Analytical solutions for potential flow past two unequal circular cylinders using the BIE in conjunction with degenerate kernels of bipolar coordinates

Abstract

In this study, an analytical solution for the potential flow past two unequal circular cylinders is derived using the bipolar degenerate kernel within the boundary integral equation. The analytical solution for the two identical cylinders was shown to agree well with the results of Lebedev, Skalskaya, and Uflyand (1965) and Morse and Feshbach (1953). For the unequal cylinder case, the analytical results were also compared with those in Morse and Feshbach's book. We identified three noteworthy points. First, the velocity potential obtained by the present formulation is different from the constant term, although it does not have any effect on the velocity. Second, both formulas were found to differ by a sign, which can be explained by the different conventions used to define the angular coordinates in the bipolar coordinates. Third, even when employing the same definition of angular coordinates in our numerical tests, we find that the sign in Morse and Feshbach's book, as well as in Lebedev, Skalskaya, and Uflyand (1965), is still incorrect. To verify these three points, numerical results for the finite series terms were obtained. The velocity distribution for different angles of attack was plotted in bipolar coordinates for the case of two unequal circular cylinders.