A Novel Approach to Predicting Critical Alternating Stall in a Centrifugal Pump Impeller with Even Blades Under Transient Conditions

Abstract

A novel approach is proposed to predict alternating stall in a centrifugal pump impeller with even blades by introducing a low-pressure ratio, which is defined as the ratio of the deviation of the low-pressure zones of adjacent impeller passages. The threshold of 2/3 is shown to be a good quantity with which to accurately and quantitatively predict alternating stall and even critical alternating stall (CAS). The effectiveness of this new approach is validated by comparison with previous findings obtained under quasi-steady conditions. Large eddy simulation data for a six-blade centrifugal pump impeller are used to predict the CAS under transient conditions, with the transient conditions corresponding to a sinusoidal flow rate with an equilibrium value of 0.5Q_d (where Q_d is the design load) and an initial phase of zero combined with different oscillation amplitudes. The low-pressure ratio frequency equals the flow rate frequency, approximately 2 Hz. The phase of the low-pressure ratio lags behind the flow rate. When the oscillation amplitude is larger than 0.15Q_d, a non-stall state occurs during the dropping stage of the flow rate. The flow rates corresponding to the CAS during the dropping and rising stages, respectively, increase and decrease as the oscillation amplitude increases.

1. Introduction

Alternating stall in centrifugal pumps with even blades is a key issue associated with the unstable flow phenomenon, which often occurs under low flow rates (typically when the flow rate is below 0.7Q_d [1,2,3]). This phenomenon usually leads to pump instability, low efficiency, high vibration, and strong noise and could potentially result in significant damage to the entire pump system. Therefore, a detailed understanding of alternating stall can not only help reveal the internal flow mechanism of centrifugal pumps [4,5] but can also help with the pump design [6,7].

Pedersen et al. [8] and Byskov et al. [9] observed a two-channel phenomenon in a centrifugal pump impeller with six blades operating at a quarter load. Subsequently, a volute was added to the centrifugal pump impeller, and the velocity distributions were measured using laser Doppler velocimetry by Johnson et al. [10]. Their findings revealed that the reduced flow rate experienced significant changes, exhibiting alternating stable patterns of stationary stall and non-stall passages, which aligned with the results obtained for a centrifugal pump impeller. Ran and Wang [11] observed alternating stall in an investigation of flow streamlines, whose pressure fluctuations were lower in amplitude and higher in frequency than those seen in rotating stall [12,13]. Cheah et al. [14] simulated the complex internal flow in a centrifugal pump impeller with six twisted blades and identified the alternating stall at a low flow rate. Zhou et al. [15] investigated the unsteady flow structures and evolution under alternating stall and rotating stall in centrifugal pump impellers. According to the characteristics of stall cells in a compressor [16] or a centrifugal pump [17], alternating stall means that the stall cells and non-stall cells are alternately distributed in the impeller passages, forming a periodic symmetry, and the circumferential propagation speed is zero. Although the stall cells play an important role in the process of alternating stall, the definition of alternating stall has not yet been unified. Stall cells can often be represented by a significant pressure drop region in the impeller passage [18], a large back-flow region caused by the increase in the attack angle at the inlet guide vane [19], a low-speed region in the inlet impeller close to the suction side [14], and stall vortex structures in the impeller identified using the Q-criterion [1].

Although existing stall prediction methods helped with understanding the mechanism behind alternating stall, they have mainly involved qualitative analyses, and quantitative predictions are more difficult. Therefore, a major goal of the current study is to propose a novel approach to quantitatively predict alternating stall in centrifugal pump impellers.

Centrifugal pumps often operate under transient conditions wherein the flow parameters dramatically change over time, potentially leading to negative effects from pressure fluctuation [20]. Many investigations have been performed on the conditions of pump startup/shutdown [21,22,23,24,25,26] and rapid valve adjustments [27,28,29]. Considering transient conditions with a sinusoidal flow rate, Kuang et al. [30] investigated the pressure fluctuation characteristics in stall and non-stall passages and compared the results with the corresponding results under quasi-steady conditions. Subsequently, Kuang et al. [31] and Li et al. [32], respectively, discussed the influence of the oscillation frequency and oscillation amplitude on the internal flow characteristics. They found that the impeller may be in a stall state and was in a non-stall state at different moments. The critical alternating stall (CAS) under transient conditions is not yet known. Therefore, another main purpose of this study is to determine the CAS under such conditions.

This manuscript is organized as follows: An introduction of the geometric models and numerical methods is given in Section 2. An introduction and validation of the novel approach are presented in Section 3. In Section 4, the novel approach is applied to the transient conditions. The main conclusions are summarized in Section 5.

2. Geometric Models and Numerical Methods

2.1. Physical Model

The physical model of the centrifugal pump impeller used in this work represents a low-specific speed centrifugal pump operating with water as the working medium, which is an experimental model at the Technical University of Denmark [8]. The centrifugal pump impeller inlet radius measures 35.5 mm, and the outlet radius is 95 mm. The main parameters of the impeller are summarized in

Table 1. Geometric parameters of the centrifugal pump impeller.

Parameter	Symbol	Numerical Value	Units
Specific speed	ns	26.3	rev/min
Blade radius of curvature	Rb	70	mm
Inlet placement angle	λ1	19.7	deg
Outlet placement angle	λ2	18.4	deg
Number of blades	Z	6	—
Blade thickness	d	3	mm
Inlet radius	R1	35.5	mm
Outlet radius	R2	95	mm
Inlet height	b1	13.8	mm
Outlet height	b2	5.8	mm

. The structural layout of the impeller domain features six blades that are both swept back and vertical, along with inlet and outlet pipes, as shown in Figure 1. The specific impeller parameters are as follows: a design flow rate of Q_d = 3.06 kg/s, a design head of H_d = 1.75 m, and a design rotational speed of n = 725 rev/min.

2.2. Governing Equations

Advancements in computational fluid dynamics have led to the extensive application of numerical simulations in the investigations of centrifugal pump impellers. In this study, steady turbulence is simulated using the Shear Stress Transport k-ω model. Following this, the steady-state results serve as the initial flow field for large eddy simulations, which are employed to capture the unsteady turbulence characteristics of the centrifugal pump impeller.

(1)

Shear Stress Transport k-ω model

The governing equation of the Shear Stress Transport k-ω equation can be expressed as follows [33]:

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho k\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho u_{j}k\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{k3}}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+P_k-{\beta }^{\prime }\rho k\omega ,$$

(1)

$${\displaystyle \frac{\partial }{\partial t}}(\rho \omega )+{\displaystyle \frac{\partial }{\partial x_j}}(\rho u_j\omega )={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\omega 3}}}\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+(1-F_1)2\rho \times {\displaystyle \frac{1}{{\sigma }_{\omega 2}\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}+{\alpha }_3{\displaystyle \frac{\omega }{k}}{P}_k-{\beta }_3\rho {\omega }^2,$$

(2)

where t and x denote the time and spatial coordinates, respectively; u denotes the velocity; ρ denotes the density; p denotes the pressure; k denotes turbulent kinetic energy; ω denotes specific dissipation rate; μ denotes dynamic viscosity; and μ_t denotes turbulent viscosity. $\beta ’$ is an empirical constant, with a value of 0.09. ${\alpha }_3$, ${\beta }_3$, ${\sigma }_{k3}$, and ${\sigma }_{\omega 3}$ denote the turbulent coefficients. F₁ denotes the blending function, and P_k denotes the generation term of turbulent kinetic energy, which are written as follows [34]:

$$F_1=\tanh (ar{g}_1^4),$$

(3)

$$ar{g}_1=\min \left(\max \left({\displaystyle \frac{\sqrt{k}}{{\beta }^{\prime }\omega y}},{\displaystyle \frac{500\upsilon }{y^2\omega }}\right),{\displaystyle \frac{4\rho k}{C{D}_{k\omega }{\sigma }_{\omega 2}{y}^2}}\right),$$

(4)

$$C{D}_{k\omega }=\max (2\rho {\displaystyle \frac{1}{{\sigma }_{\omega 2}\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}},1\times {10}^{-10}),$$

(5)

$$P_k={\mu }_t\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right){\displaystyle \frac{\partial u_i}{\partial x_j}}.$$

(6)

where y is the distance to the grid center away from the wall.

The coefficients are a linear combination of the corresponding coefficients of the underlying models, as follows:

$${\varphi }_3=F_1{\varphi }_1+(1-F_1){\varphi }_2.$$

(7)

The coefficients of the underlying models are listed for completeness: ${\alpha }_1$ = 5/9, ${\beta }_1$ = 0.075, ${\sigma }_{k1}$ = 1.176, ${\sigma }_{\omega 1}$ = 2, ${\alpha }_2$ = 0.44, ${\beta }_2$ = 0.0828, ${\sigma }_{k2}$ = 1, ${\sigma }_{\omega 2}$ = 1/0.856.

Moreover, the specific turbulence dissipation rate is calculated using the following:

$${\omega }_{\mathrm{inlet}}=\rho {\displaystyle \frac{k_{\mathrm{inlet}}}{{\mu }_t}},$$

(8)

$$k_{\mathrm{inlet}}={\displaystyle \frac{3}{2}}{I}^2{u}_{\mathrm{inlet}}^2,$$

(9)

$$I={\displaystyle \frac{U_{\mathrm{inlet}}}{u_{\mathrm{inlet}}}},$$

(10)

$${\mu }_t=CI\mu ,\hspace{1em}C=1000.$$

(11)

where U denotes fluctuating velocity. The turbulence intensity is set to 5%, and the corresponding turbulent viscosity ratio is 50.

(2)

Dynamic Smagorinsky model

The turbulence scales that can be directly computed are referred to as solvable or resolvable turbulence scales, and those that cannot be directly computed are termed non-resolvable or subgrid-scale turbulence. The subgrid-scale stress tensor resulting from the filtering process remains unknown and requires a modeling approach. The specific formulations employed in this modeling are detailed below:

$${\tau }_{ij}-{\displaystyle \frac{1}{3}}{\delta }_{ij}{\tau }_{\overline{k}k}=-2v_T{S}_{ij},$$

(12)

$$S_{\mathrm{ij}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)$$

(13)

$${\nu }_T={(C_s\mathsf{\Delta })}^2{(2S_{\mathrm{ij}}{S}_{ij})}^{1/2}$$

(14)

Germano et al. [35] introduced the dynamic Smagorinsky model, which was an improvement based on the model proposed by Lilly [36]. The core concept of this dynamic approach is to incorporate local turbulence structure information into the subgrid-scale stress through a pair of filtering steps, allowing for the adjustment of model coefficients during the computation. The stress tensor can be represented as follows:

$$L_{\mathrm{ij}}=\widetilde{u_i}\widetilde{u_j}-\widetilde{u_i{u}_i}=T_{ij}-{\tau }_{\mathrm{ij}}$$

(15)

$$L_{ij}-{\displaystyle \frac{1}{3}}{\delta }_{ij}{L}_{kk}={(C_D)}^2{M}_{ij}$$

(16)

$$M_{ij}=2{(\tilde{\mathsf{\Delta }})}^2|\widetilde{S_{ij}}|\widetilde{S_{ij}}-2{(\mathsf{\Delta })}^2\widetilde{|S_{\mathrm{ij}}|S_{\mathrm{ij}}}$$

(17)

The tilde represents the filtering operation with scale, and u_i is the resolved velocity component:

$$C_D={\displaystyle \frac{L_{ij}{M}_{ij}}{M_{ij}{M}_{ij}}}$$

(18)

2.3. Numerical Methods and Boundary Conditions

ANSYS ICEM 18.0 was employed to discretize the computational domain into structured grids, with a finer mesh specifically applied to the near-wall regions that exhibit significant pressure and velocity gradients. Figure 2 illustrates the grid layout for the centrifugal pump impeller. The total cell count for the centrifugal pump impeller system amounts to 11 million, comprising 7.67 million cells for the impeller, 2.55 million for the inlet pipeline, and 0.78 million for the outlet pipeline.

The simulations were conducted using ANSYS CFX 20.0, with consistent boundary conditions applied across the centrifugal pump impeller. The finite volume method was employed to resolve the system, utilizing the SIMPLE algorithm for the coupling of velocity and pressure. High-resolution techniques were implemented to enhance the overall accuracy, while the second-order backward Euler method was applied to address the unsteady components. For the transient calculations, a transient rotor–stator interface was utilized. At the inlet duct, a velocity boundary condition was enforced, while a Neumann boundary condition was chosen for the outlet regarding velocity, with pressure specified at the outlet. All walls were subject to defined boundary conditions of no-slip and zero penetration velocity. The time step is set at Δt = 2.3 × 10⁻⁴ s, with a sampling frequency of 4.35 kHz, ensuring that the accuracy requirements for subsequent data analysis are met. The absolute residual criterion is defined as 10⁻⁶, ensuring a high level of convergence accuracy. As several experimental measurements and numerical simulations have already been reported for quasi-steady conditions, data verification is based on these conditions. The specific validation results of the numerical simulations are detailed in previous articles, particularly in the studies by Kuang et al. [30].

2.4. Flow Rate Conditions

Numerical simulations are performed under both quasi-steady and transient conditions. Firstly, under quasi-steady conditions, three flow rates (0.25Q_d, 0.5Q_d, and 0.75Q_d) are selected to analyze. The transient conditions are considered to be a sinusoidal flow rate with an initial phase in the present study, whose mathematical expression is as follows:

$$Q=0.5Q_d+A\cdot \sin ({\displaystyle \frac{\pi \cdot n}{3}}\cdot \mathrm{t})$$

(19)

where 0.5Q_d is the equilibrium flow rate. The frequency in an oscillation period is n/6. The oscillation amplitudes A for the five different schemes are set to 0.1Q_d, 0.15Q_d, 0.2Q_d, 0.25Q_d, and 0.3Q_d, respectively. Additionally, the duration of each cycle for the inlet boundary conditions across all schemes is standardized to span six complete rotations of the impeller. When the oscillation amplitude is 0.25Q_d, respectively, the relationship between the flow rate and time is as shown in Figure 3. Under transient conditions, four instantaneous flow conditions were selected: Q₂ represents the maximum flow of 0.75Q_d, Q₄ represents the minimum flow of 0.25Q_d, and Q₁ and Q₃ represent 0.5Q_d during the rising and dropping stage, respectively.

3. Introduction and Validation of the Novel Approach

3.1. Detailed Approach Descriptions

Alternating stall is a common phenomenon in centrifugal pump impellers with even blades at low flow rates. During alternating stall, there is a significant difference in the low-pressure zones of adjacent passages. Such a phenomenon can be used to intuitively predict alternating stall, but its accuracy is relatively weak.

The low-pressure zone refers to a static pressure region with a pressure value of less than a certain critical value. It can be better reflected by the pressure coefficient, which is defined as follows:

$$C_p={\displaystyle \frac{p-p_2}{0.5\rho u_2^2}}$$

(20)

The subscript “2” indicates an impeller outlet condition. The low-pressure zone is observed in the geometric model of the present study under quasi-steady conditions at 0.25Q_d, as shown in Figure 4.

A novel parameter, called the low-pressure ratio, is proposed to quantitatively predict the alternating stall. The parameter is expressed as follows:

$$C_S={\displaystyle \frac{S_a-S_b}{\frac{1}{2}(S_a+S_b)+\epsilon }}$$

(21)

where S is the area of the low-pressure zone, and ε is a small positive number used to avoid division by zero, which is suggested to be 10^ࢤ6. A value of 0.0 indicates that Sa equals S_b, and all other values indicate that S_a is larger than S_b. Moreover, C_s ≥ 2/3, which indicates that S_a is at least twice S_b, is adopted to show that the impeller is in a stall state.

The steps used to obtain the C_s based on numerical results are as follows:

Step 1: The pressure coefficient is calculated using Equation (20). The region in which the pressure coefficient is less than −0.75 is considered the low-pressure zone. The low-pressure zone is obtained for each impeller passage.

Step 2: The C_s is calculated using Equation (21). The number of C_s is equal to the number of impeller blades.

Step 3: The six values are averaged to obtain the instantaneous coefficient.

It should be noted that, when analyzing unsteady numerical results, the ensemble average should also be considered.

3.2. Approach Validation Under Quasi-Steady Conditions

Figure 5a shows that the C_s is obviously affected by the flow rate. The C_s is approximately 0.18 at 0.75Q_d, which is smaller than 2/3. This indicates that the impeller is in the non-stall state. However, the C_s is approximately 1.47 at 0.25Q_d and 1.15 at 0.5Q_d, which means that the impeller is in the stall state.

The stall and non-stall states can also be investigated through the distributions of the pressure coefficient in the impeller mid-height plane (z/b2 = 0.5), as shown in Figure 5b. It is shown that a low-pressure zone can be observed in the suction front edge of the impeller passages for all flow rates. When the flow rate is 0.25Q_d or 0.5Q_d, there is a significant difference in the low-pressure zones of adjacent passages. These differences are not obvious when the flow rate is 0.75Q_d. A similar conclusion was reached by Zhou et al. [29].

Although the distributions of the low-pressure zones have a certain relationship with the rotational time, C_s is relatively insensitive to the rotational time, as shown in Figure 6. The fluctuation difference of the C_s at different rotational times is within 3.2%. Therefore, the accuracy of this novel approach is very high under quasi-steady conditions.

4. Application of the Novel Approach to Transient Conditions

4.1. Determination of the Critical Alternating Stall

Figure 7 shows the distributions of the pressure coefficient at four instantaneous flow rates—the maximum (Q₂), minimum (Q₄), and equilibrium (Q₁ and Q₃) flow rates. The low-pressure zones of adjacent passages are similar and different at the instantaneous flow rates of 0.75Q_d and 0.25Q_d, respectively. For the instantaneous flow rate of 0.5Q_d, the low-pressure zones of adjacent passages vary during the rising and dropping stages. These phenomena cause C_s to deviate significantly from the corresponding flow rates under quasi-steady conditions. For example, the C_s during the rising and dropping stages are 1.01 and 0.68 for the instantaneous flow rate of 0.5Q_d, respectively.

To further investigate the evolution of C_s with the rotational time, thirty-six instantaneous times are chosen in an oscillation period. Two adjacent instantaneous times are separated by 0.0138 s, corresponding to a blade rotation of 60 degrees. The frequency of the low-pressure ratio is equal to the frequency of the flow rate, which is approximately 2 Hz. Figure 8 shows that C_s cannot be represented by a value and that it shows obvious periodicity. The maximum and minimum values are approximately 1.31 and 0.20, respectively, which are smaller and larger than those under the quasi-steady conditions at 0.75Q_d and 0.25Q_d. The phase of the low-pressure ratio lags behind the flow rate by fifteen thirty-sixths of a period. Moreover, the stall and non-stall states alternate in a transient period of the flow rate. The flow rate corresponding to the CAS during the dropping and rising stages, named Q_cd and Q_cr, are approximately 0.72Q_d and 0.5Q_d, respectively.

Fluctuating characteristics play an important role in LES results. For a fully developed turbulent flow, the internal flow field exhibits a certain similarity for different flow rate periods. Three oscillation periods are considered in this study, as shown in Figure 9. It can be seen that the mean Q_cd and Q_cr, respectively, are 0.71Q_d and 0.49Q_d. Although the fluctuating CAS during the dropping stage is larger than that during the rising stage, it is still within 4%.

Regardless of whether the instantaneous flow rate equals Q_cd or Q_cr, a significant double channel phenomenon can be observed in the distributions of the pressure coefficient and streamlines, as shown in Figure 10. However, when the instantaneous flow rate is slightly increased, the double channel phenomenon weakens sharply until it disappears.

It is indicated that the novel approach presented here can be used to quantitatively predict the CAS, which reduces subjective factors compared to using cloud image prediction. Moreover, its deviation is small, the results are accurate, and it is suitable for various conditions.

4.2. Oscillation Amplitude Effects

Figure 11 shows the data with an oscillation amplitude of 0.15Q_d. A periodic distribution can be observed, which is similar to that for A = 0.25Q_d. However, many different characteristics require further investigation. Compared to A = 0.25Q_d, the phase lag of the low-pressure ratio is greater when A = 0.15Q_d, being twenty-one thirty-sixths of a period.

Figure 12 shows that the three characteristics of C_s, including the average, maximum, and minimum values, are all significantly affected by the oscillation amplitude. As the oscillation amplitude increases, the mean and median values, respectively, decrease and increase. The maximum value is larger than 2/3 for all cases. The minimum value is approximately 0.96 for A = 0.1Q_d, indicating that the impeller is always in a stall state. The minimum value decreases overall as the oscillation amplitude increases and is smaller than 2/3 for the other four oscillation amplitudes. Therefore, the impeller is not only in the stall state, and part of the period is also in the non-stall state.

Figure 8 and Figure 11 show that the phase of C_s lags behind that of the flow rate and has a certain relationship with the amplitude. The relationship between the CAS and oscillation amplitude is shown in Figure 13. The non-stall state occurs only during the dropping stage of the flow rate, at least under the working conditions of this study. For oscillation amplitudes from 0.15Q_d to 0.3Q_d, Q_cd increases from 0.617Q_d to 0.767Q_d and Q_cr decreases from 0.550Q_d to 0.433Q_d. The phase of the former is insensitive to the oscillation amplitude, especially when the oscillation amplitude is larger than 0.2Q_d. The phase of the latter lags as the oscillation amplitude increases.

5. Conclusions

When the number of blades is even, alternating stall often occurs in centrifugal pump impellers under low flow rates. A novel approach involving the introduction of the low-pressure ratio (C_s), has been proposed in this work to accurately and quantitatively predict alternating stall, including the critical alternating stall (CAS). The C_s is defined as the ratio of the deviations in the low-pressure zones of adjacent impeller passages. When the difference in the low-pressure zones is a factor of two or more, that is, C_s is equal to or greater than 2/3, the internal flow of the impeller can be considered to be in a stall state.

This novel approach was validated for a centrifugal pump impeller under quasi-steady conditions. The Cs was approximately 1.47 at 0.25Q_d (where Q_d is the design load), 1.15 at 0.5Q_d, and 0.18 at 0.75Q_d, respectively, indicating that the first two and the latter were in a stall state and were in a non-stall state. This aligns well with the results of Zhou et al. [15].

This novel approach has also been applied to the centrifugal pump impeller under transition conditions, conditions of which mean a sinusoidal flow rate with an equilibrium value of 0.5Q_d and an initial phase of zero. A periodic pulsation of C_s is observed, whose frequency and phase are the same as and lag behind that of the flow rate, respectively. The fluctuating flow rate corresponding to the CAS is within 4%.

C_s is significantly affected by the oscillation amplitude. The impeller is always in a stall state for an oscillation amplitude of 0.1Q_d, and part of the period is in a non-stall state for an oscillation amplitude of 0.15Q_d or larger during the dropping stage of the flow rate. The flow rate corresponding to the CAS during the dropping and rising stages of the C_s, respectively, increases and decreases with increasing oscillation amplitude.

Therefore, C_s is useful for revealing the internal flow characteristics in a centrifugal pump impeller. The threshold value of 2/3 is a good quantity with which to quantitatively predict alternating stall and even CAS. In the future, this approach will be extended to the study of the alternating stall of a centrifugal pump to verify its universality.