V

of approximation. For the first level of approximation, normally a simple second-order approximation is used to replace the surface or volume integrals. Therefore, a surface integral is approximated as a product of the integrand at the cell face center and the cell face area. Similarly, a volume integral is approximated as the product of the mean value of the integrand over the CV (approximated as the value at the cell center) and the CV volume. To illustrate these approximation practices, consider the CV centered on a node P. Flux through the CV boundary denoted 'e' (east face of CV) in Fig. 6.2, Fe, can be written:

JSe where f is the component of the convective or diffusive vector in the direction normal to face e and Se is the area of face e. Invoking the second-order approximation, Fe can be written:

where fe is the value of the integrand at the center of face e. Alternative second-order approximations for Fe are possible, which may write Fe in terms of fne and fse (fne and fse are values of the integrand at the north-most point and south-most point of the east face, respectively). Simpson's rule may then be used to represent Fe in terms offe, fne andfse as a fourth-order approximation. It must be noted that values of the integrand are normally not available at cell faces and, therefore, need to be obtained by interpolation. To preserve the accuracy of the above approximation, interpolation should also use at least the same or higher order approximation. Considering this, in most cases, it is sufficient to use the second-order approximation given by Eq. (6.4).

The volume integrals can be approximated as follows:

where SpP is the value of source term Sp at the cell center P, and VP is the volume of the computational cell centered around node p. A higher order approximation of this volume integral will require the values of p at more locations than just the center. Since only the values at cell centers will be available, correspondingly higher order interpolation schemes need to be used to retain the accuracy of these volume integrals. Here again, normally it is sufficient to use the second-order approximation represented by Eq. (6.5). These approximations allow one to write Eq. (6.2) in the following form (steady state equation for the two-dimensional Cartesian grid shown in Fig. 6.2):

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