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Chapter
6
Results of
sensitivity
Analysedesensibilit´e
analysis
Lecodeadjointde´critauchapˆtre§4evtlaapˆtreid´eauch§5esedanylilitpe´se´aue´teriseaunrroual´e sensibilit´epourdeuxcongurationsd'e´coulement.Lapremierecongurationestuncastestacade´mique simple,celuidel'´ecoulementdecanalplanlaminaire,identiqueaceluiutilise´pourlavalidationducode adjoint,etimple´mente´enmono-bloc.Lasecondecongurationestlecaspluscomplexedel'´ecoulement au-dessusd'unecavit´e,quine´cessiteuntraitementmulti-blocs.
Toutaulongdecechapˆtre,lestermes'inow'(ent´ered'´ecoulement),'outow'(sortied'´ecoule-ment),'upstream'(amont)et'downstream'(aval)serontutilise´senre´f´erenceal'e´coulementphysique. Encons´equence,le'inow'seraalalimitegauchedudomainedecalcul,le'outow'alalimitedroite, l''upstream'd´esignelazonesitu´eealagauchedelasourcedeperturbationetle'downstream'lazone situe´easadroite.
Danscette´etudedesensibilite´,les´equationsadjointesonte´te´forc´eesalaposition(x0 y0) champ. Le adjoint (p,(ρu),(ρv)ouρtionequasl'´)quivslauesrxeihebeleveles´esples´luengirolasuonse´d directe qui doit eˆtre force´e an d'obtenir l'effet le plus important ala position(x0 y0). L'emplacement (xc yc)vslaatelelpsuesrelevlus´nous´eesqeuqidniulatnaulamp'eilountmeceitdoulohaecadmpinjo ˆetreplace´lefor¸cageenquestion.
Chaquevariableadjointerepre´senteunesensibilite´del'e´coulementaunforc¸agesp´ecique,comme l'illustre la gure6.1.(ρu)et(ρv)sont relie´s ala variation de quantite´ de mouvement, respectivement dans les directions longitudinale et normale. Physiquement, ce type de perturbation peuvent eˆtre induites dansl'´ecoulementenl'acce´l´eranttangentiellementouverticalement(parexempleavecdescontrˆoleurs plasma pour le cas tangentiel).pteˆtrcerpee´isyhemqutene´a'le´uqtaoidnecontinuit´e,peuqu,lirestie par ajout/suppression de masse (controˆleur type soufage/aspiration). Enn,ρest relie´ aux perturba-tionsdel'e´nergie,qu'ilpeutˆetredifciledecr´eerexpe´rimentalement.
R OF SENSITIVITY ANALYSISE SULTS
´ ECOULEMENT DE CANAL PLAN
La conguration de canal plan consiste en un domaine rectangulaire de demi-hauteurhet de longueur10h. Le maillage est uniforme avec101×101points, et identique pour les calculs direct et ad-joint. Pour le calcul direct, la condition initiale est la solution analytique de Poiseuille d'un e´coulement incompressibledecanalplan,l'inowetl'outowontdesconditionsauxlimitescaracet´ristiquesnon re´´echissantesdeGiles,etlesparoissolidessontimple´mente´esaveclesconditionsauxlimitesde Gloerfelt.
La simulation adjointe est initialisee avec toutes ses variables aze´ro, ses conditions aux limites sont ´ cellesd´ecritesauxchapˆtre§4omede´titnauqedetenemuvestappliqu´eal'e´uqtaoianjdiotn,eltecrofega¸ suivantx(equation6.1N.uo)is´eutilvonssn'auaL.tebaelnopmcin,ozinatentˆanesomluelsfle6.3 re´sumelesdiff´erentscastestr´ealis´ es.
R´esultatsetconclusions
Plusieurssimulationsont´et´er´ealis´eesenfor¸cantlaquantite´demouvementadjointesuivantx, pour plusieurs conditions aux limites, et plusieurs nombres de Mach et de Reynolds. La perturba-tion la plus efcace trouve´e est la perturbation de masse. Pour de faibles nombres de Mach et de Reynolds, l'emplacement le plus efcace pour agir sur l'e´coulement se trouve aux parois, et en amont del'emplacementcible´.Nousavonstrouv´equ'une´coulement´etaitplussensibleauneacce´le´ration tangentiellelorsqu'ilsede´pla¸caitentredesparoisadiabatiquesqu'entredesparoisisothermes,mais, inversement, moins sensible al'ajout de masse.
Une augmentation du nombre de Mach ou de Reynolds implique une augmentation des valeurs ad-jointes.PourdefortnombresdeReynolds,lasensibilit´eal'acc´ele´rationtangentielled´ependuniquement deladistancealazonecible´eetestlamˆemeauxparoisetaucentreducanal.Parcontre,lesr´egions oula pression adjointe est la plus forte reste localise´e au voisinage de l'axe du canal.
´ ´ ECOULEMENT DE CAVITE
Nousconside´ronsun´ecoulementau-dessusd'unecavit´ederapportd'aspectlongueursurprofondeur deLD= 2va,nuceuocelehcitimnceienidlaterideimaniass´'peeurδD= 028, anombre de Mach deM= 06(voir gure6.19 conguration est identiqueacelle pre´sente´e au paragraphe). Cette§3.3, ounousavonsmontre´quel'´ecoulementoscillaitenmodecouchedecisaillementalafre´quencefonda-mentalef0, ce qui correspond au second mode de RossiterSt2.
L'objectificiestderecherchercommentdoitˆetreapplique´unforc¸agedecete´coulementdecavit´esi l'onveutsupprimer/att´enuerlebruitqu'ile´met,c'est-a-direcommentr´eduirelesuctuationsdepres-sion.Nousavonsproc´ede´dedeuxfa¸cons:dansunpremiertempsnousavonscherche´cequipouvait agirsurlauctuationdequantit´edemouvementsuivantxau voisinage de la couche de cisaillement, etdansunsecondtempsnousavonscherch´ecequipouvaitagirsurlauctuationdepressiondansle champlointain.Parconsequent,nousavonse´tudie´desforc¸agesdes´equationsadjointesdelaquantite´ ´ de mouvement suivantxuaelbtaauest´iselmmcoetdel'´energie6.5.
Lesde´tailsdessimulationsdirectessontdonn´esauxparagraphes§3.2et§3.3. Les simulations ad-jointessontinitialis´eesavectoutesleurvariablesaze´roetenutilisantlesconditionsauxlimitesde´crites au paragraphe§4fselullecseduo,´eett´onesomtˆancaesullcdjsantoi,sutilis´eesauxpariosslodiseP.uolr
156
lemaillagedescalculsdirectsa´et´emodi´ecommesuit:lazonetamponae´t´ed´eplace´eal'inow(ala gauche du domaine de calcul au lieu de la droite). Le champ direct utilise´ dans cette zone tampon est unecopiedelaconditiond'entr´ee.LepasdetempsducalculadjointestΔtadjoint= 1tdirect.
Re´sultats et conclusions
Nousavonsr´ealis´euneanalysedesensibilit´ed'un´ecoulementau-dessusd'unecavit´epeuprofonde enfor¸cantles´equationsadjointesdequantite´demouvementsuivantxet de conservation de la masse a diff´erentespositionsetfr´equences.L'e´quationadjointedequantite´demouvementsuivantxforc´eea´et´e adiff´erentespositionsauvoisinagedelacouchedecisaillement.Lesr´esultatsmontrentquec'estl'ajout demassedanslacouchelimiteenamontdelacavit´equiagiraleplussurlaperturbationdequantite´ de mouvement suivantxontirtsutuaiacneetuoaleleme´aguo´vtnrtcefoequegeavrc¸a.Nnsvosaou couche de cisaillement, du coin amont au coin aval de la cavite´, et jusqu' aune distance de l'ordre d'une ´epaisseurdecite´au-dessusdelacouchedecisaillement. av
Lesre´sultatsobtenusenfor¸cantl'e´quationadjointedeconservationdelamasseontmontr´esque lauctuationdepressionloindelacavit´epouvaitˆetrecontroˆle´eparajoutdemassedanslacouche limite en amont de la cavite´, et plus particulierement au voisinage du coin amont de la cavite´. Ainsi, un actionneurplac´eauvoisinagedececoinagiraalafoissurlebruit´emisparlacavit´eetperc¸uauloin, etsurlesuctuationsdequantit´edemouvementsuivantxdans la couche de cisaillement.
Lare´ponsefre´quentielledusystemeadjointmontrequ'iloscillealamˆemefre´quencequelesysteme direct,qu'onleforcealafr´equencefondamentaleouaunefre´quenceharmonique.Danstouslescas,le spectre observe est assez disperse. ´ ´
Relations avec d'autres e´tudes
Cervin˜o et al. [27] et Spagnoli et Airiau [161ilae´rtn´sedse´sieva]asneslibiduteedse-ro´eit´esa acoustiques, respectivement d'un jet 2D froid et d'une couche de m e´lange 2D. Dans ces deux e´tudes, lesplusfortesvaleursdesensibilite´sonte´te´trouv´eesauvoisinagedelalabusedesortiedujetetau de´partdelacouchedeme´lange.Lesiso-contoursdepressionadjointe,repre´sent´esalagure6.31, nous montrent que les plus fortes valeurs se trouvent dans notre cas ala naissance de la couche de cisaillement (c'est- a-dire au coin amont de la cavite´). La dispersion du spectre adjoint (voir notre r´esultats§6.2.5)´anadesecsxuedute´´eetssaubsiov´erlateon˜ivreC.sedditeetecburitt.auaisnopsre faitquelescoefcientsdese´quationsadjointesnesontpasconstantsmaisqu'ilsvarientdansletempsa cause de l'instationnarit e´ du champ direct.
Akerviketal.[6heech´rceseldemoorpsserpbolgdxua'unecavit´earronidpeuerpfonoede,lstinort] ont observe´ que la fonction propre adjointe la moins stable se trouvait au coin amont de la cavite´. Ces resultatssontcoh´erentsavecceuxtrouve´sdanscettetheseetexpose´sauparagraphe§6.2.4, oules plus ´ fortesvaleursdepressionadjointeseconcentrentauvoisinageducoinamontdelacavit´e.Marquetet al. [128numesu'ddeserahcemencouldesstau-teanndce]no´ttedudaxunioj'dste´nueli´moessgdebalo arrondieplace´eal'inte´rieurd'unconduitenformedeS,etilsont´egalementtrouv´elapressionadjointe maximale au point de separation.
C'estpourquoi,danslaplupartdes´etudesconcernantlecontroˆlactifoupassifd'´ecoulementsau-dessusdecavit´es,lesactionneurssontplacesauvoisinageducoinamontdelacavite´,oulasensibilit´e ´ del'´ecoulementestlaplusforte.
157
RE SULTS OF SENSITIVITY ANALYSIS
Introduction
The adjoint algorithm described in chapter§4and validated in chapter§5has been used to perform sensitivity analysis of two ow congurations. The rst one is a simple academic test case, a laminar plane channel ow as the one used for the validation in§5, implemented in a single block. The second one is a more complex ow of industrial interest, a cavity ow with an incoming boundary layer, which requires multi-block treatment.
The sensitivity analysis is performed using a periodic in time (sinusoidal) forcing which is applied to one of the adjoint equations. Each adjoint forcing has a different physical interpretation, as outlined in table6.1 all the adjoint variables are obtained and observed. Each. Then, the results for adjoint variable represents the sensitivity of the ow to a specic direct forcing as described in table6.2and illustrated in gure6.1 position in the adjoint eld which has the highest value indicates the most sensitive. The region of the ow to that particular direct forcing.
Adjoint variable forced (ρu)(ρv)ρp
Direct sensitivity (ρu)(ρv)pρ
Adjoint equation of
x-momentum y-momentum energy mass
Table 6.1 -Interpretation of the adjoint forcing.
Table6.1gives the interpretation of each adjoint forcing. Adjointx-momentum forcing gives the sensitivity of thex-momentum uctuations. interpretation of an adjoint They-momentum source is equivalent but in the normal direction. The adjoint density is related to the energy equation, thus forcing it indicates the sensitivity of pressure uctuations. Finally, adjoint pressure corresponds to the equation of conservation of mass, hence it gives the sensitivity of density uctuations.
Adjoint variable observed (ρu)(ρv)ρp
Direct forcing
acceleration inx-direction acceleration iny-direction energy perturbations mass injection
Table 6.2 -Interpretation of the adjoint variables.
Table6.2and gure6.1show the interpretation of each adjoint eld.(ρu)and(ρv)are related to the variation of the perturbations of momentum in the streamwise and normal directions, respectively. Physically, these perturbations can be induced into the ow by accelerating it in the tangential or normal direction (technically a perturbation ofx-momentum could be performed by a plasma controller, for example). Similarly,p, related to the continuity equation, can be physically obtained by mass injection in any direction. Finally,ρrepresents variations of energy perturbations. Technically speaking, it is complex to create such a perturbation, but for example it could be done by some source of radiation in order to energize the ow at a specic position.
158
Forcing at(x0 y0)
(ρu): tangential acceleration
(ρv): normal acceleration
Figure 6.1 -Interpretation of the adjoint variables
6.1 Channel ow
ρ: energy
p: mass injection
For convenience, during the whole chapter the termsinow,outow,upstreamanddownstreamare used respect to the direct ow. That is to say, theinowis the left boundary,outowis the right boundary, upstreammeans to the left of the source anddownstreamto the right.
6.1
Channel ow
The channel ow conguration consists on a rectangular domain of half-widthhand length10h. The grid is equidistant with101×101 the direct simulation, Forfor both the direct and the adjoint.points the initial condition is the analytical solution for an incompressible Poiseuille channel ow, the inow and outow are the non-reecting characteristic boundary conditions of Giles and the solid boundaries are implemented with Gloerfelt's wall boundary conditions.
The adjoint simulation is initialized with zeros in all variables, its boundary conditions are the ones described in chapter§4the forcing is applied to the adjoint, and x-momentum equation. Neither a buffer zone nor ghost cells have been used, since it has been found in the validation chapter§5that for a forcing of(ρu) direct eld hasthey are not necessary and they do increment the computational time. The been stored at each temporal iterationΔtdirectwhich corresponds to aCF L= 07. SinceΔtadjoint= 1tdirectdirect elds are interpolated every two iterations of the adjoint simulation., the
Table6.3 In these cases,shows the test cases performed to study the sensitivity of the channel ow. different wall boundary conditions, Mach number, Reynolds number and position of the forcing are compared. The positions are identied with a label:CENTER(center of the channel),WALL(near the upper wall) andOUTFLOW They are described in more detail in section(near the outow).§6.1.2. The section(s) where the results are discussed is indicated. In all cases several periods have been computed, and the results correspond to a time when the perturbation has reached all the computational boundaries and then there are no signicant changes from one period to another.
The forcing of the adjointx-momentum equation follows the equation:
g=Asin(ωpt) exp(xx0)2σ+p2(yy0)2
(6.1)
whereA= 001u,ωp= 2π100Δtandσp= 10Δyfor all the cases, and the origin(x0 y0)is given in table6.3. Physically, it represents a variation of the perturbation of thex-momentum.
159
RE SULTS OF SENSITIVITY ANALYSIS
6.1.1
Wall condition
Isothermal
Isothermal
Adiabatic
Adiabatic
Mach
0.1
0.4
0.1
0.4
Position label
CENTER CENTER CENTER WALL OUTFLOW CENTER WALL OUTFLOW CENTER WALL OUTFLOW CENTER WALL OUTFLOW
Reynolds
14 4475 6040 14 14 58 58 58 14 14 14 58 58 58
Section
§6.1.1,§6.1.2,§6.1.3,§6.1.4,§6.1.5 §6.1.5 §6.1.5 §6.1.2,§6.1.3,§6.1.4 §6.1.2 §6.1.4 §6.1.4
§6.1.3 §6.1.3
Table 6.3 -ow. The positions are described inSensitivity test cases for a channel §6.1.2.
Interpretation of the adjoint variables
In this sensitivity analysis the adjointx-momentum equation has been forced at(x0 y0). That means that the adjoint elds will give the sensitivity of(ρu)to different kinds of forcing of the direct equations. The adjoint eld (p,(ρu),(ρv)orρ) which shows the highest values indicates the direct equation which must be forced to obtain the largest effect at(x0 y0) position with the highest value points. The out the origin of the before-mentioned forcing of the direct equation.
A channel ow atM= 01andReh= 14with isothermal walls is considered, where the origin of the forcing(x0 y0)is theCENTERof the channel as shown in gure6.2. The gures of this section display the instantaneous isocontours of each variable at four moments equally spaced in time of a period Tof oscillation.
To start with, thex-momentum eld shown in gure6.3 The rst gureis investigated.6.3(a)shows that around the source, and for a radius approximately equal to15h, the sensitivity is the same near the walls and at the center of the channel. Further from the origin of the pulse, the sensitivity is higher near the walls. At this specic time, the sensitivity of the ow upstream and downstream from the momentum source seems approximately the same.
The next gures help to understand the temporal evolution of the sensitivity. At the center of the
1 y0
-10
2
4
x
6
8
10
Figure 6.2 -Computational domain and location of the perturbation: forcing of adjoint x-momentum atCENTER.
160
y
y
551010
x (a) t = 1/4T
x (c) t = 3/4T
401010
251010
0 10101
y
y
051010
201010
x (b) t = 1/2T
x (d) t =T
6.1 Channel ow
351010501010
651010
Figure 6.3 -Instantaneous isocontours of(ρu)during 1 period. Dashed negative values. Origin of the perturbation:CENTER.M= 01,Reh= 14, isothermal wall.
channel the values of the adjointx-decrease faster than near the solid boundaries, beingmomentum almost insignicant after2h3h. It is interesting to observe in the last plot6.3(d)that the sensitivity at the center of the channel is higher upstream.
The interpretation of these results is that in order to obtain a certain effect in the center of the channel, it is more efcient to apply acceleration inx-direction at the walls than at the interior part of the channel. It is also shown that when applying forcing at the centerline of the channel, the ow will be more affected downstream than upstream.
y
y
x (a) t = 1/4
(c)
T
x t = 3/4T
y
y
270101020510101401010075101001010100551010
x (b) t = 1/2T
x (d) t =T
12010101851010
2501010
Figure 6.4 -Instantaneous isocontours of(ρv)during 1 period. Dashed negative values. Origin of the perturbation:CENTER.M= 01,Reh= 14, isothermal wall.
161