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.

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:

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

Fig.1 Schematic of axial flow compressor system.

Whenand J=0,thesteady axisymmetric flow characteristic of compressor,ψc,can be acquired,as shown in Fig.2,and is expressed as

Shimadzu LC-20AB高效液相色谱仪，数据分析软件LCsolution（日本岛津科技有限公司）；Alltech6000蒸发光散射检测器（美国奥泰科技有限公司）；DIONEX ICS-5000+离子色谱仪，Thermo 电导检测器（美国Thermo Fisher Scientific公司）；梅特勒-托利多电子天平XS205（瑞士梅特勒-托利多公司）；pH计 （瑞士梅特勒-托利多公司）；Millipore纯化水仪（美国Millipore公司）。

其次，与两代血亲的长期共同生活与紧密联系，滋长着沈从文慈孝友悌的特殊性血缘亲情。考察沈从文对自己幼年在家的生活经历与离家后谋生的断续描叙，可以看出沈从文与父母以及兄弟姐妹之间的关系十分和谐。正如《太上感应篇·友悌》说“孝悌本一。今又专言者。欲人随事而尽之也。兄友则爱而且敬。弟悌则畏而且和。兄弟乃我身同气。只此几人。人生最为难得。”沈从文在他的作品中将这种孝悌人伦寓于日常生活琐事的叙述之中。

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.

Fig.2 Rotating stall characteristic and cubic axisymmetric characteristic.

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.

Fig.3 Stalled and unstalled characteristic of compressor for different ψc0/H based on Moore-Greitzer model.

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.

定义1 对于非线性控制系统,由系统的状态量xi(t)(i = 1,2,…,N)和控制变量uj(t)(j = 1,2,…,m) 及自变量时间 t 为坐标组成的空间称为广义状态空间.根据广义状态空间概念,显然,系统状态方程的求解是一个初值问题.

Fig.4 Equilibrium points of compressor system for different ψc0/H.

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 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 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).

温室栽培番茄育苗管理的关键时期，此时期如果管理不当，会对幼苗生产造成很大影响，稍有疏忽就会造成严重的经济损失。那么应注意哪些问题呢？

Fig.5 Effect of throttle slope on stalled performance.13

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.

Fig.6 Potential function approach for hysteresis behavior of compressor rotating stall.

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

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).

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 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.

Fig.7 Topological property of state transition in cusp catastrophic model and compressor system.

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 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

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

2.Greitzer EM.Surge and rotating stall in axial flow compressors–Part II:Experimental results and comparison with theory.J Eng Power 1976;98(2):199–211.

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

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

Fig.8 Bifurcation set of compressor system and different control routes.

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.

Fig.9 Model description of hysteresis behavior of compressor along different control routes.

Fig.9(continued)

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 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).

在新的网联、信联背景之下，互联网金融平台的营运与风控能力将从本质上得到提升，做大做优平台后在资产获取、风控水平、用户体验、技术研发等方面，更具优势。最终，互联网金融会彻底揭掉它的“神秘”面纱，成为金融体系的底层设施，构成个人信用值的重要组成，并助力金融步入真正的互联网社会。

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).

References

9.Day IJ,Cumpsty NA.The measurement and interpretation of flow within rotating stall cells in axial compressors.J Mech Eng Sci 1978;20(2):101–14.

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:

3.Day IJ.Stall inception in axial flow compressor.J Turbomach 1993;115(1):1–9.

4.Garnier VH,Epstein AH,Greitzer EM.Rotating waves as a stall indication in axial compressors. J Turbomach 1991;113(2):290–302.

5.Han SB,Zhong JJ.Effect of blade tip winglet on the performance of a highly loaded transonic compressor rotor.Chin J Aeronaut 2016;29(3):653–61.

6.Camp TR,Day IJ.A study of spike and modal stall phenomena in a low-speed axial compressor.J Turbomach 1998;120(3):393–401.

7.Day IJ,Freeman C.The unstable behavior of low and high-speed compressors.J Turbomach 1994;116(2):194–201.

专业教师通过指导和参与技术服务，接触并参与企业真实项目，真正促进教师专业技能提升，并将实际工作过程相关技术知识引入教学进行真实工学结合教学改革，将职业核心技能与职业素养培养充分融入教学过程中。在专业课程体系建设中，紧扣企业人才标准，不断优化课程设置，构建“企业标准、项目导向”的课程体系。同时在技术服务中获得对等收益，促进教师不断提高自身技术水平积极性。

8.Huppert MC.Preliminary investigation of flow fluctuations during surge and blade row stall in axial- flow compressors.Washington,D.C.:NACA;1952.Report No.:NACA-RM-E52E28.

石黑一雄在这部小说里采用的倒装式身份悬念、分解式身份悬念和情景反讽式身份悬念创新性地展现了人物的身份，使读者对小说人物的认知从迷惘到清晰，从局部到整体，从宏观到微观。这些身份悬念与小说的事件悬念相辅相成，有助于读者从不同角度了解这些人物的性格特点和生存环境。

1.Greitzer EM.Surge and rotating stall in axial flow compressors–Part I:Theoretical compression system model.J Eng Power 1976;98(2):190–8.

10.Copenhaver WW,Okiishi TH.Rotating stall performance and recoverability of a high-speed 10-stage axial flow compressor.J Propul Power 1993;9(2):281–92.

1970年末，涌现出众多的乡镇农药厂，1979年后，全国农药生产厂家快速增加，农药产量也随之增加，而后，农业改革与发展直接推动了我国农药工业由弱势产业向强势产业的发展。尤其是在改革开放前20年里，农药业引起了各界的关注和投资者的青睐，基本上每个农业县都有自己的农药企业或分装厂，农药产业逐渐强势起来。

重估线下价值我们不难发现，一如马云所说，“不是实体经济不行了,是你的实体不行了；不是零售不行了,而是你们家的零售不行了”。外卖也好、打车也好、体验经济也好，并没有颠覆传统线下行业，而是在颠覆后者的业态——人的需求远不止线上，线下，实际需求空间更大。

11.Choi M,Vahdati M.Recovery process from rotating stall in a fan.J Propul Power 2011;27(6):1161–8.

12.Choi M,Vahdati M,Imregun M.Effects of fan speed on rotating stall inception and recovery.J Turbomach 2011;133(4):041013.

11月5日，美国正式恢复了对伊朗能源、金融领域单边制裁。美国制裁伊朗，最终的目的是要制造全球石油市场的紧缺，提升石油的价格，从而为美国石油打开国际市场，促进美国原油的生产，提升美国的利益。但美国的这个设想存在极大的风险，不但难以实现，还可能遭到反噬。

13.Day IJ,Greitzer EM,Cumpsty NA.Prediction of compressor performance in rotating stall.J Eng Power 1978;100(1):1–12.

14.Abed EH,Paul KH,Hosny WM.Bifurcation analysis of surge and rotating stall in axial flow compressors.J Turbomach 1993;115(4):817–24.

15.McCaughan FE.Application of bifurcation theory to axial flow compressor instability.J Turbomach 1989;111(4):426–33.

16.Liaw DC,Abed EH.Active control of compressor stall inception:A bifurcation-theoretic approach.Automatica 1996;32(1):109–15.

17.Liu XF,Zhao L.Approximate nonlinear modeling of aircraft engine surge margin based on equilibrium manifold expansion.Chin J Aeronaut 2012;25(5):663–74.

18.Dong W,Wang Y.Detecting rotating stall precursors in axial compressors via perturbations Part 1:Theory.J Propul Power 2014;30(5):1295–306.

19.Lin P,Wang C,Wang Y.A high-order model of rotating stall in axial compressors with inlet distortion.Chin J Aeronaut 2017;30(3):898–906.

20.Thom R.Mathematical models of morphogenesis.Chichester:Ellis Harwood;1983.p.75–86.

跟4S店约好的时间是上午10点，田朵怕迟到，就破天荒地没跟小宁继续掰扯下去。可到了店里，田朵的心里更不爽了，本来早上的气还没顺过来，小宁却火上浇油——田朵跟他说话他爱搭不理，只自顾自地围着店里的新车和美女导购转，还嬉皮笑脸地跟人家闲聊：“你这么瘦是不是天天减肥呀？”“这么巧咱们是校友，哈哈……”

21.Wang QS,Ping P,Sun JH.Catastrophe analysis of cylindrical lithium ion battery.Nonlinear Dyn 2010;61(4):763–72.

22.Yu DR,Cui T,Bao W.Catastrophe,hysteresis and bifurcation of mode transition in scramjet engines and its model.Sci China Ser E:Technol Sci 2009;52(6):1543–50.

23.Petraitis PS,Dudgeon SR.Cusps and butter flies:multiple stable states in marine systems as catastrophes.Mar Freshw Res 2016;67(1):37–46.

24.Cui T,Zhong L,Yu DR.Multistability and complex routes of supersonic inlet start/unstart.J Propul Power 2011;27(6):1204–17.

25.Bao JS,Yin Y,Lu YH,Hu DY,Lu LJ.A cusp catastrophe model for the Friction catastrophe of mine brake material in continuous repeated brakings.Proceed Inst Mech Eng,Part J:J Eng Tribology 2013;227(10):1150–6.

26.Moore FK,Greitzer EM.A theory of post-stall transients in axial compression systems:Part I-Development of equations.J Eng Gas Turbines Power 1986;108(1):68–76.

27.Greitzer EM,Moore FK.A theory of post stall transients in axial compression systems:Part II–Application.J Eng Gas Turbines Power 1986;108(2):231–9.

28.Wang HH,Krstic M,Larsen M.Control of deep hysteresis aeroengine compressors.J Dyn Syst Meas Contr 2000;122(1):140–52.

29.Whitney H.On singularities of mappings of euclidean spaces.I.Mappings of the plane into the plane.Ann Math 1955;62(3):374–410.