8.6 Finite Volume Methods
239
In the second case we get:
The first approach is fully conservative. The second is conservative (and
equivalent to the first one) if the derivative dpldx is calculated using Gauss'
theorem. If the pressure gradient with respect to x is transformed into gradients with respect to J and 7 for a local coordinate system at the CV center,
one obtains:
One can calculate the derivative a t CV center by differentiating a shape
function. These approaches are in general not conservative.
Treatment of pressure terms in the equations for u, (and in 3D cases u,)
are similar to those given above for u,.
8.6.4 Three-Dimensional Grids
In 3D, the cell faces are not necessarily planar. To calculate cell volumes
and cell face surface vectors, suitable approximations are necessary. A simple method is t o represent the cell face by a set of plane triangles. For the
hexahedra used in structured grids, Kordula and Vinokur (1983) suggested
decomposing each CV into eight tetrahedra (each CV face being subdivided
into two triangles) so that no overlapping occurs.
Another way to calculate cell volumes for arbitrary CVs is based on Gauss
theorem. By using the identity 1 = div(xi), one can calculate the volume as:
where 'c' denotes cell faces and St is the x-component of the cell face surface
vector (see Fig. 8.10):
Instead of xi, one can also use y j or zk, in which case one has to sum the
products of y, S, Y or zc S,*. If each cell face is defined in the same way for
both CVs to which it is common, the procedure ensures that no overlapping
occurs and that the sum of all CV volumes equals the volume of the solution
domain.
239
In the second case we get:
The first approach is fully conservative. The second is conservative (and
equivalent to the first one) if the derivative dpldx is calculated using Gauss'
theorem. If the pressure gradient with respect to x is transformed into gradients with respect to J and 7 for a local coordinate system at the CV center,
one obtains:
One can calculate the derivative a t CV center by differentiating a shape
function. These approaches are in general not conservative.
Treatment of pressure terms in the equations for u, (and in 3D cases u,)
are similar to those given above for u,.
8.6.4 Three-Dimensional Grids
In 3D, the cell faces are not necessarily planar. To calculate cell volumes
and cell face surface vectors, suitable approximations are necessary. A simple method is t o represent the cell face by a set of plane triangles. For the
hexahedra used in structured grids, Kordula and Vinokur (1983) suggested
decomposing each CV into eight tetrahedra (each CV face being subdivided
into two triangles) so that no overlapping occurs.
Another way to calculate cell volumes for arbitrary CVs is based on Gauss
theorem. By using the identity 1 = div(xi), one can calculate the volume as:
where 'c' denotes cell faces and St is the x-component of the cell face surface
vector (see Fig. 8.10):
Instead of xi, one can also use y j or zk, in which case one has to sum the
products of y, S, Y or zc S,*. If each cell face is defined in the same way for
both CVs to which it is common, the procedure ensures that no overlapping
occurs and that the sum of all CV volumes equals the volume of the solution
domain.