The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 1 Shear aligned convection, pairing, and braid instabilities

2012 ◽  
Vol 708 ◽  
pp. 5-44 ◽  
Author(s):  
A. Mashayek ◽  
W. R. Peltier
Keyword(s):  
Stagnation Point ◽  
Three Dimensional ◽  
Negative Influence ◽  
Finite Amplitude ◽  
Shear Instability ◽  
Background Flow ◽  
Shear Modes ◽  
Stratified Shear Flow ◽  
Secondary Instabilities ◽  
Background Shear

AbstractWe study the competition between various secondary instabilities that co-exist in a preturbulent stratified parallel flow subject to Kelvin–Helmholtz instability. In particular, we investigate whether a secondary braid instability might emerge prior to the overturning of the statically unstable regions that develop in the cores of the primary Kelvin–Helmholtz billows. We identify two groups of instabilities on the braid. One group is a shear instability which extracts its energy from the background shear and is suppressed by the straining contribution of the background flow. The other group, which seems to have no precedent in the literature, includes phase-locked modes which grow at the stagnation point on the braid and are almost entirely driven by the straining contributions of the background flow. For the latter group, the braid shear has a negative influence on the growth rate. Our analysis demonstrates that the probability of finite-amplitude growth of both braid instabilities is enhanced with increasing Reynolds number and Richardson number. We also show that the possibility of emergence of braid instabilities decreases with the Prandtl number for the shear modes and increases for the stagnation point instabilities. Through detailed non-separable linear stability analysis, we show that both braid instabilities are fundamentally three dimensional with the shear modes being of small wavenumbers and the stagnation point modes dominating at large wavenumber.

The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 2 The influence of stratification

2012 ◽  
Vol 708 ◽  
pp. 45-70 ◽  
Author(s):  
A. Mashayek ◽  
W. R. Peltier
Keyword(s):  
Three Dimensional ◽  
Reynolds Numbers ◽  
Shear Instability ◽  
Developed Turbulence ◽  
Wide Range ◽  
Stratified Shear Flow ◽  
High Reynolds ◽  
The Individual ◽  
Secondary Instabilities ◽  
Alternate Routes

AbstractThe linear stability analyses described in Mashayek & Peltier (J. Fluid Mech., vol. 708, 2012, 5–44, hereafter MP1) are extended herein in an investigation of the influence of stratification on the evolution of secondary instabilities to which an evolving Kelvin–Helmholtz (KH) wave is susceptible in an initially unstable parallel stratified shear layer. We show that over a wide range of background stratification levels, the braid shear instability has a higher probability of emerging at early stages of the flow evolution while the secondary convective instability (SCI), which occurs in the eyelids of the individual Kelvin ‘cats eyes’, will remain a relevant and dominant instability at high Reynolds numbers. The evolution of both modes is greatly influenced by the background stratification. Various other three-dimensional secondary instabilities are found to exist over a wide range of stratification levels. In particular, the stagnation point instability (SPI), which was discussed in detail in MP1, may be of great potential importance providing alternate routes for transition of an initially two-dimensional KH wave into fully developed turbulence. The energetics of the secondary instabilities revealed by our simulations are analysed in detail and the preturbulent mixing properties are studied.

The influence of stratification on secondary instability in free shear layers

1991 ◽  
Vol 227 ◽  
pp. 71-106 ◽  
Author(s):  
G. P. Klaassen ◽  
W. R. Peltier
Keyword(s):  
Kinetic Energy ◽  
Initial Density ◽  
Three Dimensional ◽  
Primary Source ◽  
Underlying Mechanism ◽  
Finite Amplitude ◽  
Secondary Instability ◽  
Shear Instability ◽  
Complete Analysis ◽  
Secondary Instabilities

We analyse the stability of horizontally periodic, two-dimensional, finite-amplitude Kelvin-Helmholtz billows with respect to infinitesimal three-dimensional perturbations having the same streamwise wavelength for several different levels of the initial density stratification. A complete analysis of the energy budget for this class of secondary instabilities establishes that the contribution to their growth from shear conversion of the basic-state kinetic energy is relatively insensitive to the strength of the stratification over the range of values considered, suggesting that dynamical shear instability constitutes the basic underlying mechanism. Indeed, during the initial stages of their growth, secondary instabilities derive their energy predominantly from shear conversion. However, for initial Richardson numbers between 0.065 and 0.13, the primary source of kinetic energy for secondary instabilities at the time the parent wave climaxes is in fact the conversion of potential energy by convective overturning in the cores of the individual billows. A comparison between the secondary instability properties of unstratified Kelvin-Helmholtz billows and Stuart vortices is made, as the latter have often been assumed to provide an adequate approximation to the former. Our analyses suggest that the Stuart vortex model has several shortcomings in this regard.

Dynamics of vorticity defects in stratified shear flow

2012 ◽  
Vol 694 ◽  
pp. 292-331 ◽  
Author(s):  
N. J. Balmforth ◽  
A. Roy ◽  
C. P. Caulfield
Keyword(s):  
Linear Instability ◽  
Shear Instability ◽  
Two Dimensional ◽  
Computationally Efficient ◽  
Stratified Shear Flow ◽  
Prandtl Numbers ◽  
High Reynolds ◽  
Efficient Exploration ◽  
Secondary Instabilities ◽  
Sharp Interfaces

AbstractWe consider the linear stability and nonlinear evolution of two-dimensional shear flows that take the form of an unstratified plane Couette flow that is seeded with a localized ‘defect’ containing sharp density and vorticity variations. For such flows, matched asymptotic expansions furnish a reduced model that allows a straightforward and computationally efficient exploration of flows at sufficiently high Reynolds and Péclet numbers that sharp density and vorticity gradients persist throughout the onset, growth and saturation of instability. We are thereby able to study the linear and nonlinear dynamics of three canonical variants of stratified shear instability: Kelvin–Helmholtz instability, the Holmboe instability, and the lesser-considered Taylor instability, all of which are often interpreted in terms of the interactions of waves riding on sharp interfaces of density and vorticity. The dynamics near onset is catalogued; if the interfaces are sufficiently sharp, the onset of instability is subcritical, with a nonlinear state existing below the linear instability threshold. Beyond onset, both Holmboe and Taylor instabilities are susceptible to inherently two-dimensional secondary instabilities that lead to wave mergers and wavelength coarsening. Additional two-dimensional secondary instabilities are also found to appear for higher Prandtl numbers that take the form of parasitic Holmboe-like waves.

Secondary instabilities in rapidly rotating fluids: inertial wave breakdown

1999 ◽  
Vol 382 ◽  
pp. 283-306 ◽  
Author(s):  
R. R. KERSWELL
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Small Scale ◽  
Inertial Waves ◽  
Rotating Fluids ◽  
Initial Excitation ◽  
Geostrophic Flows ◽  
Inertial Wave ◽  
Secondary Instabilities ◽  
Wave Breakdown

Inertial waves are a ubiquitous feature of rapidly rotating fluids. Although much is known about their initial excitation, little is understood about their stability. Experiments indicate that they are generically unstable and in many cases catastrophically so, quickly causing the whole flow to collapse to small-scale disorder. The linear stability of two three-dimensional inertial waves observed to break down in the laboratory is considered here at experimentally small but finite Ekman numbers of [les ]10−4. Surprisingly small threshold amplitudes for instability are found. The results support the conjecture that triad resonances are the generic mechanism for secondary instability in rapidly rotating fluids but also highlight the ability of geostrophic flows to derive energy through a finite-amplitude inertial wave. This latter finding may go some way to explaining the significant mean circulations typically observed in inertial wave experiments.

scholarly journals Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities

2017 ◽  
Vol 832 ◽  
pp. 409-437 ◽  
Author(s):  
Dan Lucas ◽  
C. P. Caulfield ◽  
Rich R. Kerswell
Keyword(s):  
Coherent Structures ◽  
Stable Stratification ◽  
Finite Amplitude ◽  
Shear Instability ◽  
Stratified Turbulence ◽  
Layer Formation ◽  
Horizontal Shear ◽  
Chaotic Flow ◽  
Background Shear ◽  
First Time

We consider turbulence in a stratified ‘Kolmogorov’ flow, driven by horizontal shear in the form of sinusoidal body forcing in the presence of an imposed background linear stable stratification in the third direction. This flow configuration allows the controlled investigation of the formation of coherent structures, which here organise the flow into horizontal layers by inclining the background shear as the strength of the stratification is increased. By numerically converging exact steady states from direct numerical simulations of chaotic flow, we show, for the first time, a robust connection between linear theory predicting instabilities from infinitesimal perturbations to the robust finite-amplitude nonlinear layered state observed in the turbulence. We investigate how the observed vertical length scales are related to the primary linear instabilities and compare to previously considered examples of shear instability leading to layer formation in other horizontally sheared flows.

Experiments on the nonlinear stages of free-shear-layer transition

1972 ◽  
Vol 56 (4) ◽  
pp. 695-719 ◽  
Author(s):  
Richard W. Miksad
Keyword(s):  
Shear Layer ◽  
Vortex Formation ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Longitudinal Vortex ◽  
Free Shear Layer ◽  
Layer Transition ◽  
Downstream Distance ◽  
Secondary Instabilities ◽  
Mode A

An experimental study is made of the instability and transition of a laminar free shear layer by sound excitation. Primary emphasis is placed on the nonlinear stages of transition. Transition from laminar instability to turbulent breakdown covers approximately five wavelengths of downstream distance. The instability has six distinct regions of behaviour : a region of exponential growth described by linear theory; a nonlinear region where critical-layer effects are important, and harmonics and subharmonics are generated; a region of finite amplitude equilibration of the fundamental mode; a region of finite amplitude triggered sub-harmonic instabilities; a region of three-dimensional longitudinal vortex formation; and a final region of weak secondary instabilities and turbulent breakdown.

scholarly journals The structure and origin of confined Holmboe waves

2018 ◽  
Vol 848 ◽  
pp. 508-544 ◽  
Author(s):  
Adrien Lefauve ◽  
J. L. Partridge ◽  
Qi Zhou ◽  
S. B. Dalziel ◽  
C. P. Caulfield ◽  
...  
Keyword(s):  
Shear Flow ◽  
Coherent Structure ◽  
Shear Flows ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Geophysical Flows ◽  
Two Dimensional ◽  
Lateral Confinement ◽  
Stratified Shear Flow ◽  
Spatio Temporal

Finite-amplitude manifestations of stratified shear flow instabilities and their spatio-temporal coherent structures are believed to play an important role in turbulent geophysical flows. Such shear flows commonly have layers separated by sharp density interfaces, and are therefore susceptible to the so-called Holmboe instability, and its finite-amplitude manifestation, the Holmboe wave. In this paper, we describe and elucidate the origin of an apparently previously unreported long-lived coherent structure in a sustained stratified shear flow generated in the laboratory by exchange flow through an inclined square duct connecting two reservoirs filled with fluids of different densities. Using a novel measurement technique allowing for time-resolved, near-instantaneous measurements of the three-component velocity and density fields simultaneously over a three-dimensional volume, we describe the three-dimensional geometry and spatio-temporal dynamics of this structure. We identify it as a finite-amplitude, nonlinear, asymmetric confined Holmboe wave (CHW), and highlight the importance of its spanwise (lateral) confinement by the duct boundaries. We pay particular attention to the spanwise vorticity, which exhibits a travelling, near-periodic structure of sheared, distorted, prolate spheroids with a wide ‘body’ and a narrower ‘head’. Using temporal linear stability analysis on the two-dimensional streamwise-averaged experimental flow, we solve for three-dimensional perturbations having two-dimensional, cross-sectionally confined eigenfunctions and a streamwise normal mode. We show that the dispersion relation and the three-dimensional spatial structure of the fastest-growing confined Holmboe instability are in good agreement with those of the observed confined Holmboe wave. We also compare those results with a classical linear analysis of two-dimensional perturbations (i.e. with no spanwise dependence) on a one-dimensional base flow. We conclude that the lateral confinement is an important ingredient of the confined Holmboe instability, which gives rise to the CHW, with implications for many inherently confined geophysical flows such as in valleys, estuaries, straits or deep ocean trenches. Our results suggest that the CHW is an example of an experimentally observed, inherently nonlinear, robust, long-lived coherent structure which has developed from a linear instability. We conjecture that the CHW is a promising candidate for a class of exact coherent states underpinning the dynamics of more disordered, yet continually forced stratified shear flows.

Three-dimensional instabilities of steady double-diffusive interleaving

2000 ◽  
Vol 418 ◽  
pp. 297-312 ◽  
Author(s):  
OLIVER S. KERR
Keyword(s):  
Stability Analysis ◽  
Dimensional Stability ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Secondary Instability ◽  
Two Dimensional ◽  
Salinity Gradients ◽  
Secondary Instabilities ◽  
Double Diffusive

A stratified body of fluid with compensating horizontal temperature and salinity gradients can undergo an interleaving instability which takes the form of almost horizontal intrusions. As the amplitude of these intrusions grows they can undergo secondary instabilities which eventually leads to the mixing of the fluid in the interior of the intrusions. A previous study of the secondary instabilities focused on two-dimensional disturbances. These corresponded to experimental observations of that time which all seemed to indicate that flows were indeed two-dimensional. Some more recent experiments have shown that the initial secondary instability can make the flow three-dimensional, with the secondary instabilities taking the form of rolls with their axes aligned with the direction of the flow in the intrusions. Here we present a three dimensional stability analysis of steady finite-amplitude intrusions and look at the circumstances which can lead to the three-dimensional instabilities being more likely to be observed.

Resonant wave interactions in a stratified shear flow

1988 ◽  
Vol 190 ◽  
pp. 357-374 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Shear Flow ◽  
Gravity Waves ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Stratified Shear Flow ◽  
Weak Resonance ◽  
Background Shear

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

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