On the nonlinear theory of stability of plane Poiseuille flow in the subcritical range

1982 ◽  
Vol 381 (1781) ◽  
pp. 407-418 ◽  
Keyword(s):  
Reynolds Number ◽  
Nonlinear Theory ◽  
Poiseuille Flow ◽  
Plane Poiseuille Flow ◽  
New Approach ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

A new approach is proposed in the application of the weakly nonlinear theory, initiated by J. T. Stuart, to the problem of stability of plane Poiseuille flow in the subcritical range. Some results of computation are presented. The calculated threshold amplitudes for Reynolds number R = 5000 and 4000 with different frequencies are compared with Nishioka’s experimental observations. The results are encouraging.

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.

Difference in Critical Reynolds Number Between Hagen-Poiseuille and Plane Poiseuille Flows

2001 ◽  
Author(s):  
Hidesada Kanda
Keyword(s):  
Experimental Data ◽  
Reynolds Number ◽  
Poiseuille Flow ◽  
Average Velocity ◽  
Critical Reynolds Number ◽  
Inlet Section ◽  
Parallel Plates ◽  
Plane Poiseuille Flow ◽  
Contraction Ratio ◽  
Significant Difference

Abstract For plane Poiseuille flow, results of previous investigations were studied, focusing on experimental data on the critical Reynolds number, the entrance length, and the transition length. Consequently, concerning the natural transition, it was confirmed from the experimental data that (i) the transition occurs in the entrance region, (ii) the critical Reynolds number increases as the contraction ratio in the inlet section increases, and (iii) the minimum critical Reynolds number is obtained when the contraction ratio is the smallest or one, and there is no-shaped entrance or straight parallel plates. Its value exists in the neighborhood of 1300, based on the channel height and the average velocity. Although, for Hagen-Poiseuille flow, the minimum critical Reynolds number is approximately 2000, based on the pipe diameter and the average velocity, there seems to be no significant difference in the transition from laminar to turbulent flow between Hagen-Poiseuille flow and plane Poiseuille flow.

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.

scholarly journals Precessional instability of a fluid cylinder

2011 ◽  
Vol 666 ◽  
pp. 104-145 ◽  
Author(s):  
ROMAIN LAGRANGE ◽  
PATRICE MEUNIER ◽  
FRANÇOIS NADAL ◽  
CHRISTOPHE ELOY
Keyword(s):  
Reynolds Number ◽  
Nonlinear Effects ◽  
Intermittent Flow ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Aspect Ratios ◽  
Image Velocimetry ◽  
Precessional Motion ◽  
Weakly Nonlinear Model

In this paper, the instability of a fluid inside a precessing cylinder is addressed theoretically and experimentally. The precessional motion forces Kelvin modes in the cylinder, which can become resonant for given precessional frequencies and cylinder aspect ratios. When the Reynolds number is large enough, these forced resonant Kelvin modes eventually become unstable. A linear stability analysis based on a triadic resonance between a forced Kelvin mode and two additional free Kelvin modes is carried out. This analysis allows us to predict the spatial structure of the instability and its threshold. These predictions are compared to the vorticity field measured by particle image velocimetry with an excellent agreement. When the Reynolds number is further increased, nonlinear effects appear. A weakly nonlinear theory is developed semi-empirically by introducing a geostrophic mode, which is triggered by the nonlinear interaction of a free Kelvin mode with itself in the presence of viscosity. Amplitude equations are obtained coupling the forced Kelvin mode, the two free Kelvin modes and the geostrophic mode. They show that the instability saturates to a fixed point just above threshold. Increasing the Reynolds number leads to a transition from a steady saturated regime to an intermittent flow in good agreement with experiments. Surprisingly, this weakly nonlinear model still gives a correct estimate of the mean flow inside the cylinder even far from the threshold when the flow is turbulent.

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