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A Review of the Accuracy of Direct Numerical Simulation Tools for the Simulation of Non-Spherical Bubble Collapses

  • Review Article
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Journal of the Indian Institute of Science Aims and scope

Abstract

Numerical methods for the simulation of cavitation processes have been developed for more than 50 years. The rich variety of physical phenomena triggered by the collapse of a bubble has several applications in medicine and environmental science but requires the development of sophisticated numerical methods able to capture the presence of sharp interfaces between fluids and solid/elastic materials, the generation of shock waves and the development of non-spherical modes. One important challenge faced by numerical methods is the important temporal and scale separation inherent to the process of bubble collapse, where many effects become predominant during very short time lapses around the instant of minimum radius when the simulations are hardly resolved. In this manuscript, we provide a detailed discussion of the parameters controlling the accuracy of direct numerical simulation in general non-spherical cases, where a new theoretical analysis is presented to generalize existing theories on the prediction of the peak pressures reached inside the bubble during the bubble collapse. We show that the ratio between the gridsize and the minimum radius allows us to scale the numerical errors introduced by the numerical method in the estimation of different relevant quantities for a variety of initial conditions.

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Notes

  1. When the numerical solution obtained is shown not to depend on the size of the grid elements.

  2. A method to obtain a numerical solution of incompressible Navier–Stokes equations which imposes the divergence free condition at the discrete level.

  3. This problem, posed by Rayleigh in 1917, consists in determining the temporal evolution of the bubble volume of a bubble collapsing in a liquid bulk by the difference between the bubble and the ambient pressure65

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Acknowledgements

Part of this work was part of the PROBALCAV program supported by The French National Research Agency (ANR) and cofunded by DGA (French Ministry of Defense Procurement Agency) under reference Project ANR-21-ASM1-0005 PROBALCAV. The authors would like to thank Antoine Llor and Stephane Zaleski for insightful discussions.

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Appendix

Appendix

1.1 Energy Conservation Equation for a Bubble in a Liquid

For a gas bubble with arbitrary initial shape inside a liquid bulk (Fig. 7), we can consider a control volume for the bubble and another one for the liquid moving with the local flow velocities. Let \(V_g\) & \(V_l\) be the volume of bubble and liquid control volumes, respectively, \(S_g\) & \(S_l\) represent the surface area enclosing these control volumes and \({\textbf{n}}_g\) & \({\textbf{n}}_l\) shows the unit normal to these surfaces pointing outward from the control volume. The total energy equation for either of the control volumes represented by the subscript \(i \in (l,g)\) is104

$$\begin{aligned} \frac{\text {d}E_{e,i} }{\text {d}t} + \frac{\text {d}E_{k,i}}{\text {d}t}{} & {} = -\int \limits _{S_i} p_i {\textbf{u}}_i \varvec{\cdot } {\textbf{n}}_i \text {d}S + \int _{S_i} \varvec{\tau }_i' {\textbf{n}}\nonumber \\{} & {} \qquad \cdot {\textbf{u}}_i {\textbf {d}}S - \int \limits _{S_i} {\textbf{q}}_i \cdot {\textbf{n}} \text {d}S, \end{aligned}$$
(13)

where the internal energy is defined as

$$\begin{aligned} E_{e,i} = \int \limits _{V_i} \rho _i e_i \text {d}V. \end{aligned}$$

Imposing that (i) the bubble expansion and collapse process is adiabatic and mass transfer effects are negligible, (ii) the effect of body force terms (e.g., gravity) is negligible, and (iii) that the bubble pressure is uniform and well represented by an ideal gas law, the total energy conservation for the bubble can be expressed as

$$\begin{aligned} \frac{p_{g,0} V_{g,0}^{\gamma }}{\gamma - 1} \frac{\text {d}V_g^{1 - \gamma }}{\text {d}t} + \frac{\text {d}E_{k,g}}{\text {d}t}= - p_b \frac{\text {d}V_g}{\text {d}t} - \frac{\text {d}D_g}{\text {d}t}, \end{aligned}$$
(14)

where \(D_g\) captures all viscous and heat transfer processes across the interface, both of which can be eventually considered as energy lost mechanisms. For the liquid phase, Eq. (13) is expressed assuming that the velocity field is well represented by an incompressible velocity field plus a small correction due to liquid compressibility effects that effectively act as a damping mechanism included in \(D_l\)

$$\begin{aligned} \frac{\text {d} E_{k,l}}{\text {d}t}{} & {} = - \int \limits _I \sigma \kappa {\textbf{u}}_I \cdot {\textbf{n}}_I \text {d}S + (p_b - p_\infty )\nonumber \\{} & {} \qquad \frac{\text {d}V_g}{\text {d}t} - \frac{\text {d}D_l}{\text {d}t} - \frac{\text {d}E_{e,l}}{\text {d}t}. \end{aligned}$$
(15)

Adding Eqs. (14) and (15), we readily obtain the total energy evolution equation for the system of a gas bubble in a liquid as

$$\begin{aligned}{} & {} \frac{\text {d}E_{k}}{\text {d}t} + \frac{p_{g,0} V_{g,0}^{\gamma }}{\gamma - 1} \frac{\text {d}V_g^{1 - \gamma }}{\text {d}t} \nonumber \\{} & {} \quad = - \int \limits _I \sigma \kappa {\textbf{u}}_I \cdot {\textbf{n}}_I \text {d}S - p_\infty \frac{\text {d}V_g}{\text {d}t} - \frac{\text {d}D_{\mathrm{{eff}}}}{\text {d}t}, \end{aligned}$$
(16)

where \(E_k = E_{k,l} + E_{k,g}\) and the effective damping of the system is defined including the internal energy of the liquid in order to account for the effective energy lost due to acoustic radiation

$$\begin{aligned} \frac{\text {d}D_\mathrm{eff}}{\text {d}t} = \frac{\text {d}D_g}{\text {d}t} + \frac{\text {d}D_l}{\text {d}t} +\frac{\text {d}E_{e,l}}{\text {d}t}. \end{aligned}$$

If \(p_\infty\) is constant, we can integrate the Eq. (16) in time and obtain

$$\begin{aligned}{} & {} E_{k} + E_s + \frac{p_{g,0} V_{g,0}^{\gamma }}{\gamma - 1} \left( V_g^{1 - \gamma } - V_{g,0}^{1 - \gamma }\right) \nonumber \\{} & {} \quad + p_\infty \left( V_g - V_{g,0}\right) = - D_{\mathrm{{eff}}} + E_0, \end{aligned}$$
(17)

where \(E_0\) is the integration constant which can be computed from the energy at the reference state and

$$\begin{aligned} E_s = \int \limits _t \int \limits _I \sigma \kappa {\textbf{u}}_I \cdot {\textbf{n}}_I \text {d}S \text {d}t \end{aligned}$$

is the surface energy. Using

$$\begin{aligned} \int \limits _I \kappa {\textbf{u}}_I \cdot {\textbf{n}}_I dS = - \int \nabla _s \cdot \varvec{u} \text {d}S + \int \varvec{u} \cdot \varvec{p} \text {d}l \end{aligned}$$

with \(\varvec{p} = \varvec{n} \times \varvec{t}\) and \(\varvec{t}\) the tangent to the surface at the interface contour considered105, it is easy to show that in the case of a bubble that is not in contact with the wall, the surface energy reduces to \(\text {d}E_s = \sigma \text {d}S_I\), where \(S_I\) is the total surface of the interface. Otherwise, the surface energy cannot be explicitly integrated and we need to account for the energy associated with the contact of both phases with the solid wall.

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Saini, M., Prouvost, L., Popinet, S. et al. A Review of the Accuracy of Direct Numerical Simulation Tools for the Simulation of Non-Spherical Bubble Collapses. J Indian Inst Sci (2024). https://doi.org/10.1007/s41745-024-00427-7

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