Page 225 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
function as
2
2
∂ ψ ∂ ψ ∂u 1 ∂u 2 217
+ = − (7.188)
∂x 2 ∂x 2 ∂x 1 ∂x 2
2 1
A solution to the above Laplacian equation is straightforward for any numerical pro-
cedure. This equation is similar to Step 2 of the CBS scheme and an implicit procedure
immediately gives the solution. Unlike the pressure equation of Step 2, the stream function
of a solution needs to be calculated only once.
7.9 Mesh Convergence

All numerical schemes are by their nature an approximation and the CBS scheme is no
exception. However, if a scheme is to be convergent, the approximate solution should
approach the exact answer as the mesh is refined. A converged solution is one that is nearly
independent of meshing errors. An extremely coarse mesh would give a very approximate
solution, which is far from reality. As the mesh is refined by reducing the size of the
elements, the solution slowly approaches an exact solution. It should be noted that, in
theory, the solution will not be exact until the mesh size is zero, which is obviously
impossible. However, it is possible to fix a tolerance to the solution error and this can be
achieved by solving the problem on several meshes.
In order to ensure that the solution obtained is as close as possible to reality, solutions
should be obtained from several meshes starting with a very coarse mesh and finishing with
a very fine mesh. Once these solutions are available, many key quantities can be compared
and plotted against mesh densities (or number of points) as shown in Figure 7.16. If the
difference between two consecutive meshes (or number of nodes) is less than a fixed
tolerance, the coarser mesh is normally accepted as a suitable mesh for the analysis.
For two-dimensional problems, it is not difficult to carry out a detailed mesh con-
vergence study for different parameters or cases. However, in large three-dimensional
problems, it is often difficult to carry out a complete mesh convergence study. In such

Converged

Nusselt number

Number of nodes
Figure 7.16 Typical convergence study