4.7 Examples
85
With same approximations applied to other CV faces, the integral equation becomes:
which represents the equation for a generic node P. The coefficients A[ are
obtained by summing the convective and diffusive contributions, see Eqs.
(4.32), (4.33) and (4.35):
where 1 represents any of the indices P, E, W, N, S. That Ap is equal to
the negative sum of all neighbor coefficients is a feature of all conservative
schemes and ensures that a uniform field is a solution of the discretized
equations.
The above expressions are valid at all internal CVs. For CVs next to
boundary, the boundary conditions require that the equations be modified
somewhat. At the north and west boundaries, where 4 is prescribed, the
gradient in the normal direction is approximated using one-sided differences,
e.g. a t the west boundary:
where W denotes the boundary node whose location coincides with the cellface center 'w'. This approximation is of first-order accuracy, but it is applied
on a half-width CV. The product of the coefficient and the boundary value is
added to the source term. For example, along the west boundary (CVs with
index i = 2), A w 4 w is added t o Qp and the coefficient Aw is set to zero.
The same applies to the coefficient AN a t the north boundary.
At the south boundary, the normal gradient of 4 is zero which, when the
above approximation is applied, means that the boundary values are equal
to the values a t CV centers. Thus, for cells with index j = 2, 4s = 4p and
the algebraic equation for those CVs is modified to:
which requires adding As to Ap and then setting As = 0. The zero-gradient
condition a t the outlet (east) boundary is implemented in a similar way.
We now turn to the results. The isolines of 4 calculated on a 40 x 40
CV uniform grid using CDS for the convective fluxes with two values of r:
0.001 and 0.01 ( p = 1.0) are presented in Fig. 4.5. We see that transport by
diffusion across the flow is much stronger for higher r, as expected.
In order to assess the accuracy of the prediction, we monitor the total flux
of 4 through the west boundary, at which 4 is prescribed. This quantity is
obtained by summing diffusive fluxes over all CV faces along this boundary,
which are approximated by Eqs. (4.34) and (4.38). Figure 4.6 shows the variation of the flux as the grid is refined for the UDS and CDS discretizations of
85
With same approximations applied to other CV faces, the integral equation becomes:
which represents the equation for a generic node P. The coefficients A[ are
obtained by summing the convective and diffusive contributions, see Eqs.
(4.32), (4.33) and (4.35):
where 1 represents any of the indices P, E, W, N, S. That Ap is equal to
the negative sum of all neighbor coefficients is a feature of all conservative
schemes and ensures that a uniform field is a solution of the discretized
equations.
The above expressions are valid at all internal CVs. For CVs next to
boundary, the boundary conditions require that the equations be modified
somewhat. At the north and west boundaries, where 4 is prescribed, the
gradient in the normal direction is approximated using one-sided differences,
e.g. a t the west boundary:
where W denotes the boundary node whose location coincides with the cellface center 'w'. This approximation is of first-order accuracy, but it is applied
on a half-width CV. The product of the coefficient and the boundary value is
added to the source term. For example, along the west boundary (CVs with
index i = 2), A w 4 w is added t o Qp and the coefficient Aw is set to zero.
The same applies to the coefficient AN a t the north boundary.
At the south boundary, the normal gradient of 4 is zero which, when the
above approximation is applied, means that the boundary values are equal
to the values a t CV centers. Thus, for cells with index j = 2, 4s = 4p and
the algebraic equation for those CVs is modified to:
which requires adding As to Ap and then setting As = 0. The zero-gradient
condition a t the outlet (east) boundary is implemented in a similar way.
We now turn to the results. The isolines of 4 calculated on a 40 x 40
CV uniform grid using CDS for the convective fluxes with two values of r:
0.001 and 0.01 ( p = 1.0) are presented in Fig. 4.5. We see that transport by
diffusion across the flow is much stronger for higher r, as expected.
In order to assess the accuracy of the prediction, we monitor the total flux
of 4 through the west boundary, at which 4 is prescribed. This quantity is
obtained by summing diffusive fluxes over all CV faces along this boundary,
which are approximated by Eqs. (4.34) and (4.38). Figure 4.6 shows the variation of the flux as the grid is refined for the UDS and CDS discretizations of