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

The secondorder time scheme cannot be used with the VOF Explicit Schemes.

The secondorder 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 25.315) may be written as
Where is either a mixture (for the mixture model) or a phase variable, is the phase volume fraction (unity for the mixture equation), is the mixture phase density, is the mixture or phase velocity (depending on the equations), is the diffusion term, and is the source term.
As a fully implicit scheme, this secondorder timeaccurate scheme achieves its accuracy by using an Euler backward approximation in time (see Equation 25.317). The general transport equation, Equation 23.25 is discretized as
Equation
23.26 can be written in simpler form:
(23.27) 
where
This scheme is easily implemented based on FLUENT's existing firstorder Euler scheme. It is unconditionally stable, however, the negative coefficient at the time level , of the threetime level method, may produce oscillatory solutions if the time steps are large.
This problem can be eliminated if a bounded secondorder scheme is introduced. However, oscillating solutions are most likely seen in compressible liquid flows. Therefore, in this version of FLUENT, a bounded secondorder time scheme has been implemented for compressible liquid flows only. For single phase and multiphase compressible liquid flows, the secondorder time scheme is, by default, the bounded scheme.