5.5.3 Numerical Considerations
In homogeneous multiphase models, without considering velocity slips, no special treatment is required for solving the mixture momentum equations and face volume flux formulation (see Numerics) since they are the same as those equations governing the variable density single-phase flows. In this section, therefore, the focus is on the construction of the pressure-correction equation, and the treatment of the phase volume fraction equations, in particular, the interface resolving schemes in the VOF model.
Volume Continuity Equation
To satisfy the continuity constraint and ensure numerical stability, the pressure-correction equation is built based on total volume continuity rather than mass continuity. Dividing the 
phase continuity /volume fraction equation 5.254 by a phase reference density, 
, and combining all the phases together, we have a total volume continuity equation which satisfies the law of mass conservation:
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where the phase reference density is usually set as the phase density, 
.
Introducing 
as the volume of a computing cell, and integrating equation 5.330 over the control volume, we have the discretized algebraic equations:
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Following the same approach as in the single phase pressure-based solver described in Numerics, and assuming the following:
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We can rearrange equation 5.331 as the following correction equation:
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Here superscripts 
and 
represent old values and corrections. 
is time-step, 
is the area at face “
” and 
is volume flux.
Following the same approach as in the single-phase pressure-based solver, we apply the SIMPLE type of algorithms (Simple, SimpleC and SimpleS) to connect the velocity and pressure corrections and to obtain the pressure correction equation for multiphase flows:
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where 
is the linking coefficient, and 
is the linearized term.
Phase Volume Fraction Equations
The transport of a phase volume fraction is governed by the phase mass conservation. Since the total volume conservation is applied in forming the pressure-correction equation, the actual equations solved for phase volume fractions are also in the form of volume conservation for numerical consistency:
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Usually, for an 
-phase system, only the (
) equations are solved, while the 
phase is obtained from the physical constraint:
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Following the same discretizing approach as for the scalar transportation equation, described in Discretization and Solution , the integral form of equation 5.336 is as follows:
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As in the momentum, energy and total volume conservation equations, the spatial and temporal discretization schemes are crucial for numerical accuracy. For the volume fraction equations, in addition to the standard implicit time schemes, it is common practice to use the explicit time marching with high resolution advection schemes so that the interfaces in the VOF models can be captured more accurately. Both the implicit and explicit VOF formulations will be presented in detail in this section.
VOF Implicit Formulation
With the VOF implicit formulation, the discretized phase volume fraction equation has the general expression:
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In above equation, the phase volume fraction 
at the current time-step is a function of other quantities at the current time-step. Therefore, as the momentum, energy and pressure correction equations, the discretized volume fraction equation 5.339 is solved iteratively at each time-step. In Simerics-MP, the implicit formulation adopted can be summarized as follows:
Advection Schemes:
In the advection term, the volumetric flux 
is computed based on the flow field at the current time-step, see Flux Formulation. The face value 
is approximated in terms of the cell center values (
,
) and gradients (
,
) of neighboring cells “
” and “
”. As in the passive scalar equation, the advection schemes have the general form:
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With the use of different values for parameters 
, 
and 
, and the schemes to calculate the volume fraction gradients, four advection schemes are developed for the volume fraction equations: First-Order Upwind, Second-Order Upwind, Centre Difference and High Resolution. More detailed descriptions can be found in Numerics.
Temporal Schemes:
To describe the implicit temporal scheme, equation 5.339 may be generalized in the following expression:
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where the variables without a superscript are the values at the current time-step, while those with the superscript "0" or "00" indicate the values at the previous time steps.
The parameters 
and 
vary between 0 and 1, determining the time schemes. Specifically, three temporal schemes are adopted for the discretization of the phase volume fraction equations:
Euler First-Order Upwind: 
Three-Level Second-Order: 
Crank-Nicolson Method: 
VOF Explicit Formulation
When the explicit formulation is used for solving the VOF equations, the phase volume fractions at the current time-step are directly calculated based on known quantities from the previous time-step. Therefore, the VOF explicit formulation does not require an iterative solution for equation 5.339 during each time-step. However, since the rest of transport equations are solved implicitly, the time-step used for the volume fraction calculation will be in general smaller than that for the other transport equations. A sub time-step thus needs to be determined for the explicit VOF formulation, which can be either calculated automatically or user-prescribed in Simerics-MP.
With the explicit formulation, the discretized phase volume fraction equation is written as:
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where both the advection and source terms are computed based on the known quantities from the previous time-step. The volumetric flux 
is computed in the same way as 
in the implicit formulation. And the face volume fraction 
can also be estimated using one of the four advection schemes: First-Order Upwind, Second-Order Upwind, Centre Difference and High Resolution.
As for the explicit time marching schemes, the following three algorithms are offered by Simerics-MP:
Euler First Order Explicit
With the first-order Euler explicit scheme, the volume fraction equation is discretized as follows:
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Runge-Kutta Second-Order
Introducing the following function:
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we can rewrite equation 5.342 in the following:
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Then the second-order Runge-Kutta explicit scheme has the form:
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Runge-Kutta Fourth-Order
For the phase- 
volume fraction equation, the fourth-order Runge-Kutta explicit scheme has the form:
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where
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Finally, it may be noted that for an 
-phase system, typically only (
) phase volume fractions are solved, and the remaining one is obtained from the physical constraint, equation 5.337. However, one can also solve all 
- phase volume fraction equations, and equation 5.337 is satisfied by scaling each phase using the sum of the computed total volume fraction, which could be lesser or greater than 1 in an iterative process.