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23.2.3 Time Schemes in Multiphase Flow

In many multiphase applications, the process can vary spatially as well as temporally. In order to accurately model multiphase flow, both higher-order spatial and time discretization schemes are necessary. In addition to the first-order time scheme in FLUENT, the second-order time scheme is available in the Mixture and Eulerian multiphase models, and with the VOF Implicit Scheme.


The second-order time scheme cannot be used with the VOF Explicit Schemes.

The second-order time scheme has been adapted to all the transport equations, including mixture phase momentum equations, energy equations, species transport equations, turbulence models, phase volume fraction equations, the pressure correction equation, and the granular flow model. In multiphase flow, a general transport equation (similar to that of Equation  25.3-15) may be written as

 \frac{\partial(\alpha \rho \phi)}{\partial t} + \nabla \cdot... ...ec V} \phi) = \nabla \cdot \overline{\overline \tau}+ S_{\phi} (23.2-5)

Where $\phi$ is either a mixture (for the mixture model) or a phase variable, $\alpha$ is the phase volume fraction (unity for the mixture equation), $\rho$ is the mixture phase density, $\vec V$ is the mixture or phase velocity (depending on the equations), $\overline{\overline \tau}$ is the diffusion term, and $S_{\phi}$ is the source term.

As a fully implicit scheme, this second-order time-accurate scheme achieves its accuracy by using an Euler backward approximation in time (see Equation  25.3-17). The general transport equation, Equation  23.2-5 is discretized as

$\displaystyle \frac{3(\alpha_{p} \rho_{p} \phi_{p} Vol)^{n+1} - 4(\alpha_{p} \rho_{p} \phi_{p} Vol)^{n} + (\alpha_{p} \rho_{p} \phi_{p})^{n-1}}{2 \Delta t} =$     (23.2-6)
$\displaystyle \sum[A{_{nb}}(\phi{_{nb}} - \phi{_p})]^{n+1} + {S{_U}}^{n+1} - {S{_p}}^{n+1} {\phi{_p}}^{n+1} \nonumber$      

Equation  23.2-6 can be written in simpler form:

 A_p \phi_p = \sum A{_nb}\phi{_nb} + S_{\phi} (23.2-7)


$A_p = \sum {A{_{nb}}}^{n+1} + {S{_p}}^{n+1} + \frac{1.5(\alpha_{p} \rho_{p} Vol)^{n+1}}{\Delta t}$

$S_{\phi} = {S{_U}}^{n+1} + \frac{2(\alpha_{p} \rho_{p} \phi_{p} Vol)^{n} - 0.5(\alpha_{p} \rho_{p} \phi_{p} Vol)^{n-1}}{\Delta t}$

This scheme is easily implemented based on FLUENT's existing first-order Euler scheme. It is unconditionally stable, however, the negative coefficient at the time level $t_{n-1}$, of the three-time level method, may produce oscillatory solutions if the time steps are large.

This problem can be eliminated if a bounded second-order scheme is introduced. However, oscillating solutions are most likely seen in compressible liquid flows. Therefore, in this version of FLUENT, a bounded second-order time scheme has been implemented for compressible liquid flows only. For single phase and multiphase compressible liquid flows, the second-order time scheme is, by default, the bounded scheme.

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