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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
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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
caSorbonneUniversité,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 Compressibleboundarylayer3D 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).
Nomenclature
M Machnumber P r Prandtlnumber Re Reynoldsnumber x Streamwisedirection y Normaltothewalldirection
z Transverse direction, supposed as homoge-neous
R
ResidualofNavier-StokesequationsF FluxofNavier-Stokesequationsalong x-direction
G FluxofNavier-Stokesequationsalong y-direction
H FluxofNavier-Stokesequationsalong z-direction
q Vectorofconservativevariables
(
ρ
,
ρ
u,
ρ
v,
ρ
E)
Tu Streamwisevelocity v Normalvelocity w Transversevelocity
ρ
Density c Speedofsound E Totalenergy e Internalenergy p Pressure T TemperatureM RelativeMachnumber
η
Dynamicviscosityγ
Heatcapacityratioκ
Thermalconductivity cp Heatcapacity∞
Far-fieldquantitiesα
r Streamwisewavenumberβ
Transversewavenumberω
AngularfrequencyAnglebetweenthewavevectorofthe pertur-bationandthebaseflowdirection
cϕ Phasevelocity
μ
Optimalgain f Forcing vector (in particular, optimal forcing vector) q Perturbationsvector(inparticular,optimal re-sponsevector) fx Streamwiseforcing fy Normalforcing fz TransverseforcingJ
JacobianmatrixR
ResolventmatrixQE 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 thicknessBlasiuslength
√
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,
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 termρ
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=
pρ
∞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, whereρ
, u= (
u,
v,
w)
T and E respectively are the fluid density, the velocity vector and the total energy. T , p,η
andλ
respectively stand for temperature, pressure, dynamic viscosity and thermalconductivity.The referencelength L may refertothe compressibleboundarylayerthicknessδ
∗ orto theBlasius length=
√
η
∞x/
ρ
∞u∞.Non-dimensionalReynolds Re,MachM andPrandtlP r numbersareintroducedasRe
=
ρ
∞u∞Lη
∞,
M=
u∞ c∞,
P r=
η
∞cpλ
∞ (2)wherec isthespeedofsoundandcp istheheatcapacityoftheflow.The compressibleNavier-Stokesequationscanthen
bewrittenas
∂
ρ
∂
t+ ∇ · (
ρ
u)
=
0,
(3a)∂
∂
t(
ρ
u)
+ ∇ ·
ρ
u⊗
u+
pI−
1 Reτ
=
0,
(3b)∂
∂
t(
ρ
E)
+ ∇ ·
(
ρ
E+
p)
u−
1 Reτ
u
−
λ
P r Re(
γ
−
1)
M2∇
T=
0 (3c)Forathermallyandcaloricallyperfectgas,thenon-dimensionalpressure p andtotalenergyE canmoreoverbeexpressed accordingto p
=
1γ
M2 ∞ρ
T,
E=
pρ
(
γ
−
1)
+
1 2u·
u (4)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]).Henceitleadstoλ
=
η
asnon-dimensionalvariablesareused.Inthefollowing,equations(3) canberecastinthedynamicalsystemform
∂
q∂
t=
R(
q),
(7)where
R
isthedifferentialnonlinearoperatorofNavier-Stokesequations. 2.2. LineardynamicsThisstudyaimstostudytheforceddynamicsof3Dperturbationsq
(
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,thesefieldsareexpandedintoFouriermodesofwavenumberβ
.Moreover, consideringaharmonicforcingatfrequencyω
,thesequantitiesarefinallywrittenasq
(
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 = (
iωI − J )
−1 is the globalresolvent operator (withI
the identity operator) which dependsboth on theforcing frequency
ω
andthetransversewavenumberβ
.Forglobally stableflows, alleigenvaluesoftheJacobianoperator havea strictlynegative realpart.Thus,theresolventoperatoriswell definedandallows tostudytheforceddynamicsof the flowby providinga relationbetweenresponseandforcing fields.Optimal gain isnowintroduced anddefinedasthe maximumratiobetweentheenergyoftheperturbationsandtheforcing.Formally,itsexpressionreadsμ
2=
supf
||
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.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
,theresidualR
∈ R
N ofthe2D nonlinear Navier-Stokesequationsisusedtoperformafirst-orderapproximationoftheJacobianmatrixasJ
v=
R(
q+
ε
v)
−
R(
q)
ε
+
O(
ε
)
(13)Here,q
∈ R
N isthevectorofconservativevariablesassociatedwiththebaseflow,v∈ R
N isan arbitraryvectorandε
∈ 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 matrixJ
. Repeating thisprocedureN times, thewholematrixisrecovered. Moreover,itispossibletotakeadvantagefromtheblockdiagonal structureofJ
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,computingthematrixJ
whenN=
600000 usinga1500×
100 meshandathirdorderaccurate scheme takesapproximatively5 minutes (on CPU:IntelXeon(R)CPUE5-2630v2@2.60GHz). As forthe choice ofε
value, its orderofmagnitudehasto besmallenough sothat secondorderterms canbeneglected but needto behighenough toavoidnumericalround-offerrors.Ifthevectorv containsonlyonenon-zeroelement locatedatthek-component,Knoll andKeyes [50] suggesttochoseε
=
b(
1+ |
qk|)
,whereb isthesquarerootofthemachineprecision.Ifv containsmultiplenon-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,thebaseflowissupposedtobehomogeneousinthez-directionsothat
∀(
x,
y)
∈
V
,
∂
qwhere
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 beturnedintoiβ
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 matrixisfinallyseparatedintothreedifferentmatricesasJ
=
J
F,G+
J
H+
J
νz (19) whereJ
F,G=
∂
∂
q−
∂
F∂
x+
∂
G∂
yq (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,GNo transverse derivative is involved in the computation of
J
F,G in equation (20). Therefore, equation (13) can be straightforwardly used by introducingR
F,G= −∂
F/∂
x− ∂
G/∂
y.The Jacobian matrixJ
F,G is thus computedaccording toJ
F,Gv=
R
F,G(
q+
ε
v)
−
R
F,G(
q)
ε
(23) 3.3.2. ComputationofJ
HThecomputationof
J
H isachievedbyfirstlyreconsideringthefollowinglinearisationJ
Hq= −
∂
H(
q+
q)
∂
z+
∂
H(
q)
∂
z (24)whereafirstorderexpansionofH
(
q+
q)
inthefirstrighthandtermgivesJ
Hq= −
∂
∂
z H(
q)
+
∂
H∂
q q q+
∂
H(
q)
∂
z (25)Finally,thefollowingexpressionremains
J
Hq= −
∂
∂
z∂
H∂
q q q (26)Here,
∂
H/∂
qqdoesnotdependonz asassumedinequation(14).Andsince∂
q/∂
z=
iβ
q,equation(26) nowreadsJ
Hq= −
iβ
∂
H∂
q q q (27)J
H= −
iβ
∂
H∂
q q (28)Again,thenumericalcomputationof
∂
H/∂
qqisbasedonafirst-orderfinitedifferenceapproximation.However,notethat the numericalflux H is hereused insteadof theflux divergencethat was previously needed inequation (13) and (23). Hence,thematrixJ
H isnumericallybuiltaccordingtoJ
Hv= −
iβ
H
(
q+
ε
v)
−
H(
q)
ε
(29)3.3.3. Computationof
J
νzBecauseeveryfluxtermnowcontains atransversederivative, aparticularcareisrequiredtocomputethematrix
J
νz. Itisnotsimplypossibletoreplaceeach∂/∂
z intoiβ
withinthefluxesFνz,Gνz andHνz.Indeed,spurious non-zerotermsassociatedwiththebaseflowderivativealongz wouldappear,thusviolatingtheassumptionsmadeinequations(14).Toget aroundthisproblem,itcanbeobservedthatbecauseofequations(14),theperturbations
q onlyappearsunderz-derivatives inthefinallinearisedequations.Forexample,introducingthelinearisationoperator
L
,linearisingthefourthcomponentof thevectorFνz (seeAppendixA)readsL
(
η
Re∂
u∂
z)
=
η
Re∂
u∂
z=
iβ
η
Reu (30)Anticipatingthefinalresultoflinearisation,weheresuggesttomodifythefluxesFνz,GνzandHνzinto
Fνz,
Gνz and
Hνz in
whicheveryfactorinfrontofaz-derivativeissettothebaseflowvalueandeach
∂/∂
z isturnedintoiβ
(whichwillappear asafactorinthefinal expressionoftheJacobianmatrix).Toillustratethisprocedure,letustake theexampleofFνz (seeAppendixBforexhaustiveexpressionsofthemodifiedfluxes)
Fνz
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0−
2 3 η Re∂ w ∂z 0 η Re ∂u ∂z η Re−
u23∂∂wz+
w∂∂uz⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
→
Fνz=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0−
2 3 η Rew 0 η Reu η Re−
u23w+
wu⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(31)Thematrix
J
νz isfinallycomputedusingtheapproachespresentedinsection3.3.1forFνz andGνzandinsection3.3.2for Hνz.IntroducingRν
z=
∂
Fνz∂
x+
∂
Gνz∂
y,
(32)thefinalpracticalexpressiontocompute
J
νz readsJ
νzv=
iβ
Rν
z(
q+
ε
v)
−
Rν
z(
q)
ε
− β
2Hνz(
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
(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)
andq
(
x,
y)
.TheresolventmatrixR
isinvolvedinthisproblemand firstrequiresthecomputationoftheJacobianmatrixintroducedintheprevioussection.Notethattheexplicitconstruction ofR
willnotberequiredsincelinearsystemsinvolvingR
−1 willbesolved.Equation(12) isanoptimisationproblemonthefunction
μ
(
f)
whichdependsonenergynorms.Thesenormsare asso-ciatedwithdiscretescalarproductsthatcanbeexpressedbynormmatricesas||
q||
2E=<
q,
q>
E=
q∗QEq (35)||
f||
2F=<
f,
f>
F=
f∗QFf (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’senergyandwrittenforconservativevariablesisde-rivedhereafter,whereasQE
=
QEK,associatedwiththekineticenergy,isdetailedinAppendixC.Startingofffromprimitivevariables,Chu’sdisturbancesenergyEChu reads[51]
EChu
=
1 2 Vρ
|
u|
2+
Tργ
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
=
1ρ
((
ρ
ui)
−
u iρ
)
(39) T=
(
γ
−
1)
γ
M 2ρ
(
1 2|
u|
2−
e)
ρ
−
u i(
ρ
ui)
+ (
ρ
E)
(40)
wheree istheinternalenergyoftheflow.Twoparameters,associatedwithbaseflowquantities,arenowintroducedas
a1
=
(
γ
−
1)
γ
M2ρ
T (41) a2=
(
12|
u|
2−
e)
ρ
(42)Equation(37) cannowberecastwithconservativevariables.Identifyingthisequationwithequation(38) andsearchingthe matrix QChu assymmetrical,itscoefficientscanbeidentifiedas
QChu
=
1 2d⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
|
u|
2ρ
+
Tργ
M2+
a1a 2 2−
u(
1+
a1a2)
ρ
−
v(
1+
a1a2)
ρ
0 a1a2ρ
−
u(
1+
a1a2)
ρ
1ρ
+
u2a1ρ
2 uva1ρ
2 0−
ua1ρ
2−
v(
1+
a1a2)
ρ
uva1ρ
2 1ρ
+
v2a1ρ
2 0−
va1ρ
2 0 0 0 1ρ
0 a1a2ρ
−
ua1ρ
2−
va1ρ
2 0 a1ρ
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 ratheramathematicalnormwhichischoseninordertoreflecttheenergyinputoftheexternalforcingfield0.1Hencetheoptimalgain 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=
Pfs.Inthispaper,wechoosetoonlyconsiderthemomentumcomponentsoftheforcing fieldin ordertosimplifytheinterpretationoftheforcingnorm.Asimilarchoicehasbeenmadeforexampleby Sartoret al. [43]. InthiscasethematrixP has,beforediscretisation,thefollowingexpressionP
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(44)IfN isthesizeofthevector
f,thenfs hasasizeM withM≤
N andthematrixP hasasize N×
M. Therelationbetween forcingandresponsefieldsthenreads q=
R
Pfs (45)Introducingenergynormmatrices,equation(12) cannowberecastas
μ
2=
sup fs(
R
Pfs)
∗QE(
R
Pfs)
(
Pfs)
∗QF(
Pfs)
=
sup fs(
Pfs)
∗R
∗QER
(
Pfs)
(
Pfs)
∗QF(
Pfs)
(46)Equation(46) canbeseenasageneralizedRayleighquotient[35] wheretheoptimalgain
μ
2 isthenthelargesteigenvalueand
fstheassociatedeigenvectoroftheHermitianeigenvalueproblem(
R
∗QER
P)
fs=
μ
2(
QFP)
fs (47)BecauseQF isinvertibleandP∗P
=
I,equation(47) reducesto(
P∗QF−1R
∗QER
P)
A
fs=
μ
2fs (48)Tosolveequation(48),onlytheinverseoftheresolventmatrix
R
,whichcanconvenientlybecomputedfromtheJacobian matrix,isactuallyrequired.Indeed,theeigenvalueproblem(48) canbesolvedbyamatrix-freealgorithmbasedonaKrylov method[52].Asetofvectors(
v0,
Av0,
A2v0,
...)
,whichcomposestheKrylovsubspaceassociatedwiththematrixA,needsto be computed. Starting froman arbitraryvector v0,each Krylov vector vi is computedfrom theprevious one vi−1 by
solvingequationvi
=
Avi−1.ThiscomputationrequirestosolvetwolinearsystemsinvolvingR
−1and(R
∗)
−1.Thedetailedstepsofthisprocedureare developedinAlgorithm1.Finally,theoptimalresponsecanbe recoveredbysolvingthelinear system(45).
1 Notethatdefininganinputmechanicalworkwouldbemorerigorous,but,toourknowledge,thiscanonlybeachievedaposteriorioftheoptimalgain
Algorithm1ComputationoftheKrylovvectorvi associatedwiththematrixA fromthevectorvi−1. 1. Computingthematrix-vectorproduct:
t
1=Pvi−12. Solvingofthelinearsystem:R−1t 2=t1
3. Computingthematrix-vectorproduct:
t
3=QEt24. Solvingofthelinearsystem:(R∗)−1 t4=t3
5. Computingthematrix-vectorproduct:
v
i=P∗Q−1t4OpenlibraryPETSc[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) suchthatB
∂
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 definedasR =
iωB − 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,∂ρ∂y=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×
Fig. 2. Supersonic boundarylayerflowatM=4:streamwisevelocity(left)andtemperature(right)profilesagainstself-similarvariabley√Rex/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 displacementthicknessδ
0∗ atinflowasthe referencelengthscale,thenumericaldomainspansover
[
0,
800] × [
0,
30]
.Asthenumericalframeworkofthepresentpaperisforcompressibleflow,theMachnumberissetto M
=
0.
3 inorder togetasolutionclosetotheincompressibleresults.First,thebaseflowcomputationisperformedusingtheCFDsolver pre-sentedinsection3.1.Subsonicboundaryconditionsareused(seeTable1)andLx0 issetto200δ
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 whereasthewavenumberβ
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 considerFig. 3. Optimal gainasafunctionofthewavenumberβatω=0 foranincompressibleboundarylayer.Bluesquares:presentresults,atM=0.3.Black line:resultsofMonokrousoset al. [34] computedforglobal3DperturbationsinaBlasiusboundarylayer,usingatime-steppingmethodassociatedwitha fringezonetechnique.RedcrossesareobtainedbyrescalingtheoptimalgainvaluesofthebluesquareswiththoseofMonokrousoset al. [34].
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 betweentheoptimalgainbehaviourstakenasafunctionofβ
.Inparticular,theoptimalwavenumberβ
=
0.
6 isretrieved. Thisagreementisthemostsignificant:indeed,theabsolutevalueoftheoptimalgainhasnophysicalmeaning[35] contrary totheforcing frequencyorwavenumberatwhichtheoptimalgainismaximum.The 3DfieldsdepictedinFig.4 further-moresupportthevalidityofourcomputationsincethesamecounter-rotatingrollsandstreakstopologiesasMonokrousos et al. [34] arerespectivelyfoundastheoptimalforcingandresponsefieldsassociatedwiththemaximumoptimalgainvalue. 4.3. OptimalgainofasupersonicboundarylayeratM=
3Toourknowledge,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
Fig. 6. Optimal gaincomputedat ω=0 withconstrainedforcingandresponsefieldsforthecompressibleboundarylayerat M=3,fortwovaluesof R=xopt/xf.Blackline:resultsfromTuminandReshotko [21].Referencelengthscaleis0=η∞xopt/ρ∞u∞.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/δ∗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.Thevalueofxoptissetattheabscissacorrespondingto Re
=
1000 and theBlasius lengthatthisspecificabscissa,0
=
η
∞xopt/
ρ
∞u∞,will beusedasthereferencelength scale.Twodifferentvaluesofxf willsuccessivelybeconsideredsuchthattheratioR=
xf/
xoptisequalto0.
2 and0.
4.Thenumericaldomainspansover x
/
0∈ [
0,
1300]
and y/
0∈ [
0,
100]
.Theregionsoverwhichtheforcing andresponse fieldsareconstrainedspanover
x
/
0=
40 inthestreamwisedirectionbutarenotrestrictedinthenormaldirection.Resultsofoptimalgaincomputationsarecompared 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.
55.1. 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∗ obtainedfromournumericalcomputationisobservedtobelinearasaconsequenceofself-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. Thecompressibledisplacementboundaryδ
0∗ atthisabscissawillbeusedasthereferencelength.Itshouldbepointedoutthatthislengthappearsbothinthenon-dimensional forcing frequency
ω
andwave numberβ
.Finally,the actual baseflowused foroptimalgain computation istruncatedin the y-direction forcomputationalsavings.ItisshowninAppendixDthat theresultsareindependentfromthischoice of domain.Fig. 7. Computed ReynoldsnumberReδ∗ basedonthecompressibledisplacementthicknessasafunctionoftheReynoldsnumberRebasedonBlasius 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
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
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.Thisfeaturehastheoreticalgroundsina 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)
eiαrxe−αix,thestreamwisewavenumberα
rcanbecomputedasα
r=
1 iq∂
q∂
x(50)
where
standsfortherealpartofthecomplexquantity.Inpractice,thestreamwisevelocityprofile
u
(
x)
aty=
1 isused inequation(50).Thewaveanglecomparedtothebaseflowdirectioncanthenbecomputedaccordingtoψ
=
tan−1β
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.