Validation and simulation of cavitation flow in a centrifugal pump by filter-based turbulence model

The simulation of cavitation flow is sensitive to the turbulence model and cavitation model. To improve the accuracy of numerical simulation, a centrifugal pump with specific speed of 56.5 is taken as the research object, and the Filter-Based Model (FBM) was used to correct the RNG turbulence model. Zwart-Gerber-Belamri (ZBG) cavitation model, Kunz model and Schnerr-Sauer model were employed for the simulation of cavitation flow in a centrifugal pump. According to the comparison, the results calculated by the ZGB model were 8% closer to the experimental results in cavitation predictions. The cavitation occurs around the impeller ring and balance holes firstly, and then the vapor develops and cover 80% of the blade suction side. By using Q-criterion, the distribution of the vortex core region under different cavitation conditions were investigated. The volume of high intensity vortex under severe cavitation condition is 30%∼50% higher than that under other conditions. Furthermore, the distribution of the radial force of the impeller corresponds to the number of blades, and the cavitation affects the resultant force direction of the impeller. With the intensification of cavitation, the radial force of impeller increases gradually, and the regularity of radial force direction will be disorder.


Introduction
Centrifugal pump is a traditional machinery of energy conversion, which has a wide range of applications in industry due to its features of low cost and compact structure (Limbach & Skoda, 2017). Especially in petrochemical, irrigation and drainage, aerospace and other fields, low-specific speed centrifugal pumps play a very important role (El-Emam et al., 2022;Jiang et al., 2022). The power consumed by the pumping system accounts for about 22% of the world's electricity supply (Arun Shankar et al., 2016). However, due to the low operating efficiency of low-speed centrifugal pumps, which cannot effectively use most of the energy, there is an urgent need to improve the performance of pumps and increase the energy conversion efficiency.
With the development of industry and the diversity of application conditions, higher requirements are put forward for a low-speed centrifugal pump. Computational fluid dynamics (CFD) is an important research method for the flow characteristic of pumps. Kan et al. (Kan et al., 2018)  The results showed that the volute tongue tip should be installed near the middle of the two vanes. Cavitation is an unavoidable phenomenon in the pump, which has significant effects on pump reliability and performance (Brennen, 2011;Han et al., 2021). During the operation of a pump, the liquid pressure conveyed by the pump is less than the saturated vapor pressure at the conveying temperature, the liquid will undergo a phase change, i.e., the cavitation phenomenon. In the formation of a cavitation phenomenon, the rapid generation, development and collapse of air bubbles lead to the collapse of the pump cavity surface; the fluid passing through the surface of the collapsed parts will form a micro-jet to make the wall of the overflow parts damaged (Wang & Zhu, 2010). Cavitation is often accompanied by complex phase change processes, instability and multi-scale phenomena.
It is important to carry out a systematic study of the impact of cavitation on the pump, as well as to improve the performance and stability of the pump (Leroux et al., 2005;Zhang et al., 2015).
The effects of cavitation on the pump performance are still an unsolved problem. Numerical simulation and experimental measurements have become an important way to reveal a cavitation characteristic. Lu et al. (2016) performed pressure pulsation tests on a closed test bench and performed numerical simulations to obtain the evolution of cavitation. The results showed that the pump pressure pulsation frequency was 30 Hz during severe cavitation. Tan et al. (Tan et al., 2015) used a combination of the re-normalization group (RNG) k-ε turbulence model and the mass transport cavitation model to study the flow field and cavitation of pumps. The results concluded that the maximum magnitude of pressure fluctuation in the volute due to the drastic effect of cavitation is twice as large as that under non-cavitation conditions. Tao et al. (Tao et al., 2018) analyzed the relationship among the critical cavitation, the vapor volume and the fluid volume below the vaporization pressure. The results showed that there is no intersection of the optimal range of the initial and critical cavitation numbers. Sun et al. (Sun & Tan, 2020) studied the cavitation vortex of a centrifugal pump by using the ZGB cavitation model based on bubble rotation correction, and found that there was error between cavitation vortex and pressure pulsation. Suo et al. (Suo et al., 2021) used a full cavitation model and a compressible model to simulate the hydraulic axial plunger pumps and found that the vaporous cavitation can be reduced by increasing the inlet angle of a valve plate. Geng and Escaler (Geng & Escaler, 2020) applied three turbulence models and the ZGB cavitation model to simulate the cloud cavitation on 2D hydrofoils. They found that the Shear Stress Transport (SST) model is sensitive to near wall grid resolution while the cavitation characteristics are sensitive to the Zwart empirical coefficient. Furthermore, many scholars have studied cavitation using different methods, such as Large Eddy Simulation (LES), Direct Numerical Simulation (DNS), and Lattice Boltzmann Method (LBM), to gain insight into the effects and evolutionary mechanisms (Fu et al., 2021;Luo et al., 2012;Moin, 2002;Zhang et al., 2018).
Many two-equation models, such as the standard k-ε and k-ω models, are commonly used to calculate cavitation problems. Al-Obaidi (Al-Obaidi, 2019) compared ten turbulence models in the centrifugal pump simulation and verified the model performance by the experiment. However, the Reynolds averaged Navier-Stokes (RANS) method has limitations for large-scale turbulent structures (Leroux et al., 2005). To solve the above limitations and to master the flow characteristics of cavitation in the pump, modifying the turbulence model or proposing a new one to obtain more accurate results is the most common approach. Pei et al. (Pei et al., 2017) used a constant mass fraction algorithm to analyze the cavitation phenomenon in the pump and the results obtained were in good agreement with the experimental data. Tang et al. (Tang et al., 2013) performed numerical simulations of a centrifugal pump using a new cavitation model by considering thermodynamic properties; the results showed that appropriate temperature reduction could suppress the degree of cavitation. On the basis of the Rayleigh-Plesset equation, Singhal et al. (Singhal et al., 2002) developed the full cavitation model by incorporating phase-transition rate expressions. Ye et al. (Ye et al., 2019) proposed a modified partial Navier-Stokes (MPANS) model, which was applied to the study of low specific speed centrifugal pumps. The results were also compared with the SST k-ω simulation results, and it is concluded that the MPANS model simulation results are better. Han et al. (Han et al., 2021) employed a new Wray-Agarwal (WA) turbulence model to predict the complex three-dimensional turbulence properties of large curvature flow in a U-shaped bend. Cheng et al. (Cheng et al., 2021) proposed a new Euler-Lagrangian cavitation model based on the Rayleigh-Plesset equation, which significantly improved the prediction of tip vortices cavitation. The cavitating flow around ALE15 hydrofoils is investigated by Liu et al. (Liu et al., 2019) through LES and modified Schnerr-Sauer cavitation model. The filtered vorticity transport equation was employed to investigate the cavitation vortex turbulence interaction. The statistical average velocity profiles obtained by simulation and experiment showed a good agreement. Zhou et al. (Zhou & Wang, 2008) proposed a modified RNG k-ε turbulence model to simulate the cavitation flow around the NACA66 hydrofoil; the results showed that the degree of pressure increase due to interface interactions was overestimated. Johansen et al. (Johansen et al., 2004) proposed an FBM-based model to calculate the wake flow of an object and obtained high accuracy results. For the cavitation flow of cylinders and hydrofoils, the FBM was validated several times and the results showed that it could better capture the cavitation flow characteristics compared to the conventional RANS turbulence model (Huang & Wang, 2011;Zhang et al., 2015). In addition, Huan et al. (2013) improved the k-ε turbulence model by considering the effect of local compressibility on the turbulent flow; the filterbased density corrected model was proposed. Zhang et al. (Zhang et al., 2020) proposed a filter-based turbulence model and compared it with the standard k-ε model and filter-based turbulence model, which showed that the predictions of the improved filter-based turbulence model results are in good agreement with the experimental results. Zhang et al. (Zhang et al., 2017) used an FBM and a cavitation model considering the density ratio to numerically simulate the cavitation flow of a hydrofoil.
In this paper, based on the FBM modification of the RNG turbulence model, a centrifugal pump with a specific speed of 56.5 is used as the research object, and the full flow field model is built up including wearing ring and balance hole. FBM has good robustness and accuracy, but it has not been possible to systematically study the effects of cavitation flow in low specific speed centrifugal pumps. Therefore, the FBM is applied in this research to verify the predictability in centrifugal pump simulation. Furthermore, the differences between the ZBG cavitation model, Kunz model and Schnerr-Sauer model are studied and analyzed by using a combination of CFD and External Characteristics Experiments. It is concluded that the ZBG model is more suitable for the study of centrifugal pump cavitation problems. Finally, steady-state and unsteady-state simulations of the whole flow field of the centrifugal pump were also carried out to analyze the effects of different cavitation models on the external characteristics of the pump and the cavitation flow characteristics.

Governing equation
The cavitation flow can be regarded as an ideal homogeneous flow. Therefore, a two-phase flow model is established by using the mixed flow model in this paper. The governing equation of such homogeneous equilibrium flow can be expressed as follows: where ρ m is the density of the mixed medium, t is the time, the subscript i and j are the coordinate direction, u i and u j are the velocity components, μ and μ t are the dynamic viscosity and turbulent viscosity of the mixed medium respectively, and δ ij is the Kronecker inner product. When cavitation occurs in the flow field, a phase transformation process occurs between the two phases of gas and liquid. The phase transfer expression is as follows: where R e and R c are respectively the vapor and liquid transfer rates in the cavitation process.

RNG k-ε turbulence model and its modification
In this paper, the FBM proposed by Johansen et al. (Johansen et al., 2004) was used to modify the RNG k-ε model. The filter turbulence model combines the advantages of the Reynolds time-mean method and LES method, and it meets the requirements of a coarser mesh in the boundary layer compared with the ordinary LES method. In this method, only the turbulent viscosity is modified, and the expressions of k equation and ε equation remain unchanged. The modified expressions are as follows: Where, F is the filtering function, whose size is determined by the filtering size λ and the turbulence length scale k 3/2 /ε. The relation is expressed as follows: In order to ensure that the filtering process can be realized, the filtering size λ should not be less than the grid size of the calculation area. This paper takes = ( x y z)1/3, where x, y and z respectively represent the length of the grid in the impeller water area in three coordinate directions. According to the above formula, the second development is carried out in CFX, and the CEL expression is written to obtain the RNG k-ε model modified by filtering.

Cavitation model
Cavitation model is used to describe the transformation between two phases of vapor and liquid in the flow field. When cavitation occurs in the flow field, a phase transformation process occurs between the two phases of gas and liquid. The phase transfer expression is shown in Equation (3). In order to accurately describe the process of cavitation development and collapse, cavitation numerical simulations of a centrifugal pump with low specific speed based on the ZGB, Kunz and Schnerr-Sauer cavitation models (Kunz et al., 2000;Schnerr & Sauer, 2001;Zwart et al., 2004) were carried out. By comparing the test results, a better cavitation model was selected to simulate, and the cavitation flow characteristics were analyzed.
(1) ZGB cavitation model (Zwart et al., 2004), this model is the default quality transfer model in CFX software. Based on the basic equations of Rayleigh-Plesset cavitation dynamics (Singhal et al., 2002), this model can be deduced as follows: where α ruc is the volume fraction of nucleation site with a value of 5×10 −4 ; ρ v is vapor phase density; α v is vapor phase volume fraction; R B is the radius of cavitation (1×10 −6 m); P and P v are the pressure and vaporization pressure respectively; F vap and F cond are the empirical coefficients of steam generation and condensation process, with the values of 50 and 0.01, respectively. The condensation process is usually much slower than the evaporation process, so F cond is much smaller than F vap .
(2) Kunz Model (Kunz et al., 2000) was proposed by Kunz et al. based on Merkle's work. Compared with other transport equation cavitation models, the most important feature of this model is that the mass transfer rate is expressed by two different methods. For the liquid phase to vapor phase transfer, the mass transfer rate is proportional to the difference between the vaporization pressure and the flow field pressure. For the vapor phase to liquid phase transfer, the simplified form of Ginzburg-Landau potential function is borrowed, and the mass transfer rate is based on the cubic polynomial of vapor phase volume fraction. The model form is as follows: (3) Schnerr-Sauer model (Schnerr & Sauer, 2001) where n 0 is the number of vacuoles per unit liquid volume. A large number of studies showed that the optimal cavitation number density is around 1×10 13 (Li et al., 2008).

Research objects
The research object of this paper is ACB13-25 type low specific speed centrifugal pump, design flow rate Q = 13 m 3 /h, head H = 25 m, rotating speed n = 2880 r/min, specific speed is 56.5. The model assembly drawing is shown in Figure 1. The main components include motor, mechanical seal, impeller and volute, etc. Centrifugal pump impeller inlet diameter D 1 = 44 mm, impeller outlet diameter D 2 = 146 mm, blade outlet width b 2 = 6 mm, blade number Z = 5, volute inlet diameter D 3 = 150 mm.

Computing domain and grids
3D modeling software CREO 4.0 was used to carry out 3D modeling of the whole flow field of the model pump. Figure 2 shows the calculation domains of the centrifugal pump with low specific speed, including front and back chambers, wearing ring, balance hole and machine seal chamber, etc. The balance hole and the impeller passage belong to the rotation domain. In order to ensure the stability of numerical simulation results, the inlet and outlet sections are extended.
The mesh division software ICEM was used to divide the hexahedral structure of the calculation domain. In the process of meshing, key parts such as the blade surfaces and the volute tongue were locally refined by changing the number of nodes, so as to meet the y + requirements of turbulence model. The grids of main components are shown in Figure 3.   Five schemes with different number of grids were designed to carry out the independence check. The rated flow point of 13 m 3 /h of the pump was selected for numerical calculation, and the head for each scheme was obtained. As can be seen from Table 1, with the increase of the number of grids, the fluctuation of head tends to be slight. Considering the accuracy and the calculation time, the fourth scheme of grid number of 2.07 million was selected for subsequent numerical simulation.

Boundary conditions and solution control
The grid was imported into computational software CFX 19.0 for pre-processing settings. The entire flow field was set as a three-dimensional incompressible steady viscous turbulent flow field, the turbulence model was set as FBM RNG model, and the inlet and outlet boundary conditions were total pressure inlet and mass flow rate outlet,respectively. The whole computing domain was divided into two parts: the rotating domain and the stationary domain. The impeller domain belongs to the rotating domain and the rotating speed is set as 2880 r/min. The other computing domains were set as the stationary domain. Frozen rotor was used at the interface between the rotating and stationary fields. According to the machining accuracy, the wall roughness of each component was set as 0.025 mm. The calculation setting of cavitation flow field is slightly different from that of non-cavitation flow field. A liquid phase and a cavitation phase were created in the material. According to the water temperature during the experiment, the physical parameters of the two phases at 17°C were set, and the parameter values are shown in Table 2. In the fluid domain, the homogeneous multiphase flow model was checked, and the turbulence model remains unchanged.
In the cavitation setting area, the saturated vapor pressure is set as 1938 Pa, and the average diameter of cavitation is set as 1×10 −6 mm. No relative slip from phase to phase, no heat transfer, no consideration of surface tension. The initial volume fraction of the gas phase was set to 0 and the volume fraction of the liquid phase was set to 1. In the solution control, the SIMPLEC algorithm was applied, and the second-order upwind scheme was adopted to discretization the difference equations. The turbulent kinetic energy term and the turbulent energy dissipation rate term were solved using the second-order upwind difference scheme, and the near-wall flow was approximated using the standard wall function method. The maximum time step is set as 1000, and the convergence accuracy was set as 10 −4 .

Test verification
In order to verify the accuracy of the simulated data, cavitation performance test was carried out on ACB13-25  low specific speed centrifugal pump. The test scheme is shown in Figure 4. During the test, the pressure change is controlled by adjusting the inlet valve to make the pump cavitation. The inlet and outlet pressure, speed and shaft power test data at the test terminal were read and recorded.
The test results and simulation results are plotted in the same coordinate system for comparative analysis, as shown in Figure 5. It can be seen from the diagram that the simulation values of head and efficiency of centrifugal pump are very close to the experimental results, and the change trends are basically consistent. The head decreases gradually with the increase of flow rate, and the efficiency increases firstly and then decreases with the increase of flow rate. The deviation between the test and simulation is less than 6%, so the simulation results are considered to be reasonable.
Comparison of cavitation performance curves, in order to facilitate the analysis of the difference between the simulation results and the experimental results, the following dimensionless parameters are defined, i.e. head coefficient ψ and cavitation number σ , which are expressed as follows: where u 2 is the circumferential velocity at the impeller outlet; H is pump head; g is the acceleration of gravity; P in and P v are pump inlet pressure and saturated steam pressure, respectively. As shown in Figure 6, under different flow rates, the calculated results of the three cavitation models are consistent with the experimental results in terms of variation trends, and the phenomenon that the pump head coefficient drops precipitated after the cavitation number decreases to the inflection point can be predicted. Under the condition of small flow rate Q = 7 m 3 /h, the Schnerr-Sauer model and ZGB model are closer to the experimental value. Under the design flow rate Q = 13 m 3 /h, there is no significant difference between the three models, and ZGB model and Kunz model is closer to the experimental values. The performance of ZGB model is better than Schnerr-Sauer model and Kunz model under large flow conditions. In engineering, it is generally considered that critical cavitation occurred in a pump when the head decreased by 3%. In this paper, the critical cavitation number was defined as the cavitation number when the head coefficient decreased by 3%. The critical cavitation numbers calculated by ZGB cavitation model are 8% closer to the experimental values, so it is selected in this paper for further analysis of the cavitation internal flow field in the centrifugal pump with low specific speed.

Cavitation flow characteristics
The direct cause of cavitation in the impeller is that the local pressure in the flow field is lower than the saturated vapor pressure of the fluid. In order to facilitate the processing of pressure data in the impeller, the static pressure coefficient C sp is defined as follows: where P is the pressure of flow field, unit is Pa; P v saturated vapor pressure in Pa. Figure 7 shows the static pressure coefficient distribution of the impeller's middle segment (Span = 0.5) with different cavitation numbers when Q = 13 m 3 /h. Span is the dimensionless distance between the back cover of the impeller and the front cover, and its value range is 0 ∼ 1. The fluid flows into the passage from the impeller inlet, and the flow changes from axial motion to radial rotation motion, forming low pressure area on the suction sides of blade. In the non-cavitation stage (σ = 0.281 and σ = 0.116), the low-pressure area on the suction sides of blade is small. With the decrease of the cavitation number, the low-pressure area continues to expand and eventually occupies most of the impeller passage. Figure 8 depicts the static pressure coefficient distribution along the streamline from the impeller inlet to the outlet under different flow rates and cavitation numbers. The abscissa S in the figure represents the dimensionless distance from the impeller inlet to the impeller outlet, with a value range of 0 ∼ 1. When Q = 7 m 3 /h, the critical cavitation number of the pump is 0.018, when Q = 13 m 3 /h, the critical cavitation number of the pump is 0.037, and when Q = 20 m 3 /h, it is 0.124. As can be seen from the figure, with the decrease of cavitation number, the static pressure coefficient curve shows a downward trend, indicating that cavitation seriously affects the working performance of the impeller. When the S value is within the range of 0 ∼ 0.2 (from the impeller inlet to the inlet of blades), the pressure almost remains unchanged. In the non-cavitation stage, the pressure increases linearly by 2 ∼ 2.5 times along the flow direction because of impeller acting. After critical cavitation occurs, the fluid pressure in the flow passage shows a trend of decreasing first and then increasing (0.3 < S < 0.5). This is because cavitation intensifies the unsteady flow in the inlet section, resulting in a large static pressure loss at the inlet edge of blades. This phenomenon is particularly obvious under the condition of large flow rate, the pressure decreases 20% ∼ 30% at the inlet of blades when cavitation number is less than 0.137. Cavitation weakens the work performance of the impeller, but the most damages are caused by the vapors in engineering. Figure 9 shows the cavitation volume fraction distribution on the suction sides of blades and the vapor isosurface distribution in the impeller when Q = 13 m 3 /h. The red part represents the vapor iso-surface when the cavitation volume fraction α is 0.1. When the cavitation number is σ = 0.281 and σ = 0.116, there is no cavitation in the impeller. When the cavitation number is σ = 0.054, the cavitation first occurs at the impeller inlet and the edge of the balance hole; this is because that one part of the fluid in front and back chamber re-enters the impeller through the balance hole and the gap of the seal ring. The orifice throttling results in local low pressure in these areas. Although the cavitation has occurred in the centrifugal pump, it has almost no effect on the mainstream in the flow channel, and pump head has no obvious change. Therefore, it is considered that the centrifugal pump can still operate safely at this condition. In Cui et al.'s study (Cui et al., 2019), it is found that vapors first occur at the leading edge of blade suction surface, and the centrifugal pump head has no decrease under this condition. With the cavitation number σ further dropping to σ = 0.042, it is observed that the vapor mainly attaches to the suction sides of blades in sheet form, and the volume fraction of the cavitation is higher at the front cover plate than other areas. At this condition, the vapor has developed from blade surfaces to a certain extent, which has an impact on the main flow, and the head reduces about 1% (Figure 6). When the cavitation number σ = 0.037 (critical cavitation), the cavitation range further expands, and the vapor fills near the leading edge of blades, expands from suction sides to pressure sides of blades, and form a vapor flow zone in the flow passage, causes the flow passage blockage and affecting the hydraulic performance of the pump. At this condition, as the head decreases by 3% (Figure 6), the critical cavitation has occurred. When the cavitation number is reduced to 0.033, the cavitation shows explosive growth and occupies almost 70% of the entire impeller flow passage, which seriously affects the operation performance of the pump. The impeller has been unable to transfer energy to the fluid, resulting in a sharp head drop around 35% ( Figure 6).  Figure 10 depicts the distribution curve of cavitation volume fractions from impeller inlet to outlet at different flow rates. It can be seen that the cavitation volume fraction increases with the decrease of the cavitation number under different flow rates. When the cavitation number is large, the cavitation volume fraction in the impeller is very small, almost close to 0. With the decrease of cavitation number, the cavitation volume fraction in the impeller increases firstly and then decreases. The most vapor is concentrated in the interval of S = 0.2 ∼ 0.8 which represents the area between the blade inlet and the rear section of the blade. When severe cavitation occurs, the vapor coverage covers almost 80% of the entire blades. The cavitation volume fraction rises over 30% and climbs to the apex when S = 0.5. This is the place with the most serious cavitation of the impeller. However, under the flow rate Q = 7 m 3 /h and Q = 13 m 3 /h, it can be seen in the figure that vapor appears in the area between impeller inlet and blades inlet, and this phenomenon is more obvious at the small flow rate condition but disappears under large flow rate conditions. The reason is that there is a high-speed jet from seal clearance impacting to the area of impeller inlet to blades inlet, the high-speed jet forms local low-pressure zones, under the condition of small flow rate, the outlet pressure of the impeller is higher, which will aggravate this phenomenon. Moreover, under low flow rate conditions, the vapor needs to occupy more flow channel space to cause the same negative effect on head with under high flow rate conditions. The cavitation volume fraction at Q = 7 m 3 /h is 80% higher than that at Q = 20 m 3 /h when the critical cavitation occurs in the pump. Figure 11 depicts the velocity and streamline distributions in the impeller at Span = 0.5 with different cavitation numbers when Q = 13 m 3 /h. With the development of cavitation, the streamline alternations mainly concentrate in the middle and rear sections between blades. When σ = 0.054, the flow in the impeller almost does not change, although cavitation has appeared in the impeller at this condition. The cavitation mostly attaches to the blade surfaces causing little interference to the main flow. With the critical cavitation number further decreasing, the high speed velocity area increases, the flow becomes extremely unstable, vortex zone migrates from the middle of pressure sides to downstream of the suction side of blades. The tail region of the flow between blades shows disorder. When σ = 0.033, the head has already shown cliff type decline; due to the cavitation extrusion effects, the effective flow passage continues to reduce, the high velocity area further expands, accounting for 70% of the entire impeller flow passage. At this stage, the movement, rupture and absorption of the cavitation increase the instability of the flow.

Effect of cavitation on vortex distribution
When cavitation occurs, strong vortexes often generate. It is necessary to analyze the distribution characteristics of vortex structure in a centrifugal pump under different cavitation conditions. Q criterion is a commonly used criterion for judging vortices, and its physical significance is very clear. According to the imaginary part of the fluid velocity gradient tensor, the vortex strength of fluid micro clusters can be determined. The formula is as follows: equation (18), where W ij is the vorticity tensor, S is the strain rate tensor and || || F is the frobenius norm. In order to distinguish from flow rate Q, the Q-level is adopted in this paper. Q-level represents the dimensionless number of the true value divided by the maximum, so the range of Q-level is 0 ∼ 1. As shown in Figure 12, the vortex core distribution at different Q-levels in different cavitation stages at flow Q = 13 m 3 /h is shown in the figure. Q-level is positively proportional to the rotation strength of fluid eddy in the pump.
Compared with the cavitation condition, the change of Q-level has a greater influence on the vortex core distribution in the impeller passage when σ > 0.037. The most vortex core distributes at low Q-level. The flow in the section of volute outlet is relatively stable and the vortex has less distribution in this area. When the Q-level is large, the vortex core mostly distributes on blades suction sides. However, with the decrease of the threshold value, the vortex core gradually increases and spreads to the blades pressure sides. The reason is that the fluid flows into the impeller inlet and firstly impacts blade suction sides; in this area, swirls and separations easily occur. Moreover, the distributions of vortex core in the volute are more obvious, and most of these vortexes are highspeed vortexes. This phenomenon is related to the volute characteristics for it needs to convert the kinetic energy of the fluid into pressure energy. The high-speed fluid flows from the impeller into the volute and interacts with the wall surface of the volute, resulting in amounts of vortexes.
When cavitation occurs, the influence of Q-level change on the shape and position of vortex core in the impeller decreases for the cavitation, increases the amount of high intensity vortexes. This phenomenon is more obvious when σ < 0.037. Comparing with those of no cavitation condition, the high-intensity vortices increase significantly and concentrate in the impeller region, cavitation will cause a large energy loss in the impeller. With the further development of cavitation, vortices also appear at the inlet of the impeller and gradually expand to nearly the whole flow passage. Figure 13 shows vortex core volumes corresponding to different Q-level at different cavitation stages. According to the Q criterion, the volume iso-surface under different Q-levels surrounded by the vortex core were extracted, and the relationship between the volume and the cavitation state of the centrifugal pump was analyzed. It can be seen from the figure that under different cavitation numbers, the distribution curves of vortex core volume show a very similar rule. When Q-level < 0.001, the vortex core volume under different cavitation numbers remains unchanged. The more serious the cavitation is, the more vortex cores will be found in the pump. When q = 0.01, the volume of vortex core under different cavitation numbers is about 1.2×10 −4 m 3 , indicating that the influence of cavitation on vortex has been gradually weakened by this time. However, at the condition of Q-level > 0.001, the vortex core volume gradually decreases with increases of Q-level and finally trends to 0. For Q-level between 0.05 and 0.08, the volume of vortex core under severe cavitation condition is about 30-50% higher than that under other conditions. It means only the severe cavitation has influence on the high intensity vortexes.

Radial force of impeller
The centrifugal pumps usually show the phenomenon of vibration and noise when operating under an unstable condition (Al-Obaidi & Towsyfyan, 2019). The vibration is often related to the imbalance of force. The unsteady simulations were carried out with different cavitation conditions in this section. The force in x axis and y axis directions on impeller's blades, front and rear covers were monitored, and the results of numerical calculation are revised by dimensionless processing with the following equation (Eq.19).
Where F is the combined module of the radial force vector received by the impeller; ρ is the medium density, the unit is kg/m 3 ; u 2 is the circumferential velocity at the impeller outlet, unit is m/s; b 2 is the width of impeller outlet edge, unit is m. Figure 14 depicts the radial force distributions of the impeller with different cavitation numbers under the design flow rate condition. With the change of cavitation number, the cycle of radial force fluctuation in the cavitation stage is similar to that in no cavitation stage. From Figure 15 (a), it can be observed that the radial force in both x and y directions is affected by cavitation. With the decrease of the cavitation number, the amplitude of radial force fluctuation gradually expands 5% ∼ 10%, especially in the critical cavitation stage. This is due to a large number of vapors filling in the impeller, radial force waves more seriously. It can be also noticed that cavitation has a slightly higher influence on the component in the x direction than that in the y direction. Figure 15 (b) shows a cycle of the force in different cavitation stages. The radial force at 85°which is near volute tongue is higher than at other directions. Therefore, it is believed that the volute tongue has influence on the rotor. However, the peak and trough are consistent with the number of blades. Figure 15 depicts the vector distribution of the impeller's radial resultant force in different cavitation stages. The angle θ Fr in the figure represents the direction of the resultant force, and the expression is as follows: where F x and F y are the component forces in x and y directions received by the impeller, respectively. In no cavitation stage (σ = 0.281), the radial force vector presents five peaks in the circumference, which is consistent with the number of impeller blades. Impeller blades number also have influence on cavitation. Similar results (Al-Obaidi, 2020) found that when a centrifugal pump blades number = 5, cavitation was affected highly at the suction of impeller compared to other number of blades, particularly at high flow rate. The force on the impeller distributes asymmetrically, the larger radial force on the impeller mostly concentrates near the tongue area, while the force on the diagonal area of the tongue is obviously smaller. This is because, when the fluid flows from the impeller into the volute, the flow around the tongue changes dramatically, and it is easy to generate low pressure vortices here, the pressure near the tongue is obviously lower than that of other flow areas, finally causing the asymmetry of the impeller's force.
When cavitation happens, vapors will generate in the flow passage, which aggravates the asymmetry of whole passage's flow. Therefore, the radial force on the impeller is larger than that in no cavitation stage. When the cavitation number decreases to 0.042, the radial force becomes irregular and its trajectory enters an unclosed state, the peak and trough of the wave have almost no correspondence with the blade number, completely destroying the periodicity and distribution position of the radial force in no cavitation state. The cavitation blockage in the flow passage reduces operation capacity of blades, local pressure in the flow passage decreases and whole offset capacity of the radial force decreases. That is to say, the cavitation intensifies the asymmetry of the flow field, makes the impeller unbalanced. With the further development of cavitation, when σ = 0.037, the pump's head has decreased significantly, the blockage of the flow passage by cavitation is more serious, and the distribution of the radial force becomes more irregular. This phenomenon is also found by Dong et al. (2019); they found the trajectory of the radial force changed from closed to open with the development of cavitation.

Conclusion
Based on the RNG k-ε turbulence model which was modified by the FBM method, the applicability of the ZGB model, Kunz model and Schnerr-Sauer model in the simulation of cavitation flow in centrifugal pumps with low specific speed was studied in this paper. According to the cavitation experimental data, the ZGB cavitation model had better predictability and was selected for numerical simulation, and the main conclusions are as follows: (1) Cavitation is the main cause of the flow instability in the impeller. After critical cavitation occurs, the fluid pressure in the flow passage shows a tendency of going down firstly and going up secondly, especially under the condition of large flow rates; the decrease is about 20% ∼ 30%. With the intensification of cavitation, the vortex on blade pressure sides gradually shifts to blade suction sides and becomes disorderly. The unstable vortex will result in backflows in the outlet of the impeller and further weaken the hydraulic performance of the centrifugal pump.
(2) Cavitation appears near the impeller seal ring and balance holes at the initial moment, and the phenomenon is more obviously under the small flow rate condition. The cavitation at the leading edge of blades is mainly in the form of sheet cavitation. With the decrease of cavitation number, the volume fraction of the cavitation increases along the main direction towards the outlet edge of blades. Under the severe cavitation condition, the vapor covers almost 80% of the blade suction side and reduces the head of the pump.
(3) After the occurrence of cavitation, the highintensity vortices in the flow field mainly concentrate in the rotation region. With the intensification of cavitation, the volume of vortex core gradually increases, and a highspeed region appears in the vortex core at the inlet of the flow passage and gradually expands to the whole flow passage. The high intensity vortexes under severe cavitation condition are about 30% ∼ 50% higher than those under other conditions.
(4) The distribution of the radial force corresponds to the number of impeller blades, and the position of volute tongue has influence on it. The cavitation affects the resultant force direction of the impeller. With the intensification of cavitation, the radial force of impeller increases gradually. In addition, cavitation will destroy the regularity of radial force direction, resulting that the trajectory is no longer closed. The unstable radial force will influence the hydraulic performance of the pump and exacerbate the operating instability. In the future, the more accurate method of cavitation flow simulation should be studied and the new techniques to analyse and diagnose cavitation need to be applied in the experiment.

Data availability
The data used to support the findings of this study are included within the article.

Disclosure statement
No potential conflict of interest was reported by the author(s).