Hysteresis behaviors of compressor rotating stall with cusp catastrophic model

Zhiping LI,Peng ZHANG,Tianyu PAN,d,*,Qiushi LI,Jian ZHANG

aNational Key Laboratory of Science and Technology on Aero-Engine Aerothermodynamics,Beihang University,Beijing 100083,China

bCollaborative Innovation Center of Advanced Aero-Engine,Beihang University,Beijing 100083,China

cSchool of Energy and Power Engineering,Beihang University,Beijing 100083,China

dDepartment of Mechanical Engineering and Materials Science,Duke University,Durham,NC 27708,USA

1.Introduction

When the flow rate through a compressor is throttled gradually and the stall limit is reached,the essentially steady,axisymmetric flow becomes unstable.The result of this instability is very often manifested as a phenomenon known as rotating stall,which is an asymmetric phenomenon with one or several stall cells rotating at a fraction of the rotor speed while the overall mass flow rate remains nearly constant once the pattern is fully developed.1,2

Before stalling,two types of stall inception are often detected:model waves and spike.3–7The modal wave is characterized by the gradual growth of small-amplitude,essentially two dimensional,long wavelength disturbances.The spike is a three dimensional disturbance,which is localized at the tip region of a specific rotor in a multistage compressor,and has a length scale on the order of several blade pitches.According to the change of whole characteristic of the compressor in stall process,the types of rotating stall can also be divided into gradual stall and abrupt stall.For the early compressors,8in the stall process,the pressure rise often progressively changes as the flow rate increases and decreases.As the design loading of compressor blade increases,the stall-type of current mainstream compressors is generally abrupt stall;the pressure rise and the flow rate can be markedly reduced,and the efficiency can also drop sharply.In addition,a larger throttle opening is required to move the operating point from rotating stall to normal working condition than throttling at stall inception.These are the catastrophe and hysteresis of abrupt stall,the potentially serious threats to engine reliability.

For the catastrophe and hysteresis of compressor rotating stall,scholars have made some related researches.Day and Cumpsty9studied the hysteresis of compressor rotating stall through changing the design value of the flow coefficient,and found that the size of hysteresis loop will gradually decrease as the design value of the flow coefficient decreases.Copenhaver and Okiishi10tested the overall recoverability of a 10-stage compressor,and the results showed that higher shaft speeds cause low recoverability.The hysteresis of compressor rotating stall was first studied through numerical simulation by Choi et al.11,12They found that rotating stall at each operating point during recovery is stable,and rotating stall becomes unstable to be a transient after stall cells become too small to block the flow.Day et al.13introduced the blockage coefficient to assess the hysteresis of compressor rotating stall and found the compressor will recover from rotating stall when the blockage coefficient is less than 30%.From the point of structure stability,Abed et al.14and McCaughan15analyzed the characteristic of stall hysteresis with the help of bifurcation theory.They regarded the hysteresis of rotating stall as the system bifurcation.Liaw and Abed16applied the bifurcation theory to the active control of compressor stall inception,and eliminated the undesirable jump and hysteresis behavior of the uncontrolled system.

A series of studies have shown that the hysteresis behaviors of the compressor stall are affected by multi parameters,and determining the contributing factors and characteristic rule of hysteresis is essential to guide the design and control of the compressor.The catastrophe and hysteresis are related to the concept of ‘bifurcation” in mathematics.Currently,with the advancement of nonlinear dynamics,bifurcation theory has been applied to analyze the compressor rotating stall.14–19However,the bifurcation theory can only consider the effects of a single parameter.In 1983,the French mathematician Thom20first proposed catastrophe theory based on the bifurcation theory.The catastrophe theory can describe the effects of more control parameters on a system and has been widely used in the field of nonlinear dynamics.21–25Therefore,an idea to buildamodeltodescribethehysteresisbehaviorsofcompressor stall under the impact of multiple parameters is proposed based on the catastrophe theory.

This paper is organized as follows.Firstly,the equilibrium points of compressor system are determined based on Moore-Greitzer(M-G)model in Section 2.Then,in Section 3,the contributing factors of the stall hysteresis are analyzed and the physical mechanism of catastrophe and hysteresis is discussed through assessing the stability of the equilibrium points by Liapunov’s theorem.Finally,according to topological invariant rules,the equilibrium surface equation of compressor is developed based on the standard cusp catastrophic model,and it is used to describe the diverse hysteresis behaviors of compressor rotating stall along different control routes in Section 4,which is then followed by conclusions(Section 5).

2.Equilibrium state of compressor system

The basic compressor system under study is shown in Fig.1.This compressor system consists of a compressor with an inlet duct upstream and outlet duct downstream,followed by a plenum of relatively big volume and an exhaust pipe with a throttle valve at the exit.The simplest model which adequately describes the dynamics of rotating stall and surge in axialflow compressor systems shown in Fig.1 is the Moore-Greitzer model.26,27The full model is described in detail in the references and so we move straight to the simpli fied model.28The differential equations are as follows:

where the variable φ is nondimensional mass flow coefficient,which has been shifted so that zero mass flow actually occurs at φ=-1,and rescaled with W.W is compressor characteristic semi-width.ψ is the nondimensional pressure rise of the compressor.Both of these variables are averaged over the annulus of the compressor.H is compressor characteristic semi-height.ξ refers to time for wheel to rotate one radian.The parameter ψc0is the shutoff head,and it is proportional to the number of stages in the compressor.J is the square of the amplitude of the first mode of the rotating stall disturbance,so it only has physical meaning when it is positive.lcrefers to the total aerodynamic length of compressor and ducts.The parameter B in Eq.(2)is Greitzer’s B parameter,1,2which determines the type of compressor instability.When the B is small,the flow usually develops into rotating stall.The variable φTrepresents the mass flow leaving the plenum and exiting through the throttle duct.m is the compressor-duct flow parameter,and a is the reciprocal time-lag parameter of blade passage.The detail descriptions of these parameters are available in the Refs.26,27

When dψ/dξ =0,dφ/dξ =0,dJ/dξ =0 and J≠0,the rotating stall characteristic of compressor,ψe,can be expressed as Eq.(5),and is also shown in Fig.2.

The pressure rise through the throttle,ψT,is modeled by a simple parabolic relationship.

where k is throttle coefficient,which is directly proportional to the cross-sectional area of the throttle,k′is the corrected throttle coefficient.The equilibrium points of compressor system are the intersections of the throttle line with the cubic axisymmetric characteristic line or the rotating stall characteristic line,such as points A′,B′,C′and D′,as shown in Fig.2.When the equilibrium point is stable,it represents the steady axisymmetric flow or the steady rotating stall,where the angle-averaged mass flow coefficient,the pressure rise coefficient,and the rotating stall amplitude are steady.

Thus the line of equilibrium points,traced out as the throttle setting is varied,represents the characteristic of compressor system.The equilibrium state of compressor system can be divided into three regions(I,II and III)by coefficientsandandare the critical corrected throttle coefficient).There is only one equilibrium point in Region I and Region III,but there are three equilibrium points in Region II and two equilibrium points at boundaries.This is a bifurcation phenomenon of the system,and the size of region II(A′-B′-C′-D′)can be used to measure the size of hysteresis loop of compressor rotating stall.

3.Analysis of catastrophe and hysteresis of rotating stall

3.1.Contributing factors of hysteresis

As discussed above,Region II represents the hysteresis loop,so that the size of hysteresis loop can be measured byBased on the Eqs.(4)–(6),the function to calculate the size of hysteresis loop can be expressed as

where the parameters Mpand Npare defined to make the equation more concise.Early in 1978,through a lot of experiments,Day et al.13has proved that the shutoff head ψc0is largely independent of the compressor and almost unchanged,so it can be found by analyzing Eq.(7)that the main contributing factor of the hysteresis is the compressor characteristic semiheight H,which is related to the design blade loading,inlet distortion,Reynolds number of air flow,design flow coefficients and so on.Within a certain range of H,the value ofwill decrease with the decrease of H.By calculation,whenthewill be less than 0,which means that when ψc0/H exceeds a certain threshold,the stall recovery process of compressor will not have the catastrophe and hysteresis,and the stall will change from ‘abrupt stall” to ‘gradual stall”,as shown in Fig.3(a)–(c).

Fig.3,the stalled and unstalled characteristic of compressor for different ψc0/H based on Moore-Greitzer model,shows the effect of ψc0/H on hysteresis through keeping ψc0unchanged and changing H.When ψc0/H=0.5,the whole process of compressor rotating stall has obvious characteristics of catastrophe and hysteresis,as shown in Fig.3(a),which represents the nonlinear characteristic of a high-loading compressor.At this time,with the increase of ψc0/H,the characteristic of compressor will change correspondingly.When ψc0/H=1,the hysteresis of compressor stall is significantly reduced,but the obvious hysteresis loop can still be found in the stall recovery process,as shown in Fig.3(b).Further increasing the ψc0/H,when ψc0/H=4.5,as shown in Fig.3(c),hysteresis loops disappear and the whole process of compressor rotating stall no longer has the hysteresis,and the pressure rise will progressively change as the throttle coefficient increases and decreases,which represents the characteristic of a low-loading compressor.

The influence laws of parameter ψc0/H on the performance of compressor obtained in this paper are consistent with the experimental results of Day and Cumpsty,9who have studied the characteristics of the compressor for different design flow coefficients.They found,as the design flow coefficient is raised,the unstalled pressure rise increases,but so does the extent of the hysteresis.When the value of design flow coefficient is 1(ψc0/H=0.67),the whole process of compressor rotating stall has obvious characteristics of catastrophe and hysteresis.But when the value of design flow coefficient is 0.35(ψc0/H=4.7),the whole process of compressor rotating stall no longer has the hysteresis.

3.2.Physical mechanism of catastrophe and hysteresis

The parameter k′can control the working conditions of compressor(stall or unstall),and the parameter ψc0/H can affect the hysteresis of compressor rotating stall and change the type of rotating stall(abrupt stall or gradual stall).In order to study the physical mechanism of the catastrophe and hysteresis,the stability of the equilibrium points of compressor system will be analyzed at different ψc0/H based on Liapunov’s theorem in this part.

The Jacobian derivative of the Eqs.(1)–(3)is expressed as Eq.(8).According to Liapunov’s theorem,the stability of the equilibrium points can be determined through eigenvalues of the Jacobian derivative.

The equilibrium points of compressor system for ψc0/H=0.5 are shown in Fig.4(a).The equilibrium point A′is taken into Eq.(8),and the associated eigenvalues can be calculated.When the throttle is opened enough,the eigenvalues of point A′all have negative real part and the steady axisymmetric flow is stable.Calculation of the eigenvalues associated with the equilibrium point B′shows that its Jacobian derivative has two negative eigenvalues and one positive eigenvalue,which means that the equilibrium point B′is unstable to the small perturbation.So,the point B′can be regarded as the unreachable point of the system,where the stall cell has not yet fully developed.The remaining fixed point is the equilibrium point C′,and its Jacobian derivative has one negative real eigenvalue and a complex pair.For small values of Greitzer’s B parameter,the complex pair is negative,and the equilibrium point C′is stable.But at larger values of Greitzer’s B parameter,the complex pair is positive and hence the equilibrium point C′is unstable.The compressor rotating stall(the value of Greitzer’s B parameter is small)is only discussed here,so the equilibrium point C′is stable,where the stall cell has fully developed.Whenthe equilibrium point B′meets the equilibrium point A′,and this point is unstable to perturbations of rotating stall,so this point is the first point where theaxisymmetric flow state firstly loststability.Whenthe equilibrium point B′meets the equilibrium point C′,and this point is unstable to perturbations of the increase of mass flow,so this point is the first point where the rotating stall state firstly lost stability.

The equilibrium points of the compressor system for ψc0/H=4.5 are shown in Fig.4(b).The eigenvalues of the equilibrium point A′all have negative real part and the steady axisymmetric flow is stable.Similarly,Calculation of the eigenvalues associated with the equilibrium point B′shows that its Jacobian derivative all have negative real part,which means that the equilibrium point B′is stable to the small perturbation.Jacobian derivative of the equilibrium point C′has one negative real eigenvalue and a complex pair.For small values of Greitzer’s B parameter,the complex pair is negative,and the equilibrium point C′is also stable.The stall cells have fully developed at points B′and C′.So whenthe equilibrium point is invariably stable to the perturbations.Thus,the catastrophe and hysteresis of compressor system stall is caused by the instability of the equilibrium points.As the parameter ψc0/H increases,the range of unstable equilibrium points will decrease,so the size of hysteresis loop will also decrease.

The stability at the equilibrium points are mainly affected by the relative magnitudes of the slope of compressor characteristic and throttle characteristic.Normal operation might take place at a point such as point A′(Fig.4(a))which is unconditionally stable.This can be seen by considering a small reduction in mass flow,and will lead to an increase in the compressor pressure rise and a fall in the pressure drop across the throttle so that the flow is accelerated and therefore increases until the original equilibrium is restored.For a small change at point B′(Fig.4(a)),a reduction in mass flow leads to a greater decrease in the pressure rise of the compressor than the throttle pressure drop so that the flow will not be accelerated back to the former equilibrium,which means the equilibrium point B′for ψc0/H=0.5 is not stable.However,for ψc0/H=4.5(Fig.4(b)),a reduction in mass flow leads to a greater decrease in the throttle pressure drop than the pressure rise of the compressor so that the flow will be accelerated back to the former equilibrium,which means equilibrium point B′is stable in this state.

The study about the effect of the slope of throttle characteristic on the catastrophe and hysteresis of compressor rotating stall has been introduced by Day et al.13The single stage compressors of three different hub/tip ratios from 0.75 to 0.875 are studied.With the usual throttle exhausting to atmosphere there is a discontinuous characteristic,as shown in Fig.5(a).The use of suction behind the throttle valve,however,changes the characteristic into a continuous curve,as shown in Fig.5(b).

Based on the analysis for the stability of the equilibrium points,a potential function approach shown in Fig.6 can be taken to interpret the hysteresis behavior of compressor rotating stall.The hysteresis curves consist of line M-P-R-Q and line N-Q-S-P.The equilibrium point B′(Fig.4(a))represents the maximum potential energy point of the system,which is an unstable point.The equilibrium points A′and C′(Fig.4(a))represent the minimum potential energy points of the system,which are stable points.When the state point is in M,the system only has a minimum potential energy,and the system potential energy is represented by the ball in Fig.6(b)(abscissa L is the location of equilibrium point and ordinate E is the potential energy of system).The compressor is in the axisymmetric flow state,and will not go into the rotating stall state under the disturbance.When the state point is in P or R,another minimum potential energy of system appeared.Because of the barrier between the two states,the compressor will still maintain the unstall state under small disturbance.However,it has the potential to switch from the current state to rotating stall state under large disturbance.When the state point is in Q,the barrier between the two states has gradually disappeared(equilibrium point B′meets equilibrium point A′as shown in Fig.4(a)),so the compressor will switch from the unstall state to rotating stall state under small disturbance.Similarly,when the state point arrives at S,another minimum potential energy of the system appeared again,because of the barrier between the two states,and the compressor will maintain the rotating stall state under small disturbance and switch to unstall state under large disturbance;when the state point is in P,the barrier between the two states will also disappear(equilibrium point B′meets equilibrium point C′as shown in Fig.4(a)),and the compressor will recover to the axisymmetric flow state under small disturbance.

Through the analysis of Fig.6,the potential function approach can be used to interpret the hysteresis behavior of general bistable systems such as compressor rotating stall.So if the potential function of the compressor system can be established,the catastrophe and hysteresis of compressor stall can be described through the model.In view of similar physical phenomena,the French mathematician Thom20first proposed catastrophe theory,which has been widely used in the field of nonlinear dynamics through developing the potential function of systems.Therefore,this paper will try to establish the model description of catastrophe and hysteresis of rotating stall with the help of catastrophe theory.

4.Model description of compressor’s hysteresis behaviors

4.1.Topological property of compressor stall

According to the theory of nonlinear dynamic systems,the catastrophe points correspond mathematically to the singular points.Whitney29has proposed that there are general rules in the distribution of singular points in the perturbation space,that is,the topological character of singular points is invariant under smooth mapping,called ‘topological invariant rules”.The invariance property is inherent to Thom’s classification theorem,20with strict mathematical analysis,and Thom has proved an important mathematical theorem:when the continuous variation factors which lead to catastrophe are fewer than four,the various catastrophic processes in nature can be described by seven elementary catastrophic models.The elementary catastrophic models can be described by ndimension Taylor expansions:

where V(x，u，v，···)is the potential function,CG(x)is the catastrophe germ,PT(x，u，v，···)is the universal perturbation function with an m dimension space of parameters,x is the state variable of system,and u，v，···are the control variables of system.The distinction among the various types of catastrophic models is based on the specific form of the catastrophe germs.The value of i in catastrophe germ xi+1is the maximum number of system’s solutions for any constant external input.This notion can be used to determine the singularity type of the present physical system.According to the above analysis,the maximum number of equilibrium points of compressor system is 3.So that the singularity type of compressor system can be chosen through the form of catastrophe germ(i=3),which can be given by

For compressor system,the conversion between the two states is affected by two parameters:k′and ψc0/H.According to Thom’s classification theorem,the corresponding perturbation term can be described as

Thus the potential function can be given by

which corresponds to the standard cusp catastrophic model.Eq.(13),which comprises all equilibrium points,namely equilibrium surface equation,is obtained by the differentiation of Eq.(12)with respect to x.

The space con figuration of the equilibrium surface is shown in Fig.7(a).The upper leaf and lower leaf respectively represent two different stable working conditions of the system.As the control variables of u and v change,the system state will transform between the upper leaf and lower leaf.When the upper leaf is transformed to the lower leaf,the system state will show characteristic of catastrophe and the boundary of catastrophe is ATCTshown in Fig.7(a);when the lower leaf is transformed to the upper leaf,the system state will show similar characteristic of catastrophe and the boundary of catastrophe isThe two-step changes taking place in different boundaries,thus called ‘hysteresis”.The boundaries ATCTand ATBTcan be acquired through the projection of boundariesandrespectively,as shown in Fig.7(a).Furthermore,the conditions for the catastrophe points shown in Fig.7(a)can be obtained by differentiating Eq.(13)once again,solving for x in terms of u and v,and substituting the result into Eq.(13).The result is Eq.(14).20

Through analyzing the topological structure of cusp catastrophic model,as shown in Fig.7(a),when the control variable u＜0,cusp catastrophic model can show the obvious catastrophe and hysteresis as the change of v,with the decrease of the absolute value of u,the size of hysteresis loop will gradually decrease.When u＞0,the cusp catastrophic model no longer shows the obvious hysteresis.Projecting the catastrophe points onto the control surface which is composed of v and u,the catastrophe boundary is a cusp-shaped curve.

The study of the contributing factors of stall hysteresis has shown that the mass flow coefficient φ or pressure rise coefficient ψ can show the catastrophe and hysteresis as the change of k′and ψc0/H.Fig.7(b)shows the change rule of φ as the change of k′and ψc0/H based on Moore-Greitzer model.When ψc0/H ＜ 4,the mass flow coefficient φ can show the obvious catastrophe and hysteresis as the change of k′,with the increase of ψc0/H,and the size of hysteresis loop will gradually decrease.When ψc0/H ＞ 4,the mass flow coefficient φ will progressively change as the parameter k′increases and decreases.Projecting the stall points and recovery points onto the control surface which is composed of ψc0/H and k′,the catastrophe boundary is also a cusp-shaped curve.Thus the compressor rotating stall and the cusp catastrophic model have exactly the same topological properties,and the compressor rotating stallcan bemodeled asa cusp catastrophic phenomenon.

4.2.Model description of compressor’s hysteresis behavior along different control routes

To simplify the research work,the hysteresis behaviors of compressor stall will be described through ‘standard” cusp catastrophic model.According to the characteristic of compressor rotating stall and cusp catastrophic model,the mass flow coefficient φ or pressure rise coefficient ψ which can represent the state of system is chosen as the state variable,and the control variables are the parameter k′and the parameter ψc0/H because both can influence the state of the system.Taking the mass flow coefficient φ as an example,the relationship between the state variable of cusp catastrophic model and the state variable of compressor system can be described as:

and the relationship between the control variables can be given by:

then the equilibrium surface equation of compressor system is described as:

and the bifurcation set equation of compressor system can also be obtained:

According to Eq.(18),Fig.8 presents the bifurcation set of compressor system(line B′-A′-C′)in the control surface which is spanned by the parameter k′and the parameter ψc0/H.The bifurcation set is composed of the stall points and recovery points,where the catastrophe and hysteresis will occur.In order to investigate the hysteresis behaviors of compressor rotating stall,several typical control routes around the bifurcation set are available as shown in Fig.8,and each route has two directions to determine whether hysteresis exists.Fig.9 present the model description of the catastrophe and hysteresis of compressor stall based on the equilibrium surface Eq.(17)along different control routes.

The hysteresis behavior of compressor rotating stall along the control Routes I and II are shown in Fig.9(a)and(b).The solid lines are model lines based on Eq.(17)and the dash lines are the characteristic of compressor based on Moore-Greitzer model which are used as reference lines.When ψc0/H=0.5,as the coefficient k′increases,the flow rate will gradually decrease,as shown in Fig.9(a).When the coefficient k′increases to the critical value k′1,the flow rate will abruptly decrease and the compressor will go into stall condition.At this time,if the coefficient k′decreases,the flow rate will not retrace its original path to the original point but will gradually reach to another critical value k′2.If the coefficient k′continues to decrease,the flow rate will abruptly increase and the compressor will recover to unstall condition.When ψc0/H=4,the flow rate will progressively change as the coefficient k′increases and decreases,as shown in Fig.9(b).The whole changing process does not have obvious characteristics of catastrophe and hysteresis.Here,a discrepancy between blue lines and black lines can be seen,and the reason for the discrepancy is that the ‘standard” cusp catastrophic model of compressor system is a simplified model.

The hysteresis behavior of compressor rotating stall along the control Routes III and IV are shown in Fig.9(c)and(d),which shows the nonliner characteristics of compressor stall through keeping the value of k′unchanged and changing the value of ψc0/H.Keeping the value of k′constant means that the position of throttle remains the same,and inlet distortion,Reynolds number of air flow,change of angle of blade or other parameters can affect the value of ψc0/H.When k′=0.6ψc0,as the coefficient ψc0/H increases,the flow rate will gradually decrease,as shown in Fig.9(c).When the coefficient ψc0/H increases to the critical value(ψc0/H)1,the flow rate will abruptly decrease and the compressor enters the stall condition.At this time,if the coefficient ψc0/H decreases,the flow rate will not retrace their original path to the original point but will gradually reach another critical value(ψc0/H)2.If the coefficient ψc0/H continues to decrease,the flow rate will abruptly increase and the compressor will recover from the stall condition.However,when k′=0.375ψc0,the flow rate will progressively change as the coefficient ψc0/H increases and decreases,and the whole process of compressor rotating stall no longer has the hysteresis,as shown in Fig.9(d).

Fig.9(e)and(f)show the hysteresis behavior of compressor rotating stall along the control Routes V and VI.It can be seen from Fig.9(e)and(f)that the change in the flow rate φ is continuous along both routes,and there is no catastrophe and hysteresis along the control routes back and forth.In Fig.9(e),the compressor is always operating in stall condition,which is equivalent to the ‘lower leaf” of cusp catastrophic model as shown in Fig.7(a).In Fig.9(f),the compressor is always operating in unstall condition,which is equivalent to the ‘upper leaf”of cusp catastrophic model as shown in Fig.7(a).

Fig.9(g)and(h)shows the hysteresis behavior of compressor rotating stall along the control Routes VII and VIII.As shown in these two figures,the hysteresis behaviors of compressor are different from what is shown in Fig.9(a)and(c),where hysteresis loops are closed.Conversely,the hysteresis curves presented in Fig.9(g)and(h)are not closed.As shown in Fig.9(g),once the operation condition of compressor transits from ustall to stall,it is not possible to transit back from the stall condition to the unstall condition along Route VII.Similarly,as shown in Fig.9(h),it is not possible for the operation condition of the compressor to transit back from unstall to stall along Route VIII if it has previously transited from the stall condition to the unstall condition.

Obviously,the cusp catastrophic model can be conveniently used to describe the diverse hysteresis behaviors of compressor rotating stall along different control routes and evaluate the size of hysteresis loop,and it may be a potential method to study and control the compressor stall phenomenon.Of course,more work should be done to validate and develop this method based on further understanding of compressor stall in the future.

5.Conclusions

This paper focuses on the catastrophe and hysteresis of compressor rotating stall.The stability of the equilibrium points is analyzed through Liapunov’s theorem,and the contributing factors of the hysteresis of compressor rotating stall are discussed.Based on this,the cusp catastrophic model is used to describe the diverse hysteresis behaviors of compressor along different control routes.The main conclusions are summarized as follows.

(1)The size of the hysteresis loop of rotating stall is influenced by the parameter ψc0/H which is mainly related to the design blade loading of the compressor.The size of the hysteresis loop will decrease as the value of ψc0/H increases,and when ψc0/H ＞ 4,the stall recovery process of compressor will no longer have the catastrophe and hysteresis,and the stall type will change from‘abrupt stall” to ‘gradual stall”.

(2)The catastrophe and hysteresis of compressor system stall is caused by the instability of the equilibrium points.The stability of the equilibrium points is related to the relative magnitudes of the slope of the compressor characteristic and throttle characteristic.

(3)The compressor rotating stall and the cusp catastrophic model does have exactly the same topological properties,and the compressor rotating stall can be modeled as a cusp catastrophic phenomenon.According to the topological invariant rules,the cusp catastrophic model can be conveniently used to describe the diverse hysteresis behaviors of compressor rotating stall under the impact of multiple parameters and evaluate the size of hysteresis loop.

Acknowledgements

This research was supported by the National Natural Science Foundation of China(Nos.51676006 and 51636001),the Aeronautics Power Foundation of China(No.6141B090315)and China Postdoctoral Science Foundation (No.2017M610742).

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