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.

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.

On the weakly nonlinear development of Tollmien-Schlichting wavetrains in boundary layers

1996 ◽  
Vol 323 ◽  
pp. 133-171 ◽  
Author(s):  
Xuesong Wu ◽  
Philip A. Stewart ◽  
Stephen J. Cowley
Keyword(s):  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Nonlinear Effects ◽  
Mean Flow ◽  
Flow Distortion ◽  
Weakly Nonlinear ◽  
Nonlinear Development ◽  
Finite Distance ◽  
Tollmien Schlichting ◽  
High Reynolds

The nonlinear development of a weakly modulated Tollmien-Schlichting wavetrain in a boundary layer is studied theoretically using high-Reynolds-number asymptotic methods. The ‘carrier’ wave is taken to be two-dimensional, and the envelope is assumed to be a slowly varying function of time and of the streamwise and spanwise variables. Attention is focused on the scalings appropriate to the so-called ‘upper branch’ and ‘high-frequency lower branch’. The dominant nonlinear effects are found to arise in the critical layer and the surrounding ‘diffusion layer’: nonlinear interactions in these regions can influence the development of the wavetrain by producing a spanwise-dependent mean-flow distortion. The amplitude evolution is governed by an integro-partial-differential equation, whose nonlinear term is history-dependent and involves the highest derivative with respect to the spanwise variable. Numerical solutions show that a localized singularity can develop at a finite distance downstream. This singularity seems consistent with the experimentally observed focusing of vorticity at certain spanwise locations, although quantitative comparisons have not been attempted.

The onset of meandering in a barotropic jet

2001 ◽  
Vol 449 ◽  
pp. 85-114 ◽  
Author(s):  
N. J. BALMFORTH ◽  
C. PICCOLO
Keyword(s):  
Nonlinear Theory ◽  
Stability Boundary ◽  
Wave Model ◽  
Inviscid Limit ◽  
Reduced Model ◽  
Mean Flow ◽  
Single Wave ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
The Stability

This study explores the dynamics of an unstable jet of two-dimensional, incompressible fluid on the beta-plane. In the inviscid limit, standard weakly nonlinear theory fails to give a low-order description of this problem, partly because the simple shape of the unstable normal mode is insufficient to capture the structure of the forming pattern. That pattern takes the form of ‘cat's eyes’ in the vorticity distribution which develop inside the modal critical layers (slender regions to either side of the jet's axis surrounding the levels where the modal wave speed matches the mean flow). Asymptotic expansions furnish a reduced model which is a version of what is known as the single-wave model in plasma physics. The reduced model predicts that the amplitude of the unstable mode saturates at a relatively low level and is not steady. Rather, the amplitude evolves aperiodically about the saturation level, a result with implications for Lagrangian transport theories. The aperiodic amplitude ‘bounces’ are intimately connected with sporadic deformations of the vortices within the cat's eyes. The theory is compared with numerical simulations of the original governing equations. Slightly asymmetrical jets are also studied. In this case the neutral modes along the stability boundary become singular; an extension of the weakly nonlinear theory is presented for these modes.

Investigation of Mean and Turbulent Flow Behaviour Over an Escarpment

2016 ◽  
Author(s):  
R. Kilpatrick ◽  
K. Siddiqui ◽  
H. Hangan ◽  
D. Parvu
Keyword(s):  
Reynolds Number ◽  
Turbulent Flow ◽  
Flow Behaviour ◽  
Scale Model ◽  
Mean Flow ◽  
Scaled Model ◽  
Energy And Environment ◽  
Flow Parameters ◽  
Image Velocimetry ◽  
Bolund Hill

Mean and turbulent flow behaviour over a 1:25 scale model of Bolund hill was investigated at Western University’s Wind Engineering, Energy, and Environment Research Institute (WindEEE) using Particle Image Velocimetry (PIV). A range of upstream flow and surface conditions were considered. Results showed almost no Reynolds dependency on the mean flow and weak dependency of Reynolds number on the upstream surface roughness conditions. However, a strong Reynolds number and upstream surface rough dependency is observed on the turbulent flow particularly in the shear layer formed in the immediate downstream region of the escarpment. It is concluded that the consideration of Reynolds number independency must be cautiously used when extrapolating the flow parameters from scaled model testing to full scale in the field.

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.

Near-surface particle image velocimetry measurements in a transitionally rough-wall atmospheric boundary layer

2007 ◽  
Vol 580 ◽  
pp. 319-338 ◽  
Author(s):  
SCOTT C. MORRIS ◽  
SCOTT R. STOLPA ◽  
PAUL E. SLABOCH ◽  
JOSEPH C. KLEWICKI
Keyword(s):  
Reynolds Number ◽  
Particle Image Velocimetry ◽  
Environmental Science ◽  
Particle Image ◽  
Shear Velocity ◽  
Mean Flow ◽  
Mean Velocity ◽  
Near Surface ◽  
Image Velocimetry ◽  
The Mean

The Reynolds number dependence of the structure and statistics of wall-layer turbulence remains an open topic of research. This issue is considered in the present work using two-component planar particle image velocimetry (PIV) measurements acquired at the Surface Layer Turbulence and Environmental Science Test (SLTEST) facility in western Utah. The Reynolds number (δuτ/ν) was of the order 106. The surface was flat with an equivalent sand grain roughness k+ = 18. The domain of the measurements was 500 < yuτ/ν < 3000 in viscous units, 0.00081 < y/δ < 0.005 in outer units, with a streamwise extent of 6000ν/uτ. The mean velocity was fitted by a logarithmic equation with a von Kármán constant of 0.41. The profile of u′v′ indicated that the entire measurement domain was within a region of essentially constant stress, from which the wall shear velocity was estimated. The stochastic measurements discussed include mean and RMS profiles as well as two-point velocity correlations. Examination of the instantaneous vector maps indicated that approximately 60% of the realizations could be characterized as having a nearly uniform velocity. The remaining 40% of the images indicated two regions of nearly uniform momentum separated by a thin region of high shear. This shear layer was typically found to be inclined to the mean flow, with an average positive angle of 14.9°.

scholarly journals Weakly nonlinear theory for oscillating wave surge converters in a channel

2017 ◽  
Vol 834 ◽  
pp. 55-91 ◽  
Author(s):  
S. Michele ◽  
P. Sammarco ◽  
M. d’Errico
Keyword(s):  
Nonlinear Theory ◽  
Period Doubling ◽  
Nonlinear Effects ◽  
Side Band ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Natural Modes ◽  
Stokes Waves ◽  
Conversion Performance ◽  
Incident Waves

We present a weakly nonlinear theory on the natural modes’ resonance of an array of oscillating wave surge converters (OWSCs) in a channel. We first derive the evolution equation of the Stuart–Landau type for the gate oscillations in uniform and modulated incident waves and then evaluate the nonlinear effects on the energy conversion performance of the array. We show that the gates are unstable to side-band perturbations so that a Benjamin–Feir instability similar to the case of Stokes’ waves is possible. The non-autonomous dynamical system presents period doubling bifurcations and strange attractors. We also analyse the competition of two natural modes excited by one incident wave. For weak damping and power take-off coefficient, the dynamical effects on the generated power of the OWSCs are investigated. We show that the occurrence of subharmonic resonance significantly increases energy production.

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.

Observation of nonlinear effects in the propagation of m=0 torsional hydromagnetic waves

1978 ◽  
Vol 19 (3) ◽  
pp. 375-385 ◽  
Author(s):  
M. H. Brennan ◽  
A. D. Cheetham ◽  
I. R. Jones ◽  
M. L. Sawley
Keyword(s):  
Magnetic Field ◽  
Experimental Data ◽  
Theoretical Calculations ◽  
Nonlinear Effects ◽  
Quantitative Agreement ◽  
Primary Wave ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Hydromagnetic Waves ◽  
Afterglow Plasma

This paper reports on the observation of nonlinear effects in the propagation of axisymmetric (m = 0) torsional hydromagnetic waves in a partially ionized helium afterglow plasma. In accordance with the predictions of a weakly nonlinear theory, magnetic field perturbations oscillating at twice the frequency of the primary wave have been observed. Good quantitative agreement between the experimental data and theoretical calculations is obtained for moderate primary wave amplitudes.

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