THREE – DIMENSIONAL NUMERICAL SOLUTIONS

Three-dimensional numerical solutions based on surface singularity distribu­tions are similar, in principle, to methods presented for the two-dimensional case. From the theoretical aspect, only the wake and the trailing edge conditions (three-dimensional Kutta condition) will require some additional attention. The most difficult aspect in three dimensions, though, is the modeling of the geometry, especially when arbitrary geometry capability is sought.

In the first part of this chapter the geometry (of the wing) is kept relatively simple and the aerodynamics of a thin lifting surface is modeled. In principle, this simple method has all the elements of the more complex panel methods and is capable of modeling the effect of wing planform shape on the fluid dynamic loads. In addition, the examples that are presented require only limited programming effort and, therefore, are suitable for classroom instruction.

In the second part of this chapter the principles of panel codes capable of solving the flow over bodies with arbitrary three-dimensional geometry will be presented. Over the years many such methods were developed and improved, but recent trends indicate an increased use of the approach which is based on the combination of surface source and doublet distributions with the inner potential boundary condition (for closed bodies). Consequently, only this approach will be presented through a brief description of one low-order and one high-order panel method.

In terms of classroom instruction it is recommended to use at this phase one of the commercially available panel codes and train the students first to use the pre- and post-processor. This graphic pre-processor generates the surface grid (panel model) which is used to define the input to the computer program. The post processor is usually a graphic utility that allows for a rapid analysis of the three-dimensional results by using extensive graphic repre­sentation. It is believed that after studying and preparing examples with the lifting surface code in this chapter (Section 12.3) the students can safely proceed to use a larger panel code since at this phase they must have a deep understanding of the formulation and the construction of these codes.