SOLUTIONS

The principles of singular element based numerical solutions were introduced in Chapter 9 and the first examples are provided in this chapter. The following two-dimensional examples will have all the elements of more refined three – dimensional methods, but because of the simple two-dimensional geometry the programming effort is substantially less. Consequently, such methods can be developed in a short time for investigating improvements in larger codes and are also suitable for homework assignments and class demonstrations.

Based on the level of approximation of the singularity distribution, surface geometry, and type of boundary conditions, a large number of computational methods can be constructed, some of which are presented in Table 11.1. We will not attempt to demonstrate all the possible combinations but will try to cover some of the most frequently used methods (denoted by the word “Example” in Table 11.1) which include: discrete singular elements, and constant strength, linear, and quadratic elements (as an example for higher – order singularity distributions). The different approaches in specifying the zero normal velocity boundary condition will be exercised and mainly the outer Neumann normal velocity and the internal Dirichlet boundary conditions will be used (and there are additional options, e. g., an internal Neumann condition). In terms of the surface geometry, for simplicity, only the flat panel element will be used here and in areas of high surface curvature the solution can be improved by using more panels.

In this chapter and in the following chapter the primary concern is the simplicity of the explanation and the ease of constructing the numerical technique, while numerical efficiency considerations are secondary. Conse­quently, the numerical economy of the methods presented can be improved (with some compromise in regard to the ease of code readability). Also, the methods are presented in their simplest form and each can be further developed to match the requirements of a particular problem. Such improve­ment can be obtained by changing grid spacing and density, location of collocation points, wake model, method of enforcing the boundary conditions, and of the Kutta condition.

Also it is recommended to read this chapter sequentially since the first methods will be described with more details. As the chapter evolves, some redundant details are omitted and the description may appear inadequate without reading the previous sections.