# An All-Mach Number HLLC-Based Scheme for Multi-Phase Flow with Surface Tension
## Appendix C Derivation of Star-State  
Consider the general formulation for the HLLC solver:  
\[\mathbf{F}_{s_{k}}=\mathbf{F}_{k}+s_{k}\big{(}\mathbf{U}_{s_{k}}-\mathbf{U}_{k}\big{)}. \tag{12}\]By substituting for known quantities \(\mathbf{U}_{k}\) and \(\mathbf{F}_{k}\), an expression for the star-state region is obtained. First, for the continuity equation, this reads  
\[\rho_{*k}\mathbf{u}_{*_{k}}\cdot\mathbf{n}=\rho_{k}\mathbf{u}_{k}\cdot\mathbf{n}+s _{k}\big{(}\rho_{*k}-\rho_{k}\big{)},\] \[\rho_{*_{k}}s_{*}=\rho_{k}\mathbf{u}_{k}\cdot\mathbf{n}+s_{k}\big{(}\rho_ {*k}-\rho_{k}\big{)}.\]  
Hence,  
\[\rho_{*_{k}}(s_{k}-s_{*}) =\rho_{k}(s_{k}-\mathbf{u}_{k}\cdot\mathbf{n}), \tag{111}\] \[\rho_{*_{k}} =\frac{s_{k}-\mathbf{u}_{k}\cdot\mathbf{n}}{s_{k}-s_{*}}\rho_{k}, \tag{112}\]  
and for the momentum equation, this is expanded as follows:  
\[\rho_{*_{k}}\mathbf{u}_{*_{k}}\mathbf{u}_{*_{k}}\cdot\mathbf{n}+p_{*_{k}}\bm {n} =\rho_{k}\mathbf{u}_{k}\mathbf{u}_{k}\cdot\mathbf{n}+p_{k}\mathbf{n}+s_{k}\big{(} \rho_{*_{k}}\mathbf{u}_{*_{k}}-\rho_{k}\mathbf{u}_{k}\big{)},\] \[p_{*_{k}}\mathbf{n} =\rho_{k}\mathbf{u}_{k}\mathbf{u}_{k}\cdot\mathbf{n}+p_{k}\mathbf{n}+(s_{k}-s_{*} )\rho_{*_{k}}\mathbf{u}_{*_{k}}-s_{k}\rho_{k}\mathbf{u}_{k}, \tag{113}\]  
From consistency Equation (23), the following expression must hold:  
\[\mathbf{u}_{*_{k}}=\mathbf{u}_{*n,k}+\mathbf{u}_{t},\]  
where \(\mathbf{u}_{*n,k}\) and \(\mathbf{u}_{t}\) are the normal and tangential velocities, respectively. Hence,  
\[\mathbf{u}_{*_{k}} =\mathbf{u}_{*n,k}+\mathbf{u}_{t},\] \[\mathbf{u}_{*_{k}} =s_{*}\mathbf{n}+\mathbf{u}_{k}-(\mathbf{u}_{k}\cdot\mathbf{n})\mathbf{n},\] \[\mathbf{u}_{*_{k}} =\mathbf{u}_{k}+(s_{*}-\mathbf{u}_{k}\cdot\mathbf{n})\mathbf{n}. \tag{114}\]  
Substituting Equations (112) and (114) into Equation (113), an expression for the intermediate pressure, \(p_{*_{k}}\), is derived as follows,  
\[p_{*_{k}}=p_{k}+\rho_{k}(s_{k}-\mathbf{u}_{k}\cdot\mathbf{n})(s_{*}-\mathbf{u}_{k}\cdot \mathbf{n}) \tag{115}\]  
Substituting Equation (115) into Equation (113), an expression for the intermediate momentum is obtained:  
\[\rho_{*_{k}}\mathbf{u}_{*_{k}}=\frac{s_{k}-\mathbf{u}_{k}\cdot\mathbf{n}}{s_{k}-s_{*}}[ \rho_{k}[\mathbf{u}_{k}+(s_{*}-\mathbf{u}_{k}\cdot\mathbf{n})\mathbf{n}]] \tag{116}\]  
Similarly, for the energy equation, the HLLC flux is written as  
\[\rho_{*_{k}}E_{*_{k}}\mathbf{u}_{*_{k}}\cdot\mathbf{n}+p_{*_{k}}\mathbf{u}_{*_{k}}\cdot \mathbf{n}=\rho_{k}E_{k}\mathbf{u}_{k}\cdot\mathbf{n}+s_{k}\big{(}\rho_{*_{k}}E_{*_{k}}- \rho_{k}E_{k}\big{)} \tag{117}\]  
Substituting the consistency Equations (23) and (115) into Equation (117) yields the expression for the intermediate star-state energy term as  
\[\rho_{*_{k}}E_{*_{k}}=\frac{s_{k}-\mathbf{u}_{k}\cdot\mathbf{n}}{s_{k}-s_{*}}\rho_{k }\bigg{[}E_{k}+(s_{*}-\mathbf{u}_{k}\cdot\mathbf{n})\bigg{[}s_{*}+\frac{p_{k}}{\rho_{ k}(s_{k}-\mathbf{u}_{k}\cdot\mathbf{n})}\bigg{]}\bigg{]}. \tag{118}\]  
Finally, using consistency Equation (24), the following expression for the contact wave speed is obtained:  
\[s_{*}=\frac{p_{L}-p_{R}+\rho_{R}\mathbf{u}_{R}\cdot\mathbf{n}(s_{R}-\mathbf{u}_{R}\cdot\bm {n})-\rho_{L}\mathbf{u}_{L}\cdot\mathbf{n}(s_{L}-\mathbf{u}_{L}\cdot\mathbf{n})-\sigma\kappa( \alpha_{L}-\alpha_{R})}{\rho_{R}(s_{R}-\mathbf{u}_{R}\cdot\mathbf{n})-\rho_{L}(s_{L}- \mathbf{u}_{L}\cdot\mathbf{n})} \tag{119}\]  
where \(s_{k}\) is computed as per [PERSON] et al. [49].