General framework
This line of research intends at solving multiphase reacting flows problems based on hyperbolic methods. The idea is to solve mass, momentum and energy conservation equations as
\begin{alignat}{4}
\frac{\partial \rho}{\partial t} &+ div( \rho \mathbf{u}) & = 0 \nonumber\\
\frac{\partial \rho \mathbf{u}}{\partial t} &+ div\left(\rho \mathbf{u}\otimes\textbf{u} + p\underline{\underline{I}} \right) &= 0 \nonumber\\
\frac{\partial \rho E}{\partial t} &+div\left((\rho E + p)\mathbf{u}\right) &= 0 \nonumber\\
\frac{\partial \rho Y_k}{\partial t} &+ div( \rho Y_k \mathbf{u} ), &= 0
\end{alignat}
where $\rho$ is the mixture density, $\mathbf{u}$ represents the centre of mass velocity, $p$ denotes the pressure, $E$ the mixture total energy ($E=e+u^2/2$), and $Y_k$ is the mass fraction of component $k$. The mixture internal energy is defined as $e=\sum Y_k e_k$. In the system of equations, diffusion, viscous and chemical source terms have been omitted for the sake of readability.
Note that, appended with our specific thermodynamic closure model, $Y_k$ may represent a liquid component or a vapor constituent within a multicomponent perfect gas phase.
The above System of equations may be resolved through a conventional hyperbolic solver (e.g. with a HLLC solver for the fluxes), and further appended with additional physical terms through appropriate splitting. In particular, diffusion fluxes, viscous stresses, phase transition or even mass transfer within the gas phase through chemical reactions (combustion) can be tackled by the approach.
A computational example obtained with our method is provided below, presenting a 3D simulation of a rocket engine ignition. 6 coaxial injectors (liquid oxygen/gaseous hydrogen) are being ignited by the presence of a central high pressure $\rm{H_2-O_2}$ igniter.