Fluctuation of Near-Wall Pressure During the Cavitation Bubble Collapse

Abstract

Cavitation bubble collapse, which generates strong shock waves and high-velocity liquid jets, is responsible for the erosive damage to hydraulic components. In order to assess the fluctuation of near-wall pressure, in this work, an open-source package OpenFOAM is utilized for solving the Navier–Stokes equation. To track the liquid–air interface, the volume of fluid (VoF) method-based compressibleInterFoam solver is selected, and its shipped dynamic contact angle model is modified to obtain better accuracy when considering the wettability of substrates. Numerical methods are first validated by comparing with experiment, and then it is extended to study the effect of bubble diameter, pressure difference, and surface wettability on the fluctuation of near-wall pressure. Simulation results show that the initial sphere bubble goes through three stages of growth, shrinkage, and collapse near the wall. A larger bubble size leads to higher impact pressure due to the higher speed of the liquid jet. The difference in initial pressure in and out of the bubble has a great effect on the collapse behaviour. In addition, a hydrophobic surface, meaning hard liquid pining, can speed up the damping of near wall pressure. The findings in this work will be a guide to designing hydraulic components for limiting the erosive damages of cavitation bubble collapse.

1 Introduction

Cavitation bubble forms as the instantaneous pressure in the hydraulic valve falling below the vapor pressure. The collapse of those bubbles often generates shock wave and the high-speed jet, which are supposed as the major reasons for mechanical damage [1]. So, it is of great importance to understand the physics of bubble collapse and explore the method for limiting the cavitation erosion.

Cavitation involves complex two-phase flow process, including formation, growth, shrinkage and collapse of bubbles in liquid field. Cavitation bubble collapses when it closes to a rigid surface, causing considerable interface and pressure fluctuation. Philipp and Lauterborn [2] found that the velocity of high-speed jet in the centre of toroidal bubbles is about 100 m/s for millimetre bubbles, which can be significantly reduced by increasing the bubble-wall distance. By an ultra-fast high-speed camera, Lauterborn and Vogel [3] obtained breakdown shock waves with velocities of up to 5000 m/s missioned by a laser-induced bubble, and the pressure around the bubble is up to 10 GPa. Kadivar [4] experimentally demonstrated that a shark skin inspired micro structured riblet surface can passively control the cavitation-induced erosion created by collapse of single bubbles. However, the physics behind has not been fully studied due to the complexity and instantaneity of collapse. In generally, it is extremely hard to explore the physics on the complex flow field by only experiments, especially for the final sharp change stage of collapse. Hence, numerical simulation should be conducted to investigate the complex process of cavitation bubble collapse.

By using Lattice Boltzmann Method (LBM), Liu and Peng [1] simulated the collapse process of cavitation bubbles in three typical boundaries, namely, an infinite domain, a straight wall and a convex wall. They found that no second collapse happened if the distance between the wall and the bubble is relatively long. Also, the distance between bubble and wall has influence on the temperature inside the bubble. Other LBM simulation results can be fund in papers [5,6,7,8,9]. Since Computational Fluid Dynamics (CFD) with the volume of fluid (VoF) model generally provides higher spatial and temporal resolution compared to LBM, Koch et al. [10] conducted numerical study using OpenFOAM for the dynamics of a laser-induced bubble on the top of a solid cylinder. The result illustrated that the bubble collapses in the shape of a mushroom. While numerous studies have investigated the deformation behavior of collapsing bubbles, there has been limited research on the pressure fluctuation on the wall caused by the collapsing process.

In this work, therefore, our purpose is to investigate the dynamics of near-wall pressure under the influence of initial bubble diameter, pressure difference, and surface wettability. This work is organized as follows: The methodology utilized is presented in Sect. 2, in which are governing equations, interface tracking method, and dynamic contact angle model. In Sect. 3, we show the problem statement to make the simulation system and cases clear. Finally, the simulation results are given including the process of bubble deformation, bubble diameter effect, pressure difference effect and surface wettability effect.

2 Methodology

2.1 Governing Equations

Considering the compressible, immiscible, non-isothermal phases in this bubble collapse system, the compressibleInterFoam multiphase solver shipped with OpenFOAM-v2212 is utilized to simulation the compressible two-phase flows. The solve includes the governing equations of continuity, momentum, and energy as,

$$\frac{\partial \rho }{\partial t}+\nabla \left(\rho {\varvec{u}}\right)=0$$

(1)

$$\frac{\partial \rho {\varvec{u}}}{\partial t}+\nabla \cdot \left(\rho {\varvec{u}}{\varvec{u}}\right)=-\nabla P+\nabla \cdot \left[\mu \left(\nabla {\varvec{u}}+{\nabla }^{T}{\varvec{u}}\right)\right]+{\varvec{g}}+{F}_{\sigma }$$

(2)

$$\frac{\partial \left(\rho {C}_{p}T\right)}{\partial t}+\nabla \cdot \left(\rho {\varvec{u}}{C}_{p}T\right)=-\nabla \left(k\nabla T\right)+{S}_{T}$$

(3)

where $\rho$ is the density, ${\varvec{u}}$ the velocity, $t$ the time, $P$ the pressure, $\mu$ the dynamic viscosity, ${\varvec{g}}$ the gravity, ${F}_{\sigma }$ the surface tension force that calculated by continuum surface force (CSF) method proposed by Brackbill et al. [11], ${F}_{\sigma }=\sigma \kappa {{\varvec{n}}}_{\alpha }$, $\sigma$ the surface tension force coefficient, $\kappa$ the surface curvature, $\kappa =-\left(\nabla \cdot {{\varvec{n}}}_{\alpha }\right)$, ${{\varvec{n}}}_{\alpha }$ the normal vector, ${{\varvec{n}}}_{\alpha }=\nabla \alpha /\left|\nabla \alpha \right|$, $\alpha$ the volume fraction of liquid in each cell, ${C}_{p}$ the specific heat, $T$ the temperature, $k$ the thermal conductivity, ${S}_{T}$ the additional sources in the energy equation. The present work does not consider the phase change, and the transport properties such as density, dynamic viscosity, and thermal conductivity are regarded as the weighted average of $\alpha$ that can be calculated as below,

$$\rho ={\rho }_{l}\alpha +{\rho }_{v}(1-\alpha )$$

(4)

$$\mu ={\mu }_{l}\alpha +{\mu }_{v}(1-\alpha )$$

(5)

$$k={k}_{l}\alpha +{k}_{v}(1-\alpha )$$

(6)

where the subscript $l$ and $v$ indicate the liquid and vapor properties, respectively.

2.2 Interface Tracking Method

The compressibleInterFoam solver uses the volume of fluid (VoF) method to track the liquid–vapor interface by 0 < $\alpha$ < 1, where $\alpha$ = 1 indicates the liquid property, $\alpha$ = 0 means vapor phase. The transport equation for volume fraction $\alpha$ for a two-phase compressible system is given by,

$$\frac{\partial \alpha }{\partial t}+\nabla \cdot \left(\alpha {\varvec{u}}\right)+\nabla \cdot \left[{{\varvec{u}}}_{r}\alpha \left(1-\alpha \right)\right]=\alpha \left(1-\alpha \right)\left[\frac{{\psi }_{v}}{{\rho }_{v}}-\frac{{\psi }_{l}}{{\rho }_{l}}\right]\frac{Dp}{Dt}$$

(7)

where $\nabla \cdot \left[{{\varvec{u}}}_{r}\alpha \left(1-\alpha \right)\right]$ is the ‘compression term’ to compress the free surface towards a sharper one ranging from 0 to 1, ${{\varvec{u}}}_{r}$ is the velocity at cell face that can be estimated by

$${{\varvec{u}}}_{r}={{\varvec{n}}}_{f}\mathrm{ min}\left[{C}_{\alpha }\frac{\left|\phi \right|}{{{\varvec{S}}}_{f}}, {\text{max}}\left(\frac{\left|\phi \right|}{{{\varvec{S}}}_{f}}\right)\right]$$

(8)

where $\phi$ is face volume flux, ${C}_{\alpha }$ is a user defined value, ${{\varvec{n}}}_{f}$ is the face unit normal flux, ${S}_{f}$ is the face area vector. $\psi$ is the compressibility of the medium and correlated with the sound speed $C$,

$$\psi =1/{C}^{2}$$

(9)

Considering the compressibility, the liquid and air density used in Eq. (4) can be a function of $\psi$,

$${\rho }_{l}={\psi }_{l}\cdot P+{\rho }_{l0}$$

(10)

$${\rho }_{v}={\psi }_{v}\cdot P+{\rho }_{v0}$$

(11)

where ${\rho }_{l0}$ and ${\rho }_{v0}$ are the initial value of liquid and vapor density.

2.3 Dynamic Contact Angle Model

For free liquid–vapor surface, the VoF method is capable to track the interface position. However, the three-phase contact line (TPCL) forms when liquid–vapor interface meets the solid wall. Hence, the contact angle, characterizing the surface wettability, should be imposed into the boundary condition at the TPCL by a notable method proposed by Brackbill et al. [11],

$${{\varvec{n}}}_{\alpha }={{\varvec{n}}}_{\perp }{\text{cos}}\theta +{{\varvec{n}}}_{\parallel }{\text{sin}}\theta$$

(12)

where, ${{\varvec{n}}}_{\perp }$ is the normal vector of the wall surface, and ${{\varvec{n}}}_{\parallel }$ is the parallel vector to the wall surface, and $\theta$ is the contact angle.

In general, contact angle for a dynamic interface is not a constant, whereas it varies and depends on the combined effect of equilibrium contact angle ${\theta }_{e}$, contact line velocity ${u}_{cl}$, and surface roughness. OpenFOAM uses the dynamic contact angle model of Eq. (13),

$${\theta }_{d}={\theta }_{e}+\left({\theta }_{a}-{\theta }_{r}\right)\times {\text{tanh}}\left(\frac{{u}_{cl}}{{u}_{\theta }}\right)$$

(13)

where ${\theta }_{a}$ and ${\theta }_{r}$ are the dynamic advancing and receding contact angles, respectively. ${u}_{cl}$ is the contact line speed near the wall, ${u}_{\theta }$ the dynamic contact angle velocity scale, its value is chosen as 0.1 in this work. In addition, our previous study shows that the original dynamic contact angle model of Eq. (13) cannot limit to ${\theta }_{a}$ and ${\theta }_{r}$ as expected at large and small velocities. Hence, a modified dynamic contact angle of Eq. (14) is implemented to this work,

$${\theta }_{d}=\left\{\begin{array}{c}\left(1-{t}_{u}\right){\theta }_{e}+{t}_{u}{\theta }_{a}, \quad if{ t}_{u}\ge 0\\ \left(1+{t}_{u}\right){\theta }_{e}-{t}_{u}{\theta }_{r}, \quad if{ t}_{u}<0\end{array}\right.$$

(14)

where ${t}_{u}={\text{tanh}}\left(\frac{{u}_{cl}}{{u}_{\theta }}\right)$. Based on our previous validation study, the modified dynamic contact angle model of Eq. (14) with ${u}_{\theta }=0.1$ is utilized in the present work [8].

3 Problem Statement

To explore the physics behind the bubble collapse near a solid wall, a simplified computational system (1/4 of a complete domain) is constructed as shown in Fig. 1, which takes advantage of the symmetric calculation, thus limiting the computational cost. Also, only a quarter of bubble with diameter ${D}_{0}$ locates above the bottom wall with stand-off distance $h$. As for the boundary conditions, left and front patches are symmetry boundary conditions (BCs). Right, back, and top surfaces are set as zero gradient BCs, as well as free slip BCs for the bottom surface, respectively. The initial pressure and temperature in and out of the bubble are ${P}_{v}$, ${T}_{v}$ and ${P}_{l}$, ${T}_{l}$, respectively. A uniform mesh with $\Delta =$ 6.25 $\upmu$ m is used in this system. Thus, there are more than 32 grids along the diameter of bubble, which is fine enough for the two-phase flow simulation. Cases studied in this work are shown in Table 1.

Fig. 1

Schematic of the computational domain for a signal bubble collapse near a wall

Table 1 Cases simulated in this study and their respective conditions

4 Results and Discussions

4.1 Numerical Validation

To validate the ability of compressibleInterFoam solver for bubble cavitation, one case is conducted and the comparison between experiment [12] and simulation is shown in Fig. 2. It is seen that simulation well reproduces the bubble shrinkage behaviour. Moreover, the high-speed liquid jet in the centre of bubble is also obtained in simulation. Therefore, simulation results are in agreement with experiment, and the compressibleInterFoam solver can be extended to the following cases.

Fig. 2

Snapshot comparison between experiment (1st row) and simulation (2nd row)

4.2 Bubble Deformation

Figure 3 shows the snapshots for bubble deformation against time. Arrows in the 1st row indicate the velocity at the interface, and 2nd row gives the pressure contour on the wall. It is clear that the initial small bubble experiences growth, shrinkage, and collapse stages. In the bubble growth stage, due to the larger pressure of vapour phase, the initial bubble grows big rapidly shown from 0.01 to 0.06 ms, when the pressure on the wall is very small. At the bubble shrinkage stage, then, a high-speed liquid jet at 0.08 ms forms at the upper interface and goes through the bubble with velocity larger than 10 m/s. Because of the high-speed impact of liquid-jet, the pressure on the wall increases with the impingement. After that, the bubble starts to touch the solid wall, and spreads around radically. With the collapse of the bubble, the pressure on the wall decreases. The high-pressure point is obtained at the centre of bubble impact area.

Fig. 3

Variations of the velocity at the interface (1st row) and the pressure on the wall (2nd row) with bubble deformation

Former papers demonstrate that the high-speed liquid jet and high-pressure during the bubble collapse are the main reasons leading to mechanical damage [1]. Therefore, in Fig. 4, we plot the average pressure on the wall. It is seen that the pressure goes up to its peak more than 250 kPa at around 0.1 ms, which corresponds to the high-speed jet impact stage around 0.09 ms in Fig. 3. Then, the value of pressure shows the damping behaviour. It happens due to the diffusion of impact energy.

Fig. 4

Variation of the average pressure on the wall against time

4.3 Bubble Diameter Effect

To evaluate the effect of bubble diameter on the wall-pressure, we conducted Case 2 with a smaller diameter of ${D}_{0}=$ 0.16 mm. Compared with the Case 1 with ${D}_{0}=$ 0.2 mm, the dynamics of averaged wall-pressure is plotted in Fig. 5, in which the black and red curves are for bubbles with ${{\text{D}}}_{0}=$ 0.2 mm and ${{\text{D}}}_{0}=$ 0.16 mm in Case 1 and Case 2, respectively. As shown in Fig. 5, larger bubble imposes the wall a larger pressure than that for smaller bubbles. In addition, wall-pressure for the larger bubble exhibits a smaller damping ratio than that for smaller bubbles after 0.2 ms. Interestingly, the frequency of oscillation for smaller bubble is larger than that for larger bubble. This suggests that reducing the bubble diameter may be a more effective method for limiting the wall-pressure, and thus decrease the mechanical damage of high-speed jet.

Fig. 5

Bubble diameter effect on the wall-pressure. The black and red curves are for bubbles with ${D}_{0}=$ 0.2 mm and ${D}_{0}=$ 0.16 mm in Case 1 and Case 2, respectively

4.4 Pressure Effect

Figure 6 shows the damping curves of wall-pressure for bubbles with ${P}_{v}=$ 1e6 Pa and ${P}_{v}=$ 2e5 Pa in Case 1 and Case 3. The purpose of this figure is to explore the pressure difference (${P}_{v}-{P}_{l}$) effect on the wall-pressure. It can be seen that smaller pressure difference generates smaller peak of wall-pressure. Also, smaller pressure difference leads to a larger frequency and damping of oscillation of pressure. This finding suggests that limiting the pressure difference in and out of the bubble can also reduce the mechanical damage of high-speed jet.

Fig. 6

Pressure difference effect on the wall-pressure. The black and red curves are for bubbles with ${P}_{v}=$ 1e6 Pa and ${P}_{v}=$ 2e5 Pa in Case 1 and Case 3, respectively

4.5 Surface Wettability Effect

Surface wettability refers to the water repellent of a solid wall. For hydrophobic surface, meaning the contact angle larger than 90$^\circ$, water cannot pin on but slid away from the substrate. For hydrophilic surface with contact angle smaller than 90$^\circ$, however, water sticks on the surface and moves hardly. Figure 7 gives the schematics of bubble interface at 0.2 and 0.4 ms for Case 1 (${\theta }_{e}=$ 10$^\circ$ in black) and Case 4 (${\theta }_{e}=$ 170$^\circ$ in red), respectively. At 0.2 ms, bubble starts touching the bottom wall, when the shape of bobbles for Case 1 and Case 4 differs due to the wettability difference. Especially at 0.4 ms, the bobble in black is away from the substrate caused by the thin liquid film. However, the bubble pins on the wall on hydrophobic surface.

Fig. 7

Wettability difference effect on the bubble shape. The black and red curves are for bubbles with ${\theta }_{e}=$ 10$^\circ$ and ${\theta }_{e}=$ 10$^\circ$ in Case 1 and Case 4, as well as the dashed lines and solid lines are for times at 0.2 and 0.4 ms, respectively

Figure 8 illustrates the impact of surface wettability on the fluctuation of near-wall pressure. Prior to 0.4 ms, both Case 1 and Case 4 exhibit similar pressure variations due to the influence of the high-speed jet impact. However, at 4 ms, the pressure for the hydrophobic surface reaches a higher peak and subsequently attenuates rapidly to a relatively constant value of approximately 100 kPa after 4 ms. These findings indicate that a hydrophobic surface restricts the fluctuation of near-wall pressure once the bubble comes into contact with the substrate.

Fig. 8

Wettability difference effect on the wall-pressure. The black and red curves are for bubbles with ${\theta }_{e}=$ 10$^\circ$ and ${\theta }_{e}=$ 10$^\circ$ in Case 1 and Case 4, respectively

5 Conclusion

In this study, an improved compressible two-phase flow solver was utilized to simulate bubble cavitation near a rigid surface. The primary focus was to investigate the dynamics of near-wall pressure under varying conditions, including initial bubble diameter, pressure difference, and surface wettability. Simulation results revealed that the near-wall pressure quickly reaches its maximum value upon the impact of the first jet, followed by a subsequent dampening effect. Notably, the highest-pressure point was observed at the centre of the bubble impact area. Furthermore, it was observed that smaller bubble diameters, lower pressure differences, and larger contact angles contribute to a more rapid dampening of the near-wall pressure. These findings hold significant implications for the design of valves aimed at mitigating cavitation collapse erosion. By considering factors such as bubble diameter, pressure difference, and surface wettability, designers can optimize valve designs to effectively control and limit the detrimental effects of cavitation.