Weakly Nonlinear Theory — Finite Reynolds Numbers

2002 ◽  
pp. 85-121
Author(s):  
Yoshio Sone
Keyword(s):  
Nonlinear Theory ◽  
Reynolds Numbers ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

A rotating fluid cylinder subject to weak precession

2008 ◽  
Vol 599 ◽  
pp. 405-440 ◽  
Author(s):  
PATRICE MEUNIER ◽  
CHRISTOPHE ELOY ◽  
ROMAIN LAGRANGE ◽  
FRANÇOIS NADAL
Keyword(s):  
Reynolds Number ◽  
Nonlinear Theory ◽  
Reynolds Numbers ◽  
Quantitative Agreement ◽  
Precession Angle ◽  
Weakly Nonlinear ◽  
Low Reynolds Numbers ◽  
Axisymmetric Mode ◽  
Mode Amplitude ◽  
Weakly Nonlinear Theory

In this paper, we report experimental and theoretical results on the flow inside a precessing and rotating cylinder. Particle image velocimetry measurements have revealed the instantaneous structure of the flow and confirmed that it is the sum of forced inertial (Kelvin) modes, as predicted by the classical linear inviscid theory. But this theory predicts also that the amplitude of a mode diverges when its natural frequency equals the precession frequency. A viscous and weakly nonlinear theory has therefore been developed at the resonance. This theory has been compared to experimental results and shows a good quantitative agreement. For low Reynolds numbers, the mode amplitude scales as the square root of the Reynolds number owing to the presence of Ekman layers on the cylinder walls. When the Reynolds number is increased, the amplitude saturates at a value which scales as the precession angle to the power one-third for a given resonance. The nonlinear theory also predicts the forcing of a geostrophic (axisymmetric) mode which has been observed and measured in the experiments. These results allow the flow inside a precessing cylinder to be fully characterized in all regimes as long as there is no instability.

scholarly journals Weakly nonlinear theory and sea state bias estimations

1999 ◽  
Vol 104 (C4) ◽  
pp. 7641-7647 ◽  
Author(s):  
Tanos Elfouhaily ◽  
Donald Thompson ◽  
Douglas Vandemark ◽  
Bertrand Chapron
Keyword(s):  
Nonlinear Theory ◽  
Sea State ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Sea State Bias

scholarly journals “Extraordinary” modulation instability in optics and hydrodynamics

2021 ◽  
Vol 118 (14) ◽  
pp. e2019348118
Author(s):  
Guillaume Vanderhaegen ◽  
Corentin Naveau ◽  
Pascal Szriftgiser ◽  
Alexandre Kudlinski ◽  
Matteo Conforti ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Stability Analysis ◽  
Linear Stability ◽  
Nonlinear Theory ◽  
Linear Stability Analysis ◽  
Water Waves ◽  
Modulation Instability ◽  
Nonlinear Effects ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

The classical theory of modulation instability (MI) attributed to Bespalov–Talanov in optics and Benjamin–Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

Extended weakly nonlinear theory of planar nematic convection

1999 ◽  
Vol 59 (2) ◽  
pp. 1747-1769 ◽  
Author(s):  
Emmanuel Plaut ◽  
Werner Pesch
Keyword(s):  
Nonlinear Theory ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

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.

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