On the stability of Kelvin waves

1989 ◽  
Vol 206 ◽  
pp. 1-23 ◽  
Author(s):  
W. K. Melville ◽  
G. G. Tomasson ◽  
D. P. Renouard
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Resonance ◽  
Approximate Solutions ◽  
Kelvin Waves ◽  
Kelvin Wave ◽  
Governing Equations ◽  
Weakly Nonlinear ◽  
Approximate Analytical Solutions ◽  
The Stability ◽  
Good Agreement

We consider the evolution of weakly nonlinear dispersive long waves in a rotating channel. The governing equations are derived and approximate solutions obtained for the initial data corresponding to a Kelvin wave. In consequence of the small nonlinear speed correction it is shown that weakly nonlinear Kelvin waves are unstable to a direct nonlinear resonance with the linear Poincaré modes of the channel. Numerical solutions of the governing equations are computed and found to give good agreement with the approximate analytical solutions. It is shown that the curvature of the wavefront and the decay of the leading wave amplitude along the channel are attributable to the Poincaré waves generated by the resonance. These results appear to give a qualitative explanation of the experimental results of Maxworthy (1983), and Renouard, Chabert d'Hières & Zhang (1987).

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

scholarly journals Linear and weakly nonlinear instability of a premixed curved flame under the influence of its spontaneous acoustic field

2014 ◽  
Vol 758 ◽  
pp. 180-220 ◽  
Author(s):  
Raphaël C. Assier ◽  
Xuesong Wu
Keyword(s):  
Acoustic Field ◽  
Numerical Solutions ◽  
Parametric Instability ◽  
Linear Instability ◽  
Weakly Nonlinear ◽  
Observed Behaviour ◽  
Hydrodynamic Field ◽  
The Stability ◽  
Sivashinsky Equation ◽  
Asymptotic Formulation

AbstractThe stability of premixed flames in a duct is investigated using an asymptotic formulation, which is derived from first principles and based on high-activation-energy and low-Mach-number assumptions (Wu et al., J. Fluid Mech., vol. 497, 2003, pp. 23–53). The present approach takes into account the dynamic coupling between the flame and its spontaneous acoustic field, as well as the interactions between the hydrodynamic field and the flame. The focus is on the fundamental mechanisms of combustion instability. To this end, a linear stability analysis of some steady curved flames is undertaken. These steady flames are known to be stable when the spontaneous acoustic perturbations are ignored. However, we demonstrate that they are actually unstable when the latter effect is included. In order to corroborate this result, and also to provide a relatively simple model guiding active control, we derived an extended Michelson–Sivashinsky equation, which governs the linear and weakly nonlinear evolution of a perturbed flame under the influence of its spontaneous sound. Numerical solutions to the initial-value problem confirm the linear instability result, and show how the flame evolves nonlinearly with time. They also indicate that in certain parameter regimes the spontaneous sound can induce a strong secondary subharmonic parametric instability. This behaviour is explained and justified mathematically by resorting to Floquet theory. Finally we compare our theoretical results with experimental observations, showing that our model captures some of the observed behaviour of propagating flames.

An Approximate Solution for Period-m Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Approximate Solution ◽  
Numerical Simulations ◽  
Analytical Solutions ◽  
Periodic Motion ◽  
Nonlinear Oscillator ◽  
Approximate Solutions ◽  
Bifurcation Tree ◽  
Approximate Analytical Solutions ◽  
Forced Oscillator ◽  
The Stability

The approximate analytical solutions of the period-m motions for a periodically forced, quadratic nonlinear oscillator are presented. The stability and bifurcation of such approximate solutions in the quadratic nonlinear oscillator are discussed. The bifurcation tree of period-1 to chaos is presented. Numerical simulations for period-1 to period-4 motions in such quadratic oscillator are carried out for comparison of approximate analytical solutions. Such an investigation provides how to analytically determine bifurcation of periodic motion to chaos.

scholarly journals Analytic Approximate Solution for Falkner-Skan Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Bogdan Marinca
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Boundary Layer Problem ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method ◽  
Good Agreement ◽  
Layer Problem

This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained results reveal that this procedure is very effective, simple, and accurate. A very good agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

scholarly journals A Generalized Analytical Model for Joule Heating of Segmented Wires

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Vivekananthan Balakrishnan ◽  
Toan Dinh ◽  
Hoang-Phuong Phan ◽  
Dzung Viet Dao ◽  
Nam-Trung Nguyen
Keyword(s):  
Temperature Distribution ◽  
Analytical Solution ◽  
Joule Heating ◽  
Numerical Solutions ◽  
Analytical Data ◽  
Experimental Studies ◽  
Governing Equations ◽  
Steady State Temperature ◽  
The One ◽  
Good Agreement

This paper presents an analytical solution for the Joule heating problem of a segmented wire made of two materials with different properties and suspended as a bridge across two fixed ends. The paper first establishes the one-dimensional (1D) governing equations of the steady-state temperature distribution along the wire with the consideration of heat conduction and free-heat convection phenomena. The temperature coefficient of resistance of the constructing materials and the dimension of the each segmented wires were also taken into account to obtain analytical solution of the temperature. COMSOL numerical solutions were also obtained for initial validation. Experimental studies were carried out using copper and nichrome wires, where the temperature distribution was monitored using an IR thermal camera. The data showed a good agreement between experimental data and the analytical data, validating our model for the design and development of thermal sensors based on multisegmented structures.

scholarly journals Analytical and numerical results for flow and shock formation in two-layer gravity currents

1998 ◽  
Vol 40 (1) ◽  
pp. 35-58 ◽  
Author(s):  
P. J. Montgomery ◽  
T. B. Moodie
Keyword(s):  
Numerical Solutions ◽  
Spatial Dimension ◽  
Gravity Currents ◽  
Numerical Results ◽  
Shock Formation ◽  
Inviscid Fluid ◽  
Weakly Nonlinear ◽  
Relaxation Scheme ◽  
Model Equations ◽  
The Stability

AbstractMany gravity driven flows can be modelled as homogeneous layers of inviscid fluid with a hydrostatic pressure distribution. There are examples throughout oceanography, meteorology, and many engineering applications, yet there are areas which require further investigation. Analytical and numerical results for two-layer shallow-water formulations of time dependent gravity currents travelling in one spatial dimension are presented. Model equations for three physical limits are developed from the hydraulic equations, and numerical solutions are produced using a relaxation scheme for conservation laws developed recently by S. Jin and X. Zin [6]. Hyperbolicity of the model equations is examined in conjunction with the stability Froude number, and shock formation at the interface of the two layers is investigated using the theory of weakly nonlinear hyperbolic waves.

Explicit Solution for Time-Fractional Batch Reactor System

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Muhammad Jamil ◽  
Asmat Ara ◽  
Subir Das
Keyword(s):  
Homotopy Perturbation Method ◽  
Analytical Procedure ◽  
Numerical Solutions ◽  
Approximate Analytical Solution ◽  
Batch Reactor ◽  
Reactor System ◽  
Time Interval ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
Good Agreement

In this paper, the new homotopy perturbation method (NHPM) has been successively applied for finding approximate analytical solutions of the fractional order batch reactor system. An approximate analytical solution for the concentration of reactants and products that is valid for a time interval. The approximate analytical procedure is depends only on two components. The behavior of the solution and effects of different parameters and fractional index are shown graphically. Numerical solutions of ordinary batch reactor system verify our approximate solution with good agreement.

The stability and transition of a two-dimensional jet

1960 ◽  
Vol 7 (1) ◽  
pp. 53-80 ◽  
Author(s):  
Hiroshi Sato
Keyword(s):  
Laminar Flow ◽  
Numerical Solutions ◽  
Two Dimensional ◽  
Eigenvalues And Eigenfunctions ◽  
Response Characteristics ◽  
Laminar Jets ◽  
The Stability ◽  
Linear Disturbance ◽  
Good Agreement ◽  
Small Disturbances

A study was made of the transition of a two-dimensional jet. In the region where laminar flow becomes unstable, two kinds of sinusoidal velocity fluctuation have been found; one is symmetrical and the other is anti-symmetrical with respect to the centre line of the jet. The fluctuations grow exponentially at first and develop into turbulence without being accompanied by abrupt bursts or turbulent spots.The response characteristics of laminar jets to artificial external excitation were investigated in detail by using sound as an exciting agent. The effect of excitation was seen to be most remarkable when the frequency of excitation coincides with that of self-excited sinusoidal fluctuations.Numerical solutions of equation of small disturbances superposed on laminar flow were obtained assuming the Reynolds number as infinity. Theoretical eigenvalues and eigenfunctions are in good agreement with experimental results, thus verifying the existence of a region of linear disturbance in the two-dimensional jet.

scholarly journals Nonlinear Disintegration of the Internal Tide

2008 ◽  
Vol 38 (3) ◽  
pp. 686-701 ◽  
Author(s):  
Karl R. Helfrich ◽  
Roger H. J. Grimshaw
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Numerical Solutions ◽  
Internal Tide ◽  
Tidal Energy ◽  
Initial Amplitude ◽  
Governing Equations ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Hydrostatic Limit

Abstract The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Since observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully nonlinear, weakly nonhydrostatic two-layer theory that includes rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertia–gravity solutions that are explored as models of the nonlinear internal tide. These long waves are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initial sinusoidal linear internal tide is closely linked to the presence of these nonlinear waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear hydrostatic solutions, with the short solitary waves superimposed. The degree of disintegration is a function of initial amplitude of the tide and the properties of the underlying nonlinear hydrostatic solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear hydrostatic inertia–gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations in the South China Sea are briefly discussed.

Geostrophic adjustment in a channel: nonlinear and dispersive effects

1992 ◽  
Vol 241 ◽  
pp. 23-57 ◽  
Author(s):  
G. G. Tomasson ◽  
W. K. Melville
Keyword(s):  
Internal Waves ◽  
Numerical Solutions ◽  
Boussinesq Equations ◽  
Kelvin Waves ◽  
Geostrophic Adjustment ◽  
Homogeneous Fluid ◽  
Layer System ◽  
Weakly Nonlinear ◽  
Resonant Interactions ◽  
Poincare Waves

We consider the general problem of geostrophic adjustment in a channel in the weakly nonlinear and dispersive (non-hydrostatic) limit. Governing equations of Boussinesq-type are derived, based on the assumption of weak nonlinear, dispersive and rotational effects, both for surface waves on a homogeneous fluid and internal waves in a two-layer system. Numerical solutions of the Boussinesq equations are presented, giving examples of the geostrophic adjustment in a channel for two different kinds of initial disturbances, both with non-zero perturbation potential vorticity. The timescales of rotational separation (that is, the separation of the Kelvin and Poincaré waves due to their dispersive properties) and that of nonlinear evolution are considered, with particular concern for the resonant interactions of nonlinear Kelvin waves and linear Poincaré waves described by Melville, Tomasson & Renouard (1989). A parameter measuring the ratio of the two timescales is used to predict when the free and forced Poincaré waves may be separated in the solution. It also distinguishes the cases in which the linear solutions are valid for the rotational separation from those requiring the full Boussinesq equations. Finally, solutions for the evolution of nonlinear internal waves in a sea strait are presented, and the effects of friction on the wavefront curvature of the nonlinear Kelvin waves are briefly considered.

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