Asymptotic evaluation of three-dimensional wave packets in parallel flows

Abstract

This paper examines the three-dimensional wave packets which are generated by an initially localized pulse disturbance in an incompressible parallel flow and described by a double Fourier integral in the wavenumber space. It aims to clear up some confusion arising from the asymptotic evaluation of this integral by the method of steepest descent. In this asymptotic analysis, the calculation of the eigenvalues can be facilitated by making use of the Squire transformation. It is demonstrated that the use of the Squire transformation introduces branch points in the saddle-point equation that links the physical coordinates to the saddle-point value, regardless of whether the flow is viscous or inviscid. It is shown that the correct branch should be chosen according to the principle of analytic continuation. The saddle-point values for the three-dimensional problem should be considered to be the analytic continuation of those for the two-dimensional case where the saddle-point values can be uniquely determined. The three-dimensional wave packets in an inviscid wake flow are examined; their behaviour at large time is calculated asymptotically by the method of steepest descent in terms of the two-dimensional eigenvalue relation.