Three-dimensional through-flow modelling of axial flow compressor rotating stall and surge

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Aerospace

Science

and

Technology

www.elsevier.com/locate/aescte

Three-dimensional

through-flow

modelling

of

axial

flow

compressor

rotating

stall

and

surge

Mauro Righi,

Vassilios Pachidis,

László Könözsy

∗

,

Lucas Pawsey

CranfieldUniversity,Cranfield,Bedfordshire,MK430AL,UnitedKingdom

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received8December2017

Receivedinrevisedform27March2018 Accepted12April2018

Availableonline17April2018 Keywords: Post-stallmodelling Through-flowcode Rotatingstall Surge Axialcompressors Godunov-typeschemes

Thispaperpresentsathree-dimensionalthrough-flowapproachbasedonthecylindricalEulerequations incorporating a body force method. Blade performance is captured through a mixture of empirical correlationsandanovelreverseflowtreatment.Thecodeisthefirstapplicationofaphysicallycorrect Godunovsolvertothree-dimensionalrotatingstallandsurgemodelling.Thissolverensurestheaccurate calculation ofinter-cellfluxesunlikeintypicalmodernCFDcodes inwhichthe non-linearconvective termsare linearised. Validationconsists ofmodelling alow speedthree-stage axialcompressorinall operatingregions,recreatingthereverseflow,rotatingstallandforwardflowcharacteristicswithgood agreement toexperimental data. Additionalcomparisonsare made againstrotating stall cellsize and speed,towhichgoodagreementisalsoshown. Thepaper endswithsomefullsurgecyclesimulations modifying boththe tank volume after the compressorand the level of inletdistortionapplied. Both tank volume and level ofdistortionhave been found toaffect the typeof instabilitydeveloped. The developmentofthiscodeisastepforwardincompressorrotatingstallandreverseflowmodellingand allowsrecreationofafullcompressormapatasignificantlylowcomputationalcostwhencomparedto commerciallyavailable3DCFDcodes.

©2018TheAuthors.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY license(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Theoperation of ajet engine is limitedby theonset of com-pressorstall.Rotatingstallandsurgearestillnotfullyunderstood, neither in the mechanism through which the instability is trig-geredand subsequentlydevelops,nor the aerodynamic load that isimposed on individual turbomachinerycomponents during the event.This generallyleads toa conservative approachduring the designofbladesandcasings,andpenalisesthenominal operabil-ityrangeoftheengine.Numeroussolutionshavebeeninvestigated tosuppresstheoccurrenceofcompressorstallandincreasesurge margin.Experimental testsincludeaxisymmetricarc-shaped slots casingtreatment, investigatedbyPanetal.[1], andbendskewed casingtreatmenttestedby Aloneet al.[2] Other recentmethods proposedtoincreasethestabilitylimitarerecessedbladetips[3], casinggrooves[4] andradialinjectors[5].Mhosenetal.[6] numer-icallyinvestigatedtheuse oftandemrotor blades toimprovethe flowturninganddiffusionandsuppresspassageseparation.Imani andMontazeri-Gh [7] proposedanimprovementontheMin–Max limitprotectioninthe controlofaero-engines toreduce the

pos-*

Correspondingauthor.

E-mailaddress:laszlo.konozsy@cranfield.ac.uk(L. Könözsy).

sibility of accidental crossing of the stability limit. Aero-engines embedded in airframesare a commonfeature ofnext-generation aircraftsbuttheunsteadyflowdistortiongeneratedbytheS-ducts greatly reducethe surgemargin.Gil-Prieto etal.[8],developeda method to predict peak levels of distortions in S-ducts in order to avoid extensive experimental testing and continuous redesign ofthecompressorwhenmatchinginletductandengine.The vari-ouspost-stallbehavioursthatamulti-stagecompressorcanexhibit weredemonstratedbyDayandFreeman[9] onaRolls-RoyceVIPER compressor.AsillustratedinFig.1,athighspeedsthecompressor ispronetosurge,whileatmidspeeditformsafullspanstallcell andatlowspeedsafrontendstall.

Fullcharacterisationofthepost-stallbehaviourofahighspeed compressor experimentally is expensive. CFD modelling of axial turbomachinery in full 3D has beensuccessfully achieved inthe past, however the computational cost required is significant. In 1999 in a joint effort NASA and GE simulated the GE90 turbine system(18bladerows)using3DCFD[10].The9millionelements modelwasrunon121processorsinparallelandittook15hours toconvergetoasingleoperatingpoint,in10000iterations. Gour-dainetal.[11] simulatedrotatingstallusingaquasi-3Dcodeand modellingastreamsurface;yetstill asingle stageofcompression required8hoursperrevolutionona32coresystem.This reduced-order CFD was able to simulatea stable stall but the predicted https://doi.org/10.1016/j.ast.2018.04.021

Nomenclature

B Greitzer’sparameter . . . – b Blockageparameter . . . – f Volumetricforce . . . N/m3 WE X Workexchange . . . W/m3

E0

,

H Internalenergy,Enthalpy . . . J/m3

E

,

F

,

G

,

S Eulerequationsfluxesandsourceterm

M Machnumber . . . – P Pressure . . . Pa RC U R Radiusofcurvature . . . m

r Radius . . . m T Temperature . . . T t

,

τ

Time,timeconstant . . . s u Airvelocity . . . m/s U Bladevelocity . . . m/s U Conservativevariables

x Axialposition . . . m

α

Airflowangle . . . rad

β

Camber-linedirection . . . rad

γ

Staggerangle . . . rad

γ

Specificheatratio . . . –

δ

Deviation . . . rad

θ

Circumferentialposition . . . rad

ρ

Density . . . kg/m3 Total-to-totalpressurerisecoefficient . . . –

φ

Flowcoefficient . . . –

Rotationalspeed . . . rad/s

ω

Pressurelosscoefficient . . . – Subscripts

r

,

x

, θ

Radial,axial,circumferentialdirection

η

, ξ

Parallel,normaltocamber-linedirection 0 Totalquantity

S S Steady-statevalue

Fig. 1.Stall inception behaviour of Rolls-Royce VIPER compressor as simulated by Wilson [15].

numberandspeedofthecells waswrong.Khaleghi modelled ro-tating stallinception on ahalf annulus [12,13]. While the model hada spanresolutionatalow computationalcost,thistechnique was able to simulateonly modesmultiple oftwo. More recently a URANS-based CFD method has been used to model stall in a modernhighspeedcompressorbyDodds[14].Modellingonlythe firstthreestagesoftheoriginal eight,usingthefullannulus,one shaftrevolutiontook48hours ofcomputationaltimeon64CPUs. An alternative to such an approach is the useof a through-flow code, which involves simplifying the problemusing coarse grids andflowmodellingbasedontheEulerequationswithblade per-formancetakenintoaccountbyempiricalcorrelations.Suchcodes areextensivelyusedinindustry,with2Dcodeslookingonlyatthe meridional plane and3D codesmodellingthe full annulus, avail-able forstudying conventional pre-stall compressor performance.

Some codes havebeenextendedandadapted tosimulatereverse flow and stalled flow, an example of whichis Wilson [15] using aseriesof1Dthrough-flowcodesinparalleltomodelthevarious stallingbehavioursoftheVIPERengine(asshowninFig.1).Gong [16] used a body force method, based on a database of correla-tions fromconventionalCFD,tomodelrotatingstall.Thetoolwas developedbyanumberofotherresearcherswithinMITwhich im-provedthemethod,allowingittobeappliedtoamulti-stageaxial compressor, as reported by Brand [17]. Longley [18] proposed a novelmethodtopredicttheregionofhighlyseparatedflowwhich occursduring compressor stall.Sincethe circumferential element densityisnotenoughtocapturetheseparatedflowbetweenblade rows, Longleyintroduced avariableb thatcapturestheflow non-uniformity.Anequationisaddedthatpermitsthecreation, trans-portandmixingofthenon-uniformity thatrepresentsthephysics

oftheflow,allowingthepressurelossanddeviationduringreverse flowtobesufficientlycaptured.

InalmostallmodernCFDcodesthenon-linearconvectiveterms are linearised during the solution process, losing some physical featuresoftheflowphysics.Thepresentpaperproposesthe appli-cationofaphysicallytruesolvertothealreadyestablished meth-ods of using body forces coupled with empirical correlations to modelthepost-stallbehaviourofan axialcompressor.Theflowis modelledusingthe3DcylindricalEulerequationsandsolvedusing aGodunovtypesolver.Theschemeusedisfirstorderintimeand space. Althoughit introduces substantial numerical diffusionthis isnecessary to maintain thesolution stable asthere is no other source of viscosity, either modelled orartificial. As described by Ekaterinaris[19] higher orderschemes offer higheraccuracy and lowernumericaldiffusionwhilestillremainingcomputational effi-cient.Thepresentcodecanbeupgradedinthefuturetoahigher order accurate scheme by increasing the order of the interpola-tion;howeverbecauseofthelowernumericaldiffusionitwill be necessarytomodeltheviscosityintheequations.Thetheoryand subsequent application and validation against low speed experi-mentalresultsarepresentedinthispaper.

2. Methodology

Thetooldevelopedisathrough-flowcodewhichsolvesthe Eu-lerequationsinthe emptyannulus,usingthebodyforce method tosimulatethepresenceofthecompressorbladingbasedon em-piricalcorrelations.

Governingequationsandsolver.Thegoverningequationsarewritten ina cylindrical coordinatesystem foran unsteady, inviscid com-pressibleflow,andthecontinuity,momentum(Eulerequation)and energyequationscanbedescribedinacompactformas

∂

∂

tU

+

1 r

∂

∂

r

(

r E

)

+

1 r

∂

∂θ

F

+

∂

∂

xG

=

S

,

(1)

wherethe scalargoverningequations canbe written ina matrix formas

∂

∂

t

⎡

⎢

⎢

⎢

⎣

ρ

ρ

ur

ρu

θ

ρ

ux

ρ

E0

⎤

⎥

⎥

⎥

⎦

= −

1 r

∂

∂

r

⎡

⎢

⎢

⎢

⎢

⎣

ρru

r

ρ

ru2_r

ρru

θur

ρ

ruxur

ρr H

0ur

⎤

⎥

⎥

⎥

⎥

⎦

−

1 r

∂

∂θ

⎡

⎢

⎢

⎢

⎢

⎣

ρ

uθ

ρu

ruθ

ρ

u2 θ

+

P

ρ

uxuθ

ρ

H0uθ

⎤

⎥

⎥

⎥

⎥

⎦

−

∂

∂

x

⎡

⎢

⎢

⎢

⎢

⎣

ρ

ux

ρu

rux

ρ

uθux

ρu

2 x

+

P

ρ

H0ux

⎤

⎥

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎢

⎢

⎢

⎣

0 ρu2 θ r

−

∂∂rP

−

ρuθur r 0 0

⎤

⎥

⎥

⎥

⎥

⎥

⎦

.

(2)

These model equations are solved transiently in the full an-nulus on a relativelycoarse grid required to maintain acceptable computational requirements. The results are obtained by using 2

∼

5

×

105 _{elements for a four-stage compressor. A structured} meshisgenerated,basedonlyontheannulusgeometry,requiring thenumberof elements ineach geometric directionasan input. AnexamplemeshisshowninFig.2.

Duringacompressor stallitisexpectedthat thepressure and flowfield discontinuities play an importantrole in thedynamics ofthestall.Itisthereforenecessarythatanysolverusedisableto robustly andaccurately model discontinuities andhigh gradients inflow properties (

ρ

,

u

,

T) typical ofreverse flow while using a coarsemesh.Finitevolumemethodsusingclassicfluxeswouldnot beoptimalasthey areknowntointroducediffusivityand numer-icaloscillationsin suchconditions[20]. TheMacCormack scheme

Fig. 2.Example grid generation for NASA TP1493 compressor rig [30].

Fig. 3.Behaviourofmonotoneandnon-monotoneschemesindiscontinuity mod-elling,basedonoriginalartworkbyHirsch[20].

Fig. 4.Riemann problem solved at inter-cell boundaries.

(ashockcapturingmethod[21])waspreviouslyusedwithin Cran-field in this same application to avoid diffusive errors but as a finite-difference method was prone to instability when high gra-dients occurred. In addition to this, artificialviscosity had to be introduced in order to stabilise the flow, which in turn was in conflictwiththehighgradients ofthebodyforce methodand in-troduced significant numerical errors. Toproperly predict shocks oncoarse gridsmonotonicity preserving(monotoneorTVD, Total VariationDiminishing)schemesareconsideredappropriateasthey donotintroduce anyspurious oscillations,withtypicalbehaviour depictedinFig.3.

The Godunovschemeis afirst-order, monotone,finitevolume method whichcalculates the inter-volumefluxesbased on states atthevolume boundaries, obtainedbysolving thelocal Riemann problem[22].

TheRiemannproblem, showninFig.4,issolved bymodelling the wave amplitudes and the state between them, allowing the

physically correct upwinded fluxesto be calculated. As a mono-toneschemetheGodunovmethodguaranteestomodelaccurately discontinuities and to avoid any unphysical behaviour (such as rarefactionshocks). Asa first-ordermethod theGodunovscheme introduces substantial diffusivity. However, it is suggestedthat it could be advantageous when simulating a compressor with the Eulerequationstointroducesomeartificialdiffusivity.Sinceshear stress is absent,the presence ofsome numericalviscosity would preventunphysicalsolutionswithextremerecirculationfrom aris-ing. The scheme is applied in 3D as an unsplit finite volume method, which means that no dimensional splitting is used and thesolution isadvancedin time usingall fluxesin asingle step. The HLLC (Harten-Lax-van Leer-Contact) Riemann solver is used tomodelthewavesattheboundaries[22].Timestepping is per-formedbyemployinganexplicitForwardEulerscheme,wherethe CFLconditioncanbeexpressedas

t

≤

x 2amax

.

(3)

The Eulerequations are modified inside ofthe blade rowsby thefollowingsteps:

1. Addingavector f containingtheforces’sourceterm.

2. Removing the fluxes inthe tangential direction asthese are notallowedinthepresenceofblades.

3. Introduceaterm,assuggestedbyGong[16],whichtranslates theflowinsideoftherotorswithmovementoftheblades. Theequationsinsidethepassagethen becomethoseshownin Equations (4)–(5),where

iszeroforstatorblades. The transla-tionaltermissolvedusinga3RD orderupwindscheme.Usingthis formulationallowstosolvethewholedomain,bothstaticand ro-tatingparts,intheabsoluteframeofreference.

∂

∂

tU

+

1 r

∂

∂

r

(

r E

)

+

r

∂

∂θ

U

+

∂

∂

xG

=

S

+

f

,

(4) f

=

⎡

⎢

⎢

⎢

⎣

0 f r fθ fx WE X

⎤

⎥

⎥

⎥

⎦

.

(5)

Bodyforcemethod.Tosimulatethepresenceofthebladesthebody force method is used. As suggested by Longley [18] the length-scale of a rotating stall cell is greater than a few blade pitches, andit isthen sufficient to modelonly a reducednumber of vir-tual passages in place of the physical ones. The virtual passages are thetangential elements ofthe grid,with thenumber of ele-mentsintheorderofthenumberofrealblades.Thereisthenno resolutionoftheflowprofileinsidethechannels,insteadtheflow insideeachgridelementisapitchwise averageoftheflowinside onechannelormore.Sincethenumberofvirtualpassagesis con-stantalongthecompressor,theratioofreal/virtualpassagesvaries betweencompressorstagesandcanbegreater orlessthanunity. ThisisdemonstratedinFig.5.

The passages are two-dimensional asthey have axial and ra-dialresolution;eventhoughtheflowcanbeinstantaneously mis-aligned with the passage direction, the tangential flux is set to zero.Thebodyforcesconsistoftwoterms:

•

Aturningforce fN turnstheflowinthedirectionimposedby theblade without changingthe total pressurein therelative frameofreference.

•

A pressure loss force fP L imposes locally the total pressure lossestimatedbytheempiricalcorrelation, P0.

Fig. 5.Circumferentialgridelements,withdifferentreal/virtualpassagesratios be-tweenstages.

Fig. 6.Flow velocities and body force decomposition.

The body forces are calculated only along the planes formed by thegrid;theyarebasedonthevelocityprojection onthegrid and applied back in the direction of the grid. Every component of velocity normal to them is not considered and is a source of error. Thevelocity along theplane isdecomposedinto avelocity parallelandnormaltothecamberline,uηanduξ respectively.The projectionof fP L and fN formsthethree f forcesinEquation (4). The workexchange termWE X iscalculated multiplyingtheforce inthecircumferentialdirectionandthebladevelocity(Fig.6).

The direction which the body force imposes is based on the local camberline modified to matchthe inlet flow angle andthe estimated deviation at the outlet, thus the virtual passage is al-waystangentto theincomingflowandthepredictedoutletflow. Theforces fP Land fN areprojectedintheinstantaneousdirection ofthe flowto avoidanyinterferenceoftheturning andpressure losseffects.Theforcesarecalculatedbasedonthevelocitiesinthe relativeframeofreferenceandthenappliedintheabsoluteframe. Thebodyforceformulationsused,Equations (6)–(8),werederived fromwhatwasproposedbyLongley[18] andBrand[17] as

fP L

= −

P0 P

/

P0

x

·

cos

β

,

(6) fN

=

1

−

M2_x cos2

(β)(

₁

−

_M2

)

·

ρu

2 η RC U R

+

tan

β(

1

+

(

γ

−

1

)

M2

)

1

−

M2 fP L

+

fC O R, (7) fC O R

= −

ρ

|

ux| uξ s

·

cos

β

·

k

.

(8)

The turning force consistsof a main component, which takes into account the change in momentum along the local radius of curvatureofthebladepassage.Thisalsoincludesthepressure gra-dientgeneratedbytheturningforceitselfandbythepressureloss force fP L.The formulationis modified when M isnear unity to avoidsingularities.

The component fC O R is acorrectiveforce whichreactsto the misalignment between the direction imposed,

β

, and the actual flow direction;itisthereforeproportional tothevelocity compo-nentnormaltothelocalimposeddirection.Thisforceisafunction

Fig. 7.Possible flow regimes.

ofthebladepitch,howeveraconstantkisnecessary.Thevalueof k issetlarge enoughto ensureuξ iszeroatthepassageoutflow ineveryregime;inthecurrentcaseavalueof10wasfoundtobe sufficient.

Thetotalpressurelossacrossthebladerowiscalculated mul-tiplying a pressure loss coefficient

ω

with the dynamic head at theinlet.Thepressurelossisthen distributedalongthechordto obtainthepressurelossineveryelement P0;asinusoidal distri-butionsimilartothatproposedbyGong[16] isused.Thepressure loss coefficient

ω

and deviation

δ

are calculated using empirical correlationsforevery virtualbladepassageatevery gridplane.If thepassage is unstalledthen

ω

and

δ

are estimated usinga set ofcorrelationsfor forwardflow reportedby Aungier [23]. If itis stalled,then thecorrelation forseparatedpassage flow, proposed byThomasonandMoses[24],isused.

Toidentifywhetherthe bladepassageis stalled, acriterion is used,proposedbyAungier[23] andbasedontheaerodynamic ve-locityratioWR E.Thepassageisstalledif

WR E

=

P0O U T

−

PO U T P0I N

−

PI N

<

WM I N (9)

where WM I N is theminimum velocity ratiobefore stallandis a functionofthecompressorgeometry.

Theempirical correlationsusedare validonlyforsteady flow. Toaccountforthelagwhichoccurswhentheflowstallsor recov-erstheempirical coefficientsare updatedusinga first-ordertime laglaw,describedinEq. (10).Thetimeconstant

τ

isthetimethat theflowtakestocrossthepassage.

∂δ/∂

t

=

(δS S

−

δ)/

τ

.

(10)

Reverseflowmodelling.Flowmodellingduring stallrequiresthe ca-pabilityto deal withdifferent regimeswhich appear asthe flow reverses direction. These are identified atevery time step, blade row,circumferentialpositionandgridplane.Theyaregroupedinto three flow regimes: forward flow, fully reversed flow and mixed flow,Fig.7.

The mixed flow regime is the most difficult to model as it requires more assumptions and uncertainty. Gamache [25] ob-servedthatinside astallcelltheflowisfullyreversed whilstthe mixed/invertingflow ispresentonly inthe thinboundary ofthe cell.Itis thenmore importanttoaccurately modeltheseparated fullyreversedflowasthishasthegreatestinfluenceonthestalled performance.

Thereversed regime ischaracterized by the separation ofthe flowinsideanddownstreamoftheblade.Modelsofseparatedflow

Fig. 8.Flowstructureinaseparatedchannelinreverseflow,basedonthemodels byLongley[18] andThomasonandMoses[24].

havebeendeveloped byThomason andMoses[24] and more re-centlybyLongley[18],bothbasedonsimilarassumptions:

•

Flowseparatesonthesuctionsurfaceandthemixingcan con-tinueafterthetrailingedge.

•

Theseparatedjetexitsatthetrailingedgewiththedirection ofthestaggerangle.

•

Frictionisnegligibleandtheflowwhichformsthejetis isen-tropic,pressurelossderivesfromtheseparation.

•

Deviationisduetothewakemixing.

In themodelproposed byLongley theseparation isproduced by the force which abruptly turns the flow when it enters the bladeata highangle.Thisforceistheintegralofthepressureon thebladesurface,andthereforeitisalwaysnormaltotheblade.As theforceisdirectedagainsttheincomingflowitintroducesa pres-sure loss.The flow does not lose total pressure immediatelybut separatesisentropicallyandthelossisnotdetectableuntilmixing occurs.

Asimilarapproachisusedinthecurrentcode.Whenmixedor reversed flow is presentthe imposed directionstops adapting to theincomingflow. Thenormalandparallelforcesarereplacedby onlyaturningforcewhichisnormaltotheblade,andnotaligned totheflowangle.Thechangeindirectionisprovided bythe cor-rection force FC O R and

ω

is not calculated, the pressure loss is introducednaturallybytheturningforcedirectedagainsttheflow. Thedeviationisimposedattheflowoutlet(i.e.theleadingedge) withthesamemethodusedforunstalledflow.

Thissystemispreferredtotheuseofempiricalcorrelationsas itcan beappliedduring mixedflow,when aclearly definedinlet oroutletdoesnotexistandstrongradialflowispresent.The cal-culationofdeviationhasbeendevelopedstartingfromthemodel of Longley.The parameter b isdefined inhis model asthe ratio betweentherealaxial momentum oftheseparatedflow andthe axial momentumofthepitchwise averaged flow. Itcanbe calcu-latedfromtheinletattackanglebyusingEq. (11)

b

=

1

+

sin

(

α

2

−

γ

)

cos

(

α

2

)

.

(11)

Byapplyinga controlvolumetotheflowbetweenstates j and 1, as shown in Fig. 8, and by solving the momentum in the axial andtangential directions,the flowangle

α1

canbe computedby Eq. (12),thus

tan

(

α

1)

=

tan

(

α

j)

·

b

.

(12)

However,whilstthemodelconsiderstheflowinthepassageto befullyisentropic,inrealitythemixingoccurspartlyinsideofthe bladeattheinterfacebetweenthebubbleandthejet.Thisis con-sideredin the methodof Longley through an entropy-generation term, whichinevery element whereseparationispresent, mixes out the flow at a prescribed ratio. The prescribed ratio is such

asthe excess energy,which is the energycontained inthe non-uniformity ofthe flow, is mixedin the time takenforthe jet to travelthedistanceofhalfapitch.

Intheformulationproposedby thisauthortheequation mod-ellingtheb parameterisnotused,thereforeitisnecessaryto in-troducethemixingdirectlyinthedeviationcorrelation.Thepoint atwhich thefull separation is reachedis setat aquarter ofthe chordandfromthispoint theprofileofb inthepassageis mod-elledbasedonamixingratiosimilartothat whichwas proposed byLongley.Inthecurrentcasethedistanceusedtosetthemixing isnothalfthepitch,buttwicethepitch;thiswasfoundtogive ap-propriateresultsona rangeofspeedsontwodifferentlow-speed testcases.Wecanthenestimatethevalueofbatthetrailingedge, fromwhichtheanglepost-mixingcanbecalculated.Thedeviation angleisthenafunctionalsoofthebladesolidityas

δ

=

f

(

α

I N

,

γ

,

σ

).

(13)

Asthedeviationiscalculatedbasedontheinletangleitcannot be estimatedifthe flowis exitingfrombothblade edges,in this casetheflowexitangleissetasthemetaloutletangle(

δ

=

0). Simulationmethod.Thegeometryinputtothecodeconsistsofthe annulus shape, the meridional position and shape of the blades, andthe profiles along the span of inlet and outlet metal angles andstaggerangleofeachbladerow.Additionaldetailsoftheblade profiles are necessarydepending on the forwardflow correlation used.

Non-reflectingboundaryconditions[26] areappliedatinletand outlet,applyingtotalpressureandtemperaturewhereflowenters thedomainandstaticpressurewhereflowexitsthedomain.With thecurrentboundaryconditionsformulation,theflowenteringthe domainisassumedtobeaxialtothecompressor.Thepresenceof a combustorandturbinefollowing thecompressor isreproduced usingatankwhichdischargestotheatmospherethroughanozzle, asis normalpractice on an experimental compressor rig. Thisis simulatedusingazero-dimensionalmodelwhichmodelspressure anddensitybasedon theflux betweenthe tank, thecompressor outletandtheatmosphere.Thevolumeofthetankisuser-defined andinfluencesthestalldynamics.Thethrottlecanbechanged dy-namicallyasrequiredduringthesimulation.

The simulation can be run with only a single tangential ele-ment,thus beingaxisymmetric;which isusefultoquicklyobtain a starting point for stall simulations, close to the surge margin. The simulationcan then be loaded andrestartedwitha number of circumferential elements and the tank throttle model behind the compressor. The compressor is throttled to move the oper-ating point beyond the stability line and the stall is allowed to develop. For the compressor to develop a physically correct ro-tating stall cell, some asymmetry must be introduced artificially. Inthiscodeasymmetricvariationsofpressurewereintroduced in theinletflowfield,whichwasfoundtobesufficienttoproduce(in thecorrectconditions)arotatingstall.Othermethodstointroduce asymmetryincludeintroducingrandomdisturbancesintotheflow field,asdonebyLongley[18],orchangingslightlythegeometryof thecompressortoreplicatemanufacturingtolerances.

3. Resultsanddiscussion

Experimentalresults froma low speed3-stage compressor, as reported by Gamache [25] and Eastland[27], were selected asa validation case. Two builds were tested by Gamache, all results presentedhereareforthemoderatereactionbuild.Steady charac-teristicsforreverse flowandrotatingstallwere obtained,aswell asflowprofiles andstall cellinformation.Mapswere reportedin

Fig. 9.Trajectory of rotating stall inception.

the form of static-to-static, total-to-static andtotal-to-total pres-surerise characteristics.In thefollowing resultsthe total-to-total ratioisused,

ψ

T T.

Fullmapcreation.Thefullmapconsistsoftheforwardandreverse flow characteristics, withthe forwardflow characteristics formed ofarotatingstallandpre-stallcharacteristic.Toobtainthereverse flow characteristic the code is run witha single element in the circumferential direction, thus modelling the flow as axisymmet-ric. Boundary conditionsare fixed andthesimulation isrun to a convergedstate.Tocheckfornumericalorphysicalhysteresis, sim-ulations were started withinitial conditions of both reverse and forwardunstalledflow, withidenticalresultsineachcase. Exper-imental dataisreportedforthreecompressorspeeds, 1170,1800 and 2400 RPM, however on a map of flow coefficient

φ

against pressure rise coefficient

ψ

the characteristics collapse onto the sameline.Thesearedefinedas

φ

=

ux/U

,

(14)

ψ

=

P0O U T L E T

−

P0I N L E T

0

.

5ρU2

.

(15)

These definitions are based on velocity and densityupstream oftheIGV. Thefull reversecharacteristicisobtainedby changing theboundaryconditionsinstepsandallowingasettlingtime.The timerequiredforeachsimulationisintheregionofthirtyminutes. AcomparisonismadeagainsttheexperimentalresultsinFig.10, with the simulation matching the trend of the experimental rig, suggestingtheformulationproposed forreverse flowis appropri-ate.

Forrotatingstallonlythehighestspeedof2400RPMwas mod-elled,withthegridcontaining64elements inthecircumferential direction. Stallwasobtainedusingthesamemethodologyas pre-viouslydescribed,startingfromasimulationclosetothestallline andreducing the throttlefurther whilstsimultaneously introduc-ing a distortion at the inlet. Again, two levels of distortion and throttleclosurearecarriedoutforthefirstpointtoensure repeata-bilityofresultsthatareindependentofthetriggeringmechanism. Subsequent points are obtained by incremental throttle changes. ThesimulationshowninFig.9hasaphysicaltimeof500 ms(20 revolutions)andwasobtainedinapproximatelytenhours.

Successive movements on the map are obtained in physical times of up to 350 ms, depending on the flow coefficient. High flow coefficient points reach stability in longer times,suggesting that in thispart ofthe map the characteristicis almost overlap-ping withthethrottlecharacteristicandthestability isweak.The full map comparing all three regions to the experimental results is shown in Fig. 10. The forward flow region of the map is not reproduced with significant accuracy, however this characteristic

Fig. 10.Full compressor map comparing simulated and experimental performance.

Fig. 11.Fulltorquecompressormapcomparingsimulatedandexperimental perfor-mance.

dependsheavily on the correlationsused, which inthis caseare takenfromopenliterature andclearlydonotrepresentaccurately thiscompressor. The matching of the forwardflow characteristic wasnot pursued further,asit was outsideoftheresearch scope. IntheoperatingconditionsreportedbyEastlandchokingnever oc-curs, and therefore the code capability of capturing the correct chokingmassflow was not addressed. Overall the map has been recreatedwithagoodmatching.

There is no data on the compressor efficiency, but the mea-suredtorquecoefficientsare available forallthe regimes[27,25]. Insteadystateconditionsthematchingofthetorquesignifiesthe correct modelling of the total temperature rise, and thus of the efficiency. The results of the modelled and measured torque are comparedinFig.11.Theresultsfromthesimulationshavea sim-ilar trend to the experimental characteristic but with a constant offset. Only atvery low positive

φ

the trendof the results from thecodedivergesfromtheexperimentalones.

Rotatingstallcellproperties.The modelsimulated asingle rotating stall cell for all flow coefficients, in line with the experimental findings.Thecompressorrigwas probedusinga traversehotwire in front of the IGV. During rotating stall the passage of the cell leavesatraceinthehotwirevoltage,whichcanbeusedto deter-minethespeed ofthestallcell basedontheperiodofthe trace. The size can be inferred from the ratio of the distorted region ofthetrace onthe overallperiod.Mimicking thepost-processing technique described by Gamache on the simulation results, the rotatingstallcellspeedandsizearecomparedagainstthe experi-mentalresultsinFigs.12and13.

Thesimulatedstallcellspeedislowerthanthemeasuredspeed butitcaptures correctlytherange andtrend,althoughthe mini-mumwaslocatedatahigherflowcoefficient.Thetrendalsoagrees withtheexperimentalrulethat partspanstalls rotatefasterthan

Fig. 12.ComparisonofsimulatedstallcellspeedatIGVinletagainstexperimental data.

Fig. 13.Comparisonofsimulatedstallcellsizeat IGVinletagainstexperimental data,overlaid withsimulatedcontourofaxialvelocityshowingreversedflowin black.

Fig. 14.Axialvelocityataconstantradius,showingtheshapeofthestallcellacross thecompressorstages.(Forinterpretationofthecoloursinthefigure(s),thereader isreferredtothewebversionofthisarticle.)

full span [28]. The size of the cell is more difficult to compare against due to the ambiguity of the experimental measurement, with datafrom the two sources, Gamache and Eastland, proving to bediscontinuous asshownin Fig.13.Despitethis, theoverall trendiscorrectlycaptured.Anexampleofthecellshapeacrossthe compressorstagesisshowninFig.14.

Fig. 15.Comparison of surge cycle simulations with different tank volumes.

Fig. 16.Types of distortion of inlet total pressure used in stall simulations.

Surgesimulation. Following the creation ofa steady characteristic map,themodelwasusedto simulatedifferenttypesofsurge cy-cle. The model, similar to that described previously to simulate rotatingstall,hastheinletdisturbancereducedwithtwo90◦ sec-torsof

−

4 kPaand

+

4 kPa,relatively,lastingfor10 msatthestart ofthesimulationalongsideeachother.Inthisway,theasymmetry isnotdriving theeventandit isonlypresentinthe background. Many simulations are run with differing volume sizes, usingthe same constant throttle setting andthe same rotational speed of 2400 RPM.AsseeninFig.15,thelargestvolumeproducesadeep surge cycle with little overshoot that follows closely the steady characteristic with an almost constant pressure ratio during the reversalandrecoveryphasesofthecycle.

Asthe volumereducesthebehaviour becomes moredynamic, including larger overshoots on the blowdown phase and a cycle that ismore obliquein shape, reflectingwhat Day found experi-mentally[29].Oncethevolumeissmallenough,inthiscase1 m3_, the flowdoes not fullyreverse andduring the recoveryphase it developsamature locked-inrotatingstall,findingequilibriumon therotatingstallcharacteristic.

The cases which surged never developed a strong asymmet-ric flow pattern. This pattern grows during the blowdown and re-pressurisationphaseofthecycle;whilethecompressoris oper-atinginfullyreverseflowinsteadit reduces,andwhenfollowing theunstalledforwardflowcharacteristicitisremovedcompletely. Asthecasewith1 m3volumeneverreachesthereverseflow char-acteristictheasymmetry continuestogrow andinthisparticular scenariothecompressorisnotabletorecover.

Tofurtherverifythecapabilityofthecodetopredictthecorrect outcome of a stall event a further simulation is carried out us-ingthesameconditionsasinthe3 m3 _{volumecase.Asdescribed} in literature [28] with the same RPM, throttle, tank volume, the systemhasthesameGreitzerparameter, B.Thesameinitial con-ditionsareexposedtoamoreaggressivepressuredistortionatthe inlet,

−

10 kPaona90◦ sectorfor10 ms,thisdistortionisshown inFig.16asPattern Bandcomparedtothedistortionused previ-ouslyforsurge(pattern A).

Fig. 17.ComparisonoftwoeventsatsameB parametertriggeredusingdifferent inletdistortions.

Pattern B distortion guarantees that a rotating stall cell is formed immediately and as a result the operating point spirals down onto the characteristic line, as shown in Fig. 17. Accord-ing to literature there should exist a value of the B parameter that characteriseswhen the compressor transitions from a surge cycletolocked-inrotatingstall,whichwasshownwiththe previ-ous simulationstooccurbetweenavolumeof1 and1.5 m3.This currentsimulationliesfaroutsidethisboundaryyetstillmanages to lock-in stall, demonstrating the dependence on the triggering mechanism ofthe stalland not justglobalperformance parame-ters.Inthisregard,thedevelopedcodeprovesusefulasitisableto predicttheoutcomeoftwodifferentstalltriggeringmechanisms.

Validation ofthe surgecyclewas not possible since the com-pressor was not tested under these conditions. Surge cycles at different B parameterswere modelledandreportedby Gamache, howeverthesearealsoafunctionofthethrottlecapacitywhichis notreported,henceacomparisonisnotpossible.

4. Conclusions

A3Dtoolcapableofsimulatingrotatingstallandsurgewas de-velopedbasedontheGodunovfinitevolumesolver,thebodyforce method andempirical correlations. Thisworkis thefirst applica-tionofaGodunovtypesolvertorotatingstallandsurgemodelling, the featuresofthisschemearedeemed particularlybeneficialfor this application. A newformulation fortreating reverse flow has been proposed, a simplified method based on the blockage/non-uniformity model.Thenewtool was testedon alow-speed three stagecompressorwiththefollowingoutcome:

1. Thesteadyreversecharacteristichasbeenreproducedfor dif-ferent rotationalspeeds, exhibitinggood agreement withthe experimentdata andenforcingtheapplicability ofthe devel-opedsimplifiedreverseflowmodel.

2. Rotatingstallwassimulatedobtainingaverygoodagreement with the entire measured characteristic. The accuracy of the simulated performance is enhanced through comparisons of thestallcellspeedandsizewithexperimentalvalues. 3. Deep surge was simulated for different system volumes in

agreement withthe expected behaviour, with validation not possible dueto lack of experimental data. Through changing the distortionlevelthe dependenceofthestall outcomewas demonstrated,allowingdifferentbehavioursforthesame Gre-itzerparameter.

Thevalidationcarriedoutdemonstratestheapplicabilityofthe proposed formulation to low-speed compressors, with successful simulation of all three operating regimesof the compressor. The developedtool iscapableofreproducingthefullcompressormap

ofalow-speedcompressorinlessthan72hours,acomputational costunrivalledbymodern3DclassicalCFDcodes.

Conflictofintereststatement Nonedeclared.

Acknowledgements

TheauthorswouldliketoexpresstheirgratitudetoRolls-Royce plc. for funding this research and for permission to publish the paper.SpecialthankstoMr.ArthurRoweandMr.RichardTunstall fromRolls-Royceplcforsupervisingthiswork.

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