Computational Methods for Fluid Dynamics by Joel H. Ferziger & Milovan Perić & Robert L. Street

Author:Joel H. Ferziger & Milovan Perić & Robert L. Street
Language: eng
Format: epub
ISBN: 9783319996936
Publisher: Springer International Publishing

(9.49)

Then we can consider the derivative as the divergence of the vector and transform the volume integral in the above equation using Gauss’ theorem into a surface integral:

(9.50)

This shows that one can calculate the derivative of with respect to x at the CV center by summing the products of with the x-components of the surface vectors at all faces of the CV and dividing the sum by the CV volume:

(9.51)

For we can use the values used to calculate the convection fluxes, although one need not necessarily use the same approximation for both terms. For Cartesian grids and linear interpolation, the standard central difference approximation is obtained (only east and west face contribute to the derivative in x-direction because x-components of surface vectors at other faces are zero; the cell volume can be expressed as ):

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