On the cavitation-induced collapse erosion of a turbofan fuel pump

ABSTRACT Cavitation is a challenging flow phenomenon that can adversely affect the hydraulic performance of centrifugal pumps. In this study, numerical investigations were carried out incorporating the Zwart cavitation model derived from Rayleigh–Plesset equation with the Navier–Stokes equations to study a damaged turbofan pump. Pressure fluctuation was analyzed with fast Fourier transform (FFT) in both non-cavitation and cavitation flow fields. The entropy production theory was used to further analyze the positions of vortex cores and energy-loss behaviors inside the pump. The results demonstrate that the numerically predicted positions of vortex cores in high-pressure regions agree well with the erosion positions determined under real working conditions. Herein, through combined analysis of the pressure contours and the vortex-core locations, we demonstrate that the erosion failure of the surface is caused by fatigue failure due to the collapse of cavities at vortex cores. The percentages of different kinds of entropy production rates under different mass flow rates in the whole calculation domain were calculated, and it was found that the impeller suffered the most energy loss. The results of the entropy production analysis can provide guidance for subsequent optimization of turbofan pumps.


Introduction
Hydraulic machinery is frequently encountered in various applications in areas including the petroleum and aerospace industries. In the turbofan industry, they are considered as crucial components that are related to efficiency and stability. However, in the process of use, they can be damaged for several different reasons. The main causes of damage are cavitation problems, sand erosion, material defects, and fatigue . Among the causes of damage mentioned above, cavitation is the most common, and it is also the main reason for the fatigue failure of the research object in this study.
Cavitation and its accompanying erosion problems have been paid much attention in the application of hydraulic machinery (Lin et al., 2021), which is a special liquid-flow phenomenon. When the local pressure in a liquid is lower than that corresponding to the saturated vapor pressure of the liquid at a certain temperature, a cavitation core in the liquid will rapidly grow and form a cavity with a certain size. These cavities flow to highpressure regions, and under the action of the surrounding fluid, they contract, rebound, and collapse (Liu, Zhang, et al., 2020). When the collapse occurs, high-pressure liquid jet and shock wave emission will exert high stress on nearby boundary (Mahravan & Kim, 2021).
CONTACT Yadong Wu yadongwu@sjtu.edu.cn For a centrifugal pump, stable and safe operation is essential; cavitation will not only cause erosion of the flow channel, but it will also lead to flow blockage in the flow channel and cause violent vibration Sun et al., 2021;Tao et al., 2018). Cavitation has become an important factor affecting the longevity of centrifugal pumps, and it usually occurs near the leading edge of the impeller (Bouziad, 2006). When the flow rate is lower than the optimal operating point, the angle of attack at the inlet will be positive, and flow separation will occur easily on the suction surface, forming a lowpressure area in which cavities can appear (Friedrichs & Kosyna, 2002). Conversely, when the flow rate is greater than the optimal operating point, the inlet angle of attack will be negative, and cavitation will occur at the leading edge of the blade's pressure surface (Friedrichs & Kosyna, 2002). When the pump runs near the design point, cavitation areas will appear on the pressure and suction surfaces of the blade; this is also known as 'alternating blade cavitation' (Friedrichs & Kosyna, 2002). Because the collapse of cavitation will produce noise (Cernetic, 2009) and pressure fluctuations (Lu et al., 2016), detection of cavitation is possible from the analysis of noise signals and pressure fluctuations (Cudina & Prezelj, 2008).
Due to the complexity of the flow field inside a highspeed centrifugal pump, it is difficult to use the pressure drop to evaluate hydraulic losses, because researchers cannot know exactly where the energy losses are concentrated. However, the energy distribution characteristics of the flow in a centrifugal pump can be used as a criterion to evaluate its hydraulic performance and dynamic characteristics and to guide optimal pump design . In recent studies, many researchers (Chang et al., 2019;Gong et al., 2013;Su et al., 2021;Wang, Zhang, Hou, et al., 2019;Yu et al., 2020) have revealed that energy losses due to turbulent flow and friction are often closely related to local entropy generation rates, and they have begun to analyze the energy loss inside different flow fields based on the entropy production theory.
An entropy production diagnostic model for the cavitation flow was established (Wang, Zhang, Hou, et al., 2019), indicating that entropy production theory can be applied to cavitation flow analysis. The energy loss due to cavitation was analyzed (Yu et al., 2020), and the study found that the entropy production rate induced by velocity gradients provides the main contribution to the total entropy production rate when compared to that induced by temperature gradients or wall shear stress. Energy dissipation in a hydro turbine was analyzed using the entropy production theory (Gong et al., 2013), and the study concluded that the theory is useful for evaluating the performance of such a turbine. The optimal thickness distribution of blades was obtained through systematic theoretical analysis based on entropy production theory (Chang et al., 2019). The distributions of energy loss in different areas inside a high-speed centrifugal pump with straight blades were evaluated (Su et al., 2021), and the researchers found that, except for the volute, most of the flow parts were insensitive to the flow rates. However, there has been little examination of the possibility of combining entropy production theory to the study of cavitation in high-speed centrifugal pumps.
Excessive vortices will cause energy loss inside the flow field. The shedding of the large-scale cavity in cavitation flow violently changed the vorticity and pressure gradients in the flow field . Because the pressure in the central part of the vortex is lower than that in the surrounding fluid (Cucitore et al., 1999;Hunt et al., 1988), the small bubbles embedded in the fluid will get closer to the low-pressure region in the vortex center and collapse there (Green & Acosta, 1991), causing erosion on the surface. Although the use of numerical simulations greatly reduces the time cost in the process of analyzing the flow field and optimizing the pump structure, it is still difficult to comprehensively reproduce the microbubbles . Thus, the numerical simulations in this study aimed to identify the vortices and energy loss inside the flow field so as to predict the positions of the potential microbubbles and establish the causes of the surface damage, as well as to provide guidance for future optimization.
In this study, we focus on the erosion failure inside a turbofan fuel pump, which works in an extreme situation where the rotating speed is very high (around 20,000 rpm). In general, the field working conditions can be divided into 3 main situations. They are (1) the fuel supplying condition during taking-off; (2) the fuel supplying condition during acceleration; and (3) the normal flight condition without fuel-supply requirement. The special operating situations where the inlet and outlet valves only open for fuel supply leads to the fact that it will work under a small flow state with high rotating speed for a long time. This extreme situation makes the mechanism and position of erosion difficult to predict. The problem under this situation is also rarely involved in the previous studies. Herein, a vortex-identification method and multi-phase entropy production theory were combined to study non-cavitation and cavitation flow fields inside the turbofan centrifugal fuel pump, which provides a method for predicting the potential cavitation erosion that can be hardly detected through simple numerical models. The simulated vortex core positions are compared with the actual damage to the research object. The distribution of the entropy production rate inside the pump revealed the specific locations where energy loss is likely to occur. This can provide guidance for the prediction and prevention of erosion, as well as future optimization of the turbofan fuel pump. The rest of the paper is organized as follows. In Section 2, the description of the numerical simulations is presented. In Section 3, the simulation results and discussions are described. Finally, the conclusion is presented in Section 4.

Governing equations and cavitation model
The turbulence model adapted in this study was the k-ε turbulence model (Launder & Spalding, 1983), which is based on the eddy viscosity concept proposed by Boussinesq in 1877 (Boussinesq, 1877). Ansys R CFX, Release 2020 R1 was used to simulate the three-dimensional single-phase flow and cavitation flow inside the pump, the cavitation process was expressed by the transport equation of vapor volume fraction derived from the Rayleigh-Plesset equation (Plesset, 1949), which was derived from the Navier-Stokes equations under the assumption of spherical symmetry (Lin et al., 2002).
The Navier-Stokes equation is given by (White & Majdalani, 2006): (1) where U is velocity, g is acceleration of gravity, p is the pressure, ρ is the density, λ = − 2 3 μ, and μ is the viscosity. For two-phase cavitation flow, the density is defined as where α v is the vapor volume fraction, ρ l is the density of the liquid phase, and ρ v is the density of the vapor phase. The cavitation process can be described by the vapor volume fraction mass transfer equation. The transport equation of vapor volume fraction is given by: where ρ i and α i are the density and the volume fraction of two phases, respectively; S α is the source term indicates the phase-change rate, which can be defined as (Ahuja et al., 2001): whereṁ e andṁ c are the evaporation rate and condensation rate when the phase changes, respectively. The Rayleigh-Plesset equation is given by (Plesset, 1949): where R B represents the bubble radius, p v is the pressure in the bubble, p l is the pressure in the liquid surrounding the bubble, and σ is the surface-tension coefficient between the liquid and the vapor. Neglecting the viscous effect and the surface tension, Equation (5) can be reduced to the Rayleigh-Plesset equation for bubble dynamics: The change rate of bubble volume and bubble mass is given by: and respectively. If the number of the bubbles per unit volume is N B , then the vapor volume fraction can be expressed as: Then the source terms in the Equation (4) can then be defined as (Zwart et al., 2004): where C e and C c are the empirical coefficients for vaporization and condensation rates, respectively; α nuc is the volume fraction of the nucleation site; p v is the pressure in the bubble. C e = 50, C c = 10 −3 , α nuc = 5 × 10 −4 , R B = 10 −6 m.

Entropy production theory
Entropy production theory provides a new method for improving the performance of centrifugal pumps and guiding their hydraulic optimization, because it can reflect the location of irreversible losses in the fluid and the spatial distribution of energy consumption directly. For single-phase-flow, the entropy production rate can be defined as:Ṡ whereQ represents the energy transfer rate. For turbulent flows, the entropy production rate can be divided into two parts (Bejan, 1982): whereṠ D andṠ D are the entropy production rates generated by viscosity dissipation and turbulent dissipation, respectively. As it is not easy to measure the fluctuating velocity, a model has been established to replace the fluctuating velocity with turbulent dissipation (Callenaere et al., 2001;Kock & Herwig, 2004): where ρ is the density of the fluid and ε represents the turbulent dissipation rate. The total entropy production rate can be calculated by the volume integration of the specific entropy production rate: Because of the viscosity effect, an obvious velocity gradient exists on the wall surface when the fluid transitions from the turbulent core to the laminar boundary layer. Thus, the obtained entropy production rate through calculating Equation (15) will cause considerable deviation due to the error of ε in the near-wall low-Reynoldsnumber regions. A method to calculate the wall entropy production rate was established (Zhang et al., 2011): where S W is the wall entropy production rate, τ is the wall shear stress, v is the average velocity of the fluid near the wall, and A is the area of the wall. For two-phase-flow, μ, ρ in Equations (13-15) can be replaced by (Wang, Zhang, Hou, et al., 2019): where v stands for vapor and l stands for liquid. In the two-phase-flow, vapor area shares the same velocity with the liquid based on the assumption of the steady cavitation flow and homogeneous mixture model (Wang, Zhang, Hou, et al., 2019). The accuracy of the homogeneous entropy production model was verified in a recent research (Yu et al., 2021) by comparing the results obtained by either simulation and experiment. Figure 1 shows the structure of the turbofan fuel pump. The pump is composed of the impeller, volute, and shaft sleeve. The inlet valve is placed on the impeller chamber and the outlet valve is located at the exit of the diffuser. The ejector is also designed on the volute. The impeller is constructed from five slightly forward-curved straight blades. Some of the key parameters of the pump are listed in Table 1.

Research model and boundary conditions
As mentioned in Section 1, the pump works in two states: with the inlet and outlet valves open or closed. The fuel working inside the pump is liquid state and vapor state aviation kerosene. When the pump is used for fuel  supply, the inlet and outlet valves are opened. The fuel enters the impeller domain from the inlet valve and flows out from the volute outlet valve. When the pump stops supplying fuel to the aircraft, the inlet and outlet valve are closed, and a small flow rate of the fuel is then jetted by the nozzle inside the shaft sleeve to cool down the shaft seal. At the same time, the ejector helps to discharge the fuel and the impeller still maintains high-speed rotation.
The nozzle can be seen in Figure 1 as a nozzle pipe inside the shaft sleeve. Due to the special operating conditions of a turbofan fuel pump (the inlet and outlet valves only open for fuel supply), it will work under a small flow state for a long time.
When simulating the valve-open condition, the pressure inlet and mass flow rate outlet condition are adopted at the inlet and outlet valve boundaries, respectively. When simulating the valve-closed condition, the pressure inlet (nozzle) and mass flow rate outlet (ejector) are adopted as the inlet and outlet boundaries, respectively. All walls are set as no-slip wall condition. The rotating speed of the pump is 20,100 rpm. Time-averaged transient results are used to analyze the flow characteristics. The frozen rotor model is adopted for steady calculations, while the transient rotor-stator model is used for the unsteady calculations. The steady-state frozen rotor results file is used as the initial value of the transient simulation. The timestep length is set as the time for the impeller to rotate 3°, the total calculation time is the time for the impeller to rotate 10 cycles. The density of the liquid state aviation kerosene is 783 kg · m −3 , and the dynamic viscosity is 2.4 × 10 −3 kg · m −1 · s −1 . The density of the vapor state aviation kerosene is 7.1 kg · m −3 , and the dynamic viscosity is 7 × 10 −6 kg · m −1 · s −1 .   The verification of grid independence is shown in Figure 3. Since the local variables are more sensitive to mesh, pressure distribution at the volute outlet obtained by simulating four grid systems was compared. The number of the unstructured elements was 7.8, 10.9, 16.3 and 21.8 million, respectively.

Computational domain and mesh independence verification
The head coefficient and flow coefficient of a centrifugal pump can be defined as follows: where ψ is the head coefficient and ϕ is the flow coefficient, H is the head of the centrifugal pump, u 2 is the circumferential velocity at the impeller outlet, g is the acceleration of gravity, Q is the mass flow rate, D 2 is the diameter of the impeller, b 2 is the blade outlet width.
In the mesh independence verification test, the flow coefficient was set as 1.61, and the rotating speed was 20,100 rpm.
As can be seen, when grid number changed from 7.8 million to 10.9 million, the simulated local pressure values showed non-negligible deviations. However, with the mesh becoming finer, the local pressure values show negligible deviation. Therefore, the grid systems with 10.9 million, 16.3 million and 21.8 million elements performed almost equally well. In order to show more accurate details of the flow field, the grid system which contains 21.8 million unstructured elements was chosen. Figure 4 shows the experimental setup adopted to test the performance of the pump. All the experimental instruments have been calibrated. The test was carried out under the non-cavitating valve-open condition, where the rotating speed of the pump was 20,100 rpm. Figure 5 shows a comparison between the performance of the turbofan pump as obtained from experimental data and the results of the computational fluid dynamics (CFD) calculations at different flow coefficients ϕ. The CFD results are basically consistent with the experimental data. The accuracy of CFD simulation is not high at either end of the performance curve, but the error is still within acceptable limits.

Results and discussion
In the simulations, monitoring points were arranged along the circumference of the volute; their locations are shown in Figure 6.

Cavitation performance under the valve-open condition
Numerical simulations of the cavitation condition were performed, and Figure 7 shows the resulting cavitation characteristic curve of the pump at an outlet mass flow rate of ϕ = 0.71 and a rotating speed of 20,100 rpm.
The available net-positive suction head is defined as: where p inlet and p v represent the inlet pressure of the pump and the saturated vapor pressure of the working medium at the working temperature, respectively. The Y-axis is the static head of the pump, and is defined as: where p outlet is the outlet pressure of the pump, v outlet and v inlet represent the average outlet and inlet velocity at the cross sections of the pump, respectively. In engineering practice, the critical cavitation point is usually the 3% head drop point (Grote & Feldhusen, 2011), so when the head drops 6.5%, cavitation inside the centrifugal pump will have already occurred. Figure 8 shows the development of bubbles in the pump. The blue region in the figure is iso-surfaces of the saturated vapor pressure. As can be seen, vapor bubbles first appear on the suction sides near both the leading edges of the blades and the hub. With decreasing inlet pressure, the volumes of the bubble increase, and they gradually block the flow passage. Due to the impeller-volute interactions, the unsteady flow inside the pump will cause several kinds of unsteady flow phenomena, including pressure pulsation and flow separation. Figure 9 shows a frequency-domain plot of the pressure fluctuation at the different monitor locations, the frequency has been non-dimensionalized by the blade passing frequency (BPF). Figure 9(a,b) shows the pressure fluctuation in a non-cavitation flow field and a cavitation flow field at monitor positions: M14, M1, M5, M9, respectively. Figure 9(c,d) shows the pressure fluctuation in a non-cavitation flow field and a cavitation flow field at monitor positions: T-1, T-2, T-3, T-4, T-5, respectively. The peaks in the spectra are the BPF and its harmonics. The red lines marked in the figure indicate 0.4 BPF, and the blue lines marked in the figure are the amplitude of the pressure fluctuation at 0.4 BPF. When cavitation occurs, a peak at a frequency of 0.4 BPF can be discovered; this is because of the instantaneous pressure fluctuation caused by the change of the wake shape of the cavitation (Wang, Zhang, Li, et al., 2019;Wang et al., 2020). Due to the cavitation phenomenon, the existence of the cavitation phenomenon generates broadband noise and has an impact on the periodicity of the flow. It can be found that the impact at points Volute M14 (also called Volute-T-4), Volute M1, and Volute-T-5 are particularly obvious. While at other positions, the peaks in the region lower than 3 BPF can still be observed easily. Figure 10 shows the locations of the inlet nozzle and the outlet ejector. The fuel is jetted from the nozzle, flows through the gap between the shaft and the impeller, enters the impeller area, eventually flows into the volute, and is finally discharged by the ejector. The fuel flow into the impeller area is very small (ϕ = 0.003), and due to the effect of centrifugal force, it is thrown to the impeller outlet, forming a low-pressure area in the central region of the impeller. Figure 11 shows the fuel volume fraction under both valve-open condition and valve-closed condition. As can be seen from Figure 11, compared to the valve-open condition, under the valve-closed condition, the fuel enters the low-pressure area then cavitates and forms a large bubble cavity in the central area of the impeller. In contrast to the usual understanding that cavitation will cause damage to the materials, under this special working condition, the large bubbles formed by cavitation will envelop most areas of the impeller, and this will prevent the erosion damage that would be caused by the collapse of bubbles in the liquid. However, due to the large bubble volume and long-time operation under this condition, the large bubble will break up at high rotational speeds (Long et al., 2021). The small bubbles generated by this break-up will move downstream with the high-speed liquid in the flow passage and collapse near the pressure surface of the trailing edge due to high pressure, causing erosion to the blade (Franc & Michel, 2006).   Figures 12 and 13 show the positions of the vortex cores and photographs of the test object erosion. In Figure 12, the vortex in the impeller is expressed by the second invariant of the velocity gradient tensor Q c = 3 × 10 7 s −2 . As can be seen from the contours, the pressure near the volute tongue is much higher than that near the center of the impeller. The positions of the vortex cores (blue dashed circles) are similar to those of the erosion areas on the volute of the test object (white dashed circles). Likewise, in Figure 13(a), the positions of the vortex cores near the blades (red dashed circles) are also similar to those of the erosion locations on the blades of the test object; a photograph of the actual damage is shown in Figure 13(c). In Figure 13, Q c = 7 × 10 7 s −2 . As it was discussed in the introduction part, the low pressure in the central part of the vortex will draw the small bubbles in and the bubbles get closer to the vortex center will collapse there, causing erosion on the surface. Comparing Figure 13 Figure 13(b) do not represent any variable.

Characteristics of flow loss distribution inside the pump
To analyze the distribution of the entropy production rate in the flow field, three types of entropy production rate were compared. Figure 14 shows the distributions of three main parcels of the overall entropy production rate in the pump model under different mass flow rates. In actual operation, for the most fuel supply conditions, the pump works under a flow coefficient of 0.71 (the second working condition mentioned in Section 1), and cavitation phenomenon under this working condition is much obviously than those under flow coefficients of 1.78 and 2.50 (the first working condition mentioned in Section 1). As can be seen, the entropy production rate generated by turbulent dissipation (S TD ) accounts for the largest proportion. In both the non-cavitation and cavitation flow fields, turbulent dissipation causes the most losses. As the mass flow rate decreases, the proportion of S TD gradually decreases. That is because for the most fuel supply conditions, the pump has to work under a flow coefficient at 0.71 to provide fuel to the aircraft, and it is near the design flow coefficient. When ϕ = 0.71, the proportion of loss caused by turbulent dissipation in the cavitation flow field is higher than that in the non-cavitation flow field. In addition, for the turbofan pump, the wall entropy production rate (S W ) is also a non-negligible source of energy loss.
As Figure 14 shows, S W is the second-largest source of energy loss, and in the non-cavitation flow field, it increases as the flow coefficient decreases, while it decreases slightly when cavitation occurs. However, the energy loss generated by viscosity dissipation is negligible compared to that generated by turbulent dissipation   and the wall effect, that is because of the small effect of the mean velocity gradient at high Reynolds numbers (Re > 10 5 ) (Wang, Zhang, Hou, et al., 2019). Figure 15 shows the distributions of the entropy production rates in different domains under different flow coefficients, and the total entropy production rates under each flow coefficients have been marked on the top. In the non-cavitation flow field, as the flow coefficient decreases to that of the design flow coefficient, the total energy loss decreases. When cavitation occurs in the flow field, the total energy loss increases significantly and the ratio of energy loss in the impeller domain reduced slightly. The increase of the energy loss due to the cavitation development was discussed in several studies (Li et al., 2018;Liu et al., 2021). The reasons for this can be found in Figures 16 and 17. Figure 16 shows the distribution of vortex cores inside the pump at different times. The tip-leakage, leading-edge, and trailing-edge vortices can be clearly seen. Figure 17 shows the vortex core regions and the distribution of the entropy production rate inside the pump, in which Q c = 7 × 10 7 s −2 . In Figure 16, it can be seen that the vortex cores are not only distributed in the impeller, but also in the volute, especially the region near the ejector and the volute tongue. When the blade passes through the volute tongue, the vortex intensity near the tongue becomes smaller. More details can be found in Figure 17: when the flow coefficient decreases, the strength of the vortex in the impeller passage decreases significantly.
The energy loss inside the impeller is mainly concentrated at the inlet and the outlet. When the liquid enters the impeller, it first has an impact on the flow near  the inlet of the impeller blade, and the impact is more obvious under large flow rate conditions. The large flow coefficient also intensifies the impeller-volute interaction effect near the volute tongue, so the entropy production rate near the volute tongue is much higher. When cavitation occurs, it can be seen that the entropy production rate reaches significant levels at the trailing edge of the blades, near the tongue of the volute and around the bubbles, respectively. The entropy production rate around the bubbles can be observed from the zoomed-in views of (c) and (d), which is because the phase transition interface produces high entropy production rate. This phenomenon proved that under the valve-open condition, the homogeneous energy production model can predict the distribution of the energy loss near the cavitation interface. Note that the bubbles are located inside the white dashed circle in Figure 17. Figure 18 shows the entropy production rate generated by turbulent dissipation (S TD ). Under both conditions, the entropy production rate is higher at the pressure side near the trailing edge of the blades, which corresponds to the jet-wake flow. Compared with the valve-open condition, the entropy production rate is much higher under the valve-closed condition. It can be learned from Figure 17 that the distribution of the entropy production rate can reflect the vortex core regions to a certain extent, which is also supported by a recent study . This view can be further verified by combining the vortex core distributions under the two working conditions in Figure 13. Due to the special working conditions, the turbofan pump spends more time in the valve-closed condition, so the vortex cores near the blades will exist for a long time. However, whether under the valve-open or valve-closed condition, vortex cores always exist near the volute tongue. The bubbles that get into the center of the vortex will collapse due to the high pressure, and they will then cause fatigue failure on the surface of the test object. This is a special phenomenon due to the valveclosed working condition of the test object, and attention should be paid at the design stage. Figure 19(a) shows a given instantaneous relative position of the impeller and the volute. Every blade has been given a number, which aids understanding of Figure 20. Figure 19(b) shows the polyline which is created as an intersection between the plane at 0.5 blade outlet width and the blade surface. Figure 20 shows the section lines of the blades at 0.5 blade outlet width and the polylines of pressure and entropy production rate, in which the pressure and entropy production rate are along the section lines. The valve-open condition is shown in column (a) and the valve-closed condition is shown in column (b). Panels (i) to (v) (from top to bottom) represent the curves of blades 1-5, respectively. The flow coefficient of the valve-open condition is 0.71, the flow coefficient of the valve-closed condition is 0.003. Solid lines present the value along the blade pressure side; dashed lines present the value along the blade suction side.
From Figure 20, it can be seen that both the pressure value and entropy production rate reach their peak at the trailing edge of the blade pressure side (the positions of erosion), and the peak values under the valve-open condition are lower than those under the valve-closed condition. Under both working conditions, the value of  the entropy production rate on the pressure side is higher than that on the suction side.
We take blade 5, the closest blade to the volute tongue, as an example. The peak of the pressure value here is the highest among the five blades, which means that blade 5 suffers the highest load due to the impeller-volute interaction near the volute tongue. Under the valve-open condition, the pressure value reaches 3.14 MPa, while the value under the valve-closed condition reaches 5.85 MPa. It can be found from 20 a(v) and b (v) that there are two pressure peak values on the pressure side and the suction side. The peak value on the pressure side is higher than that on the suction side, which also leads to a higher pressure gradient on the pressure side. The peak value of the entropy production rate under valve-closed condition is the highest among the five blades, which means there are more vorticities in the flow near the volute tongue under this condition.
Under the valve-open condition, the entropy production rate on the pressure side has two upward trends along the direction from the impeller hub to the blade trailing edge. The first one is near the cavitation interface due to the bubble-induced turbulence, and the second one is near the blade trailing edge due to the jet-wake flow. Under the valve-closed condition, the energy loss due to the vortices near the blade trailing edge reach significant levels. However, compared to the energy loss near the blade trailing edge, the bubble-induced energy loss is not significant. The reason is that the homogeneous entropy production model may underestimate the energy loss near the cavitation interface, since the vapor pocket is so large under this working condition (Yu et al., 2021). Because the energy loss near the vapor pocket region under the valve-closed condition is not the main focus of this study, the limitation mentioned above is ignored. To the research focus on the energy loss details near the cavitation interface, a method raised by Yu et al. to calculate the energy loss near the cavitation interface will provide a more accurate result (Yu et al., 2021).

Conclusions
In this paper, we studied the mechanism of the erosion inside a turbofan fuel pump under an extreme condition. This study considered the flow field inside a turbofan pump and the causes of erosion damage on the surfaces of the impeller and the volute. CFD simulations were performed using the cavitation model derived from the Rayleigh-Plesset equation to analyze the positions and volume fractions of cavity bubbles under two different working conditions, corresponding to the valve-open and valve-closed conditions. The main conclusion can be drawn as follows: (1) The energy loss distributions in both non-cavitation and cavitation flow were simulated by homogeneous entropy production model. The distribution of the entropy production rate can reflect the vortex core regions in the flow field. The simulated vortex core positions fit well with the positions of the actual damage, which provided a method to predict the potential cavitation-induced collapse erosion in the pump design stage. (2) The energy loss behavior under the valve-open condition was analyzed. The impeller suffers the most energy loss, and the existence of the cavitation will increase the entropy production rate in the flow field. The energy losses are generally located near the cavitation interface, the blade trailing edge, and the volute tongue, respectively. In the future optimization of centrifugal pumps, researchers can focus on locations at which more energy loss occurs. For example, from the perspective of improving the flow in the flow field, the possibility of surface erosion can be reduced through optimizing the blade shape and the fillet radius of the volute tongue. To accurate predict the energy loss near cavitation interface zone will be a challenge, and the improvement of the entropy production model near the cavitation interface can be a potential research direction.

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

Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.