Linear and nonlinear barotropic instability of geostrophic shear layers

1991 ◽  
Vol 224 ◽  
pp. 49-76 ◽  
Author(s):  
L. J. Pratt ◽  
J. Pedlosky
Keyword(s):  
Shear Layer ◽  
Nonlinear Evolution ◽  
Shear Layers ◽  
Linear Instability ◽  
Barotropic Instability ◽  
Reasonable Estimate ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Geostrophic Shear

The linear, weakly nonlinear and strongly nonlinear evolution of unstable waves in a geostrophic shear layer is examined. In all cases, the growth of initially small-amplitude waves in the periodic domain causes the shear layer to break up into a series of eddies or pools. Pooling tends to be associated with waves having a significant varicose structure. Although the linear instability sets the scale for the pooling, the wave growth and evolution at moderate and large amplitudes is due entirely to nonlinear dynamics. Weakly nonlinear theory provides a catastrophic time ts at which the wave amplitude is predicted to become infinite. This time gives a reasonable estimate of the time observed for pools to detach in numerical experiments with marginally unstable and rapidly growing waves.

scholarly journals Models for strongly-nonlinear evolution of long internal waves in a two-layer stratification

2007 ◽  
Vol 14 (1) ◽  
pp. 31-47 ◽  
Author(s):  
T. Sakai ◽  
L. G. Redekopp
Keyword(s):  
Internal Waves ◽  
Nonlinear Evolution ◽  
Phase Speed ◽  
Environmental Parameters ◽  
Exact Expression ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Polynomial Fit ◽  
Nonlinear Relation ◽  
Nonlinear Extensions

Abstract. Models describing the evolution of long internal waves are proposed that are based on different polynomial approximations of the exact expression for the phase speed of uni-directional, fully-nonlinear, infinitely-long waves in the two-layer model of a density stratified environment. It is argued that a quartic KdV model, one that employs a cubic polynomial fit of the separately-derived, nonlinear relation for the phase speed, is capable of describing the evolution of strongly-nonlinear waves with a high degree of fidelity. The marginal gains obtained by generating higher-order, weakly-nonlinear extensions to describe strongly-nonlinear evolution are clearly demonstrated, and the limitations of the quite widely-used quadratic-cubic KdV evolution model obtained via a second-order, weakly-nonlinear analysis are assessed. Data are presented allowing a discriminating comparison of evolution characteristics as a function of wave amplitude and environmental parameters for several evolution models.

Nonlinear dynamics of the elliptic instability

2010 ◽  
Vol 646 ◽  
pp. 471-480 ◽  
Author(s):  
NATHANAËL SCHAEFFER ◽  
STÉPHANE LE DIZÈS
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulations ◽  
Shear Layer ◽  
Rapid Growth ◽  
Vortex Core ◽  
Nonlinear Evolution ◽  
Core Size ◽  
Layer Formation ◽  
Vortex Loop ◽  
Weakly Nonlinear

In this paper, we analyse by numerical simulations the nonlinear dynamics of the elliptic instability in the configurations of a single strained vortex and a system of two counter-rotating vortices. We show that although a weakly nonlinear regime associated with a limit cycle is possible, the nonlinear evolution far from the instability threshold is, in general, much more catastrophic for the vortex. In both configurations, we put forward some evidence of a universal nonlinear transition involving shear layer formation and vortex loop ejection, leading to a strong alteration and attenuation of the vortex, and a rapid growth of the vortex core size.

scholarly journals Shear flow instabilities in shallow-water magnetohydrodynamics

2016 ◽  
Vol 788 ◽  
pp. 767-796 ◽  
Author(s):  
J. Mak ◽  
S. D. Griffiths ◽  
D. W. Hughes
Keyword(s):  
Magnetic Field ◽  
Shear Layer ◽  
Shallow Water ◽  
Froude Number ◽  
Water System ◽  
Shear Layers ◽  
Linear Instability ◽  
Horizontal Shear ◽  
Shallow Water Magnetohydrodynamics ◽  
The Magnetic Field

Within the framework of shallow-water magnetohydrodynamics, we investigate the linear instability of horizontal shear flows, influenced by an aligned magnetic field and stratification. Various classical instability results, such as Høiland’s growth-rate bound and Howard’s semi-circle theorem, are extended to this shallow-water system for quite general flow and field profiles. In the limit of long-wavelength disturbances, a generalisation of the asymptotic analysis of Drazin & Howard (J. Fluid Mech., vol. 14, 1962, pp. 257–283) is performed, establishing that flows can be distinguished as either shear layers or jets. These possess contrasting instabilities, which are shown to be analogous to those of certain piecewise-constant velocity profiles (the vortex sheet and the rectangular jet). In both cases it is found that the magnetic field and stratification (as measured by the Froude number) are generally each stabilising, but weak instabilities can be found at arbitrarily large Froude number. With this distinction between shear layers and jets in mind, the results are extended numerically to finite wavenumber for two particular flows: the hyperbolic-tangent shear layer and the Bickley jet. For the shear layer, the instability mechanism is interpreted in terms of counter-propagating Rossby waves, thereby allowing an explication of the stabilising effects of the magnetic field and stratification. For the jet, the competition between even and odd modes is discussed, together with the existence at large Froude number of multiple modes of instability.

Numerical study of the motions within a slowly precessing sphere at low Ekman number

2001 ◽  
Vol 437 ◽  
pp. 283-299 ◽  
Author(s):  
JÉRÔME NOIR ◽  
D. JAULT ◽  
P. CARDIN
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Numerical Study ◽  
Ekman Layer ◽  
Shear Layers ◽  
Nonlinear Interactions ◽  
Geostrophic Velocity ◽  
Ekman Number ◽  
Geostrophic Shear ◽  
Good Agreement

A geostrophic circulation and a pair of oblique oscillating shear layers arise in a spherical uid cavity contained in a slowly precessing rigid body. Both are caused by the breakdown of the Ekman boundary layer at two critical circles. We rely on numerical modelling to characterize these motions for very small Ekman numbers. Both the O(E1/5) amplitude of the velocity in the oscillating shear layer and the width (also O(E1/5)) of these oblique layers are the result of in ux into the interior from the regions where the Ekman layer breaks down. The oscillating motions are confined to narrow shear layers and their amplitude decays exponentially away from the characteristic surfaces. Nonlinear interactions inside the boundary layer drive the geostrophic shear layer attached to the critical circles. This steady layer, again of O(E1/5) thickness, contains O(E−3/10) velocities. Our results are in good agreement with the experimental measurement by Malkus of the geostrophic velocity arising in a slowly precessing spheroid.

scholarly journals Experimental observation of the origin and structure of elastoinertial turbulence

2021 ◽  
Vol 118 (45) ◽  
pp. e2102350118
Author(s):  
George H. Choueiri ◽  
Jose M. Lopez ◽  
Atul Varshney ◽  
Sarath Sankar ◽  
Björn Hof
Keyword(s):  
Reynolds Number ◽  
Reynolds Numbers ◽  
Shear Layers ◽  
Viscoelastic Flows ◽  
Inertial Forces ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Wide Range ◽  
Maximum Drag Reduction ◽  
Striking Contrast

Turbulence generally arises in shear flows if velocities and hence, inertial forces are sufficiently large. In striking contrast, viscoelastic fluids can exhibit disordered motion even at vanishing inertia. Intermediate between these cases, a state of chaotic motion, “elastoinertial turbulence” (EIT), has been observed in a narrow Reynolds number interval. We here determine the origin of EIT in experiments and show that characteristic EIT structures can be detected across an unexpectedly wide range of parameters. Close to onset, a pattern of chevron-shaped streaks emerges in qualitative agreement with linear and weakly nonlinear theory. However, in experiments, the dynamics remain weakly chaotic, and the instability can be traced to far lower Reynolds numbers than permitted by theory. For increasing inertia, the flow undergoes a transformation to a wall mode composed of inclined near-wall streaks and shear layers. This mode persists to what is known as the “maximum drag reduction limit,” and overall EIT is found to dominate viscoelastic flows across more than three orders of magnitude in Reynolds number.

Experiments on subharmonic resonance in a shear layer

1995 ◽  
Vol 304 ◽  
pp. 343-372 ◽  
Author(s):  
Hyder S. Husain ◽  
Fazle Hussain
Keyword(s):  
Shear Layer ◽  
Phase Difference ◽  
Initial Phase ◽  
Subharmonic Resonance ◽  
Resonance Phenomenon ◽  
Hot Wire ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Initial Phase Difference ◽  
Laminar Shear

The subharmonic resonance phenomenon is studied using hot-wire measurements and flow visualization in an initially laminar shear layer forced with two-frequencies for various choices of the fundamental frequency f and its subharmonic f/2 with controlled initial phase difference ϕin between them. We explore the effects of the controlling parameters, namely: (i) forcing frequencies and their initial amplitudes, (ii) initial phase difference ϕin, and (iii) detuning (i.e. when the second forcing frequency is slightly different from f/2). While several of our experimental observations support predictions based on weakly nonlinear theory, others do not. We explain our data in terms of vortex dynamics concepts.

Long–Wave Morphologies in Directional Solidification

1990 ◽  
Vol 43 (5S) ◽  
pp. S85-S88
Author(s):  
D. S. Riley
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Stability Limit ◽  
Two Dimensional ◽  
Segregation Coefficient ◽  
Directionally Solidified ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Long Wave

Long–wave instabilities in a directionally–solidified binary mixture may occur in several limits. Sivashinsky identified a small–segregation–coefficient limit and obtained a weakly–nonlinear evolution equation governing subcritical two–dimensional bifurcation. Brattkus and Davis identified a near–absolute–stability limit and obtained a strongly–nonlinear evolution equation governing supercritical two–dimensional bifurcation. In this presentation these previous analyses are set into a logical framework, and a third distinguished (small–segregation–coefficient, large–surface–energy) limit identified. The corresponding strongly–nonlinear, evolution equation equation links both of the previous and describes the change from sub– to super–critical bifurcations.

Linear instability characteristics of wake‐shear layers

1992 ◽  
Vol 4 (1) ◽  
pp. 189-191 ◽  
Author(s):  
D. Wallace ◽  
L. G. Redekopp
Keyword(s):  
Shear Layers ◽  
Linear Instability

Exploitation of Subharmonics for Separated Shear Layer Control on a High-Lift Low-Pressure Turbine Using Acoustic Forcing

2013 ◽  
Vol 136 (5) ◽  
Author(s):  
Chiara Bernardini ◽  
Stuart I. Benton ◽  
Jen-Ping Chen ◽  
Jeffrey P. Bons
Keyword(s):  
Shear Layer ◽  
Separation Bubble ◽  
Boundary Layer Separation ◽  
Low Pressure ◽  
Linear Instability ◽  
Laminar Separation ◽  
Significant Control ◽  
Low Pressure Turbine ◽  
Separated Shear Layer ◽  
Sound Control

The mechanism of separation control by sound excitation is investigated on the aft-loaded low-pressure turbine (LPT) blade profile, the L1A, which experiences a large boundary layer separation at low Reynolds numbers. Previous work by the authors has shown that on a laminar separation bubble such as that experienced by the front-loaded L2F profile, sound excitation control has its best performance at the most unstable frequency of the shear layer due to the exploitation of the linear instability mechanism. The different loading distribution on the L1A increases the distance of the separated shear layer from the wall and the exploitation of the same linear mechanism is no longer effective in these conditions. However, significant control authority is found in the range of the first subharmonic of the natural unstable frequency. The amplitude of forced excitation required for significant wake loss reduction is higher than that needed when exploiting linear instability, but unlike the latter case, no threshold amplitude is found. The fluid-dynamics mechanisms under these conditions are investigated by particle image velocimetry (PIV) measurements. Phase-locked PIV data gives insight into the growth and development of structures as they are shed from the shear layer and merge to lock into the excited frequency. Unlike near-wall laminar separation sound control, it is found that when such large separated shear layers occur, sound excitation at subharmonics of the fundamental frequency is still effective with high-Tu levels.

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