A characteristic inlet boundary condition for compressible, turbulent, multispecies turbomachinery flows

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A characteristic inlet boundary condition for

compressible, turbulent, multispecies turbomachinery

flows

Nicolas Odier, Marlène Sanjosé, Laurent Gicquel, Thierry Poinsot, Stéphane

Moreau, Florent Duchaine

To cite this version:

Nicolas Odier, Marlène Sanjosé, Laurent Gicquel, Thierry Poinsot, Stéphane Moreau, et al.. A

charac-teristic inlet boundary condition for compressible, turbulent, multispecies turbomachinery flows.

Com-puters and Fluids, Elsevier, 2019, 178, pp.41-55. �10.1016/j.compfluid.2018.09.014�. �hal-02094393�

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This is an publisher’s version published in:

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To cite this version:

Odier, Nicolas and Sanjosé, Marlène and Gicquel, Laurent

and Poinsot, Thierry and Moreau, Stéphane and Duchaine,

Florent A characteristic inlet boundary condition for

compressible, turbulent, multispecies turbomachinery flows.

(2019) Computers and Fluids, 178. 41-55. ISSN 0045-7930

Official URL:

https://doi.org/10.1016/j.compfluid.2018.09.014

Erratum

/

Corrigendum

A

characteristic

inlet

boundary

condition

for

compressible,

turbulent,

multispecies

turbomachinery

flows

Nicolas

Odier

a,∗

_,

_Marlène

_Sanjosé

b

_,

_Laurent

_Gicquel

a

_,

_Thierry

_Poinsot

a

_,

_Stéphane

_Moreau

b

_,

Florent

Duchaine

a

a CFD Team, CERFACS, Toulouse 31057, France

b University of Sherbrooke, Sherbrooke QC J1K 2R1, Canada

Keywords:

Characteristic inlet boundary condition Turbomachinery inflow conditions Compressible flow

Turbulence injection Acoustic reflectivity

a

b

s

t

r

a

c

t

Amethodologytoimplementnon-reflectingboundaryconditionsforturbomachineryapplications,based oncharacteristicanalysisisdescribedinthispaper.Forthesesimulations,inletconditionsusually corre-spondtoimposedtotalpressure,totaltemperature,flowanglesandspeciescomposition.Whiledirectly imposingthesequantitiesontheinletboundaryconditionworkscorrectlyforsteadyRANSsimulations, thisapproachisnot adaptedforcompressible unsteady LargeEddy Simulationsbecauseit isfully re-flectingintermsofacoustics.Derivingnon-reflectingconditionsinthissituationrequirestoconstruct characteristic relationsforthe incomingwaveamplitudes.Theserelations mustimposetotalpressure, totaltemperature,flowangleandspeciescomposition,andsimultaneouslyidentifyacousticwaves reach-ingtheinlettoletthempropagatewithoutreflection.Thistreatmentmustalsobecompatiblewiththe injectionofturbulenceattheinlet.The proposedapproachshowshowcharacteristic equationscanbe derivedtosatisfyallthesecriteria.Itistestedonseveralcases,rangingfromasimpleinviscid2Dduct toarotor/statorstagewithturbulenceinjection.

1. Introduction

Specifying inlet andoutletboundary conditionsfor compress-ible simulationstill remains a key issue (Colonius [8]) especially for unsteady flows where wave reflections must be controlled. Inthisfield,characteristicboundaryconditionshaveprogressively become standard. Initially introduced by Thompson [48], Euler Characteristic Boundary Conditions (ECBC) wasthen extended by Poinsot and Lele [34] to viscous flows by proposing the Navier– Stokes CharacteristicBoundary Conditions(NSCBC)approach.This methodspecifiesagivennumberofquantities–forexamplestatic pressure foran outlet, velocity andtemperature foran inlet– on the boundary condition, andallowing the outgoing waves, com-puted by the numerical scheme, to leave the domain with min-imum reflection. The NSCBC strategy has been later extended to multi-speciesreactingflows andtoaeroacoustics (Baumetal.[3], Okong’oandBellan[31],Moureauetal.[29],PoinsotandVeynante

[33],Yooetal.[53],YooandIm[52],Freund[13],Colonius[7]). TheNSCBCoriginalpaper[34]providesexamplesof implemen-tations forseveral boundary conditions. A subsonic inflow

spec-∗ _{Corresponding author.}

E-mail address: nicolas.odier@cerfacs.fr (N. Odier).

ifying the three velocity components and the static temperature is detailed, and remains today a very common boundary condi-tionformostapplications.Later,GuézennecandPoinsot[19] pro-poseda modification ofthisinlet conditionto yield the vortical-flowcharacteristic boundarycondition(VFCBC), basedon aMach numberexpansiondetailedinProsser[38].ThisVFCBCallows vor-ticitywaveinjectionswhileacousticswavespropagateoutsidethe domainwithoutreflection.

Whilecharacteristicsboundariesare requiredforLESandDNS ofreactingflows oraeroacousticapplications,they are muchless usedforturbomachinerysimulations.

• At outlets,mostturbomachinery simulationsusetodayan im-posed static pressure profile satisfying an approximate radial momentumequation alsotermedsimplifiedradialequilibrium. This boundary conditionishowever andby construction fully reflecting, andthereforenotadequate forproperLESandDNS ofsuchflows.In2014,Koupperetal.[21]haveshownthatthe NSCBC methodology remains fully compatible with turboma-chinerycomputationsinstatorvanesallowingbothradial equi-libriumtooccurwhilebeingnon-reflective.

• Atinlets,turbomachineryconditionsusuallycorrespondto im-posedtotalpressurePt=Ps



1+γ−12 M2



γ γ−1 ,totaltemperature https://doi.org/10.1016/j.compfluid.2018.09.014

Tt=Ts



1+γ−12 M2



,where Ps isthe static pressure,Ts is the statictemperature, MtheMachnumber,

γ

theadiabatic coef-ficient, as well as flow direction, and species mass fractions. The use of total temperature and pressure for turbomachin-ery applications lies in the fact that the total quantity losses correspond to exchanged work and heat transfer ofsuch sys-tems. Thesequantities arealso commonlymeasured at differ-ent sections in test engine using Pitot tubes and thermocou-ples.ForEulerequationsandsteadyflows,Giles[16]andSaxer and Giles [40] proposed a total quantity inflow formulation, laterextendedto3DflowsbyAnkeretal.[2],andtounsteady flows by Schluß et al.[41]. Forunsteady Navier–Stokes equa-tions, Struijs etal. [45] also reporteda NSCBC formulationto imposetotal pressure,total temperature,andflowangle.Their chosen formulation howeverrelies on an incompressible def-initionof the total pressure whichis invalid for compressible flows.

• Aside from adequately prescribing the total quantities while treating acoustic, an inletturbomachinery boundary condition must be able to handle vorticity injection. Indeed turbulence may havea significant impact onturbomachinery flows (Choi et al. [5], Carullo et al.[4]). For a fan or a compressor com-putation, the level of turbulence is likely to modify the suc-tion side transitionmode, which will influence losses predic-tions(Jahanmiri [20],Wissinketal.[50],Michelassietal.[27], Scillitoeetal.[43]).

The aimofthispaperisthustoproposeaNSCBCinlet bound-aryconditionimposing totalpressure,total temperature, flow di-rection,andcompositionin acompressible context,andwhich is compatible with turbulence injection for both Direct Numerical Simulation (DNS)and Large EddySimulation (LES).The paper is organizedasfollows: Section 2 describesthe NSCBC formulation forthis inlet condition. Section 3 evaluates the methodology on severaltest-cases,fromacademictest-cases tocomplexindustrial configurations.For all cases,the outletconditionis a fixed static pressure,imposedthroughaNSCBCmethodology.

2. NSCBCstrategytoimposePt,Tt,directionsandspecies composition

2.1.TheNSCBCmethodology

ThissubsectionsummarizesthemainideasoftheNSCBC strat-egy. For more details, the reader is referred to Thompson [48], Giles [15] (Euler equations), Poinsot and Lele [34], Poinsot and Veynante [33] (Navier–Stokes equations), Nicoud [30] (Euler and Navier–Stokesequations).

ThecompressibleNavier–Stokesequations,usingEinstein nota-tion,are:

∂ρ

∂

t +

∂

∂

xi

(

ρ

ui

)

=0 (1)

∂ρ

ui

∂

t +

∂

∂

xj



ρ

uiuj



+

∂

Ps

∂

xi =

∂τ

i j

∂

xj (2)

∂ρ

E

∂

t +

∂

∂

xi

(

ρ

E+Ps

)

ui=

_∂

∂

xi



uj

τ

i j



−

∂

qi

∂

xi (3) where

ρ

isthe localfluid density,uithe velocitycomponents, Ps the staticpressure, Ts the statictemperature, E the total energy, and

τ

ijthestresstensordefinedas:

τ

i j=

μ



∂

ui

∂

xj +

∂

uj

∂

xi −2 3

δ

i j

∂

uk

∂

xk



(4)

Fig. 1. Rotation from cartesian basis to the normal patch basis.

with

δ

ijtheKroneckerdeltaand

μ

thedynamicviscosity.qiisthe heatfluxalongthexidirectionandisdefinedasqi=−

λ

∂_∂Txs_i,where

λ

isthethermalconductivity.Thesystemisfinallyclosedusingthe idealgaslaw:

Ps=

ρ

rTs (5)

whereristhespecificgasconstantofthemixturer= R

W,withW themeanmolecularweightofthemixtureandR=8.3143J/mol.K theuniversalgasconstant.

The present paper focuses on Direct Numerical Simulations (DNS), where all energetic scales are resolved, or on LargeEddy Simulations(LES)(Leonard[24],Germano[14]).Intermsof bound-aryconditions,using DNSorLESleadsto thesame solution.The first step to build characteristic boundary conditions is to write allequationsinareferenceframelinked totheboundarysurface, wherethenormalvectorisnotedn_,andthetwotangentialvectors arerespectivelynotedast1 andt2(Fig.1).

The characteristic analysis of Thompson [48] for Euler equa-tions consists in transforming the conservative variables U₌

(

ρ

u,

ρ

v,

ρ

w,

ρ

E

)

T _{into primitive variables in the reference frame}

(

n,t1,t2

)

so that we get: V=

(

un,ut₁,ut₂,Ps,

ρ

k

)

T. Then

trans-formingthose primitivesvariablesintocharacteristics variables,it results fromboth operationsconservative variables U which sat-isfy:

∂

U

∂

t +AU

∂

U

∂

x +BU

∂

U

∂

y +CU

∂

U

∂

z +S=0 (6)

whereAU,BU,CUaretheJacobianmatricesoftherespectivefluxes

inthex,y,andzdirections,andSisthediffusionterm.Similarly, theprimitivesvariablesVsatisfy:

∂

V

∂

t +N

∂

V

∂

n+T1

∂

V

∂

t1 +T2

∂

V

∂

t2 +S=0 (7)

whereNisthenormalJacobian,T1 andT2 arethetwo tangential

Jacobianalongt1 andt2.Thetransformationoftheprimitives

vari-ablesintonormalcharacteristicsvariablesconsistsindiagonalizing theJacobian N andwritingthebalance equationsforthe charac-teristicvariablesW:

∂

W

∂

t +D

∂

W

∂

n =SW− TW (8)

Disthusadiagonalmatrixcontainingthecharacteristic propaga-tionvelocities

λ

_i(eigenvaluesofN),andSW− TW isthesumofall

thenon-hyperbolic,non-normaltotheboundaryconditionterms: reaction,diffusion,andtangentialterms.

Thenotation_Li=

λ

_i∂W

∂n isintroducedtocharacterizethewave amplitudeassociatedwitheachcharacteristicvelocities

λ

i,wherei istheindexofthecorrespondingwave.Thecharacteristicanalysis appliedtotheNavier–Stokesequationsfinally leadstothe charac-teristicLiwaves(Fig.2),writtenintheboundaryreferenceframe (n,t1,t2)(PoinsotandLele[34]):

⎛

⎜

⎜

⎝

L+ L− Lt1 Lt2 LS

⎞

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎝

(

un+c

)



_∂un ∂n +ρ1c∂ Ps ∂n



(

un− c

)



−∂un ∂n +ρ1c∂ Ps ∂n



un∂ ut1 ∂n un∂ ut₂ ∂n un



_∂ρ ∂n − 1 c2∂ Ps ∂n



⎞

⎟

⎟

⎟

⎟

⎠

(9)

Fig. 2. Inlet and outlet boundary conditions, and respective Li waves.

L+andL−are respectively the inward andthe outward

acous-ticwaves,while_Lt1 andLt2 aretransverseshearwaves,andLSis

theentropicwave.SpecieswavesLkmayalsobeintroducedas: LS= N  k=1 Lk with Lk=

∂ρ

_∂

k n − Yk c2

∂

Ps

∂

n (10)

where

ρ

_k isthedensityofspeciesk, andY_k themass fractionof speciesk.

The NSCBC strategy consists in considering a locally one-dimensional inviscid (LODI)flow on the boundary to specifythe amplitude of ingoing waves. The characteristic system for the Navier–Stokes equations hence becomes after the LODI assump-tions:

∂ρ

∂

t +



LS+

ρ

c 2

(

L++L−

)



=0 (11)

∂

Ps

∂

t +

ρ

c 2

(

L++L−

)

=0 (12)

∂

un

∂

t + 1 2

(

L+−L−

)

=0 (13)

∂

ut1

∂

t +Lt1=0 (14)

∂

ut2

∂

t +Lt2=0 (15)

NotethattheseLODIrelationsmaybecombinedtoexpressthe time derivativesof other quantities(Appendix5.1). Waves ampli-tudesarededucedusingLODIrelations(PoinsotandLele[34]),and then used onthe boundary inEq.(7) to advancethe solutionin time. In terms of treatment, specifying the ingoing wave ampli-tudes is actually the crucial part of most NSCBC extensions and LODI formulations havebeen improved to account fortransverse terms byYoo etal.[53],YooandIm [52],Lodato etal.[26],and Granetetal.[18].

2.2. LODIrelationsforPt,Tt,flowdirection,andspeciescomposition

When theobjectiveofthe boundaryconditiontreatment isto impose Pt,Tt,flowdirectionandspeciescompositionattheinlet, theuseofNSCBCrequiresLODIexpressionsforPtandTt.Thiswas not doneinPoinsotandLele[34] anditrequiressome algebraas shownbelow.Toaccountforcompressibilityeffects,totalpressure

Pt andtotal temperatureTt need to beexpressed asfunctionsof

Fig. 3. Velocity vector → U , and corresponding θand φangles defining the flow di- rection.

thelocalflowMachnumberM.Thisimpliesthattemporal deriva-tionof Pt andTt willinvolvethe Mach numbertemporal deriva-tive.ThisMach numberderivationitselfinvolvesthederivationof thekineticenergyequation(Appendix5.2),duetotherelation: M2₌un2+ut1 2_+u t2 2

γ

rTs = 2ec

γ

rTs (16) whereec isthekinetic energy,

γ

theadiabaticcoefficient,risthe specificgasconstant, andTsthe statictemperature.The adiabatic coefficient

γ

isconsideredtobeconstant.Writing

β

=

(

γ

− 1

)

, af-teralgebraicmanipulations(Appendix5.3),itstemporalderivative isexpressedas:

∂

M2

∂

t = 2 c2



L+·



β

ec 2c − un 2



+L−·



β

ec 2c + un 2



−ut1Lt1− ut2Lt2− ec

ρ

·LS



(17) Using Eq. (17), the total pressure LODI equation becomes (Appendix5.4):

∂

Pt

∂

t = L+·



−

ρ

₂cP_Pt s+ Pt rTt ·



β

ec 2c − un 2



+L−·



−

ρ

₂cP_Pt s +Pt rTt ·



β

ec 2c + un 2



−Lt1ut1· Pt rTt −Lt2 ut2· Pt rTt −ec

ρ

·LS· Pt rTt (18) Similarly, the equation for total temperature reads (Appendix5.5):

∂

Tt

∂

t = L+·



−

β

Tt 2c + 1 Cp



β

ec 2c − un 2



+L−·



−

β

Tt 2c +1 Cp



β

ec 2c + un 2



−Lt1 ut1 Cp −Lt2 ut2 Cp +Tt

ρ

r N  k=1 rkLk− ec

ρ

Cp·LS (19) whereCp= _γγ₋₁r applies. The flow direction is fixed by imposing sin(

θ

) andsin(

φ

),where

θ

and

φ

arerespectivelytheflow angles inthe

(

O_,→n_,→t1

)

plane andinthe (O,→n,→t2) plane (Fig.3).These

twospecificvariablescanbelinkedtothelocalflowvelocity vec-torsince sin

(

θ

)

= ut1



→U



, sin

(

φ

)

= ut2



→U



(20) using:



→U



=



un2+ut12+ut22 (21)

Table 1

Summary of characteristic equations to consider to impose total pressure P t , total temperature T t , flow direction through imposition of θand φ, and species compo- sition Y k . ∂Pt ∂t = L+ ·−ρc2 PPst + rTtPt · _βec 2c −u2n  Total pressure + L−·−ρc2 Pt Ps + Pt rTt· βec 2c + un 2  LODI equation −Lt1 u t1 · Pt rTt−Lt2 u t2 · Pt rTt −ec ρ·LS ·rTPtt ∂Tt ∂t = L+ ·  −βTt2c+ C1p βec 2c −u2n  Total temperature + L−·  −βTt2c + C1p βec 2c + u2n  LODI equation −Lt1 ut1 Cp −Lt2 ut2 Cp + Tt ρr Nk=1 r kLk −ec ρCp·LS Flow direction ∂ut1

∂t = −Lt1 LODI equation ∂ut2

∂t = −Lt2 Species composition LODI equation ∂Yk

∂t = −ρ1(Lk − Y kLS)

The latterisequivalentto imposethetwo shearwaves,since:

∂

ut1

∂

t =

∂

∂

t





→U



×

(

sin

(

θ

)

)



=−Lt1 (22)

∂

ut2

∂

t =

∂

∂

t





→U



×

(

sin

(

φ

)

)



=−Lt2 (23)

Note thatwiththe formalism athand,thevelocity magnitude



→U



is not imposed,only

θ

and

φ

are imposed throughthe use oftransverse shear waves.Note that Albin etal. [1]proposed to re-writethecharacteristic equationsinadifferentframe of refer-ence.This approachshould not differfromourmethod sincethe momentumequationsarethesameinallframes.

The target Pt andTt may changewithtime, sothat terms ∂_∂P_tt and ∂Tt

∂t are kept in the LODI equations of Table 1. Finally, the speciescompositionisimposedas:

∂

Yk

∂

t =−

1

ρ

(

Lk− YkLS

)

(24)

2.3.ThePt− Tt NSCBCtreatment

Having identified the evolution equations to be considered in the context of the LODI assumptions, the remaining step con-sistsinimposingthedesiredinformationswhilesatisfyingspecific propertiesfortheboundarycondition.Asalreadystated,outofthe 5Li inthe system, the two shear wavesare fixed when impos-ingtheflowangle(Eqs.(22)and(23)).Theremaining wavesL+, LS andLk can then be determined solving the system given by Eqs.(18),(19),(24): L+·



−

ρ

₂cP_Pt s+ Pt rTt



β

ec 2c − un 2



−LS· ec

ρ

rTPtt =

∂

Pt

∂

t + Pt rTt ·

(

Lt1 ut1+Lt2ut2

)

−L−·



−

ρ

c 2 Pt Ps+ Pt rTt



β

ec 2c + un 2



(25) L+·



−

β

Tt 2c + 1 Cp



β

ec 2c − un 2



− ec

ρ

Cp·LS =

∂

Tt

∂

t + 1 Cp·

(

Lt1 ut1+Lt2ut2

)

− Tt

ρ

r N  k=1 rkLk

Fig. 4. 2D test case.

−L−·



−

β

Tt 2c + 1 Cp



β

ec 2c + un 2



(26) ∂Yk ∂t =− 1 ρ

(

Lk− YkLS

)

(27)

2.4. Finalwavesexpression

The resolution of the system of Eqs. (25)–(27) is detailed in

Appendix 5.6. It yields the following expressions, termed Pt-Tt-NSCBC-R: Lt1=−

∂

ut1

∂

t (28) Lt2=−

∂

ut2

∂

t (29) Lk=YkLS−

ρ ∂

Yk

∂

t (30) L+ =F1 ∂Tt ∂t +F2∂Pt_∂t +_rTtPt · F2F3+C1p· F1F3+F1· Tt r N k=1rk∂Y_∂tk− L−·(F2F6+F1F7) F4F2+F5F1 (31) LS = ∂Tt ∂t + 1 CpF3+ Tt r N k=1rk∂_∂Ytk− F5·L+− F7·L− F2 (32) where: F1= ec

ρ

·rTPtt (33) F2= Tt

ρ

−

ρ

eCcp (34) F3=Lt1ut1+Lt2ut2 (35) F4=



−

ρ

c 2 · Pt Ps+ Pt rTt ·



β

ec 2c − un 2



(36) F5=



−

β

Tt 2c + 1 Cp



β

ec 2c − un 2



(37) F6=



−

ρ

c 2 · Pt Ps+ Pt rTt ·



β

ec 2c + un 2



(38) F7=



−

β

Tt 2c + 1 Cp



β

ec 2c + un 2



(39)

Eq. (31) provides a relation between the right-going acoustic wave L+ to impose andthe incomingleft-going L− wave from

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, , :, , : : • • to +5,46e a 1;3 ., •

r

_• <1' _; ,, i ' i • ... "o +1,2.1.-•. � _-0.10 ··•' ····_··+' ····_;"-, .\, ··········_;'-···+···: ····+: ·········· •-+ "o +8,ooe· •.. ,-: 1 1 i • • i i +-+ f.o +9,46e-.ta ' • � j \ . li ' ' i • • . _. . -0.15

···/��····r--··�y

'.

···r···

··-r··· ... ·-r···

0.00 0.02 0.04 0.06 0.08 0.10 X

Fig. 10. Temporal evolucion of the ingoing acouscic w;we. for the Pt-Tc-NSCBC-NR formulation and for ax = 0 (rop). ax = 1000 (middle). ax = 10,000, ax = 100,000 (bonom).

ax = a Pr = an = a sin(Ol = a mC;_J· The acouscic w;we maximum crosses the inler boundary ac r = r0 + 8.41e-•s.

"non-reflecting" but instead becomes "partially non-reflecting", and

therefore can be characterized by a reflecting coefficient

R

function

of ax.

3. Validation and discussion

Two boundary conditions that impose Pr, Tr, flow direction and

species have been derived in

Section 2.4

, one being reflecting (Pt­

Tt-NSCBC-R), the other being non-reflecting (Pt-Tt-NSCBC-NR). The

aim of this paper is to evaluate the non-reflecting formulation, re­

quired for turbomachinery applications. Tests have also been per­

formed to evaluate the reflecting formulation, but not shown here

for the sake of concision. The current section allows evaluating

the Pt-Tt-NSCBC-NR boundary conditions on several test-cases, and

provides a best-practice to deal with relaxation coefficients which

are the only adjustable parameters of the method. The first test­

case (

Section 3.1

)

consists in a 2D square-box with constant inlet

Pr, Tr, flow angles and constant static outlet pressure, and allows to

investigate the flow establishment toward the imposed values for

bath conditions, depending on the relaxation coefficients. Once the

flow is established for an axial case (0

=

<P

=

0),

Section 3.2

evalu­

ates the effect of modifying the flow direction. The acoustic

reflec-tian coefficient of each conditions is then studied as a function of

the relaxation coefficients ax in

Section 3.3

. The compatibility with

a synthetic turbulence injection is finally evaluated in

Section 3.4

.

To conclude, the derived boundary condition is used in a Noz­

zle Guide Vane flow simulation in

Section 3.5

. These test-cases

are summarized in

Table 2

. Computations are performed using the

3D unstructured solver AVBP, detailed in Sch0nfeld et al.

(42]

and

Gourdain et al.

(17]

, using a two-step Taylor Galerkin TIGC (Colin

and Rudgyard,

[6

])

numerical scheme.

3.1. Convergence of the mean flow to imposed targets

The first academic test-case is a 2D test-case, with

[Lx x Ly] =

(100 mm x 100 mm], discretized with

[nx

x

_ny]

= (128 x 128] cells

(

Fig. 4

). The inlet condition corresponds to a total pressure Pr =

98, 803 Pa

,

a total temperature Tr = 281 K, a normal flow direction

(0 =

<P

= O), for air. A static pressure

Ps = 71,000 Pa is imposed at

the outlet boundary, and periodic conditions are applied on the

other boundaries. For this case, the Mach number must reach a

Table 2 Test-cases.

Test case Schematic Object of the study

2D square box Flow setting up toward target values of P t , T t , θ,φ( subsections 3.1, 3.2 ).

2D square box Acoustic wave impacting on inlet condition:

reflectivity ( subsection 3.3 ).

3D rectangular box Turbulence injection: Turbulent characteristics, and non-reflectivity in case of turbulence injection ( subsection 3.4 ).

Actual rotor-stator turbomachinery

Actual turbomachinery: Flow setting up toward target values, turbulence injection

( subsection 3.5 ). valuegivenby(

γ

=1.4): M=









2

γ

− 1



_

Pt Ps



γ−1 γ − 1



=0.7036 (46)

The initial solution is arbitrarily chosen, and characterized by a static pressure Ps=98,803 Pa, a static temperature Ts=281 K, and an initial velocity U=10 m/s. All relaxation coefficients are equal(

σ

X=

σ

Ptot =

σ

Ttot=

σ

sin(θ )=

σ

sin(φ)).

Aprobeislocatedattheinletboundarycondition,andthe tem-poral evolutions ofPt,Tt are given inFigs. 5 and6 forthe non-reflecting formulation.Figs. 7 and8 display the inletstatic pres-surePs andthe inletMach number. As theconfiguration is peri-odic inthetransverse direction,no lossoccursandtheflow may be considered isentropic: the staticpressurein the domainmust be equal to the prescribed outlet staticpressure. The inlet Mach numbershouldreachthetheoreticalvalueofEq.(46).

Figs.5to8showthat

σ

X=0doesnotallowreachingthetarget valueforthisspecifictest-caseforthenon-reflectingformulation. Whenconsidering thisphaseofflowestablishment,startingfrom aninitialsolutionthatdiffersfromthefinalstate,theusershould use significant relaxation coefficient

σ

X to rapidly converge the flowto theadequate anddesiredstate. Asevidenced bythistest case,establishingameanoperatingconditionforaflowwhose ini-tialsolutionisapureguesswhileimposingchosenboundary con-ditionisnot a simpleproblem. Clearlysuch a processis strongly dependanton theboundary conditionformulationandtheinitial solution.Oncetheflowisestablished,testsshowthattheinlet re-laxationcoefficients

σ

X can be broughtback tozero andthat no drift occurs. The corresponding established meanfield is used in thefollowing subsections asan initial condition: Ps=71,000Pa, Ts=255.711K,u=225.99m/s,M=0.7036.

P t-Tt-NSC BC-N R formulation: Reflected wave w+ at t he lnle t

1 ·: ••

:

n

:

:

.

:1

••

:-ZE

•••••••••••••

r

••

:

:

:

:

.

1 -so .. ... .. ... .. ...

.l .... .. ... .. ... .

l ...

..l. ... .. ... .

-100 ... : ... : ... : ... . 0.0545 0.0550 0.0555 Time (s) 0.0560

1-

au_ ... O'x=O ....,.. <l'x=lOO ..,_. O'x=lOOO H <'):=1000001

0.0565

Fig. 11. Reflected wave W+ depending on the relaxation coefficients ax, for non­ reflecting formulation. The incoming left-going wave w_ is also depicted.

3.2. Imposition of a flaw

direction

Starting from the previously established flow, a velocity angle of

0

= 15

°

_{is imposed at the boundary condition.}

_{Fig. 9}

_{, showing the}

time evolution of the flow direction, confions that the proposed

approach provides the correct evolution of

sin(0)

toward the de­

sired value (sin(0) = 0.258) when a non zero relaxation coefficient

is used.

3.3.

Acoustic

properties of the inlet boundary conditions

The acoustic reflectivity of the proposed formulation is evalu­

ated in this subsection, for various values of the relaxation coef­

ficient crx. To do so, a Gaussian left-going acoustic wave is super­

imposed to the flow established in

Section 3.1, initially centered at

x0 =

f;

=

¾,

and for which the following perturbation reads:

p'

=

-peu'

with

(X X₉)2

u'

= Ae-

r

(47)

with

A

= 0.001 and

r

= 0.01.

Fig. 10

shows pressure and veloc­

ity fluctuation evolutions as the acoustic wave crosses the inlet

boundary for several values of the crx coefficients for the non­

reflecting formulation: very small reflections appear.

To quantitatively evaluate the reflection coefficient

R,

static

pressure, velocity and density signais are recorded at the inlet

probe and decomposed into mean and fluctuating components,

p(t)

=

p

+

p'

(t),

U(t)

=

Ü

+

U' (t),

p(t)

= p +

p'

(t). (

48)

The inward and backward acoustic waves W+ and w_ are then re­

constructed using:

{w+ =

_{w_ =}

p'

_{p' - peu'}

+

peu'

(49a)

_(49b)

Those waves are then recast into frequencies through a Fast Fourier

Transform operator -, and the reflection coefficient

R

is finally ob­

tained using:

R = W+

_{\tl_}

(50)

Fig

.

11

depicts the reconstructed reflected wave W+ at the inlet

for several relaxation coefficients crx. Note that for a purely non­

reflecting condition, although w_ is present, W

+

should remain

zero. Any signal in W+ reconstruction is therefore indicating of a

reflection. For low values of crx (less than 100), the boundary be­

haves as expected: no reflection is observed for this non-reflecting

formulation.

Fig. 13

depicts the evolution of the reflection coefficient for the

considered working point, depending on the relaxation coefficient

50�_Pt-'---'-n'--N'-S""Cc..cB,cC'-·N""R-'-fo'--'-'-'rm-'--"ula"' ,,iot_'"'nr-: --'1-'nl"'e-'-t p"-r-"e"'ssc...cuccrc,.fleccu'--' c-" tuc.caccti-'-o'-n -�

40 ... ; ... . � 30 ··· ···i· ··· ··· i ,0

!

i

� . . i � 20

i

i

i

10 ···:,··· ···:· .. ,

!

; Q ....... -... � ... _,. 0.0545 0,0550 0,0555 Time (s) 0,0560 0,0565

o,o5�_Pt_-_:rt_-_N-'-SC -'-_{�C'-·-N_R _foc....rm_u}B _{'-la_ti.c.,on�:_lc....n.c..c}le_t _v_""el"-o c-'-ity

_,_,_f'-lu-'-ct_u...c.a_tio-'--� n : : !

!���

-0.25 ···:···:···:···

.

:u

·

:·

·

r

·

,� �

1

·

:

:H:

-0.30 ···�···+···!···

i i i 0.0550 0.0555 Time (s) 0.0560 0.0565

Fig. 12. Evolution of flucruating pressure (rop) and velocicies (bonom) depending on the rela xation coefficients ax. for the non-reflecting formulation.

crx. In this Figure, the maximum

R

is plotted for the Pt-Tt-NSCBC­

NR formulation. For a zero relaxation coefficient, the proposed for­

mulation behaves as expected. As crx increases, it behavior deteri­

orates.

The evolutions of

Pr

as the acoustic wave crosses the inlet are

presented in

Fig. 14

. For crx = 0, the proposed non-reflecting for­

mulation recover the target

Pr

value as soon as the acoustic wave

has passed the inlet. The same conclusion holds for the total tem­

perature Tr (not shown).

3.4. Compatibility with turbulence injection

As stated in the introduction, an inlet boundary condition for

turbomachinery computations must handle turbulence injection.

Reviews of turbulence injection methodologies for LES may be

found in Tabor and Baba-Ahmadi

(46)

, Dhamankar et al.

(11)

and

Wu

(51 )

. The compatibility of the proposed non-reflecting bound­

ary condition with a synthetic turbulence injection is assessed in

this subsection as it is indeed a key condition for many turboma­

chinery computations.

The three unsteady velocity components (u;, u2

,

_{u3) at the in­}

let are specified using a Kraichnan's approach

(22)

. Following the

Vortical-Flow Characteristic Boundary Condition {VFCBC) proposed

in Guézennec and Poinsot

(19)

, these fluctuations are added to the

inlet acoustic waves derived in

Section 2.4

and governing the mean

flow, so that:

(51)

(52)

a:

l,o,�---�A-'-co'--u-'-s _tic'--'re-iffc.c.ec.c...t....cio_n--'c-'-oe-'-ff_i.,...ci-'-en_t_,(_zo'--o_m_,)�---�

0.8

0.6

0.4

; ; ; ;

i

i

i

i

0,2 ···'···'···'···'···

...---20000

40000

60000

Relaxation coefficient

"x

1

+-+

Pt·Tt·NSCBC•NR 1

80000

100000

a:

l.O·�----A.,.co'--u-'-s t_ic'--'re--'fl-'-ec.c...t....c io�nr'c-'-oe.c...ffi_c.c..i .c..en_ t_,(_zo'--o�m_,) ____ �

0.8 ···'···'···'···

0,6 ···>···· ··· ···· ··· ···· ····:···i·· ···· ··· ···· ···· ··· ····

0.4 ···:-·· ···· ··· ···· ··· ···· ····:···:-· ···· ··· ···· ···· ··· ····

l

l

!

0.2 ···'···'···:···

:

500

1000

Relaxation coefficient

"x

1

+-+

Pt•Tt•NSCBC•NR 1

1500

2000

Fig. 13. Evolution of the reflection coefficient R depending on the relaxation coefficient

ax

for the reflecting and the non-reflecting formulation.

98840�_--'-Pt-'-·Tt.c...c....·N.c..cS,.cCccB-=C-'--N.c.c.R.ccf.coc..c.r m-'u""l.c.,at""'loc.c. nc...: -'- lnc.ccl..c.et'-llc.c.o.c..ctaccl

,...Pr-=e.=.css""u-'-re"--�

::::: ..•.... 111 .1 . .HW � .............

4

1

'

1 ;

j

,

f • • ...•....

',f

98780 ... ... 1 ... . .... .... ... .1 ... ... ... 1 .. ... .... ... ... .

ô:_ 98760 ···'··· ···· ···· ···· ··· ···-'···:· ···· ···· ···· ··· ····

98740 ···�···�···-�···

98720

0.0550

0.0555

nme (s)

0.0560

0.0565

1

---

'1x•O ,._. <1x•lOO .,_.. "x•lOOO t-t "x•lOOOOOI

Fig. 14. Evolution of total pressure

P,

as the left-going wave w_ crosses the inlet,

for the non-reflecting formulation.

For the validation of the proposed methodology, a turbulent

convected flow in a rectangular box is computed, for which the

inflow mean inlet Pr, Tr, as well as an additional synthetic in­

let turbulence are imposed (

Fig. 15

). The computational domain

is a [Lx x

Ly

x

Lz)

= (4 mm x 1 mm x 1 mm) rectangular box, dis­

cretized with [nx x

ny

x nz) = (392 x 98 x 98) cells. Total pressure

and temperature Pr and Tr are imposed at the inlet, static pressure

P

5

is imposed a the outlet, so that the expected mean velocity is

u

= 95 m/s. Ail other boundaries are periodic conditions. The inlet

velocity fluctuation is u' = 5.0 m/ç

1 ,

_{(TKE = �u'}

2

_{= 37.5 m}

₂

_s-2).

The target integral lengthscale is À =

1;

=

if;

= 0.56 mm, with

ke

the most energetic wavenumber in the Passot-Pouquet spec­

trum, and

Àe

the most energetic lengthscale. 1000 modes are used

to build the inlet velocity fluctuation field.

Fig

.

15

shows the injec­

tion of vortical structures near the inlet (on the left), whose sizes

are spatially increasing when convected to the exit (on the right),

as expected from a spatially decaying turbulent flow.

First, the integral turbulent timescale

trurb

and lengthscale À are

evaluated as a function of the channel axial length using the two­

time correlation

Ruu(x,

-r) (Pope

(36

1)

:

trurb(X)

= r

=rm

"'

Ruu (X,

-r )d-r .

fr=0

(54)

Where -r

max

is such that

Ruu (x,

•max) = O. From this turbulent

timescale, an integral turbulent lengthscale is deduced using Tay­

lor's hypothesis (Taylor,

(47

1)

. This hypothesis consists in con­

sidering a "frozen turbulence" advected by the mean flow, and

enables the integral lengthscale evaluation through the knowl­

edge of the integral timescale. lt is valid for an established mean

flow, and if � << 1. This hypothesis is very often used in ex­

perimental works and has been shown to fairly predict turbu­

lence scales (Dahm and Southerland

(9

1)

. Sorne authors how­

ever raised limitations for some flows configurations, as Lin

(25)

who showed this hypothesis is no longer valid for high

shear flows, or Lee et al.

(23)

who showed that this hypoth­

esis does not apply for the dilatational part of turbulence.

More recent works also show this hypothesis breaks down for

wall-bounded flows (Del Alamo and Jiménez

(

10)

, Moin

(28

1)

.

Finally:

À=

U

·

trurb

(55)

Fig. 16

displays the turbulent kinetic energy decrease expected

within the domain (a), and the turbulent integral lengthscale (b).

At the inlet, a value of

TKE

= 35 m

2

_{.s-2 is reached, very close to}

the imposed value

TKE

= 37.5 m

2

_{.s-2. A similar conclusion can be}

drawn for the integral turbulent lengthscale at the inlet. It can be

noticed that À starts decreasing, then increases as expected. The

primary decrease in À is associated to the adaptation length re­

quired to reach a physical turbulent spectrum.

.,

\ Vorticity Magnitude (s·'J 7.roJe<-05 7e+S 4.96+5 42e+5 3.SOOe<-05

Fig. 16. Turbulent kinetic energy (left) and integral lengthscale λ(right) along the x -axis (Fig 15 ).

Fig. 17. Interaction between synthetic turbulence and acoustic waves: instantaneous contours of Q criterion colored by vorticity magnitude, and cut colored by the pressure field. The flow is going from the left to the right, acoustic waves travel from right to left. Inlet: Pt-Tt-NSCBC-NR.

Fig. 18. FACTOR turbine stage configuration.

As statedinEqs. (51)to(53),fluctuationsare addedto waves imposingthemeanflow, onwhichtherelaxationmethodology is applied. In practice, velocity fluctuations modify the local mach number,andthuslocalvaluesofPtandTt.Increasing

σ

Xwill con-sequentlyinduce a damping in the resulting average kinetic en-ergy.Thismightbeareasonforthesmalldifferencefoundabove

(TKE=35m2_.s−2 _{insteadofTKE=37.5m}2_.s−2_{).Notethatthe}

ef-fect of

σ

X ontheresultingturbulencewill increaseasthe turbu-lentvelocityfluctuationswillresultinlargeMachnumber fluctua-tions.

Toevaluatetheacousticbehaviorinthecaseofturbulence in-jection,aharmonicpulsationisimposedattheoutletofan

estab-Fig. 19. Temporal evolution of integrated variables over the inlet boundary condition toward the target values, together with the relaxation coefficients used during the simulation. The last value used for σX is σX = 10 0 0 .

lishedmeanflow, leadingtoapressureperturbationthatvariesin time,reading:

pac

(

t

)

=pac0sin

(

2

π

· fac· t

)

(56)

The chosen frequency is fac=260 kHz, and the pressure ampli-tude is pac0=1000 Pa, and

σ

X=0. An instantaneous contour of Q-criterion, the second invariant of the velocity gradient tensor, anda longitudinalplane colored by thepressure field are shown in Fig.17.Harmonic acoustic wavestravel fromtheoutlet tothe inlet,andtheevaluatedacousticreflectioncoefficientR atthe in-letEq.(50)isR=0.06,provingtheabilityofthePt-Tt-NSCBC-NR toinjectturbulencewhileremainingquasinon-reflecting.The tur-bulentcharacteristicsarefoundtobethesameasthoseshownin

Fig.16.

Since the Pt-Tt-NSCBC-NR formulation is satisfying on these academictest-cases,itisthentestedinanindustrialconfiguration inthefollowingsubsection.

3.5.Turbomachineryconfiguration

Theturbinestage belongingtothenon-reactiveaxial combus-torsimulator FACTOR (FullAerothermal Combustor-Turbine inter-actions Research) configuration (Fig. 18), is computed using the Pt-Ttnon-reflectingboundarycondition(Pt-Tt-NSCBC-NR),andthe LESsolver TurboAVBP describedinWang etal.[49] enablingthe rotor relative motionina LES context. Theconfiguration consists in 2 stator vanes and 3 rotor blades, with a rotating speed of 8500rpm, on a 76 millions cells mesh. The initial solution con-sistsinauniformfield,withastaticpressureofPs=149,000Pa,a statictemperatureTs=450K,andnoinitialvelocity.A2D-mapof Pt,Ttandflowdirectionisimposedattheinlet.Moredetailsabout theFACTORconfigurationmaybefoundinDuchaineetal.[12].

ThetemporalevolutionsoftheaveragedvaluesofPtandTtover theinletpatchareshowninFig.19,aswellastherelaxation coef-ficients

σ

Xusedduringthesimulation.High

σ

X areneededduring theflowsettingup,andthesecoefficientsaredecreased oncethe targetvaluesarereached.

Fig. 20. Inlet total pressure (a), total temperature (b) and flow direction (c) for an actual turbine stage. Imposed solution on left, time averaged solution over 1.9ms on right.

Fig. 21. Isosurface of Q-criterion colored by vorticity.

FieldsofPt,Tt,andflowdirectionsin(

θ

)imposedbythe bound-ary condition and fields obtained by the LES are compared in

Fig.20,showingtheabilityofthePt-Tt-NSCBC-NRtocorrectly han-dle such spatial 2D-maps prescriptions. Finally, a turbulence in-jectionis added.To doso, a velocity fluctuation ofu=5m/s is imposed,withan integrallengthscale

λ

₌√2π

ke =

λe √

2π =1.23mm.

The resulting mean fluctuation measured at the inlet is u= 4.33m/s,andtheintegrallengthscaleis

λ

=1.17mm.Theresulting

flowisdepictedinFig.21,showingturbulentstructuresinteracting withtheinletguidevane.

4. Conclusions

AnextensionoftheNavier–StokesCharacteristicBoundary Con-dition(NSCBC)isproposedinthispapertoimpose totalpressure

Pt, total temperature Tt, flow angles

θ

and

φ

, and multi-species compositionYkattheinletofacompressibleLESorDNS.

Following the NSCBC methodology, the compressible expres-sionsoftotalpressureandtotaltemperaturearederivedtobe ex-pressed intermsof characteristicwaves_Li.Those _Li amplitudes are then obtained usingLocally One-DimensionalInviscid (LODI) relationsforsituationswherePt,Tt,

θ

,

φ

andYkareimposed.Two formulations are proposed:the firstis reflecting (Pt-Tt-NSCBC-R), the other onenon-reflecting (Pt-Tt-NSCBC-NR). Bothformulations areimplementedusingalinearrelaxationmethodology.

Thesetwoformulationsareevaluatedonseveraltest-cases,for a large rangeof relaxationcoefficients. Results show that a large range ofrelaxation coefficients enableto recover a quasi reflect-ingoraquasinon-reflectingbehavioraswellasnon-driftingmean values. The compatibility of the Pt-Tt-NSCBC-NR condition with turbulence injection is then demonstrated for a turbomachinery simulation. Asynthetic turbulence injectionis added to the pro-posedformulation,andvalidatedonaturbulentperiodic rectangu-larbox.The inletboundaryconditionleadstofairresults regard-ing the expected turbulent kinetic energy, andturbulent integral lengthscale attheinlet.Thenon-reflectivityincaseofturbulence injectionisalsodemonstrated.

Finally, this boundary condition is tested for a turbomachin-eryconfiguration.TheFACTORturbinestageischosen andspatial 2Dmapoftotalquantitiesandflowdirectionisimposed,together withasynthetic turbulencefield.This test-caseshowstheability oftheproposed Pt-Tt-NSCBC-NRtoreachtheinletPt,Tt,andflow angletargetvalues,andtohandlethe2D-mapprescriptionandthe turbulentcharacteristicsattheinletofacomplexgeometry.

Acknowledgments

TheauthorswishtoacknowledgethesupportoftheSTAE foun-dation via theRTRA projectSIMACO3FI forfinancial support.The computationalresources fortheactualturbomachinerysimulation wereprovidedbytheGENCInetwork,andwerepartofthe alloca-tion no.A0022A06074 (CINES-OCCIGEN).The firstauthoralso ac-knowledgesLuis-MiguelSeguiforfruitfuldiscussionsabout turbu-lenceinjection,andJérômedeLaborderieforcodeverification.

Appendix

5.1. Usefulrelations

Eqs. (57)to (64) are useful relations resulting from LODI rela-tionsandtheircombinations.Thoserelationsmaybefoundin[37].

r= N  k=1 rkYk (57)

∂

un

∂

t + 1 2

(

L++L−

)

=0 (58)

∂

ut1

∂

t +Lt1=0 (59)

∂

ut2

∂

t +Lt2=0 (60)

∂

Ps

∂

t =−

ρ

c 2

(

L++L−

)

(61)

∂

r

∂

t =− 1

ρ



−rLS+ N  k=1 rkLk



(62)

∂

Ts

∂

t =−

β

Ts 2c

(

L++L−

)

+ Ts

ρ

r N  k=1 rkLk (63)

∂

Yk

∂

t =− 1

ρ

(

Lk− YkLS

)

(64)

5.2. Kineticenergyderivation

ThetemporalderivationofPt andTtinvolvestheMachnumber Mderivation, whichitself involvesthe kinetic energyec equation derivation: ec= un2+ut1 2_+u t2 2 2 (65)

Thekineticenergytemporalderivativeis:

∂

ec

∂

t =un

∂

un

∂

t +ut1

∂

ut1

∂

t +ut2

∂

ut2

∂

t (66)

Using Eqs.58–60,the kinetic energytemporalderivative is fi-nally:

∂

ec

∂

t =−

un

2

(

L+−L−

)

− ut1Lt1− ut2Lt2 (67)

5.3.SquareMachnumberderivation

ThesquareMachnumberyields: M2=un2+ut1 2_+u t2 2

γ

rTs = 2ec

γ

rTs (68) Using Eqs.(67),(62), (63), thesquare Mach number temporal derivativefinallywrites:

∂

M2

∂

t = 2 c2



L+·



β

ec 2c − un 2



+L−·



β

ec 2c + un 2



− ut1Lt1 −ut2Lt2− ec

ρ

·LS



(69)

5.4.Totalpressurederivation

ThetotalpressurePtisdefinedby:

Pt=Ps



1+

β

2M 2



γ β  (70) Itstemporalderivativesgives:

∂

Pt

∂

t =

∂

Ps

∂

t



1+

β

2M 2



γ β  + Ps

γ

β

β

2 ·

∂

M2

∂

t



1+

β

2M 2



γ β−1  (71) ThetotaltemperatureTt expressionis:

Tt Ts =



1+

β

2M 2



(72) UsingEq.(61)andEq.(69),thetotalpressurederivative, writ-tenintermsofLi,isfinally:

∂

Pt

∂

t =L+·



−

ρ

c 2 Pt Ps+ Pt rTt



β

ec 2c − un 2



+L−·



−

ρ

c 2 Pt Ps +Pt rTt ·



β

ec 2c + un 2



−Lt1ut1· Pt rTt −Lt2 ut2· Pt rTt − ec

ρ

·LS· Pt rTt (73)

5.5.Totaltemperaturederivation

Thetotaltemperaturetemporalderivativeis:

∂

Tt

∂

t =

∂

Ts

∂

t · Tt Ts + Ts·

β

2 ·

∂

M2

∂

t (74)

ThisequationisdevelopedusingEqs.(63)and(69):

∂

Tt

∂

t =L+·



−

β

Tt 2c + 1 Cp



β

ec 2c − un 2



+L−·



−

β

Tt 2c + 1 Cp



β

ec 2c + un 2



−Lt1 ut1 Cp −Lt2 ut2 Cp +Tt

ρ

r N  k=1 rkLk− ec

ρ

Cp·LS (75) 5.6.Systemresolution

L+, LS and Lk are determined solving the system of un-knownsgivenby Eqs.(73),(75)and(64).The termN_k=1rkLk in Eq.(75)needstobeexpressed.Eq.(64)gives:

Lk=YkLS−

ρ ∂

Yk