Stability Of Stratified Shear Flows Research Articles

A Physical Interpretation of the Wind-Wave Instability as Interacting Waves

AbstractOne mechanism for the growth of ocean surface waves by wind is through a shear instability that was first described by Miles in 1957. A physical interpretation of this wind-wave instability is provided in terms of the interaction of the surface gravity wave with perturbations of vorticity within the critical layer—a near-singularity in the airflow where the background flow speed matches that of the surface gravity wave. This physical interpretation relies on the fact that the vertical velocity field is slowly varying across the critical layer, whereas both the displacement and vorticity fields vary rapidly. Realizing this allows for the construction of a physically intuitive description of the critical layer vorticity perturbations that may be approximated by a simple vortex sheet model, the essence of the wind-wave instability can then be captured through the interaction of the critical layer vorticity with the surface gravity wave. This simple model is then extended to account for vorticity perturbations in the airflow profile outside of the critical layer and is found to lead to an exact description of the linear stability problem that is also computationally efficient. The interpretation allows, in general, for the incorporation of sheared critical layers into the “wave interaction theory” that is commonly used to provide a physical description and rationalization of results in the stability of stratified shear flows.

Stability of shear flows with multilayered density stratification and monotonic velocity profiles having no inflection points

The stability of stratified shear flows with multilayered density distributions and monotonic velocity profiles without inflection points is studied using a class of flows of an ideal incompressible three–layer medium as a model. The flow velocity is assumed to be increasing from zero at the bottom () to its maximum value (when ), with everywhere. It is shown that such flows have two modes of eigenoscillations, fast and slow, which are both unstable for all wavelengths when the bulk Richardson number is small enough, , and are stabilized at sufficiently high J. For each mode, the configuration of its instability domain and the growth rate behaviour are studied in detail, as well as their dependence on the wavelength, J, relative magnitudes of density jumps, and the distance between the jumps. Physical mechanisms of the instability and the possibility of extending the results obtained to flows with a greater number of density interfaces are discussed.

On Formal Stability of Stratified Shear Flows

On Formal Stability of Stratified Shear Flows

Stability of stratified shear flows in channels with variable cross sections

For the instability problem of density stratified shear flows in sea straits with variable cross sections, a new semielliptical instability region is found. Furthermore, the instability of the bounded shear layer is studied in two cases: (i) the density which takes two different constant values in two layers and (ii) the density which takes three different constant values in three layers. In both cases, the dispersion relation is found to be a quartic equation in the complex phase velocity. It is found that there are two unstable modes in a range of the wave numbers in the first case, whereas there is only one unstable mode in the second case.

Instability in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves

Instability in homogeneous and density stratified shear flows may be interpreted in terms of the interaction of two (or more) otherwise free waves in the velocity and density profiles. These waves exist on gradients of vorticity and density, and instability results when two fundamental conditions are satisfied: (I) the phase speeds of the waves are stationary with respect to each other (“phase-locking“), and (II) the relative phase of the waves is such that a mutual growth occurs. The advantage of the wave interaction approach is that it provides a physical interpretation to shear flow instability. This paper is largely intended to purvey the basics of this physical interpretation to the reader, while both reviewing and consolidating previous work on the topic. The interpretation is shown to provide a framework for understanding many classical and nonintuitive results from the stability of stratified shear flows, such as the Rayleigh and Fjørtoft theorems, and the destabilizing effect of an otherwise stable density stratification. Finally, we describe an application of the theory to a geophysical-scale flow in the Fraser River estuary.

Hydromagnetic stability of stratified shear flows in the presence of cross flow

Hydromagnetic stability of stratified shear flows in the presence of cross flow

Bounds on the phase speed and growth rate of the extended Taylor–Goldstein problem

In this paper we consider the extended Taylor–Goldstein problem of hydrodynamic stability dealing with the stability of stratified shear flows in sea straits of arbitrary cross section. For this problem we have obtained the upper and lower bounds on the phase velocity of neutral modes and estimates for the growth rate of unstable modes. Furthermore, we have found a semielliptical instability region depending on the minimum Richardson number and parabolic instability regions which intersect the semielliptical instability region for a class of basic flows.

Philip Gerald Drazin

Philip Gerald Drazin

The stability of stratified shear flows of an inviscid compressible fluid in MHD

By the generalized progressing wave expansion method, the effect of a magnetic field on the linear stability of the stratified horizontal flow of an inviscid compressible fluid has been studied. It is found that the magnetic field stabilizes the system.

The stability of a sheared density interface

This study investigates the stability of stratified shear flows when the density interface is much thinner than, and displaced with respect to, the velocity interface. Theoretical results obtained from the Taylor–Goldstein equation are compared with experiments performed in mixing layer channels. In these experiments a row of spanwise vortex tubes forms at the level of maximum velocity gradient which, because of the profile asymmetry, is displaced from the mean interface level. As the bulk Richardson number is lowered from a high positive value the effects of these vortex tubes become more pronounced. Initially the interface cusps under their influence, then thin wisps of fluid are drawn from the cusps into asymmetric Kelvin–Helmholtz billows. At lower Richardson numbers increasingly more fluid is drawn into these billows. The inherent asymmetry of flows generated in mixing layer channels is shown to preclude an effective study of the Holmboe instability. Statically unstable flows (negative Richardson numbers) are subject to the Rayleigh–Taylor instability. However, if the absolute value of the Richardson number is sufficiently small the Kelvin–Helmholtz instability dominates initially.

Stability of magnetohydrodynamic stratified shear flows

A study is made of the stability of a stratified shear flow in a perfectly conducting fluid in the presence of an external magnetic field aligned with the flow. A semi-circle theorem for the present hydromagnetic case is proved. The magnetic field is found to have a stabilizing effect on the flow. The Rayleigh-Taylor instability problem in a stratified conducting fluid is discussed. Finally, a study is made of the absorption of wave energy by the mean flow in the hydromagnetic case by considering a shear flow with an anti-symmetric velocity profile given byU=tanhz. Unlike the hydrodynamic case, it is found that, in the critical layer atU=0, the transfer of the wave energy to the mean flow occurs for any value of the Richardson number. This result implies again the stabilizing effect of the magnetic field on the shear flow.

Richardson’s criterion for the stability of stratified shear flow

Chandrasekhar’s derivation, from energy considerations, of the sufficient condition R>1/4 for stability of a stratified shear flow, in which R is the local Richardson number, is amended to obtain R>1, in agreement with the earlier arguments of Richardson, Taylor, and Prandtl(L. F. Richardson, Proc. R. Soc. London Ser. A 97, 354 (1920); G. I. Taylor, Rapp. P. V. Reun. Cons. Permanent Int. d’Exploration Mer 76, 35 (1931); Sci. Pap. 2, 240; L. Prandtl, Vorträge aus dem Gebiete der Aerodynamik und verwandten Gebieten (Springer, Berlin, 1930), pp. 1–7; L. Prandtl, Führer durch die Strömungslehre (Vieweg, Braunschweig, 1942), p. 337; Essentials of Fluid Dynamics [English translation of 1949 edition of Führer durch die Strömungslehre (Hafner, New York, 1952), pp. 381 and 382]).

The stability of buoyancy-driven rolls aligned with a shear flow when the temperature gradient is nonlinear

The buoyancy driven instability of rolls aligned with a shear flow of a Boussinesq fluid is considered. It is supposed that the temperature varies not only with height but also in the direction of the flow thus allowing the vertical temperature profile to be nonlinear. Conditions are derived for the neglect of the horizontal temperature gradient in the perturbation equations. These are (i) that the vertical temperature gradient varies sufficiently slowly in the direction of the flow and (ii) that Pr h/d≫1, where Pr is the Prandtl number and h and d are the length scales for the vertical variation of velocity and temperature, respectively. Under these conditions, stability is governed by the equation for Bénard convection. The principle of exchange of stabilities is proved for an arbitrary temperature gradient profile, with isothermal, free-surface boundary conditions, and the equation is solved numerically and by an asymptotic method for a model thermal boundary layer. The analysis is then applied to an experimental study of spoke patterns, observed in the growth of electronic materials. This work also indicates that a previous investigation of the stability of stratified shear flow has a restricted range of validity.

Stability of stratified shear flows

For a steady plane parallel flow of an inviscid, incompressible fluid of variable density under gravity, it is shown that the complex wave velocity for any unstable mode lies in a semiellipse-type region whose major axis coincides with the diameter of Howard's semicircle, while its minor axis depends on the stratification. If kc i denotes the complex part of wave frequency and J 0 the minimum of the local Richardson number over the flow domain, it is further established that kc i → 0+ as J 0 → 1 4 − . The case of free upper surface and conditional reduction dependent on the curvature of the basic velocity of the unstable region is also studied.

Sufficient Conditions for Stability of Rotating Flows

This is a study of sufficient conditions for the stability of rotating flows and swirling flows of perfect conducting fluid with circular magnetic fields. Miles’ theorem [1] and Howard’s semicircle theorem [2] on the stability of stratified shear flows are extended to rotating flows with discontinuous swirl velocity. Some stronger sufficient conditions for stability are established. Some sufficient conditions for the stability and instability of the perfect conductive fluids subjected to a transverse circular magnetic field are also studied.

Stability of stratified shear flows

The concept of stability is discussed and results of the linear inviscid theory are reviewed. Examples are given to illustrate the point that ‘stable stratification’ can, in some circumstances, be destabilizing due to the vorticity generated by non-homogeneity. The linear initial value problem is discussed and related to the more usual normal mode approach. Finally, the weakly non-linear theory and the non-linear critical layer theory are outlined.

The stability of stratified shear flows

Small perturbations of a parallel shear flow U(z) in an inviscid, incompressible, stably stratified fluid of density ρ(z) are considered, for which the principal measure of stability is the Richardson number, R. For an arbitrary velocity and density profile we discuss the problem of determining whether a curve of neutral stability has adjacent unstable regions in an (α, R) plane, where α is the wave-number of the disturbance. Neutral curves bounding unstable regions are then obtained for a triangular jet flow in conjunction with various density distributions. A comparison is also made between the stability characteristics of jet and shear flows with corresponding density structures.

Experiments on the Stability of Stratified Shear Flows

Some experiments are described in which a shear flow is developed in a stratified brine solution contained in a long, closed tube of rectangular cross section. The fluid motion is produced by tilting the tube from an initial horizontal position, and the flow thus generated is found to be uniformly accelerating and easily expressed in terms of the initial density profile in the tube until disrupted by instability. (The details of the method are given in an earlier paper.) The particular case of a diffuse interface between two homogeneous layers of different densities that move with equal and opposite speeds is discussed in some detail, and the transition to turbulence is described. The transition is characterized by the formation of waves that develop into rolls of a very beautiful form. Measurements at the onset of the wave motion are compared with theoretical predictions.