Prediction of Cavitation Erosion and Residual Stress of Material Using Cavitating Flow Simulation with Bubble Flow Model

Abstract

We have developed a numerical simulation code that can predict the cavitation erosion and the residual stress of a material, which are both closely related to plastic deformation. The shock wave generated at the cavitation bubble collapse hits the material surface and the impact energy causes plastic deformation which changes the residual stress. When the impact energy is high, mass loss (i.e., erosion) occurs after the plastic deformation stage. We numerically simulated impulsive bubble pressures that varied on the order of a microsecond in the cavitating flow with the ‘bubble flow model’. The bubble flow model simulates the abrupt time-variations in the radius and inner pressure of bubbles based on the Rayleigh-Plesset equation. Although the shock wave propagation was not simulated, the impact energy was estimated based on the bubble pressure. The predicted impact energy was compared with the distribution of plastic deformation pits, which were observed on the impeller blade surface of a centrifugal pump. The predicted impact energy was also compared with the distribution of residual stress measured on a stainless steel plate after a cavitating jet impinged on the plate. The distribution of the impact energy corresponded qualitatively to that of the residual stress improvement caused by the plastic deformation. High correlation between the predicted impact energy and the plastic deformation of material was confirmed, and we found that our numerical method is relevant for the prediction of cavitation erosion and residual stress on a material surface.

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1 Introduction

Cavitation has both advantages and disadvantages for industrial products. On one side, cavitation causes performance reduction, vibration, noise, and erosion in turbomachinery such as pump, water turbine, and ship propeller. On the other side, cavitation is applied to apparatus for sterilization, purification, material cutting, and material modification as work hardening and residual stress improvement. Water Jet Peening (WJP) is a preventive maintenance technology for nuclear power plants [1, 2]. A cavitating jet is injected and impinges on the weld surfaces of structures in a nuclear reactor. The shock wave generated at the cavitation bubble collapse hits the material surface and the impact energy causes plastic deformation of the weld surface, and changes the residual stress from tensile to compressive. Compressive residual stress prevents the occurrence of stress corrosion cracking (SCC) on the weld surface.

CFD (Computational Fluid Dynamics) is commonly used to predict the influence of cavitation on industrial products. However, the prediction of cavitation erosion by CFD has only recently been attempted. Dular et al. pointed out the high correlation between the void fraction variation and pits on a thin hydrofoil [3]. The erosion area was predicted based on the numerically predicted void fraction [4]. Nohmi et al. proposed indexes for predicting cavitation erosion area. The indexes consist of void fraction, void fraction variation, pressure, and pressure variation [5, 6]. Miyabe and Maeda predicted the cavitation erosion area by using CFD and the indexes proposed by Nohmi et al. in a double-suction centrifugal feedpump [7]. Ochiai et al. numerically simulated the cavitating flow around a hydrofoil with a locally homogeneous model of a gas–liquid two phase medium [8]. The bubble dynamics was solved along the bubble moving path in the simulated cavitating flow field, and the bubble collapse pressure, frequency and energy were estimated for predicting the erosion area. Zima et al. numerically simulated the cavitating flow in a mixed-flow pump, and solved the bubble dynamics along the bubble moving path [9]. The predicted area of high cavitation intensity corresponded with the actual erosion area.

While the above mentioned studies focus on the prediction of cavitation erosion, the prediction of residual stress improvement has hardly been reported. We developed a simulation code with the bubble flow model to solve the bubble dynamics in the whole cavitating flow field. The volumetric and transitional bubble motion were modeled, and the distribution of bubble nuclei was also taken into account [10, 11]. The impact energy at the bubble collapse was estimated based on the bubble pressure and the bubble number density. In this paper, we report high correlation between the impact energy estimated by our simulation and the plastic deformation of material, which was caused by the bubble collapse. The simulation code was applied to predict the erosion [11] and the residual stress improvement [12], which are both closely related to plastic deformation of material.

2 Numerical Method

2.1 Governing Equations

We adopted the bubble flow model [10] in the simulation code to predict the cavitating flow. The following assumptions concerning bubbles were made in the model.

• The liquid phase is water, which is incompressible.

• The gas phase consists of spherical bubbles. The bubbles are filled with vapor and non-condensable gas. The evaporation and condensation on the bubble surface are modeled assuming the non-condensable gas follows an isothermal transformation during expansion and an adiabatic one during contraction [13]. Thus the gas phase is compressible.

• No collision, coalescence or breakup of bubbles occurs.

• The density and momentum of the gas phase are sufficiently small to be negligible.

The governing equations are described in generalized coordinates as follows [10, 11]. The momentum conservation equation of bubble flow is:

$$ \begin{aligned} & \frac{{\partial \left( {\rho_{L} f_{L} u_{Li} } \right)\,}}{\partial t} + J\frac{{\partial \left( {\rho_{L} f_{L} u_{Li} U_{Li} /J} \right)\,}}{{\partial \xi_{j} }} \\ & = - \nabla_{i} p_{L} + \nabla_{j} \left( {\mu \nabla_{j} u_{Li} } \right) + \frac{1}{3}\left( {\mu \nabla_{k} u_{Lk} } \right) - \rho_{L} \left\{ {2\varepsilon_{ijk} \varOmega_{j} u_{Lk} + \varepsilon_{ijk} \varOmega_{j} \left( {\varepsilon_{klm} \varOmega_{l} r_{m} } \right)} \right\}, \\ \end{aligned} $$

(13.1)

where $\rho_{L}$ (= 1,000 kg/m³) and f _L are the density and the volume fraction of the water. u _L is the water velocity and U _L is the contravariant velocity of the water. p _L and $\mu$ (= 1.0 × 10⁻³ Pa s) are the static pressure and the viscosity of the water. J is the Jacobian and ε is the Eddington’s epsilon.

In Eq. (13.1), the water velocity represents the relative velocity in a rotating coordinate system defined by the rotation speed vector $\varOmega$. In the case $\varOmega = 0$, this represents the absolute velocity in a static coordinate system. The fourth term on the right hand side includes the Coriolis force and the centrifugal force. Here, r is the distance from the rotating axis.

The conservation of the bubble number density, n _G , is:

$$ \frac{{\partial {\kern 1pt} n_{G} \,}}{\partial t} + J\frac{{\partial {\kern 1pt} \left( {n_{G} U_{Gi} /J} \right)\,}}{{\partial \xi_{j} }} = 0. $$

(13.2)

The pressure equation is:

$$ \frac{1}{{c^{2}}}\,\frac{{\partial {\kern 1pt} p_{L} \,}}{\partial t} + \nabla_{j} f_{L} u_{Lj} + \nabla_{j} f_{G} u_{Gj} - 4\pi {\kern 1pt} r_{G}^{2} n_{G} \frac{{D{\kern 1pt} r_{G} }}{Dt} = 0, $$

(13.3)

where c is the coefficient for pseudo-compressibility, f _G is the void fraction, u _G is the bubble velocity, r _G is the bubble radius, and D/Dt is the substantial derivative.

Equation (13.3) is based on a pseudo-compressibility method, which is derived from the conservation of volumetric fractions (f _L  + f _G  = 1) and the conservation of bubble number density [10]. The water pressure is influenced by the bubble behavior through the fourth term on the left hand side.

The volumetric motion of an isolated bubble is described by the Rayleigh-Plesset equation [14],

$$ r_{G} \frac{{D^{2} r_{G} }}{{Dt^{2} }} + \frac{3}{2}\left( {\frac{{Dr_{G} }}{Dt}} \right)^{2} = \frac{{p_{B} - p_{L} }}{{\rho_{L} }}\, + \,\frac{1}{4}(u_{Li} - u_{Gi} )^{2}, $$

(13.4)

where p _B is the bubble pressure. Here, i means the x, y and z directions. The second term on the right hand side accounts for the effect of the reduction of the surrounding pressure of the bubble due to the velocity difference between the bubble and its surrounding water. The bubble pressure is:

$$ p_{B} = p_{G} + p_{v} - \frac{2T}{{r_{G} }}\, - \,4\mu \,\frac{1}{{r_{G} }}\,\frac{{Dr_{G} }}{Dt}, $$

(13.5)

with:

$$ p_{G} r_{G}^{3} = const.\,{\text{if}}\;\frac{{Dr_{G} }}{Dt} > 0, $$

(13.6)

or:

$$ p_{G} r_{G}^{3\kappa } = const.\,{\text{if}}\;\frac{{Dr_{G} }}{Dt} < 0, $$

(13.7)

where p _G is the non-condensable gas pressure. Both the surface tension, T (= 0.072 N/m) and the vapor pressure, p _v (= 2,300 Pa) are constant. The viscosity, $\mu$ is assumed to be the same as that of water $(\mu = 1.0 \times 10^{-3}\,\text{Pa}\,\text{s}).$ The specific heat ratio, $\kappa$, is 1.4. Eqs. (13.6) and (13.7) are simplified equations deduced from a more precise simulation [13]. In Ref. [13], the motion of one point-symmetric bubble in incompressible liquid was simulated including the heat exchange and the phase change at the interface, the temperature distribution, the diffusion between vapor and non-condensable gas, the homogeneous nucleation of mist and its growth within the bubble. It was concluded that the cavitation bubble behaves isothermally when expanding and adiabatically when shrinking.

The validation of the present simplified model is shown in Fig. 13.1. The line labeled “DNS” was the result of the precise simulation [13]. Other lines in Fig. 13.1 were the results of the Rayleigh-Plesset equation with different models of non-condensable gas. The line labeled “Switched” corresponded to the present model and follows the DNS line well.

Fig. 13.1

Bubble dynamics models

The translational motion of a bubble is solved by taking into consideration the force balance of the bubble,

$$ F_{Ai} + F_{Pi} + F_{Di} + F_{Li} + F_{Coi} + F_{Cei} = 0, $$

(13.8)

where F _Ai is the added mass force,

$$ \begin{aligned} F_{Ai} = &\; \frac{4}{3}\pi \beta \left\{ {\left( {\frac{{\partial {\kern 1pt} \left( {\rho_{L} \,\,r_{G}^{3} \,u_{Gi} } \right)}}{{\partial {\kern 1pt} t}} + U_{Gj} \frac{{\partial \left( {\rho_{L} \,\,r_{G}^{3} \,u_{Gi} } \right){\kern 1pt} }}{{\partial {\kern 1pt} \xi_{j} }}} \right)} \right\} \\ & - \left\{ {\left( {\frac{{\partial {\kern 1pt} \left( {\rho_{L} \,\,r_{G}^{3} \,u_{Li} } \right)}}{{\partial {\kern 1pt} t}} + U_{Lj} \frac{{\partial \left( {\rho_{L} \,\,r_{G}^{3} \,u_{Li} } \right){\kern 1pt} }}{{\partial {\kern 1pt} \xi_{j} }}} \right)} \right\}, \end{aligned} $$

(13.9)

Here, β is a constant of 0.5 for a spherical bubble, U _G is the contravariant velocity of the bubble. $F_{Pi}$ is the force of the acceleration of the surrounding fluid,

$$ F_{Pi} = - \frac{4}{3}\pi \;\rho_{L} r_{G}^{3} \left( {\frac{{\partial \,u_{Li} }}{\partial t} + U_{Lj} \frac{{\partial \,u_{Li} }}{{\partial \xi_{j} }}} \right). $$

(13.10)

F _Di and F _Li are the drag and lift forces [15, 16],

$$ F_{Di} = \frac{1}{2}\pi \,\rho_{L} \,r_{G}^{2} {\kern 1pt} C_{D} |u_{G} - u_{L} |(u_{Gi} - u_{Li} ), $$

(13.11)

$$ F_{Li} = \frac{1}{2}\pi \,\rho_{L} r_{G}^{2} \,C_{L} \,|u_{G} - u_{L} |\varepsilon_{ijk} \,\omega_{Lk} (u_{Gj} - u_{Lj} )/|\omega_{L} |, $$

(13.12)

$$ C_{D} = \frac{24}{{\text{Re}_{bub} }}(1\, + \,0.15\,\text{Re}_{bub}^{0.687} ), $$

(13.13)

$$ \text{Re}_{bub} = \frac{{2\,r_{G} \,\rho_{L} \,|u_{G} - u_{L} |}}{{\mu_{L} }}, $$

(13.14)

$$ C_{L} = 0.59\,\left( {\frac{{|\omega_{L} |r_{G} }}{{|u_{G} - u_{L} |}}} \right)^{0.25}, $$

(13.15)

where $\omega_{L}$ is the vorticity vector. F _Coi and F _Cei are the Coriolis force and centrifugal force,

$$ F_{Coi} = \frac{8}{3}\pi \;\rho_{L} r_{G}^{3} \left\{ {\beta \varepsilon_{ijk} \varOmega_{j} u_{Gk} - \left( {1 + \beta } \right)\varepsilon_{ijk} \varOmega_{j} u_{Lk} } \right\}, $$

(13.16)

$$ F_{Cei} = - \frac{4}{3}\pi \;\rho_{L} r_{G}^{3} \left\{ {\varepsilon_{ijk} \varOmega_{j} \left( {\varepsilon_{klm} \varOmega_{l} r_{m} } \right)} \right\}. $$

(13.17)

The velocity of the bubble relative to its surrounding water, i.e., the slip velocity, is computed with Eqs. (13.8)–(13.17). The details on the calculation algorithm are described in Ref. [10]. No turbulence model was used in the simulation code to reduce the calculation time.

The void fraction in this simulation is calculated from the bubble radius and the bubble number density by:

$$ f_{G} = \frac{4}{3}\,\pi r_{G}^{3} n_{G}. $$

(13.18)

Equation (13.18) means that the void fraction increases when the bubble expands due to the pressure difference at the liquid–gas interface or where the bubble nuclei have accumulated.

2.2 Cavitation Impact Energy

In the actual bubble collapse phenomenon, a bubble breaks up into minute bubbles after an abrupt shrinking of the bubble, and high energy is released from the bubble in the form of high pressure. The energy released at the bubble collapse was evaluated in the accelerated cavitation erosion tests with a shock sensor as:

$$ E = k\sum\limits_{i = 1,2,3 \ldots } {p_{ci}^{2} }, $$

(13.19)

where p _c is the measured bubble collapse pressure and k is a proportionality constant [17, 18]. Although the collapse pressure was not calculated in the present simulation, the abrupt shrinking of the bubble and the impulsive bubble pressure were calculated. We assumed that the predicted maximum bubble pressure was qualitatively related to the bubble collapse pressure. We therefore numerically investigated the cavitation impact energy by defining:

$$ E = k\sum\limits_{i = 1,2,3 \ldots } {p_{Bi}^{2} (n_{G} {\kern 1pt} \varDelta V)}. $$

(13.20)

The bubble collapse pressure in Eq. (13.19) is replaced with the bubble pressure, p _B , in Eq. (13.20). The n _G ΔV represents the number of bubbles in each numerical cell.

2.3 Simulated Objects and Calculation Conditions

There are two simulated objects in this paper. We predicted the cavitation erosion area in a centrifugal pump and the residual stress improvement by a cavitating jet.

Figure 13.2 shows the numerical mesh and boundary conditions in the centrifugal pump. The impeller had six blades and a shroud, which was made of aluminum for the accelerated cavitation erosion test. The maximum diameter of the impeller was 302 mm, and the hub diameter was 78 mm. The region between the pressure and suction sides of the impeller blades was investigated using periodical boundaries.

Fig. 13.2

Simulated region and boundary conditions for pump. Based on [11], reprinted with permission from JSME

A cylindrical suction channel, which was 180 mm long axially, was connected to the region including the impeller blades. The flow rate of the water, Q, was 4.56 m³/min, and Q/Q _ηmax was 0.60 where Q _ηmax is the flow rate at the highest efficiency. The liquid velocity at the inlet boundary was uniform and equal to 2.12 m/s. This pump had a volute-type casing positioned downstream of the impeller. However, a fan-shaped discharge channel, which was 50 mm long radially, was connected downstream of the impeller blades region instead of the volute-type casing. The static pressure at the outlet boundary was varied to change the NPSH conditions.

The initial void fraction was 0.001. The initial bubble radius was 1.0 × 10⁻⁵ m. The initial bubble number density and the bubble number density at the inlet boundary were 2.39 × 10¹¹ m⁻³ based on Eq. (13.18).

No-slip conditions were assumed for the solid surface of the blade, hub, and shroud of the impeller. No-slip conditions were also assumed for the wall surface of the suction and discharge channels.

Figure 13.3 shows the numerical mesh and boundary conditions of the cavitating jet. The simulated region was limited between the nozzle and the flat plate, where the submerged cavitating jet from the nozzle impinged vertically on the plate. We assumed an axisymmetric flow field and conducted calculations within a range of 2 degrees in the $\theta$ direction using periodic boundary conditions since the nozzle and the flat plate shapes were axisymmetric. The nozzle had a cylindrical throat and a horn-shaped flow passage downstream of the throat. The nozzle throat diameter was 2 mm. The length and the expansion angle of the horn-shaped flow passage were 11 mm and 30 degrees. The cartesian mesh was prepared in the r–z plane except for the horn-shaped flow passage.

Fig. 13.3

Simulated region and boundary conditions for cavitating jet. Based on [12], reprinted with permission from ASME

The pressure boundary conditions are plotted on the dotted line in Fig. 13.3, and a constant pressure of 3.0 × 10⁵ Pa or 1.0 × 10⁵ Pa was assumed. When WJP is conducted in a nuclear reactor, the water depth above the weld and the nozzle is over 20 m in some cases. Then, the boundary pressure was 3.0 × 10⁵ Pa. On the other hand, the pressure boundary condition was fixed at 1.0 × 10⁵ Pa since the water depth at the plate was below 1 m in the laboratory experiment.

There are uniform velocity conditions at the nozzle inlet. The inlet velocity and the distance between the nozzle edge and the flat plate, i.e., the standoff distance, were changed as listed in Table 13.1. The inlet velocity and the standoff distance were different in Cases 2–5 while the pressure boundary condition was fixed at 1.0 × 10⁵ Pa. There are no-slip velocity conditions on the solid surfaces of the nozzle and the flat plate.

Table 13.1 Boundary and initial conditions

The initial void fraction was 0.001. The initial condition of the bubble radius was assumed to be 1.0 × 10⁻⁵ m under the pressure of 1.0 × 10⁵ Pa, and the initial bubble radius of 7.0 × 10⁻⁶ m was estimated under the pressure of 3.0 × 10⁵ Pa. The boundary and initial conditions are summarized in Table 13.1.

We carried out the simulation in two stages to reduce the calculation time. First, the flow field was computed using local time stepping. At this stage the bubbles were assumed in quasi-static equilibrium at the local water pressure to avoid instability in the calculation. Next, we started to calculate the bubble behavior in detail, without assuming a quasi-equilibrium condition. The time step in the second stage of the simulation was 0.02 microseconds in the pump simulation, and 0.08 microseconds in the cavitating jet simulation.

3 Results and Discussion

3.1 Centrifugal Pump

3.1.1 Cavitation Performance

Figure 13.4 compares the calculated and experimental cavitation performance of the centrifugal pump under the flow rate condition Q/Q _ηmax  = 0.60. The NPSH (Net Positive Suction Head) and total head were divided by U ² _t /(2g) to make them dimensionless: NPSH’ and $\psi$, respectively. The NPSH was obtained from the total pressure at the outlet of the suction pipe. The total head was equivalent to the total pressure increase between the outlet of the suction pipe and the outlet of the impeller.

Fig. 13.4

Cavitation performance of centrifugal pump (Q/Q _ηmax  = 0.60). Based on [11], reprinted with permission from JSME

The experimental and predicted total heads remained nearly constant when NPSH’ was high, and both decreased when NPSH’ was below a certain value. NPSH _R is the ‘Required NPSH’ at the 3 % drop of total head. While the experimental NPSH _R ’ was 0.094, the predicted NPSH _R ’ was 0.068 having a prediction error of –27 %. The underestimation of NPSH _R was due to an underestimation of the cavitation volume since, for high vapor volume fractions, bubble coalescence and sheet cavitation were not sufficiently modeled in the simulation.

The total head in the experiment included the effect of loss caused by flow in the volute-type casing; however, the flow in the volute-type casing was not calculated in the present simulation. Therefore, the predicted total head exceeded the experimental one, and the predicted total head had an error of +28 %. The predicted cavitation performance was not in quantitative agreement with that in the experiments; however, the pressure and velocity variations caused by cavitation in the impeller were qualitatively simulated from a macroscopic viewpoint.

The discussion in the following sections is based on the numerical results from Case 0 (NPSH _R ’ = 0.073), shown in Fig. 13.4.

3.1.2 Static Pressure, Void Fraction and Bubble Number Density

Figure 13.5 shows the predicted instantaneous distributions of the static pressure of water, the void fraction, and the bubble number density close to the impeller blade. Figure 13.5a shows the static pressure distribution. The average pressure decreased inside the region surrounded by the dotted line on the suction side of the impeller blade near the leading edge (LE). High pressure regions locally occurred as spots in the low-pressure region since the static pressure was affected by the bubble behavior, i.e., the bubble radius and the bubble number density through the fourth term of Eq. (13.3).

Fig. 13.5

Distributions of static pressure of water, bubble number density and void fraction close to impeller blade, a. static pressure, b. bubble number density, c. void fraction. Based on [11], reprinted with permission from JSME

Figure 13.5b shows the distribution of bubble number density. The bubble nuclei are accumulated in the low-pressure region in Fig. 13.5a. Flow separation near the blade surface occurred in the low-pressure region, and the bubble nuclei were trapped there. Bubble residence also appeared near the hub in the low energy flow.

Figure 13.5c shows the distribution of the void fraction. The high void fraction means the occurrence of cavitation. The void fraction depended on the bubble radius and the bubble number density based on Eq. (13.18). The void fraction increased in the same region as the low-pressure region in Fig. 13.5a since bubble expansion was caused by the decrease in the static pressure. The void fraction also increased in the region of bubble nuclei accumulation shown in Fig. 13.5b.

Figure 13.6 compares the distributions of bubble number density and void fraction close to the impeller blade at two different times. Both of these were obtained along the line between the leading edge (LE) and the trailing edge (TE) of the impeller blade shown in Fig. 13.5c. The line was located between the shroud and the mid-span of the blade. The distance, s, from the LE was nondimensionalized by the total line length between the leading and trailing edges (s = 0.0 at LE and s = 1.0 at TE). The bubble number density was obtained as the ratio against the density at the inlet boundary, n _G0 .

Fig. 13.6

Bubble number density and void fraction close to impeller blade along line in Fig. 13.5c. Based on [11], reprinted with permission from JSME

The bubble number density distributions at the different times were nearly same. This result means that the unsteadiness of the bubble number density distribution was not strong. The bubble nuclei accumulated in a range from s = 0.05 to 0.15 at times t ₁ and t ₂ . The void fraction also increased in a range from s = 0.05 to 0.15; however, the distributions were different at the two times. This was because the bubble radius varied as will be explained by Fig. 13.7 in the next section. The high dependence of the void fraction on bubble accumulation and bubble expansion and contraction was numerically confirmed.

Fig. 13.7

Bubble radius and bubble pressure close to impeller blade along line in Fig. 13.5c. Based on [11], reprinted with permission from JSME

3.1.3 Bubble Radius and Bubble Pressure

Figure 13.7 compares the evolutions of the bubble radius and pressure close to the impeller blade at the same times t ₁ and t ₂ as in Fig. 13.6. The bubble radius and pressure were calculated in each cell along the line shown in Fig. 13.5c. The number of cells between the leading edge (LE) and the trailing edge (TE) of the impeller blade was 76. The bubble radius was obtained as the ratio against the initial value at the inlet boundary, r _G0 .

While the bubble radius increased on the whole from the leading edge (s = 0) to location s = 0.1 due to the local reduction of water pressure, it decreased beyond s = 0.1 since pressure was increased by impeller rotation. Figure 13.7 shows the irregular variation of bubble radius between the leading edge and location s = 0.6. This behavior caused the non-uniform distribution in the void fraction visible in Fig. 13.5c. At several locations between s = 0.1 and s = 0.4, the bubble radius locally decreased, and the bubble pressure distribution exhibited simultaneously peaks of large amplitude due to the abrupt contraction of the bubbles. Figure 13.7 shows that location and height of the bubble pressure peaks changed unsteadily. The transient behavior of the bubble radius caused the different distributions of void fraction visible in Fig. 13.6.

3.1.4 Transient Bubble Pressure

Figure 13.8 shows the instantaneous bubble pressure close to the impeller blade. Scattered local high-pressure regions appeared especially between LE and the throat, which is the narrowest location between the impeller blades. These high-pressure spots show the locations of the bubble collapse. The location of the high-pressure regions unsteadily changed. Figure 13.9 plots the transient bubble pressure at a fixed point (s = 0.19) on the line in Fig. 13.5c. The bubble pressure had impulsive peaks. The predicted duration of the peaks was about one microsecond.

Fig. 13.8

Bubble pressure close to impeller blade. Based on [11], reprinted with permission from JSME

Fig. 13.9

Transient bubble pressure at a fixed point (s = 0.19) along line in Fig. 13.5c. Based on [11], reprinted with permission from JSME

The duration of the impulsive force measured using a piezoelectric ceramic sensor attached to the impeller blade surface was actually below one microsecond [19]. In the present study, the breakup of bubbles and the final stage of the bubble collapse were not simulated. However, bubble motion was simulated, which changed on the order of microseconds, in the same manner as the actual phenomenon.

Figure 13.10 plots the bubble pressure distributions along the line in Fig. 13.5c at ten different times. The time-averaged static pressure of water was also shown as the solid line in Fig. 13.10. The bubble pressure had peaks mainly between s = 0.15 and s = 0.5 while it hardly changed and was equivalent to the static pressure in ranges from s = 0.05 to 0.13, and s > 0.6. The bubble pressure became maximum near the location of s = 0.15, and decreased with the increase in s. The maximum value of the bubble pressure was about 1 MPa, which was much smaller than the general bubble collapse pressure of over 10 MPa [20]. The collapse pressure of bubbles increased in turn as a chain reaction toward the center of the bubble cloud. If the dynamics of the cloud were included in the simulation, bubble pressure may reach 1 GPa as calculated in Ref. [21]. The bubble pressure was underestimated mainly because the behavior of the bubble cloud was not modeled in the simulation. A strong unsteady flow with vortices was supposed to occur in the region where the bubbles collapsed in the present model pump under the flow rate condition of Q/Q _ηmax  = 0.60. If the large static pressure fluctuations resulting from the highly unsteady flow were accurately simulated, it is expected that the bubble collapse pressure deduced from Eqs. (13.3) to (13.5) would be larger.

Fig. 13.10

Bubble pressure close to impeller blade along line in Fig. 13.5c. Based on [11], reprinted with permission from JSME

The void fraction increased in a range from s = 0.05 to 0.15, as shown in Fig. 13.6. Unstable bubble behavior appeared just downstream of the cavitating region in Fig. 13.10. The results had a correlation with the experimental data where the impact loads were measured at a location where the cavitation bubble collapsed [19].

3.1.5 Cavitation Intensity

The cavitation intensity, I, was defined by the following equation,

$$ I_{c} = \sum\limits_{\begin{subarray}{l} p_{Bi} \ge p_{{B{\kern 1pt} th}} \\ i = 1,2,3 \ldots \end{subarray} } {{{\frac{{p_{Bi}^{2} n_{G} {\kern 1pt} \varDelta V\varDelta t}}{{2\rho_{L} C}}} \mathord{\left/ {\vphantom {{\frac{{p_{Bi}^{2} n_{G} {\kern 1pt} \varDelta V\varDelta t}}{{2\rho_{L} C}}} {\;\varDelta T}}} \right. \kern-0pt}\!\!\!\!{\:\varDelta T}}}, $$

(13.21)

where p _B is the bubble pressure, n _G is the bubble number density, ΔV is the volume of the numerical cell, $\rho_{L}$ (= 1,000 kg/m³) is the water density, C (= 1,500 m/s) is the sound velocity in water, Δt is the time step, and ΔT is the elapsed time in the simulation. Equation (13.21) was derived from Eq. (13.20). The bubble pressure in Eq. (13.21) represents the bubble collapse pressure, and n _G ΔV means the number of bubbles in each numerical cell.

Material deformation and mass loss only occurs when the impact force applied to the material surface caused by the bubble collapse pressure exceeds a threshold value [17]. A threshold of the bubble pressure, p _Bth , was thus also assumed in Eq. (13.21). Since the appropriate value of p _Bth has not been investigated, p _Bth  = 0.6 MPa or p _Bth  = 0.2 MPa was temporarily assumed based on the numerical results in Figs. 13.9 and 13.10. The cavitation intensity was only estimated when the bubble pressure exceeded the threshold pressure, p _Bth .

Figure 13.11 shows the visualized cavitation erosion area. A blue dye was painted on the suction surface of the impeller blade. After the pump had been operated for four hours, the dye was removed due to cavitation bubble collapse in two different areas. A great deal of plastic deformations in the same areas, i.e., tiny pits, was observed on the aluminum blade surface. Cavitation erosion area A was distributed between LE and the throat of the blade, and between the shroud and mid-span of the blade. The other cavitation erosion area B was distributed between LE and the throat, and between the hub and mid-span of the blade. The dye was more severely peeled off in area A than in area B. The surface roughness caused by the plastic deformations in the area A was also remarkably higher than that in the area B.

Fig. 13.11

Visualized cavitation erosion area in experiment. Based on [11], reprinted with permission from JSME

Figure 13.12 shows the predicted cavitation intensity under different conditions of p _Bth . Figure 13.12a shows the results at p _Bth  = 0.6 MPa. The high cavitation intensity area agreed with area A. The high cavitation intensity area at p _Bth  = 0.2 MPa agreed with area B, as seen in Fig. 13.12b.

Fig. 13.12

Predicted area of high cavitation intensity. a. Cavitation intensity distribution (p _Bth  = 0.6 MPa), b. Cavitation intensity distribution (p _Bth  = 0.2 MPa). Based on [11], reprinted with permission from JSME

The peak value of the bubble pressure in area A was larger than that in area B; however, the bubble number density in area B was larger, as seen in Fig. 13.5b. This means that there is a small number of strong collapses in area A, and in contrast, a large number of weak collapses in area B. The numerical results corresponded qualitatively to the above experimental states for the dye removal and surface roughness.

The cavitation intensity in area A was lower than that in area B. This is because the bubble pressure was not sufficiently high, as explained in Sect. 13.3.1.4, and the bubble number density was dominant in the calculation of Eq. (13.21). The cavitation intensity in area A can be larger than that in area B when a higher bubble pressure is simulated and an appropriate threshold bubble pressure is set up.

Although the threshold of the bubble pressure, p _Bth , should be investigated in detail in future studies, the present simulation and the method of estimating the impact energy and the cavitation intensity effectively predicted cavitation erosion area around the impeller of a centrifugal pump.

3.2 Cavitating Jet

3.2.1 Flow Pattern and Bubble Behavior

The flow pattern and the bubble behavior under the condition of Case 1 in Table 13.1 are explained in this section by using Figs. (13.13)–(13.16). The velocity difference between the liquid and the bubble, i.e., the slip velocity, was not taken into account in this cavitating jet simulation. The flow pattern and the bubble behavior were unsteady; however, Figs. (13.13)–(13.16) show the instantaneous results. The calculation of bubble pressure diverged around the nozzle. We set a quasi-equilibrium region surrounded by the dotted line in Figs. (13.13)–(13.16) to avoid the divergence of calculation. In the numerically unstable flow field, the quasi-equilibrium between the bubble pressure and its surrounding pressure of water was assumed instead of solving Eqs. (13.4)–(13.7).

Fig. 13.13

Velocity and static pressure of water in Case 1, a. velocity, b. pressure. Based on [12], reprinted with permission from ASME

Fig. 13.14

Bubble number density and void fraction in Case 1, a. bubble number density, b. void fraction. Based on [12], reprinted with permission from ASME

Fig. 13.15

Bubble pressure in Case 1, a. bubble pressure, b. time variation in bubble pressure at point A. Based on [12], reprinted with permission from ASME

Fig. 13.16

Cavitation impact energy in Case 1. Based on [12], reprinted with permission from ASME

Figure 13.13 shows the velocity and the static pressure of water in Case 1. The water jet injected from the nozzle impinged on the flat plate, and the water flowed along the plate while rolling up. The unsteadiness was not strong since the local vortex structure formed in the region where the shear stress is high was not sufficiently resolved in this simulation. The static pressure varied with a strong unsteadiness affected by the bubble behavior through the fourth term of Eq. (13.3). This result was similar with Fig. 13.5a.

Figure 13.14 shows the bubble number density and the void fraction. The bubble nuclei were distributed in the jet and near the flat plate within a radial range from the jet center axis. The void fraction depends on the distribution of the bubble number density and the bubble radius, as shown in Eq. (13.18). The void fraction increased in the jet, which means that a cavitating jet was generated.

Figure 13.15 shows the bubble pressure abruptly fluctuated in and around the main flow. Figure 13.15b shows the transient bubble pressure near the flat plate at point A in Fig. 13.15a. The bubble pressure impulsively increased in about 5 microseconds during the bubble collapse.

Sato et al. observed bubble cloud behavior in a cavitating jet impinging vertically on a wall, which was captured with a high-speed video camera [22]. The measured collapse time of the bubble cloud near the wall surface was about 50–150 microseconds, which was analyzed from successive image frames. The predicted time for the bubbles to shrink was shorter than the measured cloud collapse time since (i) we simulated isolated bubble behavior, and (ii) the jet velocity and the surrounding water pressure were higher than those in the experiment conducted by Sato et al. However, the time scale of microseconds for the bubble pressure fluctuations in the simulation corresponded with the experimental data.

The duration of the impulsive peak in Fig. 13.9 was shorter than that in Fig. 13.15b since the expansion and contraction of the bubble became quicker due to the pressure gradient caused by the impeller rotation.

Figure 13.16 shows the density of the cavitation impact energy obtained from Eq. (13.20), i.e., E/ΔV where ΔV is the volume of numerical cell, and k is assumed to be 1.0. The cavitation energy was high in the region (indicated by arrow B) around the jet center axis and in the other peripheral region (indicated by arrow C) away from the jet center axis near the plate surface. When the cavitating jet impinged vertically on the flat plate, the occurrence of ring-like erosion has been observed on the material surface (e.g., in Ref. [22]). The peripheral region indicated by arrow C has a correlation with the ring-like erosion.

3.2.2 Cavitation Impact Energy and Compressive Residual Stress

The correlation between the estimated cavitation impact energy and the measured compressive residual stress on the flat plate after WJP was investigated in Cases 2–5. The inlet velocity and the standoff distance were changed in Cases 2–5 while the pressure boundary condition was fixed at 1.0 × 10⁵ Pa. We used a stainless steel plate, on which tensile residual stress was introduced by grinding before WJP. The X-ray residual stress measurement was conducted on the flat plate before and after WJP.

Although the bubble does not release the impact energy only in one direction at the collapse, the impact energy that acted vertically on the plate surface was estimated in the present study as follows:

$$ E_{surf} = \alpha \frac{E}{\varDelta S}\left( {\frac{{T_{\exp } }}{\varDelta T}} \right) = \sum\limits_{j = 1,2,3 \ldots } {\sum\limits_{\begin{subarray}{l} p_{Bi} \ge p_{Bth} \\ i = 1,2,3 \ldots \end{subarray} } {\alpha {\kern 1pt} (p_{Bi}^{2} n_{G} \varDelta z)_{j} } \left( {\frac{{T_{\exp } }}{\varDelta T}} \right)}, $$

(13.22)

where E is the impact energy defined in Eq. (13.20) where k is assumed to be 1.0, p _B is the bubble pressure, n _G is the bubble number density, ΔT is the elapsed time in the simulation, and T _exp is the jet injection time. T _exp was 2 min without moving the nozzle in the experiment. ΔS is the cross-section of each numerical cell parallel to the plate surface, and Δz is the height of each cell (Δz = ΔV/ΔS). Equation (13.22) gives the energy per unit cross-section, which was summed up by j in the direction of the jet center axis, i.e., in the z direction since the cartesian mesh in the r–z plane was used. The $\alpha$ is a damping coefficient of the energy released at the bubble collapse, which is transferred as the pressure wave [23]:

$$ \alpha = e^{{ - \left( \frac{4}{3} \right)\left( {\frac{{f^{2} }}{{\rho {\kern 1pt} C'^{3} }}} \right)\mu {\kern 1pt} d_{j} }}, $$

(13.23)

where f is the pressure wave frequency, C’ is the sonic velocity, i.e., the velocity of a pressure wave in water including bubbles, and d is the vertical distance between the bubble collapse location and the plate surface. When the pressure wave is transferred with high frequency and low speed in a media with small density and high viscosity, the energy damping is large. The energy damping also depends on the distance from the collapse location. Equation (13.23) was adopted since the collapse pressure propagation from the bubble was not numerically simulated in the present study. f = 1.0 × 10⁶ Hz and C’ = 30 m/s were temporarily assumed; then, the cavitation impact energy was damped to about 10 % when d was 50 mm.

The adding up in Eq. (13.22) was conducted when the bubble pressure exceeded a threshold value since only high impact energy caused plastic deformation of the material surface [17]. The bubble pressure threshold was temporarily assumed to be 1.02 × 10⁵ Pa in the simulation.

Figure 13.17 compares the cavitation impact energy calculated from Eq. (13.22) and the measured residual stress in Cases 2–5. The radial location, r, and the residual stress, σ, were simultaneously nondimensionalized by the nozzle radius, r _n, and the absolute value of the minimum residual stress in Case 2, |σ _2min |. Compressible residual stress was introduced into the stainless steel plate by WJP, and this was distributed within a radial range from the jet center axis. The distribution of compressible residual stress had a peak, indicated by arrow D, away from the jet center axis. The peak indicates the strong impacts caused by cavitation bubble collapses and the correlation with ring-like erosion (e.g., in Ref. [22]). The other peak indicated by arrow E was caused not only by bubble collapse impacts but also by the impingement of water. No peak appeared in Case 5 since the jet velocity was too low.

Fig. 13.17

Comparison between predicted cavitation impact energy (red symbols) and measured residual stress (black symbols) in Cases 2–5. Based on [12], reprinted with permission from ASME

The distribution of impact energy also had a peak indicated by arrow F similar to that indicated by arrow D. Figure 13.18 compares the radial locations and the values of peaks indicated by arrows D and F in Fig. 13.17. The radial peak location of compressive residual stress in Case 5 was closer to the jet center axis compared with the other Cases 2–4. The peak value of compressive residual stress was the lowest in Case 4. The cavitation impact energy also had similar tendencies.

Fig. 13.18

Radial location and value of distribution peak of compressive residual stress and cavitation impact energy in Fig. 13.17. Based on [12], reprinted with permission from ASME

The radial range of compressive residual stress from the jet center axis, R _exp , is one of the most important measures of performance of WJP. WJP can cover the wider area of the weld surface of structures and is more efficient when R _exp is larger. R _exp ranged from 1.5 to 3.7, as shown in Fig. 13.17. Figure 13.19 compares R _exp and R _cal . R _cal is the radial range of the cavitation impact energy in Fig. 13.17, and ranged from 1.6 to 3.0. R _exp and R _cal decreased when the standoff distance was shorter in Cases 2 and 3. R _exp and R _cal also decreased when the inlet velocity was decreased in Cases 3–5. R _cal corresponded to R _exp with a prediction error of ± 20 % in Cases 2–5.

Fig. 13.19

Radial range of compressive residual stress and cavitation impact energy in Fig. 13.17. Based on [12], reprinted with permission from ASME

The simulation however could not predict the values of peaks indicated by arrows E in Fig. 13.17. The cavitation impact energy was exceedingly overestimated when the radial location was below about 1.4 in Cases 2–4 or below 0.6 in Case 5. The main cause is the following; the static pressure of water around the flow stagnation point on the flat plate and the jet center axis was much too high since turbulent flow diffusion was not taken into account in the simulation. The overestimated water pressure surrounding the bubble enormously increased the bubble pressure through Eqs. (13.3)–(13.7).

The above results demonstrated that the numerical method we developed for predicting the region of compressive residual stress after WJP was valid except near the jet center axis.

4 Conclusion

The cavitating flow simulation with the bubble flow model was applied to a centrifugal pump and a cavitating jet that impinged vertically on a flat plate for predicting the erosion area and the residual stress improvement. The impulsive bubble pressure having a peak duration on the order of a microsecond and the distribution of bubble number density were simulated. We proposed a method to estimate the cavitation impact energy based on the computation of the product of the square of the bubble pressure and the bubble number density, i.e., p ² _B n _G .

The distribution of the cavitation intensity derived from the impact energy was compared with the actual erosion area in the centrifugal pump. Computed high cavitation intensity areas correspond to the eroded areas identified experimentally from both dye removal tests and the distribution of plastic deformation pits.

Regarding the cavitating jet, the distribution of the cavitation impact energy that acted vertically on the plate surface was compared with the measured compressive residual stress on the stainless steel plate surface after WJP. The radial range of the cavitation impact energy from the jet center axis corresponded with that of compressive residual stress with a prediction error of ±20 %.

To conclude, the proposed technique of computation of the cavitation impact energy accurately reflects the distributions of erosion damage and residual stress, which are both closely related to plastic deformation of the material.