Numerical analysis of vortex and cavitation dynamics of an axial-flow pump

ABSTRACT This study focuses on the correlative mechanism of the ambient pressure and inflow uniformity on the vortex and cavitation dynamics of an axial flow pump. The shear stress transport k – ω turbulence model and Schnerr–Sauer cavitation model are applied in the unsteady detached eddy simulation. The results show that the vortex merging between the primary and secondary tip leakage vortices (TLV) happens earlier with the growth of the cavity at a lower ambient pressure. The contact position of the merged TLV to the adjacent blade moves upstream with the decrease in the cavitation number. As the uniformity of the axial inflow decreases, TLV merging and vortex shedding are also promoted. The nonlinear variation of the initial angle of attack of the impeller blade leads to the compression or expansion of sheet cavitation under non-uniform inflow conditions. The evolution process and energy transfer of the vortices are verified quantitatively using a power spectral density analysis of kinetic energy fluctuations, and the short-wave instability leads to the fast decline of spectrum peaks at a higher frequency in the cavitating flow. It is crucial to avoid severe changes of ambient pressure and inflow uniformity to ensure the design performance of pump in actual working environment.


Introduction
Tip leakage flow (TLF) is an inevitable and vital flow pattern in mixed or axial flow pumps (He et al., 2021;Horiguchi et al., 2006;Miorini et al., 2010;Wu et al., 2011). The three-dimensional tip leakage vortex (TLV) rolls up along the suction side (SS) of the blade tip corner, and the induced cavitation may reduce the efficiency and enhance the noise/vibration of the pump (Hsiao & Chahine, 2005;Luo, Ji, et al., 2020). As an essential component required for energy conversion and transportation in hydraulic systems (Guan, 2009;Kadivar et al., 2020), the flow mechanism behind the performance breakdown, especially the unsteady vortex/cavitation behaviors of the pump under various operating conditions, requires careful study.
Extensive experimental studies have been conducted to study the TLV and cavitation of the pumps. Particle image velocimetry (PIV) measurement is an effective way to uncover the vortex structures, and high-speed photography is widely adopted to investigate the cavitation morphology (Gong et al., 2022;Wu et al., 2011). Twodimensional PIV measurement was performed to reveal the TLV structures and associated turbulence of an axial water-jet pump by Wu et al. (2011). The TLV rolls up CONTACT Wan-zhen Luo luowzh5@mail.sysu.edu.cn near the suction side of the impeller blade and connects to the vortex filaments generated by the leakage flow. The interaction between the TLV and the attached cavitation affects the performance breakdown of the axial flow pump, and it was concluded that the flow rate played a significant role in delaying the tip vortex cavitation (Laborde et al., 1997;Tabar & Poursharifi, 2011). The TLV trajectory and dynamics in a multiphase pump at the off-design condition were discussed (Shi et al., 2020), and a wavy TLV was observed and verified using high-speed photography. The cavitation flow development from the initial cavitation to the first critical cavitation of the waterjet pump was analyzed via high-speed photography (Long et al., 2021). With the development of computational fluid dynamics (CFD), numerical methods are being widely employed in simulating the vortex dynamics and cavitating flow in turbomachinery (Cheng et al., 2020;Li et al., 2021;Qin et al., 2021;Zhu et al., 2020). The effects of the tip gap size on the TLF in a turbomachinery were analyzed with large eddy simulations (LES) (You et al., 2006), and it was found that the tip leakage jet and TLV produced considerable mean velocity gradients that resulted in the generation of vorticity and turbulent kinetic energy. The characteristics of horn-like vortices in an axial flow pump impeller under off-design conditions were studied by Zhao et al. (2021). Several Reynoldsaveraged Navier-Stokes (RANS) simulations were carried out to determine the TLV trajectory, TLV dynamics, and cavitation-vortex interaction of the axial and mixedflow pumps Huang et al., 2015). Han and Tan (2020) investigated the influence of the speed of rotation on TLV in a mixed flow pump functioning in the turbine mode. The generation of turbulent kinetic energy (TKE) and diffusion of the Reynolds stress were found to be the major causes of energy loss in the waterjet pump (Luo, Ye, et al., 2020). The unsteady vortical cavitation behavior in nonuniform inflow was discussed by Huang et al. (2021), and it was found that cavitation caused larger pulsations in the hydrodynamic characteristics. As a hybrid LES/RANS method, Detached Eddy Simulation (DES) was also widely adopted in the simulation of complex vortex structures (Cao et al., 2021;Gong et al., 2020;Wang et al., 2021a). The cavitating flow and pressure fluctuation in the tip region were simulated based on the delayed Detached Eddy Simulation (DDES) by Xi et al. (2021). The results demonstrate that the pressure fluctuates significantly in the flow passage caused by the suction-sideperpendicular cavitating vortices . The vortex dynamics of a ducted propeller under different working conditions were investigated with the DES method (Gong et al., 2018;Gong et al., 2021), and it was noted that the boundary layer vortices of the duct accelerate the evolution of the wake vortices. Based on a brief review of the above literature, the inflow of the pump may be inhomogeneous in several practical scenarios, and the radial loading, vibration, cavitation, and noise of the pump will deviate from the design range. However, the flow mechanism analysis of performance degradation under typical off-design conditions is lacking in current studies. To avoid the potential performance breakdown or mechanical damage to the pump, it is important to study the physical behavior of the vortex and cavitation evolution under off-design conditions.
The variation of the ambient pressure and inflow uniformity are two typical off-design scenarios for the pump in its actual working environment. In this study, we investigate the correlative mechanism between the ambient pressure/inflow uniformity and the vortex/cavitation dynamics of an axial-flow pump. The paper is structured as follows: Section 2 describes the numerical method, setup of inflow conditions, as well as validation of the numerical simulation. Section 3 details the effect of ambient pressure and inflow uniformity on the vortex/cavitation dynamics. Finally, a summary of the findings is detailed in Section 4.

Governing equations
Unsteady numerical simulations are conducted on the three-dimensional flow field of an axial-flow pump based on the DES model and the Schnerr-Sauer cavitation model. Based on the homogeneous flow assumption, the cavitating flows in the axial-flow pump are treated as a single mixed fluid in which the vapor and liquid phases share the same pressure and velocity. The incompressible continuity equation and momentum equations for the vapor-liquid mixture are as follows: where p denotes the pressure, u i denotes the velocity component in the i direction, ρ denotes the density, μ denotes the laminar viscosity, and the subscript m means the vapor-liquid mixture. Equations (3) and (4) describe the density ρ m and laminar viscosity μ m of the liquid-vapor mixture, the subscript v indicates the vapor phase, l indicates the liquid phase, and α denotes the volume fraction of each phase. The continuity and momentum conservation equations are solved using the shear stress transport k -ω turbulence model (Menter, 1994), which was employed to resolve the Reynolds stress. DES covers the boundary layer in the RANS mode and switches to the LES mode in highly separated regions. A new wall distance is defined to distinguish the domains evaluated by RANS and LES (Shur et al., 2008). The DES method combines the advantages of RANS and LES and offers good predictive accuracy in the unsteady vortex and cavitation simulations (Gong et al., 2021;Li et al., 2020;Wang et al., 2021b).
The solution procedure and pressure-velocity coupling are both based on the SIMPLEC algorithm, and the SIMPLEC second-order scheme is employed for the pressure and convection terms. A second-order central difference scheme is used for diffusion terms. The volume of fluid (VOF) method is utilized to capture the phase interface with a high resolution interface capturing scheme (Hirt & Nichols, 1981). The finite volume-based segregated flow solver available in Simcenter STAR-CCM + software is applied here.

Cavitation model
The Schnerr-Sauer cavitation model (Schnerr & Sauer, 2001) is used in the current study to predict the unsteady cavitation of the axial flow pump. The transport equation-based model is coupled to reflect the phase change process. The mass transfer equation related to cavitation is described as follows: where the two source terms,ṁ + andṁrepresent the evaporation and condensation effects during the phase change. The two source terms are derived from the generalized Rayleigh-Plesset equation. The mixture cavitation model based on the homogeneous flow assumption is applied in the current study. The two source terms are described as follows: where R b denotes the bubble radius defined as: where N b is the bubble number density.

Computational details
The main parameters of the axial flow pump are illustrated in Table 1. The diameter of the impeller (D) is 300 mm, with a tip gap of 0.3 mm. The computational domain is a cylinder with the same diameter as the casing wall. The inlet and outlet boundaries of the computational domain are at distances of 3D and 9D upstream and downstream from the impeller hub, respectively ( Figure  1). The computational domain consists of the rotational and static domains, and the two interfaces are set to enable the exchange and iteration of information between the two sub-domains. The boundary conditions are set as follows: the inlet is defined as the velocity inlet and the outlet is the pressure outlet, which enables the option of target mass flow. All physical surfaces (impeller, guide vane, shaft and casing) are set as no-slip walls. An unstructured grid system is generated and meshed using a prism layer mesh and trimmed mesh in STAR-CCM + . Figure 2 shows the volume mesh distribution of the computational domain. In order to provide a higher resolution for the cavitating flow simulation, several refinement regions are defined to improve the mesh quality near the leading and trailing edges of the blades, as well as the tip gap area. There is a total of 8.6 million cells, with 4.02 million in the rotational domain and 4.58 million in the static domain. A constant rotational speed corresponding to n = 1450 rpm is used to ensure consistency with the experimental condition. The fluid density is kept constant at ρ = 998.2 kg m −3 as the temperature equals 20°C. The dynamic viscosity is 8.887 × 10 −4 Pa s and the saturation pressure equals 2338.4 Pa. To satisfy the Courant-Friedrichs-Lewy (CFL) number requirement (CFL < 1.0), the time step is set as t = 1/8700 s, corresponding to one degree rotation of the impeller with an inner iteration number of 10. All the variables are assumed to converge when the residuals drop to 10 −5 .
Different cavitation conditions are achieved by changing the reference pressure. The cavitation number σ is defined as σ = (p intp v ) / (ρn 2 D 2 ), where p int is the total pressure at the inlet plane and p v is the saturation vapor pressure of water at room temperature. The nonuniform axial inflow is the most common situation in the actual work of the pump. Herein, two non-uniform inflow conditions with different velocity gradients are considered. The axial velocity profile and inlet velocity contours under uniform (U0) and non-uniform conditions (N1 and N2) are shown in Figure 3. The axial velocity v z /v 0 increases linearly along the radius direction from the case wall (r/R = 1) to the axial center (r/R = 0). The non-dimensional inflow velocity values are 0.6 and 0.8 near the case wall (r/R = 1) for N1 and N2 situations, respectively. The inlet flow rates of non-uniform inflow cases are equal to that of the uniform inflow case to create the same input condition. The uniformity coefficient ζ is introduced as follows: where Q is the flow rate, A is the area of the inlet. Herein, the uniformity coefficients correspond to 1 (U0), 0.928 (N1), and 0.857 (N2) for the uniform and non-uniform inflow cases, respectively.

Validation of numerical simulations
The numerical simulations are validated by comparing the hydraulic performance of the experimental and  The hydraulic performance coefficients of the pump are defined in Equations (9)-(11), where H is the pump head and M is the torque input to the pump shaft.
NPSH a is the dimensionless available net positive suction head defined as follows: where p in and v in are the static pressure and velocity at the inlet plane, respectively. The experimental and numerical curves corresponding to five flow rates are compared in Figure 4. The numerical head coefficients (K H -CFD) match well with the experimental results (K H -EFD), and the predicted hydraulic efficiency (η-CFD) is slightly larger ( ≈ 2%)  than the tested efficiency (η-EFD) within the flow range. The difference in efficiency is caused by the mechanical loss which is inevitable in pump tests, while it is not considered in the numerical simulation.
Furthermore, without the cavitation test, the convergence analysis of the grid is performed using the K H and NPSH a of the pump to evaluate the accuracy and reliability of the numerical results under uniform and non-uniform conditions. A two-grid assessment procedure (Roache, 1997). Detailed procedures can refer to the literature by Di Mascio et al. (2014) and Gong et al. (2021).
The results of the numerical simulation using medium and fine grids and the linear extrapolation (denoted by Extr.) are listed in Table 2. The current non-uniform inflow condition has little impact on the hydraulic performance of the pump as the variation of head coefficient K H is negligible, while the increasing NPSH a stands for a slight improvement of cavitation resistance under the non-uniform inflow with the same flow rate. The uncertainties of the head coefficient K H and NPSH a for nonuniform inflow cases are larger than that of the uniform inflow case. This indicates that the non-uniform inflow cases (N1 and N2) are more sensitive to the grid resolution. However, based on the results shown in Table  2, the numerical solver and grid strategy adopted can be regarded as consistent and reliable as all the numerical uncertainty U N values are less than 4%. Herein, the numerical results obtained from the fine grid are used for subsequent discussion.

TLV dynamics
The flow mechanism of the TLV evolution, including its generation, development and breakdown, determines the cavitation and pressure fluctuation characteristics in the  internal turbulent flow of the pump. Figure 5 demonstrates the transient cavitation and vortex structures at different cavitation numbers. Four cavitation numbers (σ = 2.84, 1.97, 1.78, and 1.40) are adopted to represent the absence of cavitation and situations of mild, moderate, and severe cavitation, respectively. The transient vortex structures in the cavitating flow are depicted using the Q-criterion method (q = 1.5×10 5 /s 2 ). A spiral primary tip leakage vortex (PTLV) forms from the blade tip corner and extends to the pressure side (PS) of the adjacent blade in the flow passage ( Figure 5(a)). As labeled in the dashed triangle in Figure 5(b), numerous secondary TLV (STLVs) are formed by the flow separation in the clearance region and the interaction between the leakage flow and the main flow. The tip separation vortex (TSV) is mainly located near the junction between the PS and the tip end, while the trailing edge vortex (TEV) forms a narrow low pressure area near the trailing blade ( Figure 5(a)). The induced cavitation blocks the flow passage, and thus modifies the flow characteristics near the blade surface. The vortex instability is enhanced near the cavitation boundaries due to the imbalance in pressure between the water phase and vapor phase. The first merging between the PTLV and the STLV always happens when σ ≤ 1.97 ( Figure 5(b-d)), while the second merging occurs when σ = 1.40 ( Figure  5(d)). Figure 6 shows the vortex evolution of TLV for σ = 1.40 with the time interval of 10 t. It is noted that the two-step vortex merging process presents a relatively stable state in the time domain when it reaches numerical convergence.
The contact position (labeled A, Figure 5) of the merged TLV to the adjacent blade changes due to the vortex evolution pattern under different cavitation numbers. The statistical results of the transient contact position are presented in the form of a box chart in Figure 7. The vertical axis λ/c represents the dimensionless position along the length of the chord, where c represents the chord of the blade tip. The contact position moves upstream from the trailing edge (λ/c = 0.92) to the middle blade (λ/c = 0.53) when the cavitation number decreases. The accelerated vortex evolution results in larger vortex structures and faster energy dissipation. The pressure field variation, which is caused by the upstream vortices, could be another reason for the hydraulic performance breakdown except for the vapor distribution on the suction side of the blade.
The vortex evolution alters the energy transfer process in the flow passage. The stretched out view of the vorticity (ω) distribution on the impeller casing is shown in Figure  8 to analyze the interaction between the TLV and the turbulent boundary layer. Related to the intensity of the TLV, the PTLV trajectory I is observed to start from the blade tip corner. Higher vorticity appears along the trajectory (Label II in Figure 8) where the vortex merging happens. The intensity of the vortex structure increases with the  decrease in the cavitation number, and the area of influence stretches even further to the downstream flow near the blade tip.

Cavitation dynamics
The stretched out view of the vapor volume distribution on a 90% spanwise impeller blade is shown in Figure 9.
Depending on the pressure distribution at the blade surface, the starting point (label A, Figure 9(a)) of the vapor is located near the leading edge of the blade. The starting point of the vapor moves downstream gradually along the blade chord when the cavitation number decreases. In addition, detached cavitation occurs near the middle of the blade for σ = 1.40 (seen in circle B, Figure 9(a)). The unstable detached cavitation accelerates the momentum exchange and results in a higher vorticity distribution as seen in circle C, Figure 9(b).
The three-dimensional cavitation morphology of the axial flow pump is shown in Figure 10. The iso-surface of vapor volume fraction (α v ) is depicted at α v = 0.1 in Figure 10 and subsequent figures. Sheet cavitation is the dominant component of pump cavitation. It starts near the leading edge of the impeller blade and attaches to the suction side from the blade tip to the blade root in a relatively stable manner. Due to the local high-speed TLF in the tip clearance, the tip corner vortex cavitation is distributed on the surface of the blade tip. The TLV cavitation is observed near the blade tip region, connecting to the shear layer cavitation derived from the turbulent boundary layer of the casing. The cavitation unstability is reflected near the tail closure region of the sheet cavitation under a minimum cavitation number (seen in Figure 10(c)). A re-entrant jet flow is formed here due to the imbalance in pressure between the water phase and vapor phase, and this results in an irregular trailing boundary and detached cavitation.
Furthermore, the instantaneous contours of the TKE of the impeller blade surface are shown in Figure 11. No megascopic cavitation is observed in the case of σ = 2.84, and this provides a clearer comparison of the TKE characteristics between the non-cavitating and cavitating cases. The high TKE is mainly observed near the regions with a strong adverse pressure gradient, such as the tip and leading edge of the blade (Figure 11(a), label A and B), as well as the water-vapor interface when the cavitation happens (Figure 11(b)-(d), label C). The high TKE is also observed in a spanwise narrow strip between the sheet and the TLV cavitation areas (Figure 11(b)-(d), label D). It is concluded that TKE production is the major source of energy loss in the pump (Luo, Ye, et al., 2020). This implies that the energy loss mainly happens near the areas where the phase transition occurs. The energy loss characteristics of the vortex system are directly related to the generation and development of pump cavitation.

Cavitation-vortex interaction
A strong correlation is proved between the vortex and cavitation (Huang et al., 2015;Luo, Ji, et al., 2020). The relative vorticity transport equation is introduced into the analysis of cavitation evolution to provide a deeper understanding of the flow mechanism as follows: where ω denotes the vorticity, v denotes the velocity.
The left-hand side in Equation (16) represents the variation rate of vorticity, and the four terms on the right hand denote the vortex stretching term, vortex dilatation term, baroclinic torque term, and viscous diffusion  term, respectively. The viscous diffusion term is negligible when compared to the other terms in partial flow fields.
The source terms of the vorticity transport equation on three radial positions (r/R = 0.85, 0.9 and 0.95) of the impeller blade are compared in Figure 12. The vortex stretching term is related to the velocity gradient, and it is the dominant component in the trailing area of sheet cavitation (label A, Figure 12). The TLV distortion reduces the momentum and increases the angular momentum of the flow near the wall, and it promotes the generation of vorticity as well. The vortex dilatation term represents the effect of the expansion or contraction of fluid clusters on the vorticity, and it is in direct proportion to the interphase mass transport rate. Most of the vortex dilatation distributes on the suction side of the blade leading edge (label B, Figure 12) and the middle chord (label C, Figure  12), which is consistent with the interphase boundary of the sheet cavitation. In addition, the baroclinic torque is caused by the non-parallel gradient between the pressure and velocity, and is mainly located near the area where detached cavitation occurs (label D, Figure 12). Overall, the vortex stretching and vortex dilatation are the dominant factors for vorticity transportation in the cavitating flow, and the interaction between the vortex and cavitation becomes stronger when it approaches the tip region.

Pressure fluctuation characteristics
To avoid structural resonance, studying the influence of the induced cavitation on the pressure fluctuations for the pump is important. The pressure coefficient is defined as follows: wherep is the time-averaged pressure.
Here, the pressure fluctuations at the tip clearance (P1-P3) are monitored and analyzed. P1, P2, and P3 are located at the tip gap between the leading edge, middle, and trailing edge of the blade and the casing wall. The pressure fluctuations at P1, P2, and P3 in the time domain are plotted in Figure 13(a)-(c). Similar variation trends are obtained for σ = 1.97, σ = 1.78, and σ = 1.40: six peaks and six valleys are presented within one period of rotation, which equals to the number of impeller blades. The flow field in the tip clearance changes dramatically near the leading edge while it becomes moderate in the downstream region. It is observed that the pressure fluctuation amplitude ( C p ) decreases gradually from P1 to P3. Due to the complex vortex evolution near the middle blade tip region, the pressure fluctuation amplitude at P2 for σ = 1.40 is obviously larger than that for σ = 1.97 and 1.78.
The frequency spectrum of the transient pressure at P1, P2, and P3 via the fast Fourier transform is shown in Figure 13(d)-(f). The wavenumber, k, is standardized based on the blade passing frequency (BPF), namely, k = f /nZ, where f denotes the frequency. The pressure pulsation induced by cavitation mainly exists in the low frequency range. The dominant frequencies are clearly observed at the BPF (k = 1) and its harmonics (k = 2, 3, . . . ), and the amplitude decreases gradually at higher frequencies. In addition, it is noted that the dominant frequency disappears when k ≥ 5 for P3 corresponding to three cavitation numbers.

Effect of inflow uniformity
Non-uniform axial inflow is a common situation in the actual working of pumps. The following computations are carried out under the same optimum flow rate (K Q−opt = 0.705) and cavitation number (σ = 1.78). The radial force, transient flow features, vortex and cavitation dynamics, and KE fluctuation characteristics are analyzed in turn.

Radial force
An evaluation of the radial loading of the impeller is crucial for the service life of the pump. The imbalanced radial force aggravates the bearing wear and induces significant vibration and noise of the pump in non-uniform flow. The radial loading coefficient K Tr is defined as follows: where Tr is the radial force. The radial loading coefficients of the impeller and single blade during one rotational period (0°≤ θ ≤ 360°) are compared in Figure 14. The dynamic balance of the impeller under the design condition is changed inevitably when the local surface loading is changed in the nonuniform inflow. Although the averaged K Tr under the three working conditions is close, the range of fluctuation of K Tr is obviously larger under non-uniform inflow condition (Figure 14(a)). A sinusoidal pattern is observed for the variation of the radial load of a single blade during one rotation period under the three conditions ( Figure  14(b)). The single blade also experiences a larger radial force fluctuation under non-uniform inflow conditions (N1 and N2, Figure 14(b)).

Transient flow features
The axial velocity and vorticity distribution of the longitudinal central section (x = 0) are shown in Figure 15. The axial velocity contours are divided as the impeller (I), stator (II), and hub (III) segments in sequence. The induction effect of the blades is stronger under heavier loading near the blade tip in non-uniform conditions, and thus, larger low velocity areas are observed near the tip gap in segment I. The non-uniform inflow along the radial direction leads to more complex rotor-stator interaction of the pump. The rectification effect of the stator is weakened, and the axial uniformity of outflow gradually worsens with the decrease of ζ in segment II and III.
The distribution of the out-of-plane vorticity ω x reflects the vortex characteristics. The maximum/mini mum ω x appears near the blade tip gap and casing wall (Figure 15(b)). To better analyze the details of the secondary flow, partially enlarged drawings of the target area (dotted box, Figure 15(b)) are provided in Figure 16. The velocity vector is shown in the line integral convolution mode. The pressure difference between the SS and PS of the blade is enlarged under non-uniform conditions and the turbulent leakage flow is strengthened. Recognizable vortex structures appear in the region with large vorticity, and it should be noted that the distribution of the vortex structures moves upstream (Figure 16, position A) with the decrease in the inflow uniformity.

Vortex and cavitation dynamics
In non-uniform inflow, the blade tip is subjected to heavier loads due to the lower incoming velocity. The trajectory of the TLV is obviously altered due to the variation of the vortex evolution process. The transient vortex morphology under uniform and non-uniform inflow conditions is illustrated in Figure 17. It is noted that the merging between the STLV and PTLV happens earlier with the decrease in the inflow uniformity. The contact   (Figure 17(c)).
The vortex evolution is related to the local velocity characteristics. The velocity triangle of the blade section is shown in Figure 18(b), where V r , V A , and V c denote the resultant, advance, and circumferential velocity, and u n , u a, and u t denote the resultant, advance, and circumferential induced velocity, respectively. Based on the attributes of inlet flow, the advance velocity near the casing wall under the non-uniform inflow condition is smaller than that under the uniform condition. As illustrated in Figure 18, the decrease in axial velocity (V A1 to V A2 ) results in the increase of the angle of attack (AoA, α 1 to α 2 ) for the same circumferential velocity (V c = 2π nr). The increase in the initial AoA in the blade tip region results in the intensification of flow separation in the tip gap and thus accelerates the evolution of the leakage vortex structures. Additionally, the relative change in AoA is inverse near the blade root area because the advance velocity is relatively larger under non-uniform inflow.
The vapor volume distribution on the blade surface under different inflow uniformity conditions is further analyzed. The three-dimensional, instantaneous cavitation morphology of the impeller is compared in Figure  19. The pump cavitation behavior deviates from the design condition as the vortical flow pattern is altered under non-uniform inflow. The flow passage between blades is blocked by the vapor, and the change of flow brings potential vibration and noise. Sheet cavitation covers the leading area of the suction side of the blade. Its distribution is obviously altered from the tip to the root due to the increasing inflow velocity under non-uniform conditions (Figure 19(b,c)). Vapor expansion is observed  near the tip area (label A, Figure 19) while vapor compression is observed near the blade root (label B, Figure  19) under non-uniform conditions. The sheet cavitation gradually concentrates towards the tip area along the radial direction. The relatively higher velocity increases the impact pressure and depresses the sheet cavitation near the blade root. According to the evolution of the vortices, the vapor volume of the TLV cavitation diminishes when the inflow uniformity ζ drops. The flow separation pattern in the tiny gap remains similar even though the inflow uniformity varies. Thus, the influence of inflow uniformity on the vortex cavitation of the tip corner can be ignored. Finally, from the turbulent boundary layer of the casing, the shear layer cavitation connects to the TLV    cavitation and its development depends strongly on the dynamics of the TLV cavitation.
The instantaneous contours of the pressure and TKE of the impeller blade surface are shown in Figure 20. In non-uniform inflow conditions, the higher advance velocity produces a higher impact on the blade surface and thus suppresses the water-vapor transition near the blade root. The negative pressure area is concentrated around the blade tip area, corresponding to the position of sheet cavitation. Figure 20(b) provides a clearer comparison of the TKE characteristics between the uniform and non-uniform situations. Similar to the TKE distributions in Figure 11, the high TKE is mainly located near the leading edge of the blade (labeled A, Figure  20(a)), the water-vapor boundary (labeled B, Figure 20), and a spanwise narrow strip (labeled C, Figure 20).

KE fluctuation characteristics
The KE ( = u 2 + v 2 + w 2 ) fluctuations at three probing points (P1-P3) are monitored and analyzed. The PSD analysis of KE fluctuations helps to interpret the energy transfer process in the cavitating flow. Figure  21(a) presents the averaged KE at three positions in uniform and non-uniform flow. The averaged KE increases gradually from P1 to P3 for ζ = 1 and 0.928, while it reaches its maximum at P1 for ζ = 0.857. This indicates that the increase in non-uniformity accelerates the energy dissipation in the tip gap flow near the leading edge. Figure 21(b)-(d) shows the frequency spectrum of the transient KE at P1, P2, and P3. The results indicate that the higher peak value of the spectrum curve occurs under greater non-uniformity. The TLV dominates the vortex system near P1 and its main contributions are the fundamental frequency at k = 1 (BPF) and its harmonics (k = 2, 3, 4) ( Figure 21(b)). The spectrum peaks are observed at non-integer multiples (e.g. k = 1.75) for ζ = 0.857, which denotes the vortex merging between the preliminary TLV and secondary TLV. As observed in Figure 17, the vortex merging happens slightly downstream for ζ = 1 and 0.928, and the above spectrum characteristic is not outstanding. The secondary TLV plays the major role when the flow develops to P2, and the leading frequencies are k = 1 and 2 (Figure 21(c)). The short-wave instability leads to the fast decline of the spectrum peaks at higher frequency in all cases (shortwave means that the wavelength of unstable wake mode is comparable to or smaller than the vortex core size). Additionally, the spectrum peaks at k = 4 appear at P3 ( Figure  21(d)), which is related to the vortex realignment near the trailing edge of the blade.

Conclusions
This paper investigates the correlative mechanism bet ween the ambient pressure/inflow uniformity and the vortex/cavitation dynamics for an axial-flow pump. The main conclusions are drawn as follows: (1) The PTLV forms from the blade tip corner and extends to the PS of the adjacent blade. Vortex merging happens between the primary TLV and secondary TLV, and the vortex instability is enhanced near the water-vapor interface. Lower ambient pressure promotes the evolution of TLV and results in larger vortex structures and faster energy dissipation in the flow passage.
(2) The cavitation instability is enhanced at a lower ambient pressure, resulting in the irregular interface near the tail closure region of the sheet cavitation. Vortex stretching and vortex dilatation are the dominant factors for vorticity transportation in the cavitating flow. The dominant frequencies of the pressure fluctuations are clearly noted at the BPF and its harmonics. (3) The axial non-uniform inflow alters the effective AoA of the airfoil at different radius of the impeller blade. The axial non-uniform inflow enlarges the amplitude of the radial loading fluctuation, and promotes the merging between the preliminary and secondary TLV as well. Associated with the spatial evolution of the vortices, the compression or expansion of sheet cavitation occurs under the non-uniform inflow conditions. (4) The PSD analysis of kinetic energy (KE) fluctuations gives a quantitative verification of the vortex evolution process and energy transfer in the cavitating flow. The increase in non-uniformity accelerates the KE dissipation in the tip gap flow near the leading edge, and the short-wave instability leads to a fast decline in the spectrum peaks at higher frequency.

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

Funding
Financial support was provided by the National Natural Sci-

Data availability statement
The data that support the findings of this study are available within the article.