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Science Arts & Métiers (SAM)

is an open access repository that collects the work of Arts et Métiers Institute of

Technology researchers and makes it freely available over the web where possible.

This is an author-deposited version published in: https://sam.ensam.eu

Handle ID: .http://hdl.handle.net/10985/17986

To cite this version :

B. BUGEAT, Jean-Camille CHASSAING, Jean-Christophe ROBINET, P. SAGAUT - 3D global

optimal forcing and response of the supersonic boundary layer - Journal of Computational

Physics - Vol. 398, p.108888 - 2019

Any correspondence concerning this service should be sent to the repository

Administrator : archiveouverte@ensam.eu

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Journal

of

Computational

Physics

3D

global

optimal

forcing

and

response

of

the

supersonic

boundary

layer

B. Bugeat

a

,

,

J.-C. Chassaing

a

,

J.-C. Robinet

b

,

P. Sagaut

c

aSorbonneUniversité,CNRS,InstitutJeanLeRondd’Alembert,F-75005Paris,France bDynFluidLaboratory- ArtsetMétiers- 151,Bd.del’Hôpital,75013,Paris,France cAixMarseilleUniv.,CNRS,CentraleMarseille,M2P2UMR7340,13451,Marseille,France

a

b

s

t

r

a

c

t

Keywords: Optimalforcing Globalresolvent Convectiveinstability Non-modalinstability Compressibleboundarylayer

3D optimal forcing and response of a 2D supersonic boundary layer are obtained by computing the largest singular value and the associated singular vectors of the global resolvent matrix. This approach allows to take into account both convective-type and component-typenon-normalities responsiblefor thenon-modalgrowth ofperturbations innoise selectiveamplifierflows.It ismoreoverafullynon-parallelapproachthatdoes not requireanyparticular assumptions onthe baseflow.Thenumerical methodisbased on theexplicit calculation ofthe Jacobian matrix proposed byMettot et al. [1] for2D perturbations.ThisstrategyusesthenumericalresidualofthecompressibleNavier-Stokes equationsimportedfromafinite-volumesolverthatisthenlinearisedemployingafinite differencemethod.Extensionto3Dperturbations,whichareexpandedintomodesofwave number,ishereproposedbydecomposingtheJacobianmatrixaccordingtothedirectionof thederivativescontainedinitscoefficients.ValidationisperformedonaBlasiusboundary layer and a supersonicboundary layer, in comparison respectively to global and local results.ApplicationofthemethodtoaboundarylayeratM=4.5 recoversthreeregions ofreceptivityinthefrequency-transversewavenumberspace.Finally,theenergygrowth ofeachoptimalresponseisstudiedanddiscussed.

1. Introduction

Dependingontheirdynamics,openflowscanbedividedintooscillator andnoiseselectiveamplifiers [2].Whereasthefirst oneshaveanintrinsicdynamicsrelatedtothephysicalparametersofthebaseflow,thesecondonesonlyamplify perturba-tionsinspecificrangesoffrequencies,whichgrowinspaceandadvecteddownstream.Intermsofstabilityanalysis,these considerations lead to distinguish absolute from convective instabilities. Local stability analysis [3] have extensively been employed to studythe dynamicsofvarious open flows (boundary layer [4], wakes [5], jet flows [6], etc.). Thisapproach allows,inparticular, todiscriminateabsoluteandconvective instabilitiesby computingthe growthrateofzerogroup ve-locitywaves[7].Theassumptionofa(weakly)parallelbaseflowishoweverrequiredinordertoexpandperturbationsinto Fourier-Laplacemodesalongthestreamwisedirection.

*

Correspondingauthor.

E-mailaddresses:benjamin.bugeat@dalembert.upmc.fr(B. Bugeat),jean-camille.chassaing@sorbonne-universite.fr(J.-C. Chassaing), jean-christophe.robinet@ensam.eu (J.-C. Robinet),pierre.sagaut@univ-amu.fr (P. Sagaut).

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Nomenclature

M Machnumber P r Prandtlnumber Re Reynoldsnumber x Streamwisedirection y Normaltothewalldirection

z Transverse direction, supposed as homoge-neous

R

ResidualofNavier-Stokesequations

F FluxofNavier-Stokesequationsalong x-direction

G FluxofNavier-Stokesequationsalong y-direction

H FluxofNavier-Stokesequationsalong z-direction

q Vectorofconservativevariables

(

ρ

,

ρ

u

,

ρ

v

,

ρ

E

)

T

u Streamwisevelocity v Normalvelocity w Transversevelocity

ρ

Density c Speedofsound E Totalenergy e Internalenergy p Pressure T Temperature



M RelativeMachnumber

η

Dynamicviscosity

γ

Heatcapacityratio

κ

Thermalconductivity cp Heatcapacity

Far-fieldquantities

α

r Streamwisewavenumber

β

Transversewavenumber

ω

Angularfrequency

Anglebetweenthewavevectorofthe pertur-bationandthebaseflowdirection

Phasevelocity

μ

Optimalgain



f Forcing vector (in particular, optimal forcing vector)



q Perturbationsvector(inparticular,optimal re-sponsevector) fx Streamwiseforcing fy Normalforcing fz Transverseforcing

J

Jacobianmatrix

R

Resolventmatrix

QE Normmatrixassociatedwiththeenergyofthe

perturbations

QF Normmatrixassociatedwiththeforcingfield

dChu Chu’senergydensityprofile

dF Forcingdensityprofile

yChum OrdinatewhereChu’senergyismaximum yKm Ordinate where the kinetic energy is

maxi-mum

δ

∗ Boundary layer compressible displacement thickness

Blasiuslength

η

x

/

ρ

u

Focusing on theconvectively unstable compressibleboundarylayer, first stability computationswere based ona local approach [8–10]. Along withtheoretical developments[11], these seminalstudies established the main features of com-pressible instabilities,especially notingtheirinviscid nature causedby theexistence ofageneralised inflectionpoint and the prevailinggrowthof3D perturbations(Squiretheorem[12] doesnot holdforcompressibleflows[9]).Later,local sta-bilityanalysisallowed tosuggesttheexistence ofanadditionalunstable mode[13] (generally referred toassecondmode, orMackmode)inthecaseofsufficientlyhighsupersonicMachnumbers(M

3

.

8),soonconfirmedbyexperimentalwork [14,15]. Afterwards,moresophisticated localstability analysistakinginto accountthe weaknon-parallel effectsproduced moreaccurateresults[16,17].FollowingtheworkofFarrell [18] forincompressibleflows,severallocalanalysisthenfocused oncomputingnon-modalgrowthforcompressibleboundarylayer.Optimalgrowthinatemporalformulationwasfirst pro-posedbyHanifiet al. [19] whowereabletoobservethenon-modalgrowthofcompressiblestreaks.Aspatialversionofthis analysis wassuggested byTuminandReshotko [20], afterwards improvedby consideringnon-parallel effects [21,22] and 3Dbaseflows[23].TheseapproacheswerecoupledwithaPSEmethod[24],resultinginamoregeneralframeworktostudy non-modalgrowthinweaklynon-parallelflows[25,26].However,theseapproachescannotbeconsideredasuniversalasit doesnotallowtostudyfullynon-parallelflows.

Withtheincreaseofcomputationalresources,globalstabilityanalysis(inthesenseofTheofilis [27])becameaffordable. In thisframework,thestreamwise directionissolvedasan eigen-directionwhich authorisetoconsiderfullynon-parallel baseflows. Itoffers arelevanttool tostudyglobally unstableflows suchasthe bifurcationsoccurringin cavityflows[28] and shockwave/boundary layerinteractions [29] or theonset of thetransonic buffeton an airfoil[30]).Global stability analysisishowevernot suitedtodescribethedynamicsofconvectively unstableflows,whichareglobally stable. Instead, characterisingtheresponseoftheseflowssubjecttoanexternalforcingconstitutesamorerelevantanalysisasitisdirectly relatedtotheirnoiseamplifiernature[31].Inpractice,thisapproachisrelatedtotheresolventoperatorandanoptimisation framework isemployed tocompute the optimalforcingandresponse fordifferentfrequencies.Such an analysiswasfirst implemented foran incompressibleboundary layerby performing a projection ofthe responseonto a restrictednumber of global modes[32,33]. Another strategy was afterwards developed by Monokrousos et al. [34] usinga time-stepping technique associatedwithan adjoint-basedoptimisationmethod.Morerecently,Sipp andMarquet [35] suggestedtosolve a singular value problem associated withthe globalresolvent operator and showedthat, additionally, the left and right singular vectorsconstituted anorthonormalbasis ontowhichtheforcing andresponse fieldscouldbe expanded.Besides,

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itshouldbepointedoutthattheseoptimalresponseandforcingapproachesarenon-modalinnature.Indeed,theoptimal response resultingfroman optimisationproblemcan be seenasa superpositionofglobalmodes:both modalresonance and non-modal pseudo-resonance are thus taken into account [36]. These non-modal effects are a consequence of two typesofnon-normalities,associatedwiththe non-normalnatureofthelinearisedNavier-Stokesequations[37,38]. Onthe one hand, the convective-type non-normality (the term



ρ

U

· ∇



u in the linearised momentum equation), ubiquitousin convectively unstable flows, stems from theadvection of perturbations by the baseflow. It was furthermore observed to cause a spatial separation of the forcing andresponse fields, respectively upstream and downstream [35]. On the other hand,thecomponent-type orlift-up non-normality (the term



ρ

u

· ∇



U inthe linearised momentumequation)iscaused bythetransportofbaseflowmomentumbytheperturbations.Itwasshowntoproducecomponent-wisetransferofenergy betweentheforcingandresponsefieldsasinthecaseofthelift-upmechanism[39] ortheOrrmechanism[40].

Incompressibleflows,aglobalapproachtakingintoaccountnon-modaleffectswasfirstimplementedforjetflowsasan optimalgrowthproblemwhereanoptimalinitialconditionwaslookedfor[41,42].Globaloptimalforcingbasedonresolvent computationwas thendevelopedandappliedtothereceptivityofaturbulentshockwave/boundarylayerinteraction[43]. However,toourknowledge,noworkdealingwithnon-modalgrowthof3Dglobalperturbationsincompressibleflowshas beenpublishedtodate.Giventhat3Dconvectiveinstabilitiesareespeciallyprevailinginthisregime,anefficientnumerical frameworkappearstobemissingtotacklethisproblem.

Inthispaper,weproposeanumericalmethodtostudy3Dgloballinearperturbationsdevelopinginconvectively unsta-ble,fullynon-parallel,compressible2Dbaseflows.Thisapproachisbasedonthecomputation oftheoptimalgainandthe associatedoptimalforcing andresponse,whichisachievedbysolvingasingular valueproblemassociatedwiththeglobal resolventoperator[35].TheexplicitnumericalcomputationoftheJacobianmatrix- thefirststepofthenumericalmethod -usesthediscreteframeworkpresentedbyMettotet al. [1] whichishereextendedto3Dperturbation.Thispointconstitutes themainoriginalpointofthepresentworkandthemathematicalderivationwillbefullydetailed.Anapplicationtothe3D receptivityofthesupersonicboundarylayeratM

=

4

.

5 ispresentedinordertodemonstratethepotentialofthemethod.

The paper is organised as follows.Governing equations and the theoretical approach involved in optimal gain com-putations are introduced in section 2. The numerical framework is developed in section 3, especially emphasising the computationofthe3DJacobianmatrix(section3.3).Validationofthenumericalframeworkisgiveninsection4.Finally,a detailedstudyofthe3Dreceptivityofthesupersonicboundarylayerispresentedinsection5.

2. Theoreticalapproach

2.1. Governingequations

TheflowisgovernedbythecompressibleNavier-Stokesequations.Variablesaremadenon-dimensionalaccordingto



x

=

x L

,



t

=

t L

/

u

,



ρ

=

ρ

ρ

,



u

=

u u



p

=

p

ρ

u2 ∞

, 

T

=

T T

,

E

=

E u2 ∞

,



η

=

η

η

,

λ

=

λ

λ

∞ (1)

Inthefollowing,the

symbolwillbedroppedinordertolightennotations.The

symbolreferstofar-fieldquantities. Conservative variables q

=

[

ρ

,

ρ

u

,

ρE]

T are used, where

ρ

, u

= (

u

,

v

,

w

)

T and E respectively are the fluid density, the velocity vector and the total energy. T , p,

η

and

λ

respectively stand for temperature, pressure, dynamic viscosity and thermalconductivity.The referencelength L may refertothe compressibleboundarylayerthickness

δ

∗ orto theBlasius length

=

η

x

/

ρ

u.Non-dimensionalReynolds Re,MachM andPrandtlP r numbersareintroducedas

Re

=

ρ

uL

η

,

M

=

u c

,

P r

=

η

cp

λ

(2)

wherec isthespeedofsoundandcp istheheatcapacityoftheflow.The compressibleNavier-Stokesequationscanthen

bewrittenas

ρ

t

+ ∇ · (

ρ

u

)

=

0

,

(3a)

t

(

ρ

u

)

+ ∇ ·



ρ

u

u

+

pI

1 Re

τ



=

0

,

(3b)

t

(

ρ

E

)

+ ∇ ·



(

ρ

E

+

p

)

u

1 Re

τ

u

λ

P r Re

(

γ

1

)

M2

T



=

0 (3c)

Forathermallyandcaloricallyperfectgas,thenon-dimensionalpressure p andtotalenergyE canmoreoverbeexpressed accordingto p

=

1

γ

M2 ∞

ρ

T

,

E

=

p

ρ

(

γ

1

)

+

1 2u

·

u (4)

(5)

TheviscousstresstensorofaNewtonianfluidisgivenby

τ

=

η



∇ ⊗

u

+ (∇ ⊗

u

)

T

2 3

(∇ ·

u

)

I



(5)

ThedynamicviscosityisafunctionoftemperatureandisdescribedbySutherland’slaw[44]

η

(

T

)

=

T3/21

+

Ts

/

T

T

+

Ts

/

T

,

(6)

where Ts

=

110

.

4 K and T

=

288 K.The thermalconductivitycoefficientalso dependsonthetemperatureinthe same wayasdynamicviscositydoes(

λ(

T

)

η

(

T

)

[45]).Henceitleadsto

λ

=

η

asnon-dimensionalvariablesareused.

Inthefollowing,equations(3) canberecastinthedynamicalsystemform

q

t

=

R(

q

),

(7)

where

R

isthedifferentialnonlinearoperatorofNavier-Stokesequations. 2.2. Lineardynamics

Thisstudyaimstostudytheforceddynamicsof3Dperturbationsq

(

x

,

y

,

z

,

t

)

addedtoa2Dbaseflowq

(

x

,

y

)

.Thelatter is asolutionofthe steadynonlinearcompressibleNavier-Stokesequations(3).Considering smallamplitudeperturbations andintroducingaforcingtermf

(

x

,

y

,

z

,

t

)

,thegoverningequationoftheperturbationsislinearandwrittenas

q

t

=

J

q



+

f (8)

where

J = ∂R/∂

q

|q

is the Jacobian operator. The numerical computation of the Jacobian matrix will be described in section 3.3.In equation (8), theperturbations q cannow be seenasthe response oftheflow to theexternal forcing f. As z-directionissupposedtobehomogeneous,thesefieldsareexpandedintoFouriermodesofwavenumber

β

.Moreover, consideringaharmonicforcingatfrequency

ω

,thesequantitiesarefinallywrittenas

q

(

x

,

y

,

z

,

t

)

=

q

(

x

,

y

)

ei(βz+ωt)

+

c

.

c

.

(9)

f

(

x

,

y

,

z

,

t

)

=

f

(

x

,

y

)

ei(βz+ωt)

+

c

.

c

.

(10)

Equation(8) canthenberecastas



q

=

R



f (11)

where

R = (

I − J )

−1 is the globalresolvent operator (with

I

the identity operator) which dependsboth on the

forcing frequency

ω

andthetransversewavenumber

β

.Forglobally stableflows, alleigenvaluesoftheJacobianoperator havea strictlynegative realpart.Thus,theresolventoperatoriswell definedandallows tostudytheforceddynamicsof the flowby providinga relationbetweenresponseandforcing fields.Optimal gain isnowintroduced anddefinedasthe maximumratiobetweentheenergyoftheperturbationsandtheforcing.Formally,itsexpressionreads

μ

2

=

sup

f

||

q

||

2E

||

f

||

2F (12)

wheretheenergynorms

||.||

E and

||.||

F andthenumericalcomputationofequation(12) willbedescribedinsection3.5.

3. Numericalstrategy

3.1. CompressibleNavier-Stokessolver

The baseflow is computed by means of a finite volume CFD solver as a steady solution of the nonlinear equations (3).Spatialdiscretisation ofconvective fluxesisperformedusingAUSM+ scheme[46] associatedwitha fifth-orderMUSCL extrapolation [47].Viscous fluxesatcellinterfacesare obtainedby asecond-order centeredfinite differencescheme.The unsteady equationsare marchedintimeuntilasteadystateisreached. Animplicitdualtime steppingmethodwithlocal time stepisused [48].Thissolvershowedsuccessfulresultsinshockwave/boundarylayerinteractioncomputations[47]. Inthepresentwork,boundarylayerbaseflowsarecomputedinarectangularnumericaldomain(Fig.1).Acartesianmesh is setwithageometricalprogression fromthewall.Boundary conditionsaregatheredinTable1.DirichletandNeumann conditionsareemployedasonlystationarysolutionsarecomputed.Besides,notethatanadiabaticflateplateisconsidered. Thelength Lx0 upstreamfromtheleadingedge(seeFig.1)issettozerowhensupersonicflowareconsidered.Insubsonic computations,thislengthissetsothatresultsdonotdependonitsvalue.IndependencefromtheheightLyofthedomain

is also checked in every baseflow and optimalgain computations.Validation of thissolver for the caseof a supersonic boundarylayeratM

=

4 isprovidedinsection4.1.

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Fig. 1. Numerical domain.

Table 1

Boundaryconditionsusedinbaseflowcomputations.

Boundary Supersonic conditions Subsonic conditions

1 u=1, v=0,ρ=1, p= 1 (γM2) u=1, v=0,ρ=1, ∂p ∂x=0 2 ∂u ∂y=0, ∂v ∂y=0, ∂ρ ∂y=0, p= 1 (γM2) ∂u ∂y=0, ∂v ∂y=0, ∂ρ ∂y=0, p= 1 (γM2) 3 ∂u ∂x=0, ∂v ∂x=0, ∂ρ ∂x=0, ∂p ∂x=0 ∂u ∂x=0, ∂v ∂x=0, ∂ρ ∂x=0, p= 1 (γM2) 4 u=0, v=0,∂ρ ∂y=0, ∂p ∂y=0 u=0, v=0, ∂ρ ∂y=0, ∂p ∂y=0 5 ∂u ∂y=0, v=0, ∂ρ ∂y=0, ∂p ∂y=0 ∂u ∂y=0, v=0, ∂ρ ∂y=0, ∂p ∂y=0 3.2. ComputationoftheJacobianmatrixfor2Dperturbations

Before presentingthenumericalstrategy tocompute theJacobian matrixfor3D perturbations insection 3.3, thecase of2D perturbations q

(

x

,

y

,

z

,

t

)

= 

q

(

x

,

y

)

eiωt

+

c

.

c.is firstconsidered. Indeed,the formeris actually an extension ofthe latterthatwas presentedbyMettot [49].Themethodisbasedonthelinearisationofthe2D discretisedequations.When dealingwithcompressibleflows,providedthatoneownsanonlinearCFDsolver,thismethodallowstobypassthetedious linearisationandthendiscretisationofthecompressibleNavier-Stokesequations,thusreducingtheriskoferrors.Moreover, ifonewantstocomputeadjointquantities,workinginadiscretisedframeworkmaybemoreconvenient[49].

Fromthediscretisationofthesystem(7),whosedimensionisN

∈ N

,theresidual

R

∈ R

N ofthe2D nonlinear Navier-Stokesequationsisusedtoperformafirst-orderapproximationoftheJacobianmatrixas

J

v

=

R(

q

+

ε

v

)

R(

q

)

ε

+

O

(

ε

)

(13)

Here,q

∈ R

N isthevectorofconservativevariablesassociatedwiththebaseflow,v

∈ R

N isan arbitraryvectorand

ε

∈ R

is anumerical parameter.The vector v is carefullychosen so that the coefficientsofthe matrix

J

can beconveniently recovered [49]. A simple approach consistsin settingto zero every component ofv except thek-component that isset to 1 (here, k

∈ [

1

,

N

]

). Then, equation (13) allows to get every coefficient of the k-column of the matrix

J

. Repeating thisprocedureN times, thewholematrixisrecovered. Moreover,itispossibletotakeadvantagefromtheblockdiagonal structureof

J

inordertospeed upthemethod.Asdescribed byMettot [49],thevector v canbefilledwithmorethan onecomponentequalto1.Thisleadstoreducethenumberofcallstoequation(13) byapproximatively100.Thisefficiency actually depends onthe order of the numericalscheme andon the proportion ofpoints inthe normal andstreamwise direction.Asanexample,computingthematrix

J

whenN

=

600000 usinga1500

×

100 meshandathirdorderaccurate scheme takesapproximatively5 minutes (on CPU:IntelXeon(R)CPUE5-2630v2@2.60GHz). As forthe choice of

ε

value, its orderofmagnitudehasto besmallenough sothat secondorderterms canbeneglected but needto behighenough toavoidnumericalround-offerrors.Ifthevectorv containsonlyonenon-zeroelement locatedatthek-component,Knoll andKeyes [50] suggesttochose

ε

=

b

(

1

+ |

qk

|)

,whereb isthesquarerootofthemachineprecision.Ifv containsmultiple

non-zero components, then an average of thisexpression can be used. However, in practice, setting a fixed value of

ε

between10−6 and10−8 wasobservedtobearobustchoice,probablythankstotheuseofnondimensionalquantities[50].

3.3. Extensionto3Dperturbations

ThemethodemployedtocomputeaJacobianmatrixinadiscretised-then-linearised frameworkpresentedintheprevious sectionisnowextended to3Dperturbations q

(

x

,

y

,

z

,

t

)

= 

q

(

x

,

y

)

ei(βz+ωt).Inthepresentwork,thebaseflowissupposed

tobehomogeneousinthez-directionsothat

∀(

x

,

y

)

V

,

q

(7)

where

V

isthe computationaldomain.The proposed approach liesinthe useof3D Navier-Stokes residual(providedby a finite-volume solver) toperform thefirst-order approximationin equation (13). However, the z-direction mustnow be treatedasaFourierdirection, whichforbidsthedirectuseofthisapproximationbecausetransversederivatives

∂/∂

z must beturnedintoi

β

terms.Inordertoapplyaspecialtreatmentinthisdirection,letusfirstwriteequation(3) withfluxesas

q

t

=

R(

q

)

= −

F

x

G

y

H

z (15)

whereeachfluxtermisnowseparatedintwopartsasfollows

F

=

F

Fνz (16)

G

=

G

Gνz (17)

H

=

H

Hνz (18)

The partwith subscript

ν

z includes every transverse derivatives

∂/∂

z, which are only ofviscous nature.The other part (superscript ) contains the remaining terms. Explicit expressions ofthese fluxes are givenin Appendix A. The Jacobian matrixisfinallyseparatedintothreedifferentmatricesas

J

=

J

F,G

+

J

H

+

J

νz (19) where

J

F,G

=

q





F

x

+

G

y



q (20)

J

H

=

q



H

z



q (21)

J

νz

=

q



Fνz

x

+

Gνz

y

+

Hνz

z



q (22)

Eachmatrixisthennumericallycomputedbyaspecificmethodthatisdetailedinthefollowingsubsections. 3.3.1. Computationof

J

F,G

No transverse derivative is involved in the computation of

J

F,G in equation (20). Therefore, equation (13) can be straightforwardly used by introducing

R

F,G

= −∂

F

/∂

x

− ∂

G

/∂

y.The Jacobian matrix

J

F,G is thus computedaccording to

J

F,Gv

=

R

 F,G

(

q

+

ε

v

)

R

F,G

(

q

)

ε

(23) 3.3.2. Computationof

J

H

Thecomputationof

J

H isachievedbyfirstlyreconsideringthefollowinglinearisation

J

Hq

= −

H

(

q

+

q

)

z

+

H

(

q

)

z (24)

whereafirstorderexpansionofH

(

q

+

q

)

inthefirstrighthandtermgives

J

Hq

= −

z

H

(

q

)

+

H 

q

q q

+

H

(

q

)

z (25)

Finally,thefollowingexpressionremains

J

Hq

= −

z

H

q

q q

(26)

Here,

H

/∂

q

qdoesnotdependonz asassumedinequation(14).Andsince

q

/∂

z

=

i

β

q,equation(26) nowreads

J

Hq

= −

i

β

H

q

q q (27)

(8)

J

H

= −

i

β

H

q

q (28)

Again,thenumericalcomputationof

H

/∂

q

qisbasedonafirst-orderfinitedifferenceapproximation.However,notethat the numericalflux H is hereused insteadof theflux divergencethat was previously needed inequation (13) and (23). Hence,thematrix

J

H isnumericallybuiltaccordingto

J

Hv

= −

i

β

H

(

q

+

ε

v

)

H

(

q

)

ε

(29)

3.3.3. Computationof

J

νz

Becauseeveryfluxtermnowcontains atransversederivative, aparticularcareisrequiredtocomputethematrix

J

νz. Itisnotsimplypossibletoreplaceeach

∂/∂

z intoi

β

withinthefluxesz,z andz.Indeed,spurious non-zeroterms

associatedwiththebaseflowderivativealongz wouldappear,thusviolatingtheassumptionsmadeinequations(14).Toget aroundthisproblem,itcanbeobservedthatbecauseofequations(14),theperturbations



q onlyappearsunderz-derivatives inthefinallinearisedequations.Forexample,introducingthelinearisationoperator

L

,linearisingthefourthcomponentof thevectorz (seeAppendixA)reads

L

(

η

Re

u

z

)

=

η

Re



u

z

=

i

β

η

Re



u (30)

Anticipatingthefinalresultoflinearisation,weheresuggesttomodifythefluxesz,zandzinto



z,



z and



z in

whicheveryfactorinfrontofaz-derivativeissettothebaseflowvalueandeach

∂/∂

z isturnedintoi

β

(whichwillappear asafactorinthefinal expressionoftheJacobianmatrix).Toillustratethisprocedure,letustake theexampleofz (see

AppendixBforexhaustiveexpressionsofthemodifiedfluxes)

Fνz

=

0

2 3 η Re∂ w ∂z 0 η Re ∂u ∂z η Re



u23wz

+

w∂uz



→ 

Fνz

=

0

2 3 η Rew 0 η Reu η Re



u23w

+

wu



(31)

Thematrix

J

νz isfinallycomputedusingtheapproachespresentedinsection3.3.1for



z and



zandinsection3.3.2for



z.Introducing



z

=

∂

Fνz

x

+

∂

Gνz

y

,

(32)

thefinalpracticalexpressiontocompute

J

νz reads

J

νzv

=

i

β



z

(

q

+

ε

v

)

− 

z

(

q

)

ε

− β

2



Hνz

(

q

+

ε

v

)

− 

Hνz

(

q

)

ε

(33)

3.4. Conclusionabout3Dextension

Itshouldfinallybenotedthatthenumericalimplementationofthe3Dextensionfromthe2Dmethodisstraightforward, providedthat3DNavier-Stokesresidualisavailable.Indeed,3Dextensionneedsminormodificationsofthenumericalfluxes butrecoveringthecoefficientsoftheJacobianmatrix,whichconstitutesthemaineffortofnumericalimplementation(see section 3.2), isachievedby directlyusing the2D computation routine.Here isa summary ofthemain stepsto compute the3DJacobianmatrixfroman alreadyimplemented2D method(asidefromredefinitionsofnumericalarraystotakeinto accounttheadditionaltransversemomentumcomponent).

1. Import3DNavier-Stokesresidualroutines.

2. Modifythefluxeswithintheseroutines accordingtoAppendixB. 3. Usingequations(23),(29) and(33),compute

J

v

=

J

F,Gv

+

J

Hv

+

J

νzv (34)

4. Recoverthecoefficientsusingthemethodpresentedinsection3.2.

Itshouldbefinallypointedoutthatadditionalcomputationalcosts,intermsofrandom-accessmemory,iskept afford-ablecomparetothe2D method(inpractice,a50% increase tendstobe observed).Indeed,introducingafifthcomponent

(9)

(transversevelocity)automaticallyincreasesthestorage,butbecauseaFourierexpansionisusedinthetransversedirection, no discretisationisperformedinthisdirection.Conversely, implementingafully3Dmethod(necessary for3Dbaseflows) woulddramaticallyincrease computationscosts. Indeed,thenumberofcoefficientsintheJacobianmatrixwouldincrease linearly withthenumberof points Nz in thetransverse direction. Worst,the storageneededtosolve the linearsystems

that willbe introduced in section 3.5would scaleas N2

z.If itseems today possibleto achieve such computations using

largeclustersandlimitednumbersofpoints,themethodpresentedinthispaperoffersasignificantlymoreefficientwayto compute3Dperturbationswhenconsidering2Dbaseflows.

3.5. Optimalforcingandresponsecomputation

Thissectionpresentsthenumericalapproachproposedby[35] tocomputetheoptimalgaindefinedinequation(12) and theassociatedoptimalforcingandresponsefields



f

(

x

,

y

)

and



q

(

x

,

y

)

.Theresolventmatrix

R

isinvolvedinthisproblemand firstrequiresthecomputationoftheJacobianmatrixintroducedintheprevioussection.Notethattheexplicitconstruction of

R

willnotberequiredsincelinearsystemsinvolving

R

−1 willbesolved.

Equation(12) isanoptimisationproblemonthefunction

μ

(

f

)

whichdependsonenergynorms.Thesenormsare asso-ciatedwithdiscretescalarproductsthatcanbeexpressedbynormmatricesas

||

q

||

2E

=<

q

,



q

>

E

=

qQE



q (35)

||

f

||

2F

=<

f

,

f

>

F

=

fQF



f (36)

where ∗ stands forthetransconjugateoperator. Thechoice ofQE in equation(35) isrelatedtotheenergyofthe

pertur-bations thatone wants to optimise.Forincompressibleflows, considering thekinetic energy appearsasa naturalchoice [18].Forcompressibleflows,Chu’senergynorm[51] (alsocalledMack’snorm)iswidelyusedtostudythenon-modal be-haviourofcompressibleflowdynamics[19,20,25].Itcontainsthekineticenergyoftheperturbationsandastrictlypositive termrelativetothermodynamicalperturbations.Assuch,Chu’senergyisnecessarilygreaterthanorequaltokineticenergy. ExplicitexpressionofthenormmatrixQE

=

QChuassociatedwithChu’senergyandwrittenforconservativevariablesis

de-rivedhereafter,whereasQE

=

QEK,associatedwiththekineticenergy,isdetailedinAppendixC.Startingofffromprimitive

variables,Chu’sdisturbancesenergyEChu reads[51]

EChu

=

1 2



V



ρ

|

u

|

2

+

T

ργ

M2

(

ρ



)

2

+

ρ

(

γ

1

)

γ

M2T

(

T 

)

2



d



(37)

ThenormmatrixQChu isthendefinedsuchthat



V

q∗QChuq

=

EChu (38)

whereq isthestatevectorofperturbationswrittenasconservativevariables.Letusrecallthatphysicalvariablesarehere madedimensionlessfollowingequations(1). Then,primitivevariablescanbetranslatedintoconservativevariablesby the followingrelations ui

=

1

ρ

((

ρ

ui

)



u i

ρ



)

(39) T

=

(

γ

1

)

γ

M 2

ρ



(

1 2

|

u

|

2

e

)

ρ



u i

(

ρ

ui

)



+ (

ρ

E

)



(40)

wheree istheinternalenergyoftheflow.Twoparameters,associatedwithbaseflowquantities,arenowintroducedas

a1

=

(

γ

1

)

γ

M2

ρ

T (41) a2

=

(

12

|

u

|

2

e

)

ρ

(42)

Equation(37) cannowberecastwithconservativevariables.Identifyingthisequationwithequation(38) andsearchingthe matrix QChu assymmetrical,itscoefficientscanbeidentifiedas

(10)

QChu

=

1 2d



|

u

|

2

ρ

+

T

ργ

M2

+

a1a 2 2

u

(

1

+

a1a2

)

ρ

v

(

1

+

a1a2

)

ρ

0 a1a2

ρ

u

(

1

+

a1a2

)

ρ

1

ρ

+

u2a1

ρ

2 uva1

ρ

2 0

ua1

ρ

2

v

(

1

+

a1a2

)

ρ

uva1

ρ

2 1

ρ

+

v2a1

ρ

2 0

va1

ρ

2 0 0 0 1

ρ

0 a1a2

ρ

ua1

ρ

2

va1

ρ

2 0 a1

ρ

2

(43)

Numerically,afirst-orderintegrationoverthenumericaldomain



isperformed.Todoso,ablock diagonalmatrixisbuilt fromthematrixinequation(43),takingcareofsettingd



i,jandbaseflowvaluesforeachelementaryvolume.

The matrix QF in equation (36) is defined from the canonical scalar product

||

f

||

2F

=







f



fd



(see Appendix C for

explicitexpression). Thismatrixispositivedefinite.It canbe notedthat

||

f

||

2F is nothomogeneousto anenergy,butitis ratheramathematicalnormwhichischoseninordertoreflecttheenergyinputoftheexternalforcingfield0.1Hencethe

optimalgain is not strictly definedas aratio oftwo energies, andits absolute value hasno physicalmeaning. However, detecting maximumvaluesof

μ

relative todifferentwave numbersandfrequenciesstill allowstofindout resonanceand pseudo-resonance of the flow response to a harmonic forcing. As such, it remains a relevant tool to analyse the linear dynamicsofanoiseselectiveamplifierflow.

Itispossibletoconstraintheforcingfieldbothtoalocalisedregionoftheflowandtospecificcomponentsbyintroducing amatrixP suchthat



f

=

P



fs.Inthispaper,wechoosetoonlyconsiderthemomentumcomponentsoftheforcing fieldin ordertosimplifytheinterpretationoftheforcingnorm.Asimilarchoicehasbeenmadeforexampleby Sartoret al. [43]. InthiscasethematrixP has,beforediscretisation,thefollowingexpression

P

=

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0

(44)

IfN isthesizeofthevector



f,then



fs hasasizeM withM

N andthematrixP hasasize N

×

M. Therelationbetween forcingandresponsefieldsthenreads



q

=

R

P



fs (45)

Introducingenergynormmatrices,equation(12) cannowberecastas

μ

2

=

sup fs

(

R

P



fs

)

QE

(

R

P



fs

)

(

P



fs

)

QF

(

P



fs

)

=

sup fs

(

P



fs

)

R

QE

R

(

P



fs

)

(

P



fs

)

QF

(

P



fs

)

(46)

Equation(46) canbeseenasageneralizedRayleighquotient[35] wheretheoptimalgain

μ

2 isthenthelargesteigenvalue

and



fstheassociatedeigenvectoroftheHermitianeigenvalueproblem

(

R

QE

R

P

)

fs

=

μ

2

(

QFP

)

fs (47)

BecauseQF isinvertibleandPP

=

I,equation(47) reducesto

(

PQF−1

R

QE

R

P

)







A



fs

=

μ

2



fs (48)

Tosolveequation(48),onlytheinverseoftheresolventmatrix

R

,whichcanconvenientlybecomputedfromtheJacobian matrix,isactuallyrequired.Indeed,theeigenvalueproblem(48) canbesolvedbyamatrix-freealgorithmbasedonaKrylov method[52].Asetofvectors

(

v0

,

Av0

,

A2v0

,

...)

,whichcomposestheKrylovsubspaceassociatedwiththematrixA,needs

to be computed. Starting froman arbitraryvector v0,each Krylov vector vi is computedfrom theprevious one vi−1 by

solvingequationvi

=

Avi−1.Thiscomputationrequirestosolvetwolinearsystemsinvolving

R

−1and

(R

)

−1.Thedetailed

stepsofthisprocedureare developedinAlgorithm1.Finally,theoptimalresponsecanbe recoveredbysolvingthelinear system(45).

1 Notethatdefininganinputmechanicalworkwouldbemorerigorous,but,toourknowledge,thiscanonlybeachievedaposteriorioftheoptimalgain

(11)

Algorithm1ComputationoftheKrylovvectorvi associatedwiththematrixA fromthevectorvi−1. 1. Computingthematrix-vectorproduct:

t

1=Pvi−1

2. Solvingofthelinearsystem:R−1t 2=t1

3. Computingthematrix-vectorproduct:

t

3=QEt2

4. Solvingofthelinearsystem:(R)−1 t4=t3

5. Computingthematrix-vectorproduct:

v

i=PQ−1t4

OpenlibraryPETSc[53] interfacedwithMUMPS[54] isusedtosolvethelinearsystemsbyadirectsparseLUalgorithm. Thematrix-freeeigenvalueproblemissolvedusingSLEPclibrary[55] byaKrylov-Schuralgorithm[56].

3.6. Boundaryconditionsonperturbations

Table 2 gathers the Dirichet-Neumann conditionsthat are applied on perturbations at each boundary. Numerically, a matrix

B

isintroducedintheequations(8) suchthat

B

q

t

=

J

q



+

f (49)

The matrix

B

isdefinedasthe identitymatrix exceptthatdiagonal coefficientsare setto zeroatlinescorresponding to boundary points. Note that, in practice,the globalresolvent matrixis hence definedas

R =



B − J



−1

.In order to finally implementtheconditionsgiveninTable2,thecoefficientsoftheJacobianmatrixaredirectlysetwithoutusingthe proceduregivenbyequation(13).

Table 2

Boundaryconditionsappliedonperturbationsinoptimal gaincomputationsaccording tothenumerical domain showninFig.1. Boundary Conditions 1 u=0, v=0,ρ=0, p=0 2 u=0,∂v ∂y=0,ρ=0, p=0 3 ∂u ∂x=0, ∂v ∂x=0, ∂ρ ∂x =0, ∂p ∂x =0 4 u=0, v=0,∂ρy=0,py=0

4. Validationofthepresentmethod

In thissection,solvers presentedinsection 3 arevalidatedagainst datafromexisting studies.Solutionsfromthe CFD solver are compared to theself-similar solution ofthe compressibleboundary layer. Optimal gain computations are first validatedagainst 3D globalresultsfora Blasiusboundary layer.Afterwards,a validationagainst3D non-globalresultsfor asupersonicboundarylayerisperformedsincenoresultsfor3Dglobaloptimalperturbationsareknownforcompressible flows (as opposedto 3D globalstability resultsforcompressible flows,for examplepresentedby Hildebrandet al. [57]). Note that because threedifferent testcases havebeen considered,Mach numbers andReynolds numbersvary fromone casetoanotheraccordinglywiththeexistingdatafoundintheliterature.TheseconfigurationsaregatheredinTable3.

Table 3

Flowsstudiedinsection4tovalidateCFDandoptimalgainsolvers.Reynoldsnumbersaregivenaspresentedin theexistingstudies.

§ Validation purpose Reference Mach Reynolds at inflow/outflow 4.1 Non-linear solution [58] M=4 self similar solution 4.2 3D global perturbations [34] M=0.3 Reδ∗=1000/1836 4.3 3D non-global perturbations [21] M=3 Re =0/1000

4.1. CFDsolvervalidation

A computationofa supersonicboundary layerflowisperformedat M

=

4 in ordertoassessthebaseflowsolver pre-sented insection 3.1.The numericaldomainstarts atthe leading edgeof theassumedadiabaticflat plate. The Reynolds numberatoutflowissetto Rex

=

2

×

106.ThesteadynonlinearNavier-Stokesequations(3) aresolvedonameshof800

×

(12)

Fig. 2. Supersonic boundarylayerflowatM=4:streamwisevelocity(left)andtemperature(right)profilesagainstself-similarvariableyRex/x atdifferent x-location.ResultsfromÖzgenandKırcalı [58] areshowninblackcirclesymbols.

collapse,thusrecoveringtheexpectedself-similarcharacterofthisflow(Fig.2).Furthermore,theseprofilesareinverygood agreementwiththeresultsofÖzgenandKırcalı [58] obtainedbysolvingthecompressibleboundarylayerequations.

4.2. OptimalgainofBlasiusboundarylayer

This section provides a validation of the optimal gain solver developed for 3D globalperturbations (section 3.3 and

3.5). Comparison to theglobal resultsof Monokrousos et al. [34] is proposed. Theseresults were obtained by means of anincompressiblesolverbasedonan adjointformulation,usingatime-steppingmethodandafringe zonetechnique.The physicalconfigurationherebystudied isaBlasius boundarylayerdevelopingover aflatplate. Reynoldsnumberatinflow (resp.outflow) isset at Reδ

=

1000 (resp. Reδ

=

1836). Usingthe displacementthickness

δ

0∗ atinflowasthe reference

lengthscale,thenumericaldomainspansover

[

0

,

800

] × [

0

,

30

]

.

Asthenumericalframeworkofthepresentpaperisforcompressibleflow,theMachnumberissetto M

=

0

.

3 inorder togetasolutionclosetotheincompressibleresults.First,thebaseflowcomputationisperformedusingtheCFDsolver pre-sentedinsection3.1.Subsonicboundaryconditionsareused(seeTable1)andLx0 issetto200

δ

0∗.Anewnumericaldomain,

whichis hereafterused foroptimalgaincomputations,isthen obtainedby truncatingthe fieldscomputedfromtheCFD solver sothat they matchthe domain described inthe aboveparagraph. Optimal gain computationsare then performed withtheforcingfrequencysetto

ω

=

0 whereasthewavenumber

β

variesover

[

0

,

1

.

2

]

.Kineticenergyisusedasthenorm matrixinequation (12). NotethatMonokrousoset al. [34] do nottake intoaccount the1

/

2 factor inthekinetic energy; therefore,theseresultsarerescaledtomatchthekineticenergydefinitionherebyused.Thepresentcomputationproduces slightlyloweroptimalgainvalues(around5%)comparedtotheresultsofMonokrousoset al. [34] (Fig.3).Nocompressible effectormeshinfluencehavebeenfoundtoaccountforthisdiscrepancy.Itissuggestedthatthissmalldiscrepancymight stemfromthefundamentallydifferentapproachesusedbetweenthereferenceandthepresentwork.Indeed,Monokrousos et al. [34] computed the time evolution of linearised perturbations on a greater numerical domain x

∈ [

0

,

1000

]

, where perturbations are damped for x

>

800 by means of a fringe zone technique. The authors thereby suggested to consider

Fig. 3. Optimal gainasafunctionofthewavenumberβatω=0 foranincompressibleboundarylayer.Bluesquares:presentresults,atM=0.3.Black line:resultsofMonokrousoset al. [34] computedforglobal3DperturbationsinaBlasiusboundarylayer,usingatime-steppingmethodassociatedwitha fringezonetechnique.RedcrossesareobtainedbyrescalingtheoptimalgainvaluesofthebluesquareswiththoseofMonokrousoset al. [34].

(13)

Fig. 4. Real partofoptimalforcingcomponent fz (left)andoptimalresponsestreamwisevelocityu(right)atβ=0.6 andω=0 foraboundarylayerat M=0.3.Iso-surfacesat−10%and10%ofthemaximumabsolutevalueareshowninredandblue.Thenumericaldomainistruncatedinthey-direction toeasevisualisation.

Fig. 5. Real partofuoftheoptimalresponsecomputedfortwovaluesofR=xopt/xf.Here,theforcingfieldisconstrainedtobelocalisedbetweenthe

twoverticaldotlines.Inbothcases,theenergyoptimisationdomainoftheresponseislocatedbetweentheverticalsolidlineatxopt/ 0=1000.

x

∈ [

0

,

800

]

astheoptimisationdomain,whichweusedinthepresentcomputation.However,residualperturbationsmight persistforx

>

800 andcouldaccountforaslightlyhigheroptimalgainvaluethanacomputationperformedonanumerical domain strictly limitedto x

∈ [

0

,

800

]

.Nevertheless, rescaling the presentresults allows to observe a perfect agreement betweentheoptimalgainbehaviourstakenasafunctionof

β

.Inparticular,theoptimalwavenumber

β

=

0

.

6 isretrieved. Thisagreementisthemostsignificant:indeed,theabsolutevalueoftheoptimalgainhasnophysicalmeaning[35] contrary totheforcing frequencyorwavenumberatwhichtheoptimalgainismaximum.The 3DfieldsdepictedinFig.4 further-moresupportthevalidityofourcomputationsincethesamecounter-rotatingrollsandstreakstopologiesasMonokrousos et al. [34] arerespectivelyfoundastheoptimalforcingandresponsefieldsassociatedwiththemaximumoptimalgainvalue. 4.3. OptimalgainofasupersonicboundarylayeratM

=

3

Toourknowledge,noworkdealingwith3Dglobaloptimalforcingincompressibleflowshasbeenpublishedtothisdate. Inordertovalidatethemethodproposedinsection3.3- whichallowstocompute3Doptimalglobalperturbations,a com-parisonwiththe“non-global”resultsfromTuminandReshotko [21] obtainedforasupersonicboundarylayerisperformed. This methodliesonthe optimisationofan energeticratiobetweenperturbationprofiles attwo differentlocations, using Chu’senergynorm[51].Hence,thisapproachisaspatialequivalentofanoptimalinitialconditioncomputationwhose so-lutionisbasedontheparabolisedboundarylayerequationsbyassumingtheexpectedvelocity scalesofstreaks(u

O

(

1

)

etv

,

w

O

(

ε

)

[21]).

Inordertocomparetheseresultstotheglobalperturbationsapproach,wesuggest toperformacomputation inwhich theforcing fieldisconstrainedinaregionarounda locationxf (Fig.5) thatcorrespondstotheupstream locationusedby

TuminandReshotko [21].Anywhereoutsidethisregion,theforcingfieldisequaltozero.Thisisachievedbymeansofthe

P matrixintroducedinsection3.5.Moreover,theenergyoftheresponseisoptimisedinarestrictedregionatlocationxopt

(Fig.5)whichcorrespondstothedownstreamlocationusedbyTuminandReshotko [21].Thereby,therelationbetweenour globalapproachandthe“non-global”approachusedasareferenceisthefollowing:thecomputedforcingfieldsatxfplays

the role oftheoptimalcondition, theresponse fieldsatxopt isthe perturbationthat growsdownstream andtheoptimal

(14)

Fig. 6. Optimal gaincomputedat ω=0 withconstrainedforcingandresponsefieldsforthecompressibleboundarylayerat M=3,fortwovaluesof R=xopt/xf.Blackline:resultsfromTuminandReshotko [21].Referencelengthscaleis 0=ηxoptu∞.Squaresymbols:presentresults.

Table 4

Physicalandnumericalparametersofthebaseflow. Adiabatic flat plate

M=4.5 Rex(out)=2×106⇐⇒Re (out)=1414⇐⇒ Reδ(out)=11770 Rex(opt)=1.75×106⇐⇒Re (opt)=1323⇐⇒Reδ(opt)=Reδ0∗=11000 x/δ0∗∈ [0,182]et xopt∗0=159

y/δ∗0∈ [0,45](domain used for the computation of the baseflow) y/δ∗0∈ [0,9](domain used for the computation of the optimal gain)

Mesh: 1600×180

Thebaseflow computation isperformedwiththe Machnumberset atM

=

3 andtheReynoldsnumber, basedonthe Blasiuslengthscale

=

η

x

/

ρ

u,setat Re

=

1140 atoutflow.Thevalueofxoptissetattheabscissacorresponding

to Re

=

1000 and theBlasius lengthatthisspecificabscissa,

0

=



η

xopt

/

ρ

u,will beusedasthereferencelength scale.Twodifferentvaluesofxf willsuccessivelybeconsideredsuchthattheratioR

=

xf

/

xoptisequalto0

.

2 and0

.

4.The

numericaldomainspansover x

/

0

∈ [

0

,

1300

]

and y

/

0

∈ [

0

,

100

]

.Theregionsoverwhichtheforcing andresponse fields

areconstrainedspanover



x

/

0

=

40 inthestreamwisedirectionbutarenotrestrictedinthenormaldirection.Resultsof

optimalgaincomputationsarecompared tothoseofTuminandReshotko [21] byrenormalisingtheoptimalgain. Indeed, the definitions of thesetwo quantities are different andtheir absolutevalue cannot be directly compared. A very good agreementisobservedforthetwo ratios R considered(Fig.6).The optimalresponsefieldsassociatedwiththemaximum optimalgain showsthatthegrowthofstreaksstartsfromtheforcing location(Fig.5).Theconvectivelyunstablenatureof thesestructuresisobservedastheirgrowthgoesondownstreamfromtheregionwheretheyareforced.

5. OptimalforcingandresponseofthesupersonicboundarylayeratM

=

4

.

5

5.1. Baseflow

Thebaseflow usedforoptimalgaincomputations (section5.2)ispresented inthissection.Aboundary layer develop-ing over an adiabaticflat plateis consideredat M

=

4

.

5,atwhich localstability analysis show that Mackmode reaches its maximumgrowthrate[59].Physical andnumericalparametersarereportedinTable4wherethe Reynoldsnumberis computedaccordingtodifferentreferencelengthscales.LocalstabilitystudiesusuallytakeBlasiuslength

=

η

x

/

ρ

u asreference,whichisassociatedwiththeplateabscissax accordingtoRe

=

Rex.Here,compressibledisplacement

thick-ness

δ

(

x

)

willbe considered.Asexpected, therelationbetween Re and Reδ∗ obtainedfromournumericalcomputation

isobservedtobelinearasaconsequenceofself-similarity(Fig.7).ThecorrespondinglocalMachnumberfieldisdepicted in Fig. 8. Notethat the domain over which Chu’s disturbances energyis integrated during optimalgain computation is definedoverx

<

xopt,where xoptis theabscissawhere Reδ

=

11000. Thecompressibledisplacementboundary

δ

0∗ atthis

abscissawillbeusedasthereferencelength.Itshouldbepointedoutthatthislengthappearsbothinthenon-dimensional forcing frequency

ω

andwave number

β

.Finally,the actual baseflowused foroptimalgain computation istruncatedin the y-direction forcomputationalsavings.ItisshowninAppendixDthat theresultsareindependentfromthischoice of domain.

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Fig. 7. Computed ReynoldsnumberReδ∗ basedonthecompressibledisplacementthicknessasafunctionoftheReynoldsnumberRe basedonBlasius lengthscale.a istheslopeofthelinearcurve,obtainedbylinearregression(r2>0.99).

Fig. 8. Local MachnumberfieldofthebaseflowatM=4.5.Optimisationdomainusedforoptimalgaincomputationislocatedupstreamfromthevertical dottedline.Notethatthenumericaldomainistruncatedinthey-directiontoeasethevisualisation.

5.2. Optimalgain

Resultsfromoptimalgain computationsfor3Dperturbationsdevelopingoverthebaseflowpresentedinsection5.1are showninFig.9.MeshconvergenceandcomputationalcostaregiveninAppendixDandAppendixE.Inthesecalculations, theforcingfieldisnotconstrainedtobelocalisedinanyzonesoftheflow.Threeregionsofmaximumgaincanbeidentified inthe

ω

− β

space, wheretheflowisthereforeespeciallyreceptive toan externalforcing.Optimal forcingfrequencyand wave numberassociatedwiththesethree regionsareshowninTable5.Astheforcing frequencygoestozero,anoptimal wavenumberisfoundtofavourthenon-modalgrowthofstreaks.Atmediumfrequency,apeakofoptimalgainisdetected fornon-zerowavenumbers(approximatelyhalfthevalueofthestreaksone),whichimpliesthattheassociatedperturbation hasanobliquewavestructure.Thisisthefirstmode instabilityofthecompressibleboundarylayer[11].Athighfrequency, a maximumofoptimalgain isreachedforzerowavenumberandpertainstothegrowthofthesecondmode(Mackmode). Spatialstructuresofeachoptimalforcingandresponsecorrespondingtothethreeoptimalgainmaximumswillbeanalysed insection5.3.

Theinterpretationofthevaluesreachedbythegainpeaksmustbedonecautiously.Indeed,aspointedoutinsection3.5, theabsolutevalueoftheoptimalgainhasnophysicalmeaning.Relativevaluescanhoweverbecomparedinordertoassess the efficiencyofthedifferentreceptivity mechanisms.Here,themaximumoptimalgain value isassociatedwiththe first mode instability whereas the streaks has a lower value within the same order of magnitude. The second mode hasan optimalgainoneorderofmagnitudelowerthanthefirstmode.Notethatthisisnotincontradictionwiththefactthatthe second modegrowthrateistwicehigherthanthefirstmodeone obtainedfromlocalstabilitycomputations[11].Indeed, the optimalgain isa globalquantity that accountsfortheenergygrowth ofperturbations overa givenphysicaldomain. Thus it dependson both the growthrate ofthe instability andthe length over which it grows, that is thewidth ofits neutralstabilitycurveatagivenfrequency.Theanalysisoftheenergygrowthprofilesplottedinsection5.4willshedmore lightonthismatter.

5.3. Analysisoftheoptimalforcingandresponse

Spatial structuresoftheoptimalforcingandresponseassociatedwiththethreeregions ofmaximal gainareexamined in thissection. Atlow frequency(Fig. 10),thelift-up mechanism isrecovered.Theoptimalforcing ismadeofstreamwise counter-rotating rollsthat initiatethetransport ofstreamwise momentumofthebaseflowbythe perturbation.Streaksof

(16)

Fig. 9. (b) OptimalgainofthecompressibleboundarylayeratM=4.5 for3Dperturbationsisplottedintheω− βspace.Threeregionsoflocallymaximum gainaredetectedandareassociatedwiththreelinearinstabilities.(a)Optimalgainassociatedwithfirstmodeinstabilityisplottedagainstωatβ=1.0, β=1.2 andβ=1.4.(c)Optimalgainassociatedwithstreaksisplottedagainstβatω=0.002.(d)Optimalgainassociatedwithsecondmodeinstability isplottedagainstωatβ=0.

Table 5

Optimalforcingfrequencyandwavenumberforeachinstability.

Instability ωopt βopt

Streaks →0 2.2

First mode 0.32 1.2

(17)

Fig. 10. Optimal forcing(left)andresponse(right)atω=0.002 andβ=2.2.Top:Iso-surfacesat10% offz (left)andu (right).Notethatthespanwise axisisnormalisedusingthewavenumberoftheperturbations.Bottom:Profilesatthestreamwiselocationcorrespondingtothemaximumforcingdensity (left)andChu’senergydensity(right).Profilesatx/δ∗0=35 (left)andx/δ0∗=159 (right). Forcingcomponentsarenormalisedbythemaximumvalueoffz.

Blackdottedlineindicatesthegeneralisedinflectionpoint.

highstreamwisevelocity, spanninginthestreamwisedirection, arethusgeneratedwhichcorrespondtothelocalanalysis resultspredictingazerostreamwisewavenumber[60].Thesefieldsareactuallysimilartothoseobtainedinan incompress-ibleboundarylayer[34],showingthatthelift-up effectcanbegeneralisedtocompressibleflow[19].Note,however,thata peak ofdensityappearsintheresponseprofileabovethestreamwisevelocitypeakandclosetothegeneralisedinflection point yiofthebaseflow,whichisdefinedforeachstreamwisestationas

∂/∂

y[

ρ

u

/∂

y]

(

yi

)

=

0.Thisfeaturehastheoretical

groundsina local,modalinstability framework[59] (itwillindeedbeobservedinthe optimalresponses associatedwith thefirstandsecondmodes,asdescribedfurtherinthissection).Itisherenotedthatthenon-modalgrowthofstreaksalso sharesthisbehaviour.

Atmediumfrequency,optimalforcingandresponsefieldsappearasobliquewaves(Fig.11).Assumingawavelike struc-tureoftheperturbationfields



q

(

x

,

y

)

=

q

(

y

)

eiαx

=

q

(

y

)

eiαrxeαix,thestreamwisewavenumber

α

rcanbecomputedas

α

r

= 



1 i



q



q

x

(50)

where



standsfortherealpartofthecomplexquantity.Inpractice,thestreamwisevelocityprofile



u

(

x

)

aty

=

1 isused inequation(50).Thewaveanglecomparedtothebaseflowdirectioncanthenbecomputedaccordingto

ψ

=

tan−1



β

α

r

(51)

Here,theangleoftheoptimalresponseisfoundtobeequalto72◦ whichcanbecomparedtotheangleof60◦ ofthe mostunstablefirst modeobtainedfromalocalstability analysis[11]. Sincetheoptimalresponseis notonthe onehand strictly modalinnature andontheother handis basedona globalandnot a globalanalysis, there isnoreasonto find the60◦ valueofafirstmodewavecomputedwithalocalapproach(acomparisonwithresultsfromaneN methodwould

herebeinteresting, butisbeyondthescopeofthepresentstudy).Nevertheless,bothapproachesdoshow thatgrowthof the first modeis strongerasan oblique wave. From Fig.11, itis observedthat the forcing fieldstend tobe localised in the upstream region ofthe numericaldomainwhereas the response growsdownstream asa consequenceof streamwise non-normality [37]. Moreover, the iso-surfaces of the forcing field are tilted upstream which suggests the action of the non-modalOrrmechanism.Theexaminationofthedisturbancesprofile(Fig.11)showsthattransverseforcing isthemost efficient andthat it mainlygenerates a streamwisevelocity. Comparedto incompressibleTollmien-Schlichting waves,the velocityprofileappearsfurtherawayfromthewall,closetothegeneralisedinflectionpointwhichpertainstotheprevalent inviscidmechanismofthefirstmodeinstabilityasMachnumberincreases.

Figure

Fig. 1. Numerical domain.
Table 2 gathers the Dirichet-Neumann conditions that are applied on perturbations at each boundary
Fig. 2. Supersonic boundary layer flow at M = 4: streamwise velocity (left) and temperature (right) profiles against self-similar variable y √
Fig. 4. Real part of optimal forcing component f z  (left) and optimal response streamwise velocity u  (right) at β = 0
+7

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