Analysis of Non Recoverable Stall & Other

Instabilities Using Moore Greitzer Model

Karan Govil ∗

Department of Aerospace Engineering IIT Madras, Chennai, 600 036, India

Sudarshan Kumar †

Department of Aerospace Engineering IIT Bombay, Powai, 400 076, India

A simple compression system model, described by a set of three nonlinear differentialequations (the Moore Greitzer model) is studied. These governing equations are solvedusing a fourth order Runge Kutta method. Greitzer parameter (B) and Throttle parameter(KT ), are used to investigate the behavior of the compression system at various operatingpoints. These investigations show the existence of various instabilities in certain zones of Band KT parameters. A map of these unstable regions has been constructed between theseparameters. Numerical results are presented to show that besides rotating stall and surge,there exists other instabilities which may result from a combination of rotating stall andsurge and appear only at lower values of throttle parameter (KT ).

I. Introduction

Axial compressors are widely used in numerous applications such as aircraft engines, industrial com-pressors, air conditioners, gas turbine based power plants etc. As the flow through the axial compressoris throttled from the design point to stall limit, the steady axisymmetric flow pattern that exists in thecompressor becomes unstable. A detailed description of a general compression system instabilities has beenprovided by Greitzer.1 The unstable behavior can be broadly classified into two parts: Rotating stall, andSurge.

In rotating stall there are one or more cells of stalled flow which rotate around the circumference whilethe annulus averaged mass flow becomes constant once the pattern of rotating stall is fully developed. Whilesurge is a high amplitude and low frequency oscillation of total annulus averaged mass flow through thecompressor. The frequency of surge oscillations is typically an order of magnitude lower than those associatedwith rotating stall cells. The two types of oscillations differ fundamentally in that pure rotating stall issteady (in the proper rotating frame of reference) but not axisymmetric, while pure surge is axisymmetricbut unsteady.2 These oscillations can cause a lot of structural damage to the compressor blades and hencethey should be avoided by operating the compressor in the safe region.

At low flow rates, all compressors exhibit zones of stalled flows which propagate tangentially, with bladepassages cyclically stalling and recovering in the manner as described by Emmons et al.3 Greitzer4 developeda non linear model to describe post stall transients and introduced parameter “B” which determines the modeof compressor instability, rotating stall or surge. Stenning5 discussed the properties of rotating stall and surgeand presented a simple criteria for determining system stability. Greitzer and Cumpsty6 also formulateda model for full span stall cell propagation. The problem of rotating stall was dealt in detail by Moore.7

He analysed the conditions under which a flow distortion can occur and the results were compared withthe experiments of Day and Cumpsty.8 Moore and Greitzer2 presented a theory of post stall transientsand developed a set of third order partial differential equations for pressure rise, and average and disturbedvalues of flow coefficient, as functions of time and angle around the compressor.

In Moore Greitzer model9 it was concluded that lower values of B leads to rotating stall and higher valueslead to surge. In this paper we investigate the behavior of system at throttle values close to equilibrium

∗Undergraduate student, Dept of Aerospace Engineering, IIT Madras†Assistant Professor, Dept of Aerospace Engineering, IIT Bombay

1 of 9

American Institute of Aeronautics and Astronautics

44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit21 - 23 July 2008, Hartford, CT

AIAA 2008-4583

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

values and numerical results are presented which shows the existence of a wide variety of instabilities atthese throttle values. A map is then constructed which shows the transition from various instabilities as thegoverning parameters (B and KT ) are changed.

The paper is organised as follows:

• Section II describes the Moore Greitzer model.

• Section III describes the Galerkin approximation of governing equations and their solution procedure.

• Section IV describes the results and discussions.

II. Moore Greitzer model

The complete derivation of this model can be found in Moore and Greitzer2 and only the final equationsare presented here:

Ψ(ξ) + lcdψ

dξ= ψc(Φ − Yθθ) −mYξ +

1

2a(2Yξθθ + Yθθθ) (1)

Ψ(ξ) + lcdψ

dξ=

1

2π

∫ 2π

0

ψc(Φ − Yθθ)dθ (2)

lcdψ

dξ=

1

4B2[Φ(ξ) − F−1

T (Ψ)] (3)

The dependent variables of this system of equations are the annulus averaged pressure rise coefficient Ψ(ξ),the annulus averaged axial flow coefficient Φ(ξ), and the upstream disturbance potential Y, whose axial andcircumferential partial derivatives give the local flow disturbance in the axial and circumferential directions.Independent variables include the time in wheel radians ξ and the circumferential coordinate θ. Equation (1)is a PDE in ξ and θ obtained from the momentum balance of the system evaluated at the compressor face, (2)is an ODE in ξ which results from averaging out the circumferential dependence in (1) and (3) is an ODE inξ which results from a mass balance of the plenum. Note the subscripts ξ and θ denote partial differentiation.The compressor characteristic ψc(φ) is the response of the compressor for steady axisymmetric flow. For ouranalysis, we will use the general Moore Greitzer2 cubic form shown in Fig. 1 where the cubic charateristicis shown along with the throttle curves for different KT .

ψc(φ) = ψc0 +H[

1 +3

2

( φ

W− 1

)

−1

2

( φ

W− 1

)3]

(4)

The throttle characteristic FT (Φ) represents the pressure loss across the throttle, assumed parabolic

ΨT =1

2KT Φ2

T (5)

In our analysis we will focus on the effect of two parameters, KT and B, on the performance characteristicsof a compressor in post stall transients. The throttle coefficient KT adjusts the position of the throttle,

hence the operating point of the system. B (= U2as

√

Vp

AcLc) is the Greitzer stability parameter defined in ref.4

which determines whether a given compressor is more likely to enter surge or rotating stall.

III. Galerkin Approximation and Solution Procedure of Governing Equations

In a Galerkin procedure, the solution of a differential equation is represented by a suitable sequence ofbasic functions. In this study, we wish to emphasize on the amplitudes of disturbances and their integratedeffects. Thus the information that is of most interest to us is just that information which should be at leastsensitive to the the higher harmonics of the stall cell desciption. We then proceed to develop a single termGalerkin method for solving the governing equations.

2 of 9

American Institute of Aeronautics and Astronautics

φ

ψ

−0.5 0 0.5−0.25 0.25 0.75 10

0.2

0.4

0.6

0.8

1

Cubic characteristic

KT=6.5

KT=5.28

Figure 1. Cubic compressor characteristicadopted from2

We begin by representing Y by a single harmonic function ofunknown amplitude A

Y = WA(ξ)sin(θ − r(ξ)) (6)

By using this approximation, equations (1), (2) and (3) become

dΦ

dξ=

[

−Ψ − ψc0

H+ 1 +

3

2

( Φ

W− 1

)(

1 −2

2J)

−1

2

( Φ

W− 1

)3]H

lc(7)

dJ

dξ= J

[

1 −( Φ

W− 1

)2−

1

4J] 3aH

(1 +ma)H(8)

dΨ

dξ=W/H

4B2

[ Φ

W−

1

WF−1

T (Ψ)]H

lc(9)

J(ξ) ≡ A2(ξ)This is a system of 3 simultaneous differential equations in ξ

which can be solved using a Runge Kutta method. Hence the sys-tem of equations (7), (8) and (9) were solved in MATLAB usingODE45 solver which uses a fourth order Runge Kutta method. Thevalue of parameters used is given as follows:

W = 0.25, H = 0.18, ψc0 = 0.3, a = 0.2857,m = 1.75, lc = 8

B and KT were kept as variable parameters. The equilibrium values of Ψ and Φ were taken as 0.66 and0.5 respectively. These values were taken from2 and9 where most of the calculations were carried out at athrottle setting of KT = 6.5. At this KT value, the compressor exhibited rotating stall for lower values of Band surge at higher B values, an observation similar to that of classical Moore Greitzer model.2 The detailedresults are discussed in following sections.

0 500 1000 1500 20000

1

2

3

4

Non Dimensional time ξ

Ψ

Φ

J

(a)

0 500 1000 1500 2000−1

0

1

2

3

4

Non dimensional time ξ

ΨΦJ

(b)

Figure 2. Time evolution of axial flow coefficient (Φ), pressure coefficient (Ψ) and non axisymmetric disturbance (J).(a) Rotating stall at B=1.0, KT = 6.5, (b) Surge at B=1.53, KT = 6.5

3 of 9

American Institute of Aeronautics and Astronautics

0.1 0.2 0.3 0.4 0.50.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Φ

Ψ

(a)

−0.5 0 0.5 10.2

0.3

0.4

0.5

0.6

0.7

Φ

Ψ

(b)

Figure 3. Phase plots of flow coefficient (Φ) and pressure coefficient (Ψ). (a) Rotating stall at B=1.0, KT = 6.5, (b)Surge at B=1.53, KT = 6.5

Figure 4. Map showing zones of instabilities

4 of 9

American Institute of Aeronautics and Astronautics

IV. Results & Discussion

IV.A. Normal Rotating Stall and Surge

Typical results for a disturbance amplitude of 1% of Φ are shown in Fig. 2. It is clear that in rotating stall(Fig. 2(a)), the pressure rise coefficient (Ψ) and flow coefficient (Φ) become steady after some time while theamplitude of non axisymmetric oscillation (J) becomes constant which is indicative of the stall cell rotation.In case of surge (Fig. 2(b)), the oscillation does not die out over a period of time and it is a low frequencyoscillation while the amplitude of non axisymmetric disturbance (J) dies out quickly. Phase plots of Ψ andΦ corresponding to Fig. 2 are shown in Fig. 3 which show that surge (Fig. 3(b)) exhibits a limit cyclebehavior while rotating stall (Fig. 3(a)) settles down to a fixed point.

When the solutions were obtained at lower values of KT , a different kind of behavior was observed. Theequilibrium value of KT is 5.28 (calculated from eqn.(9)) and as the throttle is gradually closed i.e. KT isincreased, the compressor shows a variety of instabilities. This type of behavior is seen for values of KT

ranging from the equlibrium value i.e. 5.28 to approximately 5.87. As B parameter is increased from 0.5to 1.2 for a particular value of KT , there exists many different types of instabilites which were observed todisappear at higher values of KT . Some of the instabilities show up when the equations are solved for alonger time. They include a variety of phenomena like “non recoverable stall” in which the oscillations of Ψ,Φ and non axisymmetric disturbance (J) do not die out as time progresses.

These instabilities have been observed and investigated for the first time. Some instances of non re-coverable stall type of behavior have also been reported by Alan et al,10 McCaughan11 and Zheng at al.12

However a detailed investigation of these modes of instabilities has not been reported in the literature. Adetailed parametric study between B and KT gives a clear insight into these new modes of instabilities andhow the transition from one mode to other takes place.

0 500 1000 1500 20000

1

2

3

4

Non dimensional time ξ

Ψ

Φ

J

(a)

0 500 1000 1500 20000

1

2

3

4

Non dimensional time ξ

ΨΦJ

(b)

Figure 6. Time evolution of Φ, Ψ and J showing the transition from normal rotating stall in (a) to non recoverable stallin (b)(a) B=0.687, KT = 5.7, (b) B=0.689, KT = 5.7

In some of these modes, random spikes are seen in J evolution whose amplitude can sometimes be ofthe order of 109. These spikes in J have a fine structure as discussed in following sections. As B is furtherincreased from the large amplitude oscillations boundary (Fig. 4) surge like oscillations start to occur.However J oscilations do not die down and are still very large compared to flow coefficient (Φ) and pressurecoefficient (Ψ). Ocassionally some unusual peaks are observed in flow coefficient (Φ). In Fig. 4, a map ofB and KT is constructed which shows different zones of instabilities. Table 1 shows the operating pointsand the corresponding modes of instability are marked on the map. We can clearly see that for lower valuesof KT , there exists many instabilities which may be a combination of surge and rotating stall and they

5 of 9

American Institute of Aeronautics and Astronautics

disappear for higher values of KT and there is a single boundary between rotating stall and surge. Theseinstabilities are discussed in detail in following sections.

Table 1. Table of operating points

B KT Instability Mode Reference

� 1.0 6.5 Rotating Stall 9

/ 2.0 6.5 Surge 9

� 1.0 6.5 Rotating Stall Fig. 2(a)

♦ 1.53 6.5 Surge Fig. 2(b)

5 0.75 5.6 Non Recoverable Stall Fig. 5

? 0.687 5.7 Rotating Stall Fig. 6(a)

? 0.689 5.7 Non Recoverable Stall Fig. 6(b)

4 0.85 5.7 Large Amplitude Oscillation Fig. 8

4 0.86 5.7 Large Amplitude Oscillation Fig. 9

IV.B. Instability Map

0 500 1000 1500 20000

0.5

1

1.5

2

2.5

3

3.5

4

Non dimensional time ξ

ΨΦJ

Figure 5. A typical behavior of Non recoverablestall at B=0.75 and KT = 5.6

A map shown in Fig. 4 was constructed which shows the zonesof various instabilities based on KT and B. The time evolutionof Ψ, Φ and J was studied with B and KT variation. Movieswere made to study the change in evolution with a change inB at different values of KT . The B parameter and KT werevaried in intervals of 0.05 and 0.01 respectively and the map wasconstructed according to the change in the system behavior.

Four zones of different types of instabilities were identifiedas follows:

• Rotating stall

• Non recoverable stall

• Large amplitude oscillations in J

• Surge

Besides these zones of instabilities two more zones of stableoperation and indistinguishable boundary region are also indi-cated in Fig. 4. Indistinguishable boundary region correspondsto the zone where the boundaries for different instabilities arevery difficult to mark.

The transition from rotating stall to non recoverable stall was noted as the J evolution changed frombeing steady after some oscillations to indefinite oscillatory behavior. The value of B corresponding totransition increases gradually as KT increases.a This type of behavior continues for some range of B andthen large oscillations in J start to occur. This region has random spikes in J and Φ. After this region thenormal surgelike behavior starts and J oscillations start dying. The map clearly shows the non recoverablestall region and large oscillation region merging together and there is a direct transition from rotating stallto surge after KT = 5.87. KT values very close to equilibrium value i.e. 5.28 are not included in this mapbecause the behavior in that region is random and it is very difficult to distinguish and mark the boundariesfor transition from one mode to other.

Most of the previous studies (e.g. KT = 6.5 in ref.9) on post stall transients have been carried out athigher throttle values i.e. KT around 6 or more than that. In that regime of KT , these modes of instabilities

aFor lower values of KT , this transition is not clear cut value of B instead there is a narrow intermix of the two zones. Athigher values this zone vanishes and there is a clear transition from non recoverable stall to high oscillation region.

6 of 9

American Institute of Aeronautics and Astronautics

do not exist. However in practical situations as the compressor is throttled down from equilibrium value.Therefore, there is a possibility that these modes can appear during the operation of a compressor. Hencethe operation of the compressor should be avoided in this zone.

IV.C. Non Recoverable stall

0.2 0.3 0.4 0.5 0.6

0.4

0.5

0.6

0.7

0.8

Φ

Ψ

(a)

0.2 0.3 0.4 0.5 0.6

0.4

0.5

0.6

0.7

0.8

Φ

Ψ

(b)

Figure 7. Phase plots of Φ and Ψ showing the transition from fixed point evolution in (a) to a limit cycle behavior in(b)(a) B=0.687, KT = 5.7, (b) B=0.689, KT = 5.7

In rotating stall, the annulus averaged flow coefficient (Φ), pressure rise coefficient (Ψ) and the amplitudeof non axisymmetric disturbance (J) become constant after some time. However in non recoverable stall Φ,Ψ and J do not become steady, they keep on oscillating indefinitely.

A typical behavior of non recoverable stall is shown in Fig. 5. This plot indicates that during a cycle, theamplitudes of Φ and Ψ decrease to a very low value when J begins to rise and as J goes down, Φ and Ψ againrises to the equilibrium values and oscillate around those values till J is zero and then again decrease as Jincreases. This feature of non recoverable stall appears when it is fully developed. At lower values of KT , asthe transition to high amplitude oscillations approaches, a marked decrease in the frequency of oscillationsis also observed.

0 500 1000 1500 2000−5

0

5

4

3

2

1

−1

−2

−3

−4

Non dimensional time ξ

Ψ

Φ

J

4205.3

−5.7x106−25.8

−21475

−3.8x109

Figure 8. Time evolution of Φ, Ψ and J in theregion of Large Oscillations instability mode atB=0.85 and KT = 5.7. The values of spikes areshown besides the corresponding spikes.

This instability disappears at approximately KT = 5.88and after that normal rotating stall and surge occurs. Fig. 6shows the change from normal rotating stall (Fig. 6(a)) to nonrecoverable stall ((Fig. 6(b))). At B = 0.687 the amplitude ofall the quantities become constant after some time while atB =0.689 all the quantities keep oscillating. The respective phaseplots of Φ and Ψ are also shown in Fig. 7. These phase plotsshow how the system evolution goes to a limit cycle (Fig. 7(b))from a fixed point (Fig. 7(a)) as it goes into non recoverablestall.

IV.D. Large Amplitude Oscillation in J

After the limit of non recoverable stall is crossed, another inter-esting phenomena is observed. The time evolution of J showsvery large spikes in J. These spikes can sometimes be of theorder of 109. J suddenly rises or falls down very steeply. Theflow coefficient Φ also shows some spikes at some instances.Typical curve at B = 0.85 and KT = 5.7 for various quantities

7 of 9

American Institute of Aeronautics and Astronautics

0 500 1000 1500 2000−1.5

−1

−0.5

0

0.5

1

Non dimensional time ξ

Ψ

Φ

(a)

0 500 1000 1500 2000−80

−60

−40

−20

0

20

Non dimensional time ξ

(b)

Figure 9. Another type of behavior of Large oscillations instability mode at B=0.86 and KT = 5.7(a) Oscillations in Φ and Ψ, (b) Oscillations in J

are shown in Fig. 8. The magnitudes of spikes are indicated inthe figure. Another example of this behavior is shown at same KT and B = 0.86 in Fig. 9.

−1.5 −1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Φ

Ψ

Figure 10. Phase plot of Φ and Ψ

showing a strange limit cycle behav-ior at B=0.86 and KT = 5.7

We can see from the plots that Ψ goes down almost to zero whenthe spikes appear in J while Φ can even become negative which is acharacteristic of surge. Phase plot at B = 0.86 is shown in Fig. 10 whichshows that system exhibits a limit cycle behavior. It is evident fromthese figures that how the behavior changes with a small increment of 0.1in B. After B = 0.86, complex solutions were obtained till B = 0.88 andthen the spikes in J start going down and normal surgelike behavior isseen after that. This complex solution regime occurs near the boundaryof large oscillation regime and surge. It is not shown in the map becauseit occurs for a very small range of B and KT and a clear distinctioncannot be made to draw a boundary.

These spikes arise at random time intervals and these time intervalsare of the order of milliseconds. After that all the quantities oscillatenormally. If we see the fine structure of these peaks, we find that it risesvery sharply and goes off to zero in a step manner. After this intervalof B, normal surgelike behavior is seen with J amplitude going down slowly and finally after some initialoscillations, it dies down completely.

V. Conclusions

This study has presented a detailed investigation of the Moore Grietzer model with the first order Galerkinapproximation at throttle values close to the equilibrium value. It was found that there exists some newtype of instabilities at certain values of KT which disappear at higher throttle values. Numerical resultsare shown which show various instabilities like non recoverable stall and other high amplitude oscillationsexist in some range of B and KT values. At throttle settings very close to the equilibrium value, there areno clear boundaries for these instabilities. After a particular value of KT , these instabilities disappear andthere exists a clear boundary between rotating stall and surge. A map was constructed which shows variouszones of instabilities.

8 of 9

American Institute of Aeronautics and Astronautics

Acknowledgments

The authors would like to thank IRCC (Industrial Research & Consultancy Centre), Indian Institute ofTechnology Bombay, for funding this work through the summer fellowship program. We also like to thank Dr.S.K. Sane (ex Professor, Department of Aerospace Engineering, IIT Bombay) for his fruitful discussions.

References

1E.M. Greitzer. The stability of pumping systems – the 1980 freeman scholar lecture. Journal of Fluids Engineering, 103,1981.

2F.K.Moore and E.M. Greitzer. A theory of post-stall transients in axial compression systems: Part 1 –development ofequations. Journal of Engineering for Gas Turbines and Power, 108, 1986.

3H.W. Emmons, R.E. Kronauer, and J.A. Rockett. A survey of stall propagation – experiment and theory. Journal of

Basic Engineering, 1959.4E.M. Greitzer. Surge and rotating stall in axial flow compressors: Part –1,2. Journal of Engineering for Power, 98, 1976.5A.H. Stenning. Rotating stall and surge. Journal of Fluids Engineering, 102, 1980.6E.M. Greitzer and N.A. Cumpsty. A simple model for stall cell propagation. Journal of Engineering for Power, 104,

1982.7F.K. Moore. A theory of rotating stall of multistage axial compressors part –1,2,3. Journal of Engineering for Power,

98, 1976.8I.J. Day, E.M.Greitzer, and N.A. Cumpsty. Prediction of compressor performance in rotating stall. Journal of Engineering

for Power, 100, 1978.9F.K. Moore and E.M. Greitzer. A theory of post-stall transients in axial compression systems: Part 2 –application.

Journal of Engineering for Gas Turbines and Power, 108, 1986.10Alan Champneys, Csaba Hos, and Laszlo Kullmann. Bifurcation analysis of surge and rotating stall in the moore greitzer

compression system. IMA Journal of Applied Mathematics, 68, 2003.11F.E. McCaughan. Bifurcation analysis of axial flow compressor instability. SIAM Journal on Applied Mathematics, 50,

1990.12Minghong Li and Qun Zheng. Wet compression system stability analysis part –1,2. Proceedings of ASME Turbo Expo

2004 Power for Land, Sea, and Air June 14-17, 2004, Vienna, Austria, 2004.

9 of 9

American Institute of Aeronautics and Astronautics