SINGULARITY ELEMENTS AND INFLUENCE COEFFICIENTS

It was demonstrated in the previous chapters that the solution of potential flow problems over bodies and wings can be obtained by the distribution of elementary solutions. The strengths of these elementary solutions of Laplace’s equation are obtained by enforcing the zero normal flow condition on the solid boundaries. The steps toward a numerical solution of this boundary-value problem are described schematically in Section 9.7. In general, when increas­ing the complexity of the method, mostly the “element’s influence” calculation becomes more elaborate. Therefore, in this chapter, emphasis is placed on presenting some of the typical numerical elements upon which some numerical solutions are based (the list is not complete and an infinite number of elements can be developed). A generic element is shown schematically in Fig. 10.1, and it requires the information on the element geometry, and strength of singularity, in order to calculate the induced potential and velocity increments at an arbitrary point P (xP, yP, zP).

xP, yP, zP Panel geometry Singularity strength

FIGURE 10.1

Schematic description of a generic panel influence coefficient calculation.

For simplicity, the symbol A is dropped in the following description of the singularity elements. However, it must be clear that the values of the velocity potential and velocity components are incremental values and can be added up according to the principle of superposition.

In the following sections some two-dimensional elements will be pre­sented, whose derivation is rather simple. Three-dimensional elements will be presented later and their complexity increases with the order of the polynomial approximation of the singularity strength. Also, the formulation is derived in the panel frame of reference and when these formulas are used in any other “global coordinate system,” the corresponding coordinate transformations must be used (for rotations and translations).