An Accurate Two-Dimensional Panel Method

A new panel technique which produces a globally accurate velocity field has been developed. Traditional panel techniques panelise the body into a set of linear panels and distribute some kind of singularity on the surface of these panels. The singularities normally used are sources, doublets and vorticity. Such panel methods produce high accuracy at the control point but do not produce accurate results globally and as one approaches the edges of the panels the velocity field diverges. This is known as the edge effect. Most computations that use traditional panel methods are not affected by this edge effect since these panel techniques are typically used to find the velocity/pressure of the fluid on the panelised body at certain control points where the no penetration (or no tangency) boundary condition is applied and at these points there is very little error. However, there are certain applications, like particle based flow solvers, where a globally accurate velocity field is required. It can be shown that however high the order of the singularity distribution on the panels, the edge effect is due to the mismatch of the slope between two adjacent panels. Hence, the natural approach to this problem is to panelise the body using cubic panels rather than linear panels by matching the slopes at the edge of each panel. The fundamental equations for the velocity field and potential due to such a cubic panel having a linear vorticity distribution are derived in this article. Also addressed are certain issues involved in the derivation. The accuracy of the method is then demonstrated by comparing the results with that obtained by traditional methods.