. TWO-DIMENSIONAL CONSTANT – STRENGTH SINGULARITY ELEMENTS

The discretization of the source, doublet, or vortex distributions in the previous section led to discrete singularity elements that are clearly not a continuous surface representation. A more refined representation of these singularity element distributions can be obtained by dividing the solid surface boundary into elements (panels). One such element is shown schematically in Fig. 10.4, and both the surface shape and the shape of the singularity strength distribution are approximated by a polynomial. In this section, for the surface representation, a straight line will be used. For the singularity strength, only the constant, linearly varying, and quadratically varying strength cases are presented, but the methodology of this section can be applied to higher-order elements.

In this section, too, three examples will be presented (source, doublet, and vortex) for evaluating the influence of the generic panel of Fig. 10.4 at an arbitrary point P. For simplicity, the formulation is derived in a panel-attached coordinate system, and the results need to be transformed back into the global coordinate system of the problem.