SUMMARY: METHODOLOGY OF SOLUTION

In view of Eq. (3.13) ((3.17) in two dimensions), it is possible to establish a fairly general approach to the solution of incompressible potential flow problems. The most important observation is that/the solution of У2Ф = 0 can be obtained by distributing elementary solutions (sources and doublets) on the problem boundaries (SB, Sw). These elementary solutions automatically fulfill the boundary condition of Eq. (3.3) by having velocity fields that decay as
r—»oo. However, at the point where r = 0, the velocity becomes singular, and therefore the basic elements are called singular solutions.

The general solution requires the integration of these basic solutions over any surface S containing these singularity elements because each element will have an effect on the whole fluid field.

The solution of a fluid dynamic problem is now reduced to finding the appropriate singularity element distribution over some known boundaries, so that the boundary condition (Eq. (3.2)) will be fulfilled. The main advantage of this formulation is its straightforward applicability to numerical methods. When the potential is specified on the problem boundaries then this type of mathematical problem is called the Dirichlet problem (Kellogg,13 p. 286) and is frequently used in many numerical solutions (panel methods).

A more direct approach to the solution, from the physical point of view, is to specify the, zero normal flow boundary condition (Eq. (3.2)) on the solid boundaries. This problem is known as the Neumann problem (Kellogg,13 p. 286) and in order to evaluate the velocity field the potential is differentiated

w – c 1 *0 л+s L/ 0]e + ,3’18)

Again, the derivative d/dn for the doublet indicates the orientation of the element as will be shown in Section 3.5. Substituting this equation into the boundary condition of Eq. (3.2) can serve as the basis of finding the unknown singularity distribution. (This can be done analytically or numerically.)

For a given set of boundary conditions, the above solution technique is not unique, and many problems can be solved by using only one type of singularity element or any linear combination of the two singularities. Therefore, in many situations additional considerations are required (e. g., the method that will be presented in the next chapter to define the flow near sharp trailing edges of wings). Also, in a particular solution a mixed use of the above boundary conditions is possible for various regions in the flowfield (e. g., Neumann condition on one boundary and Dirichlet on another).

Prior to attempting to apply this methodology to the solution of particular problems, the features of the elementary solutions are analyzed in the next sections.