Applications of different vortex identification methods in cavitation of a self-priming pump | Scientific Reports

Mar 10, 2025

Scientific Reports volume 15, Article number: 7458 (2025) Cite this article

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To investigate the internal flow mechanisms during cavitation in self-priming pumps, this study employs numerical simulations based on the RNG k-ε turbulence model coupled with the Schnerr–Sauer cavitation model. The vapor–liquid two-phase flow characteristics at the critical cavitation condition are analyzed under various flow rates and rotational speeds. Based on the vorticity method, Q criterion, λ2 criterion, Δ criterion, λci criterion, and Ω criterion, predictions and comparative analyses of the vortex structures during cavitation in self-priming pumps have been conducted. Additionally, the energy loss in various regions of the pump before and after cavitation onset has been analyzed using the theory of entropy production. The results demonstrate that the Ω criterion, Q criterion, and λ2 criterion yield consistent vortex identification outcomes across different regions. In contrast, the Δ criterion and λci criterion produce more fragmented vortices, most notably in the gas–liquid separation chamber area, whereas the vorticity method is most susceptible to disturbances from strong shear layers; Energy losses are not directly related to vorticity but show a significant correlation with vortex intensity; Changes in flow rate and the onset of cavitation have the greatest impact on average entropy production in the reflux hole region, particularly under high flow conditions; The difference in energy loss of the impeller is the largest before and after cavitation occurs; When the cavitation coefficient is constant, the vapor volume fraction in the impeller region exhibits a positive correlation with both flow rate and rotational speed.

As a category of fluid machinery, pumps play a pivotal role across the spectrum of industrial, agricultural, and everyday applications1,2,3,4. Self-priming centrifugal pumps, characterized by their unique design that allows for the storage of a sufficient quantity of liquid within the pump chamber, are a specialized form of pump. They utilize distinctive components such as gas–liquid separation chambers to achieve the mixing and subsequent separation of gas and liquid phases, culminating in the expulsion of gas and completion of the self-priming process. Self-priming pumps are widely employed in sectors such as chemical engineering, irrigation, and municipal services. Extensive scholarly research has already been dedicated to investigating the self-priming performance and hydraulic characteristics of self-priming pumps.

Wang et al. conducted numerical calculations of unsteady pressure fluctuations in a centrifugal pump before and after the onset of cavitation5. They observed that severe cavitation exacerbated the vortical flow within the impeller and gave rise to significant low-frequency pressure pulsations with large amplitudes. Through visualization experiments and noise measurements, they further investigated the noise characteristics inside the pump under cavitating conditions6. Their findings revealed that cavitation bubbles within the impeller passage exhibited a "congregation-extension-detachment-recongregation" pattern, and the broadband sound pressure level increased as the severity of cavitation intensified. Hu et al. conducted experimental investigations and numerical calculations on leakage flow in an axial-flow pump7, exploring the relationship between cavitation development and the intensity of TLV (Tip Leakage Vortex) under varying cavitation conditions. Qu et al. proposed an adaptive cavitation flow model8, and based on the Omega vortex identification theory, they discovered that compared to the ZGB model, the prediction accuracy for tip leakage vortex cavitation by the adaptive model improved by 181% under large clearances and approximately 27% under small clearances. Moreover, the predictions for attached cavitation by the adaptive model were found to be closer to experimental results. Sun et al. utilized an improved transient computational model based on the bubble rotation-enhanced BRZGB cavitation model to investigate the interplay among cavitation, vortex, and pressure fluctuation in a centrifugal pump under partial load conditions9. They observed that the peak intensity of pressure fluctuations occurred in the vicinity of the gas–liquid interface at the rear of the cavity, shifting downstream as cavitation progressed. A lagging effect was noted among the vapor volume fraction, vorticity, and pressure fluctuations, with variations in vapor volume fraction serving as the driving force behind the interactions among cavitation, vorticity, and pressure. Liu et al. found that vortex stretching is the main promoting factor for the development of cavitation10, while vortex expansion links the cavitation cavity and vortex together. A distinct baroclinic torque was observed at the liquid-vapor interface, and turbulent stresses were found to be associated with the genesis of cavitation. Gong et al. found that under lower ambient pressures, as the cavity enlarges, the merger of primary and secondary tip leakage vortices (TLV) occurs earlier11. The contact location of the merged TLV with the adjacent blade moves upstream as the cavitation number decreases. Long et al. analyzed the development process of the cavitating flow from initial cavitation to the first critical cavitation in a water jet pump through high-speed cinematography12.

Laborde and Tabar observed that the TLV rolls up near the suction side of the impeller blades and connects with the vortices generated by the leakage flow13,14. The interaction between the TLV and attached cavitation impacts the performance degradation of axial-flow pumps, leading them to conclude that the flow rate plays a significant role in delaying the onset of tip vortex cavitation. Xu et al. employed several distinct vortex identification theories to forecast the cavitating flow field within a water jet pump15. All vortex identification methodologies accurately predicted the tip separation vortices in the blade tip region; however, only the Ω and Liutex iso-surface approaches were capable of predicting the weaker vortices present in the cavitation zones. They also deliberated upon the influence of the small parameter ε in the Ω method on the identification of tip vortices. Jia et al. observed that when cavitation occurs, an omega vortex with relatively higher velocity appears at the leading edge of the impeller16. As cavitation progresses, the vortex velocity and flow velocity increase gradually. The major factors contributing to vorticity changes are the stretching term caused by variations in velocity gradients, the compression-expansion term due to changes in vorticity, and the baroclinic torque term resulting from imbalances in strength gradients. AI-Obaidi and Qubian employed a developed CFD code to perform transient numerical simulations of the flow field in centrifugal pumps with different impeller outlet diameters under single-phase and cavitating conditions17. The study results indicated that the CFD model accurately simulated the performance and characteristics of the centrifugal pumps. Lu et al. identified that tongue cavitation is a form of vortex cavitation through numerical calculation of the internal flow in a pump operating under overloaded conditions18. Xue and Piao elucidated the shedding mechanism of tongue cavitation when the pump is operated under overload conditions through numerical calculation19, and Hu et al.'s experiments verified this conclusion20. Chang et al. proposed a new self-priming pump21, and based on the numerical solution approach of the Q-criterion and vorticity transport equation, they investigated the spatial structure of vortices within the pump. Using entropy production analysis, they studied the influence of blade thickness on the pump’s energy characteristics, finding that increasing the blade thickness from leading to trailing edge contributes favorably to enhancing pump performance. Yang et al. studied a typical short jet self-priming pump and analyzed the influence of key structural parameters inside the injector on cavitation performance and energy characteristics22. The authors of this article had previously conducted an exhaustive investigation into the self-priming performance of self-priming pumps23. A recirculation piping system was constructed, encompassing components such as self-priming pumps, with a prescribed volume of water stored both in the reservoir tank and within the pump itself. The tank’s liquid surface was left open to atmospheric pressure. Concurrently, the acceleration phase of the pump’s rotational speed was incorporated into the computations through user-defined functions. Consequently, both the computational physical model and the characteristics of rotational speed variation were in strict accordance with practical circumstances, thus enabling an accurate representation of the genuine self-priming process.

Liu et al. proposed a new method for optimizing the performance of multi-stage multiphase pumps based on Oseen vortex theory. This method accurately predicted the momentum flux with an average deviation of 6.45%. It was applied to optimize the inlet blade angle of the subsequent impeller, resulting in average improvements of 0.29% in head and 0.19% in efficiency24. Sun et al. investigated the flow mechanism of hub corner separation vortices and developed a method to suppress these vortices. Using Computational Fluid Dynamics (CFD) simulations and Central Composite Design (CCD), they optimized the skew and sweep of the diffuser blades25. Zhou et al. proposed an intelligent recognition method based on Multiscale Permutation Entropy (MPE) and Random Forest (RF) for real-time detection of spiral cavitation vortex ropes (VR) in draft tubes. This method achieves 100% accurate identification of different VR states, which is of significant importance for ensuring the safe and stable operation of pump-turbines26. Zhang et al. found that the head performance is significantly strengthened by the longitudinal vortex enhancing the momentum exchange between the impeller and side channel. The axial and radial vortex structures restricted in the impeller flow passage impedes the positive momentum exchange rate resulting in high entropy loss and pressure pulsation intensities in the impeller27. Shi et al. found that under braking conditions, unstable vortices exist between the guide vanes of a pump-turbine due to the sudden change in radial velocity, which can easily lead to severe blockage between the guide vanes28. Zhang et al., based on an axial flow pump, compared the vortex identification performance of two methods: the Q-criterion and the Liutex method. They also found that the inlet vortices and the impeller interact with each other. Under the influence of the rotating blades, the vortices transition from low frequency to blade frequency, and the vortices significantly alter the tangential velocity within the impeller29.

In summary, it can be observed that numerical simulation technology plays a crucial role in the design and optimization of self-priming pumps. Vortex identification is a key aspect of this process. Through vortex identification, a deeper understanding of the vortex structures within the flow field can be achieved, which helps designers comprehend the complexity of fluid flow. It is essential for reducing energy losses and improving the efficiency of the pump. It also aids in identifying vortices that could lead to the formation of low-pressure regions, allowing for measures to be taken to avoid or mitigate cavitation phenomena.

In the operation of a self-priming pump, the liquid level in the suction reservoir decreases continuously, the self-priming height increases, and the pressure at the pump inlet drops. Upon reaching a particular operating condition, cavitation will occur within the self-priming pump. The initiation of cavitation not only exacerbates the unstable flow within the pump, leading to a decline in hydraulic performance, but it can also induce vibrations that affect the stability of the pump’s operation and may even generate noise pollution30,31,32,33. Therefore, investigating the cavitation performance of self-priming pumps holds significant engineering relevance. However, current research on the cavitation characteristics of self-priming pumps is quite limited. Cavitation performance in self-priming pumps is directly related to unstable vortex flows within the pump, hence the accurate identification of internal vortex structures during cavitation is of significant importance. This paper focuses on a horizontal self-priming pump as the subject of study. It employs CFD methods to numerically simulate the full three-dimensional, incompressible flow inside the self-priming pump during cavitation. Simultaneously, utilizing vorticity methods, the Q criterion, λ2 criterion, Δ criterion, λci criterion, and Ω criterion, the internal vortex structures of the self-priming pump during cavitation are predicted and comparatively analyzed. Additionally, an energy loss analysis of the cavitation flow field in the self-priming pump is conducted based on the theory of entropy production.

The full three-dimensional computational domain of the internal structure of the self-priming pump is illustrated in Fig. 1, primarily comprising the inlet extension, S-pipe, front pump chamber, impeller, clearance, volute, rear pump chamber, reflux hole, gas–liquid separation chamber, and outlet extension. The design operating conditions for the self-priming pump are: flow rate Qd = 120m3/h, rotational speed nd = 2950r/min, and head Hd = 75 m.

Three-dimensional computational domain of the self-priming pump.

Key structural parameters include: suction diameter Ds = 157 mm, discharge diameter Dd = 120 mm, number of blades z = 5, impeller inlet diameter D1 = 113.5 mm, impeller inlet width b1 = 5.1 mm, impeller outlet diameter D2 = 251 mm, impeller outlet width b2 = 15.2 mm, blade outlet angle β = 35°, blade coverage angle φ = 156°, balance hole diameter D3 = 5 mm, and base circle radius r = 47 mm. The calculation shows that the self-priming pump has a specific speed ns = 77, which belongs to the category of low specific speed centrifugal pumps.

The computational model of the self-priming pump was meshed using a hybrid structured-unstructured grid approach with ICEM software, and grid refinement was applied at critical locations such as the tongue, reflux hole, and balance holes. To enhance mesh quality, a more adaptable tetrahedral unstructured grid was used in the gas–liquid separation chamber area, whereas hexahedral structured grids were employed for the impeller and volute sections, as shown in Fig. 2. Figure 3 shows the local meridian grid distribution of the self-priming pump.

Self-priming pump grid.

Local meridional plane grid.

The dependence of the self-priming pump head on the grid is shown in Fig. 4. Numerical computations were performed on the self-priming pump with varying grid counts at the design condition, with grids number of 1.32 × 106, 2.13 × 106, 2.51 × 106, 2.84 × 106, 3.25 × 106, 4.14 × 106 and 4.63 × 106 respectively. As the grid density increased, the pump head tended towards stabilization, indicating that the grid independence criterion was met. It was found that when the grid count was less than 2.84 × 106, the pump head had not yet stabilized, failing to meet the requirements. Conversely, with a grid count exceeding 2.84 × 106, the pump head reached stability, fulfilling the computational requirements. While theoretically, a greater number of grid points translates to higher accuracy in the results, computational efficiency must also be considered. Therefore, for the subsequent internal flow field computations, a grid scheme with a total of 2,843,374 cells was selected as the optimal choice.

Grid independence verification.

To ensure that the flow characteristics near the wall are accurately captured, a y-plus (y+) value analysis of the near-wall region mesh was conducted based on the aforementioned grid scheme. y+ represents the dimensionless distance of the nearest node to the wall. According to the y+ values, the near-wall flow layer can be roughly divided into three regions: the viscous sublayer (y+ < 5), the buffer layer (5 < y+ < 30), and the logarithmic layer (30 < y+ < 200). To ensure computational accuracy, the boundary layer mesh should not be too dense. For the RNG k-ε turbulence model used in this study, it is appropriate for the first layer grid value to be within the logarithmic layer, where the near-wall flow is handled using wall functions.

To ensure that the flow characteristics near the wall are accurately captured, a y+ value analysis of the near-wall region mesh was conducted based on the aforementioned grid scheme. The results show that the y+ values on the blade surface range from 24 to 66, indicating that the treatment of the near-wall region meets the requirements for numerical simulation of the flow field within the self-priming pump.

In the model based on the homogeneous multiphase transport equations, the flow control equations are34:

The aforementioned set of equations consists sequentially of the continuity equation for the gas–liquid mixture, the conservation of momentum equation, and the transport equation for the gas phase volume fraction. The introduction of the volume fraction transport equation is aimed at resolving the distribution of the two phases within the flow field. Where $t$ denotes time in seconds; ${\text{i}}$ and ${\text{j}}$ represent coordinate directions; $u_{i}$ is the velocity component; $\rho_{m}$, $\rho_{v}$, $\rho_{l}$ correspond to the mixture density, gas phase density, and liquid phase density, measured in kg/m3; $\delta_{ij}$ is the Kronecker number; $\alpha_{v}$ represents the vapor phase volume fraction; $\mu$, $\mu_{t}$ denote the mixture dynamic viscosity and turbulent viscosity in kg/(m s); $R$ denotes the interphase mass transfer rate in kg/(m3 s).

$\rho_{m}$ and $\mu$ are the volume weighted averages of the vapor and liquid phases respectively:

where $\mu_{l}$、$\mu_{v}$ are the dynamic viscosities of the liquid and vapor phases respectively.

The interphase mass transfer rate $R$ can be simulated using a suitable cavitation model:

where $R_{e}$、$R_{c}$ represent the vapor generation rate and vapor condensation rate respectively.

This study employs the RNG k-ε turbulence model to close the Reynolds-averaged35. The RNG k-ε model has been widely applied in various industrial and scientific research fields, and has been well validated in many studies36,37. The transport equations for turbulent kinetic energy k and dissipation rate ε are given by:

where $u_{i}$ denotes the velocity; $\alpha_{k}$ is a constant equal to 1.39; $G_{k}$ denotes the production term of turbulent kinetic energy k caused by the average velocity gradients; $G_{{\text{b}}}$ denotes the production term of k due to buoyancy effects; $Y_{{\text{m}}}$ denotes the contribution of pulsating expansion in compressible turbulence; $S_{{\text{b}}}$ and $S_{\varepsilon }$ are user-defined source terms; $C_{1\varepsilon }$,$C_{2\varepsilon }$,$C_{3\varepsilon }$ are empirical constants; $\mu_{{\text{t}}}$ denotes the turbulent viscosity coefficient:

where $C_{\mu }$ is a model constant, assumed to be 0.9; $f(\rho )$ denotes the density function:

The turbulent viscosity in the cavitation region is restricted through modification, the exponent n was set to 10.

This study employs the Schnerr-Sauer model to simulate cavitation in the self-priming pump42:

where $R_{B}$ is the bubble radius, m; $P$ and $P_{v}$ are the field pressure and vapor pressure respectively, Pa; $\rho_{m}$, $\rho_{v}$ and $\rho_{l}$ denote the mixture density, vapor phase density, and liquid phase density respectively, kg/m3; $n_{0}$ is the number of bubbles per unit volume of liquid. The mass transfer rate in the model is proportional to $\alpha_{v} \left( {1 - \alpha_{v} } \right)$, and a significant characteristic of function ${\text{f}}\left( {\alpha_{v} ,\rho_{v} ,\rho_{l} } \right) = \rho_{v} \rho_{l} \alpha_{v} \left( {1 - \alpha_{v} } \right)/\rho_{m}$ is that when $\alpha_{v} = 0$ or $\alpha_{v} = 1$, ${\text{f}}\left( {\alpha_{v} ,\rho_{v} ,\rho_{l} } \right)$ approaches 0, while when $\alpha_{v}$ is between 0 and 1, ${\text{f}}\left( {\alpha_{v} ,\rho_{v} ,\rho_{l} } \right)$ reaches its maximum value. The sole parameter to be determined in this model is the bubble number density $n_{0}$, and extensive research has indicated that an optimal bubble number density resides around 1013.

Vorticity method38

The vorticity method calculates the intensity of vortices through the computation of the curl of the velocity field:

where $u$, $v$, $w$ are the velocities corresponding to the X, Y, and Z directions respectively; $\omega_{x}$, $\omega_{y}$, $\omega_{z}$ are the vorticities corresponding to the X, Y, and Z directions respectively; $|\omega |$ is the total vorticity.

Q criterion39,40

Conventional vortex identification methods are based on the velocity gradient tensor $\nabla v$, with the characteristic equation as presented in Eq. (16):

where P, Q, R are the three invariants of the velocity gradient tensor. The velocity gradient tensor can also be expressed as:

where $A$ represents the symmetric part known as the strain-rate tensor, $B$ is the anti-symmetric part known as the vorticity tensor.

Based on the second invariant Q of the velocity gradient tensor provided in Eq. (16), the Q criterion is directly derived, and its expression is as follows:

where Q > 0 signifies the presence of vortices. According to the definition, the Q criterion defines vortices as regions where the vorticity is greater than the strain rate. The symmetric tensor $A$ has the effect of counteracting the rigid-body rotation imparted by the anti-symmetric tensor $B$;thus, the physical significance of Q lies not only in the requirement for vorticity (represented by the anti-symmetric tensor $B$) to exist within the vortex structure but further necessitates that the anti-symmetric tensor $B$ overcome the cancellation effect represented by symmetric tensors $A$.

λ2 criterion41

When the flow field exhibits strong unsteady and viscous effects, the λ2 criterion employs the method of identifying vortices through the detection of local minima in the pressure within planes. Therefore, neglecting the unsteady and viscous terms in the incompressible Navier–Stokes equations and decomposing the velocity gradient tensor into its symmetric and anti-symmetric parts, one arrives at:

where $p$ denotes the pressure, $\rho$ denotes the density. Based on this equation, when $A^{2} + B^{2}$ has two negative eigenvalues, the pressure attains a minimum value within the plane spanned by the eigenvectors corresponding to these two negative eigenvalues. If the eigenvalues are arranged in order of $\lambda_{1} > \lambda_{2} > \lambda_{3}$, the condition for $A^{2} + B^{2}$ to have two negative eigenvalues is equivalent to $\lambda_{2} < 0$, and vortices are defined as regions where $\lambda_{2} < 0$.

Δ criterion42

Perry and Chong employed critical point theory to define regions with complex eigenvalues of $\nabla v$ as vortex cores43. In a non-rotating frame of reference that moves with the fluid particles, the instantaneous streamline pattern—derived from the Taylor series expansion of the local velocity field—is governed by the eigenvalues of $\nabla v$. When two of these eigenvalues are complex conjugates, the streamlines become closed loops or helical shapes, applicable to both compressible and incompressible flows. In unsteady flows, utilizing instantaneous streamlines implies assuming a frozen velocity field for that instant. The discriminant of the characteristic equation of the velocity gradient tensor is given by:

When the flow is incompressible, $P = 0$, thus $\tilde{Q} = Q$ and $\tilde{R} = R$, leading to $\Delta = \left( {Q/3} \right)^{3} + \left( {R/2} \right)^{2}$. It follows that when $Q > 0$, $\Delta > 0$. However, there remains a possibility of $\Delta > 0$ even if $Q < 0$. Therefore, based on the Δ criterion, points where $Q < 0$ may still reside within vortices, indicating an inconsistency between the Q and Δ criterion.

λci criterion42

Zhou et al. proposed the λci criterion based on the Δ criterion44. This criterion posits that when the velocity gradient tensor possesses a pair of conjugate complex eigenvalues, streamline patterns can effectively delineate the shape of vortices. The feature decomposition of $\nabla v$ is shown in Eq. (25):

where $\lambda_{r}$ is the real eigenvalue, $v_{r}$ is the corresponding eigenvector; whereas $\lambda_{r} \pm \lambda_{ci}$ denotes a pair of conjugate complex eigenvalues, and $v_{cr} \pm v_{ci}$ denotes their eigenvectors. Under the curvilinear coordinate system $(c_{1} ,c_{2} ,c_{3} )$, where instantaneous streamlines coincide with tracing lines, the following relation holds:

where $t$ represents a similar time parameter, while $c_{1} (0)$, $c_{2} (0)$ and $c_{3} (0)$ are constants dependent on initial conditions. From Eqs. (27) and (28), it becomes evident that this represents the trajectory of a helix, with a pitch determined by $2\pi /\lambda_{ci}$. The λci criterion presents certain limitations: the imaginary part of the eigenvalue $\lambda_{ci}$ might not exist in some exceptional scenarios.

Ω criterion42

Based on the foregoing discussion, vorticity does not necessarily represent the rotational motion of the fluid. A classic counterexample is the laminar boundary layer, where despite the presence of significant vorticity near the wall, there is no actual rotational motion in the flow. Therefore, it is necessary to further decompose the vorticity into its rotational and non-rotational components:

where $R$ is the rotational part of the vorticity, $S$ is the non-rotational part, which is essentially pure shear. Typically, the directions of $R$ and $\omega$ differ. Here, a parameter Ω is introduced, representing the ratio of the magnitude of the rotational vorticity component to the total vorticity magnitude. Liu et al. proposed a formula for estimating Ω:

The definitions of a and b are identical to those in Eq. (18). In practical application, a small positive number ε is added to the denominator in Eq. (30) to avoid division by zero issues:

Obviously, the value range of Ω is 0 ~ 1, which can be interpreted as the concentration of vorticity, or more specifically, the rigidity of the fluid motion. When Ω = 1, it signifies that the fluid is undergoing rigid body rotation. Ω > 0.5 indicates that the anti-symmetric tensor $B$ dominates over the symmetric tensor. Therefore, Ω slightly greater than 0.5 can be used as a criterion for vortex identification. In this paper, the threshold of Ω is selected as 0.52, which can better identify both strong and weak vortices simultaneously.

This article is based on FLUENT commercial computing software and adopts the Mixture multiphase flow model and RNG k-ε turbulence model. A pressure-based Coupled solver is selected for the simultaneous solution of pressure and velocity. The impeller rotation method adopts reference frame moving. The initial computational medium is pure water, with the saturation vapor pressure set at 3540 Pa. The outlet boundary of the computational domain is designated as a pressure outlet, with the reference pressure set to 0 Pa. The inlet is specified as a velocity inlet boundary, and cavitation is induced within the pump by progressively decreasing the outlet pressure. The initial liquid phase volume fraction at the inlet is 1, and the vapor phase volume fraction is 0. Surfaces in contact with the rotating domain, such as the front and rear cover plates, are set as rotating walls with the same angular velocity as the impeller. All walls in the computational domain are set to be adiabatic and no slip; The standard wall function is used in the near wall area. To enhance computational accuracy, turbulence models and convective diffusion terms are configured for high resolution, with convergence residuals set to 10−5. The momentum secondary relaxation term、the turbulent kinetic energy secondary relaxation term and turbulent dissipation rate term all adopt first-order upwind scheme. The explicit-relaxation coefficients for the momentum and pressure equation are both 0.5. The under-relaxation coefficients for the equations for the turbulent kinetic energy and its dissipation are both 0.8.

In numerical calculations, to improve the accuracy of cavitation calculations and accelerate convergence speed, numerical calculations without cavitation are performed first, and then the results are used as the initial conditions for cavitation numerical calculations. Cavitation is triggered by setting and gradually reducing the outlet pressure, thereby inducing cavitation within the self-priming pump.

Experimental research was conducted on the self-priming pump described in this paper to obtain data on flow rate, head, and efficiency. The hydraulic performance curves derived from numerical calculations were compared with those from actual experiments to ensure the accuracy of the simulation results. The experimental system utilized a range of instruments, including valves and pressure transmitters at the inlet and outlet, an electromagnetic flow meter, a prime mover, and a torque meter. Specifically, the electromagnetic flow meter model used was LZNDB-100, which has a wide measurement range of 0 ~ 210 m3/h. The pressure transmitters had a measurement range of 0 ~ 2.5 MPa. The pump test bench is shown in Fig. 5.

Hydraulic performance experiment site.

The hydraulic performance results of the self-priming pump obtained from experiments were compared with numerical simulation outcomes for validation, as illustrated in Fig. 6. With an increase in flow rate, both the experimental and numerical head decreased continuously. At the rated flow rate of 120 m3/h, the experimental head was 73.93 m, while the numerical head was 71.64 m, resulting in a relative error of 3.10%. The shaft power increased with the rise in flow rate; at the rated flow rate of 120 m3/h, the experimental shaft power was 45.67 kW, and the numerical shaft power was 44.11 kW, leading to a relative error of 3.41%. The efficiency of the self-priming pump initially increased and then decreased with increasing flow rate, with the increment diminishing progressively. At the rated flow rate of 120 m3/h, the experimental efficiency was 54.68%, whereas the numerical efficiency was 55.48%, yielding a relative error of 1.46%.

External characteristics calculated values and experimental values comparison.

It can be observed that under the rated flow condition, the discrepancy between experimental and calculated external characteristics is minimal, falling within an acceptable range. In the lower flow rate range, the calculated head and shaft power were below the experimental values; however, in the higher flow rate range, the calculated head and shaft power slightly exceeded the experimental values. Regarding efficiency, except for the significant difference between the experimental and calculated values at a flow rate of 166 m3/h, the error was within reasonable limits for other flow conditions. In summary, the discrepancies between experimental and numerical results fall within acceptable ranges, indicating a high degree of conformity, thus substantiating the reliability of the numerical calculation method.

Through steady-state numerical simulations conducted at three different flow conditions—0.6Qd, 1.0Qd, and 1.4Qd—and three different rotational speed conditions—0.8nd, 1.0nd, and 1.2nd—the aNPSH values and heads under various inlet pressures could be obtained using the aNPSH calculation formula45. Figure 7a,b show the curves of head variation with cavitation coefficient under different flow conditions and different speed conditions respectively.

where ${\text{p}}_{{{\text{in}}}}$ is the inlet pressure of the self-priming pump, Pa; ${\text{p}}_{{\text{v}}}$ is the saturation vapor pressure, ${\text{p}}_{{\text{v}}}$ = 3540 Pa; $\rho$ is the fluid density, $\rho$ = 998.2 kg/m3; ${\text{g}}$ is the gravitational acceleration, ${\text{g}}$ = 9.81 m/s2.

Head variation curves under different operating conditions.

Computations reveal that when the cavitation coefficient is sufficiently large, the head of the self-priming pump remains essentially constant, unaffected by alterations in internal energy characteristics. Under the rated rotational speed, the stable heads for the conditions at 0.6Qd, 1.0Qd, and 1.4Qd are 78.46 m, 71.64 m, and 61.88 m, respectively. At the rated flow rate, the stable heads for conditions at 0.8nd, 1.0nd, and 1.2nd are 42.09 m, 71.64 m, and 108.10 m, respectively. As the inlet pressure decreases progressively, so does the cavitation coefficient. When the cavitation coefficient diminishes to a certain level, the head begins to decline. The point at which this decline initiates is designated as the incipient cavitation point of the self-priming pump. Through computation, the incipient cavitation coefficient for each condition can be determined: 6.95 m, 5.61 m, and 8.22 m for 0.6Qd, 1.0Qd, and 1.4Qd, respectively, corresponding to inlet pressures of 71,612 Pa, 58,462 Pa, and 84,031 Pa; and 4.84 m, 5.61 m, and 10.00 m for 0.8nd, 1.0nd, and 1.2nd, respectively, corresponding to inlet pressures of 50,902 Pa, 58,462 Pa, and 101,447 Pa. As the inlet pressure declines further, the cavitation coefficient decreases, and the rate of decrease in head accelerates. When the head drops by 3%, the critical cavitation point of the self-priming pump is identified: the critical cavitation coefficient for 0.6Qd, 1.0Qd, and 1.4Qd are 1.26 m, 2.01 m, and 3.78 m, respectively, corresponding to inlet pressures of 15,835 Pa, 23,241 Pa, and 40,530 Pa, with heads of 76.11 m, 69.49 m, and 60.02 m, respectively, at critical cavitation; and for 0.8nd, 1.0nd, and 1.2nd, they are 1.85 m, 2.01 m, and 2.38 m, respectively, corresponding to inlet pressures of 21,616 Pa, 23,241 Pa, and 26,872 Pa, with heads of 40.83 m, 69.49 m, and 104.86 m, respectively, at critical cavitation. When the cavitation coefficient falls below the critical cavitation coefficient, the rate of decrease in head becomes markedly more pronounced. It can be concluded that both the incipient cavitation coefficient at lower and higher flows are greater than at the rated condition, whereas the critical cavitation coefficient of the self-priming pump increases with rising flow rates. As the rotational speed gradually increases, both the incipient and critical cavitation coefficient also increase. Moreover, the rates at which the incipient and critical cavitation coefficient change rise to varying degrees as the flow rate and rotational speed increase.

Figure 8 illustrates the variation of vapor volume fraction within the impeller against cavitation coefficient under different operating conditions. It is evident that minor vapor bubbles form prior to the onset of incipient cavitation across all conditions, yet the pump’s head remains unchanged, indicating that the generation of a small number of cavitation bubbles has no impact on the internal energy characteristics of the pump. As the cavitation coefficient decreases, the number of vapor bubbles increases, and the rate of increase in their volume fraction accelerates. At the incipient cavitation points, the vapor volume fractions for 0.6Qd, 1.0Qd, and 1.4Qd are 0.03%, 0.35%, and 0.03%, respectively, and at the critical cavitation points, the vapor volume fractions are 9.63%, 12.13%, and 6.86%, respectively. For conditions at 0.8nd, 1.0nd, and 1.2nd, the vapor volume fractions at the incipient cavitation points are 0.04%, 0.35%, and 0.07%, respectively, and at the critical cavitation points, they are 8.75%, 12.13%, and 11.72%, respectively. It is observable that, regardless of whether at the incipient or critical cavitation points, the vapor volume fraction at the rated condition is consistently the highest. Furthermore, when the cavitation coefficient is held constant, the vapor volume fraction increases as the flow rate and rotational speed increase.

Vapor volume fraction variation curves under different operating conditions.

To analyze the cavitation evolution process, blade sections were extracted from the Turbo module, and comparisons of the vapor volume fraction under different cavitation coefficient were made for the incipient cavitation, transition and critical cavitation stages at various flow and rotational speed conditions, with the volume fraction ranging from 0 to 1, as depicted in Fig. 9. It can be observed that during the incipient cavitation stage, Vapor bubbles first appear at the suction side leading edges of the blades under all conditions. Notably, not all leading edges generate vapor bubbles at this stage, and the cavities are relatively small, with areas where the volume fraction reaches 0.5 being significantly larger than those where it is 1.0 near the wall region. There are variations in the cavity volumes at the leading edges of each blade, for instance, at low-speed conditions, the bubbles occupy approximately 1/10 and 1/4 of the channel length for blades 3 and 4, respectively. This is due to differences in the pressure distribution influenced by the volute cross-sectional shape and area.

Vapor phase distribution on the unfolding surface of impeller blade cascade.

During the intermediate stage of cavitation, cavities emerge on the suction sides of all impeller blades, moving along the blade profile, and the cavitation area increases while gradually extending into the channel interior, indicating evident cavitation development. The regions with a volume fraction of 1.0 near the wall rapidly expand, with the volume fraction decreasing sharply to around 0.5 at the cavity boundary. Distinct differences still exist in the cavitation locations and cavity sizes among different blades under various conditions; under low-flow conditions, cavities on blades 2 and 4 extend most into the channel, occupying 2/3 of the channel width. As the cavitation coefficient decreases, the formation rate of cavities accelerates, and in the critical cavitation stage, cavities expand further, with the cavity lengths on most blades exceeding half the blade length and further spreading into the channel. Cavities on blades 2, 3, 4, and 5 account for over 2/3 of the channel width. Cavities detach from the cavity cluster and enter the channel under the effect of reverse jetting, forming cavity trails that extend to the pressure side, causing a small number of cavities to appear at the leading edge of the pressure side, mostly with volume fractions less than 0.5. At critical cavitation, the ratios of cavity area percentages for the 0.6Qd, 1.0Qd, and 1.4Qd conditions are approximately 1.67:1.72:1, and for the 0.8nd, 1.0nd, and 1.2nd conditions, the ratios are roughly 1:1.46:1.55.

It is evident that the expansion of cavities is significantly influenced by changes in flow rate and rotational speed. Under 0.6Qd and 1.2nd, cavities tend to intrude most extensively into the channel. Notably, during the critical cavitation phase at 0.6Qd, the cavity on the suction surface of blade 2 almost extends to the pressure surface of blade 3. However, as the flow rate increases and the rotational speed decreases, the expansion of cavities on the suction surfaces of individual blades is progressively inhibited. Due to the presence of cavitation, the effective flow area within each channel is substantially reduced. Moreover, after cavitation develops to a certain extent, there occurs a rapid truncation at the periphery of the cavitation zone, forming a triangular pattern. The intrusion of cavities into the channel leads to a sharp rise in fluid velocity within the channel, resulting in enhanced losses.

Figure 10 presents the three-dimensional distribution of cavities at the critical cavitation point for each condition, with the volume fraction range selected from 0.5 to 1. There are extensive regions with a volume fraction of 0.5 at the periphery of the cavity areas. Numerous bubbles with a volume fraction of 1 are present at the rear cover plate walls of the impeller channels, and the cavity areas decrease gradually with an increase in flow rate and a reduction in rotational speed. Notably, the balance hole influences the distribution of bubbles on the rear cover plate, generating cavities adjacent to the impeller outlet side, whose shape is affected by the direction of fluid flow within the channel, spreading in the direction of outflow, and whose size is significantly influenced by flow rate and rotational speed. Under the 1.2Qd condition, the cavity area beside the balance hole is the smallest and only occurs in two channels, while the maximum areas are found under the 0.6Qd and 1.0Qd conditions. An increase in rotational speed causes the cavity area next to the balance hole to gradually expand and merge with other cavities on the rear cover plate. As the cavitation coefficient decreases, cavities on the rear cover plate also intrude into the channel, and cavities in some conditions have begun to spread onto the pressure surfaces of adjacent blades. Additionally, it is observed that cavitation is most severe slightly downstream of the blade inlet, at the intersection of the blade and the rear cover plate wall. This is because this location is situated on the inner bend of the channel and features a significant twist in the blade and rear cover plate wall, influenced by the centrifugal effect of the fluid. Simultaneously, the pressure on the backside of the blade is lower, making it more prone to cavitation.

Three-dimensional distribution of vapor bubbles inside the impeller during critical cavitation.

Figure 11 illustrates the distribution of vortices in the impeller domain predicted based on the Ω criterion under initial field conditions for different flow rates, with a threshold value set at 0.52. It can be observed that there are weak reflux vortices at the impeller inlet, with velocities of approximately 16 m/s, 5 m/s, and 7 m/s under the 0.6Qd, 1.0Qd, and 1.4Qd flow conditions, respectively. As the flow rate increases, these reflux vortices move closer to the wall and adhere to it. A periodic pattern of attached vortices exists at the axial center position of the rear cover plate, and as the flow rate increases, their position shifts upwards and transitions from a dispersed state to a connected one. Vortices within the blade channels are connected to the inlet reflux vortices, covering a substantial area, nearly 2/3 of the channel area. The velocity range of these channel vortices is approximately 5–30 m/s, 6–29 m/s, and 7–25 m/s under the 0.6Qd, 1.0Qd, and 1.4Qd conditions, respectively.

Vortices distribution in initial impeller flow field.

Under the 0.6Qd condition, the suction side of the blade is nearly dominated by vortex cores, with numerous irregularly shaped vortices present within the channel. The intensity of the vortices tends to increase as they progress toward the impeller outlet. At the 0.6Qd condition, the velocity demarcation points of channel vortices occur at about 1/4 and 3/4 of the channel length, with a noticeable increase in vortex velocity at these demarcation points. The velocity of the trailing vortices near the outlet is the highest, ranging from approximately 25 to 30 m/s. As the flow rate increases, the morphology and distribution of vortices within each channel become more regular. At 1.0Qd, vortices within the channel split into two primary vortex cores at the leading edge, located near the suction and pressure sides, respectively. The velocity demarcation point is approximately halfway along the channel length. Vortices near the pressure side are shorter, occupying about 1/2 to 2/3 of the blade length, with velocities around 21 to 25 m/s; meanwhile, the suction side is entirely occupied by vortices, and at the impeller outlet, a trailing vortex forms, extending the entire length of the channel exit, connecting to the suction-side vortex. The velocity range of the trailing vortex is 21 to 25 m/s, with the maximum velocity occurring at the junction between the suction side and the outlet, approximately 29 m/s. At 1.4Qd, the vortex morphology resembles that of the rated condition, with the distinguishing feature being the formation of an additional bifurcated vortex core between the suction-side and pressure-side vortices, with a length intermediate to both. The velocity of the bifurcated vortex tube is substantial, around 23 m/s, and the velocity near the outlet on the suction side reaches a maximum of approximately 25 m/s. Unlike 0.6Qd, the distribution of vortex intensity in the impeller channels at 1.0Qd and 1.4Qd is relatively uniform, without a marked increase in intensity near the outlet.

Figure 12 shows distribution of vortices in the impeller domain at critical cavitation conditions for various flow rates, based on multiple vortex identification criteria. It is visually evident that the occurrence of cavitation significantly influences the morphology of vortices within the impeller domain. As indicated by Figs. 8 and 9, cavitation leads to the formation of substantial vapor blockage near the suction side of the blades, resulting in the disruption and fragmentation of vortices at the leading edges of the blades. The blockage by vapor causes an increase in fluid velocity within the passage, hence the overall intensity of vortices at critical cavitation is higher than in the initial field. Due to the vapor blockage, a significant number of weak vortices accumulate at the inlet, which decreases with increasing flow rate, accompanied by a slight increase in their intensity; at flow rates of 0.6Qd, 1.0Qd, and 1.4Qd, the velocities of these weak vortices at the inlet are approximately 4 m/s, 5 m/s, and 7 m/s, respectively. It can be observed that the Ω criterion, Q criterion, and λ2 criterion effectively identify the form and quantity of weak vortices. Owing to the accumulation of vapor in the 1/4 of the passage and the interaction between vapor and liquid, a large number of vortex cores gather in the subsequent 3/4 of the passage.

Vortices distribution inside the impeller at critical cavitation with different vortex identification criteria.

In the 0.6Qd condition, vortices identified by the Ω criterion, Q criterion, and λ2 criterion exhibit considerable similarity, with the velocity range of vortices within the passage being approximately 9 to 33 m/s. The vortices within the passage predominantly consist of irregular vortices at the mid-to-rear trailing edge and wake vortices, both of which have high intensities with velocities around 31 m/s. The highest velocity of vortices is near the outlet on the suction side of the blade, reaching about 33 m/s. The Δ criterion, λci criterion, and vorticity method do not clearly distinguish between wake vortices and vortices within the passage, and vortices identified by the λci criterion exhibit more fragmentation. At the 1.0Qd condition, results from the Ω criterion, Q criterion, λ2 criterion, Δ criterion, and λci criterion are similar, with vortices occupying almost the entire blade passage, having a velocity range of approximately 8 to 36 m/s. These primarily comprise pressure-side vortices, suction-side vortices, and wake vortices, which although mostly interconnected, still maintain relatively distinct boundaries. It is also noted that with increasing flow rate, the distribution of vortex intensity between passages becomes uneven. Vortex tubes near the suction side are thinner but stronger than those near the pressure side, with average velocities near the suction side of passages 1, 2, and 5 being about 29 m/s, compared to only 18 m/s near the pressure side. The maximum velocities occur near the outlet on the suction side and 1/4 along the passage, reaching approximately 36 m/s. The vorticity method does not clearly delineate the boundaries of the three vortex parts and shows fewer vorticities on the suction side. In the 1.4Qd condition, the velocity range of vortices within the passage is approximately 8 to 38 m/s. There are significant differences in the vortex tubes over the pressure side across different passages, with the vortex tube lengths in passages 2, 3, and 4 covering only about 2/3 of the blade’s pressure side. The vortices near the suction side of passages 1 and 5 have the highest intensities, with average velocities of about 32 m/s. Vortices within the passage exhibit pronounced bifurcations, which are discernible by the Ω criterion, Q criterion, λ2 criterion, and Δ criterion, whereas vortices identified by the λci criterion lack clear demarcation.

Figure 13 shows distribution of vortices in the volute domain at critical cavitation conditions for various flow rates, based on multiple vortex identification criteria. For regions other than the impeller, the effect of cavitation onset on the vortex distribution is negligible; therefore, this study focuses solely on the distribution of vortex cores within the volute at critical cavitation conditions. In the volute region, most of the area is occupied by long vortex tube with generally lower overall intensity. The velocity range of vortices in the volute under the 0.6Qd, 1.0Qd, and 1.4Qd flow conditions is approximately 2 to 21 m/s, 9 to 22 m/s, and 12 to 22 m/s, respectively. It is observable that the intensity of vortices increases clockwise and there is a slight increase in vortex intensity with rising flow rate. By comparison, it is not difficult to ascertain that among the six criteria, the Ω criterion yields the best depiction of the distribution and shape of individual vortex tubes, allowing for a clear visualization of the evolution of vortices within the volute. Vortex tubes primarily undergo disruption in three distinct zones depicted in the figure: zone 1 is situated near the 90° cross-section of the volute, zone 2 is located at the boundary between the volute and the reflux hole, and zone 3 is in proximity to the tongue of the volute.

Vortices distribution inside the volute at critical cavitation with different vortex identification criteria.

Under the 0.6Qd condition, due to the backflow through the reflux hole disrupting flow continuity, vortices break up at zone 2 and generate attached backflow vortices on the reflux hole wall. The velocities of vortices in the reflux hole are approximately 9 m/s, 7 m/s, and 4 m/s at the 0.6Qd, 1.0Qd, and 1.4Qd conditions, respectively, with intensity decreasing slightly as flow rate decreases. Additionally, small areas of weak vortices exist to the right of the reflux hole in the volute region, with velocities of 4 m/s, 5 m/s, and 9 m/s, respectively. Due to the rotor–stator interaction with the volute tongue, some vortices break at zone 3, but larger-diameter vortices undergo a knuckle-like deformation instead of breaking, continuing as circular vortex tube extending towards the volute outlet. At the 0.6Qd condition, the intensity of vortex tubes within the volute passage decreases gradually from below the tongue, with velocities declining from 12 m/s to 2 m/s at the outlet. Changes in vortex intensity in the volute region primarily occur in the outlet area, with vortex velocities at the volute outlet being 9 m/s and 17 m/s under the 1.0Qd and 1.4Qd conditions, respectively. Small-scale fragmented vortices with velocities around 6 m/s appear at the tongue across all conditions. As the flow rate increases, the diameter of vortices grows, and larger-diameter vortices do not break up at zone 2. The backflow vortices on the reflux hole wall become more fragmented. The knuckle-like structure of vortices at zone 3 disappears, with diameters gradually increasing towards the outlet, and boundaries of vortex tubes merge with each other.

The Q criterion, λ2 criterion, Δ criterion, λci criterion, and vorticity method all misidentify strong shear layers on the volute as vortices to varying degrees, with the vorticity method being the most severe, making it nearly impossible to discern internal vortex distributions. Results from the Q criterion and λ2 criterion are similar, featuring internal vortex structures akin to those identified by the Ω criterion. However, erroneously classifying shear layers as vortices diminishes their effectiveness compared to the Ω criterion, and they additionally detect button-like clusters of vortices between zones 2 and 3. The Δ criterion and λci criterion yield comparable outcomes, but errors stemming from shear layer misidentification are more pronounced, leading to fragmented internal vortex shapes and less distinct boundaries between vortex tubes. Notably, at the volute outlet section, the number of vortex tubes identified by the λci criterion is markedly lower than that detected by other criteria.

Figure 14 shows distribution of vortices in the S-pipe domain at critical cavitation conditions for various flow rates, based on multiple vortex identification criteria. It is evident that the vortices in the S-pipe predominantly feature a hairpin vortex structure, consisting mainly of the vortex heads in region 1, the vortex necks in region 2, and the vortex legs in region 3. The vortex legs are close to but not in direct contact with the wall, pointing in a direction similar to the main flow. The vortex heads are toroidal in nature, with directions largely in the spanwise orientation. It is observed that the hairpin vortices have a fairly uniform intensity distribution, which is minimally affected by the flow rate, with velocities averaging around 1 to 2 m/s. A backflow vortex with velocities of approximately 10 m/s exists at the outlet wall, however, the Ω criterion fails to identify this backflow vortex under the 1.0Qd and 1.4Qd conditions.

Vortices distribution inside the S-pipe at critical cavitation with different vortex identification criteria.

The thresholds for the Q criterion, λ2 criterion, Δ criterion, λci criterion, and vorticity method have all been adjusted. The Ω criterion, Q criterion, λ2 criterion, Δ criterion, and λci criterion all manage to identify the fundamental structure of hairpin vortices. Results from the Ω criterion, Q criterion, and λ2 criterion are similar, presenting the hairpin vortex structures in a relatively complete manner. The notable difference lies in the vortex in region 4. Results from the Δ criterion and λci criterion are alike, showing a shrinkage or even disappearance of the vortex neck, while the vortex leg regions retain a clear structure, though numerous fragmented vortices near the wall interfere with the observation of internal structures. The vorticity method identifies extensive wall shear layers as vortices. As the flow rate increases, the tail of the vortex in region 4 gradually shrinks, while the volume of the vortex head progressively enlarges. Concurrently, the intensity in the vortex leg region also escalates; specifically, under the 1.4Qd condition, the velocity of the vortex legs accelerates from 2 m/s to 6 m/s.

Figure 15 shows distribution of vortices in the separation chamber domain at critical cavitation conditions for various flow rates, based on multiple vortex identification criteria. The gas–liquid separation chamber is predominantly filled with weak vortices, with velocities ranging from approximately 1 to 3 m/s. Fluid entering the separation chamber from the volute carries high kinetic energy, thus resulting in stronger vortex intensity at the entrance to the separation chamber. An increase in flow rate leads to a greater quantity of intense vortices at the separation chamber’s inlet, which extend further into the chamber. Under the 0.6Qd condition, the velocity at the separation chamber’s inlet is around 9 m/s, gradually decelerating to 4 m/s as the flow reaches the top of the chamber. At the 1.0Qd condition, the vortex velocity at the inlet is approximately 10 m/s, diminishing to 6 m/s by the time it flows to the 1/4 height mark on the opposite side after passing the top. Under the 1.4Qd condition, the vortex velocity at the inlet is about 12 m/s, reducing to 7 m/s by the time it reaches the quarter height mark on the opposite side after traversing the top. The distribution of vortices can roughly be divided into two sections, as illustrated. Region 1 consists primarily of a circulation formed by numerous vortex clusters, with the center of the circulation interlaced by smaller vortices. Region 2 features a ring-shaped vortex composed of large-volume vortex clusters, with fewer vortices at the center of the ring. With increasing flow rate, the cluster-like vortices in region 2 evolve into broader vortices with increasingly well-defined boundaries. Notably, under rated conditions, the vorticity decreases at the central location of region 1, creating a distinct void.

Vortices distribution inside the separation chamber at critical cavitation with different vortex identification criteria.

The outcomes for the Ω criterion, Q criterion, and λ2 criterion remain consistently aligned, though the Q criterion identifies fewer vortices at the center of region 1 under low-flow conditions compared to the Ω and λ2 criteria, which identify a vortex blob covering the central position. The results from the Δ criterion and λci criterion are similar to each other but differ from the previous three by identifying fewer vortices, with unclear vortex boundaries and a higher number of small, fragmented vortices. The vorticity method yields the least amount of vorticity, failing to identify all weak vortices at a single threshold, and the vortex shapes significantly diverge from those identified by other criteria, performing poorly in recognizing extensive areas of weak vortices.

Figure 16 shows the surface area of vortex bands in various fluid domains identified at critical cavitation conditions across different flow rates, based on the Ω criterion. Given that threshold selection for other criteria is heavily reliant on empirical judgment and may not universally apply across all fluid domains, the Ω criterion was selected for analysis due to its established insensitivity to threshold variations, as evidenced by numerous studies. Hence, this study exclusively examines the surface area of vortex bands identified by the Ω criterion. The fluid domains, ranked by ascending vortex band surface area, are as follows: reflux hole, impeller, volute, front and rear chambers, S-pipe, and gas–liquid separation chamber. It is evident that changes in flow rate have little impact on the magnitude of vortex band areas, indicating a minimal influence of flow rate on the macroscopic structure of vortex bands. The smallest vortex surface area is found in the reflux hole, ranging from 0.001 m2 to 0.003 m2, primarily due to the limited volume of this domain. The vortex band surface areas in the impeller and volute domains are comparable, with values of 0.152 m2, 0.141 m2, and 0.139 m2 for the impeller, and 0.150 m2, 0.149 m2, and 0.149 m2 for the volute, under the 0.6Qd, 1.0Qd, and 1.4Qd conditions respectively. The variation range of vortex surface area in the front and rear chambers and S-pipe is 0.215m2 ~ 0.220 m2 and 0.258m2 ~ 0.270m2, respectively. The gas–liquid separation chamber exhibits the largest vortex band surface area, notably exceeding those of other domains, with values of 1.685 m2, 1.543 m2, and 1.722 m2 under the 0.6Qd, 1.0Qd, and 1.4Qd conditions, respectively. The vortex band surface area in the gas–liquid separation chamber is smallest at the rated operating condition. In summary, there is a trend toward decreased vortex band surface area in the impeller and volute domains with increased flow rate, with the most pronounced change occurring under the 0.6Qd condition for the impeller. The vortex band surface area in the gas–liquid separation chamber is also noticeably affected by flow rate variations.

Surface area of vortex bands in various domains at critical cavitation under different flow rates.

The entropy production rate (EPR) signifies the inevitable dissipative effects accompanying various energy conversion processes. Per the second law of thermodynamics, any practical irreversible process invariably entails entropy production. For flow within a centrifugal pump, neglecting heat transfer, viscous forces within boundary layers irreversibly convert fluid mechanical energy into internal energy, engendering entropy production; turbulence fluctuations in high Reynolds number regions also induce hydraulic losses, contributing to entropy production. Thus, from a thermodynamic perspective, the energy dissipation of fluid flow within a centrifugal pump can be evaluated through entropy production.

For Newtonian fluids, the entropy production of particles in laminar flow can be calculated using Eq. (33):

where μ is the fluid dynamic viscosity, Pa s; u, v and w are the components of the local velocity of the particle in three directions in a Cartesian coordinate system; T is the local temperature of fluid particles, K.

In turbulence, the entropy production of particles can be calculated in two parts: one is the entropy production caused by time averaged motion, and the other is the entropy production caused by velocity fluctuations. The calculation formula is as follows:

where $\dot{S}_{{\overline{D}}}^{\prime \prime \prime }$ is the entropy production generated by average velocity; $\dot{S}_{{D^{\prime } }}^{\prime \prime \prime }$ is the entropy production generated by pulsating velocity.

The entropy production generated by the average velocity can be calculated by Eq. (35):

The entropy production generated by the pulsating velocity can be calculated by Eq. (36):

Due to the RANS numerical computation methodology, the turbulent fluctuating velocity is represented through the ε equation without directly solving for fluctuating velocity magnitudes, precluding direct computation via partial differentiation of fluctuation quantities. Following the approach proposed by Kock and Herwig46, the entropy production attributed to pressure fluctuations can be calculated using Eq. (37):

where ρ is the density of fluid; ε is the turbulence dissipation.

The total entropy production of the entire flow field can be calculated using the volume fraction of particle entropy production:

Through the aforementioned equations, CFD methods can be utilized for numerical computations of the entire flow field in self-priming pumps. Post-processing of computational outcomes includes the development of a UDF program for the entropy production formula. Upon importing this UDF during FLUENT’s post-processing phase, entropy production for any fluid particle and the total entropy production across the flow field can be determined. This facilitates the analysis of energy loss distribution throughout the flow process, thereby evaluating the hydraulic characteristics of the self-priming pump and the performance of each flow component.

Figure 17 displays the comparison of average entropy production across various flow domains under initial and critical cavitation conditions at differing flow rates. The solidline curves are under critical cavitation conditions and the dashedline curves are under initial field conditions. The average entropy production rate signifies the propensity for energy loss generation within each domain. It is observable that the S-pipe and the gas–liquid separation chamber exhibit lower average entropy production rates, varying within the range of 4.61 W (m−3 K−1) to 43.53 W (m−3 K−1), with the S-pipe under the 0.6Qd condition demonstrating notably higher values than other conditions. The average entropy production in the reflux hole and impeller domains is significantly influenced by flow rate changes. Regardless of whether under initial or critical cavitation conditions, the average entropy production rate in the reflux hole at 0.6Qd and 1.4Qd exceeds that at the rated condition. During critical cavitation, the average entropy production rate in the reflux hole under the 1.0Qd and 1.4Qd conditions decreases to varying degrees, whereas at 0.6Qd, it increases. It is also noted that the average entropy production in the reflux hole at the initial field under 1.4Qd is substantially higher than at other flow conditions. Specifically, the average entropy production rates in the reflux hole at the initial field under the 0.6Qd, 1.0Qd, and 1.4Qd conditions are 421.97 W (m−3 K−1), 342.91 W (m−3 K−1), and 1084.51 W (m−3 K−1), respectively. During critical cavitation, these rates stand at 547.24 W (m−3 K−1), 298.81 W (m−3 K−1), and 743.66 W (m−3 K−1), respectively. For the impeller region, the average entropy production under initial field conditions at 0.6Qd and 1.4Qd is similar and higher than at the rated flow rate. During critical cavitation, the average entropy production increases to varying degrees across all flow conditions, with the least increase at 0.6Qd and the most significant at 1.0Qd, surpassing the high-flow condition. Specifically, the average entropy production rates in the impeller at the initial field under 0.6Qd, 1.0Qd, and 1.4Qd are 377.81 W (m−3 K−1), 297.66 W (m−3 K−1), and 395.71 W (m−3 K−1), respectively. During critical cavitation, these rates are 389.64 W (m−3 K−1), 639.12 W (m−3 K−1), and 612.94 W (m−3 K−1), respectively. The average entropy production in the volute is less affected by flow rate magnitude, increasing with rising flow before cavitation, and further rises upon reaching critical cavitation. The average entropy production rates in the volute at the initial field under 0.6Qd, 1.0Qd, and 1.4Qd are 744.93 W (m−3 K−1), 735.66 W (m−3 K−1), and 810.42 W (m−3 K−1), respectively. During critical cavitation, these rates are 801.41 W (m−3 K−1), 813.97 W (m−3 K−1), and 869.31 W (m−3 K−1), respectively. The trend in average entropy production for the front and rear chambers is opposite to that of the volute, decreasing with increasing flow rate both initially and during critical cavitation, albeit with a rise in average entropy production upon critical cavitation compared to the initial field. The average entropy production rates in the front and rear chambers at the initial field under 0.6Qd, 1.0Qd, and 1.4Qd are 1308.29 W (m−3 K−1), 1205.06 W (m−3 K−1), and 1154.17 W (m−3 K−1), respectively. During critical cavitation, these rates are 1323.05 W (m−3 K−1), 1232.69 W (m−3 K−1), and 1194.46 W (m−3 K−1), respectively.

Average entropy production in various flow domains under different flow conditions.

In summary, despite the S-pipe and gas–liquid separation chamber having larger vortex band surface areas, both the average and total entropy production rates are the lowest, indicating no direct correlation between vortex size and length and energy loss generation. Changes in flow rate and the onset of cavitation have the greatest impact on the average entropy production rates in the reflux hole and impeller. The volute and front and rear chambers exhibit higher average entropy production rates, suggesting these areas are most prone to energy loss.

Figure 18 depicts the distribution of vortex cores in the impeller region under initial field conditions at different rotational speeds, with a threshold value of 0.52 set for the Ω criterion. It is evident that changes in rotational speed have a minor effect on the morphology and quantity of weak vortices at the impeller inlet, yet significantly impact the strength and shape of vortices within the flow channel. The velocities of inlet vortices at 0.8nd, 1.0nd, and 1.2nd conditions are approximately 4 m/s, 5 m/s, and 7 m/s, respectively; the velocity ranges of channel vortices are approximately 6 to 20 m/s, 6 to 29 m/s, and 7 to 35 m/s, respectively. Similarities in vortex core distributions are observed between 0.6nd and 1.0nd conditions, where trailing vortices connect with suction-side vortices, while pressure-side vortices are shorter, forming two vortex branches in the channel. Vortex core intensity marginally increases as the flow approaches the impeller outlet. Under the 1.2nd condition, vortex intensity notably increases, accompanied by a rise in vortex quantity within the channel. Consistent with observations under varying flow conditions, the strongest vortices appear near the impeller outlet on the suction side, with peak velocities reaching 35 m/s. The velocity demarcation point for impeller channel vortices is at one-quarter of the channel length under the 0.8nd condition, at half the channel length under the 1.0nd condition, and at three-quarters of the channel length under the 1.2nd condition. Notably, the velocity range of trailing vortices shows the most variation with increasing rotational speed, being 17 to 20 m/s at 0.8nd, 21 to 30 m/s at 1.0nd, and 29 to 35 m/s at 1.2nd conditions.

Vortices distribution in initial impeller flow field.

Figure 19 illustrates the distribution of vortex cores in the impeller region under critical cavitation conditions at different rotational speeds, identified using various vortex criteria. It is visually apparent that the occurrence of cavitation significantly impacts the quantity and intensity of vortices within the impeller channel, whereas changes in rotational speed and cavitation onset have lesser effects on vortex quantity and intensity at the impeller inlet, with vortex velocities at the inlet ranging approximately from 6 to 7 m/s across conditions. Contrary to the effects of flow rate, alterations in rotational speed exert minor influences on the distribution of vortices within the channel but considerably affect vortex intensity. The velocity ranges of channel vortices during critical cavitation at 0.8nd, 1.0nd, and 1.2nd conditions are approximately 7 to 22 m/s, 8 to 36 m/s, and 8 to 40 m/s, respectively. With increasing rotational speed, especially at the 1.2nd condition, the most noticeable rise occurs in the intensity of vortices near the suction side and trailing vortices, with velocity ranges of approximately 30 to 40 m/s and 30 to 36 m/s, respectively. The maximum velocity of channel vortices is observed near the outlet on the suction side, increasing from about 22 m/s at 0.8nd to approximately 40 m/s at 1.2nd. The intensity variation of vortices at the pressure side trailing edge is relatively modest, with velocities escalating from roughly 19 m/s at 0.8nd to about 29 m/s at 1.4nd. Leading-edge vortex intensity remains virtually unchanged, suggesting that higher rotational speed conditions result in greater energy loss in the vicinity of the suction side.

Vortices distribution inside the impeller at critical cavitation with different vortex identification criteria.

At the 0.8nd condition, the outcomes from the Ω criterion, Q criterion, and λ2 criterion are similar; the Δ criterion and vorticity method predict fewer weak vortices at the inlet, while the λci criterion identifies weaker attached vortices at the axial center position of the rear cover plate; vorticity method tends to misidentify shear layers as vortices, hence the identified vortex surfaces in the flow passage are relatively smooth. At the rated speed condition, the vorticity method loses more suction-side vortices compared to the other five criteria. At the 1.2nd condition, the results from the Ω criterion, Q criterion, λ2 criterion, and Δ criterion are alike, the vorticity method can identify vortices near the wall, but it misses many vortices in the middle of the flow passage. Notably, except for the rated condition, at other varying speeds and flow rates, the vortices predicted by the λci criterion display distinct characteristics, although their quantities and distributions resemble those from the Ω criterion, Q criterion, and λ2 criterion, they feature a multitude of tiny vortex core distributions on the surface.

Figure 20 shows the distribution of vortex cores in the volute region at critical cavitation under various rotational speeds, based on multiple vortex identification criteria. The trend of vortex intensity variation under different rotational speed conditions is opposite to that under different flow rate conditions; the vortex intensity gradually increases with the increase in impeller rotational speed. At critical cavitation, the vortex velocities in the volute region for the 0.8nd, 1.0nd, and 1.2nd operational conditions are approximately 2 ~ 18 m/s, 2 ~ 22 m/s, and 2 ~ 26 m/s, respectively. The overall distribution of vortex intensity in the volute is relatively uniform. However, there are significant changes in the vortex tube intensity near the reflux hole, as shown by the cross-section position in the figure, with velocities increasing from 9 m/s at the 0.8nd condition to 23 m/s at the 1.2nd condition. Weak vortices mainly appear near the reflux hole and at the outlet. Changes in the impeller rotational speed have a minor effect on the number and shape of vortices in the volute, with the vortex distributions under the three rotational speed conditions being largely similar. At the 0.8nd and 1.0nd conditions, there are three vortex tubes at the volute outlet, but when the rotational speed continues to increase, the vortex tube closest to the tongue will shrink. It can also be observed that as the rotational speed increases, the block-like vortices at the boundary between the volute and the reflux hole gradually disappear, and the shape of the return vortices on the reflux hole wall surface is basically unaffected, with only a slight increase in intensity. The vortex velocities at the reflux hole for the 0.8nd, 1.0nd, and 1.2nd conditions are approximately 2 m/s, 6 m/s, and 8 m/s, respectively. The prediction results of vortices within the volute under different rotational speed conditions still show the best performance using the Ω criterion, allowing for a more intuitive observation of the vortex shape and quantity without interference from strong shear layers.

Vortices distribution inside the volute at critical cavitation with different vortex identification criteria.

Figure 21 depicts the distribution of vortex cores in the S-pipe region at critical cavitation under various rotational speeds, based on multiple vortex identification criteria. It can be found that the results identified by each criterion are essentially the same as those under different flow rate conditions. The shape of the vortex in region 4 and the intensity in the vortex leg area are largely unaffected by the rotational speed. The hairpin vortex velocities for the 0.8nd, 1.0nd, and 1.2nd operational conditions are approximately 2 m/s. It can also be observed that there exists an attached vortex above the wall surface in the vortex leg area, which is identified by the Ω criterion as a double-loop vortex, gradually transforming into a single loop with the increase in rotational speed. The Q criterion and λ2 criterion results do not exhibit loop characteristics, and the shapes are unaffected by the rotational speed. Notably, the vortex intensity in the S-pipe is significantly lower than that in the impeller and volute regions, and the vortex intensity distribution across various sections is relatively uniform. The thresholds for the Q criterion, Δ criterion, λci criterion, and vorticity method all require corresponding adjustments according to the operational conditions, indicating that the Ω criterion and λ2 criterion have advantages in identifying weak vortices under different conditions in the same flow domain.

Vortices distribution inside the S-pipe at critical cavitation with different vortex identification criteria.

Figure 22 illustrates the distribution of vortex cores in the gas–liquid separation chamber region at critical cavitation under various rotational speeds, based on multiple vortex identification criteria. The overall distribution of vortices under different rotational speed conditions is similar to that under different flow rate conditions. What differs is that the vortex circulation center in region 1 is not covered by a cluster of vortices but rather distributed by longer vortex tubes intersecting each other, with the structure of the intersecting vortex tubes changing as the rotational speed increases. In terms of intensity, it can be divided into two parts: the vortex intensity near the inlet is higher due to the influence of the volute, with velocities ranging from about 6 to 10 m/s, decreasing gradually as it extends inward; the rest of the region is occupied by weaker vortices with velocities of 1 to 2 m/s. Changes in rotational speed have a significant impact on the distribution of vortices with velocities around 6 to 10 m/s near the inlet of the gas–liquid separation chamber: at the 0.8nd condition, they almost occupy one-third of the separation chamber, gradually reducing to the top of the separation chamber as the rotational speed increases. Notably, all criteria identify a grid-like vortex feature neatly arranged close to the wall surface. The results from the Ω criterion, Q criterion, and λ2 criterion are similar, while the Δ criterion and λci criterion still yield a large number of fragmented vortices, with unclear boundaries for the cluster of vortices in region 1, making it difficult to discern changes in the structure and morphology of the vortices.

Vortices distribution inside the separation chamber at critical cavitation with different vortex identification criteria.

Figure 23 presents the surface area of vortex bands identified by the Ω criterion in each fluid domain under various rotational speed conditions. The vortex band surface areas in the fluid domains, from smallest to largest, are in the order of reflux hole, impeller, volute, front and rear chambers, S-pipe, and gas–liquid separation chamber. It is clearly evident that changes in rotational speed have no significant impact on the size of the vortex band area, indicating that the effect of rotational speed on the macroscopic structure of vortex bands is also minimal. The vortex surface area of the reflux hole is the smallest, varying within the range of 0.002 m2 to 0.003 m2. The vortex band surface areas in the impeller and volute regions are similar, but both exhibit noticeable changes at the 1.2nd condition. The vortex band surface areas in the impeller region for the 0.8nd, 1.0nd, and 1.2nd conditions are 0.143 m2, 0.141 m2, and 0.150 m2, respectively; in the volute region, the vortex band surface areas are 0.148 m2, 0.149 m2, and 0.142 m2, respectively. The variation ranges of vortex band surface areas in the front and rear chambers and the S-pipe region are 0.213 m2 to 0.217 m2 and 0.258 m2 to 0.260 m2, respectively. The vortex band surface area in the gas–liquid separation chamber is the largest and far greater than in other regions, being 1.640 m2, 1.543 m2, and 1.650 m2 for the 0.8nd, 1.0nd, and 1.2nd conditions, respectively, with the smallest vortex band surface area in the gas–liquid separation chamber at the rated condition. To summarize, changes in flow rate and rotational speed have relatively minor effects on the surface area of vortex bands in self-priming pumps. High-speed conditions have a noticeable impact on the vortex band surface areas in the impeller and volute regions, causing a significant increase in the impeller’s vortex band surface area and a marked decrease in the volute’s; the vortex band surface area in the gas–liquid separation chamber is also significantly affected by rotational speed.

Surface area of vortex bands in various domains at critical cavitation under different rotational speed.

Figure 24 compares the average entropy production rates in each domain under initial field conditions and critical cavitation conditions for various rotational speed conditions. The solidline curves are under critical cavitation conditions and the dashedline curves are under initial field conditions. The average entropy production rates in the S-pipe and gas–liquid separation chamber are relatively low, with the S-pipe varying within the range of 2.66 W (m−3 K−1) to 11.44 W (m−3 K−1), and the gas–liquid separation chamber within the range of 8.46 W (m−3 K−1) to 12.95 W (m−3 K−1). Under initial field conditions at 0.8nd, 1.0nd, and 1.2nd, the average entropy production in the reflux hole is 135.81 W (m−3 K−1), 342.90 W (m−3 K−1), and 613.50 W (m−3 K−1), respectively; under critical cavitation conditions, they are 97.78 W (m−3 K−1), 198.81 W (m−3 K−1), and 593.58 W (m−3 K−1), respectively. It can be seen that with the occurrence of cavitation, there is a slight decrease in the average entropy production within the reflux hole. The occurrence of cavitation has the most noticeable effect on the average entropy production within the impeller. Under initial field conditions at 0.8nd, 1.0nd, and 1.2nd, the average entropy production in the impeller is 196.40 W (m−3 K−1), 297.66 W (m−3 K−1), and 423.55 W (m−3 K−1), respectively; under critical cavitation conditions, these values rise to 364.29 W (m−3 K−1), 639.12 W (m−3 K−1), and 969.64 W (m−3 K−1), representing increases of 85%, 115%, and 129%, respectively. As the rotational speed increases, it becomes easier to generate losses within the impeller. Both the impeller and the volute experience increases in average entropy production with the onset of cavitation, but the magnitude of the increase is less pronounced than in the impeller. Under initial field conditions at 0.8nd, 1.0nd, and 1.2nd, the average entropy production in the volute is 417.30 W (m−3 K−1), 735.66 W (m−3 K−1), and 1216.73 W (m−3 K−1), respectively; under critical cavitation conditions, these values become 447.68 W (m−3 K−1), 813.97 W (m−3 K−1), and 1325.00 W (m−3 K−1), respectively. Under initial field conditions at 0.8nd, 1.0nd, and 1.2nd, the average entropy production in the pump chamber is 632.69 W (m−3 K−1), 1205.06 W (m−3 K−1), and 2048.15 W (m−3 K−1), respectively; under critical cavitation conditions, these values are 649.83 W (m−3 K−1), 1232.69 W (m−3 K−1), and 2112.07 W (m−3 K−1), respectively. In summary, the average entropy production in each domain of the self-priming pump increases to varying degrees with the increase in rotational speed; with the occurrence of cavitation, except for the reflux hole, the average entropy production in the remaining domains rises, with the impeller showing the most significant increase. Relative to flow rate, rotational speed has a greater impact on the average entropy production within the self-priming pump, and the trend of change is more pronounced with variations in rotational speed.

Average entropy production in various flow domains under different rotational speed conditions.

This paper presents a numerical simulation of the internal flow field of a self-priming pump under steady operating conditions at different flow rates and rotational speeds, both before and after cavitation onset. The distribution of vortex cores using different vortex identification methods and the energy loss characteristics within the self-priming pump were obtained. However, the current study focuses only on steady-state conditions. Future work will involve unsteady calculations to investigate the temporal variations in vortex core distribution and energy losses within the self-priming pump during cavitation.

For the impeller and volute domains, the Ω criterion, Q criterion, and λ2 criterion all yield comparable outcomes through threshold adjustment. Results from the Δ criterion and λci criterion are alike, yet feature a higher incidence of fragmented vortices, particularly prominent in the gas–liquid separation chamber region; the vorticity method performs poorly in recognizing weak vortices and tends to misidentify strong shear layers near walls as vortices. Regarding the Ω criterion, selecting a value of 0.52 allows for the concurrent identification of most strong and weak vortices, whereas the Q criterion and λ2 criterion, although capable of achieving equivalent results through threshold modification, rely excessively on empirical judgment, thereby possessing certain limitations.

The S-pipe predominantly consists of low-intensity hairpin vortices with stable strength and morphology; the gas–liquid separation chamber contains the highest vortex quantity, mostly comprised of weak vortices. The occurrence of cavitation significantly affects the shape and intensity of vortices in the impeller channel, yet has minimal impact on other domains. Increased rotational speed chiefly influences the intensity of vortices in the impeller channel, with strong vortices mainly distributed near the blade suction side. The volute region is chiefly composed of longer vortex tubes, where changes in flow rate primarily affect the intensity and morphology of exit vortices; at low flow rates, structures like the reflux hole and tongue can cause fragmentation of vortex tubes.

At critical cavitation, flow rate and rotational speed significantly impact bubble morphology; a minor accumulation of vapor also occurs at the leading edge of the blade’s pressure side; the balance hole influences the distribution of cavitation regions on the rear cover plate’s wall.

Energy loss does not directly correlate with the number of vortices but rather associates significantly with vortex intensity. During critical cavitation, energy loss in the reflux hole region decreases under the 1.4Qd condition, conversely to other flow conditions; escalation in rotational speed and cavitation onset augment energy loss in the impeller domain, with average entropy production in the impeller increasing by 85%, 115%, and 129% at critical cavitation under 0.8nd, 1.0nd, and 1.2nd conditions, respectively.

Both flow rate and rotational speed will affect the cavitation coefficient at which critical cavitation occurs in self-priming pump. When the cavitation allowance is constant, the vapor volume fraction correlates positively with both flow rate and rotational speed.

The data used to support the findings of this study are available from the corresponding author upon reasonable request.

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The research was financially supported by the “Pioneer” and “Leading Goose” R&D Program of Zhejiang (Grant No. 2022C03170).

College of Mechanical Engineering, Quzhou University, Quzhou, 324000, China

Hai-Bing Jiang, Shao-Han Zheng, Yu-Liang Zhang & He-Chao Guo

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Yu-Liang Zhang proposed the innovative idea. Hai-Bing Jiang wrote the manuscript. Shao-Han Zheng carried out the numerical simulation. He-Chao Guo revised the manuscript. All authors have agreed to the published version of the manuscript.

Correspondence to Yu-Liang Zhang.

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Jiang, HB., Zheng, SH., Zhang, YL. et al. Applications of different vortex identification methods in cavitation of a self-priming pump. Sci Rep 15, 7458 (2025). https://doi.org/10.1038/s41598-025-85662-3

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Received: 17 August 2024

Accepted: 06 January 2025

Published: 03 March 2025

DOI: https://doi.org/10.1038/s41598-025-85662-3

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