**Numerical Solutions Applied To Heat
Transfer And Fluid Mechanics Problems**

Heat conduction simulations shifted to non-rectangular grids many years ago. CFD is also rapidly moving in that direction, because of the relative ease constructing a difference mesh for a complex geometry. In two-dimensions the mesh elements tend to be triangles or mixtures of triangles and quadrilaterals. In three dimensions, tetrahedrons are common, but other forms with mixtures of triangular and quadrilateral sides are also used.

A lecture by Frank Schmidt discusses error analysis and construction of finite differences in such geometries. Finite volume approaches rely on appropriate order interpolation, and the Divergence Theorem to convert volume integrals to surface integrals. But what do you do with the pressure gradient term in the momentum equation.

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It turns out that the Divergence Theorem is just a special case of a more general integral property. For any scalar, say “p”, the volume integral of any spatial derivative is equal to the integral over its bounding surface of the scalar multiplied by the cosine of the angle between the direction associated with the derivative and the outward normal to the surface.

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For example the x component of the finite volume momentum equation becomes:

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Yes it’s a mess, but can be applied to tetrahedrons or other annoying shapes in a fairly straightforward manner.