Equation 23–39 may be integrated over the control volume in Figure 23.3: Control Volume Used to Illustrate Discretization of a Scalar Transport Equation to yield the following discrete equation
 | (23–45) |
where 
is the mass flux through face 
, 
.
In order to proceed further, it is necessary to relate the face
value of velocity, 
, to the velocity at the centers of the two adjacent cells. Linear
interpolation of cell-centered velocities to the face results in unphysical checkerboarding of
pressure. Ansys Fluent uses high-order interpolation methods to calculate face flux,
, and a pressure gradient correction is applied following a similar method
outlined by Rhie and Chow [554] to prevent checkerboarding. Two
interpolation methods are available for calculating the face value of velocity: distance-based
linear interpolation and momentum-based averaging [418]. Using this
procedure, the face flux, 
, may be written as
 | (23–46) |
where
 | (23–47) |
The definition of 
depends on the method:
 | (23–48) |
 | (23–49) |
The term 
is a function of 
, the average of the momentum equation 
coefficients (from Equation 23–41) for the cells on either
side of face 
. 
, 
and 
, 
are the pressures and velocities, respectively, within the two cells on either
side of the face. 
is a high-order correction for non-uniform and/or non-orthogonal meshes based
on distances between the cell centers and the face center. Similarly, 
is a high-order correction for non-uniform and/or non-orthogonal meshes based
on the momentum coefficients of cells adjacent to the face.
For incompressible flows, Ansys Fluent uses arithmetic averaging for density. For compressible flow calculations (that is, calculations that use the ideal gas law for density), Ansys Fluent applies upwind interpolation of density at cell faces. Several interpolation schemes are available for the density upwinding at cell faces: first-order upwind, second-order upwind (default), QUICK, MUSCL, and when applicable, central differencing and bounded central differencing.
The first-order upwind scheme (based on [290]) sets the density at the cell face to be the upstream cell-center value. This scheme provides stability for the discretization of the pressure-correction equation, and gives good results for most classes of flows. Although this scheme provides the best stability for compressible flow calculations, it gives very diffusive representations of shocks.
The second-order upwind scheme provides stability for supersonic flows and captures shocks better than the first-order upwind scheme. The QUICK scheme for density is similar to the QUICK scheme used for other variables. See QUICK Scheme for details.
Important: In the case of multiphase flows, the selected density scheme is applied to the compressible phase and arithmetic averaging is used for incompressible phases.
Important: If solution instability is encountered for compressible flow, it is recommended that you achieve a solution with a first order scheme and then switch to a higher order schemes.
For recommendations on choosing an appropriate density interpolation scheme for your compressible flow, see Choosing the Density Interpolation Scheme in the User's Guide.