sse

sss.

Also here, the free-stream turbulence level and a pressgradient parameter are the only required inputs. For flosimilar to the ones for which these empirical correlationscalibrated they often give reasonable predictions. Howethe large degree of empiricism also implies that their genality is rather limited.

Turbulence models such ask–e and Reynolds strestransport~RST! models have to some extent also been uto predict transition in environments with high free-streaturbulence levels, see Savill—Refs. 37 and 38. The baidea is that the diffusion of free-stream turbulence intoboundary layer will cause an increased turbulent productAt some position the production will sufficiently exceed thlocal dissipation, leading to a rapid growth of one of tReynolds stresses, with a subsequent modification ofmean velocity profile. However, these models fail to accofor a phenomenon such as transient growth and hencevide a rather limited description of the physical mechanisinvolved in transition, especially for lower free-stream turblence levels. A contribution to the evaluation of the useRST models for transition prediction can be found in Wesand Henkes—Ref. 39. Their study reveals significant shcomings in the modeling of the dissipation, with a laroverprediction in the pretransitional boundary layer.

This paper is organized as follows. The governing eqtions are presented in Sec. II. In Sec. III, an operator formism is introduced to aid in the definitions of optimal distubances in Sec. IV and adjoint equations in Sec. V.

In Sec. VI, we state a step-by-step description ofoptimization algorithm. All the results are collected in SeVII, where we give the fundamental results from the calclations. We compare our numerical results, from the calcutions of optimal disturbances and transient growth, withexperiments of Westinet al.40 We also propose a simpltransition prediction model, which is correlated and testfor different free-stream turbulence levels, against the expments of Roach and Brierley,41 Matsubara,42 and the directnumerical simulations of Yang and Voke.43 The main con-clusions from this study are summarized in Sec. VIII. Tderivation of the adjoint equations are given in Appendixand in Appendix B we discuss the numerical methods.

II. GOVERNING EQUATIONS

We consider the linear stability of a high-Reynoldnumber flow of a viscous, incompressible fluid over a flplate. The geometry of the problem is shown in Fig. 1.

In particular we want to study model disturbances thoccur at moderate and high levels of free-stream turbulenThese disturbances are known to be elongated in the strewise direction and to vary only slowly with time.40 Thismotivates the use ofboundary-layer approximationsto thesteady, incompressible Navier–Stokes equations. Thequations are linearized to obtain equations for the spaevolution of three-dimensional disturbances (u,v,w,p)5(u(x,y,z),v(x,y,z),w(x,y,z),p(x,y,z)) around a two-dimensional base flow (U,V,0)5(U(x,y),V(x,y),0),

ux1vy1wz50,
11 Nov 2013 14:01:57

p

te-

n-

er-

tesonrfo

caeer

tha

ll-e

ne

s it

s-

ngand

n

-l-

er

-

o-

y-

136 Phys. Fluids, Vol. 11, No. 1, January 1999 Andersson, Berggren, and Henningson

This ar

Uux1uUx1Vuy1vUy5uyy1uzz,~1!

Uvx1uVx1Vvy1vVy1py5vyy1vzz,

Uwx1Vwy1pz5wyy1wzz.

The scalings, originating from the boundary-layer aproximations, are as follows. The streamwise coordinatex isscaled by a fixed distance to the leading edge, here denol,and the streamwise velocitiesU andu are made dimensionless with respect to the free-stream velocityU` . The Rey-nolds number Re is defined by Re5U`l/n, wheren is thekinematic viscosity of the fluid. The wall-normal and spawise coordinatesy and z are made dimensionless with respect to the representative boundary-layer thicknessd5Re21/2 l , while the wall-normal and spanwise velocitiesV,v, andw are scaled with Re21/2 U` . The base flowU andVare obtained from the Blasius solution for the unperturbflat-plate boundary layer. Herep represents the pressure peturbation, which is scaled by (nU` / l ). The equations~1!corresponds exactly to the Go¨rtler equations, with the Go¨rtlernumber zero, as introduced by Floryan and Sario44,45 andHall.46 In order to obtain the equations we have neglecterms of relative orderO(Re21). Note that there are no termof O(Re21/2) to neglect, as opposed to the case with a nzero Gortler number. Van Dyke47 treats the boundary-layetheory as the leading term in an asymptotic expansionhigh Reynolds number. He shows that Prandtl’s theorybe embedded as the first step in a scheme of successivproximations. Van Dyke48 also shows that the second-ordboundary-layer terms, which are ofO(Re21/2), vanish forthe flat plate; for other shapes, the local curvature ofsurface would give a contribution at this order of approximtion.

We are interested in solutions to~1! satisfying a no-slipcondition at the plate and vanishing at infinity in the wanormal direction. In practice, we will truncate the unboundwall-normal interval to (0,ymax), with ymax well outside theboundary layer, and introduce artificial boundary conditioat y5ymax. We use the following boundary conditions in thwall-normal direction;

u5v5w50 at y50,~2!

u5w5p50 at y5ymax.

FIG. 1. Flat plate boundary layer flow with free-stream velocityU` . Thecoordinate system has directionsx, y, and z with corresponding velocitycomponentsu, v, andw.


128.143.23.241 On: Mon,

-

d

-

d

d

-

rnap-

e-

d

s

Thez dependence is taken to be periodic, which makepossible to assume solutions of the form

~u,v,w,p!5~u~x,y!cosbz, v~x,y!cosbz,

w~x,y!sin bz, p~x,y!cosbz!,~3!

whereb is the spanwise wave number. Introducing this asumption yields

ux1vy1bw50, ~4a!

~Uu!x1Vuy1Uyv5uyy2b2u, ~4b!

~Vu1Uv !x1~2Vv !y1bVw1py5vyy2b2v, ~4c!

~Uw!x1~Vw!y2bp5wyy2b2w, ~4d!

where we also have written the equations inconservativeform. Equations 4~a!–4~d! have a singularity atx50, sinceV5O(x21/2), and one would expect problems when startithe integration close to the leading edge. We discuss thisthe validity of Eqs. 4~a!–4~d! close to the leading edge iSec. VII B.

Equations 4~a!–4~d! are parabolic inx for the (u,v,w)variables, so that, given an initial disturbance (u0 ,v0 ,w0),we may impose the condition

u~x0 ,y!5u0~y!,

v~x0 ,y!5v0~y!, ~5!

w~x0 ,y!5w0~y!,

at a givenx0.0 and solve the initial-boundary-value problem ~2!, ~4!, ~5! for x.x0 to obtain the downstream deveopment of the given initial disturbance.

III. THE MAPS A AND B

We adopt an input–output point of view and considthe ‘‘output’’

uout5~u~x,y!, v~x,y!, w~x,y!!T ~6!

at x.x0 as given by the solution of the initial-boundaryvalue problem~2!, ~4!, ~5! with the ‘‘input’’ data

uin5~u0~y!, v0~y!, w0~y!!T. ~7!

Since the initial-boundary-value problem is linear and homgeneous, we may write this

uout5Auin , ~8!

whereA is a linear operator.Note that, if we combine Eqs.~4a! and~4b! and use the

fact thatUx1Vy50, we obtain

w51

bU~~b22Vy!u1Vuy2uyy1Uyv2Uvy! ~9!

for eachx.x0 . Thus, the three components ofuout are re-lated through the ordinary differential equation~9!. Hence, toavoid discontinuities in the solution to the initial-boundarvalue problem~2!, ~4!, ~5! at the starting pointx5x0 , weneed to constrain the initial data (u0 ,v0 ,w0) to satisfy rela-tion ~9!. This may be written
11 Nov 2013 14:01:57

ie

fnlw

e

wiof

ehi

ot

trti

ananll,u-

io

rized

nd

n

a-

dver,tec-

u-

thep-

erm

pec-ave

ee

se


This ar

uin5Bq, ~10!

whereq5(u0 ,v0)T and

Bq

5S u0

v0

1

bU@~b22Vy!u01Vu0,y2u0,yy1Uyv02Uv0,y#

D .

~11!

IV. OPTIMAL DISTURBANCES

The downstream development of disturbances is studby observing how the outputuout—in the terminology of Sec.III—changes with the inputuin . For this, we need a way omeasuring the ‘‘size’’ of disturbances. The most commoused measure in investigations of optimal growth is whatwill define as thedisturbance energyat a specific streamwislocationx,

E~u~x!!5E0

ymax~Re u21v21w2!dy, ~12!

where the purpose of scaling the square of the streamvelocity disturbance with Re is to obtain similar weightingall velocity components.@Recall that the wall-normal andspanwise velocity disturbances are scaled with Re21/2 U` inEq. ~4!.# The square root of the disturbance energy is anorm,given by an~real! inner product, on the space of disturbancat a fixed streamwise location, and we will acknowledge tby using the notationE(u)5iui25(u,u) in the general dis-cussion of the present section.

To calculate the optimal disturbance, we pick twstreamwise locations 0,x0,xf and maximize the outpudisturbance energy atx5xf among all suitable inputs atx5x0 with fixed ~unit! energy. ‘‘Suitable’’ means here thawe have constrained the inputs to the range of an operatoB

for the reasons discussed in Sec. III. The maximized quanwill be denoted themaximum spatial transient growth,

G~xf !5 maxiuini5iBqi51

iuout~xf !i2. ~13!

Note that other norms than the one inferred by~12! are pos-sible; different norms could even be used on the inputsthe outputs. The choice of norms is a modeling issuemust be made out of physical considerations as it criticaaffects the result of the maximization. The choice abovedisturbance kinetic energy density, is the traditional measwhen considering optimal growth, which will simplify comparisons with other calculations.

Using the notation of the previous section, express~13! may also be written


128.143.23.241 On: Mon,

d

ye

se

ss

ty

ddyare

n

G~xf !5 maxiBqi51

iABqi2

5 maxBqÞ0

iABqi2

iBqi2

5 maxBqÞ0

~ABq,ABq!

~Bq,Bq!, ~14!

Recalling some basic facts from operator theory,49 we havethe following. If the maximum of (ABq,ABq)/(Bq,Bq)is attained for some vectorq, this vector is an eigenvectocorresponding to the largest eigenvalue of the generaleigenproblem

B* A* ABq5lB* Bq, ~15!

andG(xf) is the maximum eigenvalue, necessarily real anon-negative. The operatorsA* andB* in Eq. ~15! denotethe adjoint operators toA andB with respect to the choseinner products.

The maximizerq has another, equivalent, characteriztion in terms of the generalized singular-valuedecomposition50 of the operator pairAB,B; namely,G(xf)is the square of the largest generalized singular value, anqis the corresponding generalized singular vector. MoreoBq, andABq are~proportional to! the corresponding righand left singular vectors, respectively. The left singular vtor, Bq, where q maximizesG(xf), will be denoted theoptimal disturbance, and the most natural attempt to calclate it is bypower iterations,

Bqn115rnB~B* B!21B* A* ABqn , ~16!

wherern is an arbitrary scaling parameter, used to scaleiterates to unit norm, for instance. If we make the assumtion, as above, that the maximizerq actually exists, the iter-ates will converge to the optimal disturbance as long as~i!the maximum eigenvalue issimpleandwell separatedfromthe rest of the spectrum and~ii ! the starting vectorq0 has anonzero component in theq direction, that is, (q0 ,q)Þ0.

In the numerical experiments reported below, the powiterations converged quickly, indicating that the maximueigenvalue is indeed well separated from the rest of the strum for the parameter settings of interest. However, we hno rigorous justification for such a compactness result.

Note that the operatorB(B* B)21B* in algorithm~16! is the orthogonal projector on the range ofB; that is, ifwe define un115A* ABqn , we have that un11

5rnB(B* B)21B* un115Bqn11 where qn11 solves thelinear least-squares problem

minq

iBq2un11i2. ~17!

In other words, the new candidate optimal disturbanceun11 ,obtained after applyingA* A to the previous disturbancun , is replaced in algorithm~16! by the closest disturbancun11 , in a least-squares sense, satisfying Eq.~9!.

Remark.The governing equations for the parallel caare of the general form,

ut5Lu, u~0!5u0 , ~18!
11 Nov 2013 14:01:57

ftso

op

s

s

-o

w

ron

at

e

-e

-r,

-

-of

tur-ed

hedse-

butta-

i-

are.

setaryd-on


This ar

whereL is the Orr–Sommerfeld operator independent ot~andx!. Here,u denotes the vector of velocity componenfor instance. The solution to the above system can be cstructed by the sum of the eigenmodes to operatorL. Theoptimal disturbance can thus be found by calculating antimal set of coefficients in the sum of eigenmodes toL. Thisis the standard method to obtain the optimal disturbanceparallel flows.

The nonparallel case studied here, that is problem~2!,~4!, ~5!, is of a general form similar to~18!, namely,

ux5 L ~x!u, u~0!5u0 . ~19!

However, the operatorL depends in this case onx. Thisimplies that there does not exist a single set of eigenmodeL that can be used to represent a solution to equations~19!in contrast to the parallel case~18!. However, the more general method developed in this paper is applicable to the nparallel case.

V. THE ADJOINT EQUATIONS

The operatorB and the action of the operatorA on avector, an initial disturbance, were defined in Sec. III. Hoever, to be able to perform the power iterations~16!, we alsoneed to know the action of the adjoint operatorsA* , B* ,and we need to be able to solve the linear least-squares plem ~17!. In this section, we state the relevant expressioderived in Appendix A.

A. Equations for A*

The operatorA* is the adjoint of the operatorA, in-troduced above, with respect to the inner product associwith the disturbance energy~12!. Given any square-integrable vector functionC5(c1(y),c2(y),c3(y))T, ofthe wall-normal coordinatey, the action ofA* on C is thevector F, whose components are given by expression~23!;that is, A* C5F. Details regarding the derivation of thequations in this section are given in Appendix A.

The adjoint equationsare

vy* 1bw* 50,

2px* 2Uux* 2Vvx* 2Vuy* 5uyy* 1~Vy2b2!u* ,~20!

2Uvx* 22Vvy* 1Uyu* 2py* 5vyy* 2b2v* ,

2Uwx* 2Vwy* 1bVv* 1bp* 5wyy* 2b2w* ,

whereu* (x,y), v* (x,y), w* (x,y), p* (x,y) are functionsdefined onx>x0 , y>0.

Boundary conditions for Eq.~20! are

u* 5v* 5w* 50 at y50,~21!

p* 12Vv* 1vy* 5u* 5w* 50 at y5ymax.

Note that in Eq.~20!, thex derivative has opposite sign compared to Eq.~4!. The parabolic nature of the equations dmands therefore Eq.~20! to be integrated ‘‘backwards’’ withinitial conditions specified atx5xf . If we impose the initialconditions


128.143.23.241 On: Mon,

,n-

-

in

of

n-

-

b-s,

ed

-

U~xf ,y!u* ~xf ,y!1V~xf ,y!v* ~xf ,y!1p* ~xf ,y!

5Re c1~y!,

U~xf ,y!v* ~xf ,y!5c2~y!, ~22!

U~xf ,y!w* ~xf ,y!5c3~y!,

the action of the adjoint operator is given by

f1~y!51

Re@U~x0 ,y!u* ~x0 ,y!1V~x0 ,y!v* ~x0 ,y!

1p* ~x0 ,y!#,

f2~y!5U~x0 ,y!v* ~x0 ,y!, ~23!

f3~y!5U~x0 ,y!w* ~x0 ,y!.

B. Equations for B*

The operatorB in ~10! may be written

B5S I 0

0 I

~bU !21C 1 C 2

D ,

where the differential operatorsC 1 andC 2 are defined by

C 1u5~b22Vy!u1Vuy2uyy ,~24!

C 2v5b21S Uy

Uv2vyD .

To derive an expression forB* , we need to calculate adjoints of (bU)21C 1 andC 2 . This is a more delicate mattesurprisingly enough, than the derivation ofA* . The reasonfor this is our choice of disturbance energy~12! togetherwith the fact that thew component of the velocity disturbance is related to theu and v components through thesecond-order differential expression~9!, that is, the third rowof operatorB. Recall that the definition of an adjoint operator is subordinate to the choice of domain and codomainthe operator, that is, the choice of inner products. The disbance energy~12! determines the inner product associatwith elements in the range ofB, namely the same~L2-like!one as in expression~A2!. However, equalities like~9! aremost naturally interpreted in so-calleddual norms, and re-placing, in the definition of the disturbance energy, tintegral-square ofw with the appropriate dual norm woulmake the calculation of the adjoints straightforward. The uof the integral square ofw in the disturbance energy is nevertheless the most natural from a physical point of view,it introduces some complications in the adjoint computions.

Theexistenceof the adjoints is of no concern: By specfying any domain and codomain in which (bU)21C 1 andC 2 are bounded linear operators, corresponding adjointsdefined through a variational form by Rietz representation51

However, for the kind of numerical approximations we u~spectral collocation!, we would prefer to have an expliciexpression as a differential operator together with boundconditions, not only the variational form. For our inner proucts, it is straightforward to obtain a differential expressi
11 Nov 2013 14:01:57

ind-ts

ng

ppo2

ro

rc

sc

in

ed

-

y

eat

nsalt tork-

teroveialnt isosi-ed,iz-

a-s.

lvedon-nges-

-igh

an-


This ar

for C 2* using integration by parts, but not so for the adjoof (bU)21C 1 ; the problem is in the treatment of the bounary terms that occurs when attempting integration by par

We have investigated two different ways of obtainiexpressions for the adjoint of (bU)21C 1 other than thevariational-form definition. In the first approach weapproxi-mate, in the second one, we use aduality technique. The twoapproaches give very similar results for Re&105, but theduality technique seems unstable for Re about 106 and up,whereas the approximation technique appears to be acable for all Reynolds numbers of interest. We give a shdescription of the approximate technique in Appendix ABoth techniques are presented in Anderssonet al.52 wheremore emphasis has been put on the mathematics of the plem.

C. The least-squares problem

At each iteration ~16!, we need to multiply with(B* B)21B* , which is equivalent to solving the linealeast-squares problem~17!. For the reasons discussed in SeV B, we approximate the operatorB with Be , as discussedin Appendix A 2, and solve, instead of~17!, theapproximateproblem

minq

iBeq2ui2, ~25!

with Be defined as in~A11!. For problem~25!, the normalequations are

S Re1C 1* Re~bU !22ReC 1U C 1* Re~bU !21C 2

C 2* ~bU !21ReC 1 I 1C 2* C 2D S ue

veD

5S Re u1C 1* Re~bU !21w

v1C 2* w D ~26a!

together with the boundary conditions

ue5ve50 at y50,~26b!

ue5]ve

]y50 at y5ymax.

Since the size of the problem is quite small, we chooseexplicitly assemble the discrete versions of the operatorEq. ~26a! and solve the resulting linear system with a diremethod.

VI. THE OPTIMIZATION ALGORITHMS

This section describes in algorithmic form the stepsvolved in the power iterations~16!.

~1! Solve Eqs. 4~a!–4~d! using the boundary conditions~2!and the initial conditions~10!, where (u0 ,v0) can bechosen arbitrarily at the first iteration, but is obtainfrom step~4! at later iterations.

~2! Solve the adjoint equations~20! using the boundary conditions ~21! and the initial conditions~22!, wherec1(y)5u(xf ,y), c2(y)5v(xf ,y) and c3(y)5w(xf ,y), and where (u,v,w) are obtained from step~1!.


128.143.23.241 On: Mon,

t

.

li-rt.

b-

.

toint

-

~3! From expressions~23! we obtain u(y)5f1(y), v(y)5f2(y) and w(y)5f3(y).

~4! Given (u,v,w) Eq. ~26a! is solved using the boundarconditions~26b!.

~5! Scale, if needed, the new initial disturbances and rep~1!–~4! until convergence.

In the numerical results, the number of power iteratiodepends onx0 , xf , and the Reynolds number. For the typiccases reported below, about five iterations was sufficienobtain good convergence. Running time on a desktop wostation is 5–10 min.

Luchini27 solves an optimization problem, similar bunot identical to ours. This formulation is Reynolds-numbindependent and constitutes a limiting case of the abproblem as the Reynolds number goes to infinity. Only initperturbations such that the streamwise velocity componeidentically zero are considered, and at the downstream ption xf , only the streamwise velocity component is observnot all three as in the above formulation. Instead of optiming growth in the energy norm~13!, the optimization prob-lem becomes

maximize E0

ymaxu2 dy over all initial data such that

u050 and E0

ymaxv0

21w02 dy51. ~27!

If ( u0n ,v0

n ,w0n) represents initial conditions at one iter

tion leveln, the next step in the power iteration is as follow

~1! Solve Eq.~4! using the boundary conditions~2! and theinitial conditions

u~x0,y!50, v~x0,y!5v0n, w~x0,y!5w0

n.

~2! Solve Eq.~20! using the boundary conditions~21! andthe initial conditions

U~xf ,y!u* ~xf ,y!1p* ~xf ,y!5u~xf ,y!,

v* ~xf ,y!50, w* ~xf ,y!50.

~3! The new initial disturbance is then obtained as

u0n11~y!50, v0

n11~y!5U~x0 ,y!v* ~x0 ,y!,

w0n11~y!5U~x0 ,y!w* ~x0 ,y!.

Note that there are no least-squares problem to be soin the Reynolds-number-independent formulation. This cstitutes a major practical simplification when implementithe algorithm. However, this formulation explicitly assumspecifics of the optimal disturbances~no streamwise component! and corresponding downstream response~no spanwiseand wall-normal components!. These properties are not presumed, but emerge from the complete formulation for henough Reynolds numbers.

VII. RESULTS

A. Fundamental results

We study the dependence of the maximum spatial trsient growth on the final positionxf , the spanwise wave
11 Nov 2013 14:01:57

this

laa

m

ld

somt

andss

-mgeve

o-in

ha

iegarghea

te

xi-y-

ent

n

ng

urntberow

be ber


This ar

numberb, and the Reynolds number Re; that is, we studyfunctionG5G(xf ,b,Re). Here the Reynolds number, Re,based on the distance from the leading edge of the flat pl. First, we fix the downstream position to which the equtions were integrated toxf51, and we calculateG(1,b,Re)/Re for several values of the spanwise wave nuberb between 0.1 and 1. The initial position isx050, that is,right at the leading edge of the flat plate~see Sec. VII B!.The calculations are repeated for five different Reynonumbers Re5103, 104, 105, 106, and 109, and once with theReynolds-number independent~27! version. Figure 2 showsG(xf)/Re vsb.

The three curves corresponding to Reynolds number100 000, 1 mil, and 1 bil collapse at the curve obtained frthe Reynolds-number-independent formulation, whereascurves corresponding to Reynolds numbers at 100010 000 deviate from the other four. The large-Reynolnumber graphs of Fig. 2 agree with corresponding graphLuchini.27

Thus we can conclude from this thatthe maximum spatial transient growth scales linearly with the Reynolds nuber for Reynolds numbers larger than 100 000. In the larReynolds-number limit the spanwise wave number that gimaximum amplification isb50.45.

Second, we fix the spanwise wave number tob50.45,and calculateG(xf ,0.45,Re)/Re varying the downstream psition,xf , between 0.1 and 11. The initial position was agax050. The results are shown in Fig. 3. Figure 3 shows tthe maximum spatial transient growth, as a function ofxf hasa maximum at some downstream position. The transgrowth curves, scaled with the Reynolds number, once ashow to be Reynolds-number independent in the laReynolds-number limit. The downstream position giving tmaximum amplification, for this spanwise wave number, wxf52.7 in the large Reynolds-number limit.

In fact, the graphs of Figs. 2 and 3 are intimately relain the large Reynolds-number limit. This is a consequence

FIG. 2. Maximum spatial transient growth divided by the Reynolds numvs spanwise wave number. Herex050 and xf51. ~-•- Re5103; --- Re5104; n Re5105; 1 Re5106, s Re5109,—Re independent!.


128.143.23.241 On: Mon,

e

te,-

-

s

at

hed-in

--s

t

ntine

s

dof

the result that, for high enough Reynolds number, the mamum spatial transient growth scales linearly with the Renolds number. Let us define

G~xf ,b!5 limRe→`

G~xf ,b,Re!

Re. ~28!

Assume the same physical problem scaled with two differlength scales,l and l 1 . We have

G~xf ,b,Re!5G~xf1,b1,Re1! ~29!

given that

xf* 5xf l 5xf1l 1 , b* 5bAU`

n l5b1AU`

n l 1,

Re5U`l

n, Re15

U`l 1

n;

here xf* and b* are the dimensional downstream positioand spanwise wave number, respectively. Introducingc2

5 l / l 1 and rewriting the right-hand expression in~29!, in thevariablesxf ,b,Re we obtain

G~xf ,b,Re!5G~c2xf ,b/c,Re/c2!.

Multiplying both sides byc2/Re, we obtain

c2G~xf ,b,Re!

Re5

G~c2xf ,b/c,Re/c2!

Re/c2 .

Now letting the Reynolds numbers tend to infinity and usi~28! we get

c2G~xf ,b!5G~c2xf ,b/c!,

for eachc.0.Using the results given in Fig. 2, we construct a conto

plot, containing isolines of constant maximum transiegrowth as function of dimensionless spanwise wave numand Reynolds number. The spanwise wave number is n

rFIG. 3. Maximum spatial transient growth divided by the Reynolds numvs downstream position. Herex050 and b50.45. ~-•- Re5103; --- Re5104; n Re5105; 1 Re5106; s Re5109;—Re independent!.


11 Nov 2013 14:01:57

e

e

e

or

hee

n

ldstur-

the

wiseam-onsll-hin

berd-

theand

iththe

loc-

ve-e-

pe-re

ur-h at

f. 8ent

t

l


This ar

made dimensionless with the length scalen/U` and denotedb. The results are given in Fig. 4, together with a dashcurve, describing for whichb one obtains the maximumtransient growth given specific Reynolds numbers. Theplicit expression describing this curve isb50.45/ARe,which is revealed by the scalingb5b* n/U`5b/ARe to-gether with the fact that the maximum in Fig. 2 was obtainfor b50.45.

Thev andw components of the optimal perturbation, fthe spanwise wave numberb50.45 and optimized with re-spect to downstream positionxf51, are given in Fig. 5 at thehigh-Reynolds-number limit. The correspondingu compo-nent of the response at the downstream positionxf51caused by this optimal perturbation is given in Fig. 6. Tpresence of the Reynolds number Re in the disturbanceergy~12! penalizes theu component in the input perturbatio

FIG. 4. Contour plots of maximum transient growth vsb and Reynolds

number. The dashed line describesb for which the maximum transiengrowth occurs given a specific Reynolds number. Herex050.

FIG. 5. The optimal perturbations atx5x0 . Here x050, xf51, and b50.45. Theu component is zero.


128.143.23.241 On: Mon,

d

x-

d

n-

and forces it to practically vanish, compared to thev andwcomponents, in the optimal perturbation for high Reynonumbers. On the other hand, the same scaling in the disbance energy will cause thev and w components in thedownstream response to practically vanish, compared tou component.

Note also that, because of the periodicity property~3!,the upstream disturbance in Fig. 5 corresponds to streamvortices and the downstream response in Fig. 6 to strewise streaks. It should be mentioned that the power iteraticonverge quickly, which indicates the existence of a weseparated, dominating mode. The conclusion is that, witthe limits of this model,almost any initial disturbance willdevelop into a streamwise streak given a Reynolds numlarger than about105, based on the distance from the leaing edge.

To further visualize the upstream disturbance andcorresponding downstream response, as given in Figs. 56, the spanwise periodic dependence~3! was introduced. InFig. 7~a! we give the upstream disturbance from Fig. 5, wthe spanwise dependence, plotted as velocity vectors inz–y plane. The corresponding downstream response~fromFig. 6! are shown as contours of constant streamwise veity in the z–y plane in Fig. 7~b!. Note how the low velocitystreaks are produced by thelift-up of low velocity fluid ele-ments near the wall and correspondingly how the highlocity streaks are produced by the introduction of high vlocity fluid elements pulled down from the free stream.

Four different initial perturbations were used, atx050,chosen to maximize the energy growth at four different scific downstream positions. The forward equations wemarched once with each perturbation. The four initial pertbations were chosen to generate maximum energy growtpositionsxf51, xf52.7, xf55, andxf510. The spanwisewave number was set tob50.45. The energy growth othese specific perturbations downstream is given in Figtogether, for reference, with the maximum spatial transi

FIG. 6. Downstream response atx5xf51 corresponding to the optimaperturbations in Fig. 5, that isx050 andb50.45. Thev andw componentsare zero.


11 Nov 2013 14:01:57

er-ber.

enlcu-on-

the

osi-

beeirtheing

t

aelu

-


This ar

FIG. 7. ~a! Velocity vectors in thez–y plane of the optimal disturbance ax5x0 . Herex050 andb50.45. Theu component is zero.~b! Contours ofconstant streamwise velocity representing the downstream responsex5xf51 corresponding to the optimal disturbance shown in Fig. 6. Thvandw components are zero. Here, the solid lines represent positive vaand the dashed lines represent negative values, respectively.

FIG. 8. Energy growth vs downstream position. Herex050 andb50.45.


128.143.23.241 On: Mon,

growth curve from Fig. 3, giving the envelope of these pturbations, calculated with the same spanwise wave num

B. The leading edge

As mentioned in Sec. II we could expect problems whusing the leading edge as the starting position for our calations. The problems are twofold. First, the equations ctain a discontinuity in the coefficients atx50 and second,the boundary layer approximations are not valid close toleading edge, or more precisely, a distance ofO(1/Re) in ourscaling.

We have checked the dependence on the starting ption, x0 , by varying its location, starting with a value ofx0

large enough for the boundary layer approximation tovalid. We find that the optimal disturbances, as well as thdownstream response, vary slowly and continuously withvalue of the starting position as it approaches the lead

t

es

FIG. 9. The optimal perturbations atx5x0 for three different values ofx0 .Herexf51 andb50.45.

FIG. 10. Downstream response atx5xf corresponding to the optimal perturbations in Fig. 9, that isxf51 andb50.45.


11 Nov 2013 14:01:57

th

fo

cm

ioth

threhaaris

amm

e

od

onolddfoen

uredcu-eri-s toaryam

ure-

ur-

sedtedcal-

ight

h-

ee-

uswn-e

calestinry

-

thta. ues

les


This ar

edge. Figures 9 and 10 show optimal disturbances anddownstream response at positionxf51. The spanwise wavenumber wasb50.45 and the calculations were repeatedthree different starting positionsx050, 0.01, and 0.05. Ascan be seen nothing drastic happens whenx0 approacheszero. Atx0 , we haveU51, V50.

It should be noted that the calculated initial disturbanat x050 is outside the boundary layer, while its downstrearesponse is inside the boundary layer. Thus, our calculatmodel the receptivity at the leading edge in the sensethey predict that the disturbances~at the leading edge! caus-ing the largest growth of streamwise streaks insideboundary layer are vortices aligned in the streamwise dition outsidethe boundary layer. For the record we state tthe receptivity mechanism is a vast and open field of resein transition modeling and we do not claim to model thmechanism in a wider sense.

C. Comparison with experiments

In Fig. 11 the results from Fig. 6, that is, the downstreresponse of the optimal perturbations in Fig. 5 with paraetersxf51, x050, b50.45, and Re.106, are plotted to-gether with experimental data from Westinet al.40 All thestreamwise velocity components have been normalizedunit maximum value. The normal coordinate is normalizwith the measured value of the displacement thickness,d* .For our calculations we have usedd* 51.72d, whered is therepresentative boundary-layer thickness,d5Re21/2l , definedin Sec. II. The experimental data represent profiles of romean-square streamwise velocity perturbations measurea flat-plate boundary layer at different downstream positiand free-stream velocities corresponding to various Reynnumbers. The Reynolds numbers Re of Fig. 11 are based* , that is, Re5U`d* /n. In the experiments a grid was useto generate a free-stream turbulence level at 1.5%, therethe rms streamwise velocity perturbations do not tend to zfor large values ofy/d* , whereas the calculated perturbatio

FIG. 11. Comparison between the streamwise velocity component ofdownstream response of an optimal perturbation and experimental dathe calculations the parameters used werexf51, x050, b50.45, and Re.106.


128.143.23.241 On: Mon,

eir

r

e

nsat

ec-tch

-

tod

t-insdson

rero

does. The remarkably good agreement between the measand calculated velocity profiles, and the fact that the callations contained an optimization procedure while the expments did not, indicate that the shown profile correspondsome fundamental mode triggered in the flat-plate boundlayer when subjected to high enough levels of free-streturbulence.

The five stars in Fig. 12 represent experimental measments, carried out by Westinet al.,40 of the wall-normal-maximum amplitude of the rms streamwise velocity pertbations on a flat plate with free-stream velocityU`58 m/sand free-stream turbulence level Tu51.5%. The experimen-tally measured streamwise rms values are mainly compoof low-frequency fluctuations, corresponding to elongastreaky structures, and can therefore be modeled by ourculations of spatially growing steady streaks.

Note that the stars appear approximately on a straline. Recall that the maximum spatial transient growth,G,scaled linearly with the Reynolds number in the higReynolds-number limit. We defineEout5G(1,bmax)ReEin ,where bmax50.45 ~from Fig. 2! and G is defined in~28!.ChoosingEin59.331026 in order to match the slope of thexperimental values, we obtain the solid line in Fig. 12. Rcall that

b* 5U`b

nARe5

U`b

n, ~30!

whereb* is the dimensional spanwise wave number. Ththe dimensional spanwise wave number decreases dostream ifb is constant, implying a downstream growth of thspanwise scale. A downstream growth of the spanwise shas also been observed in experiments. The results of Weet al.40 indicate that this growth is slower than the boundalayer thickness, that is, slower than the Re1/2 growth impliedby the scaling~30! and the solid line in Fig. 12. The remain

eInFIG. 12. Energy growth vs Reynolds number: experimental rms val~stars!, computed optimal growth for downstream-growing spanwise sca

~the solid line!, computed optimal growth for fixed spanwise scalesb

58.731024 ~the dashed line!, b51.331023 ~the dash-dotted line!, b51.931023 ~the dotted line!.


11 Nov 2013 14:01:57

g

oerre

e

of

Hing

hescthanu

th.in

yunc

eee

ngone

mm

ce

th

thelds

put

elsdel

umor-cal

al

fn-, weaneer

ownurs

ro-tionthe-eri-nceri-


This ar

ing three curves in Fig. 12 show the maximum energrowth possible for threefixed spanwise wave numbers,b58.731024, 1.331023, 1.931023, that is, Eout,fix

5G(1,bARe)ReEin ~Ein59.331026 also here!. We see thatthe three curves show a transient behavior, growing tmaximum and then decaying for higher Reynolds numbwith the solid line representing an envelope for the thcurves. UsingU`58 m/s and the value ofn for air the non-dimensional wave numbersb58.731024, 1.331023, 1.931023 corresponds to the distanceslz* 513.6, 9.1, and 6.0mm, between two streaks. The spanwise distance betwtwo streaks in the experiment, measured 1 m downstream ofthe leading edge~corresponding to a Reynolds numberRe55.33105), was roughly 13 mm.

Bertolotti29 makes comparisons with the same data.generates an inflow condition for steady vortices by seekself-similar solutions in a neighborhood of the leading edof the plate~but excluding the actual leading edge!. Thisinflow condition is then used in a PSE~parabolic stabilityequation! solver to find the downstream development of tstreaks. However, he seems to have misinterpreted theings used in the experiments, and would have to lowervalues ofb* used in his calculations in order to achieveagreement with the correctly scaled measurements. If wethese misinterpreted scalings in our calculations, we findthe results of Bertolotti29 are remarkably similar to oursNote that he does not attempt to calculate optimally growstreaks.

D. Transition prediction

We aim to find a correlation between the transition Renolds number, ReT and the free-stream turbulence level, TAssuming isotropic turbulence, the free-stream turbulelevel is

Tu51

U`~u82!1/2,

whereu8 is the fluctuating streamwise velocity in the frestream, and where the overbar denotes the temporal mExperiments indicate that for free-stream turbulence levbelow 0.08%, transition is not associated with growistreamwise streaks, and for levels above 5%, transitioncurs at the minimum Reynolds number where self-sustaiboundary layer turbulence can exist~see Arnal—Ref. 53—for a review!. We are interested in a model for free-streaturbulence levels at about 1%–5%. We make three assutions in order to obtain our model.

~1! We assume that the input energyE(uin), as definedin ~12! and ~7! is proportional to the free-stream turbulenenergy,

E~uin!}Tu2. ~31!

This is an assumption about the receptivity process.~2! We assume that the initial disturbance grows with

optimal rate,

E~uout!5GE~uin!5G Re E~uin!, ~32!
128.143.23.241 On: Mon,

y

as,e

en

ege

al-e

seat

g

-.e

an.ls

c-d

p-

e

where G is Reynolds-number independent. Recall thatlast equality was found to hold for large enough Reynonumbers.

~3! We assume that transition occurs when the outenergy reaches a specific value,ET ,

E~uout!5ET . ~33!

Combining assumptions~31!–~33!, we obtain

AReT Tu5K,

whereK should be constant for free-stream turbulence levat 1%–5%. The experimental data used to verify this moare given in Table I. As can be seen,K is approximatelyconstant for a variety of free-stream turbulence levels.

A similar model was given by van Driest and Blumer,34

who postulated that transition occurs when the maximvorticity Reynolds number reaches a critical value to be crelated with the free-stream turbulence level. Their empiritransitional formula becomes

1

AReT

5a1bAReT Tu2,

wherea and b were calibrated with available experimentdata to bea51024 and b562.531028. This model wascriticized by White54 in his textbook: ‘‘Since the concept oa critical vorticity Reynolds number is obviously questioable and not related to any fundamental rigorous analysiscan regard the van Driest–Blumer correlation simply asexcellent semiempiricism.’’ Note that our model yields thsame formula as the one found by van Driest and Blumwith a50 and b51/K2. A comparison between the twmodels and the experimental data given in Table I is shoin Fig. 13. As can be seen their model agrees well with ofor free-stream turbulence levels at 1%–6%.

Variations inK can have several causes. First, inapppriate leading-edge conditions can create a strong sucpeak resulting in a substantial upstream movement oftransition point, see Westinet al.40 Second, we assumed isotropic free-stream turbulence. In many cases, the expments contain anisotropies. Third, the free-stream turbuleenergy spectrum could differ between the different expe

TABLE I. Comparisons of different experimental studies.

Tu~%! ReT K

Roach and Brierleya

T3AM 0.9 1 600 000 1138T3A 3.0 144 000 1138T3B 6.0 63 000 1506

Yang and Vokeb 5.0 51 200 1131

Matsubarac

Grid A 2.0 400 000 1265Grid B 1.5 1 000 000 1500

aReference 41.bReference 43.cReference 42.


11 Nov 2013 14:01:57

th

eean.apa

ss

adglelir

ece

dadTh-fe

wwp

theof

nosell-

at aeak

oryheThetheakisaksedallycespre-orc-he

cep-

de-rsn a

d asr-eeondd-rge

ed,estuc-theofug-

r-

nsfor

l toen-

m-nk

d

sith


This ar

ments. In particular, the position of the spanwise peak inenergy spectrum could influence the growth rate.

VIII. DISCUSSION AND CONCLUSIONS

The maximum spatial transient growth has here bcalculated for steady disturbances in the Blasius boundlayer. Note, however, that our method is general andrestricted to the Blasius boundary-layer equations onlycomplication, emanating from the use of boundary-layerproximations together with our choice of energy norm, wthe derivation of the adjoint of the operatorB ~giving theconstraint on the input velocity!. This difficulty can avoidedby considering the problem directly in the large Reynoldnumber limit ~see Sec. VI!. The two different approacheagree for sufficiently high Reynolds numbers.

Another difficulty is related to the streamwise positionwhich the integration is started. We start at the leading e(x050), which numerical investigations indicate is possibHowever, the boundary-layer approximations are not vaclose to the leading edge; thus, it is not clear how to interpthe present solution in that region, particularly with respto the receptivity process. However, we conclude the largenergy growth in the streamwise streaks, inside the bounlayer, is triggered by vortices aligned in the streamwiserection at the leading edge outside of the boundary layer.experiments of Westinet al.40 show that the transition location is influenced by the shape of the leading edge, alsoboundary layers subjected to free-stream turbulence. Thisfect is not considered in our model. Nevertheless, the dostream response of the optimal disturbances agree wellthe measured wall-normal dependence of the streak amtude.

FIG. 13. Transitional Reynolds number based on the distance to the leaedge vs free-stream turbulence level~given in percent!, for two transitionprediction models and experimental data.~—The model suggested in thipaper withK51200,* The model proposed by van Driest and Blumer wa51024 andb562.531028, s The experimental data from Table I.!


128.143.23.241 On: Mon,

e

nryotA-

s

-

te.dettstry

i-e

orf-

n-ithli-

In fact we capture both the fundamental shape ofstreaks,urms ~in Fig. 11! and the downstream developmenttheir maximum,urms,max ~in Fig. 12!. Worth emphasizing isthat the perturbations applied in the experiments were inway optimized. In the calculations of Sec. VII A, it wafound that the streamwise streaks corresponded to a wseparated dominating mode, which led us to conclude thrandom disturbance would develop into a streamwise strgiven a Reynolds number larger than about 105, based on thedistance to the leading edge. However, the present thedoes not give a complete prediction of the variation of tspanwise scale of the streaks with downstream distance.curve which fits the experimental values the best, namelysolid line of Fig. 12, predicts a faster growth of the strespacing, Re1/2, than observed experimentally. However, itworth noting that the average spanwise size of the strefound in the experiment is consistent with the curve for fixspanwise wave number which best fits the experimentobtained growth, the dashed line of Fig. 12. The differenbetween the experiments and theoretical predictionssented here could possibly be related to the continuous fing of the boundary layer by the free-stream turbulence. Tpresent theory does not take this into account. Such a retivity mechanism is discussed in the thesis of Berlin.55

Finally, the results of the present work are used tovelop a transition prediction criterion for boundary layesubjected to free-stream turbulence. The model rests onumber of strong assumptions and cannot be regardeanything but a first try in this direction. However, it is suprising to find that, not only does the simple model agrfairly well with available experimental data, but it alsagrees well with the earlier model by van Driest aBlumer.34 The model could be further improved by consiering the anisotropy of the free-stream turbulence. Laeddy simulations by Voke and Yang,56 in which differentanisotropies of the free-stream turbulence were imposshowed that the wall-normal fluctuations had the stronginfluence on the transition onset, while the streamwise fltuations proved quite harmless. To calibrate and tunemodel further a more careful investigation of the naturethe free-stream turbulence would be appropriate. One sgestion could be to just use thev component of the free-stream turbulence when correlating it with the initial distubance energy, i.e.,

Ein;Tv251

U`2 v82.

ACKNOWLEDGMENTS

We wish to thank Johan Westin for fruitful discussioon transition influenced by free-stream turbulence andproviding us with experimental data. We are also gratefuMasaharu Matsubara who made his unpublished experimtal data available to us and to Stellan Berlin for helpful coments and aid with producing Fig. 1. Last, we also thaStefan Wallin and Nick Trefethen for helpful comments.

ing


11 Nov 2013 14:01:57

e-

e

a

t

n

th

n

s.


This ar

APPENDIX A: THE ADJOINT OPERATORSA* AND B*

1. The adjoint operator A*

The adjoint of any bounded linear operatorA betweentwo inner-product~Hilbert! spaces is defined through the rlation

~C,Au!5~A* C,u!, ~A1!

where u is an element in the domain ofA, and C is anelement in the space associated with the inner product57 onthe left-hand side in Eq.~A1!. In our case, the disturbancenergy~12! determines the choice of inner products in~A1!and the left-hand side of equality~A1! will be, using thenotations of previous sections,

~C,Auin!5~C,uout!

5E0

ymax@Re c1~y!u~xf ,y!1c2~y!v~xf ,y!

1c3~y!w~xf ,y!#dy

5E0

ymaxC~y!TMu ~xf ,y!dy, ~A2!

where

C5~c1~y!,c2~y!,c3~y!!T

is any square-integrable vector function of the wall-normcoordinatey and

M5S Re 0 0

0 1 0

0 0 1D .

We use the same inner product for the inner products onright-hand side of equality~A1!. Thus, if we can find a func-tion

F5~f1~y!,f2~y!,f3~y!!T,

such that

E0

ymaxC~y!TMu ~xf ,y!dy5E

0

ymaxF~y!TMu ~x0 ,y!dy,

~A3!

we know thatF may be identified withA* C with respectto our choice of inner products@compare with equality~A1!#.

We will derive an expression for the action ofA* byintegration by parts on the governing equations. To condethe notation, we start by writing Eq.~4! as a system

~Af !x5B0f1B1fy1B2fyy , ~A4!

where

f5~u,v,w,p!T5S upD , A5S 1 0 0 0

U 0 0 0

V U 0 0

0 0 U 0

D ,


128.143.23.241 On: Mon,

l

he

se

B05S 0 0 2b 0

2b2 2Uy 0 0

0 22Vy2b2 2bV 0

0 0 2Vy2b2 b

D ,

B15S 0 21 0 0

2V 0 0 0

0 22V 0 21

0 0 2V 0

D ,

B25S 0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

D .

For later use, we also define the matrixA as the first threecolumns ofA, that is

A5S 1 0 0

U 0 0

V U 0

0 0 U

D .

A remark on notation: Most objects in this section,f, A,and B1 for instance, are vector or matrix functions of bothe streamwise and wall-normal coordinatesx andy, and wewill sometimes usef(x) as a shorthand for the functioy°f(x,y) and f(y) for the functiony°f(x,y).

Let

g5~p* ,u* ,v* ,w* !T

5~p* ~x,y!,u* ~x,y!,v* ~x,y!,w* ~x,y!!T5S p*u* D

be a smooth vector function defined onx>x0 , y>0. We willimpose further restrictions ong as we proceed, allowing theidentification of the components ofg with pressure- andvelocity-like ‘‘adjoint’’ variables, as the notation suggestFor now, however,g could be any suitable function.

Taking the scalar product of the vectorg with Eq. ~A4!,integrating over the domain (x0 ,xf)3(0,ymax), and applyingintegration by parts yields

05E0

ymaxEx0

x1gT@~Af !x2B0f2B1fy2B2fyy#dx dy

5E0

ymaxEx0

x1fT@2ATgx2B0

Tg1~B1Tg!y2B2

Tgyy#dx dy

1E0

ymaxfT~xf !A

T~xf !g~xf !dy2E0

xmaxfT~x0!AT~x0!

3g~x0!dy2Ex0

x1@ fT~y!B1

T~y!g~y!1fyT~y!B2

Tg~y!

2fT~y!B2Tgy~y!#dxU

y50

y5ymax

. ~A5!


11 Nov 2013 14:01:57

b

-

-

ge

geebyad-

wee; ithe

a-


This ar

Now we demand thatg satisfies theadjoint equation

2ATgx5B0Tg2~B1

Tg!y1B2Tgyy ,

whose components are given in Eq.~20!. Appropriate bound-ary conditions for the adjoint equations can be deducedexpanding the last integral in~A5!,

2Ex0

x1@ fT~y!B1

T~y!g~y!1fyT~y!B2

Tg~y!

2fT~y!B2Tgy~y!#dxU

y50

y5ymax

5Ex0

xf@v~ymax!~vy* ~ymax!12V~ymax!v* ~ymax!

1p* ~ymax!!2uy~ymax!u* ~ymax!

2wy~ymax!w* ~ymax!#1Ex0

xf@uy~0!u* ~x,0!

1vy~0!v* ~0!1wy~0!w* ~0!2p~0!v* ~0!#dx, ~A6!

where the boundary conditions~2! have been used. Expression ~A6! vanishes if the boundary conditions~21! are en-forced. Thus, ifg satisfies Eq.~20! with the wall-normalboundary conditions~21!, expression~A5! reduces to

E0

ymaxfT~xf !A

T~xf !g~xf !dy5E0

ymaxfT~x0!AT~x0!g~x0!dy,

which is equivalent to

E0

ymaxuT~xf !A

T~xf !g~xf !dy5E0

ymaxuT~x0!AT~x0!g~x0!dy,

~A7!

since the last column ofA is zero.Imposing the initial conditionAT(xf)g(xf)5MC—an

initial condition atx5xf is appropriate by the parabolic nature of the adjoint equations—expression~A7! becomes

E0

ymaxuT~xf !MCdy5E

0

ymaxu~x0!TMM 21AT~x0!g~x0!dy.

~A8!

Comparing expressions~A8! and ~A3!, we find

F5M21AT~x0!g~x0!. ~A9!

Thus, the action of the operatorA* on an elementC isequal to the vectorF defined in~A9!. The components ofCis given in expression~23!.

2. The adjoint operator Be*

To derive an expression forBe* , we need to calculateadjoints of (bU)21C 1 and C 2 . We approximatew in ex-pression~9! by

we5~bU !21ReC 1u1C 2v, ~A10!

whereC 1 and C 2 are defined as in~24! and the linear op-erator Re is a smoothing approximator with the followinproperties. It is symmetric, positive definite and satisfi


128.143.23.241 On: Mon,

y

s

lime→0 Rez5z @all of this in L2(0,ymax)#. Moreover,Re isrequired to return twice differentiable functions satisfyinlimy→0 Rez5 limy→ymax

Rez50 for each square-integrablfunction z5z(y) and eache.0. The smoothness and thhomogeneous boundary conditions will allow integrationparts with vanishing boundary terms when deriving thejoints.

A very natural choice ofRe is Rez5ze whereze is thesolution to the elliptic equation

ze2eze95z for yP~0,ymax!, ze50 at y50,ymax,

with e.0. However, since we use spectral collocation,may use an even simpler approximator in the discrete casis described in Appendix B after we have introduced tappropriate notation.

Using the approximation ofw in ~A10!, we may definethe following approximation ofB:

Be5S I 0

0 I

~bU !21ReC 1 C 2

D . ~A11!

To compute an expression for Be* , let C5(c1(y),c2(y),c3(y)) be an arbitrary, vector function ofywith appropriate integrability properties.58 We have, by thedefinition of the adjoint

~C,Beq!5~Be* C,q!. ~A12!

Expanding the left-hand side of Eq.~A12! in our‘‘disturbance-energy’’ inner product and applying integrtion by parts yield

~C,Beu!5E0

ymaxRe c1u dy1E

0

ymaxc2v dy

1E0

ymax~c3~bU !21ReC 1u1c3C 2v !dy

5E0

ymaxu~Re c11C 1* Rec3~bU !21!dy

1E0

ymaxv~c21C 2* c3!dy, ~A13!

whereC 1* andC 2* are defined by

C 1* m5~b22Vy!m2~Vm!y2myy ,~A14!

C 2* n5b21S Uy

Un1nyD .

Note that we use the properties ofRe to make the boundaryterms vanish when performing integration by parts.

Recalling~A12!, we identify in ~A13!

Be* 5S Re 0 C 1* Re~bU !21

0 I C 2*D . ~A15!


11 Nov 2013 14:01:57

et

d

ndiio

a

-

-

y

ard-by-y

s.roxi-

l

ison-

c.ny

.

ty

-is-iza-

edrgy

za-A

s ofwsr-wnn

p-

nsas-

ob-os-f anotInthe

ose


This ar

APPENDIX B: THE NUMERICAL SCHEME

The forward equation~A4! is approximated in thestreamwise direction with a fully implicit finite-differencscheme. The scheme we use for most of the calculation isfollowing:

~Af !12~Af !05Dx@~B0f!11~B1fy!11~B2fyy!1#,

~B1!32~Af !n1122~Af !n1 1

2~Af !n21

5Dx@~B0f!n11

1~B1fy!n111~B2fyy!n11#

for n51,...,N21,

where the step length isDx andNDx5xf2x0 . Thus, exceptfor the first backward-Euler step, we use a second-orbackward-difference scheme.

For each streamwise position, we need to solve a odimensional boundary-value problem in the wall-normalrection. This is solved numerically using spectral collocatbased on Chebyshev polynomials.

Consider the truncated Chebyshev expansion

f~h!5 (k50

K

fkTk~h!,

where

Tk~h!5cos~k cos21~h!!

is the Chebyshev polynomial of degreek defined in the in-terval 21<h<1. We use as collocation points the extremof the Kth-order Chebyshev polynomialTK plus the endpoints of the interval,

h j5cosp j

K, j 50,1,...,K.

The Chebyshev interval21<h<1 is transformed to thecomputational domain 0<y<ymax using the conformal mapping in Hanifiet al.59 Any of the unknown functions in~B1!,sayun5un(y), may be approximated with

uKn ~y!5 (

k50

K

uknTk~y!,

where Tk(y)5Tk(h) with h°y being the conformal mapping mentioned above.

For each n, the Chebyshev coefficients fkn

5(ukn ,vk

n ,wkn ,pk

n), k50,...,K are determined by replacingfn

with fkn and requiring Eq.~B1! to hold at the collocation

points yk , k51,...,K21. The boundary conditions~2! areenforced by adding the equations

(k50

K

uknTk~0!5 (

k50

K

vknTk~0!5 (

k50

K

wknTk~0!50,

(k50

K

uknTk~ymax!5 (

k50

K

wknTk~ymax!5 (

k50

L

pknTk~ymax!50.


128.143.23.241 On: Mon,

he

er

e--n

In this way we obtain, at each time step, a 4(K11)-by-4(K11), dense linear system which is solved bGaussian elimination.

Next the adjoint equations~20! may be discretized simi-larly as the forward equations, using second-order backwdifference scheme in the streamwise direction and Cheshev polynomials in the wall-normal direction. In this wawe obtain thediscretized adjointequations which is the typeof discretizations used in the majority of our calculationHowever we have also tested other approaches to appmate problem~14! ~cf. below!.

The least-squares problem~26! is solved using spectracollocation methods. Bothue andve are approximated usingK11 Chebyshev coefficients, both equations of~26a! areenforced atyj , j 51,...,K21, and this system of equationscompleted with four equations enforcing the boundary cditions ~26b!. Also, for the least-squares problem~26! weneed an operatorRe with the properties described in SeV B. In the discrete case at hand, we may simply set anonzero value of the functions at the boundaries to zero

The disturbance energy~12! is approximated with

EK~uKN!5(

j 50

K

@Re W~yj !~ujN!21W~yj !~v j

N!2

1W~yj !~wjN!2#, ~B2!

whereujN , v j

N , wjN are the values of the computed veloci

disturbances atx5xf and at the~transformed! collocationpoint yj . The integral weightsW(yj ) are given by the for-mula in Hanifiet al.59

A distinctly different approach to discretization of problem ~14! than the one outlined above, is to start with a dcretization of the state equation and introduce a discrettion of the disturbance energy~12!, say the one in~B2!.From the discretization of the forward equations describabove and the definition of the discrete disturbance eneEK , the discrete counterpart to problem~14! is well definedwithout the need of introducing any independent discretition of the adjoint equations. The calculations of Appendixmay now be repeated toderive the discrete adjointequationand the discrete counterpart to the least-squares problemSec. V C. Note that this discrete adjoint equation folloautomatically from the choice of discretization to the foward equation and the disturbance energy. It is well-knoto researchers working with optimal control/desigproblems24,22,60 that this is the most consistent way of aproximating problem~14!. Both for finite-element and finite-difference approximations, computing the adjoint equatiobecomes essentially a matter of transposing the matricessociated with the discretizations. Some care has to beserved, though, especially with finite differences. Transping the matrix associated with a finite-difference stencil odifferential operator sometimes yields a stencil that isconsistent with the adjoint of the differential operator.fact, an inconsistency of this kind appears when derivingadjoint of the second-order streamwise discretization~B1!.Thus, when using this approach, it might be better to choother schemes than the one in~B1!. A good candidate is thescheme discussed in Lions and Glowinski.26 In contrast to
11 Nov 2013 14:01:57

riva

ne

umalo

ato

t-

ingisthth

iddill-n

ewuorer

ot tr

emhe

e-

ree,

.u

’ in

m

nd

id

o-

ar

ha-

l

es

h.,

in

,’’

ns

nel

smear

-

al

yity,2.

ch-m-

chhe

r

r

ryJ.

ry

am

,’’

nd6.of

of

e-

ery

ry’’ J.

nR-

e

.am


This ar

the case with finite element/difference discretizations, deing discrete adjoints is less natural for spectral approximtions using Chebychev polynomials, since the natural inproduct for this discretization is not theL2-like inner productof interest here.

The approach of using the derived, discrete adjoint eqtions together with first-order finite differences in the streawise direction and second-order finite differences in the wnormal direction were tested on a few cases for comparis

We also tried asemidiscrete adjoint61 equation ap-proach, that is, we derived the adjoint equations associwith the streamwise-only discretized forward equations~B1! and the disturbance energy~12!. The fact that the firststep in the forward equations of~B1! is treated differentlythan the rest of the steps will be reflected in the treatmenthe adjoint equations.62 Then, the forward and adjoint equations are both discretized in the wall-normal direction usChebyshev collocation. At least a partial motivation for thapproach is that the accuracy of the discretization instreamwise direction is much lower than the accuracy inwall-normal direction.

Both approaches, the discretized adjoint and the semcrete adjoint, will possibly introduce small errors in the ajoint map. This type of errors can be disastrous forconditioned optimization problems. However, the curreproblem appears to be extremely well conditioned—very fpower iterations are needed for convergence—so it islikely that the inconsistencies introduced are of any imptance. In fact, we found very small differences in convgence as well as results~up to ‘‘graphical accuracy’’ at least!between all the different approaches we tried. The mprobable reason for the robustness of the problem is thalargest eigenvalue appears to be well-separated from theof spectrum, which means that the maximization probl~14!, is very well conditioned, as opposed to many of ttypical problems of optimal control/design.

1P. S. Klebanoff, K. D. Tidstrom, and L. M. Sargent, ‘‘The thredimensional nature of boundary layer instability,’’ J. Fluid Mech.12, 1~1962!.

2J. M. Kendall, ‘‘Experimental study of disturbances produced in a ptransitional laminar boundary layer by weak free-stream turbulencAIAA Pap. 85-1695~1985!.

3A. N. Gulyaev, V. E. Kozlov, V. R. Kuznetsov, B. I. Mineev, and A. NSekundov, ‘‘Interaction of a laminar boundary layer with external turblence,’’ Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza5, 55 ~1989! ~in Rus-sian!; English transl. in Fluid Dyn.~USSR! 24, 700 ~1990!.

4P. H. Alfredsson and M. Matsubara, ‘‘Streaky structures in transition,’Proceedings of the Transitional Boundary Layers in Aeronautics, editedby R. A. W. M. Henkes and J. L. van Ingen, Royal Netherlands Acadeof Arts and Sciences~Elsevier, Amsterdam, 1996!, pp. 373–386.

5P. S. Klebanoff, ‘‘Effect of free-stream turbulence on the laminar bouary layer,’’ Bull. Am. Phys. Soc.10, 1323~1971!.

6M. V. Morkovin, ‘‘The many faces of transition,’’ inViscous Drag Re-duction, edited by C. S. Wells~Plenum, New York, 1969!.

7M. V. Morkovin, ‘‘Bypass transition to turbulence and research deserata,’’ NASA Conf. Publ.2386, 161 ~1984!.

8P. Bradshaw, ‘‘The effect of wind-tunnel screens on nominally twdimensional boundary layers,’’ J. Fluid Mech.22, 679 ~1965!.

9S. C. Crow, ‘‘The spanwise perturbation of two-dimensional boundlayers,’’ J. Fluid Mech.24, 153 ~1966!.

10T. Ellingsen and E. Palm, ‘‘Stability of linear flow,’’ Phys. Fluids18, 487~1975!.


128.143.23.241 On: Mon,

--r

a--l-n.

edf

of

ee

is--

t

n---

stheest

-’’

-

y

-

-

y

11M. T. Landahl, ‘‘Dynamics of boundary layer turbulence and the mecnism of drag reduction,’’ Phys. Fluids20, 55 ~1977!.

12M. T. Landahl, ‘‘A note on an algebraic instability of inviscid paralleshear flows,’’ J. Fluid Mech.98, 243 ~1980!.

13L. S. Hultgren and L. H. Gustavsson, ‘‘Algebraic growth of disturbancin a laminar boundary layer,’’ Phys. Fluids24, 1000~1981!.

14L. Boberg and U. Brosa, ‘‘Onset of turbulence in a pipe,’’ Z. NaturforscA: Phys. Sci.43, 697 ~1988!.

15L. H. Gustavsson, ‘‘Energy growth of three-dimensional disturbancesplane Poiseuille flow,’’ J. Fluid Mech.224, 241 ~1991!.

16B. F. Farrell, ‘‘Optimal excitation of perturbations in viscous shear flowPhys. Fluids31, 2093~1988!.

17K. M. Butler and B. F. Farrell, ‘‘Three-dimensional optimal perturbatioin viscous shear flow,’’ Phys. Fluids A4, 1637~1992!.

18S. C. Reddy and D. S. Henningson, ‘‘Energy growth in viscous chanflows,’’ J. Fluid Mech.252, 209 ~1993!.

19D. S. Henningson, A. Lundbladh, and A. V. Johansson, ‘‘A mechanifor bypass transition from localized disturbances in wall bounded shflows,’’ J. Fluid Mech.250, 169 ~1993!.

20L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, ‘‘Hydrodynamic stability without eigenvalues,’’ Science261, 578 ~1993!.

21J. L. Lions,Optimal Control of Systems Governed by Partial DifferentiEquations~Springer, New York, 1971!.

22M. Berggren, ‘‘Optimal control of time evolution systems: Controllabilitinvestigations and numerical algorithms,’’ Ph.D. thesis, Rice UniversHouston, TX, 1995.~Discrete adjoint equations are discussed in Chap.!

23M. Berggren, R. Glowinski, and J. L. Lions, ‘‘A computational approato controllability issues for flow-related models. II. Control of twodimensional, linear advection-diffusion and Stokes models,’’ Int. J. Coput. Fluid Dyn.6, 253 ~1996!.

24M. Berggren, R. Glowinski, and J. L. Lions, ‘‘A computational approato controllability issues for flow-related models. I. Pointwise control of tviscous Burgers equation,’’ Int. J. Comput. Fluid Dyn.7, 237 ~1996!.~Discrete adjoint equations are discussed in Rem. 2.4.!

25R. Glowinski and J. L. Lions, ‘‘Exact and approximate controllability fodistributed parameter systems. I,’’ Acta Numerica, 269~1994!.

26R. Glowinski and J. L. Lions, ‘‘Exact and approximate controllability fodistributed parameter systems. II,’’ Acta Numerica, 159~1996!.

27P. Luchini, ‘‘Reynolds-number-independent instability of the boundalayer over a flat plate surface. 2. Optimal perturbations,’’ submitted toFluid Mech.

28P. Luchini, ‘‘Reynolds-number-independent instability of the boundalayer over a flat plate surface,’’ J. Fluid Mech.327, 101 ~1996!.

29F. P. Bertolotti, ‘‘Response of the Blasius boundary layer to free-strevorticity,’’ Phys. Fluids9, 2286~1997!.

30J. M. Kendall, ‘‘Boundary layer receptivity to freestream turbulenceAIAA Pap., 90-1504~1990!.

31A. M. O. Smith and N. Gamberoni, ‘‘Transition, pressure gradient astability theory,’’ Tech. Rep. No. ES 26388, Douglas Aircraft Co., 195

32J. L. van Ingen, ‘‘A suggested semiempirical method for the calculationthe boundary layer transition region,’’ Tech. Rep. No. VTH-74, FacultyAerospace Eng., Delft University of Technology, 1956.

33L. M. Mack, ‘‘Transition prediction and linear stability theory,’’ AGARDConference Proceedings No. 224, NATO Paris, 1977~unpublished!, p.1-1.

34E. R. van Driest and C. B. Blumer, ‘‘Boundary layer transition: Frestream turbulence and pressure gradient effects,’’ AIAA J.1, 1303~1963!.

35J. Dunham, ‘‘Prediction of boundary-layer transition on turbomachinblades,’’ AGARD AG164, ~1972!.

36B. J. Abu-Ghannam and R. Shaw, ‘‘Natural transition of boundalayers—The effects of turbulence, pressure gradient, and flow history,Mech. Eng. Sci.22, 213 ~1980!.

37A. M. Savill, ‘‘A summary report on the COST ERCOFTAC transitioSIG project evaluating turbulence models for predicting transition,’’ ECOFTAC Bull. 24, 57 ~1995!.

38A. M. Savill, ‘‘One point closures applied to transition,’’ inTurbulenceand Transition Modeling, edited by~M. Hallback, D. S. Henningson, A.V. Johansson, and P. H. Alfredsson~Kluwer, Dordrecht, 1996!, pp. 233–268.

39K. J. A. Westin and R. A. W. M. Henkes, ‘‘Application of turbulencmodels to bypass transition,’’ J. Fluids Eng.119, 859 ~1997!.

40K. J. A. Westin, A. V. Boiko, B. G. B. Klingmann, V. V. Kozlov, and PH. Alfredsson, ‘‘Experiments in a boundary layer subject to free-stre


11 Nov 2013 14:01:57

ch

amTed--

en

erity

1.

tri

ndr-a

cesmalVA.s

is,lm,

,’’

e to

in

theintem

rgy

Seehe


This ar

turbulence. I. Boundary layer structure and receptivity,’’ J. Fluid Me281, 193 ~1994!.

41P. E. Roach and D. H. Brierley, ‘‘The influence of a turbulent free-streon zero pressure gradient transitional boundary layer development. I.cases T3A and T3B,’’ inNumerical Simulation of Unsteady Flows anTransition to Turbulence, edited by O. Pironneau, W. Rodi, I. L. Ryhming, A. M. Savill, and T. V. Truong~Cambridge University Press, Cambridge, 1992!, pp. 303–316.

42M. Matsubara~private communications! ~unpublished data received fromexperiments conducted in 1995 in the MTL-windtunnel at the Departmof Mechanics at the Royal Institute of Technology in Sweden.!

43Z. Y. Yang and P. R. Voke, ‘‘Numerical simulation of transition undturbulence,’’ Tech. Rep. No. ME-FD/91.01, Dept. Mech. Eng., Universof Surrey, 1991.

44J. M. Floryan and W. S. Saric, ‘‘Stability of Go¨rtler vortices in boundarylayers with suction,’’ AIAA Pap., 79-1479~1979!.

45J. M. Floryan and W. S. Saric, ‘‘Stability of Go¨rtler vortices in boundarylayers,’’ AIAA J. 20, 316 ~1982!.

46P. Hall, ‘‘The linear development of Go¨rtler vortices in growing boundarylayers,’’ J. Fluid Mech.130, 41 ~1983!.

47M. D. Van Dyke, Perturbation Methods in Fluid Mechanics~Parabolic,Stanford, 1975!.

48M. D. Van Dyke, ‘‘Higher approximations in boundary-layer theory.General analysis,’’ J. Fluid Mech.14, 161 ~1962!.

49We assume that all operators involved are bounded in the chosen me50The generalized singular value decomposition~GSVD! for matrices is

discussed in Sec. 8.7.3 of G. H. Golub and C. F. Van Loan,Matrix Com-putations, 2nd ed. ~John Hopkins University Press, Baltimore, 1989!.More on the role of left and right singular vectors for SVD can be fouin M. T. Chu, R. E. Funderlic, and G. H. Golub, ‘‘On a variational fomulation of the generalized singular value decomposition,’’ SIAM J. Mtrix Anal. Appl. 18, 1082~1997!.

51A. W. Naylor and G. R. Sell,Linear Operator Theory in Engineering


128.143.23.241 On: Mon,

.

st

t

cs.

-

Science~Springer, New York, 1982!, Chap. 5, Sec. 22.52P. Andersson, M. Berggren, and D. S. Henningson, ‘‘Optimal disturban

in boundary layers,’’ presented at the A.F.O.S.R Workshop on OptiDesign and Control, 30 September 1997–3 October 1997, Arlington,To appear in proceedings published in Birkha¨user’s Progress in Systemand Control Theory series~1997!.

53D. Arnal, ‘‘Transition description and prediction,’’ inNumerical Simula-tion of Unsteady Flows and Transition to Turbulence, edited by O. Piron-neau, W. Rodi, I. L. Ryhming, A. M. Savill, and T. V. Truong~CambridgeUniversity Press, Cambridge, 1992!, pp. 303–316.

54F. M. White,Viscous Fluid Flow~McGraw–Hill, New York, 1974!.55S. Berlin, ‘‘Oblique Waves in Boundary Layer Transition,’’ Ph.D. thes

Department of Mechanics, Royal Institute of Technology, StockhoSweden, 1998.

56P. R. Voke and Z. Y. Yang, ‘‘Numerical study of bypass transitionPhys. Fluids7, 2256~1995!.

57Note that the inner products on the left- and right-hand sides in~A1! maybe different, and that the definition of an adjoint operator is subordinatthe choice of inner products.

58c1 ,c2 ,U21c3PL2(0,ymax).59A. Hanifi, P. J. Schmid, and D. S. Henningson, ‘‘Transient growth

compressible boundary layer flow,’’ Phys. Fluids8, 826 ~1995!.60G. Chavent, ‘‘Identification of distributed parameter systems: About

output least square method, its implementation, and identifiability,’’Proceedings of the Fifth IFAC Symposium on Identification and SysParameter Estimationedited by R. Iserman~Pergamon, New York, 1979!,Vol. 1, pp. 85–97.

61With respect to the inner product inferred from the disturbance ene~B2!.

62The same effects are ubiquitous in unsteady optimal-control problems.Ref. 24 where a mixed explicit-implicit temporal discretization affects tfirst step in the adjoint equation.


11 Nov 2013 14:01:57

Optimal disturbances and bypass transition in boundary layers

Documents

Transcript of Optimal disturbances and bypass transition in boundary layers