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23.5.11 Solution Method in FLUENT

For Eulerian multiphase calculations, FLUENT uses the phase coupled SIMPLE (PC-SIMPLE) algorithm [ 380] for the pressure-velocity coupling. PC-SIMPLE is an extension of the SIMPLE algorithm [ 277] to multiphase flows. The velocities are solved coupled by phases, but in a segregated fashion. The block algebraic multigrid scheme used by the density-based solver described in [ 395] is used to solve a vector equation formed by the velocity components of all phases simultaneously. Then, a pressure correction equation is built based on total volume continuity rather than mass continuity. Pressure and velocities are then corrected so as to satisfy the continuity constraint.

The Pressure-Correction Equation

For incompressible multiphase flow, the pressure-correction equation takes the form

 \sum_{k=1}^n { \frac{1}{\rho_{\rm rk}} \left\{ \frac{\partia... ... - (\sum_{l=1}^n (\dot{m}_{lk}-\dot{m}_{kl})) \right\} } = 0 (23.5-130)

where $\rho_{\rm rk}$ is the phase reference density for the $k^{\rm th}$ phase (defined as the total volume average density of phase $k$), $\vec{v}_{k}'$ is the velocity correction for the $k^{\rm th}$ phase, and $\vec{v}_{k}^*$ is the value of $\vec{v}_{k}$ at the current iteration. The velocity corrections are themselves expressed as functions of the pressure corrections.

Volume Fractions

The volume fractions are obtained from the phase continuity equations. In discretized form, the equation of the $k^{\rm th}$ volume fraction is

 a_{p,k} \alpha_k = \sum_{nb} {(a_{nb,k}\alpha_{nb,k})} + b_k = R_k (23.5-131)

In order to satisfy the condition that all the volume fractions sum to one,

 \sum_{k=1}^n {\alpha_k} = 1 (23.5-132)

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