diff --git a/docs/source/FluidTimeDiscretization.rst b/docs/source/FluidTimeDiscretization.rst index 931275ab86ce6c42423d2eb89f56aeb3f2e01aa8..c538637f21eb7e7e84c50d1e118703cb7558c0b9 100644 --- a/docs/source/FluidTimeDiscretization.rst +++ b/docs/source/FluidTimeDiscretization.rst @@ -1,6 +1,6 @@ Time Discretization -=============== +=================== In the absence of reactions, we assume that the fluid density is unchanged. @@ -38,12 +38,12 @@ In the corrector - Define a new approximation to the new-time state, :math:`(\varepsilon_g \rho_g U)^{\ast \ast \ast}` by setting -.. math:: (\varepsilon_g \rho_g U)^{\ast \ast \ast} = (\varepsilon_g \rho_g U)^n + - \Delta t \left( (-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)^{\ast \ast} - - \varepsilon_g \nabla {p_g}^{n+1/2,\ast} +.. math:: (\varepsilon_g \rho_g U)^{\ast \ast \ast} &= (\varepsilon_g \rho_g U)^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} + \nabla \cdot \tau^n + \nabla \cdot \tau^{\ast \ast} \right) \\ &+ \Delta t \left( - \varepsilon_g \nabla {p_g}^{n+1/2,\ast} + \Pi + \Gamma \right) -.. math:: + (1/2) \nabla \cdot \tau^n + (1/2) \nabla \cdot \tau^{\ast \ast} + - \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast}) + \rho_g g \right) +where the external forcing terms are given by: + + .. math :: \Pi = \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast}) + .. math :: \Gamma = \rho_g g - Project :math:`U^{\ast \ast \ast}` by solving diff --git a/docs/source/Fluids.rst b/docs/source/Fluids.rst index 84294e7a72111ebe5c045eb2badb03dce0e16d4b..5a8a100fc5c6143f537684d08c8c904a9b29cf65 100644 --- a/docs/source/Fluids.rst +++ b/docs/source/Fluids.rst @@ -1,4 +1,4 @@ -.. _Chap:Fluids +.. _Chap:Fluids: Here we describe the fluid variables, the governing equations, and the time discretization of the fluid evolution. diff --git a/docs/source/GettingStarted.rst b/docs/source/GettingStarted.rst index 06d5f6847159b20a3acd05c42acfe92851f629b3..ea78ff5ad5d52dd5f6e5df4336f4cc798e14bf11 100644 --- a/docs/source/GettingStarted.rst +++ b/docs/source/GettingStarted.rst @@ -1,7 +1,7 @@ -.. _Chap:GettingStarted +.. _Chap:GettingStarted: Getting Started -================== +=============== .. toctree:: :maxdepth: 1 diff --git a/docs/source/Introduction.rst b/docs/source/Introduction.rst index 7be9080e10227684e856ea9217eb0583f7d2f976..fd7c5a6abf611f4d4534e3b516e9d05bdda555ed 100644 --- a/docs/source/Introduction.rst +++ b/docs/source/Introduction.rst @@ -1,5 +1,5 @@ MFiX-Exa Introduction -================== +===================== MFiX-Exa is a new massively parallel code for computing multiphase flow in which solid particles interact with the gas surrounding them. diff --git a/docs/source/Particles.rst b/docs/source/Particles.rst index eb37c575d173571e73fb443ab5eb7b3586519a7a..0fa2e7a13bbdf5b81120b5df9da2cb148bcc54c4 100644 --- a/docs/source/Particles.rst +++ b/docs/source/Particles.rst @@ -1,4 +1,4 @@ -.. _Chap:Particles +.. _Chap:Particles: Particles ========= diff --git a/docs/source/RunningTestSuite.rst b/docs/source/RunningTestSuite.rst index f383c2a1f41afd7df2a5fd1c41540db66ccf832f..ea1c2e751227f4152c96e5a1344fc32de3529c7f 100644 --- a/docs/source/RunningTestSuite.rst +++ b/docs/source/RunningTestSuite.rst @@ -1,5 +1,5 @@ Running the MFiX Test Suite -======================= +=========================== MFiX-Exa comes with several tests aimed at evaluating software functionalities. The source files as well as the required input files