Using the general phase scalar, ϕq, q for the qth phase, the generalized equation for phase q has the following form:

where,

and ϕq represents the dependent variables in a multiphase system:

where,

In equation 2.54 the second term on the right side represents the interphase exchanges. Specifically,

By using submodels for interphase species, mass, momentum and heat exchanges, you can derive the complete set of flow governing equations from the transport equations that are generalized here.

where,

The homogeneous multiphase model is a limiting case of the Euler-Euler multiphase flow in which the interphase transfer rate is large. The fundamental assumption is that all the phases share the same pressure field. With this assumption, homogeneous model further results in the simplification of the full inhomogeneous Eulerian multifluid model by assuming that all phases share a common velocity, temperature, and turbulence field. This approach is a good substitute to the full Eulerian multifluid model due to its easy implementation and computational economy. Physically, without the requirement of interphase exchange models in momentum and energy equations, the homogeneous model can perform as well as the full multifluid model in cases such as free surface flows (VOF), cavitation, or other highly mixed multiphase flows.

In Creo Flow Analysis, the current multiphase module adopts only the homogeneous modeling approach. The attention is focused on modeling free surface flow (volume of fluid model) and homogeneous liquid-gas two-phase flows (mixture model). In principle, you can apply the modeling capability for n phase flows.

In equation 2.54, setting ϕq=1, obtains the phase q volume fraction equation:

For the n phase system, a sum of phase volume fractions satisfies the physical constraint:

Or the total mass conservation:

where the mixture quantities are defined as follows:

where the mixture quantities are defined in the following:

In the homogeneous approach, you can model this drift velocity using an algebraic model. However, in the current VOF and mixture model, no-slip between phases is assumed:

Therefore, both the momentum exchange terms are zero.

Without the velocity slip, the energy equation for the mixture takes the following form:

where the mixture variables are defined in the following:

The last term on the right side is the interface heat transfer caused by mass transfer. With the assumption that the phases share the same temperature, Lgp depends on the definition of Hq and Hp in the solved energy equation.

As described in the Heat module, the static enthalpy of a material consists of two parts: standard state reference enthalpy and sensible enthalpy. If you assume that phase-q is liquid, and phase-p is vapor, then the phase total static enthalpies are as follows:

where,

The difference of the reference enthalpies

is the latent heat at the reference temperature of Tref and pressure pref.

In equation 2.70, if the enthalpy H is the total mixture enthalpy, you have

Then the difference due to the phase formation enthalpies or the latent heat Lpq has already been included in the energy equation. The quantity is set to zero:

Lpq=0

And the heat transfer due to mass transfer, the last right-side term in equation 2.70, is zero in the mixture energy equation.

In a CFD solver, the total enthalpy is not solved directly. Instead only the sensible enthalpy relative to the saturation temperature is included in the solved enthalpy and internal energy:

Then Lpq is not zero. It should be the latent heat:

where,

In Creo Flow Analysis the standard state reference enthalpy is automatically considered by default. No user input is required.

In the volume of fluid (VOF) and mixture multiphase models, the effect of turbulence on the mixture of phases is accounted by the extensions of the single-phase turbulence models. The turbulence models and near-wall treatments, described in Turbulence module, are extended to the multiphase flows in Creo Flow Analysis. With the mixture flow quantities, the standard and RNG k–ε models have the same general forms as in the single-phase turbulence models:

while the production of turbulent kinetic energy is calculated based on mixture turbulent viscosity and velocity gradients:

The turbulent viscosity for phase-q may be computed as:

where the first term on the right side is the turbulent diffusion term in phase-q, which has to meet the following constraint so that the total mass conservation is satisfied:

The turbulent diffusion terms are usually implemented as an option. By default, it is not included.

For (n–1) phases, the inlet volume fractions are predetermined, while the nth phase is obtained using the physical constraint:

And the volume fraction on each phase must be nonnegative.

For (n–1) phases, zero-gradient conditions apply for all the outlet, symmetry, and wall boundaries, while the nth phase is obtained using the physical constraint:

The above governing equations, turbulence models, and boundary conditions form the foundation of the homogeneous VOF and mixture multiphase models. Without external or user source terms and interphase mass transfers, they are a closed system of equations and are solved numerically using a pressure-based finite-volume multiphase solver. Many practical applications require specific submodels, such as surface tension force in VOF models and interphase mass transfers, to accurately capture the respective physical phenomena and processes. Instead of lumping the submodels into the external or user sources, it is recommended to include them directly in the built-in models.