A General Theory of Flow-Instability Inception in Turbomachinery

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A General Theory of Flow-Instability Inception in Turbomachinery

ARTICLE in AIAA JOURNAL · JULY 2013

Impact Factor: 1.21 · DOI: 10.2514/1.J052186

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Dakun Sun

Beihang University(BUAA)

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Available from: Dakun SunRetrieved on: 03 April 2016

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A General Theory of Flow-Instability Inception in Turbomachinery

Xiaofeng Sun∗ and Xiaohua Liu†

Beihang University, 100191 Beijing, People’s Republic of China

Ruiwei Hou‡

Shenyang Aeroengine Research Institute, 110015 Liaoning, People’s Republic of Chinaand

Dakun Sun§

Beihang University, 100191 Beijing, People’s Republic of China

DOI: 10.2514/1.J052186

A general eigenvalue theory on flow stability in turbomachinery is proposed with the emphasis on flow-instability

onset. Based on this theory, a stall-inception model including the effects of complex solid geometry is developed for a

multistage fan/compressors system. The capacity of the present model to predict the stall-inception point is assessed

against experimental data of both a low-speed and transonic single rotor. Comparisons with a simplified two-

dimensional modelare performedto identify the nonnegligibleeffects of spanwisedistribution of flowfieldin a general

configuration on the unstable mode of the concerned fan/compressors. It is verified that this model is capable of

predicting mass flow at the stall-onset point of both subsonic and transonic flow with a reasonable accuracy, and it is

sustainable in terms of computation cost for industrial application.

Nomenclature

A, B, C, D,G,H,Q,X

= coefficient matrix

cj = constant coefficient cv = specific heat at constant volumeDr = radial partial derivativeDrr = radial second-order partial derivativeDrz = radial and axial mixed second-order partial

derivativeDs = partial derivative on streamline coordinateDz = axial partial derivativeDzz = axial second-order partial derivative

E = total energyF = body force scalar F = body force vector F t = body force component parallel to the mean

camber surface of blades in blade-to-bladesurface

F v = body force component normal to the meancamber surface of blades in blade-to-bladesurface

f , g = smooth function on computational plane f z = unsteady force in axial direction f θ = unsteady force in circumferential directioni = imaginary unit K = dimensionality of matrixk = heat transfer coefficient

kc = undetermined body force coefficient M , N = integer m = circumferential mode number n = normal to streamline coordinate in meridian

plane

p = static pressurept = stagnation pressurept = rotary stagnation pressure

q̂ = heat volumic source by unit of mass, W∕kgR = gas constant r, θ , z = cylindrical coordinatesrt = radius at the tip of the leading edges = streamline coordinateT = temperatureT N = N -order Chebyshev polynomialt = timeU = column-orthogonal matrixU 0 = axial velocity of incoming flow at inlet

V = orthogonal matrixV = velocity scalar V = velocity vector v 0 = fluctuation velocity scalar W = diagonal matrix β = circumferential metal angle ofthe mean camber

surface in surface (θ ; z)γ = specific heat ratio λ = loss coefficient ε = body force coefficient ζ , η = rectangular coordinates in computational plane μ = dynamic viscosity coefficient ρ = densityτ = viscous stress tensor τ = time-delay constant Φ = column vector of perturbation amplitudesΩ = rotational speed of rotor, rpm ω = eigenfrequency of the fan/compressors

system ωr = real part of eigenfrequencyωi = imaginary part of eigenfrequency

Subscripts

a, b = points along one streamlinei, j = integer r, θ , z = radial, circumferential, and axial component t = stagnation property

Superscripts

T = transpose of matrix

Presented as Paper 2012-4156 at the 48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Atlanta, Georgia, 30 July–1 August 2012;received 1 July 2012; revision received 14 December 2012; accepted for publication 17 December 2012; published online 9 April 2013. Copyright ©2012 by the American Institute of Aeronautics and Astronautics, Inc. Allrightsreserved. Copiesof this paper maybe made forpersonalor internal use,on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1533-385X/13 and $10.00 in correspondence with the CCC.

*Professor, School of Jet Propulsion; [email protected].†Ph.D. Candidate, School of Jet Propulsion; [email protected].‡Ph.D., Compressor Department; [email protected].§Ph.D., School of Jet Propulsion; [email protected].

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http://dx.doi.org/10.2514/1.J052186http://dx.doi.org/10.2514/1.J052186


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I. Introduction

F LOW instability is one of the most severe challenges for turbomachinery. As one major type, rotating stall is a naturallimit to the performance of the fan/compressors, which could causecatastrophic damage to the whole fan/compressors system. Muchprogress has been achieved on studying this issue during the past fewdecades. In 1976, Greitzer [1] presented a stability model of compressor system, which can not only be used to predict the onset condition of stall and surge but also to study the nonlinear

development of stall cells. Garnier et al. [2] demonstrated that modaldisturbance rotates around the annulus during the stalling process.Experiments performed by McDougall et al. [3] and Day [4] first confirmed the existence of two types of stall precursors: modal waveand spike. In the following years, considerable experimental work (Camp and Day [5], Day et al. [6], Wilson and Freeman [7],Tryfonidis et al. [8], etc.) was conducted to investigate the evolutionprocess of rotating stall. Day et al. [6] experimentally studied on four high-speed compressor rigs and observed two new phenomena for the first time: fixed-location stall and high-frequency stall. It is of great interest that multiple types of stall precursors exist at thesame time or trigger reciprocally during the stalling process [5].Furthermore, inlet distortion, Reynolds number, and stagemismatching also affect the route to stall. In a word, it is revealed

that a variety of stall routesare involved in different compressors, andthe actual stalling patterns of one specific fan/compressors indifferent operating conditions are also various and complicated.

Although a unified theory that could explain all varieties of detailed stalling process remains some way off, it is evident that alltypes of stall precursor grow from a small oscillation to developedstall cell. Despite the significance of research on the detaileddevelopment process of stall precursor, it is the position of theinstability onset point that concerns engineers primarily. There aregenerally two different types of work to study the stall inception of the fan/compressors. The first, known as analytical models, mostlydescribe flow stability as an eigenvalue problem by simplifyingflowfield and placing emphasis on stall-inception period when theflow perturbationis small andlinear assumption is suitable.The work of Nenni and Ludwig [9] extended the channel-flow theory proposed

by Sears [10] and resulted in an analytical expression for theinception condition of rotating stall. Takata and Nagashima [11]studied the rotating stall in three-dimensional (3-D) blade rows withemphasis on the effects of nonuniform flow or shear flow on the stallinception. Gordon [12] presented a 3-D incompressible stabilitymodel to study the stall onset for general fan/compressors structure.Sun [13] developed a 3-D compressible stability model including theeffects of casing treatment, and this work was recently extended totransonic compressor stability [14]. It is verified that most analyticalmodels can predict the instability inception point with satisfactoryaccuracy as long as sufficient loss and performance characteristics of the concerned fan/compressors are given. The second type of work,numerical calculation, which is developing rapidly in recent yearswith the increasing availability of computing power, is to directly

solve unsteady Euler or Navier –Stokes (N–S) equations as an initialboundary-value problem to obtain the information related to stallingevolution. Owing to multiple time and length scales involved inrotating stall, direct time-accurate numerical calculation by unsteadyReynolds-averaged Navier –Stokes (RANS) equations on fullannulus grids is unacceptable for industrial applications in terms of unsustainable computational cost. He [15] conducted a computa-tional study on stall inception using a quasi-three-dimensionaltime-marching N–S equations method and successfully simulatedsingle-cell and multiple-cell pattern stall. Escuret and Garnier [16],Longley [17], and Chima [18] employed the body-force approach torepresent the effects of blades on the flowfield and computed thestalling process. Hoying et al. [19] performed numerical calculationson an eight-blade passage rotor to compute the flowfield before stall,and the simulation gave an insight into the physical origin of stallprecursor. Gong et al. [20] made the first effort to simulate 3-Dnonlinear development of stall precursor in multistage compressors.Gourdain et al. [21] solved quasi-three-dimensional N–S equations

on a stream surface and identified the spatial mode of flowdisturbance before stall inception by using Fourier analysis.

Numerical approach has an advantage over analytical models inthat it can consider the effects of more aerodynamic and geometryparameters on the physical process involved in rotating stall.However, it is vital for an initial boundary-value problem todetermine how to introduce the initial perturbations outside or insidethe computational domain to stimulate the stall precursor. Because of the unsteadiness, turbulence, and complicated configuration, there

are extremely plentiful of disturbance in terms of frequency,amplitude, and length scale in the actual flowfield of the fan/ compressors during the whole stalling process. This diversity of disturbance makes an inevitable trouble to this approach, and thereare not common rules yet to be followed to overcome this challenge.Nevertheless, it is noted that the stability of a dynamic system depends on its response to any initial small perturbation inside andoutside thesystem ratherthan some specific perturbation. It is evident that flow perturbation always grows via a development process from small to large in most flow-stability problems. If emphasis is placedon the inception period of flow instability, linearization is a reliabletechnique, and theoretically all flow-instability inception problemscan be described as an eigenvalue equation based on N–S equations.Take the research on the boundary layer transition for instance. Thereexists a variety of routes to deal with this problem, and the Orr –

Sommerfeld equation is one resultful theory. An integrated procedurefor solving this issue is provided as follows. First, the flow velocityprofile with some certain Reynolds number is computed by steadycomputational fluid dynamics (CFD) calculation. Under theassumptions of small perturbation and uniform shear flow, a four-order ordinary differential equation is established. After solving theOrr –Sommerfeld equation with appropriate boundary conditions, themost unstable disturbance can be judged by the imaginary part of resultant complex eigenfrequency. Meanwhile, the most unstablemode may provide an appropriate initial perturbation to the initialboundary-value problem for further inquiry on the nonlinear stage of boundary flow transition. As a matter of fact, most of earlier analytical models in studies of rotating stall were to establish variouseigenvalue equations with emphasis on stall inception. To facilitate

the analytical solution of the flowfield, a variety of simplification andassumptions are made in these models. The prediction result of thesemodels depends on the accuracy of the fan/compressors character-istics, which is not generally known with great precision, especiallyduring the design phase of new fan/compressors without sufficient empirical correlations. Numerous requirements for lossand deviationangles restrict the practical application of these simplified models.

Many researchers (Jacobs and Sherman [22], Abbott and vonDoenhoff [23],Rusak[24], Rusak and Morris [25], etc.) have made a great deal of progress on stall onset of an airfoil. It is without doubt that the study on the stall inception of compressors will benefit from the physical understanding of the airfoil stall problem. However,rotating stall of compressors is related to not only the sudden flowseparation and loss of lift on the blades but also the interactive effectsof adjacent blade rows and the propagation of flow disturbance

around the annulus. Under this circumstance, many flow details oncascade have to be neglected in therotating stall-inceptionmodel. Analternative and simplified approach is therefore required. Inconclusion, although a great deal of progress has been made on themechanism of flow stability, little research has been conducted oninstability inception of compressible flow including the effects of complex configuration of the fan/compressors. Additionally, there iscurrently not yet an effective methodology for providing anunambiguous judgment on the stall-onset point during the fan/ compressors design phase. It is the lack of such a capability that motivated the authors to develop a new flow-instability inceptionmodel in this paper. The main purpose of this investigation issummarized as follows.

1) Propose a methodology for stall-onset prediction duringthe fan/ compressors designstage, andassess thefeasibility for both subsonicand transonic flow.

2) Identify the necessity of developing 3-D compressible flow-stability model for getting an accurate stall margin prediction.

1676 SUN ET AL.

https://www.researchgate.net/publication/245354008_Rotating_Waves_as_a_Stall_inception_Indication_in_Axial_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/258225423_Stall_Inception_in_Axial_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354206_Stall_Inception_in_Axial_Flow_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/236429743_A_study_of_spike_and_modal_stall_phenomena_in_a_low-speed_axial_compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354621_Stall_inception_and_the_prospects_for_active_control_in_four_high-speed_compressors_Trans_ASME_J_Turbomach_121118-27_ISSN_0889-504X?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400165_Stall_inception_and_development_in_an_axial_flow_aeroengine?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354157_Prestall_Behavior_of_Several_High-Speed_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354621_Stall_inception_and_the_prospects_for_active_control_in_four_high-speed_compressors_Trans_ASME_J_Turbomach_121118-27_ISSN_0889-504X?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/236429743_A_study_of_spike_and_modal_stall_phenomena_in_a_low-speed_axial_compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/269210466_A_theory_to_predict_the_inception_of_rotating_stall_in_axial_flow_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/226223602_Rotating_stall_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/268457287_On_the_relation_between_the_inception_of_rotating_stall_and_casing_treatment?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/257440097_A_Model_to_Predict_Stall_Inception_of_Transonic_Axial_Flow_FanCompressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245434616_Computational_Study_of_Rotating-Stall_Inception_in_Axial_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/268455862_Numerical_simulations_of_surge_and_rotating-stall_in_multi-stage_axial-flow_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/228646391_A_Three-Dimensional_Unsteady_CFD_Model_of_Compressor_Stability?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400123_Role_of_Blade_Passage_Flow_Structurs_in_Axial_Compressor_Rotating_Stall_Inception?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354558_A_Computational_Model_for_Short-Wavelength_Stall_Inception_and_Development_in_Multistage_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239344375_Numerical_Simulation_of_Rotating_Stall_in_a_Subsonic_Compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400123_Role_of_Blade_Passage_Flow_Structurs_in_Axial_Compressor_Rotating_Stall_Inception?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400165_Stall_inception_and_development_in_an_axial_flow_aeroengine?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239344375_Numerical_Simulation_of_Rotating_Stall_in_a_Subsonic_Compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245434616_Computational_Study_of_Rotating-Stall_Inception_in_Axial_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/268455862_Numerical_simulations_of_surge_and_rotating-stall_in_multi-stage_axial-flow_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354206_Stall_Inception_in_Axial_Flow_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354621_Stall_inception_and_the_prospects_for_active_control_in_four_high-speed_compressors_Trans_ASME_J_Turbomach_121118-27_ISSN_0889-504X?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354621_Stall_inception_and_the_prospects_for_active_control_in_four_high-speed_compressors_Trans_ASME_J_Turbomach_121118-27_ISSN_0889-504X?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/236429743_A_study_of_spike_and_modal_stall_phenomena_in_a_low-speed_axial_compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/236429743_A_study_of_spike_and_modal_stall_phenomena_in_a_low-speed_axial_compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/228646391_A_Three-Dimensional_Unsteady_CFD_Model_of_Compressor_Stability?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354008_Rotating_Waves_as_a_Stall_inception_Indication_in_Axial_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/257440097_A_Model_to_Predict_Stall_Inception_of_Transonic_Axial_Flow_FanCompressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/268457287_On_the_relation_between_the_inception_of_rotating_stall_and_casing_treatment?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354558_A_Computational_Model_for_Short-Wavelength_Stall_Inception_and_Development_in_Multistage_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/258225423_Stall_Inception_in_Axial_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354157_Prestall_Behavior_of_Several_High-Speed_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/269210466_A_theory_to_predict_the_inception_of_rotating_stall_in_axial_flow_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/226223602_Rotating_stall_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==


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3) Provide the most unstable mode of the fan/compressors system for further research on the nonlinear development of stall precursor.

In this paper, a general eigenvalue theory on 3-D compressibleflow stability is proposed. In considerationof rotating stall inception,the mean flowfield of the fan/compressors is computed by RANScalculation. Then, a body-force model is suggested, which representsthe effects of blades on the flowfield. Further, by applyingappropriate boundary conditions and spectral collocation method, a group of homogeneous equations will yield, from which a stability

equation can be finally derived. After solving this eigenvalueequation using singular value decomposition (SVD) over a fine gridof eigenfrequency, the onset point of flow instability can be judgedby the imaginary part of resultant eigenvalue. Finally, modelassessments are performedon twocompressors, andthe results verifythat this model is capable of predicting the stall-onset point of bothlow-speed and transonic fan/compressors flow.

II. A General Eigenvalue Theory on Flow-InstabilityInception

A general eigenvalue theory on 3-D compressible flow stability isproposed in this section, which is describedas an eigenvalue equationbased on N–S equations as follows:

∂ ρ

∂t ∇ · ρV 0 (1)

∂ ρV

∂t ∇ · ρVV ρF − ∇p ∇ · τ (2)

∂ ρE

∂t ∇ · ρV E ρF · V − ∇ · pV ∇ · τ · V ρq̂

∇ · k∇T (3)

∇ · τ μΔV 1

3 μ∇∇ · V (4)

E cvT 1

2V 2 (5)

where the viscosity coefficient is assumed to be a constant, and theeffect of the complex solid boundary on the flowfield is representedby body forceF. Because emphasis is placed on the inception periodof flow instability, the flowfield is assumed to consist of mean flowand a small disturbance:

V V v 0 (6)

p p p 0 (7)

ρ ρ ρ 0 (8)

F F F 0 (9)

where represents the mean flow, and 0 represents the flowdisturbance. After Eqs. (6–9) are substituted into Eqs. (1–5), thelinearized N–S equations are derived. The mean flow data can becomputed by computational fluid simulation, whichis generally usedto solve 3-D steady RANS equations.

The body-force scalar is assumed to be a function of flowfieldparameters:

F ρ; p;V F ρ; p; V r; V θ ; V z ∂F

∂ ρ ρ 0

∂F

∂pp 0

∂F

∂V rv 0r

∂F

∂V θ v 0θ

∂F

∂V zv 0z (10)

F 0 ∂F

∂ ρ ρ 0

∂F

∂pp 0

∂F

∂V rv 0r

∂F

∂V θ v 0θ

∂F

∂V zv 0z (11)

Although the precise formula of the body force is not straightforwardfor some specific complicated flowfield, theoretically a body-forcemodel canbe used to reproducethe main effects offorce source on theflowfield. It is noted that the body force produced by the complexconfiguration and solid boundary will definitely introduce great challenge to a general flow-stability problem during the process of establishing the eigenvalue equation. In 3-D CFD calculation, thebody-fitted grid technique is generally implemented to cope with thetwisty solid geometry. The complicated solid boundary increasesthe complexity of transforming linearized N–S equations from thephysical domain to the computational domain. The immersedboundary (IB) method possibly provides an alternative way to modelthe complex solid boundary and was first designed by Peskin [26] tosimulate blood-valve interaction. This method adopts a force term on

the right side of the momentum equations to represent the effects of the solid boundary on the flowfield, and it is proved mathematically[27] that a material boundary effect on the flowfield is equivalent to a distribution force source. In the application of the IB method, theeffects of immersed boundaries are represented as a singular force onthe fluid. The interaction between boundaries and fluid is associatedwith the Dirac delta function.

Furthermore, the small perturbation of the flowfield is assumed tobe in the form of harmonic decomposition:

ρ 0 ~ ρr; zei−ωtmθ (12)

v 0r ~vrr; zei−ωtmθ (13)

v 0θ ~vθ r; zei−ωtmθ (14)

v 0z ~vzr; zei−ωtmθ (15)

p 0 ~pr; zei−ωtmθ (16)

where ∼ represents the amplitude of flow perturbations.After substituting Eqs. (12–16) into the linearized N–S equations,

a second-order differential equation is derived:

A ∂2Φ∂r2

B ∂2Φ∂z2

Q ∂2Φ∂r∂z

C ∂Φ∂r

D ∂Φ∂z

GΦ − iωHΦ 0

(17)

Φ f ~ ρ; ~vr; ~vθ ; ~vz; ~pgT (18)

and an eigenvalue equation can be expressed as follows:

detX 0 (19)

X ADrr BDzz QDrz CDr DDz G − iωH (20)

Drr ≡ ∂2∕∂r2; Dzz ≡ ∂

2∕∂z2; Drz ∂2∕∂r∂z;

Dr ≡ ∂∕∂r; Dz ≡ ∂∕∂z (21)

SUN ETAL. 1677

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Then, the appropriate boundary conditionfor the specific flowfieldisprescribed. After applying the coordinate transformation from thephysical domain to the computational domain and using numericaldiscretization, Eq. (19) can be solved, and the imaginary part of theresultant complex eigenfrequencyω represents whether the systemisstable with a negative value or unstable with a positive value. Byapplying the IB method, the numerical discretization can beperformed straight on regular orthogonalgrids ratherthan body-fittedgrids. This would make the coordinate transformation possible for the complex configuration and flowfield.

Although the whole procedure of formulating the eigenvalueequation seems to be straightforward, Eq. (19) will be extremelydifficult to be solved, which is caused by the constructing body forceand the hugeness of the resultant eigenmatrix for a real 3-D viscouscompressible flowfield with a complex solid boundary. Consideringthe specific flow-stability problem, several types of further simplification for calculating the flowfield, which is shown in Fig. 1,could be implemented to facilitate the solution of system stability.

In the following part of this paper, the presented general theory of flow stability is applied on the study of stall inception of the fan/ compressors, whichis one typical flow-instability problem. Cumpsty[28] outlined that, until recently, most satisfactory design systems of

the fan/compressors are developed around a successful approachproposed by Wu [29] in which the flow is described by intersectingS1 and S2 stream surfaces in blade rows. To capture the main featuresof the rotating stall and to be accordance with the modern designmethodology, the meridian average flowfield with the body force isstudied in this work. Actually, the body-force model adopted in thisstudy can beregardedas a particular application of the IB methodin a simplified case. Additionally, the presented modelherein can providean alternative tool to check the overpredicted stall margin during thedesign phase of the fan/compressors.

III. Stall-Inception Model Based on the GeneralTheory of Flow Stability

A single-stage fan/compressor system, which consists of an inlet duct, blade-rows region, inter-bladed row region, and an outlet duct,is shown in Fig. 2. In this paper, the circumferentially averagedflowfield and its derivative in the axial and radial directions arederived from steady CFD, which is performed by solving a routine3-D RANS equations solver.

Then, a flow model for studying the stability of a smallperturbation in a compressor flow needs to be developed. In thispaper,the actionof bladeson theflowfield andthe flow loss of kineticenergy caused by mixing and the shock wave outside the bladesurface are representedby a body-forcedistribution, and theviscosityoutside the blade surface is neglected. Much work (Adamczyk [30],Leboeuf and Trebinjac [31], Sturmayr and Hirsch [32], Chima [18],Gong et al. [20], etc.) has been published on the rational derivation of a flow model in a multistage turbomachine by using the blade-forceapproach. In the present model, the flowfield is described by 3-D,unsteady, compressible Eulerequationswith forcesourceterms in thefixed frame of the coordinate system:

∂ ρ

∂t∂r ρvr

r∂r∂ ρvθ

r∂θ

∂ ρvz

∂z 0 (22)

∂vr

∂t vr

∂vr

∂r vθ

∂vr

r∂θ vz

∂vr

∂z−

vθ vθ

r −

1

ρ

∂p

∂r F r (23)

∂vθ ∂t

vr∂vθ ∂r

vθ ∂vθ r∂θ

vz∂vθ ∂z

vθ vr

r −

1

ρ

∂p

r∂θ F θ (24)

∂vz

∂t vr

∂vz

∂r vθ

∂vz

r∂θ vz

∂vz

∂z −

1

ρ

∂p

∂z F z (25)

∂T

∂t vr

∂T

∂r vθ

∂T

r∂θ vz

∂T

∂z

R

cvT

∂rvr

r∂r∂vθ r∂θ

∂vz

∂z

0

(26)

Theoretically, Eqs. (22–26) can be employed to study the stabilityof 3-D nonuniform flow in multistage compressors. In this work,we focus on computing the stall-inception point for a cleaninlet condition, and the circumferential inlet distortion is out of consideration. The circumferential derivatives of mean flow in

Eqs. (22–26) are assumed to be negligible. The rationality of thissimplified flow model for multistage compressors will be madeapparent duringthe model assessments. It is noted that the mean flowdata can also be extracted from the meridian stream surface data,which are calculated by the streamline curvature method during thedesign stage of the fan/compressors.

A. Body-Force Model

To consider the effects of blades, flow mixing, and shock wave within the blade passages on the flowfield and to make thecomplicated problem computationally feasible, the body-forceapproach is taken to generate thephysical sourcesof flow turning andloss. There is no unique mathematic model for the body-force term,and different sorts of body-force models developed by Longley [17],

Chima [18], and Gonget al. [20] perform well in reproducing theflowdetail in blade passages, provided that the conservation relations for mass, momentum,and energyare reflected correctly. The body-forcemodel adopted in the present study is mainly on the basis of the work of Gordon [12] and Gong et al. [20], in which they made great progress on modeling the action of blades on flow, including theeffects of the time-delay correlation and the spanwise distribution of pressure rise in the bladed region, respectively.

In the present work, a new body-force model F is proposed, whichconsists of three parts,as shown in Fig. 3. Thefirst part represents F v,which isnormal tothe mean camber line of the blades in the blade-to-blade surface and is designed to generate the flow deviation in theblade-to-blade direction. The second part is F t1, which is assumed togenerate the whole flow loss of kinetic energy within the bladepassages. The last part is F t2, which is parallel to the camber surfaceand streamline. F t2 is mainly caused by viscosity on the bladesurface. Xu [33] conducted a rigorous study to assess viscousbody forces for unsteady calculations and presented a simplified

3-D N-S

equations

2-D axisymmetric flow

3-D compressible flow

Streamline flow

S2/Meridian streamsurface flow

3-D incompressible flow

Fig. 1 Simplified calculation models of 3-D N–S equations.

Upstream duct

z

r

Rotor Stator

Hub

Tip

Downstream ductIncoming flow

Streamlinea b

Fig. 2 Sketch map of one compressor stage in the meridian plane andtwo points on one streamline.

1678 SUN ET AL.

http://-/?-http://-/?-https://www.researchgate.net/publication/281000338_A_General_Theory_of_Three-Dimensional_Flow_in_Subsonic_and_Supersonic_Turbomachines_of_Axial-_Radial_and_Mixed-Flow_TYpes?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/4716340_Model_equation_for_simulating_flows_in_multistage_turbomachinery?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/268482999_Methodology_and_Results_of_Analysis_for_Unsteady_Flows_in_Transonic_Turbine_and_Compressor_Stages_Invited?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245392296_Throughflow_model_for_design_and_analysis_integrated_in_a_three-dimensional_Navier-Stokes_solver?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/228646391_A_Three-Dimensional_Unsteady_CFD_Model_of_Compressor_Stability?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354558_A_Computational_Model_for_Short-Wavelength_Stall_Inception_and_Development_in_Multistage_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-http://-/?-http://-/?-http://-/?-https://www.researchgate.net/publication/228646391_A_Three-Dimensional_Unsteady_CFD_Model_of_Compressor_Stability?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354558_A_Computational_Model_for_Short-Wavelength_Stall_Inception_and_Development_in_Multistage_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354558_A_Computational_Model_for_Short-Wavelength_Stall_Inception_and_Development_in_Multistage_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400203_Assessing_Viscous_Body_Forces_for_Unsteady_Calculations?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/281000338_A_General_Theory_of_Three-Dimensional_Flow_in_Subsonic_and_Supersonic_Turbomachines_of_Axial-_Radial_and_Mixed-Flow_TYpes?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245392296_Throughflow_model_for_design_and_analysis_integrated_in_a_three-dimensional_Navier-Stokes_solver?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/268482999_Methodology_and_Results_of_Analysis_for_Unsteady_Flows_in_Transonic_Turbine_and_Compressor_Stages_Invited?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/4716340_Model_equation_for_simulating_flows_in_multistage_turbomachinery?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400203_Assessing_Viscous_Body_Forces_for_Unsteady_Calculations?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/228646391_A_Three-Dimensional_Unsteady_CFD_Model_of_Compressor_Stability?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/228646391_A_Three-Dimensional_Unsteady_CFD_Model_of_Compressor_Stability?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354558_A_Computational_Model_for_Short-Wavelength_Stall_Inception_and_Development_in_Multistage_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354558_A_Computational_Model_for_Short-Wavelength_Stall_Inception_and_Development_in_Multistage_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354558_A_Computational_Model_for_Short-Wavelength_Stall_Inception_and_Development_in_Multistage_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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formulation to model the viscous force. The result verified that theviscous force is relatively small compared to the total blade forces.Therefore, F t2 is assumed to be negligible in this paper. The radialcomponent of the total body force is assumed to be negligible due tothe small radial inclination of the blades in this paper. Thus, F t1 isassumed to be an axial force. It is noted that the applicability of thefollowing linearization process and this simplified body-forceapproach for reflecting the principle physical nature will be madeapparent during the model assessment part.

After the flowfield of the fan/compressors is computed by 3-Dsteady numerical calculation, F vθ , which is the circumferentialcomponent of F v, can be obtained:

F vθ vr∂vθ ∂r

vz∂vθ ∂z

vθ vr

r(27)

In a steady, circumferentially averaged, two-dimensional (2-D)flowfield, F vθ is formulated as

F vθ vz∂vθ ∂z

(28)

which is derived from the circumferential momentum equation. Inaccordance with Gordon’s analysis [12], the local circumferentialbody force of blades is assumed to turn the relative flow toward theblade angle, i.e.,

F vθ kcvzvz tan β − vθ − rΩ ε (29)

The body force coefficient kcr; z is therefore determined byarranging Eqs. (27) and (29) as

kc vr

∂vθ ∂r

vz∂vθ ∂z

vθ vrr

vzvz tan β − vθ − rΩ ε (30)

where ε is a parameterthat is large enoughto avoid thesingularityof k

due to the extreme small deviation angle of flow. Considering theorthogonality between F v and the blade camber surface, its axialcomponent can be derived directly:

F vz F vθ tan β (31)

The stagnation pressure and rotary stagnation pressure are defined as

pt p

1

γ − 1

2 ·

ρV 2z V 2r V

2θ

γ p

γ γ −1

(32)

pt pt − ρΩrvθ (33)

Because of a variety of loss caused by mixing and viscosity in theflowfield of a fan/compressors system, the rotary stagnation pressurewill decrease along the streamline. The flow loss of kinetic energyacross the rotors andstators is assumed to be proportional to the local

relative velocity magnitude. Assuming two points a and b lie on onestreamline within the blade passage, the loss coefficient is derived as

λ ptb − pta

ρv2z vθ − Ωr2a

(34)

On theotherhand, F t1 is designed to reproducethe effects offlowlossacross the blade passages, and its distribution is derived by thefollowing formula after the mean flow data are obtained by steady

CFD:

F t1 ρzb − za ptb − pta (35)

Therefore, F t can be formulatedas follows by manipulatingEqs. (34)and (35):

F t1 λv2z vθ − Ωr

2

zb − za (36)

Finally, the axial component of body force is given by Eqs. (31) and(36):

F z F vz F t1 (37)

and the circumferential component of the body force is given byEq. (29):

F θ F vθ (38)

In accordance with Gordon’s analysis [12], the unsteady action of blades on the flowfield is described by the body-force function,which responds to the local unsteady 3-D flow variations in the bladeregion, and the unsteady force disturbances f θ and f z are modeledbya first-order lag equation with a time-delay constant τ . The linearizedform of the body force is formulated as

1 − iωτ imΩτ f θ ∂F θ ∂vθ

v 0θ ∂F θ ∂vz

v 0z (39)

1 − iωτ imΩτ f z ∂F z∂vθ

v 0θ ∂F z∂vz

v 0z (40)

The time-delay constant τ , which represents the lag between theoutput response and input data, is employed to describe the unsteadyperformance of the fan/compressors. The time scale of τ is generallyassumed to be of the order of flow-through time in the bladed region.Parameter studies on this constant are conducted in the followingmodel assessments.

B. Establishment of Eigenvalue Problem

Because emphasis is placed on the inception condition of rotatingstall in this investigation, the flow perturbation is assumed to be

small, and the solutions are in accordance with Eqs. (12–16).He [15]verified that the first circumferential mode hadthe most destabilizingeffect. Gordon [12] also demonstrated that higher-order modes wereusually more stable than lower-order modes unless extreme stagemismatching and severe inletdistortion occur. Emphasis in this paper is therefore placed on the lowest-order circumferential and radialmode that dominates the stability of the fan/compressors system.

The boundaries are prescribed such that there are no inlet disturbances coming from outside the system at the upstream end of the duct and no reflection at the downstream end of the duct, and theslip condition is used at the hub and the tip of the annular duct. After substituting Eqs. (12–16) and Eqs. (39) and (40) into Eqs. (22–26),the linearized governing equations are derived as

CDr DDz G − iωHΦ 0 (41)

where C is a radial derivative matrix, and D is an axial derivativematrix. The second-order partial derivatives disappear because

θ

Streamline

Incoming flow

a b

Trailing edge

F t2

Camber line

Leading edge

z

F t1

Fv

Fig. 3 Sketch map of the body force in the blade-to-blade surface and

two points (a and b) on one streamline.

SUN ETAL. 1679

https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-http://-/?-https://www.researchgate.net/publication/245434616_Computational_Study_of_Rotating-Stall_Inception_in_Axial_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-https://www.researchgate.net/publication/245434616_Computational_Study_of_Rotating-Stall_Inception_in_Axial_Compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/37606708_Three-dimensional_rotating_stall_inception_and_effects_of_rotating_tip_clearance_asymmetry_in_axial_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


7/14

viscosity is neglected in this simplified model. Because Eq. (41) ishomogeneous, a nontrivial solution of Φ exists only if thedeterminant of the coefficients matrix is zero, i.e.,

detX 0 (42)

X CDr DDz G − iωH (43)

Solving the established eigenvalue problem of Eq. (41) or (42) leadsto the resultant complex frequencyω ωr iωi, the imaginary part of which representswhether thesystem is stablewith a negative valueor unstable with a positive value, and the real part of whichdetermines the rotating frequency of the precursor wave. Twonondimensional factors are defined as relative speed (RS) anddamping factor (DF):

RS ωr

2mπ ·

60

Ω(44)

DF rtωi

mU 0(45)

IV. Numerical Method for Solving the StabilityEquation

After establishing an eigenvalue equation for the fan/compressorsflow-instability inception,solving the mostunstablemodeof Eq. (42)is still a tricky problem. To facilitate its solution, the equation isdiscretized over the computational grid, and the numerical method isgiven in this part.

A. Numerical Discretization

To improve the accuracy of the numerical solution and to decreasethe discrete points for saving computation cost,the spectral techniquebased on Chebyshev–Gauss–Lobatto points is implemented, which

is adopted widely in the boundary-value problem. The physical gridsinthe(z; r) plane are transformed into a computational plane (ζ ; η) sothat the transformed grid lines are orthogonal and suitable for thespectrum method. The eigenvalue of Eq. (42) is solved numericallyover the spectral grids that satisfy

ξi cosπ i∕N r; i 0; 1; : : : ; N r (46)

ηj cosπ j∕N z; j 0; 1; : : : ; N z (47)

where N r and N z are the total number of nodes in each subdomain inthe ζ and η coordinates. Assuming that f ζ is a smooth function of ζ on the interval −1; 1 , f ζ is interpolated by constructing an N -order interpolation polynomial gjζ :

f ξ XN j−0

f ξjgjξ (48)

gjξ −1j11 − ξ2T 0N ξ

cjN 2ξ − ξj

i 0; 1; : : : ; N (49)

where T N ζ is an N -order Chebyshev polynomial in the form of

T N ξ cosN cos−1 ξ (50)

and its derivative is

T 0N ξ N · sinN arccos ξ

1 − ξ2p (51)

and the coefficient cj is formulated as

cj

2; j 0; N 1; 0 < j < N

(52)

The derivative of f ζ at the collocation points satisfies

d f ξi

dξ X

N

j0

Dij f ξji 0; 1; : : : ; N (53)

where the derivative matrix Dij is formulated by Don andSolomonoff [34] as

Dij

8>>>>><>>>>>:

ci−1ij

2cj sin π 2N

ij sin π 2N −ij i ≠ j; i 0; · · · ; N 2 ; j 0; · · · ; N

− cosπ iN

2sin2π N

i i j; i 1; · · · ; N 2 ; j 1; · · · ; N

2N 216 i j 0

−DN −i;N −j i N

2 1; · · · ; N ; j 0; · · · ; N

(54)

Singularity generally appears on the leading edge of the airfoil withunsteady flow. In general, any accurate computation of airfoil flowrequires a special treatment on singularities. For an isolated airfoil,Rusak andMorris [25] and Rusak [35] treated the nose singularity for subsonicand transonic flow by using matched asymptoticmethods of inner solution around the airfoil nose and the outer region describedby the small disturbance solution, whereas for the cascade case,Namba [36] used a type of mathematical treatment to deal with thesingularity on the leading and trailing edges of the compressor blade.In the present work, the body-force model is applied to consider themain effects of cascade on the flowfield without more flow details,especially on both leading and trailing edges of the blades. On onehand, such simplification has resulted in a reasonable eigenvaluestability model with high computation efficiency; on the other hand,there is no doubt that flow details on the blades should be studied infuture work for more accurate computation results. In this paper, the

domain decomposition method is applied to deal with the singularityof the body forcedue to no body forcein the blade-freeregion. Sometransmission conditions are taken into account at the interfacesbetween subdomains. The continuity conditions at the interfaceζ 1 2between subdomain 1 and subdomain 2 are

f 1ξ12 f 2ξ12 (55)

d f 1ξ12

dξ

d f 2ξ12

dξ (56)

B. Numerical Solution

InEq.(42), thecolumn vector Φof the perturbation amplitudes has

K 5N r · N zf N zb entries, where N r is the total number of radial nodes, and N zf and N zb are the total number of axial nodes inthe blade-free and blade-row regions, respectively. The K × K matrixX is very large for a modest number of grid nodes. For instance, thereare 20 radialnodesand 90 axial nodes,whichare spaced20 upstream,20 downstream, 20 in the rotor, 20 in the stator, and 10 in the inter-bladed row region in a typical one-stage compressor (i.e.,K 9000). The matrix X contains 81 million complex valueentries. It is found that matrix X will become extremely huge for multistage fan/compressors, which will make the solution of theeigenvalue equation a very tough problem. In fact, because of the numerical rounding error, it is not feasible to determine theeigenfrequency for which the determinant of X is zero. SolvingEq. (42) is also not possible using the traditional QR decompositionmethod because of the lack of accuracy. Such a difficulty exists invarious flow-stability problems, and much research work has beenconducted previously. The Newton–Raphson iteration method iswidely used to solve such an eigenvalue equation. However, there is

1680 SUN ET AL.

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-https://www.researchgate.net/publication/24299227_Accuracy_and_Speed_in_Computing_the_Chebyshev_Collocation_Derivative?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/231898891_Transonic_flow_around_the_leading_edge_of_a_thin_airfoil_with_a_parabolic_nose?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-http://-/?-https://www.researchgate.net/publication/231898891_Transonic_flow_around_the_leading_edge_of_a_thin_airfoil_with_a_parabolic_nose?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/24299227_Accuracy_and_Speed_in_Computing_the_Chebyshev_Collocation_Derivative?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


8/14

no general rule available to determine its physical solution. Malik [37] implemented an inverse iteration technique to solve theeigenvalue, and this method is very sensitive to the initial estimate of ω, and much empiricism is required to identify pseudomodes. Sunetal. [14] extended thewinding number integral approach to solve thematrix equations for a rotating stall. The essence of this approach is a smart application of the argument principle and Nyquist stabilitycriterion [38,39]. Nevertheless, it is not feasible for the present problem because of the inevitable and unsustainable computation

consumption required for a series of determinant calculation.In this paper, SVD is adopted over fine grids on the complex planetofind the solutionof Eq. (42). Actually, this approach wasadopted inmuch study [40,41] on searching the roots for which a matrix withmany entries becomes singular. As described by Woodley and Peake[40], SVD is a very powerful technique for dealing with matrices that are either singular or numerically very close to singular. Thistechnique is based on the theorem of linear algebra that any M × N matrix X can be factorized into a product of one M × N column-orthogonal matrix U , one N × N diagonal matrixW with positive or zero elements (the singular values), and the transpose of one N × N orthogonal matrix V (More details regarding the SVD method aregiven by Press et al. [42].):

X U ·0B@

ω1

. ..

ωN

1CA · V T (57)

The condition number of a matrix is formally defined as the ratio of the largest and smallest singular values. Strictly speaking, a matrix issingular as long as the condition number is infinite, and thedeterminant of a matrix is zero if only the reciprocal of the conditionnumber is zero. However, due to the rounding error accumulatedduring the numerical process, it is often unlikely to approach themachine’s floating-point precision. In accordance with the Cooper et al. [41], the authors therefore define the roots of the eigenvalueequation as occurring when the reciprocal of the condition number issignificantly smaller than elsewhere.

V. Model Validation and Comparisons withExperimental Results

In this part, four assessments are performed to validate the abilityof the present model to calculate the stall-onset point, and somediscussions are given. The first assessment demonstrates theprediction accuracy by comparing the resultant unstable mode withthe experimental data of a low-speed compressor. A simplifiedversion of this model is developed in thesecondassessment, which isestablished on a one-dimensional streamline. Comparisons with the3-D model are conducted to reveal the nonnegligible effects of spanwise distribution of the flowfield on stall margin. The last twoassessments are to check the applicability of this model on predictingthe stall-onset point of a transonic single rotor at low and high

rotational speed.

A. Numerical Prediction and Comparisons with Experimental Resultsof a Low-Speed Compressor

Nie et al. [43] experimentally studied micro air injection for stallmargin enhancement on a low-speed three-stage axial-flowcompressor rig at the Institute of Engineering Thermophysics at theChinese Academy of Sciences, and the spike-type stalling process isverified by wavelet analysis. This compressor rig has three identicalstages. Tong [44] carried out an experimental study to explore themechanism of stall suppression by micro-injection on a single-rotor system, which includes only the first rotor of this rig. Its main designspecifications are givenby Nie et al. [43], and the design mass flowis2.6 kg/s at 2400 rpm with a tip clearance of 2.8% of tip chord. A sideview and schematic can be seen in Fig. 4. Figure 5 shows themeasurement system with single rotor and motor labeled as 1 and 2,inverterlabeled as 3, staticpressure taps and3-hole probe labeledas 4and 5, transducers labeled as 6 and 7, steady sensor box labeled as 8,

cone valve labeled as 9, BNC Connectors labeled as 10 and 11,amplifiers labeled as 12, data recorder and data acquisition boardlabeled as 13 and 14, and computer labeled as 15. Model assessment in this section is made on this single rotor.

To validatethe present flow-instability inception model, a dynamicexperiment is performed with emphasis on the stalling process andeight on-shroudpressure transducers in front of the rotor are installedto record the static pressure oscillation. Figure 6 shows that a form of short scale pips wasobservedjustbeforestallin thetime revolution of the static pressure signal. After the pips propagated circumferentially

Fig. 4 Side view of the single-rotor test rig (courtesy of Nie et al. [43]).

1. Single rotor 2. Motor 3. Inverter (IPF-18.5) 4. Static pressure taps

5. 3-hole probe 6. Transducers (XCS-190) 7. Transducers (XCS-190)

8 Steady sensor box 9. Cone valve 10. BNC Connectors (BNC-2090)

11. BNC Connectors (BNC-2090) 12. Amplifiers (8300AU)

13. Data recorder (XR-700) 14. Data acquisition board (PCI-6071E)

15. Computer

Fig. 5 Schematic of the compressor rig and measurement system(courtesy of Tong [44]).

0 2 4 6 8 10 12 14 16 18 20

emerging spike

speed: 62%Sensor 7

Sensor 6

Sensor 5

Sensor 4

Sensor 3

Sensor 2

Sensor 1

P r e s s u r e a t e i g h t a n n u l a r p o s i t i o n s

Time, rotor revolutions

Sensor 0

speed: 46%

Fig. 6 Time evolution of the static pressure near the shroud in front of the rotor.

SUN ETAL. 1681

https://www.researchgate.net/publication/24384934_Finite-difference_solution_of_the_compressible_stability_eigenvalue_problem?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/257440097_A_Model_to_Predict_Stall_Inception_of_Transonic_Axial_Flow_FanCompressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/243363320_On_the_determination_of_the_roots_of_dispersion_equations_by_use_of_winding_number_integrals?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-https://www.researchgate.net/publication/245354674_Acoustic_Resonance_in_Aeroengine_Intake_Ducts?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/232012939_Resonant_acoustic_frequencies_of_a_tandem_cascade_Part_1_Zero_relative_motion?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354674_Acoustic_Resonance_in_Aeroengine_Intake_Ducts?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400228_Micro_Air_Injection_and_Its_Unsteady_Response_in_a_Low-Speed_Axial_Compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400228_Micro_Air_Injection_and_Its_Unsteady_Response_in_a_Low-Speed_Axial_Compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400228_Micro_Air_Injection_and_Its_Unsteady_Response_in_a_Low-Speed_Axial_Compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/257440097_A_Model_to_Predict_Stall_Inception_of_Transonic_Axial_Flow_FanCompressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354674_Acoustic_Resonance_in_Aeroengine_Intake_Ducts?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245354674_Acoustic_Resonance_in_Aeroengine_Intake_Ducts?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/24384934_Finite-difference_solution_of_the_compressible_stability_eigenvalue_problem?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400228_Micro_Air_Injection_and_Its_Unsteady_Response_in_a_Low-Speed_Axial_Compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400228_Micro_Air_Injection_and_Its_Unsteady_Response_in_a_Low-Speed_Axial_Compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/239400228_Micro_Air_Injection_and_Its_Unsteady_Response_in_a_Low-Speed_Axial_Compressor?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/232012939_Resonant_acoustic_frequencies_of_a_tandem_cascade_Part_1_Zero_relative_motion?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/232012939_Resonant_acoustic_frequencies_of_a_tandem_cascade_Part_1_Zero_relative_motion?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/243363320_On_the_determination_of_the_roots_of_dispersion_equations_by_use_of_winding_number_integrals?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==http://-/?-


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at 62% rotor speed, the spike-type disturbance triggers the stall. Thefinal stall cells evolved in less than five rotor revolutions and rotatedat 46% rotor speed. A pressure spectrum in frequency domain after stall is shown in Fig. 7. Thefinalinstabilityfrequencyis located on aninterval of 16 and 20 Hz, i.e., 40–50% rotational frequency of therotor, which is assumed to be themostunstableeigenfrequency of thisrotor.

The steady 3-D viscous compressible flowfield is computed on a total number of approximately 746,000 grid nodes with a tipclearance block for one single blade passage of this single rotor. Thegrid has 105points axially, 41 pointscircumferentially, and 41 pointsspanwise. The solution algorithm is based on central discretizationand a four-stage Runge–Kutta scheme coupled with the Spalart –Allmaras turbulence model, multigrid, and local time-steppingtechnique for convergence.

Figure 8 compares the static pressure rise between the CFDcalculation and experimental results, and the efficiency is not measured. The calculated pressure rise is a reasonable approximation

to the experimental data, especially nearby the design point. It isfound that the mass flow just before the numerical stall condition is1.995 kg∕s, whereas the mass flow at the measured stall point is2.309 kg∕s. The relative error is about 13.6%, and the stall margin isoverpredicted by 12.1%. Obviously, as an unsteady process, rotatingstall could not be estimated roughly by the numerical convergence of the steady flow simulation.

The calculated mean flow data on meridian plane are processedas polynomial fits of axial and radial coordinates to eliminatethe nonsignificant eigenfrequency due to the local vortex. After introducing the flowfield data into the present model, the complexeigenvalues of the compressor system accompanying the throttlingprocess are computed, which are displayed in Fig. 9. Nine

eigenmodes within the selected solution interval are obtained for everyoperation point,whichare sequencedas modes 1 to 9 accordingto the magnitude of the real part. In this assessment, several different values of the time-delay constant are tested, and no visible change of eigenvalue emerges.

It is found that the DFs of all nine modes increase gradually as therotor approaches flow instability. The first five modes, whichpropagate slowly, approach instability faster than the last four modes,which propagate rapidly. The DF of mode 1 changes first from negative to positive at a value of mass flow equal to 2.324 kg∕s. Thesign change of the DF represents the imminent instability. The

relative error of the mass flow between the predicted stall-inceptionpoint and the measured instability point is 0.65%. Although thethrottling continues, the RSs of all the modes increase, with anexception of slight decrease for mode 1 before stall. The propagationspeed of this most unstable mode at stall inceptionis 40.5% rotationalfrequency, which is within the measured dominant instabilityfrequency interval. The comparison between computed results withexperimental data at the stall-onset point and instability frequencyvalidates the prediction accuracy of the proposed flow inceptionmodel, and it makes the capacity of adopting this model for a spike-type stalling process apparent.

B. Comparisons with a Further Simplified Versionof the Proposed Model

Most 2-D stability models (for example, Nenni and Ludwig [9],Stenning [45]) analyze the spanwise average of flow nonuniformity,and the radial distribution of the flowfield is neglected. We will showin this assessment that a low-dimensional model has a lack of

0 20 40 60 80 100 120 140 160 180 2000.00

0.05

0.10

0.15

0.20

0.25

A m p l i t u d e

Frequency, Hz

Fig. 7 Pressure spectra in the frequency domain after stall.

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4300

400

500

600

700

800

900

1000

S t a t i c p r e s s u r e r i s e , p a

Mass flow, kg/s

Experiment

CFD calculation

Fig. 8 Static pressure rise of the low-speed single rotor.

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

The predicted stall inception point

R e l a t i v e s p e e d

Mass flow, kg/s

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

a)

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

b)

The predicted stall

Stable

Unstable

D a m p i n g f a c t o r

Mass flow, kg/s

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

inception point

Fig. 9 Computed eigenvalues of the low-speed rotor: a) relative speed,and b) damping factor.

1682 SUN ET AL.

https://www.researchgate.net/publication/269210466_A_theory_to_predict_the_inception_of_rotating_stall_in_axial_flow_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245355241_Rotating_Stall_and_Surge?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/245355241_Rotating_Stall_and_Surge?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/269210466_A_theory_to_predict_the_inception_of_rotating_stall_in_axial_flow_compressors?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==


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accuracy when it is compared with a 3-Dstall-inception model. In thissection, a streamline model of stall inception is developed, which isbased on the same theory presented in Sec. IV. The only twodifferences between these two models exist on the governingequations and the dimensionality of the matrix involved in theeigenvalue equation. Details of how the simplified version of thepresent model is established are given in the appendix.

Figure 10 shows thecomputedresult using thesimplified model onthreedifferent streamlines at 97%, 50%, and 3% span. The DFs of allthe three modes increase as the mass flow is throttled, and it first crosses the critical line on the 97% span streamline at 2.225 kg∕s,which is the predicted stall-inception point. It is indicated that the

relative error between this point and the measured instability point is−3.63%. This simplified low-dimensional model overpredicts thestall margin. Tong [44] conducted wavelet analysis on the pressuresignals and experimentally observed that flow separation and stallprecursor first emerge at the tipregion of this rotor. It is indicated that this simplified model successfully predicts the most unstable region,but the prediction accuracy of the stall-inception point is obviouslyless accurate than the 3-D model. Through the whole throttlingprocess, the RSs of all three modes increase, with an exception of a slight decrease for 97% span before stall, and the propagation speedat stall inceptionis 40.5% designrotational frequency. These featuresare in accordance with the 3-D model prediction. However, thecomputed frequency spectrum using this simplified mode is not asbroad as the 3-D model. In addition, the eigenmode propagates faster at 50% span than at 97% and 3% span. In conclusion, thisassessment case validates the advantage of the 3-D model over the low-dimensional model. The whole spanwise flow distribution should beconsidered to get a more accurate stall margin prediction.

C. Eigenvalues of High Subsonic Compressor Flow-InstabilityInception

In this section, the stall-inception eigenvalues of a transoniccompressor are computed based on a typical example (i.e., NASARotor 37). The purpose of this calculation is to test the predictionability of the model for high subsonic compressor flow. Thus, themean flowfield data at 60% design rotational speed is used as theinput parameters, and the spanwise distribution of the inlet relativeMach number ranges from 0.6 to 0.9. The details of the configuration

and characteristics of this rotor were given in [46], and the detailedexperimental data were provided by Suder [47].

The procedure for the steady flowfield solution of Rotor 37 isin accordance with Sec. V.A. Because the configuration is morecomplex than the low-speed single rotor, the total of grid points isapproximately 750,000, which is sufficient to get a grid-independent solution. Figure 11 shows a comparison of characteristics betweensteady RANS calculation and experimental data. The calculatedtrends are in accordance with experimental data with underpredictedpressure ratio by 2.5% and efficiency by 3% before stall.

It is shown that the mass flow at the computational convergencepoint is about 9.68 kg∕s, which ismuchlower than 10.56 kg∕s at themeasured stall-inception point. The relative error is 8.3% (i.e., thestall margin is overpredicted by about 8.0%).

After introducing themean flow data on themeridianplaneinto thepresent stall-inception model, the resultant eigenfrequency at severaldifferent operation conditions alongthe characteristicslineis displayedin Fig. 12. Six eigenmodes within the selected solution interval arecomputed for each operation point of Rotor 37, which are sequencedas modes 1 to 6 according to the magnitude of the imaginary part.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

a)


R

e l a t i v e s p e e d

Mass flow, kg/s

97% span

50% span

3% span

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

-6

-5

-4

-3

-2

-1

0

1

b)


StableUnstable

D a m p i n g f a c t o r

Mass flow, kg/s

97% span

50% span

3% span

Fig.10 Computedeigenvaluesby thesimplifiedmodel forthe low-speedcompressor: a) relative speed, and b) damping factor.

9 10 11 12 13 14 15 161.0

1.1

1.2

1.3

1.4

1.5

a)

S t a g n a t i o n p r

e s s u r e r a t i o

Mass flow, kg/s

Experiment

CFD calculation

9 10 11 12 13 14 15 160.6

0.7

0.8

0.9

1.0

1.1

b)

Experiment

CFD calculation

E f f i c i e n c y

Mass flow, kg/s

Fig. 11 Characteristics of NASA Rotor 37 at 60% design rotationalspeed: a) stagnation pressure ratio, and b) efficiency.

SUN ETAL. 1683

https://www.researchgate.net/publication/24384941_Performance_of_single_axial_flow_transonic_compressor_with_rotor_and_stator_aspect_ratio_of_119_and_126_respectively_and_with_design_pressure_ratio_of_205?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==https://www.researchgate.net/publication/24384941_Performance_of_single_axial_flow_transonic_compressor_with_rotor_and_stator_aspect_ratio_of_119_and_126_respectively_and_with_design_pressure_ratio_of_205?el=1_x_8&enrichId=rgreq-153a688e-925c-4c6b-aef1-d96a23adca0a&enrichSource=Y292ZXJQYWdlOzI2MDgyOTQ3ODtBUzoxNzc0MDM1NTI4NzA0MDBAMTQxOTMwNzcxMTE3Nw==


11/14

In this paper, the nondimensional time delay (NTD) is defined asthe time-delay constant τ divided by the flow-throughtime withinthebladed region. In some studies before, it seems that NTD affects theaccuracy of the instability prediction, and there is not yet a unifiedprinciple to determine its value. Hoying et al. [48] rigorously testedthe effects of thisconstant on stall-inception prediction. He suggestedthattwovalues ofNTD equal to0.45forloss and 1.0 for turning ofthebladed region flow yield the best agreement with data in his paper.Haynes [49] compared the measured dynamics of one compressor with predictions givenby a stabilitymodel, anda value of NTDequalto 1.5 was found to perform well. In this paper, Parameter study on

time-delay constant is conducted f

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A General Theory of Flow-Instability Inception in Turbomachinery

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