Low Mach, variable density

Equations

mixture fraction $\xi$
equation of state
continuity
momentum

Variables: $\xi$, $\rho$, $u$, $P$

$$ \frac{\partial\rho \xi }{\partial t} = \underbrace{-\nabla\cdot\rho u \xi +\nabla\cdot(\rho D\nabla\xi)}_{F}$$$$\rho = G(\xi) $$$$\frac{\partial\rho}{\partial t} = -\nabla\cdot \rho u$$$$\frac{\partial\rho u }{\partial t} = \underbrace{-\nabla\cdot\rho u u -\nabla\cdot\tau}_{H} -\nabla P$$

Discretize in time, advance $\rho$, $\xi$

Solve the following for $\rho^{n+1}$, $\xi^{n+1}$:

$$ (\rho\xi)^{n+1} = (\rho\xi)^n + \Delta tF^n,$$$$\rho^{n+1} = G(\xi^{n+1}).$$

Pressure equation

Discretize in time, continuity, momentum:

$$ \begin{align*} & \rho^{n+1} = \rho^n - \Delta t\nabla\cdot(\rho u)^{n+1} \\ & (\rho u)^{n+1} = (\rho u)^n + \Delta tH^n - \Delta t\nabla P \end{align*} $$

Take the divergence of the momentum equation and insert into the continuity equation and solve for the pressure term:

$$ \nabla^2P = \frac{1}{\Delta t}\nabla\cdot(\rho u)^n + \nabla\cdot H^n + \frac{1}{\Delta t^2}(\rho^{n+1}-\rho^n) $$

Summary

Solve the following two equations for $\rho^{n+1}, \xi^{n+1}$,

$$ (\rho\xi)^{n+1} = (\rho\xi)^n + \Delta tF^n,$$$$\rho^{n+1} = G(\xi^{n+1}).$$

Solve the pressure equation for $P$,

$$ \nabla^2P = \frac{1}{\Delta t}\nabla\cdot(\rho u)^n + \nabla\cdot H^n + \frac{1}{\Delta t^2}(\rho^{n+1}-\rho^n) $$

Advance the momentum equation to get $(\rho u)^{n+1}$
$$ (\rho u)^{n+1} = (\rho u)^n + \Delta tH^n - \Delta t\nabla P $$
- Solve for $u^{n+1} = (\rho u)^{n+1}/\rho^{n+1}$.