Development of wedge-shaped disturbances originating from a point source in a three-dimensional boundary layer

Abstract

A recent study predicted possibility of existence of a new instability due to the curvature of external streamlines in three-dimensional boundary layers, besides the familiar cross-flow instability, but no reliable evidence of this phenomenon has yet been obtained in experiments. In expectation of dispersive development of the two instabilities, the present study deals with small disturbances induced by continuous forcing from a point source in the boundary layer along a yawed circular cylinder, and attempts to describe their spatial development into wedge-shaped distribution with a linear stability theory, which is applicable to both of the above instabilities. Unlike plane-wave disturbances, the point-source disturbances have an important peculiarity that their propagation is governed by a complex group velocity, and a new method based on the complex property of the group velocity is presented to predict the paths of propagation along which growth rates of disturbances are integrated. Results of this stability calculation clarify important differences in development between the cross-flow disturbances and the streamline-curvature disturbances. These differences will make it possible to observe the new mode of disturbances separately from the other in experiments.

Multi-Instability Analysis of Swept-Wing Boundary Layers Part 2. Effects of Wall Curvature and Nonparallelism on the <I>e<SUP>N</SUP></I> Method

“…For each set of these parameters satisfying the realization condition (17), the Y location and the N factor are obtained by integrations (18) and (19). Since the maximum amplification rate of these disturbances always appears at a zero ofβ i , as reported by Itoh,15) the present study focuses on the case ofβ i = 0. This system can then be regarded as a function of only one free parameter, and the real frequencyω is hereinafter taken to be this free parameter.…”

“…These methods are convenient to get an N factor, but they are of low accuracy and of no physical meaning in the calculation. For this crucial problem, Itoh 15) recently proposed a disturbance propagation theory, referred to as the complex ray theory. This theory can uniquely determine a wave number or frequency of the most unstable disturbance by using an extended kinematic wave theory 16) with a realization condition based on physical causality.…”

Structure of Absolute Instability in 3-D Boundary Layers: Part 1. Mathematical Formulation.

“…10) Another important case is when the disturbance pattern does not vary with the space coordinate Y perpendicular to the direction of basic-flow variation, as exemplified by disturbances of the wave-packet type induced by an instantaneous jet through a line slit parallel to the leading edge on a long cylinder or through an annular slit on the surface of a rotating disk.…”

Development of Localized Disturbances from an Oscillating Point Source in Compressible Boundary Layers.

“…8,9) The method is based on Whitham's kinematic wave theory 10) and is explained below. When the point source oscillates over time at an angular frequencyω, each wave mode developing from the source takes the form…”

“…Itoh 8,9) recently proposed a simple method to describe the development of localized disturbances including the wave packets and oscillating point source or vibrating ribbon problems, and demonstrated a few examples of the development of localized disturbances in a flat-plate boundary layer. His method is an extension of Whitham's kinematic wave theory 10) to the complex domain, which he called "the method of complex characteristics."…”