LX. Expansion in series of the exact solution for compressible flow past a circular or an elliptic cylinder

Publisher Website
Google Scholar
Abstract
This problem has been treated by various writers by successive approximation from the potential flow solution. The following method is practically just as laborious as previous methods but has the theoretical advantages of (a) using the exact hodograph equations of flow, and (b) giving a sequence of equations for the coefficients in the expansion in powers of “q” (“q” is the local stream velocity with the velocity of sound at infinity taken as unity). From these equations each coefficient may be determined in terms of those preceding. The two-series giving the distribution of q along the boundary of the cylinder and along the axis of x, together with the two hodograph expressions for the rectangular co-ordinates, are all expansions of the true analytic solution, but unfortunately a parameter “b 1,” which may be expanded in powers of “ u” the Mach number, is involved. Unless, which is in fact so up to the term q 5 for the circular boundary, b 1 occurs only as a linear factor in the second of these series, the expansion used for finding b 1(u), and hence the critical Mach number, is not the analytic expansion but merely an approximation formed by neglecting higher powers of q and so the coefficients of these powers. This is not at first sight apparent. Convergence is fair but, judging by the numerical results, at least one more term in q 7 is needed to get an accurate value of the critical Mach number.

This publication has 0 references indexed in Scilit: