Nonlinear Evolution of Multiple Helical Modes in the Near-Nozzle Region of Subsonic Circular Jets: A Weakly Nonlinear Critical-Layer Theory

2021 ◽  
pp. 137-147
Author(s):  
Zhongyu Zhang ◽  
Xuesong Wu
Keyword(s):  
Nonlinear Evolution ◽  
Critical Layer ◽  
Weakly Nonlinear ◽  
Circular Jets

Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime

1995 ◽  
Vol 291 ◽  
pp. 57-81 ◽  
Author(s):  
S. M. Churilov ◽  
I. G. Shukhman
Keyword(s):  
Travelling Waves ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Critical Layer ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
New Type ◽  
Two Stages

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

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.

W-shape bright and several other solutions to the (3+1)-dimensional nonlinear evolution equations

2021 ◽  
pp. 2150468
Author(s):  
Youssoufa Saliou ◽  
Souleymanou Abbagari ◽  
Alphonse Houwe ◽  
M. S. Osman ◽  
Doka Serge Yamigno ◽  
...  
Keyword(s):  
Acoustic Waves ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Singular Function ◽  
Ion Acoustic Waves ◽  
Weakly Nonlinear ◽  
Quantum Electron ◽  
Ion Acoustic ◽  
Trigonometric Function Solutions

By employing the Modified Sardar Sub-Equation Method (MSEM), several solitons such as W-shape bright, dark solitons, trigonometric function solutions and singular function solutions have been obtained in two famous nonlinear evolution equations which are used to describe waves in quantum electron–positron–ion magnetoplasmas and weakly nonlinear ion-acoustic waves in a plasma. These models are the (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov (NLEQZK) equation and the (3+1)-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation, respectively. Comparing the obtained results with Refs. 32–34 and Refs. 43–46, additional soliton-like solutions have been retrieved and will be useful in future to explain the interaction between lower nonlinear ion-acoustic waves and the parameters of the MSEM and the obtained figures will have more physical explanation.

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.

The effect of barotropic instability on the nonlinear evolution of a Rossby-wave critical layer

1989 ◽  
Vol 207 ◽  
pp. 231-266 ◽  
Author(s):  
Peter H. Haynes
Keyword(s):  
Simple Model ◽  
Numerical Model ◽  
Time Evolution ◽  
Rossby Wave ◽  
Nonlinear Evolution ◽  
Basic Flow ◽  
Critical Layer ◽  
Barotropic Instability ◽  
Long Wavelength ◽  
Subsequent Time

A study of the flow within the critical layer of a forced Rossby-wave is made, using a high-resolution numerical model. The possibility of growth of disturbances through barotropic instability and the extent to which these disturbances modify the subsequent time evolution is of particular interest. The flow is characterized by a parameter μ, equal to the cross-stream lengthscale divided by a downstream wavelength. In the long-wavelength case, μ [Lt ] 1, where there is a clear conceptual division between the instability and the basic flow, the results of the simulation confirm the importance of the growing and saturating disturbances in rearranging the vorticity within the critical layer. When the wavelength is not so long, the distinction between the instability and the straightforward time evolution of the basic flow is less clear. Nonetheless for μ < 0.25 the ultimate evolution is still sensitive to the details of the initial perturbations and in this sense the flow may be regarded as being unstable. The time-integrated absorptivity of the critical layer may be considerably increased by the effects of the instability, sometimes to three or four times that predicted by the Stewartson-Warn-Warn solution. The nature of the flow, at least during the period in which the dynamics are essentially inviscid, does not seem to change when higher harmonics to the forced wave are resonant. The behaviour seen in Béland's (1976) numerical model is re-examined in the light of these findings. A simple model of the redistribution of vorticity by the unstable disturbances is formulated, and its predictions are shown to agree well with the numerical simulations.

Fully coupled resonant-triad interactions in a free shear layer

1994 ◽  
Vol 278 ◽  
pp. 101-121 ◽  
Author(s):  
R. Mallier ◽  
S. A. Maslowe
Keyword(s):  
Flow Direction ◽  
Mixing Layer ◽  
Adverse Pressure Gradient ◽  
Critical Layer ◽  
Free Shear Layer ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Initial Interaction ◽  
The Mean ◽  
Fully Coupled

We report the results of an investigation of the weakly nonlinear evolution of a triad of waves, each slightly amplified on a linear basis, that are superimposed on a tanh y mixing layer. The triad consists of a plane wave and a pair of oblique modes that act as a subharmonic of order 1/2. The oblique modes are inclined at approximately ±60°. to the mean flow direction and because the resonance conditions are satisfied exactly the analysis is entirely self-consistent as an asymptotic theory. The nonlinearity first occurs within the critical layer and the initial interaction is of the parametric resonance type. This produces faster than exponential growth of the oblique waves, behaviour observed recently in the experiments of Corke & Kusek (1993). The critical-layer dynamics lead subsequently to coupled integro-differential equations governing the amplitude evolution and, as first shown in related work by Goldstein & Lee (1992) on boundary layers in an adverse pressure gradient, these equations develop singularities in a finite time.

On weakly nonlinear evolution of convective flow in a passive mushy layer

2010 ◽  
Vol 11 (5) ◽  
pp. 4010-4020 ◽  
Author(s):  
Dambaru Bhatta ◽  
Mallikarjunaiah S. Muddamallappa ◽  
Daniel N. Riahi
Keyword(s):  
Convective Flow ◽  
Nonlinear Evolution ◽  
Mushy Layer ◽  
Weakly Nonlinear

A nonlinear critical layer generated by the interaction of free Rossby waves

1998 ◽  
Vol 371 ◽  
pp. 319-344 ◽  
Author(s):  
JACQUES VANNESTE
Keyword(s):  
Continuous Spectrum ◽  
Rossby Waves ◽  
Resonance Condition ◽  
Inviscid Flow ◽  
Critical Layer ◽  
Weakly Nonlinear ◽  
Critical Layers ◽  
Flow Domain ◽  
Parallel Shear Flow ◽  
Stream Direction

Two free waves propagating in a parallel shear flow generate a critical layer when their nonlinear interaction induces a perturbation whose phase velocity matches the basic-state velocity somewhere in the flow domain. The condition necessary for this to occur may be interpreted as a resonance condition for a triad formed by the two waves and a (singular) mode of the continuous spectrum associated with the shear. The formation of the critical layer is investigated in the case of freely propagating Rossby waves in a two-dimensional inviscid flow in a β-channel.A weakly nonlinear analysis based on a normal-mode expansion in terms of Rossby waves and modes of the continuous spectrum is developed; it leads to a system of amplitude equations describing the evolution of the two Rossby waves and of the modes of the continuous spectrum excited during the interaction. The assumption of weak nonlinearity is not however self-consistent: it breaks down because nonlinearity always becomes strong within the critical layer, however small the initial amplitudes of the Rossby waves. This demonstrates the relevance of nonlinear critical layers to monotonic, stable, unforced shear flows which sustain wave propagation.A nonlinear critical-layer theory is developed that is analogous to the well-known theory for forced critical layers. Differences arise because of the presence of the Rossby waves: the vorticity in the critical layer is advected in the cross-stream direction by the oscillatory velocity field due to the Rossby waves. An equation is derived which governs the modification of the Rossby waves that results from their interaction; it indicates that the two Rossby waves are undisturbed at leading order. An analogue of the Stewartson–Warn–Warn analytical solution is also considered.

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.

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