24.7.1 Coupled Solution for Multiphase Flows

In multiphase flow, the phasic momentum equations, the shared pressure, and the phasic volume fraction equations are highly coupled. Traditionally, these equations have been solved in a segregated fashion using some variation of the SIMPLE algorithm to couple the shared pressure with the momentum equations. This is attained by effectively transforming the total continuity into a shared pressure. The ANSYS FLUENT Phase Coupled SIMPLE algorithm has been successfully implemented and solves a wide range of multiphase flows. However, coupling the linearized system of equations in an implicit manner would offer a more robust alternative to the segregated approach.

One of the fundamental problems is that the resulting matrix is not symmetric and that the continuity constraint may contribute to a zero diagonal block, making the solution difficult to obtain. One way to circumvent this problem is to use direct solvers, but these are too expensive for large industrial cases. In addition, we need to avoid a zero diagonal, resulting from the continuity constraint, and like the segregated solver, we need to construct a pressure correction equation. In multiphase, we also have the additional problem of the vanishing phase, which for the coupled solver is important to ensure some continuity in the coefficients. Like the Phase Coupled, we use a Rhie and Chow type of scheme to calculate volume fluxes and to provide proper coupling between velocity and pressure, thus avoiding unphysical oscillations.

Consider a single-phase system and let us denote the velocity correction components in the three Cartesian directions by $u'$, $v'$, and $w'$ with $p'$ denoting the shared pressure correction. These are discrete variables and can be expressed in the form $p',U'$. The linear system that is generated by the single-phase coupled solver is of the form

For a notation in component form $(p',u',v',w')$

Now let us consider a multiphase system of n-phases and denote the phasic velocity correction components in the three Cartesian directions by $u'_k$, $v'_k$, and $w'_k$ where the subscript $k$ represents the phase notation, $p'$ denotes the shared pressure correction and $\alpha'_k$ denotes the volume fraction correction ( ANSYS FLUENT can solve in both correction form for velocity and volume fraction and noncorrection form). For simplicity the matrix will be shown for two phases. The vector solution is of the form $(p',u'_1,v'_1,w'_1,u'_2,v'_2,w'_2,\alpha'_2)$ or in a shorter notation $(p',U'_1,U'_2,\alpha'_2)$. The linear system would be an extension of the one generated by the coupled solver shown by Equation  24.7-1.

This system can be easily generalized to n phases. The components of this matrix are also matrices.

For large problems we need to resort to iterative solvers. The ANSYS FLUENT AMG Coupled solver with an ILU smoother has proved to be a robust method. Most coupled solvers also need a pseudo stepping method, adding more diagonal dominance to the matrix. Our method here is to use under-relaxation factors for momentum, which is equivalent to time stepping in steady flows. Similar to that of the single phase, we have introduced a steady Courant Number instead of an under-relaxation for velocities. Having this control is important when using second order numerical schemes in the convective terms.

For the sake of simplicity, input parameters for the Multiphase Coupled solver are similar to the single-phase solver. We have the options for solving the whole system including volume fraction, or to treat the volume fraction solution in a segregated manner while preserving the pressure-velocity coupling for all phases.

Multiphase is more implicit than single phase and generally may need more under-relaxation, hence using the same values as single phase may not be ideal.

See Section  24.7.5 for information about applying the various algorithms.