Pressure- and velocity-equilibrium model

Mechanical-equilibrium flows are solved in ECOGEN using the pressure-velocity-equilibrium model (previously named Kapila’s model) [KMB+01]. In the particular case of 2 phases involved and without any extra physics (surface tension, viscosity…), this model reads:

\begin{equation} \label{system_PUEq} \left\{ {\begin{array}{*{20}{l}} {\cfrac{{\partial {\alpha _1}}}{{\partial t}} + \mathbf{u} \cdot \nabla {\alpha _1}}&{ = K div( \mathbf{u} ),} \\ {\cfrac{{\partial {\alpha _1}{\rho _1}}}{{\partial t}} + div \left( {{\alpha _1}{\rho _1}\mathbf{u}} \right) } &{ = 0 ,} \\ {\cfrac{{\partial {\alpha _2}{\rho _2}}}{{\partial t}} + div \left( {{\alpha _2}{\rho _2}\mathbf{u}} \right)}&{ = 0 ,} \\ {\cfrac{{\partial \rho \mathbf{u}}}{{\partial t}} + div \left( {\rho \mathbf{u} \otimes \mathbf{u} + p \mathbf{I}} \right)}&{ = \mathbf{0} ,} \\ {\cfrac{{\partial \rho E}}{{\partial t}} + div \left( {\left( {\rho E + p} \right) \mathbf{u}} \right)}&{ = 0 ,} \end{array}} \right.\ \end{equation}

where subscripts \(1\) and \(2\) correspond to one of the two phases, respectively. \(\alpha_k\) and \(\rho_k\) are the volume fraction and density of phase \(k\).

\(\rho = \sum\limits_{k} \alpha_k \rho_k\), \(\mathbf{u}\), \(p\), \(E = e + \cfrac{1}{2} \| \mathbf{u} \|^2\) and \(e = \sum_k \alpha_k \rho_k e_k\) are the mixture density, velocity, pressure, total energy and internal energy, respectively.

The term \(K div (\mathbf{u})\) accounts for the differences in the acoustic behavior of both phases or in other words, for the differences in expansion and compression of each phase in mixture regions. \(K\) is given by:

\begin{equation*} K = \cfrac{\rho _2 s_2^2 - \rho _1 s_1^2}{\cfrac{\rho _2 s_2^2}{\alpha _2} + \cfrac{\rho _1 s_1^2}{\alpha _1}}, \end{equation*}

\(s_k\) being the speed of sound of phase \(k\).

This model is solved thanks to the numerical method presented in [SPB09].

Remark: This model can also be solved thanks to the numerical method presented in [SCazeP+21] (velocity-equilibrium model) with an infinite pressure relaxation.