Analytical solution for potential flow across two circular cylinders using the BIE in conjunction with degenerate kernels of bipolar coordinates

Abstract

In this paper, we employ the bipolar degenerate-kernel in the boundary integral equation (BIE) to derive the analytical solution of potential flow across two identical circular cylinders. Analytical results are compared with those in Lebedev et al.. Discussions of the difference between our results and Lebedev’s solution are addressed. Velocity distribution is also plotted. Both analytical solution and velocity plots are given. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using the bipolar coordinates for a case of two cylinders. The orientation, or so called the angle of attack is considered. The result shows that the degenerate-kernel approach can be an alternative tool for analytically solving some boundary value problems, although the complex variables is always employed. The extension to 3-D problems is promising and straightforward once the degenerate kernel is available since complex variable may have difficulty.