## Abstract

The blade leading edge is a design variable that can affect the local flow patterns and pressure peaks, implying a direct effect on the cavitation performance. This study was conducted to analyze the effect of the blade leading edge shape on the cavitation and noncavitation states. A total of four sets, including the square shape, were selected under the definition of ellipse ratio, and the main focus was on the cavitation state rather than the noncavitation state. In the noncavitation state, the square set denoted a remarkable negative influence, while the other three sets obtained almost the same performance despite different ellipse ratios. In the cavitation state, the square set obtained a relatively low net positive suction head required, related to the inlet flow pattern with the cloud cavity. The other three sets contained the sheet cavity, and their suction performance tended to improve as the cavity blockage decreased. As a parallel focus, an in-depth analysis of cavitation surge and pressure gain was presented with the head drop slope for the other three ellipse sets. The numerical results included the off-design flow rate points and were validated through an experimental test.

## 1 Introduction

From the viewpoint of working fluid (water) inside a pump, the blade leading edge (LE) is the first point where the flow is controlled in the streamwise direction. The flow past the blade LE has its streamline depending on the blade angle, and then the performance of a pump can be determined with unfavorable characteristics such as separation and local pressure peaks. Thus, the LE shape can be a design variable directly involved in the flow patterns. According to a reference [1] in our field, it was reported that the elliptical LE with an appropriate aspect ratio is a hydraulically advantageous factor. A semicircle (round) shape with a normal aspect ratio (1) was described as an unfavorable factor in cavitation characteristics, and it was pointed out that if applied unavoidably, it should be limited to a small-sized pump. A wedge-like shape with a relatively large aspect ratio was described as a specialized profile for the flow, which shows zero as the incidence angle. This means that it is not suitable for the LE shape of fluid machinery in which the incidence angle is not generally zero, such as a pump, and there is an additional concern on the casting quality to the thin and long shape. Meanwhile, the description is generally qualitative and comprehensive. In another reference [2], the author reviewed a study [3] about the geometry in which the LE of an inducer gradually had a thin cutoff shape and noted a remarkable effect. Simply stated, the sharper the LE of an inducer, the better the performance in noncavitation and cavitation states. However, when the LE was very thin, it was left out of focus because of flutter. The previous results above commonly indicate that the blade LE should have an elliptic profile and is directly related to the cavitation characteristics.

These previous studies generally agreed with each other and mainly discussed the cavitation characteristics. Because the LE shape is susceptible to flow patterns and local pressure peaks [7,8], in particular, the report [9] that improved the suction performance with more parabolic-shaped LE than their base one denoted there is no disagreement on the correlation between the LE shape and the cavitation characteristics. However, previous studies commonly focused on quantitative point out for cavitation characteristics (NPSHre or cavitation coefficient), and interests in internal flow were not sufficient. It is also possible to compensate for the fact that the LE shape was just distinguished from its profile without any specific definition, and the mutual expression between the square and blunt shape was ambiguous.

In this study, an in-depth analysis of the noncavitation and cavitation characteristics was carried out with different LE shapes of a mixed-flow pump. A total of four sets, including the square shape, were analyzed under the definition of ellipse ratio (ER). The focus was on the performance and internal flow phenomena in the cavitation state rather than those of the noncavitation state. The concepts of cavity inception and blockage were employed to constitute a reasonable description. In particular, regarding the cavitation characteristics, unfamiliar perspectives such as the “cavitation surge” and the “pressure gain” were dealt with the head drop slope to the decrease of the inlet pressure (NPSH). Some references to these perspectives would be described in Head Drop and Suction Performance section. The numerical results included the off-design flow rate points, and were not only compared with the tendency of previous studies but also validated with the experimental test. Meanwhile, the words “cavity” and “vapor” were used together in the same meaning to keep the commonly used idioms in our field.

## 2 Mixed-Flow Pump Model

The mixed-flow pump model in this study has an impeller that had been optimized [10] with an objective function of the hydraulic efficiency through the design of experiment and response surface method. For reference, the blade LE had an ER as “2” as defined below.

### 2.1 Design Specification.

Table 1 contains the design specifications of the mixed-flow pump model in this study. The geometry is shown in Fig. 1(a), and each window of views A, B, and C are indicated to be continuously applied in the discussion. Figure 1(b) is the meridional plane, and each curve is drawn on the corresponding outline of a blade in Fig. 1(a). The specific speed in the concept of type number ($Ns$), flow coefficient (Φ), head coefficient (Ψ), total efficiency ($η$), and shaft power coefficient ($λ$) are defined as follows:
$Ns= ωQ(gH)34$
(1)
$Φ= Cm2U2$
(2)
$Ψ= 2gHU2 2$
(3)
$η= ρgHQTω, (Tω=L)$
(4)
$λ= Lρ2A2U23$
(5)

where ω, Q, $H$, g, $Cm2$, $U2$, $ρ$, $T$, $L$, and $A2$ denote the angular velocity, volumetric flow rate, total head, acceleration due to gravity, meridional component of absolute velocity at the impeller outlet, rotational velocity at the impeller outlet, water density, torque for impeller and hub, shaft power, and area of the impeller outlet, respectively.

Fig. 1
Fig. 1
Close modal
Table 1

Design specifications of the mixed-flow pump model

 Specific speed (⁠$Ns$⁠) 2.43 Flow coefficient (⁠$Φ$⁠) 0.19 Head coefficient (⁠$Ψ$⁠) 0.51 Shaft power coefficient (⁠$λ$⁠) 0.118 Number of blade (⁠$Z$⁠) 5 Blade thickness (⁠$δ)$ 3.1, mm Tip clearance (⁠$δt)$ 0.3, mm Rotation speed (⁠$N$⁠) 2400, rev/min Hub incidence angle (⁠$i1h$⁠) 10.9, deg Shroud incidence angle (⁠$i1s$⁠) 5.5, deg Inlet radius ratio (⁠$r1h$/$r1s$⁠) 0.262
 Specific speed (⁠$Ns$⁠) 2.43 Flow coefficient (⁠$Φ$⁠) 0.19 Head coefficient (⁠$Ψ$⁠) 0.51 Shaft power coefficient (⁠$λ$⁠) 0.118 Number of blade (⁠$Z$⁠) 5 Blade thickness (⁠$δ)$ 3.1, mm Tip clearance (⁠$δt)$ 0.3, mm Rotation speed (⁠$N$⁠) 2400, rev/min Hub incidence angle (⁠$i1h$⁠) 10.9, deg Shroud incidence angle (⁠$i1s$⁠) 5.5, deg Inlet radius ratio (⁠$r1h$/$r1s$⁠) 0.262

### 2.2 Ellipse Ratio.

The concept of an ER was defined as shown in Fig. 2 to denote the LE shape numerically. As an imaginary dotted line corresponding to the blade thickness ($δ$) in the normal direction was placed at the point where the full line forms an ellipse (curvature) on each of the pressure and suction surfaces, an intersection point with the mean line could be indicated. The length ($re$) from the intersection point to the LE point compared to half of the blade thickness ($δ/2$) was defined as the ER. Therefore, in the case of a rectangular LE without an ellipse shape, the ER should be 0, and as the ER gradually increases, the LE shape should be designed to be thinner and sharper. Each one-factor-based analysis was performed for the set of ER0, 1, 2, and 5, and the ER of each set was uniformly applied to the entire span.

Fig. 2
Fig. 2
Close modal

## 3 Numerical Setup

The numerical method applied in this study was described in each section below. The validation results for the numerical method were contained in the Results and Discussion section with the experimental tests. For reference, a series of numerical methods applied in this study had been dealt with in some previous papers [1113] that analyzed the ER2 set from other perspectives.

### 3.1 Computational Domain and Grid System.

The computational domain is shown in Fig. 3(a). Based on the flow direction, stationary inlet, rotating impeller, and stationary outlet domain have interfaces for each. Only the impeller was selected as a flow passage, and the tip clearance was considered. Here, an additional stationary part, the diffuser, could be excluded to save time for the iterative analysis because it had little effect on the tendency of pressure (head drop) and efficiency curves in cavitation and noncavitation states [1113]. Instead, there were just deviations in the upper–lower levels on each curve.

Fig. 3
Fig. 3
Close modal

Figure 3(b) presents the hexahedral grid system near the blade LE. It was captured in “view B” displayed in Fig. 1(a). This grid system was selected with the grid refinement technique established by Roache [14]. This technique proposed the grid convergence index (GCI) to consistently report grid improvement studies in computational fluid dynamics (CFD). As an objective asymptotic approach, this technique could be the most formal step to quantify the uncertainty of grid convergence. Figure 4 shows the y+ distribution on the blade surface for three grid systems (N1, N2, and N3) constructed with the GCI method. Each distribution suggests that the y+ remained almost constant. Here, the averaged y+ on the blade surface was about 8.6. Figure 5 contains the results of the grid test performed with the GCI method: ER2 set in cavitation state. In terms of (a) total head and (b) efficiency, the N1 set seems sufficiently competent because stability criteria for the N1 set ($GCIfine21$) were calculated as (a) 0.00231 and (b) 0.00002. These are considerably lower than the self-proposed value [15]; in other words, it means that the result of the numerical analysis was hardly affected in the N1 set. The results for the (c) circumferential and (d) axial velocity on the imaginary plane connecting the blade LE appeared to have some unstable factors; however, they had sufficiently converged with the value of (c) 0.00808 and (d) 0.01029, respectively. Finally, the grid system corresponding to the N1 set was adopted for this study.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

### 3.2 Governing Equation and Turbulence Model.

The Reynolds-averaged Navier–Stokes equations were discretized using the finite volume method. The time variation terms were considered in the transient analysis, not in the steady-state analysis. The shear stress transport standard model based on k-ω was applied to capture the turbulence in the cavitating flow [16,17]. It also has strong reliability to be an accurate prediction method for the location and amount of the cavity (vapor) in an advanced study [18]. For the two-phase transition, the homogeneous mixture model in cfx was employed as one of the most common approaches to deal with cavitation [17,19]. It states that the velocity, temperature, and pressure between the phases are equal. In the case of the multiphase approach, the number of sets of governing equations is dependent on the number of considered phases. However, in the homogeneous approach, one set of governing equations is solved for all phases. The phase change and vapor volume fraction of a fluid having incompressibility can be calculated, and time and computer resources can be saved with high accuracy enough to predict the cavitation [17]. Thus, the flow was treated as a mixture of two incompressible phases in this study. Meanwhile, the advection scheme of a high-resolution discretization with the second-order approximation was applied to ensure the minimized numerical convergence. The maximum root-mean-square residuals from the governing equations of mass and momentum were kept below 1.0 × 10−5 and 1.0 × 10−4 for the steady-state and transient simulations.

As a cavitation model, the Rayleigh–Plesset equation (RPE) was employed to solve the two-phase transition. The constants for the RPE are listed in Table 2. In practice, the vaporization and condensation processes in the cavitation state have different time scales for each, the condensation process typically being the slower one. The empirical coefficients, $Ce$ and $Cc$, are introduced to allow for these constraints: the typical default values in Table 2. These empirical coefficients were especially recommended for the transition of LE cavitation through the experimental data of Shen, Dimotakis [21], and Gerber [22]. From the review of Luo et al. [23] and Nied$z’$wiedzka et al. [19], these empirical coefficients could show higher accuracy in complex cavitation through “fine-tunes.” Meanwhile, the focus of this study is the sheet cavity near the blade LE instead of a complex combination of cavity types. Moreover, a previous study [18], which had compared the distribution of sheet cavity before and after fine-tunes for the above empirical coefficients, concluded that the Zwart model (cfx) with the typical default values for this study shows the most negligible difference compared to the experimental data.

Table 2

Constants for the Rayleigh–Plesset equation

 Cavity radius (⁠$Rb$⁠) [16,20] $1×10−6$⁠, m Nucleation site volume fraction (⁠$αnuc$⁠) [16] $5×10−4$ Empirical coefficients for vaporization (⁠$Ce$⁠) [21,22] $50$ Empirical coefficients for condensation (⁠$Cc$⁠) [21,22] $0.01$
 Cavity radius (⁠$Rb$⁠) [16,20] $1×10−6$⁠, m Nucleation site volume fraction (⁠$αnuc$⁠) [16] $5×10−4$ Empirical coefficients for vaporization (⁠$Ce$⁠) [21,22] $50$ Empirical coefficients for condensation (⁠$Cc$⁠) [21,22] $0.01$

### 3.3 Boundary Condition.

The total pressure (stable) was given to the inlet to be adjusted in the cavitation analysis, and the mass flow rate was given to the outlet. The working fluids were water and vapor at 25 °C. The frozen-rotor and transient rotor–stator method was applied to each interface for the steady-state and transient analysis. A single passage with the periodic condition was applied to ensure less time spent in the steady-state analysis, and full passage was assumed in the transient analysis. The total time for one revolution was 0.025 s, and transient data were achieved every 3 deg. The automatic wall function was applied with the smooth and nonslip condition on the boundary wall. The turbulence intensity level was selected as medium (1–5%), which would be calculated as 3.44% in this case. The medium turbulence intensity is recommended for a flow in not-so-complex devices [20].

### 3.4 Solver Information.

The solver of the numerical simulation was a commercial code, ansyscfx 19.1. The simulations were put on parallel PCs with Intel® Xeon® CPU X5690 clocked at 3.47 GHz of dual processor. For the computational time spent, one set for steady-state and one revolution for transient-state took 1 h and 26 h, respectively.

## 4 Results and Discussion

Each subsection below contains the experimental validation in the first paragraph. In the noncavitation and cavitation states, the experimental tests were carried out with the same facility. The facility had been constructed in the Korea Institute of Industrial Technology (KITECH), South Korea [1113]. As a lab-scale, measuring instruments such as pressure gages, temperature sensors, magnetic flow meters, and torque meters formed a closed loop. The maximum standard deviation of the measuring instruments was ±0.2% for each. The international standard for pump tests, ISO 5198 and ANSI/HI 1.6 [24,25], were fully applied to obtain annual certification from a certificate authority. The flow rate was controlled using two gate-type valves. The pressure in the closed loop was regulated through the compressor and vacuum pump.

Meanwhile, for the numerical results, the noncavitation state means the condition in which the inlet pressure was infinitely high in the cavitation state. The discussion would be based on the steady-state results but included additional descriptions for the transient results with respect to “cavitation surge” and “pressure gain.” The transient results were discussed with the data from each last revolution.

### 4.1 Noncavitation State.

The results of the experimental test and numerical analysis in the noncavitation state are shown in Fig. 6, which is based on the performance curve of the ER2 set. The flow coefficient ($Φ$), head coefficient ($Ψ$), and shaft power coefficient ($λ$) were defined in Eqs. (2), (3), and (5), respectively. First, CFD′ consisted of only the impeller as a numerical domain. Second, CFD* consisted of the impeller and tip clearance, which is the base of this study. Third, CFD″ consisted of the impeller, tip clearance, and diffuser. Here, the inlet and outlet parts in the numerical analysis were extended in a hydraulically advantageous shape (straight pipe). For the last, the case of experimants (Exp.) consisted of the impeller, tip clearance, diffuser, and curved pipe in the inlet and outlet flow passage. Additional factors could not be avoided, such as leakage, roughness, and friction (bearing and seal) losses in the experimental test. Accordingly, all of the results from the numerical analysis exceeded that of the experimental test, and each performance curve showed a reasonable deviation in the upper–lower levels of the head (yellow) and efficiency (blue) with respect to each factor. That is, when the tip clearance was considered on the impeller, the performance was degraded, and it deteriorated when the diffuser was attached. Nevertheless, each performance curve's tendency (slope) did not obtain any notable differences, especially near the design flow rate. Here, from the viewpoint of efficiency (blue), the slope near the low and high flow rate range tended to be flat without a diffuser, and it was more pronounced in the higher flow rate range. This is due to the pressure rise, which should be substituted in Eq. (4). As a diffuser was attached, the pressure loss in the low and high flow rate range was greater than at the design flow rate, and the efficiency was calculated as a larger deviation in the higher flow rate range. Meanwhile, the shaft power (red) showed almost the same upper–lower levels and tendency, regardless of the factors. This means that the torque on the rotating parts was almost constant regardless of the factors.

Fig. 6
Fig. 6
Close modal

Figure 7 shows the performance curve with respect to each ER in the noncavitation state. Above all, the ER0 set denoted a remarkable negative influence compared to the other three sets (ER1, 2, and 5). From a relative point of view, based on the ER0 set, the pressure and efficiency were decreased, and the shaft power was increased as the flow rate increased. The other three sets obtained almost the same performance despite different ER. Meanwhile, in the ER5 set, which has the thinnest LE shape, the slope was slightly deformed in the low flow rate range.

Fig. 7
Fig. 7
Close modal

The internal flow characteristics for each ER set are presented as Fig. 8 in “view A.” In Fig. 1, view A was aiming at suction surfaces of two blades; however, in Fig. 8, the shroud span of one blade is spotlighted for the following reason: the limiting streamlines were drawn on imaginary planes passing through the shaft centerline, and the planes were placed to divide the blade shroud span from LE to trailing edge (TE) into ten equal parts in the circumferential direction ($θ$). The Q-criterion was also indicated as the isosurface with additional contour for the circumferential velocity. The contour denotes a higher value (red) as the circumferential velocity opposite to the rotational direction is more vigorous, where its legend is the upper one. Here, Q has a deeper meaning to the location and distribution rather than the intensity of the vortex [26]. The intensity of the vortex may depend on the unfavorable region of streamlines rather than Q. First, as a perspective with respect to the ER, the ER0 set obtained the widest vortex region compared to the other three sets, with severe separation and recirculation. The elongated vortex from the LE tip to the rotational direction was observed in all the ER sets; it was also worst in the ER0 set. The flow was separated relatively far from the blade LE and suction surface (SS) in the ER0 set. This in turn caused a recirculating flow over a very wide area near the inlet (left part in view A). On the other hand, the internal flow characteristics of the other three sets were roughly the same, with a tendency to be better than those of the ER0 set. This is the representative phenomenon for the lower performance of the ER0 set in Fig. 7. Next, as a perspective depending on the flow rate, regardless of the ER effects, the inlet recirculation was generated at the low flow rate point (0.9$Φd$) in Fig. 8(a) and disappeared in Figs. 8(b) and 8(c). This characteristic must be related to the theoretical pattern of the inlet velocity triangle; that is, as the flow rate gradually increased, the incidence angle decreased, and the recirculation was also relieved. Meanwhile, an additional vortex was identified that exhibited a very low velocity just below the vortex from the tip, which could be a reattached flow at the blade SS.

Fig. 8
Fig. 8
Close modal

In order to analyze the local flow field near the blade LE in more detail, the vector distribution is shown in Fig. 9. The vectors were put on the plane, which placed at the 0.85 (pink), 0.90 (yellow), and 0.95 (blue) span, respectively. The flow of the ER0 set floated excessively far from the blade LE surface in the whole flow rate range, especially at the low flow rate (0.9$Φd$). It looks as if a bird's nest-shaped pocket (dead region) was formed immediately after the blade LE, and it was not improved even at the high flow rate (1.1$Φd$). Here, the dead region can be defined as follows: when a flow collides with a blunt- or square-shaped solid, the flow separates from the solid surface, creating a vortex. It may have a complex unsteady flow, while its time-averaged flow velocity is very small, so the flow velocity can be idealized and assumed to be dead. Paradoxically, the inlet flow pattern due to the square-shaped LE near the design flow rate (0.9–1.1$Φd$) resembled unfavorable characteristics in the fairly low flow rate range where the incidence angle is not well matched. On the other hand, the other three sets showed a blade-friendly flow pattern compared to the ER0 set and gradually stabilized as the flow rate increased. From a strict point of view among the three sets, the flow from the shroud LE was further separated in the ER5 set as the flow rate decreased. Even at the same flow rate point, which means no change in incidence angle, the flow was more floating for the ER5 set. This could be combined with the slight deformation of the slope in the low flow rate range of the ER5 set in Fig. 7. In other words, as the incoming flow did not follow the blade surface well, the stall could be encountered at a higher flow rate point. The flow angle outside the appropriate range causes a conflict with the main flow. Here, as a precaution, the difference in ER at the same flow rate cannot affect the incidence angle (flow angle). However, even at the same flow rate, the local flow around the blade LE is clearly affected with respect to the difference in ER.

Fig. 9
Fig. 9
Close modal

### 4.2 Cavitation State.

Prior to the description for the cavitation state, experimental validation of the numerical analysis was performed. Figure 10 is the cavitation characteristics of the head drop curve obtained in the conditions of flow coefficient, $Φ=$ 0.192 (1.0$Φd$) and 0.225 (1.175$Φd$). Here, the cavitation coefficient ($σ$) is defined as follows:
$σ= 2g(NPSH)U22$
(6)
$NPSH= Pt1−Pvρg,m$
(7)

where $Pt1$ and $Pv$ denote the inlet total pressure and saturation vapor pressure. From Eqs. (6) and (7), the cavitation coefficient ($σ$) is a dimension related to the inlet pressure. Hence, Fig. 10 shows the head drop curve as the inlet pressure decreases. The left graph was presented based on the absolute value, and the right one was relatively compared with the value of each 0% head drop (noncavitation). At the design flow rate (1.0$Φd$), the head gradually decreased with a gentle slope as the inlet pressure decreased. At a higher flow rate (1.175$Φd$), the head preserved its own head level despite a decrease in inlet pressure and then rose sharply up to a certain peak near the 3% head drop point. Eventually, it contained a steep drop. More details about the suction performance and specific slope were discussed in the subsections below.

Fig. 10
Fig. 10
Close modal

Meanwhile, since water is incompressible, the internal flow is only affected by the generated vapor in the cavitation state. This is the fundamental principle of the subsections below. Moreover, the pump in this study was designed as a high-efficiency device (82.8% in experimental test at design flow rate) with a relatively large incidence angle, as listed in Table 1. In this case, the head drop would be more noticeable even with a relatively small amount of vapor [11].

#### 4.2.1 Head Drop and Suction Performance.

In this section, the suction performance with respect to the ER was primarily analyzed with the head drop curve. The flow rate range was selected near the design point (0.9–1.1$Φd$), where the pump would be mainly operated. Figure 11 shows the head drop curves for each flow rate (left) and the head drop points at each flow rate (right). First, from the left graph of Fig. 11(a), the ER0 set obtained relatively low NPSHre for the whole flow rate compared to the other three sets. The slope of the curve tended to be steep near the 3% head drop point and was hardly affected by the flow rate. The steep slope of the square-shaped LE had been reported in the previous study [6] and implied better suction performance; a consistent tendency has been confirmed. However, it might obtain noise and vibration [4].

Fig. 11
Fig. 11
Close modal

Figures 11(b)11(d) show the results for the ER1, 2, and 5 sets, respectively. Compared to the above ER0 set, a significant difference was identified in the slope with respect to the flow rate. First, as common features identified from the left graphs for the other three sets, the curves generally obtained a gentle slope at the low flow rate (0.9$Φd$), while it contained a steep slope near the 3% head drop point at the high flow rate (1.1$Φd$). In addition, strained slopes were observed near $σ≈1.0$ for low flow rate and $σ<0.3$ for high flow rate. These slope-related characteristics directly influenced the NPSHre. The slope near $σ≈1.0$ at the low flow rate was referred to as “cavitation surge” under the generic perspective [2729]. In the 1970s–1990s, cavitation surge was analyzed that occurs due to the phase delay of the internal flow near the vapor. In this case, some kind of oscillation could be confirmed from flow rate or cavity, and a strong backflow inside the inlet passage should be developed to contain the phase delay, i.e., it was a limited perspective corresponding to a case in which the flow rate is low enough to cause backflow inside the inlet passage; in this study, the point designated as the low flow rate (0.9$Φd$) is near the design point, not too low. As the other causes of cavitation surge, Horiguchi et al. [30] reported the cavity length over 65% of the blade camber length, and Tsujimoto et al. [31] pointed out a large amount of vapor blocking the flow passage in a severe cavitation state with very low sigma ($σ<0.1$). As such, cavitation surge is a phenomenon that frequently occurs in a low flow rate range with a backflow or in a low sigma range with a large amount of vapor. However, as in this study, cavitation surge could be observed even from the formation of vapor near the design flow rate without backflow [29,32]; even in this case, cavity oscillation might be confirmed on the impeller blade surface. On the other hand, the horse's saddle-shaped slope, which the curves at the high flow rate contained near $σ<0.3$, was referred to as “pressure gain” [27]. In the head drop curve under the cavitation state, a negative slope is not common sense. However, an “auto-oscillation” could be obtained, which repeats negative and positive slopes in a specific range of inlet pressure, and it is being called pressure gain instead of auto-oscillation in recent years. In this regard, there was a report that this instability is from the flow penetrating into a gap between the floating vapor and the blade SS [33]. The streamwise length of the floating vapor had to be long enough to develop the gap, where the streamwise length denotes the “cavity length” [12]. There was also a report that instability is from the backflow with a large amount of vapor, just like a surge in a low flow rate range [34]. However, as in this study, a type of pressure gain exhibiting another property could be confirmed with a sudden peak immediately before the 3% head drop in the flow rate range higher than the design point. According to the characteristics of pressure gain, the ER1 set in Fig. 11(b) obtained the 3% head drop points three times in the head drop curve as the inlet pressure decreased. In this case, the most encouraging factor, NPSHre, must be carefully determined. The 2% head drop points for the ER2 set in Fig. 11(c) and the 1% head drop points for the ER5 set in Fig. 11(d) were obtained three times for each. The slope of the head drop curve related to cavitation surge and pressure gain in this section is being exposed only as a quantitative graph in our field [16,32,35]. Meanwhile, except for the ER0 set, the suction performance could be summarized in the order of ER5, 2, and 1 from the NPSHre selection, and the larger the ER, the better the suction performance for the whole flow rate.

From the graphs of each right side in Figs. 11(b)11(d), as the ER increased, each head drop point was placed at a lower sigma range to obtain better suction performance. Regardless of the effect from ER, the suction performance was generally the worst at the design flow rate, which improved as the flow rate increased. Reportedly [1,2], the suction performance in the cavitation characteristics are good near the design flow rate point and are not good in the high flow rate range. However, if the local flow pattern at the impeller inlet is dominantly influenced by the generated vapor, the suction performance can show better characteristics in the high flow rate range [2]. Thus, obviously, the suction performance with respect to the flow rate cannot be generalized to a specific tendency and should be recognized as the own characteristics of a pump [16,32,3538].

From the data of the left graphs in Fig. 11, the head drop curves were analyzed in more detail. In Fig. 12, each graph was divided into the flow rate. The ER0 set with little difference in cavitation characteristics regarding the flow rate would be discussed separately from the other three sets showing a distinct tendency with the flow rate. In the ER0 set, NPSHre was located around $σ≈0.2$ regardless of the flow rate, and the slope was quite similar. Here, the wiggle in the range of $σ≈$ 0.2-1.0 can be classified as another type of cavitation surge [29] that can be identified in the lower flow rate range than the low flow rate (0.9$Φd$) of this study. On the other hand, the cavitation surge to be noted in this study is as follows: at the point of low flow rate (0.9$Φd$), the ellipse sets (ER1, 2, and 5) obtained a sudden head drop around $σ≈1.0$. This slope caused NPSHre to be reached at a higher sigma so that negatively affected suction performance. In the range, $σ≈0.3$ at the design flow rate (1.0$Φd$), a flat slope with almost zero gradients was commonly found, which had the effect on delaying head drop. At the high flow rate (1.1$Φd$), the head rose sharply (pressure gain) around $σ≈0.3$ and peaked around $σ≈0.2$. The dotted boxes and arrows on each graph indicate figure numbering to observe the internal flow, and each four solid symbols on the ER2 curve at the low (0.9$Φd$) and high flow rate (1.1$Φd$) denote the point for transient analysis.

Fig. 12
Fig. 12
Close modal

Figure 13 shows the efficiency drop curve at each point corresponding to Fig. 12. Most of all, it was consistent with the tendency in Fig. 12. This means that the generated vapor in the cavitation state directly affected the efficiency. On the other hand, it was difficult to confirm any strained slope that could be confirmed in a specific sigma range of the head drop curve. In particular, at the high flow rate (1.1$Φd$), the head sharply rose to a peak. However, the efficiency continued to decrease without any peak. This implies that the internal flow gradually deteriorated regardless of the increase in the head. Meanwhile, the drop in Fig. 13 is an absolute concept, and the same tendency was confirmed in a relative concept.

Fig. 13
Fig. 13
Close modal

#### 4.2.2 Cavity Inception and Propagation (Blockage).

The difference in head drop curve and cavitation characteristics with respect to the ER would be investigated in this section. Figure 14 shows the cavity inception and growth in the cavitation state. Stuparu et al. [39] proposed an analytical technique to express the cavity inception with a semilogarithmic plot for the vapor volume inside the flow passage. According to this analysis technique, the tendency of the plot was generally linear, and it could be interpreted that the steeper the slope, the higher the sensitivity to cavitation. As a premise, the authors denoted that the vapor volume defining the cavity inception could be selected in comparison with the head drop characteristics so that flexible results might be secured. Although the actual cavity inception point was slightly different from the indicated point on the graph, it could be understood as a technique that quantified cavity inception under the same criterion of vapor volume [11]. In Fig. 14, the vapor volume was normalized as the entire volume of a flow passage, and the enlarged scale was placed as each small graph at the right corner. The criterion of vapor volume defining the cavity inception was selected as 1.0 × 10−8 (about 0.000001% in the entire volume of the flow passage), which was a value with little effect on the semilogarithmic plot. As expected, the ER0 set retained almost the same slope of cavity inception and growth in whole flow rates. The sigma of cavity inception ($σ≈1.8$) was the lowest among all sets. Here, the sigma corresponding to the cavity inception point and the inflection point of the head drop curve should be noted. The ER0 set maintained its base head (0% head drop) to $σ≈0.2$, which was lower than the point of the cavity inception. There must be a “specific reason” to maintain the head despite the generated vapor inside the flow passage. On the other hand, the other three sets showed a well-matched tendency between the cavity inception and the inflection of the head drop. The particularly notable point is that the lowest cavity inception points were obtained in the order of ER5, 2, and 1 at whole flow rates. This also indicates a reasonable tendency toward better suction performance in larger ER.

Fig. 14
Fig. 14
Close modal

The “specific reason” mentioned above was presented in Fig. 15 at the sigma points (designated in Fig. 12) below the cavity inception for the ER0 set. For a more detailed comparison, Fig. 16 corresponding to Fig. 9 was additionally presented at $σ=$ 0.83. The red-colored formation in Figs. 15 and 16, which looks like a volume, is the isosurface of the vapor volume fraction. Each noncavitation state was shown in Figs. 8 and 9 so that the amount of vapor gradually increased as the inlet pressure (sigma) decreased. Here, as an important reconfirmation for Fig. 9, the ER0 set formed a dead region for the vapor immediately after the blade LE on SS in the noncavitation state. In the cavitation state, if the generated vapor was placed in the dead region, as shown in Fig. 16, the head level could be maintained while the efficiency decreased. That is, in the ER0 set, the vapor was formed in the dead region after the blade LE on SS and did not affect the internal flow compared to the noncavitation state, even in the cavitation state. In each low, design, and high flow rate in Fig. 15, the internal streamline in the range of $σ=$ 1.20–0.25, 1.20–0.31, and 1.20–0.25 had little difference compared to Fig. 8, respectively. The vector distribution at $σ=$ 0.83 (Fig. 16) was nearly the same pattern as the noncavitation state (Fig. 9). At $σ=$ 0.19, 0.23, and 0.19 in Fig. 15, where the head drop was confirmed, the vapor occupied a very wide region, which could affect the pattern of streamline. On the other hand, the other three sets were different. As described in the following sections, the internal flow pattern was affected immediately after the cavity inception. The head decreased as the inlet pressure decreased from the cavity inception. The cavity formation and thickness showed a specific tendency with different ER, which could be linked to the concept of cavity blockage on the blade surface. Here, the ER0 set would be excluded from the mutual description below. From Fig. 15, it could be further confirmed that the cavity growth of the ER0 set was from the cloud cavity, not the sheet cavity. At $σ=$ 1.20 for each flow rate, the vapor of ER0 was developed separately from the blade surface and was floating. In this case, it was not suitable to apply the concept of cavity blockage defined in this study. Moreover, it was not appropriate to compare the ER0 set, which did not affect the internal flow pattern during the development of the cavity, with the other three sets under the same viewpoint.

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal

Figure 17 shows a schematic definition of the cavity blockage ($ζv$) in this study. The vapor occupied length on the imaginary plane connecting the LEs in the tangential direction was quantified as a ratio ($t1v/t1$), where $t1v$ is the maximum circumference of the vapor at the impeller inlet, and $t1$ is the circumference of the shroud at the impeller inlet, respectively. The cavity blockage could be considered as an unfavorable thickness to the streamwise direction as the vapor was formed at the impeller inlet [11]. Figure 18 shows the cavity blockage with respect to each flow rate in the cavitation state. The y-axis was set on a semilogarithmic basis. From Fig. 18, the difference in suction performance for the ER was clearly identified. In the three sets, the ER1 set formed the thickest vapor even from the cavity inception than the ER2 and 5 sets. The ER2 set obtained thinner vapor than that of ER1, and the thinnest vapor could be formed in the ER5 set. However, the difference in cavity blockage with respect to ER was hardly seen in the severe cavitation range ($σ<0.5$), the important point of view is from the moment of cavity inception. It is obviously referred that the generated vapor in the flow passage acts as an obstacle accompanying the flow separation. The thicker the cavity blockage, the stronger the head drop, and this should be the main cause of the difference in suction performance [11]. Finally, the correlation between the head drop slope and the cavity blockage of each set was in good agreement. The head drop was dominantly affected with the cavity blockage, and it was just accompanied by the phenomena (cavitation surge and pressure gain) that could not be explained as a single effect of blockage in a temporary or specific situation while the head drop continued as the inlet pressure decreased. In this way, the cavitation should be analyzed from various perspectives. Meanwhile, the difference in cavity blockage between each ER was relatively small in terms of high flow rate point, but it was sufficient to affect the head drop. The correlation between the flow rate and the cavity blockage was difficult to be established.

Fig. 17
Fig. 17
Close modal
Fig. 18
Fig. 18
Close modal

Figures 19(a)19(c) show the vapor volume fraction of the hub (0)-to-shroud (1) span on the LE plane. The vapor was developed near the shroud and gradually propagated to the hub as the inlet pressure decreased. Here, it was possible to confirm a specific tendency that could analyze the cavitation characteristics depending on the flow rate. As the flow rate increased from the low to high flow rate, the vapor volume fraction indicated the span having the maximum value, thereby establishing a correlation between the flow rate and vapor formation inside a pump. That is, as the flow rate increased, the vapor would be formed more dominantly near the shroud tip. Figure 19(d) shows the value of the normalized span obtaining the maximum vapor volume fraction at each flow rate, where the long dotted line, full line, and short dotted line indicate the low, design, and high flow rate, respectively. As the flow rate increased, the scattered points were placed near the shroud tip.

Fig. 19
Fig. 19
Close modal

One point that could have been out of the question was pinched as the last paragraph in this section. Under the descriptions related to Figs. 14(b) and 18(b), the head drop of the ER5 set in Fig. 12(b) should be better than the ER1 and 2 sets from each base head (0% head drop). However, in the cavitation state at $σ>2.0$, the head drop of the ER5 set was lower than that of the ER1 and 2 sets. The lower head drop of the ER5 set was difficult to confirm in terms of cavity inception, blockage, and volume of hub-to-shroud span. Figure 20 shows the internal flow characteristics in the cavitation state ($σ=$ 2.61), where the ER5 set obtained the lower head drop than the ER1 and 2 sets at the design flow rate. It could be observed that the vapors of the ER5 set were not occupied from the shroud and were locally located at the lower span. On the other hand, the vapors of the ER1 and 2 sets were completely squeezed to the end of the shroud tip. The lower head drop could be confirmed with this temporary formation.

Fig. 20
Fig. 20
Close modal

#### 4.2.3 Cavitation Surge.

This section is about the sudden head drop observed at $σ≈1.0$ in Fig. 12(a). This phenomenon (referred to as cavitation surge) was confirmed in the ranges of $σ=$ 0.83–0.93, 0.83–0.87, and 1.00–1.20 for the ER1, 2, and 5 sets, respectively. Figures 21(a-1), 21(b-1), and 21(c-1) show the internal flow characteristics of each set at the sigmas, including those ranges. The red-colored value in “view C” indicates the upper endpoint of the generated vapor within the normalized span from hub (0) to shroud (1). From the results, the head drop slope associated with cavitation surge that could be observed at the low flow rate (0.9$Φd$) occurred when the propagation of the generated vapor became relatively rapid toward the shroud tip. The point at which the vapor rapidly increased toward the shroud tip appeared at $σ=$ 0.83–0.93, 0.83–0.87, and 1.00–1.20 for each ER1, 2, and 5, which coincided with the range of the sudden head drop as shown in Figs. 21(a-2), 21(b-2), and 21(c-2). In addition, the inlet recirculating flow due to vapor propagation toward the shroud tip was strengthened in this range. As such, the cavitation surge described in this study is possible when no vapor is generated near the shroud in the cavitation state. In other words, it may occur in conditions in which vapor propagation near the shroud can rapidly increase. The sigma range for cavitation surge is generally not too low in this paragraph so that the vapor can gradually grow.

Fig. 21
Fig. 21
Close modal

Figure 22 shows the transient analysis results to observe the unstable characteristics of cavitation surge; ER2 set only. In the left graph, the steady-state results (green) are the same as Fig. 12(a), and the transient results (sky-blue) from the data of each last revolution are presented as the averaged value (solid-diamond symbol) and minimum-maximum value (bar symbol; pressure fluctuation). As expected, quite strong pressure fluctuations were identified at points near the sudden head drop ($σ=$ 0.83 and 0.87), which was referred to as cavitation surge from the steady-state analysis. Although the pressure fluctuation was also contained from a state that could be understood as a noncavitation ($σ=$ 3.21); however, this is because it was in the low flow rate condition (0.9$Φd$) in which the incidence angle was increased. In addition, the range of pressure fluctuation at the point ($σ=$ 1.09) just before the cavitation surge was relatively more minor than near the cavitation surge, and the averaged values of the transient results formed the steep slope (sudden head drop) that the steady-state results showed in the similar inlet pressure range. Accordingly, the fast Fourier transform analysis was performed as the right graph. This model pump's blade passing frequency (BPF) is 200 Hertz; however, it was difficult to observe any regular pattern, including BPF, at all four sigma points. The peaks with considerable magnitude were distributed with frequent intervals at non-BPF. The maximum peak of the magnitude was found near the second BPF for all sigma points, and the maximum level was proportional to the fluctuation range of the left graph.

Fig. 22
Fig. 22
Close modal

This section is related to the flat slope in Fig. 12(b), which was confirmed in the ranges below $σ≈$ 0.47, 0.35, and 0.31 for the ER1, 2, and 5 sets, respectively. Figure 23 shows the internal flow characteristics of each set at the sigmas, including those regions. From the results, the stagnant range of the head drop that could be observed at the design flow rate (1.0$Φd$) occurred when the vapor completely propagated to the entire span (hub-to-shroud) at the LE. However, the head broke down as the inlet pressure further decreased, and then the cavitation became severe. The stagnant head drop described in this study can be observed in the sigma range, which is low enough to allow vapor to cover the blade LE completely.

Fig. 23
Fig. 23
Close modal

#### 4.2.5 Pressure Gain.

This section focuses on the sharp head rise for the ER1, 2, and 5 sets, in the range $σ<$ 0.3 of Fig. 12(c). Figure 24 shows the internal flow characteristics of each set at the sigmas, including that region. From the results, the vapor for each set just propagated as the inlet pressure decreased, and it was difficult to confirm any specific features even in the range where the inlet pressure was less than $σ≈$ 0.3. The inlet recirculating flow was just getting stronger little by little as the inlet pressure decreased. As a further schematic, Fig. 25 shows the vapor formation at the point ($σ=$ 0.25) where the head rose sharply. Since the inlet pressure was relatively low, the vapor was distributed quite long in the streamwise direction. Here, the vapor had a shape that was bent in a direction perpendicular to the blade surface near the shroud tip, and this shape was referred to as a “winglet” in our field. The winglet is a representative design technique for anticavitation [4043]. The slope, such as pressure gain, could be observed with this formation even though the inlet recirculating flow was not strong. As such, this phenomenon can be observed in the sigma range that is low enough to allow vapor to form a winglet-like shape. Besides, the vapor must be distributed generally near the shroud tip to have the shape of the winglet. In this study, the sharp head rise with a negative slope was most pronounced in the high flow rate point because the point of maximum vapor volume fraction in hub-to-shroud span was located at the highest span as in Fig. 19(d). The vapor could form a complete winglet-like shape at the high flow rate, while the vapor was empty near the shroud tip at the low and design flow rate. As an additional constraint, this phenomenon can be observed in a pump with good head drop characteristics (suction performance). If there is excessive head drop (when the internal flow becomes complicated), the slope can no longer show a negative slope. In other words, this phenomenon is more likely to be observed as the cavity blockage is low.

Fig. 24
Fig. 24
Close modal
Fig. 25
Fig. 25
Close modal

Figure 26, which has the same footnotes as Fig. 22, while the steady-state results (green) are the same as Fig. 12(c), presents the transient analysis results for pressure gain. Since it was in the high flow rate condition (1.1$Φd$) with a more stable value of incidence angle, there was almost no pressure fluctuation in the noncavitation state ($σ=$ 3.21). The pressure fluctuation was not observed until $σ=$ 0.79, where the inlet pressure was further reduced. However, it was remarkably confirmed at the points ($σ=$ 0.19 and 0.39) near the horse's saddle-shaped slope, which was confirmed as pressure gain from the steady-state analysis. The fluctuation range was intense at the deepest saddle point ($σ=$ 0.39): there was little fluctuation at the inlet pressure point ($σ=$ 0.79) just before the saddle, so pumps with this head drop characteristic may need to be operated with caution; the case where the incidence angle has a relatively stable value and the head drop curve does not include the saddle-shaped (positive) slope, it contains almost no pressure fluctuation to a fairly low sigma point [11]. Meanwhile, the averaged values of these transient results (solid-diamond symbol with sky-blue color) still formed the positive slope contained in the steady-state results, near the similar inlet pressure range. The fast Fourier transform analysis results on the right inherited the tendency of pressure fluctuations. The irregular peaks of pressure magnitude at the points ($σ=$ 0.19 and 0.39) where the pressure gain occurred made it impossible to observe a stable pattern of BPF. Here, the maximum peak of the pressure magnitude identified at the deepest saddle point ($σ=$ 0.39) coincided with the second BPF; this may mean a warning due to the overlap of cycles (superposition). The pressure magnitudes for the two higher sigma points ($σ=$ 3.21 and 0.79) are shown in Fig. 27 as the enlarged y-axis. In the noncavitation state ($σ=$ 3.21), regular BPF peaks were confirmed every 200 Hertz up to about 800 Hertz; with a slightly higher magnitude at $σ=$ 0.79. The magnitudes remaining after 800 Hertz should not be understood as instability because it was too low.

Fig. 26
Fig. 26
Close modal
Fig. 27
Fig. 27
Close modal

## 5 Conclusion

1. In the noncavitation state, the ER0 set denoted a remarkable negative influence compared to the other three sets (ER1, 2, and 5) because of the widest vortex accompanying severe separation and recirculation. The other three sets obtained almost the same performance despite different ER.

2. In the cavitation state of the ER0 set, the vapor did not affect the internal flow pattern compared to the noncavitation state. The ER0 set obtained relatively low NPSHre for the whole flow rate compared to the other three sets.

3. In the cavitation state of the ER1, 2, and 5 sets, the thinner cavity blockage was formed with a better suction performance as the ER increased. The cavity blockage, which could act as an obstacle accompanying the flow separation in the flow passage, should be a main variable for the head drop.

4. The cavitation surge described in this study could occur when no vapor was generated near the shroud in the cavitation state. In other words, it might occur in conditions in which vapor propagation near the shroud could rapidly increase. The sigma range ($σ≈1.0$) for cavitation surge was not too low so that the vapor could gradually grow.

5. The stagnant head drop described in this study could be observed in the sigma range ($σ≈0.3$), which was low enough to allow vapor to cover the blade LE completely. However, the head broke down as the inlet pressure further decreased and the cavitation became severe.

6. The pressure gain indicated in this study could be observed in the sigma range ($σ<0.3$) that was low enough to allow vapor to form a winglet-like shape. Besides, the vapor should be distributed generally near the shroud tip in order to have the shape of the winglet. As an additional constraint, this phenomenon could be observed in a pump with good head drop characteristics (suction performance).

7. The phenomena perceived as cavitation surge and pressure gain contained a significant range of pressure fluctuations. The pressure fluctuations could increase suddenly near the strained slope on the head drop curve. In particular, the phenomenon described as pressure gain in this study might require careful attention during operation.

## Funding Data

• Korea Institute of Energy Technology Evaluation and Planning (KETEP), Korea Government (MOTIE) (Grant No. 2021202080026D; Funder ID: 10.13039/501100003052).

## Conflict of Interest

There are no conflicts of interest.

## Nomenclature

• $A2$ =

area at impeller outlet, m2

•
• $Cc$ =

empirical coefficients for condensation ($Cc$$=0.01$)

•
• $Ce$ =

empirical coefficients for vaporization ($Ce$$=50$)

•
• $Cm2$ =

meridional component of absolute velocity at impeller outlet, m/sec

•
• ER =

ellipse ratio

•
• $g$ =

acceleration of gravity, m/s2

•
• GCI =

grid convergence index

•
• $H$ =

•
• HD =

•
• KITECH =

Korea Institute of Industrial Technology

•
• $L$ =

shaft power, kW

•
• LE =

•
• $N=$ =

rotational speed, rev/min

•
• $Ns$ =

specific speed (type number)

•
• NPSH =

•
• NPSHre =

•
• $Pt1$ =

inlet total pressure, Pa

•
• $Pv$ =

saturation vapor pressure, Pa

•
• PS =

pressure surface

•
• RPE =

Rayleigh–Plesset equation

•
• $Q$ =

volumetric flow rate, m3/sec

•
• SS =

suction surface

•
• $T$ =

torque for impeller and hub, J

•
• $t1$ =

circumference of shroud at impeller inlet, mm

•
• $t1v$ =

maximum circumference of vapor at impeller inlet, mm

•
• $U2$ =

rotational velocity at impeller outlet, m/sec

•
• Z =

•
• $δ$ =

•
• $δt$ =

tip clearance, mm

•
• $ζv$ =

cavity blockage

•
• $η$ =

total efficiency

•
• $λ$ =

shaft power coefficient

•
• $ρ$ =

water density, kg/m3

•
• $σ$ =

cavitation coefficient

•
• Φ =

flow coefficient

•
• $Ψ$ =

•
• $ω$ =

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