Setting up the Finite Difference Equations

In CFD, finite difference equivalents of the differential balance equations are solved in a computational grid . The computational grid consists of node points at which we wish to calculate the dependent variable, which can be the temperature, the concentration of some chemical component or, as it is in the Navier-Stokes equations, the momentum (mass times velocity). A onedimensional finite difference grid is sketched in Fig. 7.1.1. 

Differential balance equations for some quantity (/ , which could be the concentration of some chemical species, or the x-, y- or -momentum, are given in Appendix 7. A. We wish to give an idea of how finite difference equations are formulated by presenting a finite difference equivalent of the one-dimensional balance equation, Eq. (7.A.2), in its steady state form  

There are different ways of doing this. One way is to take the differential equation, and substitute estimates for the first and the second derivatives in terms of the values of the dependent variable at the grid points. The definitions of the first and second derivatives indicate that we should take  

We have thus obtained an algebraic equation for ip at node point i in terms of the values at the neighboring points. The set of equations for cp at all the node points can be solved iteratively to obtain the flow field. This is the principle of CFD. 

Another way of formulating finite difference equations is to perform the balances in (p directly on the computational cells. Eq. (7.1.4) could have been derived by performing the same balance on the cell in Fig. 7.1.1 as that on the differential element in Appendix 7.A. To obtain Eq. (7.1.4), when performing the balance, we make the following choices for the values of cp and its gradient at the cell boundary between node i-1 and 

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To evaluate a given row of the A,B oxC matrix it can readily be seen what has to be done from the equation set. The partial derivatives of one of the functions must be evaluated with respect to each solution variable, each variable first derivative and each variable second derivative. These partial derivatives allow the elements in Eq. (11.60) to be evaluated row by row, or column by column if more convenient. In fact it is more efficient to evaluate the terms on a column by column basis as one then only has to select an increment value once for each variable and derivative and then apply this incremented value to all the functions. In terms of a single variable BV problem, there is roughly as much computational effort in setting up the finite difference equations, where N is the number of coupled equations. Then there is additional time required to solve the set of coupled matrix equations as represented by Eq. (11.58). 

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