Volume of Fluid Method

The Volume of Fluid (VOF) multiphase model implementation in Simcenter STAR-CCM+ belongs to the family of interface-capturing methods that predict the distribution and the movement of the interface of immiscible phases. This modeling approach assumes that the mesh resolution is sufficient to resolve the position and the shape of the interface between the phases.

The distribution of phases and the position of the interface are described by the fields of phase volume fraction ${\alpha }_i$. The volume fraction of phase $i$ is defined as:

where $V_i$ is the volume of phase $i$ in the cell and $V$ is the volume of the cell. The volume fractions of all phases in a cell must sum up to one:

where $N$ is the total number of phases.

Depending on the value of the volume fraction, the presence of different phases or fluids in a cell can be distinguished:

• ${\alpha }_i=0$—the cell is completely void of phase $i$
• ${\alpha }_i=1$—the cell is completely filled with phase $i$
• $0<{\alpha }_i<1$—values between the two limits indicate the presence of an interface between phases

The material properties that are calculated in the cells containing the interface depend on the material properties of the constituent fluids. The fluids that are present in the same interface-containing cell are treated as a mixture:

where ${\rho }_i$ is the density, ${\mu }_i$ is the dynamic viscosity, and ${\left(C_p\right)}_i$ is the specific heat of phase $i$.

Volume Fraction Transport Equation

The distribution of phase $i$ is driven by the phase mass conservation equation:

$$\frac{\partial }{\partial t}{\int }_V^{}{\alpha }_{i}dV+\underset{A}{\oint }{\alpha }_i\mathbf{v}\cdot d\mathbf{\text{a}}={\int }_V^{}\left(S_{{\alpha }_i}-\frac{{\alpha }_i}{{\rho }_i}\frac{{D\rho }_i}{Dt}\right)dV-{\int }_V\frac{1}{{\rho }_i}\nabla \cdot \left({\alpha }_i{\rho }_i{\mathbf{v}}_{d,i}\right)dV$$

(2584)

where $a$ is the surface area vector, $v$ is the mixture (mass-averaged) velocity, $v_{d,i}$ is the diffusion velocity, $S_{{\alpha }_i}$ is a user-defined source term of phase $i$, and ${D\rho }_i/Dt$ is the material or Lagrangian derivative of the phase densities ${\rho }_i$.

Simcenter STAR-CCM+ calculates the volume fractions of phases as follows:

• When there are two VOF phases present, the volume fraction transport is solved for the first phase only. In each cell, the volume fraction of the second phase is adjusted so that the sum of the volume fractions of the two phases is equal to 1.
• When there are three or more VOF phases present, the volume fraction transport is solved for all phases. The volume fraction of each phase is then normalized based on the sum of the volume fractions of all phases in each cell.

If a non-zero sharpening factor is specified, an additional term is added to the VOF transport equation:

$$\nabla \cdot \left({\mathbf{\text{v}}}_{c_i}{\alpha }_i\left(1-{\alpha }_i\right)\right)$$

(2585)

where:

• ${\alpha }_i$ is the volume fraction of phase $i$
• ${\mathbf{\text{v}}}_{c_i}$ is defined as follows:

  ${\mathbf{\text{v}}}_{c_i}=C_{\alpha }\left|\mathbf{\text{v}}\right|\frac{\nabla {\alpha }_i}{\left|\nabla {\alpha }_i\right|}$

• $C_{\alpha }$ is the sharpening factor
• $\mathbf{\text{v}}$ is the fluid velocity

Continuity Equation

The total mass conservation equation for all phases is given by:

$$\frac{\partial }{\partial t}\left(\underset{V}{\int }\rho dV\right)+{\oint }_A\rho v\cdot da=\underset{V}{\int }SdV$$

(2586)

where $S$ is a mass source term that is related to the phase source term as follows:

$$S=\sum _i{S}_{{\alpha }_i}\cdot {\rho }_i$$

(2587)

The dependency on the volume fractions of the constituent phases of the fluid mixture is accounted for through the density, which is given by Eqn. (2581).

Momentum Equation

$$\begin{array}{c}\frac{\partial }{\partial t}\left({\int }_V\rho \mathbf{\text{v}}dV\right)+\underset{A}{\overset{}{\oint }}\rho v\otimes v\cdot d\mathbf{\text{a }}=\end{array}-{\oint }_{A}pI\cdot da+{\oint }_{A}T\cdot da+{\int }_V\rho \mathbf{g}dV+{\int }_V{f}_{b}dV-\sum _i{\int }_A{\alpha }_i{\rho }_i{\mathbf{v}}_{d,i}\otimes {\mathbf{v}}_{d,i}\cdot da+{\int }_V{S}_i^{\alpha }dV$$

(2588)

where:

• $p$ is the pressure
• $I$ is the unity tensor
• $T$ is the stress tensor
• $f_b$ is the vector of body forces
• $S_i^{\alpha }$ is the phase momentum source term

Energy Equation

$$\frac{\partial }{\partial t}{\int }_V^{}\rho EdV+{\oint }_A^{}[\mathrm{\rho H}v+\sum _i{\alpha }_i{\rho }_i{H}_i{\mathbf{v}}_{d,i}]\cdot d\mathbf{\text{a}}=-{\oint }_A^{}\dot{\mathbf{\text{q}}}″\cdot d\mathbf{\text{a}}+{\oint }_A^{}\mathbf{\text{T}}\mathbf{\cdot }\mathbf{\text{v}}d\mathbf{\text{a}}+{\int }_V^{}{\mathbf{\text{f}}}_b\mathbf{\cdot }\mathbf{\text{v}}dV+{\int }_V^{}{S}_{E}dV$$

(2589)

where:

• $E$ is the total energy
• $H$ is the total enthalpy
• $\dot{q}″$ is the heat flux vector
• $S_E$ is a user-defined energy source term

The terms that contain the diffusion velocity $v_{d,i}$ in Eqn. (2584), Eqn. (2588), and Eqn. (2589) are due to the slip between phases. See Interface Sharpening through Slip Velocity Modeling and Phase Slip.