Eulerian Models and Governing Equations

This section explains the governing equations for the volume of fluid (VOF) and mixture multiphase models, modeling of turbulence and boundaries in multiphase flows.

General Multiphase Governing Equations

In the Euler-Euler approach, the different phases or components in a multiphase system are assumed to be mathematically interpenetrating continua which share the same flow pressure. Since the physical space or 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. The Euler-Euler approach has two types of models that are used regularly:

• Inhomogeneous or Eulerian Multifluid Model—Directly solves the governing equations on each phase including the phase of momentum, energy, turbulence, species, and volume fraction equations. The phase-to-phase interactions, the interphase transfers of momentum, mass, species, and heat are modeled by physical submodels.

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

equation 2.54

where,

ρq | q density |

velocity | |

Sϕq | source term |

Tϕq | diffusion coefficient |

αq | volume fraction in the qth phase |

and ϕq represents the dependent variables in a multiphase system:

equation 2.55

where,

Uq, Vq, Wq | phase velocity components |

Hq | phase total enthalpy |

Yqi | mass fraction of species “i” in the qth phase |

k | turbulent kinetic energy |

ε | turbulent kinetic energy dissipation rate for k–ε models |

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

p | pth phase |

n | number of phases in the multiphase system |

mass transfer from the qth phase to the pth phase | |

θpq | direct phase exchange of the transporting quantities including momentum, energy, and species |

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.

• Homogeneous Multiphase Model—Simplified and economical alternative to the inhomogeneous model. The homogeneous modeling 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, ϕm, the generalized governing equation has the following expression:

equation 2.56

where,

m | mixture of phases |

all variables with m | mixture or phase averaged values |

and is the difference between the phase q velocity and the mixture velocity:

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.

VOF and Mixture Multiphase Models

The volume of fluid (VOF) and mixture multiphase models use the homogeneous modeling approach. The transport equation of the volume fraction at each phase is obtained from equation 2.54. The governing equations for the mixture momentum and energy are derived using equation 2.56 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 2.54, setting ϕq=1, obtains the phase q volume fraction equation:

equation 2.57

where the rate of mass exchange terms, and represent the magnitude of source and sink respectively for the phase q. In an interphase mass transfer process, one of the two terms are usually zero. See the following example:

In an evaporation process, the liquid phase q loses mass, and , while in the vapor phase, and .

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

equation 2.58

Or the total mass conservation:

equation 2.59

where the mixture quantities are defined as follows:

◦ Volume-Averaged Mixture Density

equation 2.60

◦ Mass-Averaged Mixture Velocity

equation 2.61

• Mixture Momentum Equation—Obtained by summing the individual momentum equations for all the phases in the system. From equation 2.56 by setting , you have

equation 2.62

where the mixture quantities are defined in the following:

◦ Volume-Averaged Mixture Viscosity:

equation 2.63

The diffusion coefficient Γ in equation 2.62 is computed using mixture dynamic viscosity μ and turbulent viscosity μt. The last two terms on the right side represent the direct momentum transfer and the mass-transfer induced momentum exchange. They are determined by the phase drift velocities, defined as:

equation 2.69

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.

◦ Mixture Energy Equation

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

equation 2.70

where the mixture variables are defined in the following:

▪ Volume-Averaged Heat Conductivity

equation 2.71

▪ Mass-Averaged Mixture Energy and Enthalpy

equation 2.72

In the mixture energy equation 2.70, the viscous heating term, is computed as in the single phase flow; and SE is the total external or user heat source.

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:

equation 2.73

equation 2.74

where,

pref | reference pressure |

Tref | reference temperature |

hq,ref | phase-q standard state reference enthalpies |

hp,ref | phase-p standard state reference enthalpies |

The difference of the reference enthalpies

equation 2.75

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

▪ Including Standard Reference Enthalpy:

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

equation 2.76

equation 2.77

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.

▪ Excluding Standard Reference Enthalpy:

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:

equation 2.79

equation 2.80

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

equation 2.81

where,

equation 2.82

equation 2.83

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

Turbulence Models

• Mixture k-ε Turbulence Models

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:

equation 2.84

equation 2.85

where, the mixture density ρ, velocity and molecular viscosity μ are computed from the respective phase values using relations in equation 2.60, equation 2.61, and equation 2.63 respectively; Sk and Sε includes both possible external and user sources and the phase-interaction sources. The turbulent viscosity for the mixture, μt is calculated directly from the expression:

equation 2.86

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

equation 2.87

where S is the modulus of the mean mixture rate-of-strain,

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

equation 2.87

• Effect of Turbulent Diffusion

For multiphase turbulent flows, a turbulent disperse force arises when you average the instantaneous interfacial drag term, which acts like that of phase diffusion. The inhomogeneous Eulerian multifluid model generally treats this turbulent effect as an additional interphase force, determined by the gradients of phase volume fractions, in phase momentum equations. However, this turbulent effect can also be modeled by directly considering it as a turbulent diffusion term in the phase volume fraction equations. By dividing and grouping all the sources as (the sum of interphase mass transfer and external mass sources), you have the following governing equation for phase-q volume fraction in turbulent flows:

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.

Modeling of Multiphase Boundaries

In the volume of fluid (VOF) and mixture multiphase models, the boundary conditions for flow and energy equations are the same as those in the single-phase flows. These are 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 (n–1) phases, the inlet volume fractions are predetermined, while the nth phase is obtained using the physical constraint:

equation 2.88

equation 2.89

And the volume fraction on each phase must be nonnegative.

• Outlet/Symmetry/Wall Boundary

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:

equation 2.90

equation 2.91

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.