Finite-Amplitude Instability of the Compressible Laminar Wake. Strongly Amplified Disturbances

Abstract

The interaction between mean flow and finite-amplitude disturbances in. certain experimentally observed unstable, compressible laminar wakes is considered theoretically without explicitly assuming small amplification rates. Boundary-layer form of the two-dimensional mean-flow momentum, kinetic energy and thermal energy equations and the time-averaged kinetic energy equation of spatially growing disturbances are recast into their respective von Karman integral form which show the over-all physical coupling. The Reynolds shear stresses couple the mean flow and disturbance kinetic energies through the conversion mechanism familiar in low-speed flows. Both the mean flow and disturbance kinetic energies are coupled to the mean-flow thermal energy through their respective viscous dissipation. The work done by the disturbance pressure gradients gives rise to an additional coupling between the disturbance kinetic energy and the mean-flow thermal energy. The compressibility transformation suggested by work on turbulent shear flows is not applicable to this problem because of the accompanying ad hoc assumptions about the disturbance behavior. The disturbances of a discrete frequency which corresponds to the most unstable fundamental component, are first evaluated locally. Subsequent mean-flow and disturbance profile-shape assumptions are made in terms of a mean-flow-density Howarth variable. The compressibility transformation, which cannot convert this problem into a form identical to the low-speed problem of Ko, Kubota, and Lees because of the compressible disturbance quantities, nevertheless, yields a much simplified description of the mean flow.

Energy integral description of the development of Kelvin-Helmholtz billows

“…The technique is essentially similar to one introduced in boundary-layer theory by von Khr- mhn (1921). Such a technique for finite-amplitude instability problems w a discussed by Stuart (1958) for parallel shear flows, and more recently it has been extended to problems in a homogeneous fluid for developing shear layers where good agreement has been found with some of the details of laboratory experiments (KO et al, 1970;Liu & Lees, 1970;Liu & Gururaj, 1974). It is thus of advantage to consider our present problem in this light, particularly because we are interested in the time evolution of the billow and mean flow interactions.…”

“…It is only the subsequent dynamics, or energetics, that we deal with. The disturbance distribution, following earlier works (for instance, Liu & Lees, 1970), is taken to be described by a local linear theory but suitably modified by an unknown amplitude function, A @ ) .…”

“…In order to study the development or life cycle of the Kelvin-Helmholtz billows, we extend the technique for homogeneous shear flows (KO et al, 1970;Liu & Lees, 1970;Liu & Gururaj, 1974) to a stratified shear flow. From detailed observations in homogeneous shear flows, the appearance of instability waves in such flows coincided with large distortions of the mean flow as the instability develops into finite amplitude.…”

“…The mean flow and the developing instability interacts strongly and a formulation h s e d on integrated energy equations of both components provide an overall description of the interaction. The details of the formulation and results for homogeneous shear flows are discussed in KO et al, (1970), Liu & Lees (1970) and Liu & Gururaj (1974). The extension of the energy integral consideration to the stratified shear layer problem, with application to observations in geophysical problems, is discussed in this paper.…”

Spatial Stability of a Reactive Wake

“…Spatial development is what is observed experimentally (Freyniuth, 1966;Michalke, 1965) and, though the two types of stability analysis can be related by a Galilean transformation (Gaster, 1968), the group velocity used in this transformation is only accurate for small amplification rates. The present renewed interest in the motion of coherent structures in flows (Crow and Champagne, 1971 ;Brown and Roshko, 1974;Cantwell, 1981 ;Hussain, 1983) has led to modeling of such organized motion as a spatially growing instability wave that develops due to its nonlinear interaction with the mean flow (Liu and Lees, 1970;Menon, 1983). Such an approach has been shown to be more accurate in describing the transition to turbulence (Liu and Gururaj, 1974;Demetriades, 1971).…”