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5.6 Discretization     continued...

Note that the governing differential equations are in flux-conservative form so that mass and energy are conserved when the equations are discretized. Finite difference approximations can now be derived by appealing to a control volume approach. To do this, it is necessary to recognize that

where and are the grid sizes in the and directions. We therefore multiply all the governing equations, both for the internal mode, equations (69) to (75), and external, equations (90) to (92), by before finite differencing. Thus the continuity equation (69) becomes

By virtue of equation (96), the finite difference equivalent of the continuity equation becomes

where , , , denote finite difference operators. Thus for any quantity :

Figure 19 makes the physical interpretation of equation (98), which describes the mass balance in the control volume shown.