Loading docs/source/FluidEquations.rst +0 −12 Original line number Diff line number Diff line Here we describe the fluid variables, the governing equations, and the time discretization of the fluid evolution. Fluid Variables =============== Loading Loading @@ -35,12 +32,3 @@ where :math:`\sum_p \beta_p (V_p - U_g)` is the drag term in which :math:`V_p` r Conservation of fluid volume: .. math:: \frac{\partial \varepsilon_g}{\partial t} + \nabla \cdot (\varepsilon_g U_g) = 0 Time Discretization =============== 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 docs/source/FluidTimeDiscretization.rst 0 → 100644 +41 −0 Original line number Diff line number Diff line Time Discretization =============== 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 :math:`U^{MAC}`, the face-centered (staggered) MAC velocity which is used for advection. #. Define an approximation to the new-time state,:math:`(\varepsilon_g \rho_g U)^* = (\varepsilon_g \rho_g U)^n + \Delta t ( -\nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g) + \varepsilon_g \nabla {p_g}^{n-1/2} + \nabla \cdot \tau^n + \sum_{part} \beta_p (V_p - {U_g}^*) + \rho_g g )` #. Project :math:`U^*` by solving :math:`\nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot (\varepsilon_g U)^*` then defining :math:(\varepsilon_g U)^{**} = (\varepsilon_g U)^{*} - \frac{\varepsilon_g}{\rho_g} \nabla \phi and :math:`{p_g}^{n+1/2,*} = {p_g}^{n-1/2} + \phi` In the corrector #. Define an approximation to the new-time state,:math:`(\varepsilon_g \rho_g U)^{***} = (\varepsilon_g \rho_g U)^n + \Delta t ( (-1/2) \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^n -(1/2) \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^{**} + \varepsilon_g \nabla {p_g}^{n+1/2,*} + (1/2) \nabla \cdot \tau^n + (1/2) \nabla \cdot \tau^{**} + \sum_{part} \beta_p (V_p - {U_g}^{**}) + \rho_g g )` #. Project :math:`U^{***}` by solving :math:`\nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot (\varepsilon_g U)^{***}` then defining :math:(\varepsilon_g U)^{n+1} = (\varepsilon_g U)^{***} - \frac{\varepsilon_g}{\rho_g} \nabla \phi and :math:`{p_g}^{n+1/2} = {p_g}^{n-1/2} + \phi` docs/source/Fluids.rst +4 −1 Original line number Diff line number Diff line .. _Chap:Fluids Here we describe the fluid variables, the governing equations, and the time discretization of the fluid evolution. Solving the Fluid Equations =========================== Loading @@ -7,4 +10,4 @@ Solving the Fluid Equations :maxdepth: 1 FluidEquations FluidTimeDiscretization Loading
docs/source/FluidEquations.rst +0 −12 Original line number Diff line number Diff line Here we describe the fluid variables, the governing equations, and the time discretization of the fluid evolution. Fluid Variables =============== Loading Loading @@ -35,12 +32,3 @@ where :math:`\sum_p \beta_p (V_p - U_g)` is the drag term in which :math:`V_p` r Conservation of fluid volume: .. math:: \frac{\partial \varepsilon_g}{\partial t} + \nabla \cdot (\varepsilon_g U_g) = 0 Time Discretization =============== 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
docs/source/FluidTimeDiscretization.rst 0 → 100644 +41 −0 Original line number Diff line number Diff line Time Discretization =============== 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 :math:`U^{MAC}`, the face-centered (staggered) MAC velocity which is used for advection. #. Define an approximation to the new-time state,:math:`(\varepsilon_g \rho_g U)^* = (\varepsilon_g \rho_g U)^n + \Delta t ( -\nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g) + \varepsilon_g \nabla {p_g}^{n-1/2} + \nabla \cdot \tau^n + \sum_{part} \beta_p (V_p - {U_g}^*) + \rho_g g )` #. Project :math:`U^*` by solving :math:`\nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot (\varepsilon_g U)^*` then defining :math:(\varepsilon_g U)^{**} = (\varepsilon_g U)^{*} - \frac{\varepsilon_g}{\rho_g} \nabla \phi and :math:`{p_g}^{n+1/2,*} = {p_g}^{n-1/2} + \phi` In the corrector #. Define an approximation to the new-time state,:math:`(\varepsilon_g \rho_g U)^{***} = (\varepsilon_g \rho_g U)^n + \Delta t ( (-1/2) \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^n -(1/2) \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^{**} + \varepsilon_g \nabla {p_g}^{n+1/2,*} + (1/2) \nabla \cdot \tau^n + (1/2) \nabla \cdot \tau^{**} + \sum_{part} \beta_p (V_p - {U_g}^{**}) + \rho_g g )` #. Project :math:`U^{***}` by solving :math:`\nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot (\varepsilon_g U)^{***}` then defining :math:(\varepsilon_g U)^{n+1} = (\varepsilon_g U)^{***} - \frac{\varepsilon_g}{\rho_g} \nabla \phi and :math:`{p_g}^{n+1/2} = {p_g}^{n-1/2} + \phi`
docs/source/Fluids.rst +4 −1 Original line number Diff line number Diff line .. _Chap:Fluids Here we describe the fluid variables, the governing equations, and the time discretization of the fluid evolution. Solving the Fluid Equations =========================== Loading @@ -7,4 +10,4 @@ Solving the Fluid Equations :maxdepth: 1 FluidEquations FluidTimeDiscretization
