Axial Compressor Stall

7/27/2019 Axial Compressor Stall

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DISTRIBUTED NONLINEAR MODELING AND STABILITYANALYSIS OF AXIAL COMPRESSOR STALL AND SURGE

Catherine A. Mansoux, Daniel L. Gysling, Joga D. Setiawan, James D. PaduanoMassachusetts Institute of Technology

Cambridge, MA 02139

Abstract

This paper presents a nonlinear formulation of the Moore-Greitzerrotating stall model that is suitable for control analysis and design.The model is validated by comparing stall inception experimentsto simulated stall inception transients. The shape of the nonlinearcompressor characteristic is shown to be a primary determinant ofstall inception transient behavior. A Lyapunov stability analysisprocedure is then presented. This procedure allows basins of

attraction to be determined for various system operating points.Examples are given, and implications for design of rotatingstall/surge control systems are discussed.

Nomenclature

lc, B, m, , Compression system geometric parameters (see[27])

A Flowfield transformation operator, Eqs. (14), (15)E Flowfield transformation operator, Eqs. (14), (16)G Fourier transform matrix, Eq.. (17)M 2N+1N Number of harmonics modeled in finite order

approximation, Eq. (17)S (1xM) matrix to take mean of a distributed vector

quantity, Eq. (20)T (Mx1) matrix to for zeroth mode of surge/stall model,

Eq. (20)V Lyapunov function() Fourier transform, Eq. (13)n Harmonic number Perturbation quantity Laplacian Size of basin of attraction Nondimensional axial distance along compressor duct,

zero at compressor face Circumferential angle around compressor annulus Nondimensional time Flow coefficient, Eq. (1) (Mx1) vector containing flow coefficient values at

distributed values of Nondimensional instantaneous local pressure rise across

compressor Nondimensional velocity potentialC Pressure rise delivered across compressor

T Pressure drop imposed across throttle

T Inverse ofTKT Throttle constant, Eq. (2)() Annulus average, Eq. (9)()ac Non-axisymmetric component, Eq. (9)

1. Introduction

Rotating stall and surge are violent limit cycle-type oscillations inaxial compressors, which result when perturbations (in flowvelocity, pressure, etc.) become unstable. Originally treatedseparately, these two phenomena are now recognized to becoupled oscillation modes of the compression system -- surge is

the zeroth order or planar oscillation mode, while rotating stall isthe limit cycle resulting from higher-order, rotating-wavedisturbances. The importance of these phenomena to the safetyand performance of gas turbine engines is widely recognized [1],and various efforts to either avoid or control both rotating stall andsurge have been studied [2, 3, 4, 5, 6].

Central to the issue of control is reliable modeling that capturesthe essential physics of the process, without being toocumbersome for control law design and analysis. For example,CFD models of rotating stall exist [7, 8], but these are clearly notsuitable for conventional control law design. On the other hand,various 1D models have been proposed for use in control lawdesign [9, 10]. These models are useful for a limited class ofcompressors in which rotating stall is recoverable and non-

debilitating (primarily centrifugal compressors). However, theyignore the essential physics of rotating stall, which is (at least) 2dimensional. Several studies [11, 12, 13] have shown that, in highspeed axial compressors common to gas turbine engines, rotatingstall is the performance limiting instability, even when iteventually initiates surging of the engine. These studies indicatethat one must include rotating stall dynamics in any study of activecontrol of axial compressors.

Thus a 2D model of limited (compared to CFD) dimensionality isdesired. The Moore-Greitzer model has proved useful in thiscontext for linear control [3, 18, 19, 20, 28]. But this model alsoincorporates various nonlinear features, which have not untilrecently been validated experimentally or exploited in a controlcontext. For instance, the model captures the essential

nonlinearities which determine rotating stall inception transientbehavior [12]. It also explicitly accounts for nonlinear couplingbetween rotating stall and surge [15].

To begin to take advantage of the nonlinear form of the model,this paper presents several results. After a brief account of thefluid-dynamic PDEs, a nonlinear ODE form for use in controlanalysis and design is derived. This model is then verifiedagainst experimental data in which perturbations are large enoughto cause nonlinearity to be important. This is an important stepbecause various assumptions are made during modeling. Aphysically motivated Lyapunov analysis is then conducted for therotating stall/surge problem. This analysis allows domains ofattraction to be determined. Examples point out the important


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physical aspects of rotating stall stabilization.

2. Rotating Stall/Surge Model

Consider the schematic diagram of an axial compressor in Fig. 1.It consists of an upstream annular duct, a compressor modeled asan actuator disk, a downstream annular duct, and a throttle.During stable operation, flow through the compressor can beassumed to be circumferentially uniform (axisymmetric), and asingle non-dimensional measure of flow through the compressordetermines the system state. One such measure is the 'flowcoefficient', which is simply the nondimensionalized value of theaxial velocity:

=(axial velocity)(rotor speed) (1)

During quasi-steady operation, the total-to-static pressure risedelivered by the compressor is simply determined by its 'pressurerise characteristic,' denoted C() (Refer to Fig. 2). The pressurerise is balanced by a pressure loss across a throttling device, whichcan be either a simple flow restriction (used for testingcompressors as components) or the combustor and turbine in a gasturbine engine. The balance between pressure rise across thecompressor and pressure drop across the throttle is depicted as anintersection between the characteristics of the two devices,C() and T(), where, for low pressure ratios, T() is usually

taken to be a quadratic function of:

T = 2KT2. (2)

depends on the degree of throttle closure. In a typicalexperiment, the throttle is slowly closed, the throttle characteristicbecomes steeper (modeled by increasing in (2)), the

intersection point between C() and T() changes, and the

equilibrium operating point of the compressor moves from highflow to low flow (see Fig. 2).

The stability of the equilibrium point represented by theintersection between C() and T() has been the topic ofnumerous studies, due to its importance in the safe, highperformance operation of gas turbine engines. In our model, thesystem state under unsteady, possibly non-axisymmetric (i.e.circumferentially varying) conditions is characterized by threeterms: the annulus averaged pressure difference across thecompressor, , the annulus averaged flow coefficient, , and thespatially distributed perturbation on , denoted :

(,,) = + (,,), (3)

where is the nondimensional axial position in the compressor (the

Fig. 2: Compressor and throttle behavior during a typical experimental test.

PressureRise(orDro

p),

Flow Coefficient,

stable equilibriaunstable equilibriathrottle characterstic

Increasing KT

XX X

X equilibrium points

origin is chosen to be at the compressor face), is thecircumferential position, and is non-dimensional time (in rotorrevolutions).

Note that evaluation ofCand T must now be conducted morecarefully due to the unsteady and non-axisymmetric character ofthe flow: Cis evaluated at = 0 (the compressor face), and

varies with both and i.e. we evaluate C(+ (0,,)). Thusthe compressor is viewed as a distributed memoryless nonlinearityoperating on the local (in ) flow coefficient. On the other hand,due to the nature of the downstream flowfield, T can beevaluated for the annulus averaged flow; thus the pressure loss

across the throttle is simply T(

).One additional variable must be introduced in order to set up thesystem of equations. The upstream flowfield, being twodimensional, admits both axial velocity perturbations () andcircumferential velocity perturbations. Rather than introduce acircumferential velocity variable, we define the perturbationvelocity potential, such that

()

=and

()

=(circ. vel.). (4)

Although necessary for the derivation, will ultimately beeliminated from the equations, along with all of the partialderivatives with respect to space, leaving an operator-theoretic

A B CD

Compression system components: A - inlet duct, B - compressor, C - downstream duct, D - throttle.

Fig. 1:


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ordinary-differential relationship.

These are the relevant system states. Derivation of a model for thedynamics of the compression system is presented in otherdocuments, and is beyond the scope of this paper. Here we willpresent the model in a format that is coherent and accessible froma control theoretic point of view, and discuss the dynamicimplications of the model. Thus we present the fluid dynamicswithout further derivation, referring the interested reader to therelevant literature [11, 14, 15, 18, 27]. Note that time, , isnondimensionalized by the rotor revolution period.

1) For the upstream flow field we assume no incoming vorticity(clean inlet conditions) and thus the flow in this region ispotential:

2 = 0 0. (5)2) The annulus averaged pressure rise across the compressor(indicated by an overbar) lags behind that imposed by the throttlecharacteristic, due to mass storage in the downstream ducting andplenum chamber:

= 14l

cB2

T = 0. (6)

3) The annulus averaged flow coefficient accelerates to balancethe difference between the annulus averaged pressure risedelivered by the compressor and the annulus averaged pressurerise that actually exists across the compressor (in unsteadysituations these can be different):

=1lc

C + = 0. (7)

4) Finally, the non-axisymmetric (indicated by the subscript 'ac')part of the pressure rise delivered by the compressor acts to

accelerate the flow through the rotor and stator passages, wherepart of this acceleration is due to the rotor moving through a non-uniform flow perturbation (this is reflected by the presence of apartial derivative with respect to ):

m

+

+

= C + ac

= 0. (8)

5) Definitions: lc, B, m, , are scalar geometric parameters defined

in [27], T() is the inverse of T, and the definitions of 'annulusaveraged' and 'non-axisymmetric' take the expected forms:

C =1

2C

0

2d

,and

C ac = C C . (9)

note that only through the nonlinearity ofCare the axisymmetric

and the non-axisymmetric dynamics coupled. Thus if one assumessmall perturbations, rotating stall and surge are decoupledphenomena and can be stabilized separately. One can also study1D nonlinear perturbations alone [10, 23]. This approach carrieswith it the implicit assumption that the effect of rotating stall isalready captured (at the 1D level) by an 'annulus averagedcompressor characteristic' (different from the C we have definedhere). This is a useful approach in some centrifugal compressorsand low-speed fans, if no abrupt drop in pressure rise isexperienced when rotating stall occurs.

2.1 Control-Theoretic FormulationThe system of Eqs. (5-8) is a coupled set of partial differentialequations, in which two space dimensions and time are present.However the space dimensions can be eliminated by introducingthe solution for the upstream velocity potential as follows:

= n eine n n = 0

0. (10)

This solution satisfies Laplace's equation, and the other boundaryconditions of clean inlet flow experiments as described in [18].Using Equation (4), we can also write a solution for in theupstream flowfield:

= n eine n

n = 0

0, (11)

where n()= n n . These relationships allow to beeliminated from the system, and Equation (8) to be written

m

n

n + n + inn ein = C + ac

n =n 0

(12)

where the partial derivatives have been eliminated, and n is thenthspatial Fourier coefficient of the axial velocity perturbation atthe compressor face ( = 0):

n() = (0,,) =12

(0,,) ein d0

2

(13)

Introduction of the Fourier series solutions to eliminate spatialderivatives is a well-known method for handling distributedsystems. The modal (or 'spectral') form of the equations allowslinear control techniques to be applied. This approach has provensuccessful in laboratory stabilization of rotating stall [18, 19]. To

study the effects of nonlinearity on the system, an operator-theoretic form is often more useful. By substituting (13) into (12),we can write the system (6,7,12) as follows:

E()

=

0 1

4lcB2

0

1lc

0 0

0 0 A()

+

1

4lcB2

T()

1lc

C(+)

C(+) ac

,(14)

where:

E( ) = 1 n+ ( )

= mn

+ 12

( )ein( ) d0

2n = n 0

(15)and

A( ) = 1 in( )

= in 12 ( )ein( ) d

0

2

n =n 0

. (16)

E and A are linear operators that represent spectrally the effect ofthe upstream flowfield, as well as allowing derivatives withrespect to and to be evaluated. The linear and nonlinear partsof Equation (14) have been separated for illustrative purposes, butthe equations are clearly in the desired form, i.e. x=F(x).


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Equations (14) can be further recast into a number of differentforms, depending on the application, by proper application of theFourier transform and its inverse. For instance, we haveimplemented an update equation for E() in our simulation, and

then used the proper Fourier relationships to recover . It is alsopossible, by choosing appropriate state variables, to reduce thesystem to one involving convolution with influence functions inthe circumferential variable , and thus eliminate the summationsover n. Finally, spatial discretization of() and substitution ofmatrix Fourier transformations in place of summations andintegrals allows the system to be written as a finite-dimensionalstate-space system, with good numerical properties for simulationand control work [18]. This is the form we will need in Section 4,so we briefly outline it here.

If we define the following as the system's state vector:

=

(1) + (

2) +

(2N+1

) +

, where n = 2n2N+1

and N is a power of 2, then we can replace the Fourier transformsand inverses in (15) and (16) with DFT and inverse DFT matrices,if we assume that modes higher than the Nyquist do not appear inthe compressor (more detailed models of unsteady compressorperformance, coupled with experiments [28], justify thisassumption). The result is a finite-order nonlinear state spacemodel, which we proceed to derive.

It will prove useful to define specially scaled, real-valued DFTmatrices, which essentially decompose and synthesize Fourier

series sine and cosine coefficients instead of exponentialcoefficients. These matrices are:

G : Re 1 Im 1 Re 2 Re N Im N

T

= 22N+1

12

12

12

cos(1) cos(2) cos(2N+1)

sin(1) sin(2) sin(2N+1)

cos(21)

sin(21

)

cos(N1) cos(N2N+1)

sin(N1) sin(N2N+1) (17)

G1 GT

where the last equality holds because the DFT matrix is scaled by2/(2N+1) instead of 2/(2N+1). Using these matrices, we can

easily define matrix versions of the linear transformations E and Ain Eqs. (14-16):

E = G1DEG, where:

DE =

lcm1

+ 0

0 m1 +

mN

+ 0

0 mN

+

, (18)

and, A = G1DAG, where:

DA =

00 0

0 N

N 0 . (19)

Note that the annulus averaged (surge) part of () is nowincorporated as part of the state, so the matrices E and A areslightly different than their continuous domain counterparts - the(1,1) entry of DE and DA now contain the surge dynamics. Thisapproach simplifies the final form of the equations, and allowssurge and rotating stall to be analyzed together. Equation (14) cannow be rewritten as follows:

E

= A + C() T

= 14lcB

2S

T

, (20)

where is the state vector representing the distributed flowcoefficient state, S is a row vector that extracts from :

S= [1/M 1/M 1/M]

and T creates a distributed (vector) representation of a zerothmode disturbance from its scalar equivalent:

T = [1 1 1]T .

The purpose of this section was to introduce the Moore-Greitzermodel in two forms that are useful to control theorists. Because ofthe assumptions on the upstream flowfield, the partial-differentialnature of the original system of equations has been eliminated, andthe dependence on axial position has been 'solved out' of the

system, leaving a simpler set of equations.Refinement of the above model to include the effects ofunsteadiness on the compressor characteristic has been conductedby Haynes et. al. [28], for the linearized case. Their refinementrecognizes that the compressor characteristic is not in reality amemoryless nonlinearity. In fact, the lags inherent in thecompressor pressure-rise response have a strong effect on both thedamping and the rotation frequency of pre-stall waves. It isstraightforward to append the dynamics introduced in [28] to theequations derived here; to avoid unnecessary complexity we willnot present this extension. Simulations indicate that, althoughcompressor pressure-rise lags change the effective inertias andstability characteristics of the system, they do not add significantextra dynamics -- in the parlance of the linearized model,


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including compressor lags adds fast dynamics, but also alters thedominant root locations. Thus if we model the change in thedominant root locations and neglect the fast dynamics, the modelwill presumably be accurate at frequencies of interest. For a priori

prediction of compressor behavior, incorporating compressor lagsis vital, as discussed in [19] and [28].

3. Comparison of Simulated and Experimental Stall Inception

The nonlinear functions Cand T govern both the stallinception and the fully developed stall behavior. Fully developedstall has been treated by other researchers [21, 22], and appears tobe less relevant to the stabilization problem, for which we areprimarily interested in the character and subsequent avoidance ofstall inception. Thus we will concentrate here on the effect ofCand T

on nonlinear stall inception behavior, showing both

experimental and theoretical results.

Stall inception - that is, transition from axisymmetric flow to fullydeveloped stall - is of interest because only during this transitionprocess are two important criteria met by the flow field: 1)relatively small axial flow perturbations, and 2) strong influenceof the nonlinearities. Meeting the first criterion is important toinsure that the assumptions inherent in the model described aboveand in [11] are reasonable. Specifically, the "semi-actuator disk"assumption used to simplify the representation of the flow acrossthe compressor is a more severe approximation during fullydeveloped stall than during pre-stall and stall-inception conditions[26]. Restricting the model validation to waves of smallamplitude relative to fully developed stall does not pose a problemto control law development, because presumably any workingcontroller would stabilize the system and avoid the fully-developed stall condition.

The second condition above is important in the present studybecause understanding and accurately modeling nonlinear stallinception behavior is our goal. In some compressors, over acertain range of flow coefficients, linearization of the dynamics in(13) is a workable approach [3]. However, to further extend theoperating range of these compressors, and to control compressorsin which nonlinearities are more severe, we must understand thenonlinear behavior.

Our approach in the present study is to carefully model threeexperimental low speed compressors with different stall inceptionbehavior. The parameters in the model are chosen on the basis ofcompression system geometry, modified by physical reasoning andcomparison between simulation and experiment (i.e. a simplistic'identification' procedure). Experimental stall inception data arethen compared to nonlinear simulation, to verify that the modelcaptures the important transient phenomena.

3.1 Compressor ModelsFor convenience we will label the three experimental compressorsused as compressors C1, C2, and C3. The details of theexperimental setup for each compressor are described inreferences [3], [28], and [30] respectively. The geometric detailsof the compressors (those necessary to perform the simulation) arepresented in Tables I and II, and the compressor and throttlecharacteristics are shown in Fig. 3. Note the followingapproximations in the choice of the model parameters:

1) The 'B' parameter [11] is set to 0.1 in the experimental

compressors - this represents in all cases a 'worst case' maximum.Even using this maximum, the surge dynamics are very stable forthe experiments we will present; this is most easily expressed byrecognizing that the following approximation to Eq. (6) applies asB0:

T . (21)

Thus the mean flow will follow the throttle characteristic veryclosely in these experiments, because the system has almost nomass storage capacity.

2) An 'unsteady loss parameter' is used to account for the com-pressor lags [29]. A priori prediction of this parameter is difficult


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TABLE I - COMPRESSOR MODEL PARAMETERS

C1 C2 C3

U (speed at mean wheel radius) 73 m/s 72 m/s 36 m/sR (wheel radius) 0.259 m 0.287 m 0.686 m at stall inception 0.4202 0.463 0.431 at stall inception 10.8 9.41 6.3lc 8.0 6.66 4.75B 0.1 0.1 0.1m 2 2 2 0.65 1.29 0.42 0.18 0.68 0.25eff(adjusted to match

frequencies)

0.44 0.68 0.4

Measurement position, -0.5 0.2 ~0Nondim. convection time (usedfor unsteady lossapproximation)

0.2 0.2 0.5

Discharge plenum pressure -0.75 0 0

TABLE II - COMPRESSOR CHARACTERISTICS

C1: C() =

1.9752 0.0987 + 0.0512 ; 0.025

12.7763 + 6.3942 0.295 + 0.0535 ; 0.025 < 0.30

5.5364 + 7.7203 4.2042 + 1.127 + 0.0719 ; > 0.30

C2: C() =

12.1172 2.423+ 0.221 ; 0.1

49.6243 + 39.5092 6.413 +0.395 ; 0.1 < 0.4

10.06952 + 9.430 1.184 ; > 0.40

C3: C() =

42 2 + 0.5 ; 0.25

143.143 + 143.042 44.683 +4.717 ; 0.25 < 0.405

13.3652 + 11.574 1.920 ; 0.405 < 0.463

5.4282 + 4.211 0.213 ; > 0.463

[28], so we have used physically reasonable values (~the convec-tion time across one blade), adjusted to improve the model fit.

3) The simulations presented here also use an 'effective ', whichis adjusted from the geometrical value to match the perturbationrotation frequency.

The final and most important model parameter is the axisymmetriccompressor characteristic, C. The characteristic shapes shown inFig. 3 are determined in two steps. First, experimental data foreach compressor is fit with polynomial segments. However, theexperimental data extends only over the portion of the curvesindicated by solid lines, because beyond this point thecompressors are unstable and cannot be operatedaxisymmetrically. To extend the compressor characteristicsbeyond their measured portion, the nonlinear simulation is used.Simulated and actual stall transients are compared, and informed

choices are made for the shapes of the unknown portions of thecharacteristics. Details of this procedure

0

0.4

0.8

1.2

0-0.2 0 0.2 0.4 0.6 0.8

Compressor and throttle characteristics used to simulate rotating stall inception in compressor: (a) C1, (b) C2, and (c) C3. Transients of annulus average (a.a.) pressure vs.

flow during stall inception are also shown.

Fig. 3:

Known part of characteristicEstimated part of characteristic

Transient into rotating stall

Equilibrium point

Throttle characteristic

x

x

x

x

x

xx

(a)

0

0.4

0.8

1.2

(b)

0.4

0.8

1.2

(c)

are given in [32], and the results are shown in Figs. 4-6, whichconstitute the nonlinear model validation.

3.2 Discussion of Simulated and Experimental Stall Inception

TransientsThe 'best fits' of the nonlinear simulation to the transient data areshown in Figs. 4, 5, and 6. In all cases the salient features aresimilar between the experiment and simulation, although the threecompressors have very different stall inception behavior.Compressor C1 has a long, slow growth of pertur-bation wavesinto rotating stall. Compressor C2, on the other hand, hasrelatively small perturbation waves, followed by a sharp inception


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wave that

0 10 20 30 40 50 60 70 80 90 100

Experimental Stall Inception Data

Simulated Stall Inception - Frequency Adjusted Using Effective

Comparison of experimental and simulated stall inceptionfor compressor C1.

Fig. 4:

SCALE:

= 0.2

Probe #1

Probe #2

Probe #3

Probe #1

Probe #2

Probe #3

, Rotor revs

SENSOR LOCATIONS:

12

3

= -0.5

Comparison of experimental and simulated stall inception for compressor C2.

Fig. 5:

, Rotor revs0 5 10 15 20 25

Probe #1

Probe #2

Probe #3

Probe #1

Probe #2

Probe #3



SCALE: = 0.2

SENSOR LOCATIONS:

12

3

-0.2

leads quickly into fully developed rotating stall. Compressor C3is even more severe in this regard: nonlinear influences are seenwhile the waves are still quite small, and the stall wave growsquickly into a violent nonlinear event. Despite these differencesin behavior, the proper choice of the shape of C allowsexperimental observations to be closely matched by the nonlinearmodel.

There is a larger discrepancy between the simulated and themeasured stall inception behavior for compressor C3 than for C1and C2. Several factors contribute to this discrepancy. Mostimportant is the effect of non-axial flow on the measurements.Hot-wire anemometers are used to measure the flow velocity inthe experiments; these devices effectively measure the absolute

value of

Comparison of experimental and simulated stall inception for compressor C3.

Fig. 6:

12

3

0 2 4 6 8 10 12 14, Rotor revs

Probe #1

Probe #2

Probe #3

Probe #1

Probe #2

Probe #3



SCALE: = 0.2

SENSOR LOCATIONS: 0

axialplus radial flow perturbations. In the C3 experiment, the hotwires are mounted very close to the compressor face ( = 0),where significant non-axial and reverse flow perturbations exist.In the C1 and C2 experiments, on the other hand, the hot wires aremounted further upstream of the compressor ( = -0.5 and = -0.2 respectively) where the measured perturbations are primarilyaxial; a 'fluid dynamic filter' exists (Eq. (11)) that smoothes outthe internal flow details, allowing the more global influences(those modeled by Moore-Greitzer) to be observed.

To understand the influence of the nonlinearity in C on stallinception, consider a sinusoidal velocity perturbation beingmapped through a compressor characteristic, shown in Fig. 7.From the figure it is clear that at the peak of the characteristic, alinear representation of C is insufficient; the slope of thecharacteristic is near zero, so higher order derivatives becomeimportant. The high velocity portion of the wave experiencesattenuating pressure forces, because it accesses the stable side ofthe characteristic, while the low flow side experiencesdestabilizing pressure forces. Because of the interaction caused bypartials with respect to and (Eq. 8), the pressure rise does notact alone to accelerate the flow. Instead there is an 'integratingaction' of, for instance, the first sinusoidal harmonic. If theintegral effect of the positive and negative parts of thecharacteristic causes a net attenuating effect, then the wave willdie away (or converge to a small amplitude limit cycle). If theintegral effect is amplifying, the wave will continue to grow, witha speed that depends at each instant on the wave shape and itsmapping through C and T.

The two extremes of this general behavior are: 1) gradualcharacteristics, whose behavior can be adequately characterizedusing a linearized analysis (with the proper choice of slope), and2) characteristics with abrupt changes in slope. In the latter case(when the unstable side is steeper as in Fig. 7), a sharp drop inpressure rise is experienced by the low-flow part of the wave.This pressure drop overrides other influences and causes a quick,localizeddeceleration of the flow (as in Fig. 6). This drop in flow


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further reduces the overall pressure rise delivered by thecompressor,

0

0.2

0.4

0 1 2 3 4 5 60

0.2

0.4

0.6

0 1 2 3 4 5 6

() c(())

Fig. 7: Effect of nonlinear characteristic on pressure forces whichact to accelerate a velocity perturbation. Pressure forces (on right) below the mean value act to decelerate the flow.Note the significant crossfeed between the first harmonic and the zeroth and second harmonics.

0

0.20.4

0.6

0.8

0 0.2 0.4 0.6 0.8

Note drop inc

which in turn moves the mean operating point towards lowervalues (in order to satisfy the throttle characteristic, Eq. 21). Thesystem thus evolves into rotating stall at a rate that is prematurewhen viewed from the linearized analysis. In fact, with the properinitial condition, a compressor can go into stall while at anannulus-averaged operating point that is still stable in the

linearized sense. In these cases, the domain of attraction of theoperating point has become very small, because of the existence ofa nearby abrupt change in the nonlinear mapping C. Section 4makes this observation mathematically rigorous.

Also important to the stall inception behavior is the slope of thethrottle characteristic, T. IfT is steep at the nominal operatingpoint ( T/ large), the annulus-averaged flow coefficient isinsensitive to changes in the pressure rise delivered by C. Thisis the case in C1 (Fig. 3a), in which the throttle discharges to aplenum below atmospheric pressure, making the throttle linesteep during stall initiation. This steep slope, combined with theshape of C, accounts for the slow transient into rotating stallshown in Fig. 4.

When T is shallow ( T/ small), on the other hand, thenonlinearity of C couples more immediately into changes in themean flow as follows: C maps energy from higher harmonicsinto the zeroth harmonic (this effect is represented by the functionC + , and can be seen in Fig. 7). This causes a loss inannulus-averaged pressure rise, which must be accompanied by arelatively large drop in ifT is shallow (Eq. (21)). Thus the rateat which the system evolves into rotating stall depends in part onthe slope ofT. Compressor C3 is the best example of this type ofbehavior - the transient from stall inception to fully developedstall is under one rotor revolution, partially because of the shapeofT.

Another important conclusion one can draw from the resultspresented here is that nonlinear effects, when they are important,

often manifest themselves as high spatial frequencies in the stalltransients. In both C2 and C3, the rotating stall precursor wavestarts as a sinusoidal wave (first spatial harmonic). During stallinception, however, significant spatial harmonic content above thefirst harmonic exists in the wave shape as it transitions intorotating stall. Figure 7 is an example of how this comes about - itshows a first harmonic perturbation being transformed into a2nd harmonic acceleration term . Figure 8 shows the result -during stall inception, what began as a 1st harmonic perturbationbecomes a multi-harmonic perturbation, eventually transitioningagain to a 1st-harmonic dominated fully developed stall cell. Thusapproximations which model only the first spatial harmonic(Galerkin approximation, [15]) may not capture the importantnonlinear effects at stall inception accurately enough to allowrealistic control law design.

It should be noted in closing this section that the philosophy of theMoore-Greitzer model is to ignore blade-to-blade flow detailswhen studying transient rotating stall behavior. This philosophy,although validated in a wide variety of both high and low speed

compressors [12] is not universally applicable. Day [30] hasmeasured stall inception behavior that mustbe observed at higherresolution to be fully understood. In such cases, the Moore-Greitzer model requires further refinement; 3-dimensional effectsneed to be at least partially accounted for to properly model thesestall phenomena.

4. Lyapunov Analysis of Rotating Stall

Having qualitatively validated the model in a nonlinear sense, wenow turn to quantitative techniques for control law analysis.Lyapunov methods are the foundation for various nonlinearcontrol design procedures (such as feedback linearization, slidingmode control, and Lyapunov control), and thus a Lyapunov

stability analysis of rotating stall is a logical first step.Furthermore, we present herein a complete method, based onLyapunov concepts, for assessing the large-amplitude disturbancerejection characteristics (domains of attraction) of axialcompression systems.

To conduct a Lyapunov analysis and subsequently deduce stabilityregions, we first translate the origin to the equilibrium point, as inFig. 9. We then find a suitable 'incremental energy' or Lyapunovfunction to characterize the system state. Although any positive


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0

2

4

6

8

10

0 1 2 3 4 5 6 7

1st Harmonic

2nd Harmonic

3rd Harmonic

4th Harmonic

Fig. 8: Participation of 1st 5 Fourier harmonics in stall inception. Note that harmonics 2 and 3 temporarily become larger than the 1st harmonic during stall inception. Simulation

results are for compressor C3 (same as Fig. 6).

Time, rotor revolutions

DFTcoefficientmagnitude

(64points)

5th Harmonic

Coordinate system for Lyapunov stability calculation.Fig. 9:

T()

c()

definite function based on the system state is a candidateLyapunov function, careful choice of the form leads to a moreelegant and physically meaningful formulation. Consider thefunction

V = 12M

TE + 2B2lc2 , (22)

where M=2N+1 and, from here on, all states and functions are

evaluated about the new origin. It is straightforward to show thatE is positive definite based on the definitions (18), so V is positivedefinite. This form of the Lyapunov function clearly accounts forall perturbations that might exist in the compressor, and thussuitably characterizes the systems 'incremental energy'.

More important, however, is the way this choice for V effects theform ofV. By taking the derivative of (22), substituting Eqs. (20),and simplifying, we have

V = M

TA + M TC T . (23)

We can eliminate the first term in this equation, because A is arotation matrix whose eigenvalues are all on the j-axis, as

follows: Using Eq. (19),

TA = G DA G = 0. (24)

The last equality holds due to the form of DA, which appears in

(19). Equation (24) reflects the fact that A

is simply

rotated by/2, and is thus orthogonal to it . Using (24), Eq. (23) simplifiesto

V =M

TC T , (25)

which is the two-dimensional analog of the "incremental powerproduction and dissipation" equation motivated and developed bySimon [23]. Taken to the 2D continuum limit (i.e. M ), (25)can be expressed as follows:

V = 12

() C () d T 0

2

, (26)

which is the annulus averaged incremental energy production of

the compressor, minus the incremental energy dissipation of thethrottle. This measure of compression system stability wasoriginally proposed by Gysling [20]. Since T is a positivedefinite function of(see Fig. 9: T lies strictly in the firstand third quadrant), the 'worst case' value ofV will occur when is zero. Thus we can consider the nonlinear stability of therotating stall system by simply analyzing the function

Tc ,(or, equivalently, the integral term in Eq. (26) ).

4.1 Basins of AttractionUsing the Lyapunov analysis presented above, it is possible todetermine a class of perturbations that are guaranteed to be stable.This 'basin of attraction' can be defined by finding the largesthypersphere:

S = :

1

2M

T

E

2

(27)

inside which V is always less than zero. An estimate of can bemotivated as follows. For a given operating point, we cancompute the function C() as a function of as shown in Fig.10. This function 'maps' a perturbation () into a value ofV, asshown in the figure. The shape ofC() shown in Fig. 10 isquite general near the local maximum ofC for stable equilibria.C() will invariably have a local maximum that goes throughthe origin, be negative for values of greater than zero, andbecome positive for values of below some value that we call -d.

Based on this discussion we see that, at stable equilibria and forperturbations () whose minima are greater than -d, V in Eq. (26)

will always be negative. Thus these perturbations will becomesmaller in magnitude, where magnitude is measured by V in Eq.(22). Stated mathematically, we have

min0


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values of that satisfy n d n. This can be accomplishedby choosing to limit the amplitude of the major axis of thehyperellipsoid defined by the transformation E. Fortunately, wehave already expressed the diagonalized form of E in Eq. (18), sothat the major axis amplitudes are known. From (18), (27), and

c()

Ma in velocit erturbations into V.Fi . 10:

()

c(())()

()c(())dV =1

2

2

0

.

(28), we deduce the required value of to guarantee stability:

2 = d2

2M min lc,

mN

+(29)

Where the min function here indicates that the size of the basin ofattraction may be limited by either surge (lc) or rotating stall (m/N

+ ) considerations. For our purpose, which is to show the effectof compressor characteristic shape on the size of the basin ofattraction, d itself will provide a sufficient measure; from Fig. 10 itis apparent that d is simply the distance along the axis from theorigin to the next zero of the function C().

4.2 ExamplesIn the preceding section we defined a parameter d which indicatesthe size of the stable basin of attraction around a given operatingpoint near stall. This measure provides a quantitative way tocompare the stability of various compressors, and allowsexperimental stall inception behavior to be linked to thecompressor characteristic nonlinear shape.

To illustrate, consider the following scenario, which is a commontest procedure. A compressor is put into operation at a stableoperating point. The throttle is then closed quasi-steadily, movingthe operating point toward stall. During this process, the slope of

the characteristic ( dC/d ) will monitonically approach zero.According to the linearized theory (see, for instance, [3]) thedamping ratio of rotating stall perturbation waves is directly

proportional to dC/d , and thus during our hypotheticalexperiment, the system will begin to resonate - random excitationswhich exist in the compressor environment will excite the rotatingstall modes, and waves will begin to appear in the compressor.The size of these waves depends both on the damping ratio of therotating stall modes and on the excitation level. Unsteadiness inthe compressor tends to increase as the peak of the compressorcharacteristic is approached, so one might expect the excitationlevel to also rise during such an experiment. When thedisturbance wave becomes large enough that it is outside thedomain of attraction for the current operating point, thecompressor will go unstable.

Based on this discussion, a compressor is more likely to stall if dis small for a given value of dC/d . Thus plotting d as afunction ofdC/d allows the stall flow coefficient to be assessedas shown

0

0 Slope, dc/d

P

erturbationMagnitudeLimit,

d

Importance of both the size of the basin of attraction and the compressor slope in determining the stall point.

Fig. 11:

Basin of attraction( 0 as dc/d0 )

Amplitude of perturbations( increases as dc/d0 )

STALL POINT

0

0.1

0.2

0.3

0.4

-10Slope, dc/d

P

erturbationMagnitudeLimit,

d

Basin of attraction variation with level of linear stability,as measured by the compressor characteristic slope.

Fig. 12:

C1

C2

C3

schematically in Fig. 11. It also indicates the damping ratio atwhich one can expect the system to go into rotating stall, whichdirectly influences the size of the "pre-stall" waves which existprior to stall inception.


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-8 -7 -6 -5 -4 -3 -2 -1 00

0.02

0.04

0.06

0.08

0.1

0.12

Slope

d

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.3

-0.35

-0.25

-0.2

-0.15

-0.1

-0.05

0

Fig. 13: Example of the effect of the unstable part of the com-pressor characteristic on the domain of attraction.

Figure 12 shows d vs. dC/d for the three example compressorsintroduced in Section 3. It clearly shows that in compressor C1we expect strong resonance to occur prior to stall inception,because only under such highly resonant conditions will thedomain of attraction be violated. On the other hand, compressorsC2 and C3 tend to go unstable more easily, and thus require onlymild resonance for stall inception. This is in fact the observedexperimental behavior in Figs. 4, 5, and 6.

Clearly, compressors which go unstable well before they arelinearly unstable are more difficult to control, especially if onelimits oneself to linear controllers. Detection of pre-stall waves,

an important area of current research, is also more difficult in suchcompressors. Thus it is of interest to understand the conditionsunder which the domain of attraction of a compressor will besmall when measured against dC/d . Figure 13 and 14 showtwo parametric studies which provide some insight into the trends.

Figure 13 shows three compressors whose stable (i.e. measurable)characteristics are identical, but which have different unstable

characteristics. A plot of d vs. dC/d for these compressorsreveals that, for a given stable characteristic, the steeper theunstable characteristic, the more difficult one can expect bothdetection and control of rotating stall to be.

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

-14 -12 -10 -8 -6 -4 -2 0

0

0.02

0.04

0.06

0.08

0.1

0.12

Slope

d

Fig. 14: Example of the effect of the narrowness of the peak ofthe compressor characteristic on the domain of attraction.

Figure 14 shows the effect of the "narrowness" of thecharacteristic peak on basins of attraction (this parameterization ofcharacteristics is similar to the 'S' parameterization used byMcCaughan [22]). Here we see that wide peaks are more benignfrom a detection and control perspective than narrow peaks.Unfortunately the latter may be more prevalent in high-speedcompressor applications, where rotating stall control is of highestinterest.

This delineation of compressor behavior provides an explanationfor two experimentally observed phenomena, both of which areinexplicable using linearized arguments: 1) some compressors

exhibit large traveling waves a prior to stall, while others stallwhile traveling waves are still relatively small, 2) theexperimentally determined damping ratios of pre-stall waves donot generally go to zero before stall inception occurs.

5. Conclusions

A nonlinear, control-theoretic form for the Moore-Greitzer modelhas been presented. This form is sufficient to accurately modelnonlinear stall-inception behavior provided a) the unstable part ofthe compressor characteristic is properly adjusted, and b) theeffects of unsteadiness on the pressure rise delivered by thecompressor are included. Stall inception, rather than fullydeveloped stall, is used as a means to verify the Moore-Greitzermodel of rotating stall.


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The effect of the shape of the pressure rise characteristic has beenelucidated through evaluation and comparison of experimental andsimulation results. The most important conclusions are:

a) The shape of the compressor characteristic determines, to alarge extent, the character of stall inception.

b) second and higher harmonics often interact strongly with thefirst harmonic during stall inception.

A Lyapunov stability analysis of the nonlinear rotating stall/surgesystem is also presented. This analysis is used to determine thesizes of domains of attraction for stable operating points, and todetermine the conditions which favor resonance of pre-stall wavesprior to stall inception. The modeling and analysis toolsdeveloped here are valuable for the evaluation of compressorstability and for the design of control laws, and will be extended tothe feedback control case in future work.

6. Acknowledgments

This work was supported by the US Air Force Office of ScientificResearch, Major Daniel B. Fant, Technical monitor. This supportis gratefully acknowledged.

7. References

1. Greitzer, E.M., Review - Axial Compressor Stall Phenom-ena, J. of Fluids and Eng., Vol. 102, 1980, pp. 134-151.

2. Ludwig, G.R., Nenni, J.P., Tests of an Improved RotatingStall Control System on a J-85 Turbojet Engine, ASME 80-GT-17, Gas Turbine Conference and Products Show, NewOrleans, March 1980.

3. Paduano, J.D. et al., Active Control of Rotating Stall in aLow-Speed Axial Compressor J. of Turbomachinery, Vol.115, January 1993 , pp. 48-56.

4. Day, I.J., "Active Suppression of Rotating Stall and Surge inAxial Compressors,"J. of Turbomachinery, Vol. 115, No. 1,January 1993.

5. Badmus, O.O. et al., "A Simplified Approach for Control ofRotating Stall, Parts I and II", AIAA 93-2229 and 93-2234,29th Joint Propulsion Conference, Monterey, CA, June 1993.

6. Greitzer, E.M. et al., "Dynamic Control of AerodynamicInstabilities in Gas Turbine Engines," AGARD LectureSeries 183, Steady and Transient Performance Prediction ofGas Turbine Engines, May 1992.

7. Takata, H., Nagano, S., "Nonlinear Analysis of RotatingStall,"J. of Eng. for Power, Vol. 94, No. 4, 1972.

8. Pandolfi, M,. Colasurdo, G., "Numerical Investigations onthe Generation and Development of Rotating Stalls," ASME78-WA/GT-5, 1978.

9. Escuret, J.F., Elder, R.L., "Active Control of Surge in Multi-Stage Axial-Flow Compressors, ASME 93-GT-39, Interna-tional Gas Turbine and Aeroengine Congress and Exposition,Cincinnati, OH, May 1993.

10. Badmus, O.O., Chowdhury, S., Eveker, K.M., Nett, C.N.,"Control-Oriented High-Frequency Turbomachinery Model-ing: Single-Stage Compression System 1D Model," ASME93-GT-18, International Gas Turbine and AeroengineCongress and Exposition, Cincinnati, OH, May 1993.

11. Greitzer, E.M., "Surge and Rotating Stall in Axial FlowCompressors, Part I: Theoretical Compression SystemModel, and Part II: Experimental Results and Comparisonwith Theory,"J. of Eng. for Power, April 1976, pp. 190-216.

12. Garnier, V.H., Epstein, A.H., Greitzer, E.M., "RotatingWaves as a Stall Inception Indication in Axial Compressors,"J. of Turbomachinery, Vol. 113, April 1991.

13. Day, I.J., Freeman, C., "The Unstable Behaviour of Low andHigh Speed Compressors," ASME 93-GT-26, InternationalGas Turbine and Aeroengine Congress and Exposition,Cincinnati, OH, May 1993.

14. Moore, F.K., "A Theory of Rotating Stall of Multistage Com-pressors, Parts I-III," J. of Eng. for Power, Vol. 106, 1984,pp. 313-336.

15. Moore, F.K., Greitzer, E.M., "A Theory of Post-Stall Tran-sients in Axial Compression Systems, Part I - Developmentof Equations, and Part II - Application, " J. of Eng. for GasTurbines and Power, Vol. 108, 1986, pp. 68-97.

16. Pinsley, J.E., Guenette, G.R., Epstein, A.H., Greitzer, E.M.,"Active Stabilization of Centrifugal Compressor Surge,"J. ofTurbomachinery, Vol. 113, October 1991, pp. 723-732.

17. Gysling, D.L., Dugundji, J., Greitzer, E.M., Epstein, A.H.,"Dynamic Control of Compressor Surge using TailoredStructures," J. of Turbomachinery, Vol. 113, October 1991,pp. 710-722.

18. Paduano, J.D., "Active Control of Rotating Stall," Ph.D. The-sis, MIT Dept. of Aeronautics and Astronautics, February1992.

19. Haynes, J., "Active Control of Rotating Stall in a Three-StageAxial Compressor," S.M. Thesis, MIT Dept. of Aeronauticsand Astronautics, September 1992.

20. Gysling, D.H. "Dynamic Control of Rotating Stall in AxialFlow Compressors Using Aeromechanical Feedback," Ph.D.Thesis, MIT Dept. of Aeronautics and Astronautics, August1993.

21. Liaw, D.C., Adomaitis, R.A., Abed, E.H., "Two-ParameterBifurcation Analysis of Axial Flow Compressor Dynamics,"American Control Conference, Boston, June 1991.

22. McCaughan, F.E., Hong, X.X, "Stability of Fully DevelopedRotating Stall," ASME 92-GT-57, International Gas Turbineand Aeroengine Congress and Exposition, Cologne,Germany, June 1992.

23. Simon, J.S., Valavani, L., "A Lyapunov Based NonlinearControl Scheme for Stabilizing a Basic Compression System

Using a Close-Coupled Valve," American Control Confer-ence, Boston, June 1991.

24. Vidyasagar, M., Nonlinear Systems Analysis, Prentice-Hall,New Jersey, 1978.

25. Fink, D.A., "Surge, Dynamics and Unsteady Flow Phenom-ena in Centrifugal Compressors," Ph.D. Thesis, MIT Dept.of Aeronautics and Astronautics, June 1988.

26. Lavrich, P.L., "Time Resolved Measurements of RotatingStall in Axial Flow Compressors," Ph. D. Thesis, MIT Dept.of Aeronautics and Astronautics, August 1988.

27. Hynes, T.P., Greitzer, E.M., "A Method of Assessing Effectsof Circumferential Flow Distortion on Compressor Stability,"

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28. Haynes, J.M., Hendricks, G.J., Epstein, A.H., "Active Stabi-lization of Rotating Stall in a Three-Stage AxialCompressor," ASME 93-GT-346, International Gas Turbineand Aeroengine Congress and Exposition, Cincinnati, OH,

May 1993.29. Boussios, C., Rotating Stall Inception: Nonlinear

Simulation, and Detection with Inlet Distortion, S.M.Thesis, MIT Dept. of Mechanical Engineering, March 1993.

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31. Chue, R. et al., "Calculations of Inlet Distortion InducedCompressor Flow Field Instability,",Int. J. of Heat and FluidFlow, Vol. 10, 1989, pp. 211-223.

32. Paduano, J.D., Gysling, D.L. "Rotating Stall in Axial Com-pressors: Nonlinear Modeling for Control," SIAMConference on Control and its Applications, Minneapolis,1992.

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