# An All-Mach Number HLLC-Based Scheme for Multi-Phase Flow with Surface Tension
## 4 Numerical Test Cases
### 2-D Problems
#### 4.2.2 Underwater Explosion  
Here, we implement the popular underwater explosion test-case as described in [20; 27] to test the two-phase compressible properties of the scheme. A confined charge is detonated in a rigid tank that is partially filled with water. The domain is rectangular where \((x,y)\in[-2,2]\times[-1.5,1]\) m\({}^{2}\). The free surface separating the air and water layer is located at \(y=0\). A high pressure bubble of a radius \(R_{0}=0.12\) m is centred at \(\mathbf{x}_{c}=(0,-0.3)\) m defining three regions in the domain as  
\[(\rho,p,\gamma,p_{\infty})=\begin{cases}\left(1.225\;\mathrm{kgm}^{-3},1.0132 5\times 10^{5}\;\mathrm{Pa},1.4,0.0\right),&\text{if $y\geq 0$,}\\ \left(1250\;\mathrm{kgm}^{-3},10^{9}\;\mathrm{Pa},1.4,0.0\right),&\text{if $y<0$ \  
Figure 8: Underwater explosion: HLLC (left) vs. HLLC-CICSAM (right). The first, second and third rows depict the evolution of the total energy field, respectively, at 0.4, 1.2 and 2.5 ms.  
Figure 7: Underwater explosion: Schlieren-type images of density field for HLLC-CICSAM.  
Here, we implement the test-case as described by [PERSON] et al. [27]. An R22 gas bubble with a radius of \(R_{0}=25\) mm is centred at \(x_{c}=(225,44.5)\) mm in a shock-tube of size \((x,y)\in[0,445]\times[0,89]\) mm\({}^{2}\). The planar wave is located behind at \(x=275\) mm, defining the pre- and post-shock regions, respectively,  
Figure 10: Shock-bubble interaction: initial setup.  
Figure 9: Underwater explosion: pressure plot across x = 0.  
\[(\rho,u,p,\gamma,p_{\infty})=\begin{cases}\begin{pmatrix}1.686\ \mathrm{kgm}^{-3},(-113.5,0) \mathrm{ms}^{-1},1.59\times 10^{5}\ \mathrm{Pa},1.4,0.0\end{pmatrix},&\text{if }x\leq 275\ \mathrm{mm}\,\\ \begin{pmatrix}3.863\ \mathrm{kgm}^{-3},(0,0),1.01325\times 10^{5}\ \mathrm{Pa},1.249,0.0 \end{pmatrix},&\text{if }|\mathbf{x}-\mathbf{x}_{c}|\leq R_{0},\\ \begin{pmatrix}1.225\ \mathrm{kgm}^{-3},(0,0),1.01325\times 10^{5}\ \mathrm{Pa},1.4,0.0 \end{pmatrix},&\text{if }x>275\ \mathrm{mm}\cap\quad|\mathbf{x}-\mathbf{x}_{c}|>R_{0}. \end{cases}\end{cases}\end{cases}\]  
The simulation is run on a 3560 \(\times\) 356 node mesh at a CFL of 0.9 for a total time of 1020 us.  
Figures 11 and 12 illustrate the evolution of total energy as a function of time and again HLLC is compared with CICSAM for the VoF equation. For the same adiabatic coefficient, the speed of sound inside the R22 gas bubble is slower compared to the air around. Hence, the wave generated inside the bubble is a rarefaction wave while the incoming wave is a shock wave. The interaction of these two waves produces two outgoing transmitted waves that get reflected at the boundary and free surfaces. These subsequent waves result in the denser fluid (R22) being decelerated by the lighter fluid (air). These lead to the appearance of Rayleigh-Taylor instabilities as illustrated at \(t=690-1020\ \upmu\)s. For the purpose of comparison with experimental data, the Schlieren-type images are shown in Figure 13.  
Figure 12: Shock-bubble interaction: HLLC **(left)** vs. HLLC-CICSAM **(right)**.  
Figure 13: Shock-bubble interaction: Schlieren-type images of the density field for HLLC-CICSAM.  
Qualitatively, the plots show flow features that are consistent with numerical results illustrated in (Figure 5) in [27] with similar improvements as previously discussed with respect to the VoF and energy field seen when using CICSAM.