Non-linear wave-number interaction in near-critical two-dimensional flows

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Bénard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.

Download Full-text

Non-Linear Dynamics of a Complex Aircraft Brake System: Experimental and Theoretical Approaches

Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2003/vib-48579 ◽

2003 ◽

Cited By ~ 2

Author(s):

Jean-Jacques Sinou ◽

Fabrice Thouverez ◽

Olivier Dereure ◽

Guy-Bernard Mazet

Keyword(s):

Parametric Design ◽

Time History ◽

Original System ◽

Linear Stability Theory ◽

Brake System ◽

Computational Time ◽

Linear Dynamics ◽

Theoretical Approaches ◽

Non Linear ◽

Non Linear System

Stability and non-linear dynamics in a complex aircraft brake model are investigated. The non-linear contact between the rotors ands the stators, and mechanisms between components of the brake system are considered. The stability analysis is performed by determining the eigenvalues of the jacobian matrix of the linearized system at the equilibrium point. Parametric studies with linear stability theory is conducted in order to determine the effect of system parameters on stability. In order to obtain time-history responses, the complete set of nonlinear dynamic equations may be integrated numerically. But this procedure is both time consuming and costly to perform when parametric design studies are needed. So it is necessary to use nonlinear analysis : the center manifold approach and the rational approximants are used to obtain the limit cycle of the non-linear system and to study the behaviour of the system in the unstable region. Results from these nonlinear methods are compared with results obtained by integrating the full original system. These non-linear methods appear very interesting in regard to computational time and also necessitate very few computer resources.

Download Full-text

On the non-linear Lamb-Taylor instability

Journal of Fluid Mechanics ◽

10.1017/s0022112069000371 ◽

1969 ◽

Vol 38 (3) ◽

pp. 619-631 ◽

Cited By ~ 28

Author(s):

Ali Hasan Nayfeh

Keyword(s):

Layer Thickness ◽

Travelling Waves ◽

Finite Thickness ◽

Standing Waves ◽

Linear Case ◽

Wave Number ◽

Multiple Time Scales ◽

Solid Boundary ◽

Multiple Time ◽

Non Linear

A non-linear analysis of the inviscid stability of the common surface of two superposed fluids is presented. One of the fluids is a liquid layer with finite thickness having one surface adjacent to a solid boundary whereas the second surface is in contact with a semi-infinite gas of negligible density. The system is accelerated by a force normal to the interface and directed from the liquid to the gas. A second-order expansion is obtained using the method of multiple time scales. It is found that standing as well as travelling disturbances with wave-numbers greater than$K^{\prime}_c = k_c[1+\frac{3}{8}a^2k^2_c + \frac{51}{512}a^4k^4_c]^{\frac{1}{2}}$where a is the disturbance amplitude and kc is the linear cut-off wave-number, oscillate and are stable. However, the frequency in the case of standing waves and the wave velocity in the case of travelling waves are amplitude dependent. Below this cut-off wave-number disturbances grow in amplitude. The cut-off wave-number is independent of the layer thickness although decreasing the layer thickness decreases the growth rate. Although standing waves can be obtained by the superposition of travelling waves in the linear case, this is not true in the non-linear case because the amplitude dependences of the wave speed and frequency are different. A mechanism is proposed to explain the overstability behaviour observed by Emmons, Chang & Watson (1960).

Download Full-text

ASYMPTOTICAL ANALYSIS OF SOME COUPLED NONLINEAR WAVE EQUATIONS

Mathematical Modelling and Analysis ◽

10.3846/13926292.2011.560618 ◽

2011 ◽

Vol 16 (1) ◽

pp. 97-108 ◽

Cited By ~ 1

Author(s):

Aleksandras Krylovas ◽

Rima Kriauzienė

Keyword(s):

Wave Equations ◽

Initial Conditions ◽

Travelling Waves ◽

Linear Equations ◽

Travelling Wave ◽

Nonlinear Wave Equations ◽

Linear Wave ◽

Resonance Case ◽

Non Linear ◽

Asymptotical Analysis

We consider coupled nonlinear equations modelling a family of travelling wave solutions. The goal of our work is to show that the method of internal averaging along characteristics can be used for wide classes of coupled non-linear wave equations such as Korteweg-de Vries, Klein – Gordon, Hirota – Satsuma, etc. The asymptotical analysis reduces a system of coupled non-linear equations to a system of integro – differential averaged equations. The averaged system with the periodical initial conditions disintegrates into independent equations in non-resonance case. These equations describe simple weakly non-linear travelling waves in the non-resonance case. In the resonance case the integro – differential averaged systems describe interaction of waves and give a good asymptotical approximation for exact solutions.

Download Full-text

A weak turbulence analysis of the two-stream instability

Journal of Plasma Physics ◽

10.1017/s0022377800005511 ◽

1971 ◽

Vol 5 (1) ◽

pp. 1-30 ◽

Cited By ~ 35

Author(s):

D. F. Smith ◽

P. C. W Fung

Keyword(s):

Energy Density ◽

Electromagnetic Radiation ◽

Particle Interaction ◽

Plasma Waves ◽

Wave Number ◽

Linear Wave ◽

Weak Turbulence ◽

Wave Particle Interaction ◽

Non Linear ◽

Stream Instability

In this investigation, we study the effects of non-linear interaction of plasma waves in the weak turbulence approximation on a stream-plasma system which is subject to the two-stream instability. The momentum distribution of the stream is assumed to be a Gaussian function and the background plasma is assumed to be a Maxwellian with a temperature T. The non-linear wave-particle interaction of induced scattering on the polarization clouds of ions may lead to stabilization of the stream—plasma waves in the resonant cone are scattered into the nonresonant region. The non-linear wave-particle interaction of induced scattering on the polarization clouds of ions may lead to stabilization of the stream; plasma waves in the resonant cone are scattered into the non-resonant region and vice versa, leading to a well-defined average energy density in plasma waves which is much less than the energy density of the stream. Non-linear scattering also causes transformation of plasma waves into electromagnetic radiation near the fundamental of the electron plasma frequency of the background plasma, and combination of two plasma waves leads to electromagnetic radiation near the second harmonic of this frequency. Expressions are given for the latter process for emission and absorption for arbitrary values of the ratio of the wave number of plasma waves to the wave number of the electromagnetic wave. The major features of the whole phenomenon are illustrated with the help of a numerical example, with conditions typical for the solar corona in which case an ion stream would be stabilized, but an electron or neutral stream (with the chosen density) would not.

Download Full-text

WAVE LOADS ON A NAVIGATION LOCK SLIDING GATE: NON-LINEAR EFFECTS

Coastal Engineering Proceedings ◽

10.9753/icce.v36v.papers.8 ◽

2020 ◽

pp. 8

Author(s):

Luca Cavallaro ◽

Claudio Iuppa ◽

Rosaria Ester Musumeci ◽

Pietro Scandura ◽

Enrico Foti

Keyword(s):

Analytical Model ◽

Cfd Simulation ◽

Analytical Models ◽

Wave Number ◽

Linear Wave ◽

Vertical Force ◽

Wave Loads ◽

Vertical Load ◽

Non Linear ◽

Navigation Lock

The wave loads on a navigation lock gate provided with an opening in the ballast tank are analyzed using a mathematical model based on the linear wave theory and the numerical integration of the Navier-Stokes Equation. The analysis focuses on the evaluation of the non-linear effect influence on the vertical load on the gate. It is shown that the numerical and analytical models agree on the identification of the value of the wave number at which the maximum value of the dimensionless vertical force on the gate is detected. However the analytical model overestimates the peak value of the vertical load with respect to the CFD simulation. To fill this gap, in this paper an easy to use procedure is developed which allows to correct the results of the analytical model.

Download Full-text

Non-Linear Wave-Current Interactions in the SWADE Research Program.

Coastal Engineering 1992 ◽

10.1061/9780872629332.030 ◽

1993 ◽

Author(s):

S. P. Kjeldsen ◽

H. C. Graber

Keyword(s):

Research Program ◽

Linear Wave ◽

Non Linear

Download Full-text

Excitation Forces Estimation for Non-linear Wave Energy Converters: A Neural Network Approach

IFAC-PapersOnLine ◽

10.1016/j.ifacol.2020.12.1213 ◽

2020 ◽

Vol 53 (2) ◽

pp. 12334-12339

Author(s):

M. Bonfanti ◽

F. Carapellese ◽

S.A. Sirigu ◽

G. Bracco ◽

G. Mattiazzo

Keyword(s):

Neural Network ◽

Wave Energy ◽

Linear Wave ◽

Network Approach ◽

Neural Network Approach ◽

Wave Energy Converters ◽

Non Linear ◽

Energy Converters

Download Full-text

Ribbing Instability Analysis of Forward Roll Coating

Journal of Mechanics ◽

10.1017/s1727719100002628 ◽

2009 ◽

Vol 25 (2) ◽

pp. 167-175

Author(s):

K. N. Lie ◽

Y. M. Chiu ◽

J. Y. Jang

Keyword(s):

Surface Coating ◽

Velocity Ratio ◽

Rolling Process ◽

Linear Stability Theory ◽

Wave Number ◽

Roll Coating ◽

Critical Capillary Number ◽

Roll Velocity ◽

Forward Roll Coating ◽

Numerical Program

AbstractThe ribbing instability of forward roll coating is analyzed numerically by linear stability theory. The velocity ratio of two rolls is fixed to be 1/4 for practical surface coating processes. The base flows through the gap between two rolls are solved by use of powerful CFD-RC software package. A numerical program is developed to solve the ribbing instability for the package is not capable of solving the eigenvalue problem of ribbing instability. The effects of the gap between two rolls, flow viscosity, surface tension and average roll velocity on ribbing are investigated. The criterion of ribbing instability is measured in terms of critical capillary number and critical wave number. The results show that the surface coating becomes stable as the gap increases or as the flow viscosity decreases and that the surface coating is more stable to the ribbing of a higher wave number than to the ribbing of a lower wave number. The effect of average roll velocity is not determinant to the ribbing instability. There are optimum and dangerous velocities for each setup of rolling process.

Download Full-text

Control of arrivals to a stochastic input–output system

Advances in Applied Probability ◽

10.2307/1426752 ◽

1980 ◽

Vol 12 (4) ◽

pp. 972-999 ◽

Cited By ~ 48

Author(s):

Søren Glud Johansen ◽

Shaler Stidham

Keyword(s):

Markov Decision Process ◽

Decision Process ◽

Input Output ◽

Distinctive Features ◽

Control Models ◽

Special Cases ◽

Non Linear ◽

Markov Decision ◽

Output System ◽

Stochastic Input

The problem of controlling input to a stochastic input-output system by accepting or rejecting arriving customers is analyzed as a semi-Markov decision process. Included as special cases are a GI/G/1 model and models with compound input and/or output processes, as well as several previously studied queueing-control models. We establish monotonicity of socially and individually optimal acceptance policies and the more restrictive nature of the former, with random rewards for acceptance and both customer-oriented and system-oriented non-linear waiting costs. Distinctive features of our analysis are (i) that it allows dependent interarrival times and (ii) that the monotonicity proofs do not rely on the standard concavity-preservation arguments.

Download Full-text