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25.4.2 Discretization of the Continuity Equation

Equation  25.4-1 may be integrated over the control volume in Figure  25.2.1 to yield the following discrete equation


 \sum_f^{N_{\rm faces}} J_f A_f = 0 (25.4-5)

where $J_f$ is the mass flux through face $f$, $\rho v_n$.

In order to proceed further, it is necessary to relate the face values of velocity, ${\vec v}_n$, to the stored values of velocity at the cell centers. Linear interpolation of cell-centered velocities to the face results in unphysical checker-boarding of pressure. FLUENT uses a procedure similar to that outlined by Rhie and Chow [ 304] to prevent checkerboarding. The face value of velocity is not averaged linearly; instead, momentum-weighted averaging, using weighting factors based on the $a_P$ coefficient from equation  25.4-3, is performed. Using this procedure, the face flux, $J_f$, may be written as


 J_f = \rho_f \frac{a_{p,c_0} v_{n,c_0} + a_{p,c_1} v_{n,c_1}... ..._1} \cdot \vec{r_1}))={\hat{J}}_f + d_f \, (p_{c_0} - p_{c_1}) (25.4-6)

where $p_{c_0}$, $p_{c_1}$ and $v_{n,c_0}$, $v_{n,c_1}$ are the pressures and normal velocities, respectively, within the two cells on either side of the face, and ${\hat{J}}_f$ contains the influence of velocities in these cells (see Figure  25.2.1). The term $d_f$ is a function of $\bar{a}_P$, the average of the momentum equation $a_P$ coefficients for the cells on either side of face $f$.



Density Interpolation Schemes


For incompressible flows, FLUENT uses arithmetic averaging for density. For compressible flow calculations (i.e., calculations that use the ideal gas law for density), FLUENT applies upwind interpolation of density at cell faces. Several interpolation schemes are available for the density upwinding at cell faces: first-order upwind (default), second-order-upwind, QUICK, MUSCL, and when applicable, central differencing and bounded central differencing.

The first-order upwind scheme (based on [ 172]) 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. The first-order scheme is the default scheme for compressible 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 Section  25.3.1 for details.

figure   

In the case of multiphase flows, the selected density scheme is applied to the compressible phase and arithmetic averaging is used for incompressible phases.

figure   

For stability reasons, it is recommended that you achieve a one-dimensional solution with a first order scheme and then switch to a higher order scheme for compressible flow calculations.

See Section  25.8.4 for recommendations on choosing an appropriate density interpolation scheme for your compressible flow.


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