scholarly journals Can weakly nonlinear theory explain Faraday wave patterns near onset?

2015 ◽  
Vol 777 ◽  
pp. 604-632 ◽  
Author(s):  
A. C. Skeldon ◽  
A. M. Rucklidge
Keyword(s):  
Nonlinear Theory ◽  
Stokes Equations ◽  
Single Mode ◽  
Theoretical Work ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Temporal Modes ◽  
Pattern Forming

The Faraday problem is an important pattern-forming system that provides some middle ground between systems where the initial instability involves just a single mode, and in which complexity then results from mode interactions or secondary bifurcations, and cases where a system is highly turbulent and many spatial and temporal modes are excited. It has been a rich source of novel patterns and of theoretical work aimed at understanding how and why such patterns occur. Yet it is particularly challenging to tie theory to experiment: the experiments are difficult to perform; the parameter regime of interest (large box, moderate viscosity) along with the technical difficulties of solving the free-boundary Navier–Stokes equations make numerical solution of the problem hard; and the fact that the instabilities result in an entire circle of unstable wavevectors presents considerable theoretical difficulties. In principle, weakly nonlinear theory should be able to predict which patterns are stable near pattern onset. In this paper we present the first quantitative comparison between weakly nonlinear theory of the full Navier–Stokes equations and (previously published) experimental results for the Faraday problem with multiple-frequency forcing. We confirm that three-wave interactions sit at the heart of why complex patterns are stabilised, but also highlight some discrepancies between theory and experiment. These suggest the need for further experimental and theoretical work to fully investigate the issues of pattern bistability and the role of bicritical/tricritical points in determining bifurcation structure.

On the instability of a three-dimensional attachment-line boundary layer: weakly nonlinear theory and a numerical approach

1986 ◽  
Vol 163 ◽  
pp. 257-282 ◽  
Author(s):  
Philip Hall ◽  
Mujeeb R. Malik
Keyword(s):  
Boundary Layer ◽  
Nonlinear Theory ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Numerical Approach ◽  
Finite Amplitude ◽  
Lower Branch ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Attachment Line

The instability of a three-dimensional attachment-line boundary layer is considered in the nonlinear regime. Using weakly nonlinear theory, it is found that, apart from a small interval near the (linear) critical Reynolds number, finite-amplitude solutions bifurcate subcritically from the upper branch of the neutral curve. The time-dependent Navier–Stokes equations for the attachment-line flow have been solved using a Fourier–Chebyshev spectral method and the subcritical instability is found at wavenumbers that correspond to the upper branch. Both the theory and the numerical calculations show the existence of supercritical finite-amplitude (equilibrium) states near the lower branch which explains why the observed flow exhibits a preference for the lower branch modes. The effect of blowing and suction on nonlinear stability of the attachment-line boundary layer is also investigated.

scholarly journals Weakly nonlinear modelling of a forced turbulent axisymmetric wake

2017 ◽  
Vol 814 ◽  
pp. 570-591 ◽  
Author(s):  
Georgios Rigas ◽  
Aimee S. Morgans ◽  
Jonathan F. Morrison
Keyword(s):  
Turbulent Flows ◽  
Bluff Body ◽  
Stokes Equations ◽  
Nonlinear Models ◽  
Navier Stokes ◽  
Turbulent Regime ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Zero Net Mass Flux ◽  
High Reynolds

A theory is presented where the weakly nonlinear analysis of laminar globally unstable flows in the presence of external forcing is extended to the turbulent regime. The analysis is demonstrated and validated using experimental results of an axisymmetric bluff-body wake at high Reynolds numbers, $Re_{D}\sim 1.88\times 10^{5}$, where forcing is applied using a zero-net-mass-flux actuator located at the base of the blunt body. In this study we focus on the response of antisymmetric coherent structures with azimuthal wavenumbers $m=\pm 1$ at a frequency $St_{D}=0.2$, responsible for global vortex shedding. We found experimentally that axisymmetric forcing ($m=0$) couples nonlinearly with the global shedding mode when the flow is forced at twice the shedding frequency, resulting in parametric subharmonic resonance through a triadic interaction between forcing and shedding. We derive simple weakly nonlinear models from the phase-averaged Navier–Stokes equations and show that they capture accurately the observed behaviour for this type of forcing. The unknown model coefficients are obtained experimentally by producing harmonic transients. This approach should be applicable in a variety of turbulent flows to describe the response of global modes to forcing.

On perturbations of Jeffery-Hamel flow

1988 ◽  
Vol 186 ◽  
pp. 559-581 ◽  
Author(s):  
W. H. H. Banks ◽  
P. G. Drazin ◽  
M. B. Zaturska
Keyword(s):  
Stokes Equations ◽  
Temporal Stability ◽  
Pitchfork Bifurcation ◽  
Critical Value ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Nonlinear Perturbations ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Rigid Plane

We examine various perturbations of Jeffery-Hamel flows, the exact solutions of the Navier-Stokes equations governing the steady two-dimensional motions of an incompressible viscous fluid from a line source at the intersection of two rigid plane walls. First a pitchfork bifurcation of the Jeffery-Hamel flows themselves is described by perturbation theory. This description is then used as a basis to investigate the spatial development of arbitrary small steady two-dimensional perturbations of a Jeffery-Hamel flow; both linear and weakly nonlinear perturbations are treated for plane and nearly plane walls. It is found that there is strong interaction of the disturbances up- and downstream if the angle between the planes exceeds a critical value 2α2, which depends on the value of the Reynolds number. Finally, the problem of linear temporal stability of Jeffery-Hamel flows is broached and again the importance of specifying conditions up- and downstream is revealed. All these results are used to interpret the development of flow along a channel with walls of small curvature. Fraenkel's (1962) approximation of channel flow locally by Jeffery-Hamel flows is supported with the added proviso that the angle between the two walls at each station is less than 2α2.

scholarly journals Theory of weakly nonlinear self-sustained detonations

2015 ◽  
Vol 784 ◽  
pp. 163-198 ◽  
Author(s):  
Luiz M. Faria ◽  
Aslan R. Kasimov ◽  
Rodolfo R. Rosales
Keyword(s):  
Stokes Equations ◽  
Period Doubling ◽  
Spatial Dimension ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Dynamical Characteristics ◽  
Reactive Euler Equations ◽  
Doubling Sequence

We propose a theory of weakly nonlinear multidimensional self-sustained detonations based on asymptotic analysis of the reactive compressible Navier–Stokes equations. We show that these equations can be reduced to a model consisting of a forced unsteady small-disturbance transonic equation and a rate equation for the heat release. In one spatial dimension, the model simplifies to a forced Burgers equation. Through analysis, numerical calculations and comparison with the reactive Euler equations, the model is demonstrated to capture such essential dynamical characteristics of detonations as the steady-state structure, the linear stability spectrum, the period-doubling sequence of bifurcations and chaos in one-dimensional detonations and cellular structures in multidimensional detonations.

scholarly journals Weakly Nonlinear Theory of Pattern-Forming Systems with Spontaneously Broken Isotropy

1996 ◽  
Vol 76 (25) ◽  
pp. 4729-4732 ◽  
Author(s):  
A. G. Rossberg ◽  
A. Hertrich ◽  
L. Kramer ◽  
W. Pesch
Keyword(s):  
Nonlinear Theory ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Pattern Forming

Precession of a rapidly rotating cylinder flow: traverse through resonance

2015 ◽  
Vol 782 ◽  
pp. 63-98 ◽  
Author(s):  
Francisco Marques ◽  
Juan M. Lopez
Keyword(s):  
Aspect Ratio ◽  
Stokes Equations ◽  
Complex Dynamics ◽  
Nonlinear Models ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Small Scale ◽  
Viscous Effects ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear

Recent experiments using a rapidly rotating and precessing cylinder have shown that for specific values of the precession rate, aspect ratio and tilt angle, sudden catastrophic transitions to turbulence occur. Even if the precessional forcing is not too strong, there can be intermittent recurrences between a laminar state and small-scale chaotic flow. The inviscid linearized Navier–Stokes equations have inertial-wave solutions called Kelvin eigenmodes. The precession forces the flow to have azimuthal wavenumber $m=1$ (spin-over mode). Depending on the cylinder aspect ratio and on the ratio of the rotating and precessing frequencies, additional Kelvin modes can be in resonance with the spin-over mode. This resonant flow would grow unbounded if not for the presence of viscous and nonlinear effects. In practice, one observes a rapid transition to turbulence, and the precise nature of the transition is not entirely clear. When both the precessional forcing and viscous effects are small, weakly nonlinear models and experimental observations suggest that triadic resonance is at play. Here, we used direct numerical simulations of the full Navier–Stokes equations in a narrow region of parameter space where triadic resonance has been previously predicted from a weakly nonlinear model and observed experimentally. The detailed parametric studies enabled by the numerics reveal the complex dynamics associated with weak precessional forcing, involving symmetry-breaking, hysteresis and heteroclinic cycles between states that are quasiperiodic, with two or three independent frequencies. The detailed analysis of these states leads to associations of physical mechanisms with the various time scales involved.

scholarly journals Solitary Waves and Undular Bores in a Mesosphere Duct

2015 ◽  
Vol 72 (11) ◽  
pp. 4412-4422 ◽  
Author(s):  
Roger Grimshaw ◽  
Dave Broutman ◽  
Brian Laughman ◽  
Stephen D. Eckermann
Keyword(s):  
Solitary Waves ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Solitary Wave Solution ◽  
Internal Gravity Wave ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Wave Radiation ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

Abstract Mesospheric bores have been observed and measured in the mesopause region near 100-km altitude, where they propagate horizontally along a duct of relatively strong density stratification. Here, a weakly nonlinear theory is developed for the description of these mesospheric bores. It extends previous theories by allowing internal gravity wave radiation from the duct into the surrounding stratified regions, which are formally assumed to be weakly stratified. The radiation away from the duct is expected to be important for bore energetics. The theory is compared with a numerical simulation of the full Navier–Stokes equations in the Boussinesq approximation. Two initial conditions are considered. The first is a solitary wave solution that would propagate without change of form if the region outside the duct were unstratified. The second is a sinusoid that evolves into an undular bore. The main conclusion is that, while solitary waves and undular bores decay by radiation from the duct, they can survive as significant structures over sufficiently long periods (~100 min) to be observable.

scholarly journals Shock-resolved Navier–Stokes simulation of the Richtmyer–Meshkov instability start-up at a light–heavy interface

2009 ◽  
Vol 642 ◽  
pp. 421-443 ◽  
Author(s):  
R. M. J. KRAMER ◽  
D. I. PULLIN ◽  
D. I. MEIRON ◽  
C. PANTANO
Keyword(s):  
Stokes Equations ◽  
Single Mode ◽  
Base Flow ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Viscous Effects ◽  
Navier Stokes Equations ◽  
Interface Growth ◽  
Start Up ◽  
Analytic Models

The single-mode Richtmyer–Meshkov instability is investigated using a first-order perturbation of the two-dimensional Navier–Stokes equations about a one-dimensional unsteady shock-resolved base flow. A feature-tracking local refinement scheme is used to fully resolve the viscous internal structure of the shock. This method captures perturbations on the shocks and their influence on the interface growth throughout the simulation, to accurately examine the start-up and early linear growth phases of the instability. Results are compared to analytic models of the instability, showing some agreement with predicted asymptotic growth rates towards the inviscid limit, but significant discrepancies are noted in the transient growth phase. Viscous effects are found to be inadequately predicted by existing models.

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