Equations
- mixture fraction $\xi$
- equation of state
- continuity
- momentum
Variables: $\xi$, $\rho$, $u$, $P$
$$ \begin{align*} \frac{\partial\rho \xi }{\partial t}& = \underbrace{-\nabla\cdot\rho u \xi +\nabla\cdot(\rho D\nabla\xi)}{F} \ \rho &= G(\xi) \ \frac{\partial\rho}{\partial t} &= -\nabla\cdot \rho u \ \frac{\partial\rho u }{\partial t}& = \underbrace{-\nabla\cdot\rho u u -\nabla\cdot\tau}{H} -\nabla P \ \end{align*} $$
Discretize in time, advance $\rho$, $\xi$
$$ \left. \begin{align*} & (\rho\xi)^{n+1} = (\rho\xi)^n + \Delta tF^n \ &\rho^{n+1} = G(\xi^{n+1}) \end{align*} \right}\text{solve for }\rho^{n+1}, \xi^{n+1} $$
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 $\rho^{n+1}, \xi^{n+1}$,
$$ \begin{align*} & (\rho\xi)^{n+1} = (\rho\xi)^n + \Delta tF^n \ &\rho^{n+1} = G(\xi^{n+1}) \end{align*} $$
- 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 $u^{n+1} = (\rho u)^{n+1}/\rho^{n+1}$.