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

2011 ◽  
Vol 64 (6) ◽  
Author(s):  
Jeffrey R. Carpenter ◽  
Edmund W. Tedford ◽  
Eyal Heifetz ◽  
Gregory A. Lawrence
Keyword(s):  
Shear Flow ◽  
Physical Interpretation ◽  
Shear Flows ◽  
Flow Instability ◽  
Phase Locking ◽  
River Estuary ◽  
Wave Interaction ◽  
The Stability ◽  
Interaction Approach ◽  
The 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.

Shear flow instability in a conducting viscous fluid

1973 ◽  
Vol 57 (3) ◽  
pp. 481-490
Author(s):  
B. Roberts
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Viscous Fluid ◽  
Flow Instability ◽  
Approximate Solutions ◽  
Viscous Fluids ◽  
Neutral Stability ◽  
Parallel Magnetic Field ◽  
The Stability ◽  
Shear Flow Instability

The effect of a parallel magnetic field upon the stability of the plane interface between two conducting viscous fluids in uniform relative motion is considered. A parameter reduction, which has not previously been noted, is employed to facilitate the solution of the problem. Neutral stability curves for unrestricted ranges of the governing parameters are found, and the approximate solutions of other authors are examined in this light.

A Buoyancy–Vorticity Wave Interaction Approach to Stratified Shear Flow

2008 ◽  
Vol 65 (8) ◽  
pp. 2615-2630 ◽  
Author(s):  
N. Harnik ◽  
E. Heifetz ◽  
O. M. Umurhan ◽  
F. Lott
Keyword(s):  
Shear Flow ◽  
Baroclinic Instability ◽  
Time Integration ◽  
Wave Interaction ◽  
Integration Scheme ◽  
Flow Phenomena ◽  
Time Integration Scheme ◽  
Stratified Shear Flow ◽  
Interaction Approach ◽  
Wave Kernel

Abstract Motivated by the success of potential vorticity (PV) thinking for Rossby waves and related shear flow phenomena, this work develops a buoyancy–vorticity formulation of gravity waves in stratified shear flow, for which the nonlocality enters in the same way as it does for barotropic/baroclinic shear flows. This formulation provides a time integration scheme that is analogous to the time integration of the quasigeostrophic equations with two, rather than one, prognostic equations, and a diagnostic equation for streamfunction through a vorticity inversion. The invertibility of vorticity allows the development of a gravity wave kernel view, which provides a mechanistic rationalization of many aspects of the linear dynamics of stratified shear flow. The resulting kernel formulation is similar to the Rossby-based one obtained for barotropic and baroclinic instability; however, since there are two independent variables—vorticity and buoyancy—there are also two independent kernels at each level. Though having two kernels complicates the picture, the kernels are constructed so that they do not interact with each other at a given level.

scholarly journals Vorticity inversion and action-at-a-distance instability in stably stratified shear flow

2011 ◽  
Vol 670 ◽  
pp. 301-325 ◽  
Author(s):  
A. RABINOVICH ◽  
O. M. UMURHAN ◽  
N. HARNIK ◽  
F. LOTT ◽  
E. HEIFETZ
Keyword(s):  
Shear Flow ◽  
Energy Flux ◽  
Critical Level ◽  
Flow Instability ◽  
Wave Interaction ◽  
Interfacial Waves ◽  
Interfacial Wave ◽  
Flux Divergence ◽  
Action At A Distance ◽  
Distance Interaction

The somewhat counter-intuitive effect of how stratification destabilizes shear flows and the rationalization of the Miles–Howard stability criterion are re-examined in what we believe to be the simplest example of action-at-a-distance interaction between ‘buoyancy–vorticity gravity wave kernels’. The set-up consists of an infinite uniform shear Couette flow in which the Rayleigh–Fjørtoft necessary conditions for shear flow instability are not satisfied. When two stably stratified density jumps are added, the flow may however become unstable. At each density jump the perturbation can be decomposed into two coherent gravity waves propagating horizontally in opposite directions. We show, in detail, how the instability results from a phase-locking action-at-a-distance interaction between the four waves (two at each jump) but can as well be reasonably approximated by the interaction between only the two counter-propagating waves (one at each jump). From this perspective the nature of the instability mechanism is similar to that of the barotropic and baroclinic ones. Next we add a small ambient stratification to examine how the critical-level dynamics alters our conclusions. We find that a strong vorticity anomaly is generated at the critical level because of the persistent vertical velocity induction by the interfacial waves at the jumps. This critical-level anomaly acts in turn at a distance to dampen the interfacial waves. When the ambient stratification is increased so that the Richardson number exceeds the value of a quarter, this destructive interaction overwhelms the constructive interaction between the interfacial waves, and consequently the flow becomes stable. This effect is manifested when considering the different action-at-a-distance contributions to the energy flux divergence at the critical level. The interfacial-wave interaction is found to contribute towards divergence, that is, towards instability, whereas the critical-level–interfacial-wave interaction contributes towards an energy flux convergence, that is, towards stability.

A laboratory study of the minimum wind speed for wind wave generation

1988 ◽  
Vol 192 ◽  
pp. 339-364 ◽  
Author(s):  
Kimmo K. Kahma ◽  
Mark A. Donelan
Keyword(s):  
Wind Speed ◽  
Shear Flow ◽  
Wind Wave ◽  
Flow Instability ◽  
Wave Generation ◽  
Wind Speeds ◽  
Instability Theory ◽  
Low Wind Speeds ◽  
Shear Flow Instability ◽  
The Waves

The minimum wind speed for wind wave generation has been investigated in a laboratory wind-wave flume using a sensitive slope gauge to measure the initial wavelets about 10 μm high. The growth at very low wind speeds was higher than predicted by the viscous shear-flow instability theory. Assuming that the growth is exponential, the inception wind speed at which the growth rate becomes positive can be defined. It occurred at (friction velocity) u* ≈ 2 cm/s, somewhat lower than the u* ≈ 4–5 cm/s predicted by shear-flow instability theory. However, the observed growth rates were close to the theory at higher wind speeds when the waves were higher than 1 mm. The effect of temperature on the wind speed at which the waves become readily visible is shown to be appreciable and in keeping with the temperature dependent viscous damping. Other sources of growth are discussed. Our estimates show that the Phillips resonance mechanism might be sufficiently effective to generate the observed growth at very low wind speeds.

A note on surface waves generated by shear-flow instability

2001 ◽  
Vol 447 ◽  
pp. 173-177 ◽  
Author(s):  
JOHN MILES
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Asymptotic Approximation ◽  
Variational Formulation ◽  
Numerical Solutions ◽  
Flow Instability ◽  
Analytical Description ◽  
Short Waves ◽  
The Stability ◽  
Shear Flow Instability

Morland, Saffman & Yuen's (1991) study of the stability of a semi-infinite, concave shear flow bounded above by a capillary–gravity wave, for which they obtained numerical solutions of Rayleigh's equation, is revisited. A variational formulation is used to construct an analytical description of the unstable modes for the exponential velocity profile U = U0 exp(y/d), −∞ < y [les ] 0. The assumption of slow waves ([mid ]c[mid ] [Lt ] U0) yields an approximation that agrees with the numerical results of Morland et al. The assumption of short waves (kd [Gt ] 1) yields Shrira's (1993) asymptotic approximation.

The Effect of Boundaries on the Stability of Inviscid Stratified Shear Flows

1976 ◽  
Vol 43 (2) ◽  
pp. 243-248 ◽  
Author(s):  
F. Einaudi ◽  
D. P. Lalas
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Shear Flows ◽  
Infinite Domain ◽  
Hyperbolic Tangent ◽  
Stratified Shear Flow ◽  
The Stability ◽  
Exchange Of Stability

The influence of the presence and position of solid boundaries on the stability of an inviscid, stratified shear flow, is examined numerically for the case of a hyperbolic tangent velocity profile and an exponentially decreasing density. The presence of solid boundaries is shown to stabilize short wavelengths and destabilize large wavelengths. Furthermore, extra unstable modes, not present in an infinite domain, are found for large wavelengths, both for symmetric and asymmetric boundaries. Finally, the validity of the principle of exchange of stability is examined, and it is shown to be unreliable even for the case of symmetric boundaries.

scholarly journals A theory of MHD instability of an inhomogeneous plasma jet

2010 ◽  
Vol 77 (3) ◽  
pp. 315-337 ◽  
Author(s):  
ANATOLY S. LEONOVICH
Keyword(s):  
Shear Flow ◽  
Plasma Flow ◽  
Flow Instability ◽  
Inhomogeneous Plasma ◽  
Plasma Jet ◽  
Wkb Approximation ◽  
Tangential Discontinuity ◽  
Radial Coordinate ◽  
Helmholtz Instability ◽  
The Stability

AbstractA problem of the stability of an inhomogeneous axisymmetric plasma jet in a parallel magnetic field is solved. The jet boundary becomes, under certain conditions, unstable relative to magnetosonic oscillations (Kelvin–Helmholtz instability) in the presence of a shear flow at the jet boundary. Because of its internal inhomogeneity the plasma jet has resonance surfaces, where conversion takes place between various modes of plasma magnetohydrodynamic (MHD) oscillations. Propagating in inhomogeneous plasma, fast magnetosonic waves drive the Alfven and slow magnetosonic (SMS) oscillations, tightly localized across the magnetic shells, on the resonance surfaces. MHD oscillation energy is absorbed in the neighbourhood of these resonance surfaces. The resonance surfaces disappear for the eigenmodes of SMS waves propagating in the jet waveguide. The stability of the plasma MHD flow is determined by competition between the mechanisms of shear flow instability on the boundary and wave energy dissipation because of resonant MHD-mode coupling. The problem is solved analytically, in the Wentzel, Kramers, Brillouin (WKB) approximation, for the plasma jet with a boundary in the form of a tangential discontinuity over the radial coordinate. The Kelvin–Helmholtz instability develops if plasma flow velocity in the jet exceeds the maximum Alfven speed at the boundary. The stability of the plasma jet with a smooth boundary layer is investigated numerically for the basic modes of MHD oscillations, to which the WKB approximation is inapplicable. A new 'unstable mode of MHD oscillations has been discovered which, unlike the Kelvin–Helmholtz instability, exists for any, however weak, plasma flow velocities.

Wave breakdown in stratified shear flows

1977 ◽  
Vol 79 (3) ◽  
pp. 481-497 ◽  
Author(s):  
M. T. Landahl ◽  
W. O. Criminale
Keyword(s):  
Shear Flows ◽  
Flow Instability ◽  
Stability Boundary ◽  
Small Scale ◽  
Mean Flow ◽  
Order Unity ◽  
Stable Regime ◽  
Density Interface ◽  
Density Interfaces ◽  
The Stability

The wave-mechanical condition (Landahl 1972) for breakdown of an unsteady laminar flow into strong small-scale secondary instabilities is applied to some simple stratified inviscid shear flows. The cases considered have one or two discrete density interfaces and simple discontinuous or continuous velocity profiles. A primary wavelike disturbance to such a flow produces a perturbation velocity that is discontinuous at a density interface. The resulting instantaneous system, defined as the sum of the mean flow and the primary oscillation, develops a local secondary shear-flow instability that has a group velocity equal to the arithmetic mean of the instantaneous velocities on the two sides of the interface. According to the breakdown criterion, the disturbed flow will become critical whenever this velocity reaches a value equal to the phase velocity of the primary wave. The calculations show that for a single density interface breakdown may occur for low to moderate wave amplitudes in a fairly narrow range of Richardson numbers on the stable side of the stability boundary. On the other hand, in the unstable regime maximum wave slopes of order unity may be reached before breakdown occurs; this conclusion is in qualitative agreement with experiments. When the system includes two density interfaces, it is found that there exists a range of high Richardson numbers far into the stable regime for which breakdown may take place even for very small and zero wave interface deflexions.

scholarly journals Combined electrokinetic and shear flows control colloidal particle distribution across microchannel cross-sections

Soft Matter ◽  
2021 ◽  
Author(s):  
Varun Lochab ◽  
Shaurya Prakash
Keyword(s):  
Shear Flow ◽  
Cross Sections ◽  
Lift Force ◽  
Shear Flows ◽  
Colloidal Particle ◽  
Particle Distribution ◽  
Particle Migration ◽  
Flow Parameters

We quantify and investigate the effects of flow parameters on the extent of colloidal particle migration and the corresponding electrophoresis-induced lift force under combined electrokinetic and shear flow.

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