Flow Characteristics and Pressure Pulsation Analysis of Cavitation Induced in a Double-Volute Centrifugal Pump

Abstract

Cavitation is a complex multiphase flow phenomenon, and the generation of transient phase transitions between liquid and vapor during cavitation development leads to multi-scale vortex motion. The transient cavitation dynamics and centrifugal pump’s rotor–stator interaction will induce pressure fluctuations in the impeller and the volute fluid of the centrifugal pump, resulting in a complex flow field structure. Based on the Schnerr–Sauer cavitation model and SST k-ω turbulence model, this paper studies the transient characteristics of the cavitation-induced unsteady flow in the centrifugal pump and the excitation response to the pressure pulsation in the volute under different flow conditions, taking the large vertical double-volute centrifugal pump as the research object. The results indicate the following: As the impeller rotates, in the external excitation response, the jet-wake flow structure at the centrifugal pump blade outlet shows an increase in the blade frequency signal. This is evident near the measurement points of the volute tongue and separator. When severe cavitation occurs, the maximum amplitude at the blade frequency in the volute shifts from the pump tongue (30°) to the downstream of the tongue (45°). The value of f_pmax is 3.1 times that when NPSHa = 8.88 m. By applying the Omega vortex identification method, it can be seen that the interaction between the vortices at the blade trailing edge and the stable vortex in the volute tongue undergoes a process of elongation, fusion, separation, and recovery. This represents the downstream influence of the impeller on the volute. When Q = 0.9Q_d, the process of the blade passage vortex tail detaching and dissipating in the impeller flow path can be observed, demonstrating the upstream influence of the volute on the impeller.

1. Introduction

In the contemporary era marked by the emergence of diverse new pumps, centrifugal pumps remain the most prevalently used among all pump types. Their widespread applications span multiple key fields, such as aerospace [1,2,3,4], agricultural irrigation [5,6,7], and petrochemical industries [8,9,10]. However, when cavitation occurs in centrifugal pumps, a series of complex phenomena take place. Along with the vapor–liquid phase conversion, the internal flow field experiences complex multiphase flow and turbulent phase transformation processes, leading to strong unstable transient characteristics, and the external manifestations are also rather intricate [11].

The instability of cavitation flow has attracted the attention of numerous scholars. Understanding the cavitation evolution process crucially depends on acquiring the overall transient details of the flow field. In the past, the flow field information during the onset and progression of cavitation was detected through stroboscopes and high-speed photography. In recent times, high-speed cameras have been commonly used for monitoring the occurrence and development of cavitation. Additionally, particle image velocimetry (PIV) has become an extensively utilized technique in fluid dynamics research [12,13,14].

For instance, Keller J et al. [15] applied PIV technology to sequentially record the wake vortex within the impeller channel and at the blade outlet of the centrifugal pump. Their research focused on examining the wake vortex detachment and its influence and development in relation to the baffle tongue under high-flow conditions. From the perspective of vortex kinetics, they preliminarily revealed the internal principle of dynamic–static interference. Matthieu et al. [16] used stereoscopic particle image velocimetry to study three-dimensional hydrofoil tip leakage cavitation vortices under different clearances, inflow velocities, and attack angles. Friedrichs et al. [17] observed the unsteady rotational cavitation process of the classical centrifugal pump and its internal cavitation distribution under certain technological conditions.

In addition to experimental research, flow field analysis techniques based on particle image velocity measurement, such as using laser-induced fluorescence tracer particles, have been used to evaluate the special physical characteristics of cavitation. Gopalan and Katz [18] employed the laser-induced fluorescence particle image velocity measurement technique to examine the velocity field in the closed-cavity area. They determined that the main factor causing cavitation collapse in the closed region is the vortex. However, cavitation experiments are costly, which restricts the amount of information that can be obtained.

With the progress of numerical simulation technology in recent years, numerical simulation of the cavitation flow field has achieved remarkable results, which has had a significant impact on cavitation research. Tan et al. [19] adopted a ZGB-modified cavitation model and RNG turbulence model to simulate the development process of cavitation in the centrifugal pump under off-design conditions. The experimental results were mainly consistent with the numerical findings, demonstrating the reliability and accuracy of the numerical model. Zheng et al. [20] used the Q criterion and Omega vortex identification method to study the pressure pulsation characteristics of cavitation under the combined action of vortex structure and periodic shedding. They found that due to the influence of cavity shedding and the flow disturbance caused by local gas–liquid flow, a high-speed vortex appears at the lower end of the main vortex. The formation and shedding of this vortex are the causes of the pressure pulsation and oscillation of the centrifugal pump. In summary, previous research on centrifugal pump cavitation has mainly focused on experimental methods like PIV and numerical simulation methods. Experimental studies have explored the flow field characteristics and vortex-related phenomena during cavitation, while numerical simulations have been used to predict and analyze the cavitation process under different conditions. These studies have laid a foundation for further understanding the complex mechanisms of centrifugal pump cavitation, but there are still aspects that need further exploration, such as more in-depth research on the interaction between different factors in the cavitation process.

The rotor–stator interaction within a centrifugal pump constitutes one of the essential causes that impact the structure of the internal flow field in the unsteady state domain, and the pressure pulsation serves as one of the vital signals for the external representation of the rotor–stator interaction [21,22,23]. Parrondo J.L et al. [24] conducted numerical computation investigations regarding the pressure pulsation traits of centrifugal pumps within diverse working conditions. They examined the spectral composition of pressure pulsation and forecasted the circumferential distribution features of the pressure pulsation amplitude at blade frequency under varying working circumstances. Based on the LES numerical method, Zhang N et al. [25] studied the phenomenon of the rotor–stator interaction in centrifugal pumps. Through the correlation analysis of flow structure and pressure pulsation, it was found that the interference was mainly a process of impact, cutting, and deformation of the blade wake shedding vortex and baffle tongue. Especially under cavitation conditions, the flow field inside the centrifugal pump is more complex, and the periodic shedding of the vortex structure and cavitation and their interaction will also affect the pressure pulsation signal. Cavitation is an important excitation source in centrifugal pumps [26]. Yang Jingjiang et al. [27] utilized the SST $k-\omega$ turbulence model along with the ZGB cavitation model to carry out simulations of unsteady cavitation flow and explored the influence of cavitation on pressure pulsation in the tongue area. The results showed that the SST $k-\omega$ turbulence model possessed a good level of simulation accuracy. The level of pressure pulsation at the septal tongue would increase as cavitation advanced. Lu et al. [28] performed a numerical emulation to investigate the inlet and outlet pressure oscillations and the internal flow instability within centrifugal pumps under unsteady conditions. The outcomes demonstrated that as cavitation evolved, the spatial distribution non-uniformity of bubbles on each blade and impeller could be evidently detected, including cavitation phenomena such as cavitation pulsation, cavitation gyration, and asymmetrical cavitation. The flow vortices within each blade passage are generated at the trailing edge of the cavity due to bubble detachment and implosion, which interact reciprocally and heighten the flow instability. An appropriate mathematical model is among the crucial elements for numerical simulation computation. Numerous scholars have proposed modifications based on fundamental models and developed many turbulence models and cavitation models that are prevalently utilized nowadays [29,30,31,32,33,34]. Sun et al. [35] utilized the enhanced BRZGB cavitation model considering bubble rotation to perform numerical simulations and experimental validations regarding the interaction between cavity pulsation and vorticity in centrifugal pumps under partial load circumstances. Through the examination of the relative vorticity transport equation, it was shown that the RVS and the RVD related to the cavity exert substantial influences on the extension and development of the cavity.

Currently, the experimental and numerical simulation investigations regarding the cavitation performance of hydrofoils and centrifugal pumps have reached a relatively advanced stage, and the research on excitation response characteristics based on the rotor–stator interaction of the centrifugal pump is also highly developed. However, scant attention has been directed towards research on the rotor–stator interaction of the centrifugal pump under diverse cavitation conditions, particularly for large vertical double-volute centrifugal pumps. For centrifugal pumps with a double-volute structure, in addition to the special flow field structure caused by impeller rotation at the tongue of volute, the separator is also an important unstable flow field area. A more in-depth exploration of the transient flow field structural traits within the double-volute centrifugal pump under varying cavitation circumstances, along with the external excitation reaction to rotor–stator interaction, is beneficial for the further enhancement of the centrifugal pump’s performance and offers a guideline for the optimal choice of hydraulic machinery apparatus like centrifugal pumps. In this study, a numerical simulation of the transient flow field structure and pressure pulsation characteristics during cavitation of a large vertical double-volute centrifugal pump under different flow rate scenarios was carried out. The analysis of the changes in pressure pulsation within the double volute, combined with the cavitation structure and vortex dynamics of the centrifugal pump, was conducted. The external response and cavitation formation mechanism of a double-volute centrifugal pump are studied comprehensively, and the cavitation vortex structure of a centrifugal pump and the combined effect of the two on pressure pulsation are further analyzed and designed to improve the safe and stable operation of hydraulic machinery.

2. Numerical Calculation Model and Method

2.1. Numerical Calculation Method

2.1.1. Basic Equations of Fluid Mechanics

• (1). Mass Conservation Equation (Continuity Equation)

In fluid dynamics, the mass conservation equation, also known as the continuity equation, is given by

(1)${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla \cdot (\rho v)=0$

${\displaystyle \frac{\partial \rho }{\partial t}}$ represents the rate of change in density ρ with respect to time t. It accounts for the unsteady behavior of the fluid.

$\nabla \cdot \left(\rho v\right)$ is the divergence of the mass flux ρv. Here, v is the velocity vector of the fluid. The divergence term describes how the mass is being transported through a control volume. For an incompressible fluid, where the density ρ is constant, the equation simplifies to $\nabla v=0$.

In Cartesian coordinates (x, y, z), the continuity equation for an incompressible fluid can be written as ${\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{\partial w}{\partial z}}=0$, where u, v, and w are the x, y, and z components of the velocity vector v.

• (2). Momentum Conservation Equation (Navier–Stokes Equation)

The momentum conservation equation, or the Navier–Stokes equation, is given by

(2)$\rho {\displaystyle \frac{Dv}{Dt}}=-\nabla p+\mu {\nabla }^{2}v+F$

(3)${\displaystyle \frac{Dv}{Dt}}={\displaystyle \frac{\partial v}{\partial t}}+\left(v\cdot \nabla \right)v$

$\rho {\displaystyle \frac{Dv}{Dt}}$ represents the rate of change in momentum of a fluid element. The left side of the equation accounts for the inertial forces acting on the fluid.

$\nabla p$ is the pressure gradient force, which drives the fluid due to pressure differences.

$\mu {\nabla }^{2}v$ is the viscous force, where µ is the dynamic viscosity of the fluid. This term accounts for the internal friction within the fluid.

F represents any external body forces acting on the fluid, such as gravity.

2.1.2. Turbulence Model

Turbulence is a complex phenomenon characterized by chaotic and irregular flow patterns. Direct numerical simulation of turbulent flows requires extremely fine meshes and small time steps, making it computationally expensive. Therefore, turbulence models are used to approximate the effect of turbulence on the mean flow. The SST $k-\omega$ turbulence model is a two-equation model that combines the advantages of the $k-\epsilon$ and $k-\omega$ models. The SST $k-\omega$ turbulence model has a high degree of accuracy in handling low Reynolds number calculations near the wall [36]. By considering the transmission of turbulent shear stress in the flow process, it can accurately predict flow separation in the case of an inverse pressure gradient, with the advantages of high computational accuracy and good convergence. The SST $k-\omega$ turbulence model postulates that the turbulent viscosity has a connection with the turbulent kinetic energy and turbulent frequency and is applied to work out the turbulent kinetic energy $k$ and dissipation rate $\omega$. The correlation is formulated as

(4)${\mu }_t=P{\displaystyle \frac{k}{\omega }}$

(5)${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_{i}k\right)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}\left(\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right)+G_k-{\beta }^{*}\rho k\omega$

(6)${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_i\omega \right)}{\partial x_i}}=\partial \left(\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\omega }}}\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right)+\alpha {\displaystyle \frac{\omega }{k}}{G}_k-\beta \rho {\omega }^2$

Ρ represents the fluid density;

β represents the auxiliary parameter;

β* represents the critical value of the auxiliary parameter;

G_k represents the turbulent production term;

σ_k represents the turbulent kinetic energy dissipation rate;

σ_ω represents the turbulent vorticity dissipation rate;

μ represents the dynamic viscosity;

μ_t represents the turbulent viscosity;

α represents the thermal diffusion coefficient.

2.1.3. Cavitation Model

The researchers Schnerr and Sauer proposed the cavitation model known as the Schnerr–Sauer model [37,38], which is used to handle the cavitation phase-transition process. Grounded in the Rayleigh–Plesset equation for single-bubble dynamics, the Schnerr–Sauer cavitation model acquires the connection between the rate of alteration of the gas-phase mass and the rate of change in the gas-phase fraction by combining the mixture continuity equation and the gas-phase integral equation. The equations for computing the evaporation term m+ and the condensation term m- are presented as follows:

(7)$m^{+}=C_v{\displaystyle \frac{3{\alpha }_v(1-{\alpha }_v)}{R_B}}{\displaystyle \frac{{\rho }_v{\rho }_l}{\rho }}\sqrt{{\displaystyle \frac{2}{3}}{\displaystyle \frac{P_v-P}{{\rho }_1}}},P<P_v$

(8)$m^{-}=C_c{\displaystyle \frac{3{\alpha }_v(1-{\alpha }_v)}{R_B}}{\displaystyle \frac{{\rho }_v{\rho }_l}{\rho }}\sqrt{{\displaystyle \frac{2}{3}}{\displaystyle \frac{P-P_v}{{\rho }_1}}},P\ge P_v$

(9)$R_B={\left({\displaystyle \frac{{\alpha }_v}{1-{\alpha }_v}}{\displaystyle \frac{3}{4\pi n_0}}\right)}^{1/3}$

ρ represents the fluid density;

ρ_l represents the density of the liquid phase;

P_v represents the saturated vapor pressure;

P represents the local pressure in the flow field;

C_v represents the condensation coefficient;

C_c represents the evaporation coefficient;

R_B represents the bubble radius;

n₀ represents the bubble number density;

α_v represents the gas phase fraction.

2.1.4. Omega Vortex Identification Method

The Omega method [39,40,41,42] further divides the vortex elements of the fluid particles into rotational and non-rotational parts, so that the vorticity can be expressed as

(10)$\omega =R+S$

R is the rotational part of the vorticity; S is the non-rotating part of the vorticity, i.e., pure shear.

Generally speaking, R and S have different directionalities. In this context, the parameter Ω is introduced, which stands for the ratio of the vorticity of the rotational part to the overall vorticity. The Omega vortex identification approach is as follows:

(11)$\mathbf{\Omega }={\displaystyle \frac{{‖\mathit{B}‖}_F^2}{{‖\mathit{A}‖}_F^2+{‖\mathit{B}‖}_F^2+\epsilon }}$

2.2. Centrifugal Pump Calculation Model and Grid Division

In this research, a vertical single-stage single-suction volute-type centrifugal pump, whose specific speed n_s is 191.3, is selected as the research object. The principal design parameters of the model pump are as follows: the flow rate is Q_d = 0.321 m³/s, the head is H_d = 40.3 m, the rotational speed is n = 1480 r/min, and there are 7 blades on the impeller. All operating conditions are pure water conditions without considering entrained air. The inlet curved inlet conduit, impeller, volute, and outlet diffuser are included in the full-flow-path computational domain of the centrifugal pump. The 3D modeling software UG NX 2000 was employed to construct the geometric model of the centrifugal pump. The geometric model of the centrifugal pump is illustrated in Figure 1.

The volute uses a tetrahedral unstructured grid, and the rest of the zone is divided using a structural grid. To remove the impact of the grid number on the accuracy of the numerical model, grid independence verification was conducted. Considering the time cost, Figure 3 was selected for subsequent calculations. At this time, the total number of grid cells in the model was 10,904,601. The grid quality of all components was greater than 0.4, and the angle was greater than 18°.

The verification of grid independence is presented in Table 1.

2.3. Comparison Between Numerical Calculation and Test

To guarantee the precision of numerical simulation results, a comparison was made between the numerical calculation outcomes of the model pump’s head and efficiency and the corresponding experimental data. The experimental data were obtained from the TP3 test bench in the Hydraulic Machinery Laboratory of the China Institute of Water Resources and Hydropower Science and Technology. As shown in Figure 2, the peak efficiency point appears to be approximately between 1.1 and 1.2 times the design operating condition, and the hydraulic efficiency lies within the range of 92.1% to 92.9%. The variation tendencies of the model pump’s H and η are in line with those of the experimental data, albeit with a slight divergence. As depicted in Figure 3 and Figure 4, the numerical prediction of the cavitation performance coincides with the cavitation performance curve of the model test, exhibiting a minimal error. Thus, it can be concluded that the numerical calculation model and approach adopted in this paper possess a high level of reliability. See Table 2 and Figure 5.

3. Numerical Simulation Results and Analysis

3.1. Energy Characteristic Analysis of Centrifugal Pump

Figure 6 depicts the variation in the blade frequency pressure at the impeller inlet detection point and the head (H) of the centrifugal pump with the cavitation number under the design operating condition. With the reduction in the cavitation number, the cavitation within the centrifugal pump gradually intensifies. When the cavitation number decreases to NPSHa = 3.36 m, a cliff-like sudden drop occurs in the H of the centrifugal pump. Incorporating the energy characteristics of cavitation-induced pressure pulsation, the cavitation of the centrifugal pump is classified into four stages: the initial stage of cavitation (Stage I: NPSHa = 10.9–25.2 m), the development stage of cavitation (Stage II: NPSHa = 6.8–10.9 m), the transition stage of cavitation (Stage III: NPSHa = 3.3–6.8 m), and the deterioration stage of cavitation (Stage IV: NPSHa ≤ 3.3 m). Different degrees of cavitation possess distinct physical properties, exerting a considerable influence on the pressure pulsation in the volute area and the impeller area.

To investigate the energy characteristics of the centrifugal pump under different flow conditions, the cavitation performance curves for the three flow conditions of 0.9Q_d, 1.0Q_d, and 1.1Q_d were plotted. As illustrated in Figure 7, with the decrease in the NPSHa, the development trends of the H and efficiency (η) of the three distinct flow conditions are identical. A sudden drop in H and η occurs at a specific NPSHa. Taking a 1% reduction in efficiency as the criterion, it is deemed that severe cavitation has emerged within the centrifugal pump at this point, with bubbles occupying the impeller flow passage. With the reduction in the flow rate, cavitation is more prone to occur in the centrifugal pump. The critical net positive suction head (NPSHc) for the three flow conditions is 5.30 m, 4.27 m, and 4.07 m, respectively. As the flow rate decreases, the angle of attack and fluid flow separation occurs on the suction side of the impeller. At this time, there may be obvious vortices within the impeller flow passage, weakening the work capacity of the pump and resulting in a sudden drop in H and η.

3.2. Characterization of Pressure Pulsations in the Volute of Centrifugal Pumps

The outlet of the centrifugal pump blade exhibits a typical jet-wake structure. Due to the rotational effect of the impeller, the detached vortices at the blade outlet causing the rotor–stator interaction between the impeller and the volute tongue present as an increase in the blade frequency signal in the external excitation response.

In order to investigate the circumferential distribution traits of the PPA within the volute at various cavitation levels, the amplitudes of the f_p signal of the PP at the outer side measurement points of the volute at three different cavitation degrees under the design operating condition were taken to plot the polar coordinate θ(x), r(y) graph, as shown in Figure 8. When no severe cavitation occurs (NPSHa > 3.36 m), the extreme point of the f_p signal within the volute appears at the volute tongue (30°). As the inlet pressure diminishes and the occurrence of severe cavitation within the pump, the extreme point of the f_p signal within the volute appears downstream of the volute tongue (45°).

Under the two operating conditions of NPSHa = 8.88 m and NPSHa = 3.76 m, with the rotation of the impeller, the pressure pulsation amplitude of the f_p in the double volute reaches the maximum at the measurement point θ = 30° (volute tongue) and then gradually decreases along the flow direction. The amplitude signal increases and remains stable at θ = 225° (downstream of the separator). When severe cavitation occurs at NPSHa = 3.36 m, the PP signal at the measurement point θ = 45° increases and becomes the maximum extreme point of the PPA signal within the volute, while the PP signals at other measurement points decrease to nearly zero. With the increase in the cavitation degree, the amplitude of the f_p signal of the PP at each measurement point within the volute shows an overall increasing trend. When NPSHa is 8.88 m and 3.76 m, f_pmax is 6143.3 Pa and 6704.3 Pa, respectively. When severe cavitation (NPSH = 3.36 m) occurs, f_pmax = 19,157.7 Pa, which is much larger than the other two cavitation conditions.

When there is no intense cavitation happening inside the centrifugal pump, the measurement points are subject to the rotor–stator interaction among the double-volute tongues, the separator, and the impeller, which leads to alterations in vortex formation. The vortices detached from the blade outlet collide with the volute tongues and the separator as the impeller rotates, thereby producing pressure pulsation indications. When severe cavitation occurs, the blade frequency signal at the measurement point θ = 45° away from the volute tongue is mainly determined by the periodic shedding of the vortices at the volute tongue. The fact that this measurement point is the maximum point of the blade frequency amplitude implies a state transition of the shed vortices at this point, affecting the stability of the flow field at the measurement point. As the fluid flows, the energy of this vortex is rapidly weakened, and the flow field at other measurement positions is stable. In conclusion, there is a significant difference in the pressure pulsation amplitude at the f_p of the measurement points before and after the double-volute tongue and the separator, especially near the volute tongue position, indicating different mechanisms of pressure pulsation at the front and rear measurement points.

To comprehensively detect the changes in pressure pulsation within the volute flow area under cavitation, the auto-power spectra of PP at three different measurement points (θ = 30°, θ = 45°, and θ = 90°) under three different cavitation degrees in the design operating condition were obtained. As shown in Figure 9, the principal peak signals of pressure pulsation emerge at the f_p and its higher harmonics 2 f_p and 3 f_p. With the increase in the cavitation degree, the PPA at the f_p is significantly affected by cavitation. The amplitude at the f_p during severe cavitation is approximately 3.1 times that of NPSHa = 8.88 m. Except for NPSHa = 3.76 m, the pressure pulsation signal at the f_p is much greater than that at other frequencies, being the dominant signal frequency of pressure pulsation. When NPSHa = 3.76 m, the low-frequency pulsation signal increases rapidly and exceeds the f_p to become the main frequency signal, but the blade frequency still occupies the second-highest position in the spectral amplitude. At this time, the bubble clusters on the impeller blades of the pump are compressible. The appearance of severe low-frequency signals within the volute indicates that the bubbles on the impeller blades shed and enter the volute flow area. As a result of the cyclic expansion and contraction oscillations of the bubble region, low-frequency pressure pulsation signals are generated.

There are significant differences in the PP signals at different measurement points. Under non-severe cavitation, the PPA at the measurement point θ = 30° is the largest, and there are obvious high-frequency signals compared to the measurement points of θ = 45° and θ = 90°. It can be considered that the pressure pulsation signal within the volute at this time is not completely dominated by the f_p signal, and other high-frequency signals such as 2 f_p and 3 f_p occupy a portion of the energy. This occurs as the volute tongue zone is subject to the impeller’s rotation, leading to rotor–stator interaction and alterations in the vortex configuration, which directly impacts the PP signal at the measurement point θ = 30°. To investigate the correlative features between PP and unsteady flow, an examination of the centrifugal pump’s internal flow field is indispensable.

3.3. Characterization of Internal Flow in Centrifugal Pumps

3.3.1. Bubble Volume Distribution at Different Flow Rates

Figure 10 shows the iso-surface diagrams with a cavitation volume fraction of 10% under the critical cavitation state of the centrifugal pump under different Q values. The critical cavitation state is determined by a 1% drop in efficiency. Influenced by the rotor–stator interaction at the double-volute tongue and the spectator, the vortices occupy the flow passage between the blades, hindering the development of bubbles, and resulting in an asymmetric distribution of bubbles in different flow conditions. When Q = Q_d and under large flow conditions (1.1Q_d), a distinct bubble ring formed by the gas–liquid mixture appears at the inlet of the impeller, and the bubble area is larger than that in the low-flow condition. In the low-flow condition (0.9Q_d), the fluid streams into the impeller at a reduced velocity, and the fluidic pressure across the entire volute and impeller zone is elevated. Under the influence of the inlet pressure, the pressure within the low-pressure region is more prone to descend beneath the saturated vapor pressure of the liquid, thereby giving rise to cavitation. Similarly, in correspondence to a 1% reduction in efficiency, at the flow rate of Q = 0.9Q_d within the low-flow-rate regime, the cavitation zone in the impeller flow passage is remarkably less extensive than that in the other two flow rate scenarios. This is because the decrease in flow rate means a reduction in the inlet velocity of the impeller, which in turn increases the angle of attack. An increase in the inlet incidence angle may cause flow separation at the impeller outlet. At this time, large-scale vortices are more likely to form in the flow passage, thereby impeding the formation of cavitation in the flow passage. Combined with the external characteristic curves of the centrifugal pump, it can be inferred that with the decline in the flow rate, the H and η of the centrifugal pump are more likely to drop sharply. However, under the same cavitation condition, the cavitation area in the impeller region of the pump is smaller under the low-flow-rate condition.

3.3.2. Time Domain Characteristics of Volute Flow Field Under Severe Cavitation Under Design Conditions

Figure 11 presents the total pressure variation diagram when the NPSHa reaches 3.76 m under the design working condition, specifically during the process in which the blades of the centrifugal pump sweep past the tongue and the separator. In each time step of the operation, a discernible low-pressure zone invariably emerges within the tongue region. When it comes to Blade I and Blade II, at the moment of t + Δt, a localized high-pressure area (total pressure > 5 × 10⁵) can be clearly observed at the outlet of the impeller. As the impeller commences its rotational motion, this high-pressure area initiates a progressive expansion towards the volute. Notably, it vanishes completely once Blade I sweeps past the tongue, signifying a dynamic transformation in the pressure distribution pattern within this critical area.

A similar phenomenon unfolds in the separator region as well. The low-pressure area situated at the leading edge of the separator, being under the influence of the impeller’s continuous rotation, is successively swept by the high-pressure area within the blade passage. This interaction leads to a sequential reduction in the size of the low-pressure area, which subsequently experiences a recovery phase. Meanwhile, the high-pressure area at the impeller outlet, as the blades sweep past the partition, undergoes an outward expansion process until it ultimately dissipates, vividly illustrating the complex and intertwined pressure dynamics within the centrifugal pump during its operation.

Figure 12 illustrates the spatio-temporal evolution process of the vortex structure in the volute tongue region when NPSHa is 3.76 m under the design working condition with the flow rate Q = Q_d. As can be seen from the figure, there exists a passage vortex adhering to the back of the blade within the blade passage. This vortex cluster remains stable and does not undergo deformation or shedding with the rotation of the impeller. At each time step, a stable vortex cluster exists at the trailing edge of the blade, and there is also a vortex cluster of a stable size in the volute tongue region. When the blade sweeps past the tongue, the vortex cluster at the trailing edge of the blade is elongated and extends towards the tongue. At the moment of t + 3Δt, the two vortex clusters merge. Subsequently, at the next moment, the vortex cluster at the trailing edge of the blade separates from the one at the tongue and is elongated as the blade rotates. It resumes its original shape at the moment of t + 6Δt and then gradually shrinks thereafter. This is due to the gradual increase in the flow passage area within the volute and the impeller, causing the vortex strength to be dissipated.

As the blade rotates, the vortex cluster at the trailing edge of the blade undergoes a process of elongation, fusion, separation, and recovery, completing the periodic shedding of the blade wake vortex, which represents the upstream effect of the volute on the impeller. The process of the periodic shedding of the blade wake vortex entering the volute region leads to significant fluctuations in the pressure values within the volute, thereby affecting the PP characteristics in the region around the volute tongue.

It is worth noting that starting from a position approximately 20° away from the volute tongue, there is a uniform vortex cluster on the outer wall of the volute flow passage. This vortex cluster does not deform with the rotation of the impeller, and the pressure values at the points where this vortex cluster attaches remain stable during the rotation of the impeller. See Figure 13.

In the low-flow-rate state where Q = 0.9 Q_d, the magnitude of the blade passage vortex close to the impeller outlet is markedly diminished in comparison to that in the design operating state. Still, it adheres stably to the blade’s rear. Conversely, the vortex near the blade tip is expanded relative to that in the design working state. This vortex will lead to the increase of pressure in the flow field and hinder the generation of cavitation in the impeller region, resulting in a small cavitation area under low flow. Similar to the design working condition, the process of elongation, fusion, separation, and recovery of the blade wake vortex can be observed. It can be captured that when the blade sweeps past the tongue, the tail of the vortex cluster within the blade passage sheds at the moment of t + 5∆t, generating a high-speed vortex cluster VD1, at which time the pressure at the vortex group falling off will increase. Subsequently, the vortex cluster VD1 gradually shrinks and disappears within the blade passage. The entire process from the generation to the extinction of the vortex cluster VD1 is completed within the impeller flow domain and does not shed into the volute region. Therefore, the vortex cluster VD1 has no significant impact on the pressure within the volute region. However, it reflects that the interaction between the volute and the blade can cause changes in the flow field structure within the impeller flow domain, which represents the upstream effect of the volute on the impeller.

To sum up, the flow structures in the regions before and after the volute tongue are significantly affected by the rotor–stator interaction between the volute and the blade. Under both the design working condition and the low-flow-rate condition, the downstream effect of the impeller on the volute can be clearly observed, with a process of elongation, fusion, separation, and recovery of the blade wake vortex. The periodic shedding of the blade wake vortex is the main cause of the pressure pulsation at the monitoring points in the volute downstream of the tongue. Under the low-flow-rate condition, the upstream effect of the volute on the impeller can be observed, with the periodic shedding of the tail of the blade passage vortex attached to the back of the blade when it sweeps past the tongue. The shedding of the vortex leads to the reduction in the overflow section area in the impeller zone, the destruction of the continuity of the flow field, and the reduction in head and efficiency in the external characteristics.

4. Conclusions

• (1). Under the rotating effect of the impeller, the jet-wake flow structure at the exit of the centrifugal pump blade shows an increase in the blade frequency signal in the external excitation response. When severe cavitation occurs under design flow conditions Q = Q_d, the maximum amplitude at blade frequency in volute shifts from the tongue of the pump (30°) to downstream of the tongue (45°). The value of f_pmax is 3.1 times that when NPSHa = 8.88 m. Before serious cavitation occurs, it can be observed that the pressure pulsation at the measuring point of the double-volute separator is also affected by impeller rotation, resulting in a rotor–stator interaction and the increase of f_p at the measuring point.

• (2). With the reduction in flow, the centrifugal pump is more likely to produce a sudden drop in head and efficiency. However, due to the increase of the inlet angle of attack, the vortex area in the impeller fluid increases, and the cavity area in the impeller fluid of the centrifugal pump is smaller under the same cavitation condition in the low-flow condition.

• (3). When serious cavitation occurs in the centrifugal pump at the two flow rates of Q_d and 0.9Q_d, the effect of rotor–stator interaction causes the impeller to produce a downstream effect on the volute, which has a great influence on the flow field structure at the tongue of the volute and then causes pressure fluctuation in the region.

• (4). The upstream effect of the volute on the impeller can be observed when the flow rate is 0.9Q_d. The tail of the blade passage vortex breaks off periodically, and a high-speed vortex group VD1 is generated. The high-speed vortex VD1 completes the process of falling off and dying in the impeller basin and does not enter the volute area to affect the pressure pulsation of the measuring point in the volute.

Author Contributions

Conceptualization, Y.T.; Validation, X.Z.; Investigation, L.L., W.Z. and J.D.; Resources, S.L. and Z.W.; Writing—review & editing, Y.Z. and C.Y. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data is contained within the article.

Conflicts of Interest

Authors Yongsha Tu, Siwei Li and Zhongzan Wang were employed by the company Hunan Water Resources and Hydropower Survey, Design, Planning and Research, Co., Ltd. Authors Lifeng Lu and Wenjie Zhou were employed by the company Hunan Provincial Water Resources Development and Investment Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

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Figures and Tables

Figure 1. Computational domain of centrifugal pump. (a) Volute zone PPA test points, (b) centrifugal pump model.

Figure 2. Unit mesh details. (a) Impeller, (b) volute, (c) outlet diffuser tube, and (d) curved elbow-shaped inlet pipe.

Figure 3. Comparison of pump’s numerical and experimental external characteristic curves.

Figure 4. Comparison of pump’s numerical and experimental cavitation characteristic curves.

Figure 5. Test pump model.

Figure 6. Cavitation evolution stages divided according to cavity-induced pressure pulsations.

Figure 7. Cavitation characteristics of centrifugal pump under different flow conditions.

Figure 8. The radial distribution characteristics of pressure fluctuation amplitude at the blade frequency under design conditions. (a) NPSHa = 8.88 m, (b) NPSHa = 3.76 m, and (c) NPSHa = 3.36 m.

Figure 9. Pressure pulsation spectrum characteristics at different measurement points under design conditions. (a) Pressure pulsation spectrum characteristics when NPSHa = 3.36 m, (b) pressure pulsation spectrum characteristics when NPSHa = 3.76 m, and (c) pressure pulsation spectrum characteristics when NPSHa = 8.88 m.

Figure 9. Pressure pulsation spectrum characteristics at different measurement points under design conditions. (a) Pressure pulsation spectrum characteristics when NPSHa = 3.36 m, (b) pressure pulsation spectrum characteristics when NPSHa = 3.76 m, and (c) pressure pulsation spectrum characteristics when NPSHa = 8.88 m.

Figure 10. Bubble volume distribution at critical cavitation points under different flow rates. (a) 0.9Qd, (b) 1.0Qd, and (c) 1.1Qd.

Figure 11. Time-varying pressure map of severe cavitation in the impeller under design conditions. (a) Time-varying pressure map of volute tongue and (b) time-varying pressure map of volute separation.

Figure 12. Variation diagram of Omega vortex in volute during severe cavitation under design conditions. (a) Three-dimensional perspective and (b) the x-y plane perspective.

Figure 13. Variation diagram of Omega vortex in volute during severe cavitation under low-flow conditions. (a) Three-dimensional perspective and (b) the x-y plane perspective.

Table 1

Grid independence analysis.

Grid Scheme Number	Mesh	Head (H)/m	Efficiency (η)/%
1	2,155,635	42.96	89.3
2	5,991,514	40.16	90.7
3	10,904,601	40.41	91.4
4	18,405,533	40.15	91.0
5	22,078,837	40.07	90.9

Table 2

Comparison and analysis of error of head and efficiency test and numerical simulation.

Flow Rate/(m3/s)	Head (H)/m			Efficiency (η)/%
	NumericalSimulation	Experiment	Error/%	NumericalSimulation	Experiment	Error/%
2.85	63.82	66.45	−3.96%	55.23	57.54	−4.01%
3.81	63.67	66.25	−3.89%	64.61	67.12	−3.72%
6.1	60.35	63.52	−4.99%	80.85	84.81	−4.67%
6.99	57.89	60.79	−4.77%	85.4	89.13	−4.18%
7.63	56.34	57.93	−2.74%	90.48	90.15	0.36%
8.39	53.93	55.17	−2.25%	91.61	90.52	1.20%
9.22	48.53	51.01	−4.86%	91.02	90.45	0.63%
9.81	44.62	46.68	−4.41	90.9	89.82	1.20%

References

1. Kye, B.; Park, K.; Choi, H.; Lee, M.; Kim, J.-H. Flow characteristics in a volute-type centrifugal pump using large eddy simulation. Int. J. Heat Fluid Flow; 2018; 72, pp. 52-60. [DOI: https://dx.doi.org/10.1016/j.ijheatfluidflow.2018.04.016]

2. Ye, C.; Wang, C.; Yan, H.; Wang, F.; Zheng, Y.; van Esch, B. Investigation on transition characteristics of laminar separation bubble on a hydrofoil. Phys. Fluids; 2023; 35, 105154. [DOI: https://dx.doi.org/10.1063/5.0170024]

3. Boghosian, M.E.; Cassel, K.W. A pressure-gradient mechanism for vortex shedding in constricted channels. Phys. Fluids; 2013; 25, 123603. [DOI: https://dx.doi.org/10.1063/1.4841576] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/24399860]

4. Choi, J.S.; McLaughlin, D.K.; Thompson, D.E. Experiments on the unsteady flow field and noise generation in a centrifugal pump impeller. J. Sound Vib.; 2003; 263, pp. 493-514. [DOI: https://dx.doi.org/10.1016/S0022-460X(02)01061-1]

5. Franc, J.; Michel, J. Fundamentals of Cavitation; Springer: Berlin/Heidelberg, Germany, 2006.

6. Wang, C.; Wang, F.; Ye, C.; Wang, B.; Zou, Z. Application of the MST turbulence model to predict the tip leakage vortex flows. Eng. Comput.; 2021; 38, pp. 344-353. [DOI: https://dx.doi.org/10.1108/EC-04-2020-0227]

7. Ye, C.L.; Wang, C.Y.; Zi, D.; Tang, Y.; van Esch, B.P.; Wang, F.J. Improvement of the SST γ-Reθt transition model for flows along a curved hydrofoil. J. Hydrodyn.; 2021; 33, pp. 520-533. [DOI: https://dx.doi.org/10.1007/s42241-021-0044-6]

8. Yan, H.; Heng, Y.; Zheng, Y.; Tao, R.; Ye, C. Investigation on pressure fluctuation of the impellers of a double-entry two-stage double suction centrifugal pump. Water; 2022; 14, 4065. [DOI: https://dx.doi.org/10.3390/w14244065]

9. Ryazantesev, V.M.; Plyasov, V.V. Axial force on the screw of a single flow high pressure submersible pump. Russ. Eng. Res.; 2010; 30, pp. 109-111. [DOI: https://dx.doi.org/10.3103/S1068798X10020024]

10. Lefor, D.; Kowalski, J.; Herbers, T. Investigation of the potential for optimization of hydraulic axial thrust balancing methods in a centrifugal pump. Proceedings of the 11th European Conference on Turbomachinery Fluid Dynamics and Thermodynamics; European Turbomachinery Society; Madrid, Spain, 23–27 March 2015.

11. Tang, Y.; Wang, F.; Wang, C.; Ye, C.; Qu, Q.; Xu, J. Investigation on the influence of seal clearance leakage on the rotating stall characteristics for a centrifugal pump. Phys. Fluids; 2024; 36, 025176. [DOI: https://dx.doi.org/10.1063/5.0197065]

12. Foeth, E.J.; Van Doorne CW, H.; Van Terwisga, T.; Wieneke, B. Time Resolved PIV and Flow Visualization of 3D Sheet Cavitation. Exp. Fluids; 2006; 40, pp. 503-513. [DOI: https://dx.doi.org/10.1007/s00348-005-0082-9]

13. Westra, R.W.; Broersma, L.; van Andel, K.; Kruyt, N.P. PIV measurements and CFD computations of secondary flow in a centrifugal pump impeller. J. Fluids Eng.; 2010; 132, 061104. [DOI: https://dx.doi.org/10.1115/1.4001803]

14. Wu, H.; Tan, D.; Miorini, R.L.; Katz, J. Three-dimensional flow structures and associated turbulence in the tip region of a waterjet pump rotor blade. Exp. Fluids; 2011; 51, pp. 1721-1737. [DOI: https://dx.doi.org/10.1007/s00348-011-1189-9]

15. Keller, J.; Blanco, E.; Barrio, R.; Parrondo, J. PIV measurements of the unsteady flow structures in a volute centrifugal pump at a high flow rata. Exp. Fluids; 2014; 55, 1082. [DOI: https://dx.doi.org/10.1007/s00348-014-1820-7]

16. Dreyer, M.; Decaix, J.; Münch-Alligné, C.; Farhat, M. Mind the gap: A new insight into the tip leakage vortex using stereo-PIV. Exp. Fluids; 2014; 55, 1849. [DOI: https://dx.doi.org/10.1007/s00348-014-1849-7]

17. Friedrichs, J.; Kosyna, G. Rotating Cavitation in a Centrifugal Pump Impeller of Low Specific Speed. J. Fluids Eng.; 2002; 124, pp. 356-362. [DOI: https://dx.doi.org/10.1115/1.1457451]

18. Gopalan, S.; Katz, J. Flow structure and modeling issues in the closure region of attached cavitation. Phys. Fluids; 2000; 12, pp. 895-911. [DOI: https://dx.doi.org/10.1063/1.870344]

19. Lei, T.; Shan, Z.B.; Liang, C.S.; Chuan, W.Y.; Bin, W.B. Numerical simulation of unsteady cavitation flow in a centrifugal pump at off-design conditions. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.; 2014; 228, pp. 1994-2006. [DOI: https://dx.doi.org/10.1177/0954406213514573]

20. Zheng, Y.; Zhao, X.; Zhou, W.; Tian, Q.; Cao, S.; Xia, K. Analysis of flow characteristics and Pressure pulsation in impeller region of a centrifugal pump induced by cavitation. Trans. Chin. Soc. Agric. Mach.; 2024; 55, pp. 244-251.

21. Spence, R.; Amaral-Teixeira, J. A CFD parametric study of geometrical variation on the pressure pulsation and performance characteristics of a centrifugal pump. Comput. Fluid; 2009; 38, pp. 1243-1257. [DOI: https://dx.doi.org/10.1016/j.compfluid.2008.11.013]

22. Spence, R.; Amaral-Teixeira, J. Investigation into pressure pulsation in a centrifugal pump using numerical methods supported by industrial tests. Comput. Fluid; 2008; 37, pp. 690-704. [DOI: https://dx.doi.org/10.1016/j.compfluid.2007.10.001]

23. Zhang, N.; Yang, M.; Gao, B.; Li, Z.; Ni, D. Experimental investigation on unsteady pressure pulsation in a centrifugal pump with special slope volute. J. Fluids Eng.; 2015; 137, 061103. [DOI: https://dx.doi.org/10.1115/1.4029574]

24. Parrondo-Gayo, J.L.; Gonzalez-Perez, J.; Fernandez-Francos, J. The Effect of the Operating point on the pressure fluctuations at the blade passage frequency in the volute of a centrifugal pump. J. Fluids Eng.; 2002; 124, pp. 784-790. [DOI: https://dx.doi.org/10.1115/1.1493814]

25. Zhang, N.; Yang, M.; Gao, B.; Li, Z.; Ni, D. Investigation of Rotor-Stator Interaction and Flow Unsteadiness in a Low Specific Speed Centrifugal Pump. Stroj. Vestn.; 2016; 62, pp. 21-33. [DOI: https://dx.doi.org/10.5545/sv-jme.2015.2859]

26. Bachert, R.; Stoffel, B.; Dular, M. Unsteady cavitation at the tongue of the volute of a centrifugal pump. J. Fluids Eng.; 2010; 132, 061301. [DOI: https://dx.doi.org/10.1115/1.4001570]

27. Kan, K.; Xu, Y.; Song, Z.; Cheng, J.; Leonce, M.A.; Ye, C. Investigation on the influence of geometric parameters in water jet propulsion systems at low cruising speeds. Phys. Fluids; 2024; 36, 115192. [DOI: https://dx.doi.org/10.1063/5.0230161]

28. Lu, J.; Yuan, S.; Siva, P.; Yuan, J.; Ren, X.; Zhou, B. The characteristics investigation under the unsteady cavitation condition in a centrifugal pump. J. Mech. Sci. Technol.; 2017; 31, pp. 1213-1222. [DOI: https://dx.doi.org/10.1007/s12206-017-0220-3]

29. Kubota, A.; Kato, H.; Yamaguchi, H. A New Modelling of Cavitating Flows a Numerical Study of Unsteady Cavitation on a Hydrofoil Section. J. Fluid Mech.; 1992; 240, pp. 59-96. [DOI: https://dx.doi.org/10.1017/S002211209200003X]

30. Kunz, R.; Boger, D.; Stinebring, D.; Chyczewski, T.; Lindau, J.; Gibeling, H.; Venkateswaran, S.; Govindan, T. A Preconditioned Navier–Stokes Method for Two-Phase Flows with Application to Cavitation Prediction. Comput. Fluids; 2000; 29, pp. 849-875. [DOI: https://dx.doi.org/10.1016/S0045-7930(99)00039-0]

31. Singhal, A.; Athavale, M.; Li, H.; Jiang, Y. Mathematical Basis and Validation of the Full Cavitation Model. ASME J. Fluids Eng.; 2002; 124, pp. 617-624. [DOI: https://dx.doi.org/10.1115/1.1486223]

32. Senocak, I.; Shyy, W. A Pressure-Based Method for Turbulent Cavitating Flow Computations. J. Comput. Phys.; 2002; 17, pp. 363-383. [DOI: https://dx.doi.org/10.1006/jcph.2002.6992]

33. Zhao, H.R.; Wang, F.J.; Wang, C.Y.; Ye, C.L.; Yao, Z.F.; Zhong, Q. A modified VLES model for simulation of rotating separation flow in axial flow rotating machinery. J. Hydrodyn.; 2022; 34, pp. 570-584. [DOI: https://dx.doi.org/10.1007/s42241-022-0058-8]

34. Kan, K.; Chen, H.; Zheng, Y.; Zhou, D.; Binama, M.; Dai, J. Transient characteristics during power-off process in a shaft extension tubular pump by using a suitable numerical model. Renew. Energy; 2021; 164, pp. 109-121. [DOI: https://dx.doi.org/10.1016/j.renene.2020.09.001]

35. Sun, W.; Tan, L. Cavitation-Vortex-Pressure Fluctuation Interaction in a Centrifugal Pump Using Bubble Rotation Modified Cavitation Model Under Partial Load. ASME J. Fluids Eng.; 2020; 142, 051206. [DOI: https://dx.doi.org/10.1115/1.4045615]

36. Menter, F.R. Influence of freestream values on k-ω turbulence model predictions. AIAA J.; 1992; 30, pp. 1651-1659. [DOI: https://dx.doi.org/10.2514/3.11115]

37. Schnerr, G.H.; Sauer, J. Physical and numerical modeling of unsteady cavitation dynamics. Proceedings of the of 4th international Conference on Multi-Phase Flow; New Orleans, LA, USA, 27 May–1 June 2004.

38. Sauer, J.; Schnerr, G.H. Unsteady cavitation dynamics—Dependence on modeling and boundary conditions. Proceedings of the 2nd Japanese-European Two-Phase Flow Group Meeting; Tsukuba, Japan, 25–29 September 2000.

39. Liu, C.; Gao, Y.; Tian, S.; Dong, X. Rortex: A new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids; 2018; 30, 035103. [DOI: https://dx.doi.org/10.1063/1.5023001]

40. Liu, C. Liutex: Vortex definition and third generation vortex identification method. J. Aerodyn.; 2020; 38, pp. 413–431, 478. (In Chinese)

41. Liu, C.; Wang, Y.; Yang, Y.; Duan, Z. New Omega vortex identification method. Sci. China Phys. Mech. Astron.; 2016; 59, 684711. [DOI: https://dx.doi.org/10.1007/s11433-016-0022-6]

42. Gao, Y.; Liu, C. Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids; 2018; 30, 085107. [DOI: https://dx.doi.org/10.1063/1.5040112]