An axial flow pump with a design volume flow rate of 0.35 m³/s, head of 6.7m, and rotation speed of 1450 rpm was chosen as the research object. The 3D model of the axial flow pump is shown in Figure 1. The whole pump section consists of five flow passage parts, namely the suction chamber, impeller, guide vane, support, and pressurized chamber. In addition, the diameter of pump impeller is 300 mm, the hub ratio is 0.5, the tip clearance is 0.2 mm, the impeller has four blades and the guide vane has seven blades.

Three governing equations, mass conservation equation, momentum conservation equation, and energy conservation equation, dominate the motion in the axial flow pump. For a general axial flow pump, its working pressure is low and the amplitude of the temperature change is small, so the density of the fluid medium could be considered constant and the heat transfer can be ignored. Eqs. 1, 2 show the tensor forms of the mass conservation equation and momentum conservation equation of the viscous incompressible medium. ∂ u i ∂ x i = 0 (1) ∂ u i ∂ t + u j ∂ u i ∂ x j = F i − 1 ρ ∂ p ∂ x i + υ ∂ 2 u i ∂ x j ∂ x j (2) where,

SST k- ω turbulence model was selected for this study, which can predict effectively the fluid separation point under the condition of reverse pressure. At the same time, when solving the low Reynolds number flow near the wall area, the SST k- ω turbulence model has higher accuracy. Its basic equations are as follows: ∂ ρ k ∂ t + ∂ ( ρ u j k ) ∂ x j = ∂ ∂ x j ( μ + μ t σ k ) ∂ k ∂ x j + τ i j ∂ u i ∂ x j − β * ρ ω k (3) ∂ ρ ω ∂ t + ∂ ρ u j ω ∂ x j = ∂ ∂ x j ( μ + μ t σ ω ) ∂ k ∂ x j + γ ν t τ i j ∂ u i ∂ x j − β ρ ω 2 + 2 1 − F 1 ρ σ ω 2 1 ω ∂ k ∂ x j ∂ ω ∂ x j (4) where, σ k =1.176, σ ω =2, β =0.075, β * =0.09, κ =0.41, γ 1 = β / β * − σ ω κ 2 / β * . F 1 = tanh arg 1 4 ⁡ (5) arg 1 ⁡ = min ( max ( k β * ω y ; 500 ν y 2 ω ) ; 4 ρ k C D σ ω 2 y 2 ) (6) C D = max ( 2 ρ 1 σ ω 2 ω ∂ k ∂ x j ∂ ω ∂ x j , 1.0 × 10 − 10 ) (7) where, F ₁ is a mixed function. F ₁ = 1 in the area near the wall, which means that the whole area uses the k- ω turbulence model, and F ₁ = 0 in the area far always from the wall, which means that the whole area uses the standard k- ε turbulence model, y is the distance from the first layer of the grid to the wall, σ ω 2 = 1.168. The limiting equation of eddy viscosity is as follows: v t = a 1 k max a 1 ω , S F 2 (8) v t = μ t / ρ (9) where, μ t is turbulent viscosity, S is an invariant measure of strain rate, F ₂ is also a mixed function, which is equal to 1 in the boundary layer and 0 in the shear layer, and a 1 is a constant, equal to 5/9. F 2 = tanh arg 2 2 ⁡ arg 2 ⁡ = max 2 k 0.09 ω y ; 500 ν y 2 ω (10)

As a kind of low Reynolds number turbulence model, the SST k-ω turbulence model requires that y + ≤ 1 , which will ensure that at least one layer of grid exists within the viscous sublayer. However, it is a requirement that is almost impossible to meet unless a large number of grids that ordinary servers cannot afford are adopted. It has been shown in relevant research (Tao, 2001)that the first layer of the grid near the wall will also fall in the viscous sublayer when y + ≤ 5 . The y + near the wall of the impeller and guide vane is shown in Figure 2, where all the y + near the blades is small than five and the final average of y + is 1.1, and the growth rate of the grids off the wall is 1.1.

All the flow passage components of the axial flow pump were meshed by hexahedral structured grids. The grid independence was judged by the change of pump efficiency with the increasing of the number of grids. When the efficiency change amplitude did not exceed 2% with the increase of the number of grids, it could be considered that the grid independence verification was passed, which was shown in Figure 3. In addition, the grid independence verification was carried out under the condition that y + meets the requirements of the SST k-ω turbulence model. The final grids of the computing domain verified by grid independence were shown Table. 1 and Figure 4.

The whole numerical calculation was completed by ANSYS-CFX 20.1 The medium is water with a density of 998.2 kg/m³ and a dynamic viscosity of 0.000889 Pas. The rotating speed of the impeller was set to 1450rpm and all the other flow passages were stationary. A mass flow boundary (350 kg/s) with a turbulent intensity of 5% was adopted at the inlet of the computational domain and a pressure boundary (1 atm) was set at the outlet of the computational domain. All walls were subject to non-slip-wall boundary condition. Since the accuracy of the unsteady calculation results is higher than that of the steady calculation results, the unsteady calculation results were used in this study. However, the unsteady calculation requires the steady calculation results as the initial condition to accelerate convergence. The interfaces between the rotating region and the stationary region adopted the interface of the frozen rotor in the steady calculation, while the sliding grids were used in the unsteady calculation. The time-step size was set as 0.000115 s, which is the time for the rotor to turn for one degree to obtain sufficient flow field solution accuracy (Lin, et al., 2021). By changing the flow rate at the inlet boundary of the computational domain, the numerical calculation of the flow field structure in the axial flow pump under different flow rate conditions was completed. In addition, to weaken the influence of the inlet and outlet boundary conditions on the calculation results of the flow field in the axial flow pump, it is necessary to keep a sufficient distance between the inlet and outlet boundary and the axial flow pump. The distance between the inlet of the suction chamber and the impeller inlet was set as 8 R _t , and the distance between the end of the diffusion section and the outlet of the pressurized chamber was set as 6 R _t .

In order to verify the accuracy of numerical simulation, we completed the external characteristic test of the axial flow pump on the closed test bench. The schematic diagram of the test bench is shown in Figure 5 and the experimental instruments are shown in Figure 6, in which the uncertainty of the flowmeter is lower than 1.5%, the speed uncertainty of the torque speed sensor is less than 0.2% and the torque uncertainty of that is less than 1.5%, and the uncertainty of the pressure sensors is less than 1%. The external characteristic test at different flow rate was completed by adjusting the regulating valve opening. All the data were converted to digital form and stored in a computer database.

The numerical calculation results of the external characteristic curves, the head, and efficiency, of the axial flow pump were compared with the test results as shown in Figure 7. The simulation results are in good agreement with the test results near the design mass flowrate (0.35 m³/s) condition. However, when the deviation from the design mass flow rate condition is too large, there is an obvious difference between the test value and the CFD value. Especially under the condition of small flow rate, the difference between the test results and the CFD results is the most obvious, which has lost reference value to the engineering practice. So, the turbulence model SST k-ω has sufficient accuracy in predicting the flow field structure under the design mass flow rate condition of the axial flow pump.

As we know, in the circumferential direction of specific axial and radial coefficient positions, the pressure, velocity, liquid flow angle, and other physical parameters in the impeller passage of the axial flow pump are periodically and symmetrically distributed. However, only the average value in space is taken as the design value in the design process of the axial flow pump. It also has been confirmed that the average value of the impeller inlet flow field structure in space is in good agreement with the theoretical design value under the design condition (Guo, et al., 2022) which proved that it is feasible to study the energy transfer and loss mechanism in the impeller passage using the weighted average treatment of the flow field structure in the impeller passage. The energy transfer and loss mechanism in the axial flow pump impeller was studied by using the method of the weighted average of mass flow rate in the circumferential direction in this study. x M C A = ∑ Δ m ⋅ x m (11) where x is the physical variable, x M C A is the weighted average value of x in the circumferential direction based on the mass flow rate at the position of the specific streamwise location and radius coefficient. So, the weighted average value based on the mass flow of the parameters at different streamwise locations and different radius coefficients could be studied on the meridional surface. The data processing diagram is shown in Figure 8.

Equation 11 is the Bernoulli equation for the ideal fluid in the relative coordinate system. For a horizontal axial flow pump, the impeller inlet height is as same as the outlet center height, namely, z 1 = z 2 . And so are the diameter of the impeller inlet and outlet. Then the circumferential velocity is the same, that is, Eq. 11 is transformed into Eq. 12, from which, it could be deduced that fluid maintains conservation of the mechanical energy from the impeller inlet to the outlet in the relative coordinate system and the dynamic energy is continuously transformed into the pressure energy. z 1 + p 1 − t ρ g + V 1 − r e 2 − U 1 2 2 g = z 2 + p 2 − t ρ g + V 2 − r e 2 − U 2 2 2 g (12) p 1 − t ρ g + V 1 − r e 2 2 g = p 2 − t ρ g + V 2 − r e 2 2 g (13)

The pressure coefficients are defined as follows to represent visually the energy change at the impeller inlet to a higher degree. C p = p − p in ¯ 0.5 ρ V i n 2 , C d = 0.5 ρ V 2 0.5 ρ V i n 2 , C t o t = C p + C d (14) where, p i n ¯ is the average pressure at the pump inlet, C p is the static pressure coefficient, C d is the dynamic pressure coefficient, C t o t is the total pressure coefficient.

The flow surface of radius 0.5 was taken as the research object. The energy coefficients on the flow surface were processed by weighted average based on the mass flow. Equation 12 is changed into Eq. 13. C 1 t − t o t − r e l − M C A = C 2 t − t o t − r e l − M C A (15) C 1 t − p − M C A + C 1 − d − r e l − M C A = C 2 t − p − M C A + C 2 − d − r e l − M C A (16) C 1 − t o t − r e l − M C A = C 2 − t o t − r e l − M C A + ∆ C 1 − 2 − t o t − r e l − M C A (17)

The coefficients are defined as follows to represent the energy change visually in the impeller passage to a higher degree. C p = p − p in 0.5 ρ V i n − r e 2 , C d = 0.5 ρ V r e 2 0.5 ρ V i n 2 , C t o t = C p + C d (18) where,

As shown in Figure 9, the total pressure coefficient in the relative coordinate system decreases slowly from the impeller inlet to the outlet and the decreasing range is very small, which proves the correctness of the theoretical analysis. That is, the mechanical energy of the fluid on the same flow surface in the impeller channel is conserved. The reason why the total pressure coefficient decreases slowly from the impeller inlet to the outlet is the hydraulic loss caused by the viscosity as shown in Eq. 15. The hydraulic loss consists of the friction loss caused by friction and the mixed loss caused by the boundary layer separation. The dynamic energy is constantly converted into pressure energy from the impeller inlet to the outlet as a whole in the relative coordinate system, which is the source of the pressure head of the axial flow pump.

However, it is easy to be noticed that the fluid dynamic pressure coefficient increases slightly before decreasing rapidly. On a specific flow surface of the impeller passage, it can be regarded as an infinite plane cascade. The aerodynamic theory has pointed out that a high speed and low pressure region will appear at the suction surface of the airfoil near the leading edge due to flow around the airfoil, so is the cascade in the impeller passage as shown in Figure 11C.

The theoretical head of the pump is defined as the sum of the dynamic head and the theoretical static head in the absolute coordinate system, in which, the theoretical static head is the difference between the theoretical pressure head from the impeller outlet to the inlet (Eq. 17), and the dynamic head is the difference of the velocity head in the absolute coordinate system between the impeller outlet and the inlet (Eq. 18). According to the relative Bernoulli equation, the theoretical pressure head difference between the impeller outlet and the inlet is equal to the velocity head difference between the impeller inlet and the outlet in the relative coordinate system, as shown in Eq. 17. The inlet and outlet velocity triangles of the impeller are shown in Figure 10. From the impeller inlet to outlet, the relative velocity gradually decreases and the dynamic energy is converted into the pressure energy. The reverse pressure gradient always exists in the whole impeller channel. The reverse pressure gradient will increase the thickness of the boundary layer and then, lead to its separation, which will cause great mixing loss. It is also the fundamental reason why the efficiency of the axial flow pump is lower than that of the axial flow turbine. H t = H P − t + H d (19) H p − t = p 2 − t ρ g − p 1 − t ρ g = V 1 − r e 2 2 g − V 2 − r e 2 2 g (20) H d = V 2 − s t n 2 − V 1 − s t n 2 2 g (21)

According to the velocity triangle, the relative velocity can be expressed as Eq. 19. V r e 2 = V m 2 + U − V u 2 (22)

The normal inlet is always adopted in the design of the axial flow pump, that is, V u 1 = 0 m/s and the impeller inlet and outlet of the axial flow pump have the same diameter, that is, V m 1 = V m 2 and U 1 = U 2 , The theoretical static head Eq. 22 can be derived from Eqs. 20, 21. V 1 − r e 2 = V m 1 2 + U 1 − V u 1 2 = V m 1 2 + U 1 2 (23) V 2 − r e 2 = V m 2 2 + U 2 − V u 2 2 = V m 2 2 + U 2 2 + V u 2 2 − 2 U 2 V u 2 (24) H p − t = 2 U 2 V u 2 − V u 2 2 2 g (25)

Equation 23 could be derived from the further derivation of Eq. 17. H d = V u 2 2 2 g (26)

The dynamic head is completely determined by the circumferential velocity component at the impeller outlet. Equation 24 is the theoretical head expression for the axial flow pump. H t = U 2 V u 2 g (27)

The actual head of the impeller could be calculated by Eqs. 25, 26. The theoretical head can also be expressed by Eq. 27. The actual head must be less than the theoretical head due to the hydraulic loss. It is not difficult to find out that the actual head and the theoretical head have the same dynamic head by comparing Eqs. 26, 27. The difference between them mainly exists in the static head. It has been described that the total energy in the relative coordinate system will decrease slowly because of the hydraulic loss in Eq. 15. The dynamic energy in the relative coordinate system could not be completely converted into the pressure energy without any hydraulic loss. During the flow process, the dynamic energy must be partially converted into internal energy in the form of friction and mixing caused by boundary layer separation, and then lost. The hydraulic loss is equal to the relative total energy reduction. H = H p + H d (28) H = p 2 ρ g − p 1 ρ g + V 2 − s t n 2 2 g − V 1 − s t n 2 2 g = p 2 ρ g − p 1 ρ g + V u 2 2 2 g (29) H t = p 2 − t ρ g − p 1 − t ρ g + V u 2 2 2 g = V 1 − r e l 2 − V 1 − r e l 2 2 g + V u 2 2 2 g (30)

The weighted average value of the performance characteristics such as H _d , H _p , H, H _t and Δ C t o t − r e l on the flow surface of R* 0.5 based on mass flow were taken as shown in Figure 11A and were recorded as H _d-MCA , H _p-MCA , H _MCA , H _t-MCA and Δ C t o t − r e l − M C A ( Δ C t o t − r e l − M C A = H _t-MCA- H _MCA ).

The static head is the main part of the axial flow pump head. It is mainly obtained by converting the relative dynamic energy into the pressure energy and the transformation is completed in the front half of the blade. The static head shows a trend of decreasing at first and then increasing on the whole flow surface, which is consistent with the changing trend of the static pressure coefficient in Figure 9. The negative static head exists near the leading edge, which is caused by that the flow around the blade will increase the relative velocity of the flow field near the leading edge of the suction surface (Figure 11D). This area is also the place most prone to cavitation. For the axial flow pump with a specific size, the key to improving its cavitation performance is to control the relative velocity at the suction surface of the leading edge. The dynamic head increases rapidly at the leading edge of the blade and then tends to be flat. It is also obtained mainly in the front half of the blade.

It is not difficult to infer that the actual head of the impeller is also mainly obtained in the front half flow passage. From the perspective of energy transmission, the torque formed by the pressure difference between the pressure surface and the suction surface is the fundamental source of the fluid energy. As shown in Figure 12, the pressure coefficient increases slowly from the leading edge to the trailing edge, while the pressure coefficient of the suction surface decreases rapidly near the leading edge and then up to be the same as that of the trailing edge gradually. The negative pressure coefficient at the front half of the whole blade is maintained on the suction surface of the blade. Therefore, the pressure difference between the pressure surface and the suction surface at the front half of the blade is far larger than that at the second half of the blade, which means that the torque on the front half of the blade is far larger than that on the second half. That is the reason why the actual head of the impeller is mainly obtained in the front half flow passage.

The friction loss caused by the velocity gradient in the boundary layer and the mixing loss caused by the separation of the boundary layer is the basic source of losses in the impeller flow passage. It can be seen from the distribution of the relative total pressure loss coefficient from the impeller inlet to the impeller outlet in Figure 11A that the gradient of the relative total pressure loss coefficient near the leading edge (Area A) and the trailing edge (Area B) is larger than that in the area between them, while the gradient near the trailing edge is greater than that near the leading edge. The relative velocity of the fluid near the leading edge at the suction surface of the blade will increase significantly due to the flow around the blade, which will result in a corresponding increase in the friction shear stress coefficient at this location as shown in Figure 11B. The greater the shear stress coefficient, the greater the friction loss gradient. Then, as the relative velocity decreases, the shear stress coefficient decreases, and the gradient of the relative total pressure loss coefficient decreases. The impeller passage of the axial flow pump, as we know, is basically under the adverse pressure gradient, and the adverse pressure gradient at the suction surface is obvious larger than that at the pressure surface. The thickness of the boundary layer on the suction surface increases gradually under the action of the adverse pressure gradient and reaches the maximum at the trailing edge as shown in Figure 11D. For the turbulent boundary layer, there is violent mixing movement in its interior. The thicker the boundary layer, the more mixing loss and therefore it can be seen that the relative total pressure loss coefficient gradient increases significantly at the trailing edge. In addition, the wake jet is formed after the fluid leaves the blade and it will result in more severe mixing loss. The gradient of the relative total pressure loss coefficient will also further increase.

Overall, on the one hand, friction loss is the main form of hydraulic loss in most areas of the impeller flow passage except the trailing edge, and the gradient of the relative total pressure loss coefficient is obviously greater because of the flow around the blade. On the other hand, the mixing loss inside the boundary layer and wake is the main form of hydraulic loss at the trailing edge. The gradient of the relative total pressure loss coefficient caused by mixing loss in the wake is the largest in the whole impeller flow passage. The point to improve pump efficiency is to avoid boundary layer separation in the flow channel and to eliminate the wake as soon as possible to prevent it from spreading in the downstream channel.

Figure 13 shows the distribution of the weighted average value of H _t , H and η based on the mass flow on the meridional surface of the impeller flow passage. The trend of the theoretical head (Figure 13A) and the actual head (Figure 13B) in the main flow area (Radial coefficient between 0.05 and 0.95) are basically consistent. They gradually increase from the impeller inlet to the outlet and the impeller has high hydraulic efficiency. However, the circumferential velocity component, V u , of the fluid near the hub (Radial coefficient between 0 and 0.05) is large due to the rotation of the hub. According to the Euler equation, the increase of the circumferential velocity component means the increase of the theoretical head. The rotation of the blade tip makes the fluid near the rim (Radial coefficient between 0.95 and 1) maintain a high circumferential velocity component, while the flange wall makes the axial velocity in the tip area attenuate rapidly. Therefore, the absolute velocity of the fluid in this area is dominated by the circumferential direction. According to the Euler equation, this area will have a high theoretical head.

The actual head is the difference between the theoretical head and the hydraulic loss head. The fluid in the pressure surface will return to the suction surface of the blade through the clearance between the tip and the rim under the effect of the pressure difference between the pressure surface and the suction surface, which is called tip leakage flow. The tip leakage flow will cause complex leakage vortex, which will damage the flow field structure at the rim seriously and cause a lot of hydraulic loss. As a result, the actual head in this area also decreases. The actual head in the rim is obviously lower than that in other locations with the same streamwise location as shown in Figure 13B.

Efficiency is the ratio of the actual head to the theoretical head as shown in Figure 13C. It has been analyzed that the fluid in the main flow area has higher hydraulic efficiency because of that the main losses in this area are caused by the friction loss of the airfoil surface and the wake mixing loss of the trailing edge, which are small, while the hydraulic efficiency drops rapidly near the hub and rim. The fluid near the hub area is mainly affected by the corner separation, that near the rim area is mainly affected by the leakage vortex.