Investigation of rotating stall in radial vaneless diffusers with asymmetric inflow

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Investigation of rotating stall in radial vaneless diffusers with asymmetric inflow Chenxing Hu a , Ce Yang a,∗ , Xin Shi a , Runnan Zou a , Lin Liu b , Hua Chen b a b

Beijing Institute of Technology, Beijing 100081, China Dalian Maritime University, Dalian 116026, China

a r t i c l e

i n f o

Article history: Received 25 July 2019 Received in revised form 27 October 2019 Accepted 9 November 2019 Available online xxxx Keywords: Centrifugal compressor Vaneless diffuser Structural sensitivity Global stability Adjoint method

a b s t r a c t Despite its simple geometry, the turbulent flow in vaneless diffusers is asymmetric and highly skewed. In the present work, a frozen eddy viscosity approach was employed to investigate the instability in a vaneless diffuser with different axial widths. A turbulent stability analysis was performed around the numerically computed mean flows with a non-uniform inflow for both isolated and full-annular vaneless diffusers. The predictions of the flow instability frequency and coherent structure were validated against experimental data. By performing a structural-sensitivity analysis corresponding to the leading eigenvalue, the instability mechanisms for the isolated and full-annular vaneless diffusers were revealed. The sensitivity analysis indicated that the interaction between the boundary layer and the main flow may have been the primary cause of the self-excited instabilities in a narrow diffuser under both axisymmetric and asymmetric inflow. The contribution of the reverse flow near the walls was relatively small. However, the influence of the separation flow near the wall on the instabilities of a wide diffuser was significant, particularly under high-skew inflow conditions. The wavemaker regions were located on the shroud side near the inlet and the hub side near the outlet. When connected to the impeller in the upstream direction, the diffuser outlet backflow was responsible for instability in the diffuser with a radius ratio of 1.53. The jet-wake flow in the diffuser inlet had little impact on the flow instability. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Many studies have been performed on the dynamics of the unsteady flow in centrifugal compressors. It is well known that an unsteady flow, such as a rotating stall in the impeller or diffuser, is a potential danger to compression systems [1–6]. In particular, the turbulent flow in the vaneless diffuser in a centrifugal compressor is dominated by the low-frequency perturbations of large-scale coherent structures. The perturbations may be detrimental to the safety margin, as they may induce unsteady side loads and vibrations. Therefore, prediction of the instability onset is often necessary in engineering design, which requires a deep understanding of the underlying mechanism. Despite the simple geometry, the flow in a vaneless diffuser is highly skewed owing to the difference in viscosity between the main flow and boundary layer. At a high Reynolds number, the main flow away from the walls usually experiences twodimensional development from the diffuser inlet to the outlet, and the turbulent boundary layer near the walls exhibits three-

*

Corresponding author. E-mail address: [email protected] (C. Yang).

https://doi.org/10.1016/j.ast.2019.105546 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

dimensional (3D) developments. As shown in Fig. 1, the main flow streamline (marked as 1) is a nearly logarithmic spiral curve. For the boundary layer, subsequent flow separation may occur owing to the high adverse pressure gradient (marked as 3). Then, the separated flow may undergo reattachment (marked as 4). The separation point is related to not only the geometric parameters, such as the axial width and radius ratio, but also the flow conditions, such as the Reynolds number and velocity distributions. These characteristics of the vaneless diffuser flow may lead to different physical explanations of the instability mechanism [7,8]. Owing to the unique flow structure of the vaneless diffuser, one of the most challenging problems involving the instability in the vaneless diffuser is the effect of the width parameter. Several experimental and numerical analyses have been performed on the influence of the axial width [10,11]. Researchers such as Senoo [12], Dou [13], and Frigne [14], who assumed that the 3D development of the wall boundary layer was responsible for the instability, believed that an increase in the axial width generally lead to more unstable condition of the diffuser. According to previous experimental studies [15–18], the reverse flow, which is significantly influenced by the axial width, generally reduces the diffuser stability. When the reverse flow occurs or develops to some extent,

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Nomenclature r Rf m bz q u p k

ε Cμ

μ μt M

= = = = = = = = = = = = =

diffuser radius radius ratio wave number width ratio flow variable velocity quantity pressure turbulence kinetic energy turbulence dissipation constant molecular viscosity eddy viscosity left eigenvalue constant matrix

J

ω Re

λ

= = = =

right eigenvalue constant matrix eigenvalue Reynolds number structural sensitivity

Subscripts r

θ z –

∼ †

= = = = = =

radial direction index circumferential direction index axial direction index time-averaged term coherent perturbation term adjoint coherent perturbation term

Fig. 1. Sketch of the main flow and boundary-layer flow in the vaneless diffuser [9].

instability perturbations may be excited. Researchers such as Abdelhamind [19], Moore [20], and Shen [21], who assumed that the inviscid main flow was the direct cause of the instability perturbations in wide diffusers, found that an increase in the axial width reduces the acceleration effect on the main flow induced by the boundary-layer thickness [22]. An overview of the studies on vaneless diffuser stability reveals that the roles of the main flow and boundary layer in the occurrence of instability in narrow and wide diffusers remains unclear. Another factor that makes the stability problem of diffuser flow complex is the influence of the non-uniform swirling inflow coming from the rotating impeller. For the self-sustained perturbations in an internal turbulent flow, the non-uniformity of the spanwise flow at the diffuser inlet plays a critical role in the formation of instability. Considering the influence of the non-uniform distributed velocity, methods based on the boundary-layer theory are often implemented by performing boundary layer calculations under a velocity distribution obtained from theoretical models. As shown in Fig. 2, Senoo proposed an analytical method for evaluating the flow behavior in a vaneless diffuser with an asymmetric inflow and validated the predicted critical inflow angles against experimental results [12]. The effect of the radial velocity distortion on the critical inflow angles was greater than that of the circumferential velocity distortion. In the author’s previous study [22], the inflow velocity with a linear or conic distribution was examined in an inviscid stability analysis, and the predicted critical inflow angles did not exhibit significant differences. This indicated that the distribution of the inflow velocity may be more crucial for a viscous flow than for an inviscid flow. In recent years, developments in adjoint-based sensitivity have provided alternative paths in the stability research of centrifugal

Fig. 2. Distributions of the radial velocity employed in Senoo’s analytical method [12].

compressors. Structural-sensitivity analysis, which was introduced by Giannetti and Luchini [23], typically quantifies the sensitivity of each global mode to internal feedback. Chomaz showed that the perturbations in a turbulent flow may significantly displace the eigenvalue of the system if they occur at the overlap region between the direct and adjoint global modes [24]. This means that structural sensitivity can be used to identify the spatial core region where the instability perturbations are induced. Relevant studies on this adjoint-based framework were performed for the cases of cylinder wakes [25], a jet flow [26], and a confined flow [27]. Marquet [28] and Luchini [29] developed a sensitivity method by considering the base flow effect, and the modification in the base flow with the greatest stabilizing effect on the eigenvalues was observed for cylinder flows. Rigas performed optimal passive control of an electrically heated Rijke tube, together with an experimental sensitivity analysis [30]. Further details regarding sensitivity

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Fig. 3. Sketch of the vaneless diffuser model.

can be found in Sipp’s [31] and Luchini’s [32] reviews. Given the successful applications of the adjoint-based sensitivity method for hydrodynamic instability, the utilization of these techniques for the turbulent flow in a vaneless diffuser may be essential to obtain insight into the instability mechanism. In the author’s previous study, a frozen eddy viscosity method employed in the vaneless diffuser was proposed, and the structural sensitivity of the vaneless diffuser under an axially uniform inflow was investigated [33]. However, the vaneless diffuser flow is generally non-uniform and skewed. The present study extends the sensitivity formalism for the global modes governed by the incompressible Navier–Stokes equations to a vaneless diffuser with an asymmetric inflow condition. The objective was to identify the origin of the instability for both narrow and wide vaneless diffusers under a non-uniform inflow. Additionally, the non-uniform flow with a jet-wake flow structure coming from the impeller was considered. The remainder of the paper is organized as follows. The numerical methods and formulations are presented in Sec. 2. In contrast to previous research on diffuser stall relying on boundary-layer calculations, the global modes were captured via a frozen eddy viscosity analysis. Then, unsteady simulations of full-annular and isolated diffusers were performed to examine the mean flow, as described in Sec. 3. The predicted results for the diffusers with different inflow conditions are validated against experimental data in Sec. 4. In Sec. 5, the direct and adjoint global modes corresponding to the leading eigenvalue are presented. Finally, a sensitivity analysis for the isolated vaneless diffuser and full-annular diffuser was performed. The origin of the instability flow under a non-uniform inflow is discussed in Sec. 6. 2. Numerical methods and formulations 2.1. Flow configuration We investigated the flow developing in the vaneless diffuser with parallel walls, as shown in Fig. 3. In particular, the highly distorted inflow coming from the impeller has a velocity field of v 1 = ( v r1 , v θ 1 , v z1 ), which represents the radial, circumferential, and axial velocities, respectively, at the diffuser inlet. The inflow angle, which can be used to measure the working condition of the diffuser, is defined as α = arctan( v r1 / v θ 1 ). When instability occurs in the vaneless diffuser, the corresponding inflow angle can be treated as the critical inflow angle αc . To avoid the influence of the impeller, the impeller back sweep angle and the slip factor related to the impeller geometry are omitted. With regard to the diffuser geometry, the radius ratio R f = r2 /r1 (defined as the ratio of the outlet radius to the inlet radius) is introduced. The width ratio bz = b/r1 is defined as the ratio of the axial width to the inlet radius.

The fluid motion is described by the state vector q = (u , p )T , where p represents the pressure and u = ( v r , v θ , v z ) represents the velocity components. The 3D incompressible flow in the vaneless diffuser satisfies the following Navier–Stokes equations:

∂u 1 2 + u • ∇ u = −∇ p + ∇ u ∂t Re ∇ • u = 0.

(1) (2)

Eqs. (1) and (2) are made nondimensional by applying the diffuser inlet radius r1 as the length scale and the inlet velocity v 1 as the velocity scale. The dimensionless pressure is defined as p /ρ1 v 21 , where ρ1 represents the flow density at the diffuser inlet. 2.2. Global stability analysis The instability of the flow is investigated within the framework of linear theory. The genetic instantaneous quantity u can be de¯ a coherent perturbation or composed into a time-averaged term u, ˜ and a stochastic turbulent motion part u  . In organized wave u, the present study, focus is placed on the coherent perturbations ˜ which are associated with the long-term stability. The mean u, flow of the vaneless diffuser is assumed to be homogeneous in the circumferential direction; thus, ∂/∂θ = 0. After substituting the decomposed velocity and pressure into Eqs. (1) and (2), the linearized Navier–Stokes equations are sought. The linearization details can be found in the author’s previous work [33]. The Reynolds number based on the molecular viscosity μ must be redefined owing to the introduction of the eddy viscosity μt . An effective Reynolds number defined as Re eff = ρ1 v 1 r1 /(μ + μt ) is employed, and the stability analysis applied here is known as the frozen eddy viscosity approach.

   ∂ u˜ 1  T ˜ ˜ ¯ ¯ ˜ ˜ ∇ +∇ u + u • ∇ u + u • ∇ u = −∇ p + ∇ • ∂t Re eff

(3)

∇ • u˜ = 0

(4)

The boundary conditions of the linearized Navier–Stokes equations for the vaneless diffuser flow are as follows.

u˜ = 0

at diffuser inlet and walls

p˜ = 0 at diffuser outlet

(5) (6)

The assumption of the circumferential mean flow allows the coherent perturbation variables to be expressed in normal modes. In the present analysis, the velocity and pressure coherent perturbations are decomposed into the normal form:

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u˜ = uˆ (r , z)e i (−ωt +mθ ) p˜ = pˆ (r , z)e

i (−ωt +mθ )

(7) (8)

,

where uˆ and pˆ represent the amplitude functions. ω = ωreal + i ωimag represents the complex frequency, and m represents the circumferential wavenumber. The imaginary part ωimag represents the growth rate of the perturbations, and the real part of ωreal represents the rotating speed of the perturbations. By substituting Eqs. (7) and (8) into Eqs. (3) and (4), the linearized Navier–Stokes equations can be transformed into a generalized eigenvalue problem:

M qˆ = ω J qˆ ,

(9)

where qˆ = (uˆ , pˆ )T represents the direct global modes. The detailed expressions of M and J can be found in the Appendix. 2.3. Adjoint method and structural sensitivity For the structural-sensitivity analysis based on the adjoint method, the adjoint quantities corresponding to the direct quantities are defined as q† = (u † , p † )T . u † represents the adjoint velocity coherent perturbation, and p † represents the adjoint pressure coherent perturbation. The derivation of the adjoint equations is based on the inner product, as follows:









q† , L q˜ = L † q† , q˜ ,

(10)

where L represents the linear operator and L † represents the corresponding adjoint linear operator. Then, the adjoint linearized Navier–Stokes equations can be obtained.

∂ u† + u † • ∇ u¯ − u¯ • ∇ u † ∂t    1  ∇ + ∇ T u† = −∇ p † + ∇ • Re eff

†

∇•u =0

† †

at diffuser inlet †

†



v θ = v z = 0, p † + v r v¯ r −

1

1

Re eff R f



†

= vθ =

† vz

= 0 at diffuser walls

†

†

(13)

= 0 at diffuser outlet

(15)

(16)

†

†

†

†

δ(r − r0 , z − z0 )C 0 .

(17)

M and J are described in the Appendix. Given the direct modes obtained from Eq. (9) and adjoint modes obtained from Eq. (17), the structural sensitivity can be defined as the overlap between the direct and adjoint global modes, as reported by Gainnetti and Luchini [20]. Once a perturbation δ H is introduced into the linear operator, it can be expressed as a localized feedback δ H = C • uˆ in terms of the force–velocity

(18)

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

|δ ω| =

|



†

D

|



ˆ | uˆ • C • udS †

D

ˆ uˆ • urdS |

≤ C 0 λ,

(19)

where λ represents the structural sensitivity of the eigenvalue to the feedback at a given point: † uˆ uˆ  λ=  † . ˆ | D uˆ • urdS |

(20)

2.4. Spatial discretization and eigenvalue calculation The spatial discretization of the calculation domain for the direct and adjoint approach is performed via the Chebyshev collation spectral method. The structured mesh is established with Chebyshev–Gauss–Labatto points along the r and z directions, and the calculation domain is defined as [−1, 1] × [−1, 1] in the Gauss–Lobatto grid. The influence of the grid number on the calculation results can be eliminated by changing the spatial resolution. In particular, for a vaneless diffuser with a radius ratio of 1.8 and a bz of 0.055, the spatial resolution with 40 collocation points in the r direction and 30 in the z direction is selected for the calculation. Owing to the size of the leading dimension of the eigenvalue problem matrices O(104 –107 ), the QZ algorithm is employed for solving the eigenvalue problem.

3.1. Numerical configuration

where qˆ = (uˆ , pˆ † )T represents the adjoint global modes. Then, the eigenvalue problem for the calculation of the adjoint stability is derived as

M † qˆ = ω† J † qˆ .

r

(12)

With a similar treatment of the perturbations to the direct approach, the adjoint quantities can be written as † † q† = qˆ e i (mθ −ω t ) ,

1

3. Mean flow

(14) † vr

C (r , z) =

(11)

The boundary conditions of the adjoint linearized Navier–Stokes equations are as follows.

vr = 0

coupling. If the feedback is localized in space (r0 , z0 ), C can be expressed as follows:

With regard to the mean flow for the stability calculation, the strategy is to use the time-averaged data of the URANS results for both the full-annular compressor and the isolated vaneless diffuser. The full-annular compressor consists of an impeller and a vaneless diffuser. As indicated by Sparkvosky [34], the numerical model of the compressor without a volute can be applied to study the onset of instability in the impeller and diffuser, although the domain limitations do not allow simulation of the fully developed rotating stall or consideration of the circumferential non-uniformity induced by the volute in the compressor. Additionally, self-excited perturbations can occur in the isolated vaneless diffuser, according to Lejvar [35] and Gao [36]. Another benefit of the isolated vaneless diffuser in the present study is that it allows the effects of different velocity distributions to be investigated by avoiding the distorted flow upstream. The geometric parameters and inflow conditions for the full-annular compressor and the isolated vaneless diffuser are presented in Table 1. The physical model of the vaneless diffuser with the structured gird is selected for the 3D URANS simulation of the mean flow, as shown in Fig. 4. After gridindependence calculations, the grid numbers for the full wheel and the vaneless diffuser are selected as 300 and 120 million, respectively. The timestep for the unsteady simulation is 1.3 × 10−5 , and the calculation time is 2 s, which is sufficient for the flow development in the diffuser. The turbulence model of k − ε is applied in the simulation, which is appropriate for the global stability analysis and prediction of the eddy viscosity [37]. The total pressure and total temperature are specified as the inlet boundary condition in

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Table 1 Geometric and flow parameters of the full-annular compressor and the isolated vaneless diffusers. Parameters

Values

Blade number Impeller inducer tip diameter Impeller exducer tip diameter Impeller speed Diffuser width ratio bz Diffuser radius ratio Rf Re at diffuser inlet Ma at diffuser inlet Turbulence model

6 full blades and 6 splitter blades 133 mm 182 mm 44198 rpm 0.055, 0.14, 0.10 1.8, 1.5 3.6 × 105 0.1 k−ε

5

After the calculation results are obtained, the time-averaged velocity and eddy viscosity at the meridian plane are exported as the mean flow for global mode analysis. For the eddy viscosity, it is assumed that the turbulent stress is proportional to the mean velocity gradient, similar to the case of the viscous stress. The eddy viscosity can be obtained using the following equation when the k − ε turbulence model is employed in calculating the mean flow:

μt = C μk2 /ε

(21)

where krepresents the turbulence kinetic energy, ε represents the turbulence dissipation rate, and C μ is a constant. To consider the effect of the asymmetric inflow, taking the vaneless diffuser with an axial width of 0.055 for instance, the axisymmetric and non-axisymmetric velocity distributions shown in Fig. 5 are employed. Three different types of non-axisymmetric radial velocity distributions are employed. The flow skewness increases as the distribution D1 turns into the distribution D3, and the radial velocity near the hub side is significantly higher than that near the shroud side under distribution D3. The circumferential velocity distribution remains axisymmetric because its distortion has little effect on the flow stability, as reported by Senoo [12]. 3.2. Simulation results of overall performance

Fig. 4. Mesh of the full wheel and isolated vaneless diffuser for the mean-flow calculation.

both steady and unsteady simulations. The static-pressure boundary condition is specified in the steady simulation at the outlet, while the opening boundary condition is employed at the outlet in the unsteady simulations, so that the backflow can be captured.

The aerodynamic performance of the full-annular compressor without a volute is quantified using the total pressure ratio. As shown in Fig. 6(a), the unsteady simulation results are validated against the experimental data reported in a previous work [38]. The magnitude of the flow rate and the total pressure ratio oscillation tend to increase as the flow coefficient decreases. The

Fig. 5. Distributions of the radial and circumferential velocities in the axial direction. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

Fig. 6. Numerical results and the static-pressure spectrum near the stall condition [37].

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Fig. 7. Radial velocity and eddy viscosity of the narrow vaneless diffuser near the instability condition (bz = 0.055). Table 2 Critical inflow angles for the narrow and wide isolated vaneless diffusers. Width ratio 0.055

0.14

Velocity distribution

Critical inflow angle (◦ )

Axisymmetric D1 D2 D3 Axisymmetric D1 D2 D3

8.53 8.59 8.74 8.80 14.31 14.66 15.20 15.45

compressor is unstable or in a stall condition when the corrected mass flow rate is <1.47 kg/s. At the mass rate of 1.47 kg/s, the frequency spectrum of the monitored pressure at the diffuser inlet after >5000 timesteps, which is long enough for the perturbation development, is shown in Fig. 6(b). A blade passing frequency and main blade passing frequency of 8846.75 and 4429.60 Hz, respectively, are observed with amplitude peaks in the spectrum. Moreover, a low-frequency signal with the value of 248.85 Hz can be observed, which may be the stall frequency. For the isolated diffusers, the critical inflow angle can be obtained when the simulation cannot converge. As shown in Table 2, the narrow diffuser with a width ratio of 0.055 and the wide diffuser with a width ratio of 0.14 both tend to be less stable when the skewness of the inflow velocity distribution increases. The narrow diffuser is far more stable than the wide diffuser, as proven by previous experiments [12]. 3.3. Mean flow The mean flow of the isolated diffuser with the width ratio of 0.055 under different inflow conditions is presented in Fig. 7. It shows time and circumferentially mass averaged velocity and eddy viscosity for the narrow diffuser. In addition to the characteristics of the radial velocity, the areas where reverse flow occurs are marked by black arrows in Fig. 7(a). Owing to the mass conservation law and friction loss, the radial velocity tends to decrease along the diffuser radius, although it increases near the inlet un-

der the effect of the increased boundary-layer thickness. Then, the boundary layer converges at the radial position of r = 1.3, and a fully developed flow is formed for all four diffusers. With an increase in the skewness of the inflow velocity, the mean flow near the diffuser inlet tends to exhibit S-shape development with stronger flow separation and a reverse flow near the shroud and hub walls. The influence of the asymmetric inflow can hardly be observed when the radius ratio is >1.4, at which the fully developed flow is dominant for the narrow diffuser. In Fig. 7(b), the eddy viscosity of the mean flow for the vaneless diffuser is illustrated. The influence of the velocity distribution D1 differs little from that of the axisymmetric distribution. For distributions D2 and D3, the eddy viscosity exhibits opposite distributions in the axial directions compared with the radial velocity. This indicates that the normal Reynolds stress or the viscosity are most effective in the diffuser inlet region. The mean flow of the wide diffuser with a width ratio of 0.14, which has an inflow condition similar to that of the narrow diffuser, is presented in Fig. 8. It shows time and circumferentially mass averaged velocity and eddy viscosity for the wide diffuser. The plot of the radial velocity indicates that the inviscid main flow dominates the wide diffuser passage, and the high-radial velocity region is mainly located at the radial position of r = 1.2–1.4. A fully developed flow is formed near the diffuser outlet under the axisymmetric and D1 inflow distributions. Under the influence of the asymmetric distributed inflow velocity, flow separation and a reverse flow occur to a large extent near the shroud wall. The eddy viscosity is closely related to the development of the boundary layer. In particular, for the diffusers with inflow distributions D3 and D4, the eddy viscosity increases significantly in the area where flow separation occurs. Compared with the eddy-viscosity distribution for the narrow diffuser, the plot for the wide diffuser indicates that the maximum value of the viscosity occurs near the diffuser outlet. 4. Stability results Once the circumferential averaged mean flow was obtained from numerical simulations, it is interpolated to a coarser mesh, on which the stability problem is solved. In the present study, the

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Fig. 8. Radial velocity of the wide vaneless diffuser near the instability condition (bz = 0.14).

Fig. 9. Eigenvalue spectrum and wavenumber prediction results for the wide diffuser with inflow distribution D3 (bz = 0.14).

MATLAB software is used to solve the eigenvalue problem. The imaginary parts of the predicted leading eigenvalues are close to zero under the critical inflow angles, and the flow is asymptotically stable, partly owing to the characteristics of the mean-flow approach. For the wide diffuser (bz = 0.14) with inflow distribution D3, the results of the stability calculations are presented in Fig. 9. To avoid the spurious eigenvalue, all the values are ensured to be independent of the grid number. The leading eigenvalue of the vaneless diffuser can be defined as the eigenvalue with the largest imaginary part. The predicted wavenumber is 4, which agrees with experimental observation of [22], for both the narrow and wide diffusers, indicating that the width ratio may not be the identifying factor that determines the perturbation number. However, the predicted rotating speed or frequency of the instability perturbations generally increases as the wavenumber increases in the range of 1–6. The predicted rotating speed of perturbation for the narrow and wide diffusers with different inflow conditions is validated against previously reported experimental data [12], as shown in Table 3. With an increase in the skewness of the inflow velocity distribution, the predicted frequency tends to increase up to D2 and decrease at D3. At D3, the predicted results become closer to the experimental results compared with that of the axisymmetric case. This is because the asymmetric distribution of the inflow velocity is similar to that of the practical flow in industrial centrifugal com-

Table 3 Predicted stall frequency and wavenumber for the isolated vaneless diffusers. Width ratio

0.055

0.14

Velocity distribution

Rotating speed

Experimental value

Wavenumber

Axisymmetric D1 D2 D3 Axisymmetric D1 D2 D3

0.225 0.233 0.241 0.238 0.205 0.220 0.223 0.214

– – – – 0.21 0.21 0.21 0.21

4 4 4 4 4 4 4 4

pressors. Thus, the frozen eddy viscosity approach can predict the frequency of instability perturbations. 5. Sensitivity analysis and instability mechanism 5.1. Isolated narrow diffuser The general dynamics for the narrow diffusers with different inflow conditions are identical to those described in Fig. 10 in the case of direct and adjoint modes. The direct velocity perturbations, which are mostly localized near the diffuser outlet for the four inflow conditions, increase in amplitude while moving toward the

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Fig. 10. Direct and adjoint velocity modes for the narrow diffuser near the instability condition (bz = 0.055).

Fig. 11. Structural sensitivity of the narrow diffuser near the instability condition (bz = 0.055).

diffuser outlet. This is physically correct not only because the velocity perturbations are limited by the inlet boundary condition but also because they are easily induced where the damping effect of the viscosity is weak. The flow receptivity is represented by the modulus of the adjoint velocity or pressure, which is used to evaluate how the flow reacts to the external disturbances of a given frequency at a certain spatial location. The adjoint velocity modes as shown in Fig. 10 (b) exhibit the opposite distribution to the direct modes, and they reach a maximum value at the diffuser inlet. The contribution of

the asymmetric inflow distribution to the adjoint modes dominates the adjoint field, suggesting that the axial velocity distribution plays a significant role in the dynamics of the flow receptivity. Furthermore, the spatially separated direct and adjoint modes indicate the large non-normality of the global mode. By combining the direct and adjoint modes, the structural sensitivity can be used to reveal the origin of instability perturbations. The structural sensitivity of the narrow diffuser, which is the inner product between the direct and adjoint velocity, are illustrated in Fig. 11. The high-magnitude regions of the structural sensitiv-

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Fig. 12. Direct and adjoint global modes of the wide diffuser near the instability condition (bz = 0.14).

ity for all four diffusers are localized at the downstream position of the intersection of boundary layers, although different inflow distribution are imposed. The main difference of the diffuser with distribution D3 from the others is that the core of the structural sensitivity is close to the hub side. This indicates that the inflow distribution asymmetry may influence the axial position of the wavemaker. According to the radial velocity shown in Fig. 7(a), there exist flow separation region at the radial position of r = 1.1 near the shroud. However, no obvious high sensitivity regions can be observed near the shroud side at the same radial position in Fig. 11. It is indicated that the contribution of the reverse flow near the shroud and hub walls to the amplification of the perturbation in the narrow diffusers is relatively small. Regarding the instability mechanism for the narrow diffusers, an inflection point of the velocity axial distribution may exist under the interaction between the main flow and the boundary-layer flow owing to the different flow directions, and instability perturbation can be easily induced at the inflection points, which is similar to the instability in crossflows. After the flow is fully developed, the flow directions for the passage tend to be the same, and the wavemaker structure no longer exists. 5.2. Isolated wide diffuser Regarding the wide diffusers, Fig. 12 shows the direct and adjoint global modes of the diffuser with a width ratio of 0.14 under four different inflow conditions. The magnitude of the direct global modes tends to be large at locations where the velocity is low and the viscosity is high, particularly near the diffuser outlet. The asymmetric distribution of the mean flow significantly influences the global modes, and the perturbations for the diffusers with distributions D2 and D3 are mainly localized on the shroud side. This indicates that the boundary layer makes greater contributions to the perturbations with an increase in the diffuser width ratio. Further evidence of this lies in the contours of the adjoint global modes in Fig. 12(b). The separation regions of the boundary layers at the radial position of r = 1–1.2 are the main origins of the adjoint perturbations. The effect of the asymmetric inflow distribution of the mean flow on the adjoint modes is similar to that on the direct modes, as the magnitude of the adjoint perturbations on the shroud side is significantly larger than that on the hub side.

Additionally, the adjoint modes tend to increase on the hub side near the diffuser outlet as the inflow velocity skewness increases. As shown in Fig. 13, the magnitude of the product of the direct and adjoint global modes determines the structural sensitivity of the wide diffuser. For the diffusers with the axisymmetric and D1 inflow conditions, the wavemaker regions are located in the core of the separation bubble on both sides of the diffuser walls. When the asymmetric inflow is imposed, the separation regions on the shroud side near the inlet and on the hub side near the outlet are responsible for the flow instability. With an increase in the inflow skewness, the effect of the reverse flow near the outlet appears to increase. Thus, it is reasonable to assume that the instability perturbations in the wide diffuser with the axisymmetric inflow condition are induced and amplified in the separation region of boundary layers. This may explain why the narrow diffuser is far more stable than the wide diffusers. With an increase in the width ratio, the stabilization effect of the viscosity tends to decrease, and the flow stability can be more easily excited with the boundary-layer separation dominating the diffuser field. To obtain further insight into the origin of the instability and flow characteristics, the flow angle of the mean flow and the streamline of the global modes for the diffuser with distribution D3 were examined, as shown in Fig. 14. According to the mean flow, negative values of the flow angle are observed near the shroud and hub walls. This indicates that boundary layer undergoes separation on the shroud side near the diffuser inlet and on the hub side near the diffuser outlet. The streamlines of the direct and adjoint modes are presented in Fig. 14(b). The introduction of self-excited perturbations is permitted by using the Dirichlet boundary condition. For the direct global modes, the backflow perturbations with the highest amplitude are observed at the outlet. Strong adjoint perturbations are localized in the separation regions in both the upstream and downstream directions. Meanwhile, strong oscillation of the direct and adjoint mode can be found at the interface between the boundary layer and main flow. When the direct and adjoint modes are combined, the separation regions marked as regions A and B act as instability perturbation amplifiers, along with the interacting region between the boundary layers and the main flow, which is marked as region C. The intrinsic nature of the structural sensitivity suggests that the instability perturbations in the wide diffuser with a high-skew inflow may be induced and amplified in the separation region and the in-

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Fig. 13. Structural sensitivity of the wide diffuser near the instability condition (bz = 0.14).

Fig. 14. Modal amplitude distribution for the wide diffuser with inlet distribution D3 (bz = 0.14).

teraction region between the boundary layers and main flow. The contributions of the different factors to the instability properties may change slightly as the axial distribution of the inflow velocity changes. 5.3. Full annular Thus far, the influence of the asymmetric inflow on the instability properties has been discussed with regard to the isolated vaneless diffuser. To obtain insight into the instability mechanism in the practical compressor, the sensitivity analysis of the full-annular compressor with the vaneless diffuser (R f = 1.53, bz = 0.1) at the mass flow rate of 1.47 kg/s is considered here. Using Fig. 15(a), the mean flows of the isolated diffusers with axisymmetric and D3 inflow distributions and the diffuser with the rotating impeller in the upstream direction are investigated. For the isolated diffuser, the boundary layer at both walls converges near the diffuser outlet, and the eddy viscosity is related to the development of the boundary layer. When connected to the impeller, the diffuser inflow is composed of an asymmetric flow in the axial direction and a jet-wake flow in the circumferential direction. Under the influence of tip clearance leakage flow, the inflow radial velocity on the

hub side is significantly higher than that on the shroud side. A reverse flow can be observed on the shroud side near the diffuser inlet. Fig. 15(b) shows the radial velocity of the diffuser from the full-annular simulation at the 95% spanwise plane. When the flow is near the stall condition, a reverse flow at the diffuser inlet is observed near the shroud wall, and a large backflow is observed near the diffuser outlet. Because the reverse flow and the jet-wake flow is in the radial range of r = 1.15–1.2, the backflow at the diffuser outlet has no interaction with the reverse flow at the diffuser inlet. The structural sensitivity of the diffusers obtained from the isolated-diffuser and full-annular simulations is presented in Fig. 16. Under an axisymmetric inflow, the wave-like perturbations are mainly induced at the shear layer where the boundary layers intersect. However, under a skewed inflow, the reverse flow at the radial position of r = 1.05–1.15 on the shroud side acts as the main wavemaker. Compared with the isolated diffuser, the wavemaker region is localized at the radial position of r = 1.25–1.53 for the diffuser connected to the impeller. This indicates that the outlet backflow has the most significant effect on the occurrence of instability perturbations. Although the inflow distribution of the

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Fig. 15. Radial velocity of the diffusers with axisymmetric, D3, and full-annular inlet distributions (bz = 0.1).

maker regions responsible for flow instability were confirmed by performing a structural-sensitivity analysis. The meaning of the present work lies at improving the predicting accuracy of the stability model proposed previously. Especially at the design stage, the influence of the inflow velocity distribution on the stability can be considered. Another application of the present work is the flow control of the vaneless diffuser. The receptivity based on the adjoint mode considers how the flow reacts to one environmental disturbance or harmonic forcing. When flow control techniques such as unsteady blowing/suction or wall roughness change are introduced, the determination of the optimal location can be directly guided by the adjoint mode. The following conclusions are drawn.

Fig. 16. Sensitivity for the diffusers with axisymmetric, D3, and full-annular inlet distributions (bz = 0.1).

diffuser with the impeller in the upstream direction is similar to the D3 distribution in the axial direction, the jet-wake flow in the circumferential direction alters the sensitivity at the diffuser inlet. For the diffuser with a radius ratio of 1.53, the reverse flow existing in the jet-wake flow does not necessarily lead to instability. According to the work of Dean [39], the effects of the jet-wake flow and the outlet backflow are related to the radius ratio, and the wavemaker region may change significantly when the radius ratio changes. 6. Conclusions and discussions The instability properties of an isolated vaneless diffuser and a full-annular vaneless diffuser were studied in the framework of the turbulent biglobal linear theory. The mean flow for the vaneless diffusers was obtained numerically using URANS, and the simulation results were validated against experimental data. Then, direct and adjoint global stability analyses were performed for a vaneless diffuser with an asymmetric inflow distribution. The wave-

1. The frozen eddy viscosity approach can be used to predict the instability frequency. With an increase in the skewness of the inflow velocity distribution, the predicted frequency tends to increase up to D2 and decrease at D3. The width ratio may not have identical effects on the circumferential wavenumber. 2. For the isolated narrow vaneless diffuser, the shear layer in the interaction region between the main flow and the boundarylayer flow may be the main wavemaker region for both axisymmetric and asymmetric inflows, although inflow distribution asymmetry may influence the axial position of the wavemaker. The reverse flow near the shroud and hub walls has little impact on the occurrence of instability perturbations. 3. For the isolated wide vaneless diffuser, instability perturbations with the axisymmetric inflow condition are induced and amplified in the separation region of boundary layers. Under an asymmetric inflow, e.g., D2 or D3, the separation regions at the shroud wall near the diffuser inlet and the hub wall near the diffuser outlet are responsible for the self-excited perturbations. The interaction region between the boundary layers and the main flow may contribute to the flow instability. 4. Compared with the isolated diffuser having a skewed inflow distribution, the reverse flow existing among the jet-wake flow does not lead to instability for the full-annular diffuser. The instability perturbations are mainly induced and excited by the outlet backflow.

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12

im r

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

vθ r

+

∂ vθ ∂r

(− ∂∂vzz −

∂ vr ∂r

+ς

This work was supported by the National Natural Science Foundation of China (Grant No. 51736001) and was undertaken at the Turbomachinery Institute of Beijing Institute of Technology, China. Appendix A

M qˆ = ω J qˆ

(A.1a)

M=

∂ vr ∂r

+

v r ∂∂r ∂ vθ ∂r

+

∂ ∂r

imv θ r

+

v z ∂∂z

+

+

vθ r

1 2im Ree f f r 2

− ∂ vz ∂r

∂

−

−η−ξ −

1 Ree f f

∂r

−

∂ Re1 ∂r

∂ Re1

im r

∂ ∂r

ef f

− 2vr θ + v r ∂∂r +

imv θ r

∂r

∂ ∂z

ς=

vr r

−η−ξ −

v r ∂∂r +

⎛

imv θ r

∂ vr ∂z

−

∂ vθ ∂z

−

∂ ∂z ∂ Re1

∂ Re1

ef f

∂r

1 r

∂ ∂r

∂ ∂r

ef f

0 0 0 0

1 r2

−ξ −

i

1

2



Ree f f

∂ Re1

ef f

∂r

0

∂2 m2 ∂ 2 1 ∂ 1 ∂ − + + − 2 2 2 2 2 r ∂r ∂r r ∂θ ∂z r

∂ Re1 ∂ ∂ ef f + ∂r ∂z ∂z

∂

1 Ree f f

∂z

∂ ∂z

∂ ∂z

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(A.1b)

⎞

⎜ i 0 0 0⎟ ⎟ J =⎜ ⎝0 i 0 0⎠ 0 0

im r

ef f

∂z

z + dv + v z ∂∂z − η − Re1 dz

(A.1c)

 (A.1d)

(A.1e)

where qˆ = (uˆ , pˆ )T . The corresponding eigenvalue problem for the adjoint linearized Navier-Stokes equations are expressed as follows: †

M † qˆ = ω† J † qˆ

†

†

M =

⎛

1 + ∂∂r r ⎜ ∂ vz θ ⎜ (− ∂ z − vrr + ς ) + ξ † ∂∂r − imv + κ ∂∂z − η† r ⎜ 2v 1 2im θ ⎜ − r − Re r 2 ⎜ ef f ⎝ ∂ Re1 ∂ 2 Re1 ∂ 1 ef f ∂ ef f ∂ vr 1 Ree f f + + + ∂z ∂ z ∂r ∂r∂ z r ∂z

−

imv θ r

0



Ree f f

1

1 Ree f f

∂z

∂ ∂z ∂ ∂z

∂r

0

+

∂

2

1 Ree f f

∂r∂ z

0

1

0 0⎟ ⎟ 0⎠ 0

+

∂ Re1

2 ∂ Ree f f r

∂r

∂ Re1

ef f

∂z

ef f

∂ z2

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

− ∂∂z

(A.2c)

ef f

+

 (A.2d)

(A.2e)

∂r

1

+

∂ 2 Re1

− ∂∂r − im r

⎞

(A.2b)

⎞

∂2 ∂2 m2 ∂ 2 − 2 2 + 2 2 ∂ r r ∂ θ ∂ z

Ree f f r

1

+ ξ † ∂∂r − η† + κ ∂∂z −

0 0 0 −i

1

Ree f f r 2

†

∂z ∂ Re1

im r

∂

∂

im r

∂ 2 Re1

ef f

∂ 2r

(A.2f)

(A.2g)

†

where qˆ = (uˆ , pˆ † )T .

0

ef f

1

κ = −v z +

0

ξ=

0

ξ † = −vr +

ef f

1 2im Ree f f r 2

+ v z ∂∂z +

vr r

⎜ −i 0 J† = ⎜ ⎝ 0 −i

η† =

im r

η=

⎛

0 1 r

+

+

− η†

1 Ree f f

0

− ∂∂vzz −

The eigenvalue problem for the linearized Navier Stokes equations can be expressed as follows:

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

∂ vz ∂r

1

κ

∂ vθ ∂z

Acknowledgements

⎛

∂

Re im ( 1 2 + ∂ er f f ) r Ree f f r θ ) + ξ † ∂∂r − imv + ∂∂z r

+

(A.2a)

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