# Morphological Analysis of a Collapsing Cavitation Bubble near a Solid Wall with Complex Geometry
## 2 Methods  
### Pseudopotential MRT-LBM  
The pseudopotential MRT-LBM is adopted to simulate the flow field. By using the MRT collision operator, the evolution of the density distribution function can be expressed as [47]  
\[f_{\alpha}(\mathbf{r}+\mathbf{e}_{\alpha}\delta_{t},t+\delta_{t})=f_{\alpha}( \mathbf{r},t)-\left(\mathbf{M}^{-1}\mathbf{\Lambda}\mathbf{M}\right)_{\alpha \beta}\left(f_{\beta}-f_{\beta}^{eq}\right)+\delta_{t}F^{\prime}_{\alpha} \tag{1}\]  
where \(f_{\alpha}(\mathbf{r},t)\) is the particle density distribution function, \(f_{\beta}^{eq}(\mathbf{r},t)\) is the corresponding equilibrium distribution function, \(\mathbf{r}\) is the particle position, \(\delta_{t}=1\) is the time step, \(\mathbf{e}_{\alpha}\) is the discrete velocity along the \(\alpha\)th direction, \(F^{\prime}_{\alpha}\) is the component of forcing term \(F^{\prime}\) along the \(\alpha\)th direction, \(\mathbf{M}\) is an orthogonal transformation matrix, and \(\mathbf{\Lambda}\) is the relaxation diagonal matrix. For the D2Q9 lattice model, \(\mathbf{e}_{\alpha}\) is defined as [48]  
\[\mathbf{e}_{\alpha}= \begin{cases}(0,0)&\alpha=0\\ \left(\cos\left[(\alpha-1)\frac{\pi}{2}\right],\sin\left[(\alpha-1)\frac{\pi} {2}\right]\right)&\alpha=1,2,3,4\\ \sqrt{2}\left(\cos\left[(\alpha-5)\frac{\pi}{2}+\frac{\pi}{4}\right],\sin \left[(\alpha-5)\frac{\pi}{2}+\frac{\pi}{4}\right]\right)&\alpha=5,6,7,8\end{cases}  \tag{2}\]\(\mathbf{M}\) is given by  
\[\mathbf{M}=\left( \begin{array}{cccccccccccc}1&1&1&1&1&1&1&1&1\\ -4&-1&-1&-1&-1&2&2&2&2\\ 4&-2&-2&-2&-2&1&1&1&1\\ 0&1&0&-1&0&1&-1&-1&1\\ 0&-2&0&2&0&1&-1&-1&1\\ 0&0&1&0&-1&1&1&-1&-1\\ 0&0&-2&0&2&1&1&-1&-1\\ 0&1&-1&1&-1&0&0&0&0\\ 0&0&0&0&1&-1&1&-1\end{array} \right) \tag{3}\]  
and \(\mathbf{\Lambda}\) is expressed as  
\[\mathbf{\Lambda}=diag\left(\tau_{p}^{-1},\tau_{e}^{-1},\tau_{g}^{-1},\tau_{j}^ {-1},\tau_{q}^{-1},\tau_{j}^{-1},\tau_{q}^{-1},\tau_{v}^{-1}\right) \tag{4}\]  
where \(\tau_{p}^{-1}\), \(\tau_{e}^{-1}\), \(\tau_{g}^{-1}\), \(\tau_{g}^{-1}\), \(\tau_{q}^{-1}\), and \(\tau_{v}^{-1}\) are the relaxation time factors for different moments. In the simulation, these relaxation times are selected as \(\tau_{v}^{-1}=\tau_{p}^{-1}=\tau_{j}^{-1}=1.0\), \(\tau_{e}^{-1}=\tau_{g}^{-1}=0.8\), and \(\tau_{q}^{-1}=1.1\). Using \(\mathbf{m}=\mathbf{M}f\) and \(\mathbf{m}^{qq}=\mathbf{M}f^{eq}\), \(f_{\alpha}\) and \(f_{\beta}^{eq}\) can be projected onto the moment space. Equation (1) can be transformed to [49]  
\[\mathbf{m}*=\mathbf{m}-\mathbf{\Lambda}(\mathbf{m}-\mathbf{m}^{eq})+\delta_{t }(\mathbf{I}-0.5\mathbf{\Lambda})\mathbf{S} \tag{5}\]  
where \(\mathbf{m}*=(\mathbf{m}_{0}^{*},\mathbf{m}_{1}^{*},\ldots,\mathbf{m}_{8}^{*})\), \(\mathbf{I}\) is the unit tensor, and \(\mathbf{S}\) is the forcing term in the moment space with \((\mathbf{I}-0.5\mathbf{\Lambda})\mathbf{S}=\mathbf{M}F^{\prime}\). For the D2Q9 lattice, \(\mathbf{m}^{eq}\) can be expressed as  
\[\mathbf{m}^{eq}=\rho\Big{(}1,-2+3\Big{|}v\Big{|}_{2},1-3\Big{|}v\Big{|}_{2},v_ {x},-v_{x},v_{y},-v_{y},v_{x}^{2}-v_{y}^{2},v_{x}v_{y}\Big{)}^{T} \tag{6}\]  
where \(\rho=\sum_{x}f_{\alpha}\) is the macroscopic density, \(\mathbf{v}\) is the macroscopic velocity and \(\left|\mathbf{v}\right|^{2}=v_{x}^{2}+v_{y}^{2}\). Then, the streaming process can be given as  
\[f_{\alpha}(\mathbf{r}+\mathbf{e}_{\alpha}\delta_{t},t+\delta_{t})=f_{\alpha}^ {*}(\mathbf{r},t) \tag{7}\]  
where \(f^{*}=\mathbf{M}^{-1}\mathbf{m}^{*}\). \(\mathbf{v}\) in Equation (6) can be obtained from  
\[v=\frac{\sum_{\alpha}\mathbf{e}_{\alpha}f_{\alpha}+\delta_{t}\mathbf{F}/2}{\rho} \tag{8}\]  
where \(\mathbf{F}=(F_{x},F_{y})\) is the total force acting on the fluid system in the two-dimensional (2D) space. \(\mathbf{F}\) includes the fluid-fluid interactive, fluid-solid interactive, and bulk forces. According to the criteria described in Ref. [50], the influence of gravity can be neglected in the investigation of the cavitation bubble collapse problem.