On a method to evaluate Fourier-Bessel series with poor convergence properties and its application to linearized supersonic free jet flow

Abstract

An analytical method based on Kummer’s series transformation is presented, which allows for the evaluation of Fourier-Bessel series with poor convergence properties and, in addition, yields the singularities of the series in closed form. This method is applied to the Fourier-Bessel series which arise as solutions of the linearized gas dynamic potential equation for the cylindrical flow field of a supersonic free jet. As an illustrative example, the presented method is applied to D. C. Pack’s classical solution for the axisymmetric free jet with initial homogeneous pressure perturbation, which in its original form cannot be evaluated directly. It is shown that in contrast to the plane jet, the flow field of the axisymmetric jet does not exhibit a strictly periodic behaviour, whereas its singularities are distributed periodically.