## 25.4.1 Discretization of the Momentum Equation

The discretization scheme described in Section  25.3 for a scalar transport equation is also used to discretize the momentum equations. For example, the -momentum equation can be obtained by setting :

 (25.4-3)

If the pressure field and face mass fluxes are known, Equation  25.4-3 can be solved in the manner outlined in Section  25.3, and a velocity field obtained. However, the pressure field and face mass fluxes are not known a priori and must be obtained as a part of the solution. There are important issues with respect to the storage of pressure and the discretization of the pressure gradient term; these are addressed next.

FLUENT uses a co-located scheme, whereby pressure and velocity are both stored at cell centers. However, Equation  25.4-3 requires the value of the pressure at the face between cells and , shown in Figure  25.2.1. Therefore, an interpolation scheme is required to compute the face values of pressure from the cell values.

Pressure Interpolation Schemes

The default scheme in FLUENT interpolates the pressure values at the faces using momentum equation coefficients [ 304]:

 (25.4-4)

This procedure works well as long as the pressure variation between cell centers is smooth. When there are jumps or large gradients in the momentum source terms between control volumes, the pressure profile has a high gradient at the cell face, and cannot be interpolated using this scheme. If this scheme is used, the discrepancy shows up in overshoots/undershoots of cell velocity.

Flows for which the standard pressure interpolation scheme will have trouble include flows with large body forces, such as in strongly swirling flows, in high-Rayleigh-number natural convection and the like. In such cases, it is necessary to pack the mesh in regions of high gradient to resolve the pressure variation adequately.

Another source of error is that FLUENT assumes that the normal pressure gradient at the wall is zero. This is valid for boundary layers, but not in the presence of body forces or curvature. Again, the failure to correctly account for the wall pressure gradient is manifested in velocity vectors pointing in/out of walls.

Several alternate methods are available for cases in which the standard pressure interpolation scheme is not valid:

• The linear scheme computes the face pressure as the average of the pressure values in the adjacent cells.

• The second-order scheme reconstructs the face pressure in the manner used for second-order accurate convection terms (see Section 25.3.1). This scheme may provide some improvement over the standard and linear schemes, but it may have some trouble if it is used at the start of a calculation and/or with a bad mesh. The second-order scheme is not applicable for flows with discontinuous pressure gradients imposed by the presence of a porous medium in the domain or the use of the VOF or mixture model for multiphase flow.

• The body-force-weighted scheme computes the face pressure by assuming that the normal gradient of the difference between pressure and body forces is constant. This works well if the body forces are known a priori in the momentum equations (e.g., buoyancy and axisymmetric swirl calculations).

 When a case contains porous media, the body-force-weighted scheme is applied only for non-porous faces, where the scheme takes into account the discontinuity of explicit body forces (e.g., gravity, swirl, Coriolis) and the discontinuity of pressure gradients for flows with rapidly changing densities (e.g., natural convection, VOF). All interior and exterior porous faces are treated with a special scheme that preserves the continuity of the normal velocity across cell faces in spite of the discontinuity of the resistance.

• The PRESTO! (PREssure STaggering Option) scheme uses the discrete continuity balance for a "staggered'' control volume about the face to compute the "staggered'' (i.e., face) pressure. This procedure is similar in spirit to the staggered-grid schemes used with structured meshes [ 277]. Note that for triangular, tetrahedral, hybrid, and polyhedral meshes, comparable accuracy is obtained using a similar algorithm. The PRESTO! scheme is available for all meshes.

See Section  25.8.3 for recommendations on when to use these alternate schemes.

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