6.2.1. Discretization of Governing Model Equations

The governing transport equations are integrated over each computational cell (employing a divergence theorem) and over the considered time interval At:

i+At j j dV + J pU$ ■ ndS = j r$V$ ■ ndS + j S$dV

where integration over V denotes volume of a computational cell and S denotes surface of a computational cell. In order to discuss the main issues of the finite volume method, let us first consider a steady state problem. For such a case, the outer integral over time and the first term of Eq. (6.2) will vanish. The second term is the net convective flux of quantity <p through the boundaries of a control volume. The third term is the net diffusive flux. To calculate convective and diffusive fluxes, one needs to know (p U0) or (r^ grad 0) everywhere on all the surfaces bounding the considered control volume. This information is not available since the values of all the variables are known only at the computational nodes. It is, therefore, necessary to make approximations while estimating the flux through the surface of computational cells. The fourth term in Eq. (6.2) represents the volumetric source (or sink). Usually all terms in the governing equations which cannot be classified as in the first three terms, are accommodated in such a source term. Approximations also need to be invoked to estimate such volume integrals.

To carry out the integration step shown in Eq. (6.2) on a computational cell, two levels of approximation need to be invoked. At the first level, the surface (or volume) integral is approximated in terms of the variable values at one or more locations on the cell face (within the CV). At the second level, the variable values at these locations are approximated in terms of the values at the computational nodes (cell centers). A judicious compromise between simplicity, ease of implementation, accuracy and computational efficiency has to be made while selecting these two levels

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