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JAST� Vol� �� No� �� pp ����c� Iranian Aerospace Society� March ����

Numerical Investigation of Rotating�Stall

in a Stage of an Axial Compressor

with Two Di�erent Approaches

N� Amanifard�� B� Farhanieh�� K� Ghorbanian

Rotating�Stall �RS� through a cascade of an axial compressor has been nu�merically investigated by employing an unsteady two�dimensional �nite�volumedensity based computer code� To validate the computer code� three test caseswere prepared� and the good agreement resulting from comparison of the resultshas given adequate assurance of the code� The cascade problem was studiedthrough di�erent geometric approaches� which were classi�ed in two maincategories� The �rst was the semi�stage �single rotor�cascade�� and the secondwas the complete stage� For each geometric approach� the number of blades wasvaried from a minimum number to a maximum value �variable cascade length��The RS modal characteristics and its development for all geometric cases wereobserved� The RS was incepted with a ��� reduction in ow coe�cient and a���� increase in the load coe�cient from their normal operating values� In allcases� the captured modal characteristics of RS varies with the variation of thecascade length to a maximum value� and this triggers the same modes of RS�The variation of the modal characteristics of the RS� in the two main geometricapproaches seems to be similar� but at di�erent levels�

INTRODUCTIONAnalysis of the �ow �eld in unstable condition� espe�cially in Gas�Turbine engines� has been mainly basedon experimental observations and studies for the lastdecade ���� Recently� several computational �uiddynamics �CFD� codes have been developed� Fromthe viewpoint of �ow instability phenomena� axialcompressors with their adverse pressure gradients inthrough �ow direction are the most critical componentin Gas�Turbine engines� Today� using CFD tools is astandard practice in the study of cascade �ow withinthe stable operating range of a compressor� but theCFD approach still needs to be established as a soundprediction method for operation in the unstable region�Notable progress in the use of CFD�type techniquesfor calculation of instability e�ects has been made

�� Assistant Professor� Dept� of Mech� Eng�� Guilan Univ��Rasht� Iran� Email� namanif�guilan�ac�ir�

�� Professor� Dept� of Mech� Eng�� Sharif Univ� of Tech��Tehran� Iran�

� Associate Professor� Dept� of Aerospace Eng�� SharifUniv� of Tech�� Tehran� Iran�

by Sisto et�al� �� and Jonnavithula et�al� ��� Theyused a two�dimensional discrete vortex model� withthe separation point being obtained by an integralboundary layer� The evolution of stall is well predictedin their experiment� although only up to six bladepassages are used in the computation� Further� anumerical study was made by He � in a single stageaxial compressor� The Navier�Stokes equations werediscretized in space by �nite volume method� andintegrated in time by a four stage Rung�Kutta scheme�The second� and fourth�order blended smoothing wasadopted in both the stream wise and circumferentialdirections for numerical damping� Baldwin�Lomax tur�bulence model was also adopted� Outa et�al� �� made anumerical simulation for stall cells in rotor�stator frameof a compressor using viscous approach� Furthermore�a numerical study was done by Saxer et�al� �� oninviscid �ow passing through axial compressors� andthe history of stall vortices were investigated for a�fteen�blade passage of a single stage� The mass �owrate �uctuations were approximately similar to thoseobtained from their experimental observations�

In previous CFD works on RS� one can not �nd

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any criterion for choosing the number of blades andthe geometric approaches� The scope of this workis to propose an adequate number of blades in CFDstudies of RS� and to compare the e�ect of using afree�rotor cascade in place of a complete stage on RSmodal shapes�

GOVERNING EQUATIONSIn Cartesian coordinate� let Q be an unknown vectorde�ned for a two�dimensional study as follows�

Q � ��� �u� �v� �ET� �q�� q�� q�� q�

T� ��

where E is the total energy �E � e � �u� � v������� is the density� u is the axial velocity componentand v is the tangential velocity component� Let V beany volume with bounding surface �V and outwardunit normal n� Assuming that the volume doesnot vary with time� Q satis�es the following integralconservation law�

d

dt

ZV

QdV �

ZV

�

�tQdV � �

I�V

#E�ndS� ���

The equivalent di�erential form of Equation ��� in aninertial reference system reads�

�

�tQ � �r�#E� ���

This accounts for the inviscid �#EE� and viscous �#EV �contributions� i�e�

#E � #EE � #EV � ���

where

#EE � ��u� �uu� pI� �u�E � P���T� ��a�

#EV � ��� ���q� u��T � ��b�

and

� ��ru�ruT�� �

��r�uI� � �

q � ��rT� ���

The governing equations are transformed to a com�putaional space for the numerical solution� Hence� theyare as follows�

�Q

����EE

� ��FE��

��Ev

� ��Fv��

� ���

Q is the premitive variables matrix� EE and FE arethe transformed convective �ux matricies in � and� directions respectively� and E and F are thetransformed viscous matrices in � and � directions in

turn� � is the dimensionless time variable� u is thevelocity vector� q is the heat �ux� I is the identitymatrix� � is the absolute viscosity and � is the thermalconductivity� The superscript T denotes the transposematrix assignment�

Regarding the two�dimensional approach inpresent work� the governing equations for relative frameare the same as those in the absolute frame becausethere is no Coriolis acceleration �no radial componentof velocity�� Additionally� the stagnation enthalpy isconstant in the relative frame and as a result there is noenergy transfer between the �uid and the blades in therelative coordinate� Consequently� the only treatmentin stationary�cascade studies �if desired� is the using ofabsolute velocities in place of relative velocities in thesolver� As evident� the absolute velocities are achievedby adding the rotation speed to relative velocities�

NUMERICAL PROCEDUREFinite�Volume FormulationReconsidering Equation ���� the time derivative isapproximated by a �rst�order backward di�erencingquotient and the remaining terms are evaluated at timelevel n� � Thus�

Qn�� �Qn

��� �

�E

� �n����

�F

���n�� �

��Ev

� �n�� � �

�Fv��

�n��� ���

Integrating Equation ��� over square ABCD shown inFigure � and using Green�s theorem provides�

�Q���

� �EEr �EEl� �

��

���FEt �FEb� �

��

� �Evr �Evl� �

��

���Fvt �Fvb�� ���

Since� in Equation ���� the �ux vectors are evaluatedin time step n � � they can be expressed in terms of�Q� by using the Taylor expansion and a �rst orderapproximation in time�����

Figure �� The Finite Volume Cell�

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Numerical Investigation of Rotating�Stall in a Stage of an Axial Compressor with Two Di�erent Approaches �

Figure �� Schematic shape of Block Boundary Condition�

The inviscid �ux vectors on cell faces were evalu�ated by Van�Leer�s Flux splitting scheme� To preventthe oscillatory behavior of the numerical results and toincrease the accuracy� the Van�Leer�s limiter was addedto the �ux splitting algorithm �����

The second�order derivatives are evaluated bycentral di�erence approximation�

Numerical Boundary ConditionsRegarding the subsonic in�ow� the in�ow and out�owboundary conditions are set for transonic �ow uponeigen vectors characteristics�

For the upper and lower boundaries of the cas�cades� the periodic boundary condition was used to givethe circumferential continuity of the cascade�

The physical domain was split to multi zones forusing the multi�block technique and each block� Figure� shows the schematics shared area of the two adjacentblocks� The common walls of the rotor and stator rowsshares their data in each time step as their adjacent wallboundary condition� However� the data exchangingfrom the relative frame to absolute frame was alsoadded to the adjacent wall condition at the interfaceof the rotor and stator for the complete stage study�

yU�vx���

u�U

Figure �� Comparing the computed velocity pro�le for �atplate with the experimental pro�le at ��� of plate length�from the leading edge of the �at plate�

PressureCoe�cient�Cp�

Percent of Chord

Figure �� Computed and experimental Pressure coe��cients over a semi NACA���� airfoil� Mach number is ���and Reynolds Number is ����

Grid GenerationEach passage �between two blades� has an individualmesh� which is generated by mesh generator programusing Partial Di�erential Equations �PDE� method�Clustering is available by related source terms as well asorthogonality� The mesh generated for each individualpassage is considered as a single block� and the solverassembles them to prepare the complete area of solutionby using the multi�block boundary condition� Theresolution is chosen upon grid dependency studies�which show that a ���� grid for each block has aminimum CPU time without any grid dependency ofthe results�

Turbulence ModelingAn algebraic model� which is not written in terms ofthe boundary layer quantities and is very robust inseparated regions� is the modi�ed Baldwin�Lomax �BL�model ����� The BL model is implemented by He � in numerical investigation of Rotating�Stall inceptionin a multi�blade cascade �ow in an axial compressor�in which the �ow may have large scale separated zones�Moreover� the comparison of other turbulence modelssuch as ��� with BL model� done by Bohn et�al� ��shows adequate assurance using BL model in cascadeproblems� Although a large amount of memory isrequired in multi�blades studies� the BL requires theleast amount of memory and CPU time with respectto higher�order turbulence models� Consequently� inthe present work the BL model is preferred�

DISCUSSION OF RESULTSCode Validations StudiesThe performance of the described methodology isassessed by comparing the computed results with otherveri�ed data�

The �rst test case is performed on �ow over a�at plate in incompressible range of the Mach number�M � ����� The Reynolds number is �� and the

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transition point is set at leading edge of the plate� Thisstudy gives adequate assurance of viscous solution ofthe solver with the implemented turbulence model�

Additionally� the good agreement of the resultswith those reported by Bohn et�al� shown in Figure� shows that the grid resolution ������ used forthe computational domain is su�cient to capture theboundary layer�

The second test study is performed on the sub�sonic viscous �ow over a NACA��� airfoil� TheReynolds number is set to � and the �ow is turbulent�The inlet Mach number is set to ���� and the angle ofattack is set to zero� Figure �� shows the computedresults� which are in good agreement with experimentalresults reported by Fletcher ���

The third test study is performed on the transonicviscous �ow over a NACA��� airfoil� The Reynoldsnumber is set to � and the �ow is turbulent� The inletMach number is set to ���� and the angle of attack is��� Figure � shows the computed results� which are ingood agreement with the experimental results reportedby Hirsch ���

The �nal test is performed to evaluate the op�timum grid� A two�passage stage which is a multiblock system similar to that used for compressor stagesimulation is chosen for the grid dependency study�Figure �� Three grids are performed for the studyand the grid ��� � �for each block� �Figure �� seemsto be adequate for the present study�

FLOW THROUGH A ROTOR OF AXIALCOMPRESSOR

The �nal studies are performed on the �ow through asingle rotor�cascade and the �ow through a complete

PressureCoe�cient�Cp�

Figure �� Computed and experimental Pressure coe��cients over a semi NACA���� airfoil� Mach number is �����the Reynolds Number is ���� and at �� angle of attack�

Figure �� The geometry of the double passage stage usedfor the mesh dependency study�

Log�ContError�

Solution Cycles

Figure �� The Mesh dependency study upon convergencehistory�

��D stage at the unstable operation point of an axialcompressor�

The geometrical characteristics of the blades usedin the cascades �rotor and stator rows� are given inTable � and the �ow characteristics in the stableand unstable operating conditions are given in tables�nd� �rd� respectively� The thermodynamic path forchoosing the unstable operating point near the stabilitylimit line is shown in Figure ��

The studies for the free�rotor and the completestage cases were prepared for a set of cascade lengths

Table �� Geometrical characteristics of the cascade�

Stagger angel

of rotor blades��

Stagger angel

of stator blades �

Rotor blade

pro�leNACA���A��

Stator blade

pro�leNACA���A��

Solidity �� �

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Numerical Investigation of Rotating�Stall in a Stage of an Axial Compressor with Two Di�erent Approaches ��

with �� �� �� �� �� � �� �� and blades� The bladesare numbered from bottom to top� Near the leadingedge of the odd blades in rotor row� numerical probesare located to indicate the axial velocity traces� Toprovide a clear imagination about the computationalarea� Figure � illustrates the ��blade cascade for thefree�rotor case and the complete stage case as a sampleof all studied cases� The solid lines show the boundariesof each block �regarding the multi�block technique��and � probes are located near the leading edge of theblades�

The total number of �nite volume cells is equalto� Number of blades ������� Figure � shows anenlarged section of the computational area for the free�rotor case to give a visual presentation of the usedmesh�

The velocity traces for RS condition of the ��blade and the �blade were nearly the same andthey were sketched against the dimensionless time�rotor revolutions� and presented in Figure for bothgeometric cases� For clarity� the time traces from eachprobe shown in Figure are shifted by a constantinterval� and the time is computed from the time ofimposing the unstable conditions� It is noticed that thiscalculated RS pattern seems to be di�erent from thatcommonly observed in experiments� where a single RScell almost always comes into existence during a stallinception process� From the modal�wave point of view�the single�cell pattern should be associated with the�rst�order modal wave with its wave�length being thecircumference� In the present case� no indication of the�rst�order modal�wave was observed� As He � studied�

Table �� The stable condition�

rh�rt ���

R ����

Pin ������ Pa

Tin �� K�

Pexit� Pin �������

�� ���

� ����

� ���

Vxin � m�s

Min ����

Ur �� m�s

RPM ����

Table �� The unstable condition�

Pexit�Pin �������

�� ���

Vxin �� m�s

� ����

RPM ����

StagePressureRatio

Corrected Mass Flow Rate

Figure �� The thermodynamic path for reaching RScondition from the normal operating point�

Figure � The ���blade geometry and the probe location�

the reason is that the background noise in calculationswas not big enough to trigger the �rst modal wave togrow�

The streamlines of ��blade studies at the timeof ���� of the rotor revolutions were shown in Figure�� The deep cells happened approximately at every

Figure �� The sample enlarged mesh generated for rotor�cascade studies�

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�a� Free�Rotor Cascade

�b� The Complete StageFigure ��� Velocity traces for ���Blades geometry� Thetraces are shifted with constant interval for clarity�

�ve blades distance from each other in both cases� andthe streamlines seem to be nearly similar� Observingthe velocity traces of the ��blades cases shows thatthe deep cell frequencies of the two cases are nearlythe same but the amplitudes are obviously di�erent�The results of r the �blade show similar behaviors forthe streamlines and the velocity traces and� therefore�are not repeated here� The velocity traces for the ��� �� �� and � blades are shown in Figures to �respectively� The comparison of the velocity betweenthe free�rotor and the complete stage approaches ineach number of blades shows that the di�erences infrequencies and the amplitudes of the velocity tracesare not negligible� This means that the modal shapesof RS for the two geometric approaches di�er from eachother when the number of blades decreases� However�this also shows that decreasing the number of bladesbelow a minimum value may a�ect the modal shapesof RS� and any chosen number of blade can probablytrigger one of the modal shapes of RS� Reconsideringthe velocity traces for various numbers of blades con�cludes � or three modal shapes for the RS� The modesof �� � and � blades�cascades �mode A� are nearly

�a� Free�Rotor �b� The Complete StageFigure ��� The unstable streamlines for �� and �� bladesgeometry at time ��� rotor revolutions�

the same in frequency and amplitude� but they di�erin stall initiation� Similarly� the modal shapes of RSfor the � and � blades cases �mode B� are nearly thesame� but the modes A and B are obviously di�erentin modal characteristics� The results of blades ofthe free�rotor case magnify the existence of a thirdmode for RS� but this does not render the same resultfor the complete stage problem� Figure � provides abrief comparison of modal characteristics of the deepcells during the RS e�ect� It illustrates the frequencyvariation against the number of blades for the stagewith and without stator cascade� The results show thatin values lower than � for the number of blades blades�the frequency of the velocity traces �deep cell repeatingfrequency� varies apparently for the complete and semistages� Figure � also shows that after � blades� thefrequencies of the two geometric approaches are nearlythe same� but the frequency varies when the number ofblades increases from a minimum value� Similar resultsare achieved when considering Figure �� where theduration time of a complete growth of a deep cell inRS �stall initiation� is sketched against the number ofblades for the complete and semi stages� The results ofFigure � again show that after about �� there is nodependency between the stall initiation and the numberof blades for both geometric approaches� This realtyvalidates the forced noise method used by He � forobserving a wide range of RS modal shapes�

CONCLUDING REMARKSA multi�block ��D solver was employed to investigatethe viscous through��ow in stage of axial compres�sor� Three test cases gave provided assurance for

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Numerical Investigation of Rotating�Stall in a Stage of an Axial Compressor with Two Di�erent Approaches ��

employment of the code� The rotating stall e�ect wasnumerically observed for the �ow through a complete�with stator� and a semi�stage axial compressor� TheRS was studied for both cases in a relatively wide

range of cascade lengths� The following conclusionsare noteworthy�

� All velocity traces show a periodic behavior similar

�a� Free�Rotor �b� The Complete Stage

Figure ��� Velocity traces for ���Blade geometry during the RS e ect� The traces are shifted with constant intervals forclarity�

�a� Free�Rotor �b� The Complete StageFigure ��� Velocity traces for ���Blades geometry during the RS e ect� The traces are shifted with constant intervals forclarity�

�a� Free�Rotor �b� The Complete StageFigure ��� Velocity traces for ��Blade geometry during the RS e ect� The traces are shifted with constant intervals forclarity�

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�a� Free�Rotor �b� The Complete Stage

Figure ��� Velocity traces for ��Blades geometry during the RS e ect� The traces are shifted with constant intervals forclarity�

�a� Free�Rotor �b� The Complete Stage

Figure ��� Velocity traces for ��Blades geometry during the RS e ect� The traces are shifted with constant intervals forclarity�

to previous reports �He � and Saxer ���� for anychosen number of blades in RS phenomena for bothmain approaches�

�� Comparing velocity traces shows that there areapproximately at least two major modes for RSduring the variation of cascade length lower than� blades� and the modal shapes of RS do notvary after � blades� The cascade length doesnot a�ect the captured modal characteristics ofthe RS� This may be related to that distance ofthe periodic boundary conditions of the lower andupper boundaries in longer cascades is for enough�

�� Both geometric approaches of the complete and thesemi stages nearly show the same behavior of RSdeep cells during the variation of cascade length�but the frequency and stall initiation times vary indi�erent levels for the two cases�

�� Even though using lower cascade lengths apparentlycauses di�erent modes of RS� the modes seem to be

arti�cial results of boundary e�ects and unstablebehavior of the �ow� As a result� the employmentof higher numbers of blades in modal studies of theRS needs an exciting noise to trigger di�erent realmodal shapes of the RS� This conclusion validatesusing forced pressure noise method in modal studiesof RS used by He � �

�� In RS modal studies� the employment of semi�stageapproach causes errors in frequency of RS� andits propagation speed� Consequently the completestage seems to be a better approach�

� All cases show that without any considerationof modal characteristic studies of the RS eventhe � blades case� with free�rotor approach �themuch simpler geometry�� can predict the instability�which is a useful help to minimize the CPU timesand the required memories in CFD investigationsduring the o��design prediction studies of the axialcompressors� The o��design prediction determines

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StallCellFrequancy�Fs�Fr�

Number of Blades

Figure ��� Variation of the frequency of the vortexoscillation in circumferential direction at the RS e ect forthe stage with and without stator cascades�

StallInitiationtime�t�T�

Number of Blades

Figure �� Variation of the RS initiation time againstthe variation of the cascade length for the stage with andwithout stator cascades�

the location of the stability limit line in compressormaps�

REFERENCES�� Inue� M�� Kuroumaru� M�� Tanino� T�� and Furukawa�

M�� �Propagation of Multiple Short�Length�Scale StallCells in an Axial Compressor Rotor�� ASME J� ofTurb�� ���� PP �����������

�� Day� I� J�� Breuer� T�� Escuret� J�� Cherrett� M��and Wilson� A�� �Stall Inception and the Prospectsfor Active Control in Four High�Speed Compressors��ASME J� of Turb�� ���� PP ������������

� Silowski� P� D�� �Measurements of Rotor Stalling in aMatched and a Mismatched Multistage Compressor��GTL Report� Gas Turbine Laboratory� MassachusettsInstitute of Technology� ���� �������

�� Sisto� F�� Wu� W�� Thangam� S�� and Jonnavithula� S���Computational Aerodynamics of Oscillating Cascadewith Evolution of Rotating Stall�� AIAA Journal� ���PP ��������������

�� Jonnavithula� S�� Thangam� S�� and Sisto� F�� �Compu�tational and Experimental Study of Stall Propagationin Axial Compressors�� AIAA Journal � ��� PP ����������������

�� He L�� �Computational Study of Rotating�Stall Incep�tion in Axial Compressors�� Journal of Propulsion andPower� ��� PP ����������

�� Outa� E�� Kato� D�� and Chiba� K�� �A N�S Simulationof Stall Cell Behavior in a ��D Compressor Rotor�Stator System at Various Load�� ASME Paper ���GT�� � �������

�� Saxer�Flelici� H� M�� Saxer� A� P�� Inderbitzen� A��Gyarmathy� G�� �Prediction and Measurement of Ro�tating Stall Cell in an Axial Compressor�� ASMEJournal of Turbomachinery� ���� PP ������������

�� Ho man� K� A�� Chiang� S� T�� Computational FluidDynamics for Engineers� A Publication of EngineeringEducation SystemTM� ������

��� Peyret� R�� Grasso� F�� and Meola� C�� Handbookof Computational Fluid Mechanics� Academic Press��������

��� Bohn D� and Emunds R�� �A Navier�Stokes ComputerCode for Theoretical Investigations on the Applicationof Various Turbulence Models for Flow PredictionAlong Turbine Blades�� Proceeding of the InternationalGas Turbine and Aero�engine Congress and Exposi�tion� Houston� Texas� �������

��� Fletcher� C� A� J�� Computational Techniques for FluidDynamics �� �������

�� Hirsch� C�� Numerical Computation of Internal and Ex�ternal Flows� A wiley�Interscience Publication� �������