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23.3.2 Volume Fraction Equation

The tracking of the interface(s) between the phases is accomplished by the solution of a continuity equation for the volume fraction of one (or more) of the phases. For the $q^{\rm th}$ phase, this equation has the following form:


 \frac{1}{\rho_{\rm q}} \left [ \frac{\partial}{\partial t} (... ...sum_{p=1}^n ( {\dot{m}_{\rm pq} - \dot{m}_{\rm qp}} ) \right ] (23.3-1)

where $\dot{m}_{\rm qp}$ is the mass transfer from phase $q$ to phase $p$ and $\dot{m}_{\rm pq}$ is the mass transfer from phase $p$ to phase $q$. By default, the source term on the right-hand side of Equation  23.3-1, $S_{\alpha_q}$, is zero, but you can specify a constant or user-defined mass source for each phase. See Section  23.7 for more information on the modeling of mass transfer in FLUENT's general multiphase models.

The volume fraction equation will not be solved for the primary phase; the primary-phase volume fraction will be computed based on the following constraint:


 \sum_{q=1}^n \alpha_q = 1 (23.3-2)

The volume fraction equation may be solved either through implicit or explicit time discretization.



The Implicit Scheme


When the implicit scheme is used for time discretization, FLUENT's standard finite-difference interpolation schemes, QUICK, Second Order Upwind and First Order Upwind, and the Modified HRIC schemes, are used to obtain the face fluxes for all cells, including those near the interface.


 \frac{\alpha_q^{n+1}\rho_q^{n+1} - \alpha_q^n \rho_q^n}{\Del... ...m_{p=1}^n ( {\dot{m}_{\rm pq} - \dot{m}_{\rm qp}} ) \right ] V (23.3-3)

Since this equation requires the volume fraction values at the current time step (rather than at the previous step, as for the explicit scheme), a standard scalar transport equation is solved iteratively for each of the secondary-phase volume fractions at each time step.

The implicit scheme can be used for both time-dependent and steady-state calculations. See Section  23.10.1 for details.



The Explicit Scheme


In the explicit approach, FLUENT's standard finite-difference interpolation schemes are applied to the volume fraction values that were computed at the previous time step.


 \frac{\alpha_q^{n+1} \rho_q^{n+1}- \alpha_q^n \rho_q^n}{\Del... ...m}_{\rm pq} - \dot{m}_{\rm qp}} ) + S_{\rm\alpha_q} \right ] V (23.3-4)


where $n+1$ = index for new (current) time step
  $n$ = index for previous time step
  $\alpha_{q,f}$ = face value of the $q^{\rm th}$ volume fraction, computed from the first-
      or second-order upwind, QUICK, modified HRIC, or CICSAM scheme
  $V$ = volume of cell
  $U_f$ = volume flux through the face, based on normal velocity

This formulation does not require iterative solution of the transport equation during each time step, as is needed for the implicit scheme.

figure   

When the explicit scheme is used, a time-dependent solution must be computed.

When the explicit scheme is used for time discretization, the face fluxes can be interpolated either using interface reconstruction or using a finite volume discretization scheme (Section  23.3.2). The reconstruction based schemes available in FLUENT are Geo-Reconstruct and Donor-Acceptor. The discretization schemes available with explicit scheme for VOF are First Order Upwind, Second Order Upwind, CICSAM, Modified HRIC, and QUICK.



Interpolation near the Interface


FLUENT's control-volume formulation requires that convection and diffusion fluxes through the control volume faces be computed and balanced with source terms within the control volume itself.

In the geometric reconstruction and donor-acceptor schemes, FLUENT applies a special interpolation treatment to the cells that lie near the interface between two phases. Figure  23.3.1 shows an actual interface shape along with the interfaces assumed during computation by these two methods.

Figure 23.3.1: Interface Calculations
figure

The explicit scheme and the implicit scheme treat these cells with the same interpolation as the cells that are completely filled with one phase or the other (i.e., using the standard upwind (Section  25.3.1), second-order (Section  25.3.1), QUICK (Section  25.3.1, modified HRIC (Section  25.3.1), or CICSAM scheme), rather than applying a special treatment.

The Geometric Reconstruction Scheme

In the geometric reconstruction approach, the standard interpolation schemes that are used in FLUENT are used to obtain the face fluxes whenever a cell is completely filled with one phase or another. When the cell is near the interface between two phases, the geometric reconstruction scheme is used.

The geometric reconstruction scheme represents the interface between fluids using a piecewise-linear approach. In FLUENT this scheme is the most accurate and is applicable for general unstructured meshes. The geometric reconstruction scheme is generalized for unstructured meshes from the work of Youngs [ 412]. It assumes that the interface between two fluids has a linear slope within each cell, and uses this linear shape for calculation of the advection of fluid through the cell faces. (See Figure  23.3.1.)

The first step in this reconstruction scheme is calculating the position of the linear interface relative to the center of each partially-filled cell, based on information about the volume fraction and its derivatives in the cell. The second step is calculating the advecting amount of fluid through each face using the computed linear interface representation and information about the normal and tangential velocity distribution on the face. The third step is calculating the volume fraction in each cell using the balance of fluxes calculated during the previous step.

figure   

When the geometric reconstruction scheme is used, a time-dependent solution must be computed. Also, if you are using a conformal grid (i.e., if the grid node locations are identical at the boundaries where two subdomains meet), you must ensure that there are no two-sided (zero-thickness) walls within the domain. If there are, you will need to slit them, as described in Section  6.8.6.

The Donor-Acceptor Scheme

In the donor-acceptor approach, the standard interpolation schemes that are used in FLUENT are used to obtain the face fluxes whenever a cell is completely filled with one phase or another. When the cell is near the interface between two phases, a "donor-acceptor'' scheme is used to determine the amount of fluid advected through the face [ 145]. This scheme identifies one cell as a donor of an amount of fluid from one phase and another (neighbor) cell as the acceptor of that same amount of fluid, and is used to prevent numerical diffusion at the interface. The amount of fluid from one phase that can be convected across a cell boundary is limited by the minimum of two values: the filled volume in the donor cell or the free volume in the acceptor cell.

The orientation of the interface is also used in determining the face fluxes. The interface orientation is either horizontal or vertical, depending on the direction of the volume fraction gradient of the $q^{\rm th}$ phase within the cell, and that of the neighbor cell that shares the face in question. Depending on the interface's orientation as well as its motion, flux values are obtained by pure upwinding, pure downwinding, or some combination of the two.

figure   

When the donor-acceptor scheme is used, a time-dependent solution must be computed. Also, the donor-acceptor scheme can be used only with quadrilateral or hexahedral meshes. In addition, if you are using a conformal grid (i.e., if the grid node locations are identical at the boundaries where two subdomains meet), you must ensure that there are no two-sided (zero-thickness) walls within the domain. If there are, you will need to slit them, as described in Section  6.8.6.

The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM)

The compressive interface capturing scheme for arbitrary meshes (CICSAM), based on the Ubbink's work [ 377], is a high resolution differencing scheme. The CICSAM scheme is particularly suitable for flows with high ratios of viscosities between the phases. CICSAM is implemented in FLUENT as an explicit scheme and offers the advantage of producing an interface that is almost as sharp as the geometric reconstruction scheme.


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