Volume Partitioning

When simulating two-way coupled Lagrangian multiphase flow or Dispersed Multiphase (DMP) flow, the actual volume that is occupied by the continuous fluid and the actual area over which fluxes are computed are reduced by the presence of the dispersed phase. For Fluid Film simulations, the available volume for the fluid neighbouring the fluid film is smaller than the cell volume. Similarly, in porous media modeling using the physical velocity formulation, a fraction of the cell volume is occupied by a solid phase, thus reducing the available volume for the fluid. For two-dimensional models, the mesh is assumed to have a unit depth (in SI units).

In the following figure, volume partitioning for a cell volume $V$ can generally be comprised by a sum of fluid volumes $V^F$, porous solid volumes $V^S$, and dispersed phase volumes, here Lagrangian particle, volumes $V^L$, each of which can be comprised of individual phases indicated by the subscript $i$:

The total cell volume is given by:

The available volume fraction for the fluid $\eta$ is then defined as the ratio of volume occupied by all fluid phases to the total cell volume:

The volume fraction for one phase of the fluid ${\alpha }_i$ is defined as the ratio of volume occupied by that phase to the total fluid volume:

where ${\displaystyle \sum _i^{}{\alpha }_i=1}$

In case of volume partitioning, the governing fluid flow equations Eqn. (664) to Eqn. (666) are scaled by using the dimensionless fluid volume fraction ${\alpha }_i$ and the available volume fraction $\eta$:

Continuity Equation

$$\frac{\partial }{\partial t}\underset{V}{\int }{\alpha }_i\eta \rho dV+{\oint }_A{\alpha }_i\eta \rho v\cdot da=\underset{V}{\int }{S}_{u}dV+\underset{V}{\int }{S}_{\mathrm{int}}dV$$

(874)

Momentum Equation

$$\begin{array}{c}\frac{\partial }{\partial t}\underset{V}{\int }{\alpha }_i\eta \rho \mathbf{\text{v}}dV+\underset{A}{\overset{}{\oint }}{\alpha }_i\eta \rho v\otimes v\cdot d\mathbf{\text{a }}=\end{array}-{\oint }_A{\alpha }_i\eta pI\cdot da+{\oint }_A{\alpha }_i\eta T\cdot da+\underset{V}{\int }{\alpha }_i\eta f_{b}dV+\underset{V}{\int }{s}_{u}dV+\underset{V}{\int }{s}_{\mathrm{int}}dV$$

(875)

Energy Equation

$$\frac{\partial }{\partial t}\underset{V}{\int }{\alpha }_i\eta \rho EdV+{\oint }_A^{}{\alpha }_i\eta \mathrm{\rho H}v\cdot d\mathbf{\text{a}}=-{\oint }_A^{}{\alpha }_i\eta \mathbf{\text{q}}\cdot d\mathbf{\text{a}}+{\oint }_A^{}{\alpha }_i\eta \mathbf{\text{T}}\mathbf{\cdot }\mathbf{\text{v}}d\mathbf{\text{a}}+\underset{V}{\int }{\alpha }_i\eta {\mathbf{\text{f}}}_b\mathbf{\cdot }\mathbf{\text{v}}dV+\underset{V}{\int }{S}_{u}dV+\underset{V}{\int }{S}_{\mathrm{int}}dV$$

(876)

where $S_u,s_u$ are user source terms and $S_{\mathrm{int}}$ are internal source terms arising from modeled interactions.

If you are solving for species or passive scalars, the corresponding transport equations, Eqn. (1871) and Eqn. (1899), respectively, are also affected by volume partitioning:

Species Transport

$$\frac{\partial }{\partial t}\underset{V}{\int }{\alpha }_i\eta \rho Y_i\mathrm{dV}+\underset{A}{\oint }{\alpha }_i\eta \rho Y_i\mathbf{\text{v}}\cdot da=\underset{A}{\oint }{\alpha }_i\eta [J_i+\frac{{\mu }_t}{{\sigma }_t}\nabla Y_i]\cdot da+\underset{V}{\int }{S}_{Y_i}dV$$

(877)

Passive Scalars Transport

$$\frac{\partial }{\partial t}\underset{V}{\int }{\alpha }_i\mathrm{\eta \rho }{\varphi }_{i}dV+\underset{A}{\oint }{\alpha }_i\mathrm{\eta \rho }{\varphi }_i\mathbf{\text{v}}\cdot da=\underset{A}{\oint }{\alpha }_i\eta j_i\cdot da+\underset{V}{\int }{S}_{{\varphi }_i}dV$$

(878)