# An All-Mach Number HLLC-Based Scheme for Multi-Phase Flow with Surface Tension
## 3 Numerical Methodology
### Inviscid HLLC-CICSAM Edge Flux  
#### 3.5.1 Numerical Energy Consistency Criteria  
The incorporation of CICSAM into HLLC begins by considering a key difference between the manner in which MUSCL and CICSAM operate. Notably, MUSCL is a monotone up-winding scheme that interpolates left and right states on either side of a face. These are then blended by HLLC based on characteristic wave-speeds to arrive at a discretised face-flux. In contrast, CICSAM employs a blend of up-wind differencing and down-winding to reconstruct a flux based on availability (amount of fluid available in the cell). As a result, CICSAM yields a more accurate discretisation of the \(\alpha\) flux in the VoF equation. As HLLC is more desirable where compressible flow characteristics are key (\(\rho,\mathbf{u},p\) etc. ), a consistent blend with CICSAM is sought. To enable this, face-fluxes are discretised in terms of primitive variables (\(\rho,\mathbf{u},p\)) in this work. This is illustrated. Consider the special case of an interface problem only involving constant pressure and velocity. Here, the momentum and energy equations reduce in 1-D to  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)=0, \tag{25a}\] \[\frac{\partial}{\partial t}(\rho e)+\frac{\partial}{\partial x}(\rho eu)=0. \tag{25b}\]  
The VoF equation is  
\[\frac{\partial\alpha}{\partial t}+\frac{\partial}{\partial x}(\alpha u)=0. \tag{25c}\]  
Since the sharp propagation of the interface is desired, the semi-discrete form of the VoF equation reads  
\[\frac{\partial\alpha}{\partial t}\approx-\frac{1}{\mathcal{V}_{l}}\sum_{f\in \partial\Omega_{l}}\alpha_{f,C}u_{ff}, \tag{26}\]  
where \(\alpha_{f,C}\) denotes the face value associated with CICSAM and \(u_{ff}\) is the face-flux given by Equation (33). If the energy field is reconstructed in terms of conserved variables using the HLLC-MUSCL approach, the thermal equation is discretised as  
\[\frac{\partial}{\partial t}(\rho e)\approx-\frac{1}{\mathcal{V}_{l}}\sum_{f\in \partial\Omega_{l}}(\rho e)_{f,H}u_{ff},\]where subscript \(H\) denotes the internal energy flux obtained from HLLC associated with the MUSCL reconstruction. Introducing the gas stiffened EOS, the above expression then reads  
\[\frac{\partial}{\partial t}\bigg{(}\frac{p+\gamma p_{\infty}}{\gamma-1}\bigg{)} \approx-\frac{1}{\mathcal{V}_{l}}\sum_{f\in\partial\Omega_{l}}\bigg{(}\frac{p+ \gamma p_{\infty}}{\gamma-1}\bigg{)}_{f,H}u_{ff}.\]  
The above equation can be split into two transport sub-equations as  
\[\frac{\partial}{\partial t}\bigg{(}\frac{p}{\gamma-1}\bigg{)} \approx-\frac{1}{\mathcal{V}_{l}}\sum_{f\in\partial\Omega_{l}} \bigg{(}\frac{p}{\gamma-1}\bigg{)}_{f,H}u_{ff}, \tag{27a}\] \[\frac{\partial}{\partial t}\bigg{(}\frac{\gamma p_{\infty}}{\gamma-1}\bigg{)} \approx-\frac{1}{\mathcal{V}_{l}}\sum_{f\in\partial\Omega_{l}} \bigg{(}\frac{\gamma p_{\infty}}{\gamma-1}\bigg{)}_{f,H}u_{ff}. \tag{27b}\]  
Since the pressure is constant and non-zero, Equation (27a) reduces to  
\[\frac{\partial}{\partial t}\bigg{(}\frac{1}{\gamma-1}\bigg{)} \approx-\frac{1}{\mathcal{V}_{l}}\sum_{f\in\partial\Omega_{l}}\bigg{(}\frac{ 1}{\gamma-1}\bigg{)}_{f,H}u_{ff},\]  
which can be further expanded in terms of the volume fraction field as  
\[\frac{\partial}{\partial t}\bigg{(}\frac{\alpha}{\gamma_{1}-1}+\frac{1-\alpha }{\gamma_{2}-1}\bigg{)}\approx-\frac{1}{\mathcal{V}_{l}}\sum_{f\in\partial \Omega_{l}}\bigg{(}\frac{\alpha_{f,H}}{\gamma_{1}-1}+\frac{1-\alpha_{f,H}}{ \gamma_{2}-1}\bigg{)}u_{ff}, \tag{28}\]  
where \(\alpha_{f,H}\) represents the face volume fraction field that is consistent with the HLLC-MUSCL reconstruction of the energy field. Equation (28) can be summarised for a phase \(i\in\{1,2\}\) as  
\[\frac{\partial}{\partial t}(\rho e)+\frac{\partial}{\partial x}(\rho eu)=0. \tag{25b}\]0  
where the same conclusion can be drawn from Equation (27b).