METHODS BASED ON QUADRATIC DOUBLET DISTRIBUTION (USING THE DIRICHLET B. C.)
To demonstrate some of the techniques needed for higher-order methods, the linearly varying elements were used in the previous section. The last method to be tested then should be a linearly varying strength vortex panel-based method using the Dirichlet boundary condition. However, such a linearly varying vortex element is equivalent to a quadratic doublet distribution, which will be used to formulate the next two methods.
Also, for the constant strength singularity distribution based elements for N panel elements, N boundary condition based equations were constructed. Combined with the Kutta condition a set of N + 1 algebraic equations were obtained, including the last unknown, which was the wake doublet strength Hw – For higher-order methods, the number of unknowns increase linearly with the order of approximation and, therefore, additional equations must be specified. In the case of the linear methods, N — 1 equations (-1 since the trailing edge point is excluded) were obtained by equating the neighbor panel strengths, which is equivalent to requiring a continuous singularity distribution strength. In this section a generic approach is provided to obtain the additional equations that are needed as the order of approximation for the singularity strength distribution increases. Usually these equations are based on requirements such as smooth first, second, and higher derivatives at the panel corner point between two neighbor panels, but there are different methods of “obtaining” these additional equations, which can be optimized for certain problems. Therefore, the approach presented here is mainly to demonstrate the method, but improvement of these methods to tailor them for specific problems is encouraged.