The numerical model and method for simulating the flow filed of single phase (non-cavitation flow) in the centrifugal pump can be found in previous work by Tan et al. [17]. The fluid in the cavitation flow field is considered as a homogeneous and compressible mixed medium of liquid and vapor. The continuity and momentum equations in the Cartesian coordinates are given as follows: (1) ∂ ρ m ∂ t + ∂ ( ρ m u i ) ∂ x i = 0 (2) ∂ ( ρ m u i ) ∂ t + ∂ ( ρ m u i u j ) ∂ x j = − ∂ p ∂ x i + ∂ ∂ x i [ ( μ m + μ t ) ( ∂ u i ∂ x j + ∂ u j ∂ x i − 2 3 ∂ u k ∂ x k δ i j ) ]where ρ and μ are the density and dynamic viscosity, calculated by weighted average of each phase volume fraction α, ρ_m = ρ_lα_l + ρ_vα_v, μ_m = μ_lα_l + μ_vα_v; subscripts l, v and m denote the liquid phase, vapor phase and the mixture, respectively; u is the velocity; p is the pressure; μ_t is the turbulent viscosity; and subscripts i, j, k denote the axes directions.

The re-normalisation group (RNG) k-ε turbulence model is widely applied in turbomachines, which is suitable to simulate the complex flow in rotating and high curvature computational domain. As to the cavitation flow, the fluid in a centrifugal pump is compressible and made of liquid water and vapor. Therefore, the RNG k-ε turbulence model should be modified to improve the prediction accuracy for the cavitation condition. Considering the influence of compressibility on the cavitation flow, the turbulent viscosity is modified by introducing the function f(ρ_m) [19,20]. The turbulent viscosity μ_t is amended by: (3) μ t = f ( ρ m ) c μ k 2 ɛwhere k and ε denote the turbulent kinetic energy and turbulent kinetic energy dissipation rate, respectively; the empirical constant is c_μ = 0.09.

The density function f(ρ_m) is defined as: (4) f ( ρ m ) = ρ v + [ ( ρ m − ρ v ) ( ρ l − ρ v ) ] n ⋅ ( ρ l − ρ v )where n is a constant and taken the value of 10 [19,20].

The liquid-vapor mass transfers due to cavitation are solved by the transport equation—based cavitation model proposed by Zwart et al. [21]: (5) ∂ ( ρ ν α ν ) ∂ t + ∇ ⋅ ( ρ ν α ν u ) = m ˙ vap − m ˙ con

In this model, a transport equation with source terms based on the homogeneous flow theory is used to solve the mass transfer between liquid and vapor phases. The fundamental assumption of this model is that the cavitation bubble does not interact with each other and the nucleation site density remains the same. However, with the development of cavitation the liquid volume fraction inevitably decreases, the nucleation site density should be modified accordingly. To further improve the modeling of the vaporization and condensation processes, the mass transfers for vaporization rate ṁ_vap and condensation rate ṁ_con are modeled as follows: (6) m ˙ vap = C vap 3 α nuc ( 1 − α v ) ρ v R b 2 3 max ( p v − p , 0 ) ρ l (7) m ˙ con = C con 3 α v ρ v R b 2 3 max ( p − p v , 0 ) ρ lwhere C_vap and C_con are the empirical calibration coefficients for vaporization and condensation rates, respectively; R_b is the bubble radius; p_v is the vapor pressure; and α_nuc is the nucleation site volume fraction.

The vapor pressure p_v in most cavitation model is chosen as a constant according to the test temperature. However, it is actually influenced by the local turbulent pressure fluctuation, so this effect is taken into consideration in the present cavitation model by giving the modified vapor pressure p_v as follows: (8) p v = p sat + 0.195 ρ m kwhere p_sat is the saturation pressure at the test temperature.

In the present calculation work, the commercial CFD code CFX 13.0 is employed. The equations solved in the calculation are the Navier-Stokes equations coupled with the RNG k-ε turbulence model and transport equation cavitation model. The spatial domain is discretized by using an element-based finite volume method.

In order to make the pump inlet pressure consistent with the experimental operation, the total pressure at the pump inlet is specified and the velocity direction is taken to be normal to the boundary. The mass flow at the pump outlet is selected. The scalable wall functions are imposed to solve the near-wall flow close to the no-slip wall over the impeller blades and sidewalls, the volute casing and the inlet and outlet pipe walls. The pressure at pump inlet decreases step by step until a convergence at each given operating condition in the cavitation flow calculations.

Figure 4 shows the computation domain and mesh of the centrifugal pump with IGVs. The computation domain is composed of three modules: suction pipe without or with IGVs, impeller and volute corresponding to the real pump in experiment. All the meshes are structural hexahedra in the computational domain, and the grids at the blade surface and volute tongue are locally refined in order to capture the flow details and improve the calculation accuracy, as shown in Figure 4b–d.

Five different mesh densities are used to calculate the pump performances at design point without cavitation, as shown in Table 2. The simulated results indicate the weak influence of mesh density on the pump head H and efficiency η in the present simulations. Therefore, the coarsest mesh with 1,804,742 elements is employed for the following calculations in order to reduce the calculation loads.

Here, H₁ and η₁ are the pump head and efficiency calculated by using the Mesh 1, respectively.

Figure 5 shows the comparison of pump head and efficiency without IGVs between the experimental and numerical results. There is a good quantitative agreement at the flow rate of 260–370 m³/h, with the maximum relative errors smaller than 5.0%, which demonstrates that numerical calculations simulate the pump performance accurately.

Figure 6 shows the experimental results of pump head and efficiency without and with IGVs for different flow rates. It can be seen from Figure 6a that the pump head is significantly affected by the prewhirl regulation of IGVs. The head increases as the negative prewhirl angle decreases, and decreases as the positive prewhirl angle increases. This can be easily explained by the Euler equation in hydraulic machineries as follows: (9) H th = ω g ( C u 2 r 2 − C u 1 r 1 )where H_th is the theoretical head of pump; ω is the rotational angular velocity of impeller; g is the gravitational acceleration; C_u and r denote the circumferential velocity and the radius, respectively; subscripts 1 and 2 denote the pump inlet and outlet, respectively. According to the Equation (9), when the IGVs are regulated to the negative angle, they will induce a negative circumferential velocity, and then cause an increase on the pump head.

For the flow rate of 150–375 m³/h, the efficiencies of centrifugal pump under prewhirl regulation are all higher than that without IGVs, as shown in Figure 6b. The maximum efficiency increase is more than 2.0% around the designed condition region. Compared to the pump without IGVs, the improvement of pump efficiency for both positive and negative prewhirl regulations is mainly attributed to the optimized flow pattern at the impeller inlet and the improved flow field in the impeller [17,18]. Another reason is that the uniformly arranged vanes in the suction pipe homogenize the local flow field which is actually non uniform due to the blend and flange upstream of the suction pipe.

Figure 7 shows numerical results of three-dimensional streamlines in the pump suction pipe and impeller without and with IGVs at Q = 260 m³/h. The three-dimensional streamlines in the pump suction pipe and impeller start from the suction pipe inlet by sampling forty equally spaced points on this surface.

Without IGVs, the streamlines are straight in the suction pipe and then gradually rotate in the impeller with the same direction as the rotational impeller. Due to the asymmetrical three-dimensional geometry of IGVs, the fluid regulated by the IGVs has positive prewhirl at an IGV angle of 0°. For the positive prewhirl regulation, the fluid has an obvious positive circulation in the suction pipe while passing the IGVs. The rotation strength increases as the prewhirl angle increases from 12° to 24°, as shown in Figure 7c,d. For the negative prewhirl regulation, the circumferential movement of streamlines at −12° in the suction pipe is not as obvious as that at 12°, because the movement of fluid in suction pipe is also affected by the flow pattern in impeller, where the rotation intensity of fluid is far stronger than that in suction pipe with the opposite direction.

Figure 8 shows the measured head drop curves without and with IGVs at flow rates of Q = 260, 315 and 370 m³/h, respectively. Net positive suction head available (NPSHA), which is the difference between the total energy and vaporization energy for unit weight of the fluid at the pump inlet, is defined in Equation (10). The cavitation more likely occurs as NPSHA deceases: (10) NPSHA = p ρ l g + u 2 2 g − p v ρ l g

In Figure 8a, the numerical results at Q = 260 m³/h are also given. The comparison between the measured and calculated results shows that the numerical simulation can accurately simulate the drop trend of pump head with the decrease of NPSHA, especially the sudden drop of H at the critical point. The value of NPSHA at which the head drops by 3% is usually defined as the net positive suction head critical (NPSHC). The IGVs installed in front of the pump induce hydraulic loss such as friction loss, impact loss and vortex shedding, so the total pressure at the inlet of pump with IGVs is lower than that without IGVs. Figure 8 illustrates that the values of NPSHC for centrifugal pumps with IGVs are larger than that without IGVs, which means that the pump with IGVs has worse cavitation performance. However, on the other hand, the influence of IGVs on the pump cavitation performance is very limited. The differences of NPSHC for the centrifugal pump without and with IGVs are less than 0.5 m at three flow rates. For this type of centrifugal pump, it still works well in most of the engineering application when the NPSHC varies 0.5 m.

Figure 9 shows the volume fraction of vapor in the impeller and the total pressure contour at the impeller inlet by numerical simulation at the cavitation critical point of NPSHA = 2.0 m for Q = 260 m³/h. Due to the interaction of the rotational impeller and stable volute, the pressure distribution in the impeller is not symmetrical, which leads to the inhomogeneous cavity in the impeller. The total pressure contour at the impeller inlet becomes uneven when the pump has IGVs installed as shown in Figure 9b–f. For the positive prewhirl angle of 12° and 24° as shown in Figure 9c,d, the high pressure region shrinks as the prewhirl angle increases. Figure 9d shows that the central region is occupied by the low pressure fluid while the high pressure fluids are separated into six discontinuous regions when the angle increases to 24°. The change of the pressure distribution at the impeller inlet for negative angle is not as significant as the positive angle, but the total pressure contours are still influenced by the six IGVs as shown in Figure 9e,f.

Table 3 lists the total pressure at suction pipe inlet and impeller inlet. The area-weighted average method is used in the CFX 13.0 in present work to obtain the total pressure at suction pipe inlet and impeller inlet. The area-weighted average of the total pressure is the average value of the total pressure on a surface when the mesh element sizes are taken into account. In the numerical simulation, the total pressure at suction pipe inlet is set to a fixed value for five cases without and with IGVs according to the experimental condition; therefore, the six values of total pressure at suction pipe inlet are nearly the same in Table 3. However, the values at the impeller inlet are different due to the various losses induced by the IGVs at different prewhirl angles. The total pressure loss between the suction pipe inlet and impeller inlet for the case without IGVs is 255 Pa, approximately 1.0% of the total pressure at suction pipe inlet. When the IGVs' prewhirl angle is 24°, the corresponding values increase to 1777 Pa and 7.6%, which are consistent with the pressure contour in Figure 9d.

Undoubtedly, the total pressure losses in suction pipe with IGVs are larger than that without IGVs. For the tested centrifugal pump in this paper, the total pressure losses under positive prewhirl regulation are larger than that under negative prewhirl regulation with IGVs. The prewhirl strength of fluid obviously increases as the angle increases when the IGVs are regulated to positive angle. This is also apparent from the elongated streamlines as shown in Figures 7c,d. The extended streamlines mean larger friction loss and probably deteriorate the flow pattern. Taking the opposite prewhirl angles 12° and −12° for example, the total pressure loss of positive regulation is about twice of that of negative regulation.

Figure 10 shows the total pressure contour at four cross sections between IGVs and impeller inlet with equal distance of 0.1 m for the IGVs' angle of 24°. Under regulation of IGVs, six divided regions of local high pressure appear near the suction pipe wall, as shown in Figure 10b–e. As the fluid flows downstream, the six high pressure regions become circumferential uneven, and the areas of those regions gradually reduce, which indicates that the velocity circulation weakens accordingly. Therefore, the distance between the IGVs and impeller has obvious influence on the effects of prewhirl regulation of IGVs.