# An All-Mach Number HLLC-Based Scheme for Multi-Phase Flow with Surface Tension
## 4 Numerical Test Cases
### 2-D Problems
#### 4.2.6 Rayleigh-Plesset Collapse Problem  
A popular test-case [12; 40; 57] to evaluate the effect of non-linear terms in multiphase compressible flow is the so called Rayleigh-Plesset bubble expansion-contraction problem [58]. Here, we consider a cylindrical bubble (2-D) in an incompressible liquid. A gas bubble of radius \(R_{0}\) at initial pressure \(p_{b,0}\) expands and contracts in an incompressible liquid due to a sudden change in the surrounding pressure. The evolution of the radius is governed by the Rayleigh-Plesset model [59]. The initial pressure field is  
Figure 16: Oscillating bubble: relative error on time period of oscillation.  
Figure 15: Oscillating bubble: computed kinetic energy evolution.  
\[p(r,0)=p_{b,0}-\frac{\sigma}{R_{0}}-\frac{\ln\left(\frac{r}{R_{0}}\right)}{\ln \left(\frac{\xi}{R_{0}}\right)}\left[p_{b,0}-\frac{\sigma}{R_{0}}-p_{S,0}\right]. \tag{45}\]  
where \(R\), \(\dot{R}\), \(\ddot{R}\) are the bubble radius, interface velocity and acceleration of the bubble, respectively. Further, \(p_{S,0}\) is the initial pressure at a finite distance \(S\).  
For the purpose of simulating the problem, a bubble of radius \(R_{0}=2\)\(\upmu\)m is first initialised at the centre of a square domain of size \([-5,5]\)\(\upmu\)m and a Cartesian mesh is employed. The driving pressure function at the outer boundary is expressed as  
\[p_{S}(t)=200(1+0.1\sin(10\omega_{c}t)), \tag{46}\]  
where \(\omega_{c}=10208967.75\) s\({}^{-1}\) as per [59; 60]. The material properties are as per [PERSON] et al. [12]:  
\[(\rho_{1},p_{b},\gamma_{1},p_{\infty_{1}}) =\left(10^{-3},100,1.4,0.0\right),\] \[(\rho_{2},p_{2},\gamma_{2},p_{\infty_{2}}) =\left(1.0,200,7.14,3\times 10^{4}\right),\]  
with \(\sigma=0.1\). At each iteration, a Dirichlet pressure boundary condition (pulse) is set at the outer boundary as per (46). The simulation is run at a CFL number of 0.5 for 0.5 s.  
Figure 17 illustrates a time lag on the numerical solution with an overshoot on rebound, which is invariant with respect to mesh refinement. This \"incorrect\" solution is due to the use of a finite Cartesian domain where such effects were first seen in [40]. To demonstrate this and also to show the applicability of the scheme developed in this paper to non-orthogonal grids, the calculation is repeated using a curvilinear mesh of radius 5 \(\upmu\)m where elements are grown in size outside of the bubble. Since height functions are limited to structured grids, the convolution method is used for the curvature calculation.  
Figure 17: Rayleigh–Plesset problem (Cartesian mesh): predicted bubble radius evolution.  
Figure 18 illustrates that the numerical solution now asymptotes to the analytical, correcting the time lag and overshoot in Figure 17. In Table 2, the relative error on the minimum radius of collapse, the time period and the amplitude of rebound are recorded. Here, a maximum error of 4.9% is noted on the minimum radius on the coarsest mesh and 1.2% on the finest mesh. Satisfactory convergence with the analytical solution is obtained on the finest mesh (\(\frac{R}{\Delta x}=80\)). Figure 19 illustrates the evolution of the bubble at the start, mid-way and end time of the simulation. As shown, the CICSAM algorithm allows for sharp capturing of the interface retaining the geometric integrity of the bubble. This test-case demonstrates that the scheme can accurately predict the non-linear collapse of a poly-tropic gas bubble.  
Figure 19: Rayleigh–Plesset problem: evolution of volume fraction field.