# An All-Mach Number HLLC-Based Scheme for Multi-Phase Flow with Surface Tension
## 3 Numerical Methodology
### Inviscid HLLC-CICSAM Edge Flux
#### 3.5.2 Intermediate Star-State  
In this sub-section, the following notation will be employed to differentiate between different scheme variants in discretising a variable \(\phi\) at a face: \(\phi_{k,M}\) will refer to MUSCL, \(\phi_{f,C}\) to CICSAM and \(\phi_{k}\) on its own will define a combination of the two aforementioned operators for an arbitrary side \(k\in\{L,R\}\). The HLLC normal flux may be re-written from Equation (22) as  
\[\mathbf{F}_{f}=\frac{1+\text{sign}(s_{*})}{2}\big{[}\mathbf{F}_{L}+s^{-}(\mathbf{U}_{*,L}- \mathbf{U}_{L})\big{]}+\frac{1-\text{sign}(s_{*})}{2}\big{[}\mathbf{F}_{R}+s^{+}(\mathbf{ U}_{*,R}-\mathbf{U}_{R})\big{]}, \tag{31}\]  
where the flux for an arbitrary side \(k\) is computed via  
\[\mathbf{F}_{k}=\begin{bmatrix}\rho_{k,M}\mathbf{u}_{k,M}\cdot\mathbf{n}\\ \rho_{k,M}\mathbf{u}_{k,M}\mathbf{u}_{k,M}\mathbf{u}_{k,M}\cdot\mathbf{n}+p_{k,M}\mathbf{n}\\ (\rho_{k,M}E_{k}+p_{k,M})\mathbf{u}_{k,M}\cdot\mathbf{n}\\ \alpha_{f,C}\mathbf{u}_{k,M}\cdot\mathbf{n}\end{bmatrix},\]  
and the intermediate star-state is derived as shown in Appendix C as  
\[\mathbf{U}_{*_{k}}=\chi_{k}\begin{bmatrix}\rho_{k,M}\\ \rho_{k,M}\mathbf{u}_{k,M}+\rho_{k,M}(s_{*}-\mathbf{u}_{k,M}\cdot\mathbf{n})\mathbf{n}\\ \rho_{k,M}E_{k}+\rho_{k,M}(s_{*}-\mathbf{u}_{k,M}\cdot\mathbf{n})\Big{(}s_{*}+\frac{p _{k,M}}{\rho_{k,M}(s_{k}-\mathbf{u}_{k,M}\cdot\mathbf{n})}\Big{)}\\ \alpha_{f,C}\end{bmatrix}.\]  
The sonic characteristic, \(\chi_{k}\), is defined as  
\[\chi_{k}=\frac{s_{k}-\mathbf{u}_{k,M}\cdot\mathbf{n}}{s_{k}-s_{*}},\]  
where \(s_{*}\) denotes the contact wave computed as per [1, 48],  
\[\begin{split} s_{*}=\overbrace{\frac{\rho_{L,M}-p_{R,M}+\rho_{R, M}\mathbf{u}_{R,M}\cdot\mathbf{n}(s_{R}-\mathbf{u}_{R,M}\cdot\mathbf{n})-\rho_{L,M}\mathbf{u}_{L,M} \cdot\mathbf{n}(s_{L}-\mathbf{u}_{L,M}\cdot\mathbf{n})}{\rho_{R,M}(s_{R}-\mathbf{u}_{R,M}\cdot \mathbf{n})-\rho_{L,M}(s_{L}-\mathbf{u}_{L,M}\cdot\mathbf{n})}}^{\text{Convective flux term}}\\ -\underbrace{\frac{\sigma\kappa(\alpha_{L,\sigma}-\alpha_{R, \sigma})}{\rho_{R,M}(s_{R}-\mathbf{u}_{R,M}\cdot\mathbf{n})-\rho_{L,M}(s_{L}-\mathbf{u}_{L,M}\cdot\mathbf{n})}}_{\text{Surface tension term}}.\end{split} \tag{32}\]  
The first term results from the 'traditional' HLLC convective wave-speeds. The second term is the result of capillary effects over the two-phase interface where \(\alpha_{k,\sigma}\) denotes the face value of alpha that is consistent with the discretisation of the surface tension source term. This is further expanded on in Section 3.5.4.  
The energy field is discretised as  
\[(\rho E)_{k}=\frac{1}{2}\rho_{k,M}\mathbf{u}_{k,M}\cdot\mathbf{u}_{k,M}+(\rho e)_{k},\]  
where\[\left(\rho e\right)_{k}=p_{k,M}\frac{1}{\gamma-1}\bigg{|}_{f,C}+\frac{\gamma p_{ \infty}}{\gamma-1}\bigg{|}_{f,C}.\]  
The wave-speeds are computed as  
\[s^{-}=\min(0,s_{L})\quad\text{and}\quad s^{+}=\max(0,s_{R}),\]  
where \(s_{L}\) and \(s_{R}\) denote the characteristic left and right wave-speed, respectively. As per [PERSON] et al. [49], the wave-speeds are estimated using the Roe average eigenvalues as  
\[s_{L}=\min(\bar{\mathbf{u}}\cdot\mathbf{n}-\bar{c},\mathbf{u}_{L,M}\cdot\mathbf{n}-c_{L})\quad \text{and}\quad s_{R}=\max(\bar{\mathbf{u}}\cdot\mathbf{n}+\bar{c},\mathbf{u}_{R,M}\cdot \mathbf{n}+c_{R}),\]  
where  
\[c_{k}=\sqrt{\frac{1}{\gamma-1}\big{|}_{f,C}\bigg{(}H_{k}-\frac{1}{2}\mathbf{u}_{k, M}\cdot\mathbf{u}_{k,M}\bigg{)}}.\]  
Further, \(H_{k}\) is the specific stagnation enthalpy computed via  
\[H_{k}=E_{k}+\frac{p_{k,M}}{\rho_{k,M}}.\]  
Finally, \(\bar{\phi}\in\{\hat{u},\hat{H}\}\) is computed using  
\[\bar{\phi}=\frac{\sqrt{\rho_{L}}\phi_{L}+\sqrt{\rho_{R}}\phi_{R}}{\sqrt{\rho_{ L}}+\sqrt{\bar{\rho}_{R}}},\]  
where the nomenclature has previously been defined.