Receptivity and structural sensitivity study of the wide vaneless diあuser flow with adjoint method

Xiocheng ZHU ,Chenxing HU ,*,Zezho LIU ,Ho LIU ,Zhohui DU

a School of Mechanical Engineering/2011 Aero-Engine Collaborative Innovation Center,Shanghai Jiao Tong University,Shanghai 200240,China

b College of Politics and Law,Hebei University,Hebei 071000,China

KEYWORDS

Abstract The stability of the flow in the vaneless diffuser of a centrifugal compressor is studied with the linear theory.The characteristics of direct and adjoint perturbation modes are investigated,and the receptivity of the instability mode to momentum forcing or mass injection is identified based on the adjoint modes.Analysis shows that the perturbation with the largest amplitude is located at the outlet of the vaneless diffuser,while the highest-receptivity region is located in the middle of the vaneless diffuser along the radial direction.The large difference between the direct and adjoint modes indicates that the instability mechanism cannot be identified from a study of either eigenmode separately.Therefore,the structural sensitivity analysis is adopted to study the feedback of the instability mode.The structural sensitivity of the eigenvalue which is proportional to the perturbation pressure and velocity is used to explain the mechanism of flow control for the vaneless diffuser.

1.Introduction

The performance of compressors at low mass flow is generally characterized by the occurrence of flow instability such as rotating stall and surge.Under these flow phenomena,deterioration in aerodynamic performance and subsynchoronous vibration may be brought upon compressors.Especially,for a centrifugal compressor widely employed in turbochargers,rotating stall not only occurs in the impeller-like axial compressor,but also may exist in the vaneless diffuser.In order to avoid the occurrence of flow instability,a centrifugal compressor is forced to be operated under a safety margin with a loss of high-pressure ratios.Under this circumstance,main efforts are focused on flow stability in the vaneless diffuser of a centrifugal compressor.

In the frame of vaneless diffuser,the nature of flow instability has been extensively investigated by numerous researchers.1–4Among many efforts made by researchers to explain the mechanism and predict the occurrence of rotating stall within the vaneless diffuser,the cause of the disturbance and its amplification which induce instability has always and still been the highlight of their studies.Usually,a contained flow such as a vaneless diffuser flow may turn into instability due to the flow in one local region,which is called the wavemaker region.5,6The rest of the flow only responds to the force from this region.From the view of Jansen7and Senoo and Kinoshita8who focused on boundary layer calculation,the separation of three-dimensional boundary layers is responsible for the occurrence of instability in the vaneless diffuser,and the disturbance from the boundary layers is amplified in the inviscid core flow as a feedback system.Considering the critical in flow angle corresponding to radial inversion flow,Senoo and Kinoshita8proposed an empirical formula to predict the onset of instability,which has been widely employed in industrial design of vaneless diffusers.Based on this theory,a wall roughness control at the hub side of a vaneless diffuser was used to suppress the three-dimensional separation in order to delay the occurrence of instability.9On the other hand,a linear stability theory was adopted by Moore,10Tsujimoto et al.,11and Chen et al.12The occurrence of stall is associated with the instability of the inviscid core flow in vaneless diffusers.A two-dimensional numerical model developed by Moore10was based on the calculation of 2D incompressible Eulers’equations,and was capable of predicting the onset of stall and the stall rotation speed.In order to verify the inviscid core flow theory,jet flow injection near the diffuser inlet was introduced into a vaneless diffuser to suppress the occurrence of stall in Tsurusaki and Kinoshita’s work.13

Besides the direct linear stability study above,a sensitivity and receptivity study based on the gradient method may provide more evidences of the instability mechanism.Adjoint operators,which are originally defined by Lagrange,have been thoroughly substantiated theoretically and broadly applied in solving many problems such as perturbation theory,optimization,and data processing.14,15The adjoint of a linear operator is indeed one of the most important and useful concepts in functional analysis.The introduction of adjoint method severs as a mathematical tool to enforce constraints in the early days.However,the interpretation of adjoint operators as sensitivity measures or cost-functional gradients gives them a physical meaning.16Under the frame of local stability,the adjoint method was firstly adopted as an optimization technique influid mechanics by Hill,17and then introduced for receptivity studies by Luchini and Bottaro.18By studying the Ginzburg-Landau model equation,Chomaz19showed that the wavemaker region could be identified as the overlapping region between direct and adjoint global eigenvectors.For the frame of global stability,the wavemaker region of the spiral vortex breakdown in swirling flows was confirmed with structural sensitivity analysis by Qadri et al.,20and two different feedback mechanism related with the instability mode were revealed.The receptivity and structural sensitivity of global modes for cylinder wake were investigated with adjoint techniques by Giannetti and Luchini.21It was noticed that the wavemaker region was similar to the region where the placement of a control cylinder suppressed the vortex shedding in the experiments by Strykowski and Sreenivasan.22A sensitivity and receptivity study not only points out the wavemaker region of a flow field,but also may guide the flow control.In Fani et al’s work,23both structural sensitivity and sensitivity to base flow modifications can present quantitative indications on designing the control of the flow in a symmetric channel with a sudden expansion.

According to the above studies on stability and sensitivity,structural sensitivity shows where an instability of a fluid flow is most sensitive to changes in internal feedback mechanisms,and is capable of optimizing the flow control technique.Applying sensitivity and receptivity analysis to a stability study of a vaneless diffuser could be beneficial on revealing the vaneless diffuser’s stability mechanism.The present study extends the previous inviscid core flow stability study of a wide vaneless diffuser24and investigates the sensitivity of the inviscid core flow in the wide vaneless diffuser.Main contents are organized as follows.A direct global eigenmode,which is focused on the large time scale growth of perturbations imposed on a base flow,is firstly sought with a direct global stability analysis on the inviscid core flow.The geometric parameter radius ratio and wave number are varied to confirm their influences on vaneless diffuser stability.Then,an adjoint approach is carried out,and the receptivity based on adjoint modes under different wave numbers is investigated.Meanwhile,the structural sensitivity analysis of the instability mode in a vaneless diffuser is applied.The wavemaker region of the vaneless diffuser with different feedback mechanisms is studied.Finally,the sensitivity to the feedback which is proportional to pressure is compared with experimental data.

2.Direct global stability

2.1.Model implementation

2.1.1.Physical model

The vaneless diffuser considered in this paper has two parallel walls,and the base flow is assumed to be circumferentially uniform,isentropic,and have no axial velocity.The flow sketch is shown in Fig.1.The diffuser outlet radius ratio Rfand width ratio bzare illustrated in Fig.1(a),and an in flow circumferential angle α is defined as α =arctan（Vr/Vθ）at the diffuser inlet as illustrated in Fig.1(b).Vrand Vθare the inlet radial and the circumferential velocity components,respectively.Meanwhile,the impeller rotational speed ωimpeller,the impeller tip velocity U,the impeller back sweep angle β and the relative velocity W at the impeller outlet are illustrated in Fig.1(b).It should be noted that the slip factor between the relative flow angle and the impeller back sweep angle is omitted for simplification.In the following cases,the impeller back sweep angles are all set as 90°by default.Then,a 3D stability model with a 2D incompressible base flow in the vaneless diffuser can be developed.

For the flow stability analysis in this paper,the viscosity and compressibility are neglected.The flow field is described by 3D,unsteady,incompressible Eulers’equations as follows:

Fig.1 Sketch of vaneless diffuser.

where vr,vθ,vzare the radial,circumferential and axial velocity.p is the pressure.These variables are all nondimensionalized as shown in Table 1.The terms with the subscript 2 represents the dimensional variables,and ρ2represents the density of the flow.

Here,Eulers’operator can be described as L,and the governing equations are expressed as follows:

The hypothesis underlying this paper is that the linear evolution of the infinitesimal perturbation to the base flows is concerned.Therefore,variable q can be decomposed into a base flow partand a perturbation part q′as follows:

The direct operator associated with the stability analysis refers to the dynamics of the small perturbations based on an equilibrium state as

where u′and p′are the velocity and pressure perturbations,respectively.Substituting the ansatz Eq.(6)into Eulers’Eq.(5),linearized Eulers’equations are obtained as follows:

Table 1 Non-dimensional variables and parameters defined in terms of dimensional variables.

2.1.2 Base flow

Based on the following hypothesis of the base flow:

incompressible Eulers’equations for the base flow are simplified as

with boundary conditions of

where P represents the inlet pressure.Considering Eq.(14)and the boundary conditions in Eqs.(15)–(19),the wall normal velocity results to be zero in the whole domain,i.e.,

The radial velocity can be derived from Eqs.(11)and(15)as

Then the radial velocity for the base flow is

Similarly,Eqs.(12)and(13)lead to the following expression for the circumferential velocity:

The circumferential velocity for the base flow is

The pressure is obtained via Eq.(12)as

where P is the inlet pressure.Then the base flow as a function of the radial position can be obtained.For instance,when the normalized inlet radial velocity is 0.3 and the radius ratio is 2.2,the distribution of radial velocity and circumferential velocity along the radius direction are illustrated in Fig.2.

2.1.3.Establishment of an eigenvalue problem

Another assumption of the global stability analysis is that the base flow is homogeneous in the circumferential direction.Then,the homogeneity of the base flow permits the perturbations to be expanded in terms of Laplace transform in time and Fourier transform in the circumferential direction.The perturbations in Eq.(7)take the following form:

Because the impeller back sweep angle β is 90°,the nondimensionalized rotational speed of the disturbances in the circumferential direction is given as

After substituting Eq.(26)into Eqs.(8),(9),a generalized eigenvalue problem can be established as

where A and B are complex matrix that satisfy the generalized eigenvalue problem.

2.2.Leading eigenvalues and eigenmodes

The numerical domain of a vaneless diffuser is discretized by the spectral collocation method with Chebyshev-Gauss-Labatto points chosen along the r and z directions.Then the generalized eigenvalue problem is calculated with the generalized Schur(QZ)decomposition algorithm,which is realized by the function eig in the numerical software MATLAB.What should be mentioned is that the singularity of matrix B may lead to the existence of spurious eigenvalues.One way to distinguish real eigenvalues from spurious ones is by varying the collocation point number.

The direct global stability of a vaneless diffuser with a radius ratio of 2.2 and a width ratio of 0.1 is predicted here.At the diffuser inlet,a uniform radial velocity distribution is applied.In the calculation,the wave number is set to be 3.In order to eliminate the influence of the numerical fault,leading eigenvalues at three different spatial resolutions are calculated as shown in Table 2.It can be seen that the leading eigenvalues remain the same under both mid-res and highres.Then,the middle spatial resolution is selected in the present calculation.

Table 2 Leading eigenvalues calculations at three different spatial resolutions.

The stability of the base flow is assessed by monitoring the behavior of the most unstable mode or the instability corresponding to the leading eigenvalues.Fig.3(a)shows the calculated eigenvalue spectrum with a uniform in flow condition when the in flow flow angle is 18.62°.It can be seen that when the largest imaginary part of the eigenvalues turns to 0,instability will occur.The in flow angle at this state is defined as the critical in flow angle αc.Corresponding to the leading eigenvalue,the velocity modulus increase linearly along the radial direction as shown in Fig.3(b).It indicates that instability may be induced in the form of a disturbance as described in Fig.3(b)superposed on the base flow with a wave number of 3 propagating along the circumferential direction.Meanwhile,the direct global stability model is capable of predicting the occurrence and rotating speed in wide diffusers as reported in our previous research.24

2.3.Influences of geometric parameters on stability

Among geometric parameters of a vaneless diffuser,the radius ratio has a quite large influence on the diffuser vaneless stability.Meanwhile,combining the wave number with the radius ratio allows a study of the most unstable wave number when instability occurs.Therefore,3 cases with different radius ratios Rf are calculated at 5 different wave numbers.Fig.4 shows the influence of the radius ratio on the critical stall angle and rotational speed of stall derived from the present model.In the calculation,the in flow velocities are all uniformly distributed in the axial direction.

In Fig.4(a),the influence of the radius ratio on the critical in flow angle αcis illustrated.It is clearly shown that the critical in flow angle decreases significantly with a decrease of Rf,and the largest critical angle is obtained when the wave number m is around 3 with a radius ratio of 2.2.It indicates that rotating stall occurs with the form of three stall cells,which is consistent to the result in Abidogun’s work.25The wave number corresponding to the most unstable mode differs among different radius ratios,indicating that the stall number is variable under different radius ratios.In Fig.4(b),the circumference rotating speed of the disturbance corresponding to the onset of instability f vs the wave number m is also illustrated.It can be seen that the rotational speed of the disturbance increases linearly with an increase of the wave number m,while the disturbance rotating speed decreases quickly with the diffuser radius ratio,which is consistent with the results of Moore10and Chen et al.12When the radius ratio Rf=2.2,counter-rotating stall or backward traveling wave is observed,which is consistent with the observation in experiments.

Fig.4 Influence of wave number m on critical angle and rotational speed of disturbances.

3.Adjoint global stability

3.1.Adjoint approach

With the calculated direct eigenmodes,the adjoint eigenmodes are essential to obtain the sensitivity of the eigenvalues to changes.For considering the gradient-based sensitivity analysis with the adjoint method,the adjoint quantities are defined as

where u?and p?denote the adjoint velocity and pressure perturbations,respectively.

Considering the axisymmetric nature of the flow configuration,the concept of a standard L2inner product in the cylinder coordinate is introduced as

where the centered dot refers to the usual scalar product of two vectors,and 〈〉represents inner product.The linearized Eulers’operator,L′,has its corresponding adjoint linearized operator L?,which is defined in terms of the inner product as follows:

Then the adjoint governing equations can be obtained from the direct governing equations by integrating by parts as follows:

where A?and B?are complex matrix that satisfy the generalized eigenvalue problem for adjoint stability analysis.The same algorithm as the direct approach can be applied to calculate Eq.(35),and the adjoint eigenvalues and eigenmodes can be obtained.In order to value the adjoint eigenmodes directly,the adjoint velocity modes are normalized with the condition

3.2.Adjoint eigenspectrum and eigenmodes

Through the calculation of the generalized eigenvalue problem in Eq.(35),the adjoint eigenspectrum relative to Fig.2 is illustrated in Fig.5.It can be seen that the adjoint eigenspectrum is almost the same except that the value of the real part of the eigenvalues is opposite.The eigenmode shown in Fig.5(b)is pretty different from the direct eigenmodes since it represents the gradient information.The region with the largest adjoint velocity modulus is located in the middle of the diffuser along the radial direction.Due to the uniform in flow condition,the distribution of adjoint velocities is axisymmetric.

4.Formation of structural sensitivity and receptivity

4.1.Derivation of structural sensitivity

The sensitivity of the direct global mode to structural changes in the linearized Eulers’equations is proportional to the overlap between the direct and adjoint global modes and defined as the structural sensitivity.The structural sensitivity mainly considers the effects on the system’s state of a variation in one of its own parameters,especially the disturbance field.The high structural sensitivity region,or the so-called ‘‘wave maker region”,indicates the area where a modification in the structure of the system produces the largest drift of the eigenvalue.

Consider the linearized Eulers’equations with the eigenvalue ω1,and the source terms δH and δR are introduced to account for a possible physical forcing mechanism as follows:

Fig.5 Adjoint eigenspectrum and eigenmode corresponding to instability.

where double primes indicate quantities satisfying the perturbed equations.Assume that the eigenvalue ω1is drifted with δω1and the eigenmodes perturbation δ^q= ［δ^u，δ^p］T.Then a simple expansion in terms of the solution of the unperturbed problem is

After inserting Eqs.(38)–(40)into Eqs.(36)and(37)and applying the Lagrange identity,the perturbation of the eigenvalue can be expressed with the adjoint modes21over the flow domain D as

Once the structural perturbations δH and δR are specified,the associated eigenvalue shift can be obtained.Here,we consider the two most significant structural sensitivities of the eigenvalue with respect to a localized feedback.The first kind of feedback is proportional to the velocity and affects the momentum equations.Then the structural perturbations are in the form of δH=C^u and δR=0,where C is the 2 × 2 matrix of the coupling coefficients.If the feedback is localized in space,C can be expressed at the position （r0，z0）as

where C0is a constant coefficient matrix, and δ（r-r0，z-z0）denotes the Kronecker delta function.

Therefore,the eigenvalue drift due to the localized feedback mechanism proportional to the velocity can be derived with the Laplace transform as follows:

where λuis the eigenvalue’s structural sensitivity to the feedback mechanism proportional to the velocity at a given point as

The concept of a wavemaker is then introduced in Giannetti and Luchini’s work21by identifying the regions of flow with the largest λu.Similarly,when another kind of feedback that is proportional to the pressure and affects the continuity equation is considered,the structural sensitivity to the feedback proportional to the pressure at a given point is illustrated as follows:

4.2.Derivation of receptivity

Receptivity analysis mainly aims at the process how one environmental disturbance produces an instability wave in the flow.External forcing such as one acoustic or vorticity wave may be introduced into the flow as momentum forcing or mass injection,and an instability wave is induced.From the mathematical view,the receptivity describes the response to additive changes to the governing equations,modelling external sources of influence.It is customary to distinguish the natural receptivity with an external forcing of very long wavelength from the forced receptivity with one harmonic forcing26With adjoint eigenmodes in hand,the effects of generic initial conditions and forcing terms on the time-asymptotic behavior of the solution of linearized Eulers’equations can be derived.

Consider Eq.(41)where the structural perturbation δH is the momentum forcing F and initial conditions uin,and δR is the mass injection Mmass.Due to the linearity,the effect of each term in the global mode can be studied separately once the direct and adjoint eigenmodes are determined.It can be seen that the effects of the forcing functions or mass injection are only associated with the adjoint variables,which stands for Green’s function for the receptivity problem.21Therefore,the receptivity to the momentum forcing and initial or boundary conditions is expressed by the adjoint velocity modulus,and the receptivity to the mass injection is expressed by the adjoint pressure modulus in the present study.

5.Receptivity analysis of the vaneless diffuser

5.1.Receptivity to initial conditions and momentum forcing

In order to compare the receptivity of the instability mode,the same vaneless diffuser with a radius ratio of 2.2 and three different wave numbers is studied.The adjoint velocity modulus,which represents the receptivity to the momentum forcing and initial conditions of the instability mode at different wave numbers,is illustrated in Figs.6(a)–6(c).It can be seen that the region where the momentum forcing and initial condition are most effective is located in the middle of the vaneless diffuser in the radial direction,and the flow fields in the inlet and outlet areas have the lowest receptivity to the momentum forcing and initial conditions in all three cases.Under a uniform distribution of inlet velocities,the distribution of receptivity is also asymmetric in the axial direction.It can be seen that the wave number has little influence on the receptivity to the momentum forcing and initial conditions,and an instability perturbation most likely occurs when an external forcing such as momentum forcing and initial condition is induced in the middle of the diffuser.

5.2.Receptivity to mass flow injection under different wave numbers

The adjoint pressure modulus,which represents the receptivity to the mass injection of the instability mode at three different wave numbers,is illustrated at Figs.6(d)–6(f).Compared to the data in Figs.6(a)–6(c),the receptivity to the mass injection of the instability mode is significantly influenced by the wave number.The distribution of receptivity to the mass injection is still axisymmetric in the axial direction.For different wave numbers,the high-receptivity region is varied in the radial direction.When the wave number is 1,the region with the highest receptivity is located at the radial position of r=1.6.When the wave number is 2,the high-receptivity region moves upstream to the radial position of r=1.5.When the wave number increases to 3,there are two high-receptivity regions,which are located at the radial positions around r=1.35 and r=1.68,respectively.It is shown that instability perturbations may be most likely induced by an external forcing such as a mass injection at different regions under different wave numbers.

Fig.6 Spatial distributions of receptivity of the instability mode at different wave numbers.

System matrices that result in non-orthogonal eigenvectors are known as non-normal matrices.For a normal system,direct and adjoint modes coincide,while for a non-normal system,these two modes may differ substantially.16In the present work,the difference between the direct and adjoint fields is a consequence of the non-normality of the linearized Eulers’equations.Thus,the linearized Eulers’operator for the inviscid core flow in the vaneless diffuser is moderately non-normal.

6.Structural sensitivity analysis of the vaneless diffuser

6.1.Structural sensitivity to feedback mechanism proportional to perturbation velocity

The overlap of the direct and adjoint global modes yields the strongest shift of the amplification rate in the vaneless diffuser.As shown in Figs.7(a)–7(c),the structural sensitivity to feedback proportional to the perturbation velocity at three different wave numbers is illustrated.It can be seen that the wavemaker regions under different wave numbers are located in the middle of the vaneless diffuser with a closer distance to the outlet in the radial direction.It indicates that under the assumption of an inviscid base flow,the instability wave mainly comes from the middle of the vaneless diffuser,which is consistent with the strongest perturbations detected in the middle of the vaneless diffuser by Tsurusaki et al.27Close to the outlet of the vaneless diffuser,the product of adjoint and direct modes becomes larger with an increase of the wave number,indicating that the downstream of the vaneless diffuser may be important for the instability dynamics under a large wave number.

6.2.Structural sensitivity to feedback mechanism proportional to perturbation pressure

Fig.7 Spatial distributions of structural sensitivity of instability mode with a uniform in flow at different m.

The structural sensitivity to feedback proportional to the perturbation pressure,on the other hand,is associated with the mass injection and pressure coupled flow field.As shown in Figs.7(d)–7(f),the structural sensitivity to feedback proportional to the perturbation pressure is illustrated under three different wave numbers.It can be seen that the distribution of the high structural sensitivity to feedback proportional to the perturbation pressure is similar to that of the adjoint pressure as shown in Figs.7(a)–7(c).With an increase of the wave number,the regions with high structural sensitivity moves closer to the diffuser inlet in the radial direction.When the wave number is 3,efficient flow control may be achieved with techniques associated with mass injection at the diffuser inlet.

7.Application of structural sensitivity on flow control with steady injection

7.1.Flow control experiments

Passive or active control may be efficiently applied to the vaneless diffuser flow with a spatial distribution of the receptivity or sensitivity of the instability mode.In the view of structural sensitivity,a flow control technique such as steady flow injection can be considered as mass injection into the pressure coupled field,leading to a negative shift of the eigenvalue.In the following study, flow control experiments with jet flow injection are studied in terms of the structural sensitivity to feedback to the perturbation pressure.

Fig.8 Jet nozzle inserted in diffuser.13

In flow control experiments conducted by Tsurusaki and Kinoshita,13the experiment rig consists of a two-dimensional centrifugal impeller,a vaneless diffuser,and an injection pipe.The vaneless diffuser has parallel walls with the radius ratio Rf=2 and the width ratio bz=0.2.Eight nozzles arranged with the same azimuthal intervals are installed at the radial position of r=1.06 to inject air into the diffuser as shown in Fig.8.The injection direction can be adjusted so that the optimal injection effect can be obtained.

With a pressure transducer located at r=1.06 and a hot wire probe located at r=1.52,pressure and velocity fluctuations can be measured.For the vaneless diffuser without nozzles,three perturbations occur at the same flow coefficient φ=0.072 with 1-cell,2-cell,and 3-cell components,respectively.Then injection flows at different axial positions are tested to suppress the rotating stall.According to the reference,when the injection flow angle γ=0°(the jet flow is opposite to the impeller rotating direction),the stall can be suppressed.

The control effect for the rotating stall with 1,2,and 3 cells under different axial jet positions is shown in Fig.9.It can be seen that all the three stall perturbation are suppressed regardless of the axial position considering the cases with nozzles but no control.Compared with the flow control applied on the 1-cell and 3-cell components,the best control effect is obtained with the 2-cell component.Moreover,the 1-cell component is only slightly suppressed considering the original fluid field without control.The result indicates that the flow near the wall is not the only reason contributing to the occurrence of rotating stall.The instability of the inviscid main flow may have also caused rotating stall,which is consistent with the direct global analysis using the present inviscid model.Therefore,the direct global analysis is applied on the vaneless diffuser under the same flow coefficient and uniform in flow distribution.The flows under wave numbers of 1,2,and 3 are all unstable according to the direct global stability model.

7.2.Application of sensitivity analysis on the vaneless diffuser of the experiment

Fig.9 Flow control effect of the injection under different axial positions.13

In order to study the physical mechanism for different flow control effects with different stall cells,the spatial distributions of the structural sensitivity to feedback proportional to the perturbation pressure when the wave number is 1,2,and 3 are illustrated in Fig.9.When the wave number is 1 as shown in Fig.10(a),the high-structural sensitivity region is located in the range of r=1.15 to r=1.5,and the structural sensitivity near the injection position r=1.06 is relative low,resulting in that the instability perturbation is not sensitive to the mass injection.Meanwhile,in Fig.10(b),the high-structural sensitivity region moves upstream compared with that of the 1-cell component.A similar sensitivity distribution can also be found in Fig.10(c),where the high-structural sensitivity region is located in the range of r=1.06 to r=1.55.For rotating stalls with 2 and 3 cells,the injection flow may induce an eigenvalue drift significantly,and a better flow control effect may be obtained compared with that of the 1-cell stall.

Fig.10 Contours of structural sensitivity to feedback proportional to perturbation pressure.

Fig.11 Structural sensitivities to feedback to perturbation pressure at radial position of r=1.06 when m is 1,2,and 3.

The structural sensitivity of the instability perturbation at the inject position may directly reveal the flow control mechanism.The structural sensitivity to feedback to the perturbation pressure at the radial position of r=1.06 is illustrated in Fig.11.When the wave number is 1,2,and 3,the structural sensitivities of all cases come to their maximums in the middle of the axial direction.The structural sensitivities of the 2-and 3-cell components are both larger than that of the 1-cell component,which may be why the instability is only slightly suppressed by the injection flow for the 1-cell component.Compared with the 3-cell component,the structural sensitivity of the 2-cell component is larger,especially the structural sensitivity between the axial height of 0.2 to 0.8.That may lead to the significant effect of the injection flow for the 2-cell component.It should be noted that when the wave number is 3,the structural sensitivity distribution takes the form of a sine-like wave along the axial direction.

8.Summary and conclusions

In this paper,a direct global stability analysis is carried out,and the onset of instability is predicted.Then,direct and adjoint eigenmodes are obtained.The spatial distribution of receptivity to momentum forcing or mass injection is investigated.The structural sensitivity of the eigenvalue is discussed,and the wavemaker region of the inviscid core flow in a vaneless diffuser is illustrated.

The following conclusions are drawn:

(1)A combination of direct and adjoint global modes can be used to describe the receptivity and sensitivity of the eigenvalue.For a vaneless diffuser,the receptivity to momentum forcing and initial conditions is hardly influenced by the wave number,while the receptivity to mass injection is highly associated with the wave number of instability perturbation.

(2)The linearized Eulers’operator for the inviscid core flow in the vaneless diffuser is moderately non-normal,considering the values difference between the direct and adjoint modes.

(3)Under the inviscid core flow assumption,the wavemaker region of the vaneless diffuser lies in the middle of the vaneless diffuser in the axial direction.

(4)The structure sensitivity of the eigenvalue to the disturbed pressure can be used to guide the flow control with the means of steady injection.In a flow control experiment,the best flow control effect is achieved for the stall with a 2-cell component of which the structural sensitivity near the injection area is relative large.