A generalized non-reflecting inlet boundary condition for steady and forced compressible flows with injection of vortical and acoustic waves

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Daviller, Guillaume and Oztarlik, Gorkem and Poinsot, Thierry

A generalized non-reflecting inlet boundary condition for

steady and forced compressible flows with injection of vortical

and acoustic waves.

(2019) Computers and Fluids, 190.

503-513. ISSN 0045-7930 .

Official URL:

https://doi.org/10.1016/j.compfluid.2019.06.027

A

generalized

non-reflecting

inlet

boundary

condition

for

steady

and

forced

compressible

flows

with

injection

of

vortical

and

acoustic

waves

G.

Daviller

a,∗

_,

_G.

_Oztarlik

b

_,

_T.

_Poinsot

b

a CERFACS, Toulouse 31100, France

b IMF Toulouse, INP de Toulouse and CNRS, France

Characteristic boundary conditions Non reflecting boundary conditions Turbulence injection

Acoustic forcing

Thispaperdescribes a new boundarycondition for subsonic inlets incompressible flowsolvers. The method uses characteristic analysis based on wave decomposition and the paper discusses how to specify the amplitude of incoming waves to inject simultaneouslythree-dimensional turbulence and one-dimensionalacousticwaves whilestill beingnon-reflectingforoutgoingacoustic waves.The non-reflectingpropertyisensuredbyusingdevelopmentsproposedbyPolifkeetal.[1,2].Theyarecombined withanovelformulationtoinjectturbulenceandacousticwavessimultaneouslyataninlet.Thepaper discussesthecompromisewhichmustbesoughtbytheboundaryconditionformulationbetween con-flictingobjectives:respectingtargetunsteadyinletvelocities(forturbulenceandacoustics),avoidinga driftofthe meaninletvelocitiesand ensuringnon-reflectingperformancesforwaves reachingthe in-letfromthecomputationaldomain.Thiswell-knownlimitofclassicalformulationsisimprovedbythe newapproachwhichensuresthatthemeaninletvelocitiesdonotdrift,thattheunsteadycomponents ofvelocity(turbulenceand acoustics) arecorrectlyintroduced intothe domainand thatthe inlet re-mainsnon-reflecting.Theseproperties arecrucial forforcedunsteady flowsbutthe sameformulation isalsouseful forunforcedcaseswhereitallowstoreachconvergencefaster.Themethodispresented byfocusingontheexpressionoftheingoingwavesandcomparingitwiththeclassicalNSCBCapproach [3].Fourtestsarethendescribed:(1)theinjectionofacousticwavesthroughanonreflectinginlet,(2) thecompressibleflowestablishmentinanozzle,(3)thesimultaneousinjectionofturbulenceand ingo-ingacousticwavesintoaductterminatedbyareflectingoutletand(4)aturbulent,acousticallyforced Bunsen-typepremixedflame.

1. Introduction

Specifying boundary conditions for compressible flow simula-tionsisstillamajorissueinmanyfieldssuchasastrophysics [4,5], aerodynamicsandaeroacoustics [6–11]orcombustioninstabilities and noise [2,12–16]. The presentpaperfocuseson a limited part of thisproblem:thespecification ofinletboundary conditionsin subsoniccompressibleflows.Its objectiveistoconstructa bound-aryconditionwhichshouldsatisfythreeproperties:

• P1-Provideawell-posedformulationfortheNavier–Stokes equationsaswellasperfectlynon-reflectinginletproperties

• P2-Allowtoinjectplaneacousticwaves

• P3-Allowtoinjectthree-dimensionalturbulence

∗ _{Corresponding author.}

E-mail address: daviller@cerfacs.fr (G. Daviller).

It is important to satisfy P1, P2 and P3 simultaneously: the capabilityto inject ingoing acoustic wavesand turbulenceat the sametimeonaninletpatchwhilelettingoutgoingacousticwaves cross the boundary without reflection is crucial in many config-urations. Well-known examples include the determination of the transferfunctionofturbulentflames [15,17],thepredictionof com-bustion noise in gas turbines [18–20] or the evaluation of the acoustic transfer matrix of singular elements in turbulent flows

[21–24]. All thesestudies requireto introduce harmonic acoustic forcing andturbulence on the sameinlet boundary while letting

acousticwaves propagate from the computational domain to the

outsidewithoutreflection(Fig.1).Similarly,studiesofcombustion noiseingasturbines [20,25,26]requiretoperformsimulations of thenoise produced by a jet forced simultaneously by turbulence andbyacousticwavesgeneratedinthecombustionchamber.

This paper uses characteristic boundary conditions

Fig. 1. An example where turbulence and acoustic waves must be introduced through the inlet of a compressible simulation while acoustic waves reflected from the computational domain must propagate without reflection through the same sur- face: the computation of the Flame Transfer Function of a turbulent flame [15] .

Fig. 2. Characteristic waves at a subsonic inlet (inlet at x = 0 ).

compressible solvers [11,28–30]. These methods use character-istic analysis to decompose the Navier–Stokes equations at the

boundary1 _{and identify waves going into the domain and waves}

leaving the domain. Wave amplitudes can be expressed as

spa-tial derivatives of the primitive variables. The amplitudes of

waves leaving the domain depend only on the flow within the

computational domain: they can be computed using one-sided

derivativesoftheresolved field inside thisdomain.Inversely,the

amplitudesof the wavesentering the computational domain can

notbe obtainedbydifferentiatingthefieldinthedomainbecause

thiswould lead to an ill-posed problem: they must be imposed

usinginformationgivenby theboundaryconditions. Fig.2shows that, for a subsonic inlet, only the outgoing acoustic wave L1 (see Section 2 forwave definitions)is leavingthe computational domain at speed u_{− c where} u is the local convection velocity andc the local soundspeed. All other waves (the acoustic wave

L5 at speed u+c, the entropy wave L2 at speed u and the

two transverse waves _L3 and L4 at speed u) are entering the

domain and must be specified using the boundary conditions.

Thispaperfocuseson the determinationofthese incomingwave amplitudes.

Section 2 recalls the basis ofthe NSCBC(Navier Stokes Char-acteristicBoundaryConditions)techniqueappliedtoan inlet.The specificationoftheincomingwavesispresentedin Section3and anovelinletconditionabletosatisfypropertiesP1–P3isdiscussed (calledNRI-NSCBCforNonReflectingInletNSCBC).Before perform-inganysimulation,asimpletheoreticalapproachisusedtopredict thereflectioncoefficientofthestandardNSCBCformulationandof thenewNRI-NSCBCconditionfortheinjectionofacousticwavesin onedimension(Section4).Theseresultsarevalidatedusinga one-dimensionalsimulationof aforcedduct in Section 5. Theimpact

1 Note that more sophisticated methods such as Perfectly Matched Layers _[31–

33]can be used to make boundaries fully non-reflecting for multidimensional flows in other fields such as electromagnetism or aeroacoustics. These methods are used mainly in infinite domains while characteristic-based methods are usually preferred in inlet/outlet/ configurations.

oftheNRI-NSCBCformulationisthenillustratedthroughthree ex-amples2_:

• Section 6 shows that, for an unforced multi-dimensional flow(anozzlecase),using theNRI-NSCBCconditionallows toeliminateacousticwavesandtoconvergemuchfasterto steadystate,acrucialpropertyforcompressibleflowsolvers

which often remain limited by small time steps and long

computationtimesbeforeconvergence.

• Section7presentsanexampleofsimultaneousacoustic forc-ingandturbulenceinjectioninathree-dimensionalchannel andshowsthat theNRI-NSCBCis ableto satisfy Properties 1–3simultaneously.

• Finally Section 8 proposes a DNSof a premixed turbulent flamewhichisforcedacousticallyusingbothboundary con-ditionsformulations.

2. Characteristicinletboundaryconditionforsubsonicflows

Consider asubsonicinlet(Fig.2)wheretheboundary planeis the (y, z) plane. The velocity components to impose at thisinlet are (ut_{, v}t_{, w}t_{). These components can be steadyor change with}

timewhenturbulenceand/or acousticwavesareinjected through theinlet.Notethatthesefieldsare“target” values:theycorrespond totheinjected velocitysignalswhichmustbeimposed atthe in-let, not necessarilyto the values (u,v, w) which will be actually reachedduringthecomputationbecauseoutgoingreflectedwaves (comingfromthecomputationaldomain)alsochangethevelocity andpressurefieldontheinletpatch(Fig.1).

TheNavier–Stokesequationsattheinletcanberecastinterms ofwavespropagatinginthexdirection,leavingtheothertwo di-rectionsunchanged:

∂ρ

∂

t +d1+

∂

∂

y

(

ρv

)

+

∂

∂

z

(

ρ

w

)

=0 (1)

∂

(

ρ

E

)

∂

t + 1 2



u2₊

_v

2_+w2



_d 1+ d2

γ

− 1+

ρ

ud3+

ρv

d4+

ρ

wd5 +

∂

∂

y[

v

(

ρ

es+p

)

]+

∂

∂

z[w

(

ρ

es+p

)

]=

∇

(

λ∇

T

)

+

∇

(

u.

τ

)

(2)

∂

(

ρ

u

)

∂

t +ud1+

ρ

d3+

∂

∂

y

(

ρv

u

)

+

∂

∂

z

(

ρ

wu

)

=

∂τ

1j

∂

xj (3)

∂

(

ρv

)

∂

t +

v

d1+

ρ

d4+

∂

∂

y

(

ρvv

)

+

∂

∂

z

(

ρ

w

v

)

+

∂

p

∂

y=

∂τ

2j

∂

xj (4)

∂

(

ρ

w

)

∂

t +wd1+

ρ

d5+

∂

∂

y

(

ρv

w

)

+

∂

∂

z

(

ρ

ww

)

+

∂

p

∂

z =

∂τ

3j

∂

xj (5) where

τ

istheviscous stresstensor.esandE arethesensibleand

totalenergiesrespectively:

es=  T To CvdT and E=es+ 1 2

(

u 2₊

_v

2_+w2

₎

₍₆₎

Thesystemof Eqs.(1)to (5)containsderivativesnormaltothex boundary(d1 tod5),derivativesparalleltothexboundary(called “transverse terms”), andviscous terms. The vector d is given by characteristicanalysis:

2 Note that the present paper focuses on inlet boundary conditions: readers are

d=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

d1 d2 d3 d4 d5

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

1 c2



L2+ 1 2

(

L5+L1

)



1 2

(

L5+L1

)

1 2

ρ

c

(

L5− L1

)

L3 L4

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

∂

(

ρ

u

)

∂

x

ρ

c2

∂

u

∂

x +u

∂

p

∂

x u

∂

u

∂

x+ 1

ρ

∂

p

∂

x u

∂v

∂

x u

∂

w

∂

x

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(7) where c isthe localspeed of soundgivenby c2₌

_γ

_p/

_ρ

_andthe Li’sare theamplitudesofcharacteristicwavespropagatingatthe

characteristicvelocitiesu− c,uandu+c:

L1=

(

u− c)

∂

p

∂

x −

ρ

c

∂

u

∂

x



(8) L2=u

c2

∂ρ

∂

x −

∂

p

∂

x



(9) L3=u

∂v

_∂

x and L4=u

∂

w

∂

x (10) L5=

(

u+c

)

∂

p

∂

x +

ρ

c

∂

u

∂

x



(11)

3. Specificationofincomingwaves

For a subsonic three-dimensional inlet, the problem is well posed if four conditions are imposed [3,36]. In a characteristic

based methodsuch asNSCBC,thismeansthat thefourincoming

waves_L2,L3,L4 andL5 mustbeimposed.The outgoingwaveL1 doesnotdependontheboundaryconditionsandcanbecomputed using one-sided derivatives ofthe field inside the computational domain. Therefore, the solution can be advanced in time on the inlet,usingthesystemof Eqs.(1)to (5)forboundaryvaluesifan evaluation forL2 to L5 can be found. The principle of NSCBCis toevaluate thesewave amplitudesasiftheflow waslocally one-dimensional andinviscid (LODI).LODI equations providean esti-mationofthewave amplitudesLiwhichisusuallychosensothat

thephysicalboundaryconditionissatisfied.UsualLODIequations are:

∂ρ

∂

t + 1 c2



−L2+ 1 2

(

L5+L1

)



=0 (12)

∂

u

∂

t + 1 2

ρ

c

(

L5− L1

)

=0 (13)

∂v

∂

t +L3=0 (14)

∂

w

∂

t +L4=0 (15)

∂

p

∂

t + 1 2

(

L5+L1

)

=0 (16)

The difficult question and the differentiating factor between

characteristicmethodsisthespecificationoftheingoingwave am-plitudesL2,L3,L4 andL5 as afunction of the chosen inlet condi-tions.Forexample,foraconstantvelocityinlet,theLODIequation

(13)wouldsuggestthat theincomingacousticwave amplitudeL5 shouldbeequaltotheoutgoingwaveL1but this approachis often

toosimpleforunsteadycases.3 _{Forthesakeofsimplicity,the}

pre-sentationislimitednowtoan isentropicinletwhere theentropy wave L2 is set tozero.Using Eqs. (13)–(15), theLODI expression forsuchanisentropicinletistowritetheincomingwavesas:

L5=−2

ρ

c

∂

ut

∂

t ,L3=−

∂v

t

∂

t andL4=−

∂

wt

∂

t (17)

which allows to inject an unsteady signal of components (ut_{, v}t_,

wt_{).Unfortunately, Eq.(17)doesnotworkinpracticeforthree}

rea-sons:

1. For a steady inlet (ut₌

_v

t_=wt_{= constant),this condition}

isperfectly reflecting as the ingoing waves L3, L4 and L5 areallexactlyzero:thesolverknowsthattheinletvelocity isconstant but it hasno information on the valuesof the targetvelocitiessothat,inmultidimensionalconfigurations, themeaninletvelocitiesusually driftbecauseoftransverse andviscous termspresentin thesystemof Eqs.(1) to (5). Thisisusuallycorrected byaddingalinearrelaxationterm to the target values ut_{, v}t _{and w}t _{asproposed initially by}

RudyandStrikwerda [36,38].Forthenormalvelocityu,the ingoingacousticwaveL5becomes:

L5=

ρ

c

−2

∂

ut

∂

t +2K

(

u− u t

₎



(18) whilethetwotransversewavesarewritten:

L3=−

∂v

t

∂

t +2 K

(

v

−

v

t

₎

_{and L} 4=−

∂

wt

∂

t +2K

(

w− w t

₎

(19) Termssuchas

(

u− ut

₎

_{arecalled“relaxation” terms [36,38].}

Theydonothaveatheoreticalbasis4_{:theyofferthesimplest}

linear correction form which can be added to the NSCBC

theorytoavoidadriftofmeanvaluesasitforcesthe instan-taneousvelocity utogoto itstarget valueut _witha

relax-ationtime1/K.Independentlyofitsexactform,thistermin

Eq.(18)issufficientto avoiddriftingmeaninletspeed val-ueswhenKis“sufficiently” largebutitalsodeterioratesthe non-reflectingcharacter oftheinletaswill beshownlater. Transversewaves(L3andL4)raisenodifficultyasthey are not associated to anyaxial acoustic wave and will not be discussedanymoreintherestofthepaper.

2. An interesting issue in the correction term

(

u− ut

₎

_of

Eq. (18) is how to choose the “target” value ut_{. A simple}

choicewouldbeut_=u¯+ut

+whereu¯isthetargetmean ve-locityandut

+ isthe target unsteadyvelocity (either acous-ticorvortical) imposed onthe inletpatch. A betterchoice wasproposedin [1,2]whopointedoutthatoutgoing acous-ticwaves(inducingvelocityfluctuationswhichwillbecalled u₋) mayalsoreachtheinletwhentheypropagatefromthe computationaldomaintotheinletandshouldbeaccounted forin ut_{. Fortunately,these outgoing waves can be}

evalu-ated in the limit of plane, low-frequency waves using the outgoingwaveamplitude_L1whichisreadilyavailableinall NSCBCmethods.Therefore,theproperwaytoaccountforu₋ istoaddittothetargetvelocitytohaveut_=u¯+ut

++u−. A last issue linked to Eq. (17) is the choice of the relax-ation coefficient K (units: s−1). The proper scaling for K is

3 One aspect of this problem which is not discussed here is the need to also

incorporate transverse terms in the ingoing wave expressions [11,30,37] . At an inlet, these terms play a limited role and they will be omitted throughout the present paper.

4 A temptative explanation for this expression was actually proposed recently by

Fig. 3. Typical behavior of the solution for the classical NSCBC inlet condition ( Eq. (18) ) as a function of the reduced relaxation coefficient σ= KL/c. L is a typ- ical domain length of the domain and c the sound speed. The shaded area is the desired operational zone.

thereducedfactor

σ

=KL/c where Landcare a character-isticlengthandatypicalsoundspeedofthedomain respec-tively [36,38].Choosinganadequate valueforKisacritical issue inmanycases.Verylarge valuesof

σ

can lead toan unstablesolutionandadivergenceofthesimulationevenin a stableflow(Fig.3) becausethewave amplitudebecomes too large as soon as the velocity deviates from its target value: such instabilities are purely numerical and are due to the boundary condition. On the other hand,low values provide non-reflecting characteristics butwilllet the mean solutiontodriftfromitstargetmeanvalueu¯becauseof vis-cousandtransversetermsaffectingthesolutioninthe sys-temof Eqs.(1)to (5).Therefore,thereusually isarangeof

σ

values which provide both non drifting and quasi

non-reflecting properties.Rudy andStrikwerda suggestthat this occursnear

σ

₌0_.25butinpractice,widerangesof

σ

have to betestedbyNSCBCusers,leadingtoinefficienttrialand errorprocedures.In somecases,largevaluesofKare used, leadingto inlets wherethe velocity istotally fixedbutthe boundaryisfullyreflecting.

3. Another and more surprising problemwas pointed out by

Prosser [39]andconfirmedbyGuezennecandPoinsot [40]. IntheclassicalNSCBCapproach,toinjectaperturbationut_,

theincomingwaveisexpressedas:

L5=−2

ρ

c

∂

ut

∂

t (20)

This is a correct formulation to inject acoustic waves but

Prosser [39] used a low Mach number expansion of the

Navier–Stokesequationstoshowthattheproperexpression to inject vortical perturbations was different and that the factor2hadtobesuppressedtohave:

L5=−

ρ

c

∂

ut

∂

t (21)

The fact that the factor 2 of Eq. (20) used for acoustic wave injection must be removed for vorticity injection in

Eq. (21) was confirmed by the analysis of Polifke et al.

[1] andtests [40] show that indeed, Eq. (21)is the proper wave expression to inject vortical perturbations (isolated vortices or fully developed turbulence) but raise a simple question: which expression (Eqs. (20) or (21)) should be used in practice?The presentpaper showsthat they actu-allymustbecombinedasdiscussedbelow.

Theserecentresultssuggestageneralizedformulationforinlet boundaryconditionswhichisthe basisfortheNRI-NSCBC condi-tiondescribedhere5_{Inthisformulation,ingoingperturbationsare}

splitinto two separate components whichare superposed tothe incomingwave amplitude:the vorticalfluctuation, corresponding

5 The reviewing phase of the present paper showed that this result could have

been obtained also by combining results proposed by the group of Pr Polifke [1] with the work of Pr Prosser[39] .

Table 1

Comparison of the classical NSCBC and the NRI-NSCBC conditions. ¯u is the mean target velocity, u t

a is the target acoustic fluctuation, u tv is the target vor-

tical fluctuation and u −is the reflected velocity fluctuation reaching the inlet

from the computational domain.

Boundary Transverse Axial condition Waves L 3 and L 4 Wave L 5

NSCBC L 3 = −∂v t ∂t and L 4 = −∂w t ∂t Lρ5c = −2 ∂ ut a ∂t − 2 ∂ut v ∂t +2 K[ u −( ¯u + u t a + u tv)] NRI-NSCBC L 3 = −∂v t ∂t and L 4 = −∂w t ∂t Lρ5c = −2 ∂ut a ∂t − ∂ut v ∂t +2 K[ u −( ¯u + u t a + u tv + u −)] toasignalut

v andtheacousticfluctuationcorresponding toa

sig-nalut

a.Eachcomponentishandledindividuallyandsuperposedin

theincomingwaveasfollows:

L5

ρ

c=−2

∂

ut a

∂

t −

∂

ut v

∂

t







I +2K[u−

(

u¯+ut a+utv+u−

)

]







II (22)

where partI of expression (22)combines a term2

∂

ut

a/

∂

t to

in-troduce acoustic waves and another one

∂

ut

v/

∂

t to inject

turbu-lence,eachofthem withthecorrectfactor(2foracousticsand1 for turbulence). Part II of expression (22) is the relaxation term. Itis introducedto avoiddriftandisnot proposed bythe charac-teristic theory.Itincludes ut

a andutv butalsou− assuggestedby Polifke etal. [1]. This formulation allows to use exact terms for 2

∂

ut

a/

∂

t and

∂

utv/

∂

t(satisfyingpropertiesP2andP3introducedin Section 1) whileproviding an expression for the relaxationterm whichshouldbezeroaslongasnonacoustictermsremainsmall attheinlet(satisfyingpropertyP1).Thisshouldallowtouselarge relaxation factors K avoiding the drift of mean values while still beingnonreflecting forallnormal acousticwaves:therelaxation termin Eq.(22)becomesnonzeroonlywhen viscousand trans-versetermsbecomenonnegligibleontheinlet.Forallothercases, the relaxationterm(II) iszero and Eq.(22)reduces to theexact NSCBCapproachforL5:L5=−2

ρ

c∂u t a ∂t −

ρ

c ∂ut v ∂t .

IntheexpressionoftheincomingwaveL5(Eq.(22)),the acous-ticvelocityu− associated tothereflectedwave reaching theinlet fromthecomputationdomainmustbeevaluated.Inthecaseofan outlet, Polifkeetal. [1,41] useda methodcalledCBF (characteris-ticsbasedfilter)toobtainaplaneaveragedvaluefortheoutgoing wave amplitude. Thisrequiresintroducinga seriesofplanesnear

theoulet ofthe computational domainwhere theoutgoingwave

can be evaluated. CBFis precise but can be difficult or impossi-bletoimplementincaseswherethedomainoutlethasacomplex shapeortypicallyataninletasstudiedinthepresentwork.Here an alternative techniqueis usedwhere u₋ is evaluated locallyat eachpointoftheinletpatchfromthetimeintegraloftheacoustic waveamplitudeL1 whichisavailableinNSCBC:

u₋= 1 2

ρ

c

t

0

L1dt (23)

Expression (23)foru₋avoidsusingthePWMapproach [1]andcan beusedinanycodeusingNSCBCboundaryconditions.6

The following sections compare the new NRI-NSCBC

formula-tion tothe classicalNSCBC conditions [3,12]. Table 1summarizes thewaveexpressionswhichwillbeusedforbothboundary condi-tions.

6 In practice, for certain cases, _{Eq. (23) must be high-pass filtered to remove any}

4. Theoreticalanalysisofreflectioncoefficientsinone dimension

Afirstmethodtoanalyzethedifferencesbetweenthestandard NSCBCandtheNRI-NSCBCconditionsistoconsiderasimplecase (Fig.4) such asthe inlet of a configuration wherewaves can be assumedtobeone-dimensional.No vorticalperturbationis intro-duced:ut

v=0.Theinletpatchhasaconstantmeanvelocityu¯and

is submittedto an acousticharmonicforcing atpulsation

ω

with a targetamplitudeut

a.Areflected waveinducing an acoustic

per-turbation u₋ associatedto a wave amplitudeL1 alsoreachesthe inlet so that the exactvelocity fluctuation at theinlet should be ut

a+u−.Atthispointu−couldbeanysignal:theonlyassumption

isthat u₋isa reflectionduetotheacousticforcingandthatitis thereforeaharmonicsignalatpulsation

ω

too.

This caseallows to derive analytically, what the inlet velocity willbeinacodeusingtheboundaryconditionsof Table1.To ob-tain thisresult,thetwo ingoingwave formulationsof Table1are gathered in a single notation for this case with acoustic forcing onlyandnoturbulenceinjection(ut

v=0): L5

ρ

c=−2

∂

ut a

∂

t +2K[u−

(

u¯+u t a+

α

u−

)

] (24)

When

α

=0,thestandardNSCBCconditionisobtainedwhile

α

= 1 yields theNRI-NSCBCcondition. The reflected wave (amplitude L1)createsanacousticvelocityu−whichreachestheinletand in-teracts withtheinletboundary condition. Ingeneral, u₋ isnever zero:reflected wavesare foundattheinlet ofmostcompressible computations.

Assuming that all quantities fluctuate atpulsation

ω

and ex-pressing all variables as f

(

t

)

₌Re[fˆexp

(

_−i

ω

t

)

]_, it is possible to combine Eqs.(24)and (13)toobtainthevelocityfluctuationsu= u− ¯uattheinlet.Todothis,L1 isobtainedfromu−using:

∂

u₋

∂

t =

1

2

ρ

cL1 so that Lˆ1=−2i

ωρ

cuˆ− (25)

Eq.(13) can thenbe used toobtain the inletvelocity fluctua-tionsuˆusing Eq.(24)forL5 and Eq.(25)forL1:

∂

u

∂

t =−

1

2

ρ

c

(

L5− L1

)

(26)

whichleadsto: ˆ

u=uˆt a+

K

α

− i

ω

K− i

ω

uˆ− (27)

Eq.(27)conveystwomessages:

• Ingeneral, the inletvelocity fluctuation uˆ isnot equal tothe acoustic forcing amplitude imposed on the boundary uˆt

a. For

both boundary conditions (

α

₌0 or 1), the only cases where ˆ

u isequal touˆt

a corresponds to situationswhere nooutgoing

wavereachestheinlet(uˆ₋=0).

Fig. 4. Characteristic waves at a subsonic inlet (inlet at x = 0 ). The acoustic forcing induces a velocity fluctuation u t

a . A longitudinal acoustic wave (amplitude u −) is

reaching the inlet from the computation domain.

• The exactsolutionisthat theinletvelocityshouldbethesum of theacoustic contributionscoming fromleft andright:uˆ=

ˆ ut

a+uˆ−. When

α

=1 (NRI-NSCBC condition), Eq. (27) shows

that this property isalways satisfied. Forthe standard NSCBC condition(

α

=0),theconclusionisopposite:theinletvelocity

ˆ

u is never equal to its theoretical value except in rare cases whereK₌0oruˆ₋₌0.

Knowinguˆ,itisalsopossibletoexpresstheinletwaveL5:

ˆ

L5=Lˆt5+R1Lˆ1 with R1=

K

(

1−

α

)

K− i

ω

(28)

where _Lˆt

5=2i

ωρ

cuˆta is the target forcing wave and R1 can be viewedasthe reflectioncoefficient of theboundary condition: it measureshowmuchofaleftgoingwaveL1reflectsintothe ingo-ingacousticwaveL5.When

α

=1(NRI-NSCBC),R1isexactlyzero: theinletistrulynon reflectingandtheinjectedwave_Lˆ₅contains onlytheimposedwave_Lˆt

5.ForthestandardNSCBCcondition(

α

= 0),R1 isneverzero:theinletisreflectingandanyoutgoingwave reachingitwillbereflectedbackintothedomain,makingtheinlet effectivelymoreandmorereflectingasKisincreased.Notethat,in thiscase,thereflectioncoefficient R1 in Eq.(28)matchesthe ex-pressionobtainedforthereflectioncoefficient bySelleetal. [34], Polifkeetal. [1]andPirozzoliandColonius [10].

5. Aone-dimensionalductwithinletacousticforcing

The two boundary conditions of Table 1 are testedfirst on a one-dimensionalductoflengthL(Fig.5) forcedacousticallyatits inlet(x₌0) andterminated by a fixed pressure outlet(p₌0at x=L).Thisisadirectapplicationoftheresultsof Section4where u₋willbespecified:theinletforcingisharmonicandcorresponds toavelocityfluctuationuˆt

a.Theoutletisfullyreflectingsothat

in-goingwaveswillreflectatx=Lintooutgoingwaves(u₋) and in-teractwiththeinletconditionatx=0.Foralltestcasespresented in thissection and the following ones, the compressible Navier– Stokes equations are solved using the fourth-order TTGC scheme (onregularmeshes [42])intheAVBPsolver [43,44].Time advance-mentisfullyexplicitandaCFLnumberof0.7isusedforallruns. Sincethe outletreflectioncoefficient is ₋₁ (to ensure p₌0), theratiobetweenL1andL5isknown(L1/L5=− exp

(

2ikL

)

where k=

ω

/c is the wave number). Therefore the ratio I between the waveamplitudewhichtheboundaryconditionshouldimpose(_Lt

5) andthewavewhichwillactuallybeimposed(L5)by Eq.(24)can alsobeexpressedas:

I= Lˆ5 ˆ Lt 5 = 1 1+R1exp

(

2ikL

)

(29)

Iisadeterioration index:whenit isequalto unity,theboundary conditionis perfect,the injectedwave istheone imposedby the userandtheinletvelocityisu=u¯+ut

a+u−whichistheexact so-lution.Anynon unityIvalueindicates thattherelaxationtermin

Eq.(24)isperturbingthe inletboundaryconditionandmaking it partiallyreflecting. Obviouslyforthe NRI-NSCBCconditionwhere

α

=1 and R1=0, Iis equal to unity for all K values while it is not for the standard NSCBC approach. To check this result, one-dimensional simulations of the configuration of Fig. 5 were per-formedfor

σ

=0,2 and5atthree forcing frequencies (100,200

Fig. 5. Tests of inlet boundary conditions for a one-dimensional duct forced by a harmonic wave at the inlet and terminated by a pressure node at x = L .

Fig. 6. Comparison of the quality index I ( Eq. 29 ) in log scale, for the pulsated duct of Fig. 5 : analytical result ( Eq. (29) ) vs compressible simulations. (a) standard NSCBC method, (b) new NRI-NSCBC method. Lines: analytical solutions; symbols: simulations.

and500Hz).TheresultsobtainedintermsoftheindexIasa func-tionofthereducedrelaxationcoefficient

σ

aredisplayedin Fig.6. Thesimulationsmatchexactlytheanalyticalresultof Eq.(29)and confirmthatfortheNSCBCstandardformulation,thedeterioration indexIcanreachlargevaluesmorethan20for

σ

=5),indicating

thatthis methodis muchlessaccurate than the newNRI-NSCBC

conditionwhich offers aunity Iindexforallvalues ofthe relax-ation coefficient

σ

. These results confirm the analysis of Polifke etal. [1] who pointed out the importance oftaking u₋ into ac-countintherelaxationterm.

6. Fasterconvergenceforunforcedcompressibleflows

The capabilities ofthe NRI-NSCBCcan be illustrated ina sec-ondtest,todemonstratethatitallowsfasterconvergencein multi-dimensional,compressible,unforcedflowsbecauseacousticwaves are perfectlyevacuated through the boundarieswhile avoiding a driftofmeanvalues,thankstoalargevalueoftheinletrelaxation coefficientK.Foranunforcedflow(ut

a=utv=0),theinletwaveof Eq.(22)becomes:

L5=2K

ρ

c[u− ¯u] forNSCBC (30)

and

L5=2K

ρ

c[u−

(

u¯+u−

)

] forNRI− NSCBC (31)

Thetest caseisatwo-dimensional subsonicnozzle(Fig.7)where theinletMachnumberis0.014,correspondingtoaninletvelocity of5m/s.Theoutletconditionisp=0.Theinitialcondition corre-spondstoazerovelocityfield andconstantpressureand temper-atureeverywhereinthedomain,includingontheinletpatch.The shapeofthenozzleisgivenby(unitm):

y=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

0.02



1.0− 0.661514e  −ln2(x/0.6)2



, x<0 0.02



1.0− 0.661514e  −ln2(x/6)2



, x≥ 0 (32)

When the simulation begins,the inlet boundary conditionstarts modifying theinletvariables. Theobjectiveofthetest isto mea-surethephysicaltimerequiredforthesimulationtoreachsteady stateandtocheckwhetheracousticmodesoftheconfigurationare triggered.

Thisflowisagoodprototypeofmanycompressiblesimulations. The initial conditions (zero velocity everywhere) combined with the inlet condition (which ramps rapidly to its target value) can generatestrong perturbationsand acousticwaves:withthe stan-dardNSCBCconditions(Fig.3), lowvaluesofKleadtomean val-ueswhichdonotconvergetothetargetvaluesordriftawayfrom them. On the other hand, large values of K avoid drifting mean

Fig. 8. Typical behavior of the solution for the new NRI-NSCBC inlet condition as a function of the reduced relaxation coefficient σ= KL/c. L is a typical domain length and c the sound speed. The shaded area is the desired operational zone.

valuesbutinducereflectionsandundampedacousticwaveswhich delayconvergence.WithNRI-NSCBC,thisproblemdisappears:itis possibletouselargevaluesofKandstillbenonreflectingsothat convergenceis reachedveryfast. Fig.8summarizes this observa-tionandcanbecomparedto Fig.3.

Fig. 9 displays a typical time evolution of inlet velocity and pressure for a reduced relaxation coefficient

σ

=KL/c=0.017. Since this coefficient is small, the inlet velocity (Fig. 9, left) in-creasesslowlyandnoacousticwavesaretriggered.However, con-vergenceisreachedafteralongtimeforbothmethods(NSCBCin solidlineandNRI-NSCBCindashedline).

To increase the convergence speed, the natural solution is to increasetheinletrelaxationfactor: Fig.10showsthesolutionsfor a reducedrelaxation coefficient

σ

=KL/c=17. The inlet velocity rapidlyreachesitstarget(5m/s)butacousticoscillationsare trig-geredusingtheclassicalNSCBCmethod(solidline)because acous-ticwavesaretrappedwithinthedomain:pressureandvelocity os-cillate at150 Hzwhich isthe frequencyofthe firstmode ofthe setup with u=0 atthe inlet and p=0 atthe outlet. Inversely, theNRI-NSCBCmethodgivesasolution(dashedline)whichis sta-bilizedafteroneacoustictime.

Increasingthe relaxationcoefficient evenmore (asoften done by NSCBC users when they observe an oscillating inlet velocity) doesstabilize theinlet velocity (Fig.11,left) butmakes theinlet even more reflecting, leading to pressure inlet excursions which actually grow in time (Fig. 11, right) and a boundary condition whichisillposedandwilleventually leadtofulldivergence. NRI-NSCBC,asexpected, is onlyweakly affectedby the increaseof

σ

andleadstoastablesolutionrapidly.Thissimpleexamplereveals anotherinterestingfeatureoftheNRI-NSCBCcondition,which pro-videsfastconvergencetosteadystate,usinglargerelaxation coef-ficients,ausefulpropertyinallcompressiblesolvers.

7. Simultaneousinjectionofturbulenceandacousticwaves throughanon-reflectinginlet

Thistest casecorrespondsto asituationwherean inlet(x=0 in Fig. 12) is used to inject both turbulence and an harmonic

acousticwaveintoasquaresectionchannel.Thedomainisa three-dimensionalparallelepipedicbox wheretheoutletisfully reflect-ing(imposedpressure: p=0atx=L).Therefore,theinletis sub-mittedtothreewaves:

• vorticitywavesassociatedtotheturbulenceinjectionut v, • aningoing acousticwave associated tothe acousticforcing

ut a,

• an outgoing acoustic wave reflected from the outlet and

propagatingbacktotheinletu₋.

The mesh isa pure hexahedragrid with 392_{× 98× 98} points correspondingtoadomainsizeofL=4× 1× 1mm.Themean in-letvelocity is homogeneousin thex=0plane:U=100 m/s. Pe-riodicityconditionsareapplied inthetwotransversedirectionsy andz.VerylargevaluesoftherelaxationcoefficientKareusedin both NSCBCandNRI-NSCBC:K=2.106_s−1 _{corresponding to a} re-ducedcoefficient

σ

₌KL_/c=23.The inletissubmittedto two si-multaneousoutsideexcitations:

1. Three-dimensional turbulence: the RMS velocity of the in-jectedturbulent field is ut

v,RMS=5 m/s and its most

ener-geticwavelengthis0.5mm.Theturbulencespectrumhasa PassotPouquetexpression [45].

2. One-dimensionalplanaracoustic wave:theacousticforcing isa longitudinal harmonic wave introduced at the domain inlet,at a frequenc f₌260kHzwith a peak amplitudeof ut

a,peak=2m/s.

Notethat the ratiobetween thetwo excitations levels canbe fixed arbitrarily and is configuration dependent. It is measured herebytheratiooftheexcitationvelocitiesut

v,RMS/uta,peak=2.5.

Fig.13displaysfieldsofQ-criterion [46]forthestandardNSCBC (left)andthe NRI-NSCBCapproaches(right), showinga usual de-caying turbulent field. The same figure displays a pressure field in one plane, revealing that the axial plane acoustic forcing can be identified on the pressure signal. These qualitative results re-quiremoreanalysistoseetheinfluenceoftheboundarycondition.

Fig.14showstheFFTsofpressure(left)andvelocity(right)atthe inlet(x=0m).

The two velocity spectra (right image) of Fig. 14 are close

and both boundary conditions (NSCBC and NRI-NSCBC) produce

inlet fluctuations which match the spectra of the target veloc-ityut very well, confirmingthat the turbulent signal is correctly

introduced. Note that no discrete peak is visible at the acoustic forcing frequency (f=260 kHz). For the conditions chosen here (ut

a,peak/utv,RMS=0.4),the acousticforcing isnot strongenough to

dominatetheturbulentforcing.Forpressure(leftimage),however, the two spectra differ: both exhibit a peak atthe acoustic forc-ing frequency(andits first harmonic) buttheNSCBCresults also

Fig. 10. Inlet velocity (left) and pressure (right) time evolutions for σ= KL/c = 17 . NSCBC: solid line. NRI-NSCBC: dashed line.

Fig. 11. Inlet velocity (left) and pressure (right) time evolutions for σ= KL/c = 170 . NSCBC: solid line. NRI-NSCBC: dashed line.

Fig. 12. Simultaneous injection of three-dimensional turbulence (velocity ampli- tude u t

v ) and one-dimensional acoustic forcing (velocity amplitude u ta ) at the inlet

of a domain with reflecting outlet ( p _{= 0 ).}

revealmultiple other peaks dueto non physicalresonances. It is interestingtocomparethepressurefield obtainedintheLESwith theanalyticalsolutioncorrespondingtoaforcedinletatfrequency fandanoutletconditionp₌0.Inalaminarflow(intheabsence

ofturbulenceinjection),takingintoaccountthecorrectiondueto non-zeroMachnumber,thissolutionis:

p

(

x,t

)

=

ρ

cua



e−ik+x− e−i(k+_L+k−_(L−x))



eiωt (33) and u

(

x,t

)

=ua



e−ik+x_+e−i(k+_L+k−_(L−x))



eiωt ₍₃₄₎

wherek+=

ω

/

(

c+u

)

,k−=

ω

/

(

c− u

)

and

ω

=2

π

f . Thevariance ofpressurep2_{canbeobtainedby p}¯2_=pap∗

a/2,withp∗athe

con-jugate complexof pa. Fig. 15 showsvariationsof



p2 _{along the} ductaxisforbothboundaryconditionsandcomparesittothe an-alyticalsolutionof Eq.(33).TheNRI-NSCBCcapturesperfectlythe analyticalsolutionshowingthatfortheseconditions(ut/ua=2.5),

theunsteady pressurefield is onlyweakly affectedby the turbu-lence injection and that the NRI-NSCBC condition doesnot alter thisproperty.Ontheother hand,similarlyto thecaseofacoustic forcinginalaminarflow(Section 5),theclassicalNSCBC formula-tion modifiestheacoustic field structure andfailsto capturethe analyticsolution.

Fig. 13. Simultaneous injection of isotropic homogeneous turbulence and acoustic wave. Isosurface of Q criterion Q = 2 . 5(U/L )2 colored by the axial velocity

Fig. 14. Simultaneous injection of isotropic homogeneous turbulence and acoustic wave. Spectra of pressure (left) and velocity (right) at the domain inlet x = 0 m, y = z = 0 . 005 m. Solid line: NSCBC, dashed line: NRI-NSCBC. The arrows indicate the frequency ( f a = 260 kHz) at which the acoustic wave is introduced. The triangles on the right

image correspond to the spectra of the injected turbulence target signal u t

v .

Fig. 15. Pressure perturbation structure along duct axis: field of( p 2_{) vs x . Solid}

line: NSCBC, dashed line: NRI-NSCBC, symbols: analytical solution.

8. Aturbulent,acousticallyforcedpremixedflame

The last example is a direct numerical simulation (DNS)of a stoichiometric,premixedturbulentflamestabilizedinaslot-burner configuration.Theinletisforcedbyturbulenceandbyanharmonic acousticwaveintroduced simultaneously,ausualsituationfor ex-ampletostudyFlameTransferFunctionsinthermoacoustics.

The DNS is performed with the explicit, compressible solver (AVBP) for the 3D Navier–Stokes equations with simplified ther-mochemistryonunstructuredmeshes [43,47].ATaylor–Galerkin fi-niteelementschemecalledTTGC [42]offourth-orderinspaceand time isused. The acoustic CFL number is 0.7. The outlet

bound-ary condition is handled using an imposed pressure, NSCBC

ap-proach [3] with transverse terms corrections [37]. The inlet is treated either withthe standard NSCBC formulation orwith the

new NRI-NSCBC approach. Other boundariesare treated as

peri-odic.

Methane/air chemistry at 1 bar is modeled using a global 2-step schemefittedtoreproduce theflamepropagationproperties suchastheflamespeed,theburnedgastemperatureandtheflame thickness [48]. This simplified chemistry description is sufficient to studythe dynamicsof premixedturbulent flames. Fresh gases arestoichiometric:thelaminarflamespeedisS0

L =40.5cm/s.The

flamethicknessesare

δ

0

L =0.34mm(basedonthemaximum

tem-perature gradient) and

δ

1

L =0.7 mm (based on the distance

be-Fig. 16. Physical domain used for the DNS. At the inlet, a double hyperbolic tan- gent profile is used to inject fresh gases in a sheet ≈ 8 mm high, surrounded by a coflow of burnt gases. Top-bottom (along y ) and left-right (along z ) boundaries are periodic. The isosurface is a typical view of T = 1600 K.

tweenreducedtemperaturesof0.01and0.99).Themeshisa ho-mogeneoushexahedragridwithaconstantelementsize



x=0.1 mm,ensuring7–9pointsinthepreheatzoneand4–5inthe reac-tionzone. Withthisresolution,thetemperatureandheat release profiles givenby the DNScode matchperfectly theresults given byaspecializedone-dimensionalflamecode(Cantera)fora lami-narpremixedflame.The domainsizeis512cells(5,12 cm)inthe xdirectionand256cells(2,56cm)intheyandz ones,foratotal of33.55millioncells(Fig.16).Thefreshgasinjectionchannelhas aheighth=8.53mm(h/

δ

0

L ≈ 25).

Theinletstreamisa centralflowofstoichiometricfreshgases surroundedbyacoflowofburntgasesatlowinjectionvelocity. In-lettemperaturesare300and2256Kinthefreshandburntgases, respectively. The temperature and composition of the burnt gas coflowcorresponds totheproducts ofan adiabaticcombustionof thefreshgases.Meaninletvelocityprofilesareimposedas:

u

(

x=0,y,z

)

=uco+

(

uin− uco

)

1+tanh

y− h 2

δ



×

1− tanh

y− 2h 2

δ



(35)

where uin=10 m/s is the maximum speed in the fresh gases,

uco=0.1 m/s is the minimum speed in the coflow of hot gases

and

δ

is the momentum thickness of the shear layer (

δ

=0.08

mm) corresponding to a vorticity thickness of 0.36 mm.

Turbu-lenceis injectedin thefreshgasesonly. The RMSvelocity ofthe incomingflowis1m/sandtheintegrallengthscaleislF =2mm.

Thespectrumoftheinjectedturbulencecorresponds toaPassot– Pouquet form [45].Acoustic forcing isintroduced atthe inlet on the fresh gas stream. The forcing frequency is fa=1 kHz and

Fig. 17. Flame response at four instants of the acoustic forcing period ( f a = 1 kHz). Temperature field and isolevels of vorticity.

Fig. 18. Pressure spectra for NSCBC (solid line) and NRI-NSCBC (dotted line) at the domain inlet. The f a arrow corresponds to the acoustic forcing at 1 kHz. The three

other arrows are the first three longitudinal eigenmodes of the computational box at 4350, 12500 and 19500 Hz. The spectral resolution is 70 Hz.

conditionsimposed at theinlet, the value ofK isthe same: K= 100000 s−1 (leading to

σ

ࣃ15). Aseries of 4snapshots showing theflame responseto turbulent andacousticforcing is displayed in Fig.17.Mushroom-shapedflamestructuresarecreatedat1Khz asexpectedforacousticallyforcedflames [49].Thesestructuresare separatedspatiallybyuin/ fa=1cmandinteractwiththeinjected

turbulence.

ThetwoDNSof Fig.17,obtainedbyNRI-NSCBCandNSCBC,are obviouslydifferentbutitisdifficultto saywhichoneis thebest. Amorequantitativeresultcanbeobtainedbylookingatthe pres-surespectraatthedomaininletin Fig.18.Thespectraobservedfor NRI-NSCBCcorresponds tothe expectedresult:a discretepeak at theacousticforcing frequencyfasuperimposed onbroadband

tur-bulentnoise.Ontheotherhand,forNSCBC,threeadditional high-levelpeaksalsoappear:theyareduetotheexcitationbytheflame oftheeigenmodes ofthe computational domain.It ispossibleto

verify that these modes are indeed acoustic modes by using an

Helmholtzsolver [50] taking intoaccount the mean temperature distributioninthedomainandtheboundaryconditions(imposed inletvelocity andimposed outlet pressure). The frequencies pre-dicted forthe firstthree acoustic modesgiven by the Helmholtz solveraremarkedby arrowsin Fig.18.The threeacousticmodes

which are excited when NSCBC is used are the longitudinal 1/4

wave(at4350Hz),3/4wave(at12500Hz)and5/4wave(at19500 Hz)modes.Theirfrequencies,computedwiththeHelmholtzsolver, matchthefrequenciesobservedintheLESwitha5percent accu-racy.Thesemodesinteractwiththeflameresponseattheacoustic forcing frequency(fa=1 kHz)and makethe NSCBC run difficult

tointerpret:measuringtheflameresponseat fa=1kHzwouldbe

impossiblefortheNSCBCrunbecausethisresponseispollutedby thethreeacousticeigenmodesforcedbytheboundaryconditions. Clearly,NSCBCfailstoinjectacousticforcingandturbulence with-out excitingthecavitymodesofthe computationaldomainwhile NRI-NSCBCsucceedsinthistask.

9. Conclusions

This paper has described a new boundary condition for

sub-sonicinlet,basedonacombinationoftheformalismproposed by Polifkeandcoworkers [1,2]toaccountforoutgoingacousticwaves andan extensionofthe methodofGuezennecetal.[40] to intro-duceboth turbulenceandacousticwavessimultaneously.The ap-proachcanbesummarizedintheexpressionoftheingoingwave:

L5

ρ

c=−2

∂

ut a

∂

t −

∂

ut v

∂

t +2K[u−

(

u¯+u t a+utv+u−

)

] (36) where ut

a is the velocity of the injected acoustic wave, utv is the

axialvelocityoftheturbulentsignal,u¯isthemeantargetvelocity, Kisarelaxationcoefficientandu₋ isthevelocityoftheoutgoing wave which is estimated locally usingthe outgoing wave ampli-tudeL1: u₋= 1 2

ρ

c t 0 L1dt (37)

Analysis and tests show that thisNRI-NSCBC condition performs better than the standard NSCBCapproach: it allows to use large values forthe relaxation coefficient K andto obtain non-drifting mean values and non reflective capabilities simultaneouslyecor. Tests were performed for one-dimensional acoustic forcing in a duct, for flow establishmentin a compressiblenozzle, for simul-taneous injection of acoustic waves and turbulence in a three-dimensionalchannelterminated byafixedpressureoutletand fi-nally for a turbulent premixed flame forced acoustically. For all cases, NRI-NSCBCcaptured the expected solution accurately sug-gestingthatthiscould becomea standardapproach in compress-iblecodes.

Acknowledgments

We thankDr L.Selle (IMF Toulouse,CNRS) forhelpful discus-sionsandtheCERFACSCFDteamstaff fortheirscientificand

tech-nical support about the CFD code AVBP. The contribution of V.

Bouillin to the developmentof NRI-NSCBC isgratefully acknowl-edged.

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