A high-order model of rotating stall in axial compressors with inlet distortion

Peng LIN,Cong WANG,*,Yong WANG

aSchool of Automation Science and Engineering,South China University of Technology,Guangzhou 510641,China

bDepartment of Mechanics and Engineering Science,Peking University,Beijing 100871,China

A high-order model of rotating stall in axial compressors with inlet distortion

Peng LINa,Cong WANGa,*,Yong WANGb

aSchool of Automation Science and Engineering,South China University of Technology,Guangzhou 510641,China

bDepartment of Mechanics and Engineering Science,Peking University,Beijing 100871,China

Available online 8 May 2017

*Corresponding author.

E-mail address:wangcong@scut.edu.cn(C.WANG).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2017.03.014

1000-9361©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

In this paper,a high-order distortion model is proposed for analyzing the rotating stall inception process induced by inlet distortion in axial compressors.A distortion-generating screen in the compressor inlet is considered.By assuming a quadratic function for the local flow total pressure-drop,the existing Mansoux model is extended to include the effects of static inlet distortion,and a new high-order distortion model is derived.To illustrate the effectiveness of the distortion model,numerical simulations are performed on an eighteenth-order model.It is demonstrated that long length-scale disturbances emerge out of the distorted background flow,and further induce the onset of rotating stall in advance.In addition,the circumferential non-uniform distribution and time evolution of the axial flow are also shown to be consistent with the existing features.It is thus shown that the high-order distortion model is capable of describing the transient behavior of stall inception and will contribute further to stall detection under inlet distortion.

©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Axial compressors;

Dynamic modeling;

Flow instability;

Inlet distortion;

Rotating stall;

Stall inception

1.Introduction

Inlet distortion often happens in aeroengines and has a significant impact on aeroengine compressor performance.Normally resulting from separation of the lip or inlet,side gusts,hot gas ingestion,etc.,inlet distortion can trigger an entire compression system’s instability known as surge and rotating stall.Therefore,accurate prediction of a system’s dynamic behavior under inlet distortion is necessary to avoid the occurrence of instability.

There has been a substantial effort to describe the effects on stability margin of axial compressors due to inlet flow nonuniformity,and further to develop prediction methods for assessing the effects associated with inlet distortion.1–6Based on the celebrated Moore-Greitzer model,7Hynes and Greitzer3developed a nonlinear method to assess circumferential inlet distortion’s effects on the stability of incompressible flow compressors,and the method was also shown to work well for a low-speed compressor with rotating inlet distortion.5The Hynes-Greitzer model predicted that small-amplitude waves would travel around the annulus,and the waves were not purely sinusoidal but changed shape as they propagated around the annulus.Longley8preformed distortion flow experiments and observed the predicted circumferential variation in a magnitude of the propagation disturbances.A heightened interest in stall inception was aroused with the attempts to understand the physical mechanisms of rotating stall,and further to extend the compressor operating range.9–13Lin and Chen14presented a model for static circumferential total pressure distortion and investigated the stall triggering mechanism of distortion-induced instability of axial compression systems.Furthermore,Lin et al.15made an experimental study of rotating stall on a three-stage low-speed axial compressor under inlet distortion.They concluded a long-to-short length-scale disturbance evolution before fully developed stall.Salunkhe and Pradeep16extended the Moore-Greitzer model to incorporate the effects of static inlet distortion and tip injection on axial compressor performance.Theoretical results showed that the compressor performance deteriorated under the influence of static inflow distortion.

It is noticed that these works mentioned above were mainly based on the Moore-Greitzer model which cannot describe the occurrence of high-order stall modes.In a unif i ed framework,rotating stall and surge can be viewed as eigenmodes of compression systems,with surge constituting the zeroth-order mode and rotating stall representing the higher-order modes.17Numerical analyses have exhibited that the severity of stall hysteresis increases with the order of modes.18Therefore,the truncation of high-order harmonics in the disturbance flow can be a feasible way to improve the modeling accuracy.e.g.,Mansoux et al.19derived a high-order discrete model(Mansoux model)through transforming the Moore-Greitzer model into an ordinary differential equation(ODE)model of order 2N+1,whereNis the number of pairs of stall modes.The Mansoux model could describe the transient behavior of stall inception and coincide well with experimental results within a certain precision.Based on the Mansoux model,Wang et al.20,21proposed a deterministic learning algorithm for modeling and rapid detection of system dynamics corresponding to stall inception,and carried out on a low-speed axial flow compressor test rig at Beihang University for online experimental verification.They concluded that the proposed approach could detect the stall inception signal of a compressor 0.3–1.0 s in advance to the onset of rotating stall at different speeds.

Motivated by these studies,we attempt to provide a highorder distortion model for analyzing the rotating stall inception process induced by circumferential total pressure distortion in low-speed axial compressors.A difficult problem in modeling of inlet distortion is how to describe a distortion quantitatively.22The distortion intensity varies over the flow as the inlet dynamic pressure varies.Although square-wave functions are used to describe pressure rise across a distortion screen,the selection of a unif i ed expression is not given since it depends on the geometric characteristics of axial compressors.Another important problem is how to describe the fluiddynamic interaction between a compressor and a distortion screen.17In the presence of circumferential nonuniform inlet flow,the steady compressor characteristic is a function of the circumferential variable θ,and the coupling between the compressor and inlet distortion plays a role in the linearized behavior of the flow field perturbations.3It is necessary to provide a quantitative function to describe the coupling.

In this paper,we present a high-order distortion model for rotating stall in a low-speed axial compressor with circumferential total pressure distortion.The circumferential total pressure distortion generated by a static screen can be viewed as a part of the compressor.A new steady compressor characteristic under inlet distortion is constructed to describe the coupling between the compressor and the distortion screen,and then dominates the compression system dynamics.The high-order distortion model in a real-valued state-space form is derived based on a generalized 2N+1 harmonics expansion of the perturbation flow and a spatial discretization in the circumferential angle θ.Considering the high-order stall modes,the wave propagating around the compressor annulus has a rich harmonic structure for a compression system with inlet distortion,and then the high-order distortion model can more accurately depict the dynamics of rotating stall,especially the transient behavior of stall inception.

The rest of the paper is organized as follows.In Section 2,we present basic assumptions and concepts about compression system modeling for low-speed axial compressors.The main results are given in Section 3,including derivation of the high-order distortion model with circumferential total pressure distortion.Section 4 demonstrates the simulation results.In Section 5,we give the conclusions of this paper.

2.Preliminaries

In this section,a brief review of two generally recognized stall inception types is presented,and relevant fundamental modeling assumptions are given.Detailed discussions on these two aspects can be obtained in Refs.23,24.

2.1.Stall inception

The objective of the high-order distortion model is to describe the transient process of rotating stall,especially stall inception.Thus,it is useful to summarize the stall inception phenomena that occur in low-speed axial compressors.

The time history in Fig.1,which results from simulation in the case of uniform inflow,serves to illustrate the terminology for different time periods in a typical transient into rotating stall or surge.The process is consistent with the description in Ref.24.Stall inception is of interest because two important criteria are fulfilled during this transition process19:(1)relatively small axial flow perturbations,and(2)strong influence of nonlinearities.It was found that there were two routes of stall inception,namely,modal waves and spikes.9The former are long length-scale disturbances with circumferential extents that are an order of magnitude or higher than a blade passage.On the contrary,the latter are short length-scale disturbances with extents of one to several blade passages.Some compressors have been found to exhibit both types of unsteady disturbances.Even in thesecircumstances,in any particular configuration,one can still point to one or the other(modal waves or spikes)as being the dominant phenomenon in the process of transition to rotating stall.

2.2.Compression system under inlet distortion

In experiments,a compressor test with inlet distortion is carried out using screens which block part of the annulus.Two cases of distortion screen distribution have been discussed,3,14,15,24one of which is at far upstream from the compressor and the other one is at the compressor inlet.In either case,the unsteady perturbations upstream from the compressor will be a potential type.That is,analyzing one of the two cases is suitable for potential perturbations.Therefore,the main considered total pressure distortion is created by a screen placed at the compressor inlet.Specif i cally,a lumped parameter compression system representation is given in Fig.2,which consists of an upstream annular duct,a sector distortion screen,a compressor,a downstream annular duct,a plenum(representing the combustor volume in an engine),and a throttle.The compressor is modeled as a high solidity semi-actuator disk with a high enough hub tip radius ratio so that the flow field can be counted as two-dimensional;the flow in the upstream duct is considered non-rotational,and the flow in the downstream duct is rotational but linearized;distortion is created by a screen placed at the compressor inlet;a plenum exists downstream from the compressor;the throttle duct is assumed to be very short.

The general approach of modeling compressor flow field dynamics is to take the steady-state axisymmetric performance characteristics of individual components and to include additional terms to represent the nonaxisymmetric and unsteady flow effects.7The state of the high-order distortion model under unsteady,possible nonaxisymmetric conditions is characterized by two terms:the non-dimensional plenum pressureppand the spatially distributed flow coefficients,denoted as

where η is the non-dimensional axial position in the compressor,non-dimensionalized by the mean rotor radiusR(η=0 at the compressor face);θ is the circumferential angle around the compressor annulus;and ξ is the non-dimensional time variable(in rotor revolutions).

3.High-order distortion model

Modeling of inlet distortion is usually based on the models for rotating stall.Mansoux et al.19and Paduano et al.24,25provided finite-dimensional state-space models for rotating stall,which will be used as a basic framework for the analysis of inlet distortion in this section.For the research of inlet distortion,Hynes and Greitzer3discussed two circumferential nonuniform flow distribution cases and provided a function to describe the pressure rise across the distortion screen.Paduano24further derived a distortion model based on the proposed state-space model.The distortion model was applied to study the effect of distortion on the compressor stability and represented in a complex-valued state-space form.

In this section,based on existing models,an external force from the distortion screen is introduced and the upstream duct flow field model is reconstructedfirstly.Then,a distortion model for rotating stall is presented by using the general Galerkin projection and a spatial discretization method.In addition,the distortion model is in the form of real-valued state-space,which can also be used for control and simulation research.

3.1.Model of distortion screen

A quadratic function of the local flow is selected for a quantitative description of the pressure loss when the flow passes across a distortion screen.The model of the distortion screen is determined by a screen loss functionptscreen:

A similar form of Eq.(1)is specif i ed when a distortion is generated by a screen placed at the compressor inlet.3,14,16The unsteady correction to the total pressure-drop across the distortion screen is neglected because the length from the screen to the first rotor is short when compared to other characteristic lengths in the system.The use of a nonlinear screen loss function has two main purposes:one is for a quantitative description of the circumferential total pressure distortion and the other is for an introduction of the non-uniform flow in a nonlinear manner.Different forms have also been proposed,24in which the pressure rise is a quadratic function of the circumferential average flow which is a constant but nonlinear.

3.2.Model of upstream duct

In the upstream region,the flow field is treated as linear,even when the flow in the compressor itself exhibits large perturbations.24The incompressible and non-rotational assumption on the flow field implies the existence of a potential flow Θ satisfying Laplace’s equation ▽2Θ =0,and the solution is given as:

where Θ is the non-dimensional potential flow,whose axial and circumferential partials are axial and circumferential flow coefficients,such as φ = ∂Θ/∂η.are the real and imaginary parts of complex Fourier coefficients of the potential flow,respectively.The final two terms of Eq.(2)represent the 0th harmonic(spatial mean part of the flow perturbations).

The flow variable is represented as a Fourier series with respect to the circumferential variable θ.A slightly nonstandard transform,which is equivalent to the standard formula as long as the complex Fourier coefficients are real,is adopted.25

Then,by superimposing the distortion on the upstream flow field,the axial momentum equation is written as an unsteady Bernoulli-type equation24:

wherecrepresents a constant and is evaluated far upstream from the compressor to obtain zero.The initial conditions of Eq.(4)are that the non-dimensional total pressureptandptscreenare approximately zero at η=-lu(luis the length of ducting upstream from the compressor).

The total pressure at the compressor face is obtained by putting Eqs.(2)and(3)into Eq.(4),yielding

Compared with the case of uniform inlet flow,there is an additional termptscreenin Eq.(4),which can be considered as an external force if the local flow remains unchanged.However,the view of distortion as an external force is not strictly correct during large oscillations of the local flow in an experimental scenario.In this paper,the coupling relationship between the distortion and system dynamics is used as well as verified in the following derivation.

3.3.System model

In Sections 3.1 and 3.2,the dynamic models for the distortion screen and the upstream duct are restructured.The other component models of the compression system were previously derived25and suitable for the problem of inlet distortion.Combining Eq.(5)and the other component models of the compression system,a set of ordinary differential equations for all spatial modes is integrated in terms of the real-value Fourier coefficients of the flow variable as follows:

wherepcnew=pc+ptscreenis the screen inlet total pressure to compressor exit static pressure rise which is viewed as a compressor characteristic function under inlet distortion at a local value of φ;λ is the non-dimensional inertia parameter for the fluid in the rotating blade passages;μ represents the overall inertia of the rotor plus stator blade passages;lcis the nondimensional overall effective length of compressor ducting;ΦT（pp）=is the throttle characteristic;KTis the exhaust characteristic parameter which depends on the degree of throttle closure andBis Greitzer’s stability parameter.The other component models and integration procedures for all spatial modes were described in detail in the Ref.23.

The new non-dimensional pressure risepcnewincludes the pressure difference associated with the distortion screen component.That is,the integrative result shows that the distortion screen can be combined with the compressor as a whole.The interaction between the compressor and the distortion screen can be described by the new non-dimensional pressure risepcnewin our model.Greitzer’s stability ‘B’plays an important role in determining the system stability.17For small values of the parameter,e.g.,B=0.1,the general trend of the compression system will operate in the characteristic of pure rotating stall.

Then,Eq.(6)is transformed into a finite-dimensional statespace form.The high-order distortion model can be obtained by performing the following steps:

(1)A high-order harmonic Galerkin expansion of the disturbance flow,which is used to provide high-order stall modes and reduce the infinite dimensional system to a finite dimensional system of ODEs.

(2)A spatial discretization in the circumferential variable θ（k=1，2，...，2N+1）,in which 2N+1 points are equally spaced around the annulus to describe the flow coefficients φ.

Then,the final form of the flow at the compressor face is obtained as:whereNis the number of pairs of stall modes,and the value ofNis a power of 2 which can replace the Fourier transforms in Eq.(7)with Fourier coefficients matrices.

Eq.(6)yields an overall system of 2N+2 equations,that is,

Simplifying Eq.(8)by a state transitionsame independent variable equations are obtained.The state transition matrix is shown as follows:

Linetal.14,15presented a distortion model in the form of partial differential equation(PDE)and the equations were solved numerically by using a pseudo-spectral method.Different from their distortion model14,15,the distortion model given in our paper is in a high-order state-space form in which the rotating stall dynamics withNpairs of stall modes corresponding toNharmonics of spatial Fourier series for the local flow are taken into account.To the best of the authors’knowledge,this is the first high-order distortion model for describing the stall inception process in axial compressors with inlet distortion.Compared with models presented in previous works,3,14the model given in this paper possesses the following three features:

(1)The state space representation form of the distortion model makes it convenient for active control design as well as the solving of Eq.(9)without using a pseudospectral method.

(2)High-order harmonics are included in the disturbance flow,making the nonlinear stall inception behavior being adequately described.

(3)Eq.(6)is accurately represented so that the situation whenNapproaching infinity is recovered.

4.Simulation

To illustrate the effectiveness of our distortion model,simulations are performed on an eighteenth-order model,which in experiments would correspond to seventeen probes evenly placed around the circumference in front of a compressor as shown in Fig.3,with seventeen probes distributed around the circumference in front of a compressor.As is approximate to 21.18°in each portion,we takeN=8，M=17 as a case of Eq.(9).

Two distortion scenarios are given:one with a 42.35°distortion screen placed at the region starting from Probe 5 and ending at Probe 7,and the other with two 21.18°distortion screens placed at Probes 1 and 11 respectively.To simplify,the two cases of distortion scenarios are named as Case 1 and Case 2,respectively.The process of rotating stall is simulated through continuous adjustment of the exhaust character-istic parameterKTfrom 7 to 9.41.Details on the two cases are discussed in the following sections.

4.1.Distortion model parameters

The input parameters required are the basic geometry of the compression system and the compressor’s performance characteristic under inlet distortion.The parameters in the model are chosen on the basis of the Mansoux-C2 compressor geometry.19Table 1 lists the dimensionless geometrical parameters of the Mansoux-C2 compressor.Greitzer’s stability parameter‘B’is set to be 0.1 in our simulations.Ideally,the inertia parameter λ,which is purely a geometrical quantity,should be modified by a multiplicative constant to allow for the fluid in the inter blade row gaps and for unsteady viscous effects.7,8Herein,a geometrical quantity is chosen for the sake of simplicity.

The axisymmetric characteristic without inlet distortion is given as follows:

Since the function φ（θ）is unknown,the steady compressor characteristicpcnew（φ）is an unknown function of the circumferential variable θ.In simulations,the loss coefficientsKS（θ）for the distortion screen is specif i ed with a square sector and fixed at 1.6.A particular value ofpcnew（φ）can be determined by numerical calculation in Case 1.Fig.4 shows the curves of the axisymmetric compressor characteristicpc（φ）and a particularpcnew（φ（θ5））(abbreviated aspcnew（5）).It is noticed that the compressor pressure rise has an apparent decline due to inlet distortion at the same flow.The result shows that inlet distortion has a remarkable effect on the compressor performance.

Table 1 Values of dimensionless parameters.

4.2.Stall characteristics induced by distortion inflow

The circumferential non-uniform distributions of the axial flow in a steady inlet distortion condition are shown in Fig.5.The results demonstrate that the important feature about the total pressure non-uniformity is caught indirectly.The total pressure distortion produces a corresponding distortion inflow in front of the compressor.In Fig.5(a),the flow coefficients gradually decreases from the distorted Probe 5 until the undistorted Probe 8 due to the nonlinear distortion loss functionptscreen.The decrease at the undistorted Probe 8 demonstrates that the distorted sectors are wider than the angles covered by the distortion screen due to the effect of the rotor.Similar results have been obtained by theoretical and experimental methods.14,15In Fig.5(b),the flow coefficients gradually decrease from the distorted Probes 1 and 11 until the undistorted Probes 3 and 13.The result also indicates that the distorted sectors are wider than the angles covered by the distortion screens.

Fig.6 shows the partial state trajectories in the time domain.The developments of stall inception are obviously different between distorted(such as Probe 5)and undistorted sectors(such as Probe 9).Compared with the initial disturbances at undistorted sectors,the initial disturbances at distorted sectors are much smaller.From the trajectory of the flow coefficient,it can be concluded that the stall inception process initiates with long length-scale disturbances.For instance,the initial disturbance from the signal of Probe 5 stretches from 198.6 to 206.5 rotor revolution which is approximately 7.9 times to that of the annulus.Moreover,as the occurrence of rotating stall,the plenum pressure significantly drops.This result demonstrates that the performance of the compression system decreases significantly due to inlet distortion and also verifies the accuracy of the distortion model.The onset of stall in Case 2 is estimated at about 577 rotor revolution while at about 200 rotor revolution in Case 1.Compared with Case 2(keeping the total distorted sector angle constant),Case 1(splitting the total distorted sector angle into two segments)induces the stall in advance,that is,splitting the total distorted sector angle into two segments has a smaller effect on the compressor stability than keeping the total distorted sector angle constant.This result agrees well with Reid’s experiment results.1

To further explain the influence of inlet distortion,signals from all probes are placed side-by-side in both cases,as shown in Fig.7.The flow coefficients from Probe 1 is placed at both the bottom and top to represent the cyclic placement of the probes.In Fig.7(a),symbol#denotes virtual probe,the initial disturbances take place in the region starting from Probe 5 and ending at Probe 8,and then the disturbances become a mature form at Probe 9.In Fig.7(b),the initial disturbances take place in the region starting from Probe 1 and ending at Probe 5 with disturbances in a mature form at Probe 6.These results show that the initial disturbances emerge out of the distorted background flow and travel around the compressor annulus in the direction of rotor rotation.Intuitively,from the flow physics point of view,the distorted sectors represent higher aerodynamic loads to the rotor blade,so these sectors with higher loads naturally become the sources of disturbances.15Moreover,the results also provide guidance for stall detection.For instance,the first probe able to detect significant stall disturbances should be Probe 9,but not Probe 5 as in Case 1.

By contrasting,Fig.7(c)shows time evolution of simulated stall inception for the compressor with a clean inflow condition.It can be found that the onset of rotating stall with inlet distortion is earlier than the case relative to the clean flow condition.It is demonstrated that the distortion screen can trigger initial disturbances that lead to the development of rotating stall.Although the initial disturbances under distorted and clean inflows conditions are long length-scale,the time evolutions of stall inception are significantly different in shape and length-scale.As the time evolution of stall inception under inlet distortion is short,there will be an increase of difficulty in stall detection.

Compared with existing simulation results,3,14our simulation results can clearly reveal the size,shape,and structure of the stall inception process.The flow disturbance which grows into the stall cell appears from the distorted background flow,and further induces the onset of rotating stall in advance.In addition,the performance of the compression system decreases significantly due to inlet distortion.

5.Conclusions

In this paper,we have made a theoretical study to demonstrate the effect of circumferential inflow distortion on the dynamic of rotating stall in low-speed axial compressors.The derived high-order distortion model not only captures the essential fluid dynamic features of rotating stall,but also depicts the developments of initial disturbances induced by inlet distortion.The following conclusions can be drawn from the simulations:(1)the distorted sectors are wider than the angles covered by distortion screens;(2)long length-scale initial disturbances emerge out of the distorted background flow and propagate in the direction of rotor rotation;(3)the total distorted sector angle being split into two segments has a significantly smaller effect on the compressor stability;(4)inlet distortion triggers the initial disturbances and induces the occurrence of rotating stall in advance.Based on simulation results,the high-order distortion model can describe the transient behavior of stall inception and provide a theoretical foundation for further research on stall detection.In addition,more distortion scenarios and experimental investigations will be undertaken to assess the validity of the high-order distortion model in our future research.

Acknowledgements

This study was co-supported by the National Major Scientific Instruments Development Project of China(No.61527811),the National Science Fund for Distinguished Young Scholars of China(No.61225014),the Guangdong Inovative Project(No.2013KJCX0009),the Guangdong Provice Natural Science Foundation(No.2014A030312005),the Guangdong Provice Key Laboratory of Biomedical Engineering,and the Space Intelligent Control Key Laboratory of Science and Technology for National Defense.

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13 April 2016;revised 9 October 2016;accepted 12 January 2017