Fluid Time Step
In the absence of reactions, we assume that the fluid density is unchanged.
We compute the fluid volume fraction directly from the particle locations.
Thus here we focus on the discretization of the momentum equation
In the predictor
Define \(U^{MAC,n}\), the face-centered (staggered) MAC velocity which is used for advection, using \(U_g^n\)
Define an approximation to the new-time state, \((\varepsilon_g \rho_g U_g)^{\ast}\) by setting
\[\begin{split}(\varepsilon_g \rho_g U_g)^{\ast} &= (\varepsilon_g \rho_g U_g)^n -
\Delta t \left( \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g) + \varepsilon_g \nabla {p_g}^{n-1/2} \right) \\ &+
\Delta t \left( \nabla \cdot \tau^n + \sum_p \beta_p (V_p - {U_g}^{\ast}) + \rho_g \varepsilon_g g \right)\end{split}\]
Project \(U_g^{\ast}\) by solving
\[\nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot \left( \frac{1}{\Delta t}
(\varepsilon_g U_g)^{\ast}+ \frac{\varepsilon_g}{\rho_g} \nabla {p_g}^{n-1/2} \right)\]
then defining
\[U_g^{\ast \ast} = U_g^{\ast} - \frac{\Delta t}{\rho_g} \nabla \phi\]
and
\[{p_g}^{n+1/2, \ast} = \phi\]
In the corrector
Define \(U^{MAC,\ast \ast}\) at the “new” time using \(U_g^{\ast \ast}\)
Define a new approximation to the new-time state, \((\varepsilon_g \rho_g U_g)^{\ast \ast \ast}\) by setting
\[\begin{split}(\varepsilon_g \rho_g U_g)^{\ast \ast \ast} &= (\varepsilon_g \rho_g U_g)^n - \frac{\Delta t}{2} \left( \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^n + \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^{\ast \ast}\right) + \\ &+ \frac{\Delta t}{2} \left( \nabla \cdot \tau^n + \nabla \cdot \tau^{\ast \ast \ast} \right) + \Delta t \left( - \varepsilon_g \nabla {p_g}^{n+1/2,\ast} + \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast}) + \varepsilon_g \rho_g g \right)\end{split}\]
Project \(U_g^{\ast \ast \ast}\) by solving
\[\nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot \left( \frac{1}{\Delta t} (\varepsilon_g U_g)^{\ast \ast \ast} + \frac{\varepsilon_g}{\rho_g} \nabla {p_g}^{n+1/2,\ast} \right)\]
then defining
\[U_g^{n+1} = U_g^{\ast \ast \ast} - \frac{\Delta t}{\rho_g} \nabla \phi\]
and
\[{p_g}^{n+1/2} = \phi\]