Fluid Variables
Variable
Definition
\(\rho_g\)
Fluid density
\(\varepsilon_g\)
Volume fraction of fluid (= 1 if no particles)
\(U_g\)
Fluid velocity
\(\tau\)
Viscous stress tensor
\(g\)
Gravitational acceleration
Fluid Equations
Conservation of fluid mass:
Unlike two-fluid modeling, \(\varepsilon_g = 1 - \varepsilon_s\), is not a solution variable. Rather, \(\varepsilon_s\) is evaluated from the particle field through volume filtering,
where \(A_i\) is a general particle property, \({\mathcal V}_i\) is the particle volume and \(\mathcal G\) is a transfer kernel with compact support–here linear hat. Setting \(A_i = 1\) gives the local void fraction.
Assuming the fluid phase is incompressible, \(\frac{D \rho_g}{Dt} = 0\), the conservation of fluid mass is equivalent to the conservation of fluid volume:
The conservation of fluid momentum is:
where \(M_{sg} = - M_{gs}\) is the generalized interfacial momentum transfer from the solid particles to the fluid-phase. Like \(\varepsilon_s\), \(M_{gs}\) is determined from the L-E transfer kernel by setting \(A_i = F_{gi}\), where \(F_{gi}\) is the force due to the fluid-phase on the ith particle. Following MFiX classic (and many other CFD-DEM codes designed for high density ratio gas-solids flows), only buoyancy (pressure gradient) and steady drag are considered:
where \(\boldsymbol{V}_{ig} = \boldsymbol{V}_i - \boldsymbol{U}_g ( \boldsymbol{X}_i )\) is the velocity of ith particle relative to the fluid-phase (at the particle position \(\boldsymbol{X}_i\)). \(F_{gi}\) is closed by the specification of a drag coefficient, \(C_D\). Currently, MFIX-Exa includes Wen-Yu, Gidaspow and BVK2 drag laws.
In chemical engineering literature, it is common to lump all drag-related terms of \(M_{gs}\) into \(\beta\). With this simplification and some re-arrangement, the fluid momentum takes the more convenient form: