Ray tracing of shear flow instability waves through a slightly inhomogeneous background flow /

Abstract

In addition to ray tracing, this work explores the mixed spatial and temporal instability problem for a Falkner-Skan-like family of velocity profiles. An analog in the mixed spatial-temporal instability problem of a well-known result due to Tollmien (1935) in the temporal instability problem is derived. The results also suggest there is a counterpart to the Rayleigh inflection point theorem for the mixed temporal and spatial instability problem.

An extension of Whitham's kinematic wave theory, which holds for modal waves in a non-conservative system, has been applied in order to find the trajectories of shear flow instability waves. Two cases of flat-plate boundary-layer transition flow are considered. In the first case, the slowly varying background flow is given by velocity profiles fitted to the experimental work of Klebanoff et al. (1962); the second case uses the recent work of Williams et al. (1984). A family of cubic-tanh profiles is fitted to the experimental instantaneous profiles thus providing the local dispersion relation which is given by a small wavenumber approximation to the Rayleigh equation. In both cases the secondary wave packets experience focussing in the location of breakdown. The results also indicate the importance of wave trapping in the breakdown process. This work is further verification of Landahl's theory of breakdown originally proposed in 1972.