In the Euler-Euler approach, the different phases/components in a multiphase system are assumed to be mathematically interpenetrating continua which share the same flow pressure. Since the physical space (volume) is shared by all the phases, the concept of phase volume fraction is introduced to describe the phase transport. The phase volume fractions are assumed to be continuous functions of space and time and their sum is equal to one. Conservation laws are applied for each phase to derive a set of governing equations, which are closed by theoretical or empirical constitutive relations. Under the Euler-Euler multiphase modelling approach, there are two types of widely-used models: Inhomogeneous and homogeneous models.

 

Inhomogeneous (Eulerian Multi-fluid) model

This model directly solves the governing equations on each phase: the phase momentum, energy, turbulence, species and volume fraction equations. The phase-to-phase interactions, the inter-phase transfers of momentum, mass, species and heat, are modelled by physical sub-models.

Introducing the general phase scalar, , subscript “” for the phase, the generalized governing equation for phase- has the following form:

 

5.251

where , , and are the phase - density, velocity, source term and diffusion coefficient respectively; and is the volume fraction in the phase, and represents the dependent variables in a multiphase system:

5.252

where , , and are the phase velocity components; indicates the phase total enthalpy; is the mass fraction of species “” in the phase; stands for the turbulent kinetic energy; and is the turbulent kinetic energy dissipation rate for models.

 

In equation 5.251, the second term on the right-hand side represents the inter-phase exchanges. Specifically, with subscript “” indicating the phase, and being the number of phases in the multiphase system, characterizes the mass transfer from the phase to the phase; is the mass transfer from the phase to the phase, and represents direct phase exchange of the transporting quantities (momentum, energy, species, etc.).

 

With the introduction of sub-models for interphase species, mass, momentum and heat exchanges, the complete set of flow governing equations can be derived from the above generalized transport equations.

 

Homogeneous Multiphase model

As a simplified, economical alternative to the inhomogeneous model, the homogeneous modelling approach averages the phase governing equations of flow, energy and turbulence to obtain a set of mixture transport equations, while the phase volume fractions are still solved. For the mixture scalar, , the generalized governing equation has the expression:

 

5.253

where the subscript “” indicates the mixture of phases; and all the variables with the subscript “” represent the mixture or phase averaged values, and is the difference between the phase - velocity and the mixture velocity: .

 

The homogeneous multiphase model can be considered as a limiting case of the Euler-Euler multiphase flow in which the interphase transfer rate is very large. On top of the fundamental assumption that all the phases share the same pressure field, the homogeneous model further results in the simplification of the full inhomogeneous Eulerian multi-fluid model by assuming that all phases share a common velocity, temperature and turbulence field. This approach is a good substitute to the full Eulerian multi-fluid 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 multi-fluid model in cases such as free surface flows (VOF), cavitation, or other highly mixed multiphase flows.

 

In Simerics-MP, the current multiphase module only adopts the homogeneous modelling approach. Specifically, the attention is focused on modelling free surface flow (VOF Model) and homogeneous liquid-gas two-phase flows (Mixture model), though in principle, the modelling capability can be applied for -phase flows.

In both the VOF and Mixture Multiphase model, the homogeneous modelling approach is adopted. The transport equation of the volume fraction at each phase can be obtained from equation 5.251, while the governing equations for the mixture momentum and energy are derived using the equation 5.253 and the conservation laws of mass, momentum and energy. The set of the governing equations are presented in this section.

Phase-q Volume Fraction Equation

In equation 5.251, setting =1, we obtain the phase - volume fraction equation:

5.254

where the rate of mass exchange terms, and represent the magnitude of source and sink respectively for the phase - . In an interphase mass transfer process, one of the two terms are usually zero. For example, in an evaporation process, the liquid phase - loses mass, and , while in the vapor phase, and .

 

For an - phase system, the sum of phase volume fractions satisfies the physical constraint:

5.255

Or the total mass conservation:

5.256

where the mixture quantities are defined in the following:

Volume-Averaged Mixture Density:

5.257

 

Mass-Averaged Mixture Velocity:

5.258

 

Mixture Momentum Equation

The momentum equation for the mixture of phases is obtained by summing the individual momentum equations for all the phases in the system. From equation 5.253, by setting , we have:

5.259

 

Where the mixture quantities are defined in the following:

Volume-Averaged Mixture Viscosity:

5.260

The diffusion coefficient in equation 5.259 is computed using mixture dynamic viscosity and turbulent viscosity . The last two terms on the right hand side represent the direct momentum transfer and the mass-transfer induced momentum exchange. They are determined by the phase drift velocities, , defined as:

5.261

In the homogeneous approach, this drift velocity can be modeled 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.

Mixture Energy Equation

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

5.262

Where the mixture variables are defined in the following:

Volume-Averaged Heat Conductivity:

5.263

Mass-Averaged Mixture Energy and Enthalpy:

5.264

In the mixture energy equation 5.262, the viscous heating term, , is computed as in the single phase flow; and is the total external/user heat source.

As for the last term on the right hand side, it is the interface heat transfer caused by mass transfer. With the assumption that the phases share the same temperature, the term depends on the definition of and in the solved energy equation.

As described in the Heat module, the static enthalpy of a material consists two parts: standard state reference enthalpy and sensible enthalpy. Assuming that phase-q and phase-p are liquid and vapor, respectively, we have the phase total static enthalpies as:

	5.265
	5.266

Where and are the reference pressure and temperature; and and are the phase-q and phase-p standard state reference enthalpies, respectively. The difference of the reference enthalpies:

5.267

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

Including Standard Reference Enthalpy:

In equation 5.262, if the enthalpy is the total mixture enthalpy, we have

	5.268
	5.269

 

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

5.270

And the heat transfer due to mass transfer, the last right-hand term in equation 5.262, is zero in the mixture energy equation.

 

Excluding Standard Reference Enthalpy:

In a CFD solver, instead of solving the total enthalpy directly, only the sensible enthalpy relative to the saturation temperature is included in the solved enthalpy and internal energy:

	5.271
	5.272

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

5.273

Where

	5.274
	5.275

In Simerics-MP, by default, the standard state reference enthalpy is automatically considered. No user input is required.

 

Mixture k-ε Turbulence Models

In the VOF and mixture multiphase models, the effect of turbulence on the mixture of phases is accounted for using the extensions of the single-phase turbulence models. In Simerics-MP, the turbulence models and near-wall treatments, described in Turbulence module, are extended to the multiphase flows. With the mixture flow quantities, the standard and RNG models have the same general forms as in the single-phase turbulence models:

	5.276
	5.277

where the mixture density ,velocity and molecular viscosity are computed from the respective phase values using relations in equation 5.257, equation 5.258 and equation 5.260 respectively; and includes both possible external/ user sources and the phase-interaction sources.

The turbulent viscosity for the mixture, is computed directly from the expression

5.278

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

5.279

 

Where is the modulus of the mean mixture rate-of-strain, .

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

5.280

Effect of Turbulent Diffusion

For multiphase turbulent flows, a turbulent disperse force arises from averaging the instantaneous interfacial drag term, which acts like phase diffusion’s. In the inhomogeneous Eulerian multifluid model, the more common approach is to treat this turbulent effect as an additional interphase force, determined by the gradients of phase volume fractions, in phase momentum equations. However, it can also be modelled by directly considering it as a turbulent diffusion term in the phase volume fraction equations. Specifically, by dividing and grouping all the sources as (the sum of interphase mass transfer and external mass sources), we have the following governing equation for phase-q volume fraction in turbulent flows:

5.281

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

5.282

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

In VOF and mixture multiphase models, the boundary conditions for flow and energy equations are the same as those in the single-phase flows, described in Flow and Heat modules. For the phase volume fractions, only fixed values and zero-gradient are applied in the following:

n-Phase Inlet Boundary

For () phases, the inlet volume fractions are pre-determined, while the phase is obtained using the physical constraint:

	5.283
	5.284

And the volume fraction on each phase must be non-negative.

 

Outlet/Symmetry/Wall Boundary

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

	5.285
	5.286

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