Turbulence of one-dimensional weakly nonlinear dispersive waves

2001 ◽  
pp. 107-116 ◽  
Author(s):  
V. E. Zakharov ◽  
P. Guyenne ◽  
A. N. Pushkarev ◽  
F. Dias
Keyword(s):  
Dispersive Waves ◽  
Weakly Nonlinear ◽  
One Dimensional

ASYMPTOTICAL ANALYSIS OF ONE DIMENSIONAL GAS DYNAMICS EQUATIONS

2001 ◽  
Vol 6 (1) ◽  
pp. 117-128 ◽  
Author(s):  
A. Krylovas ◽  
R. Čiegis
Keyword(s):  
Gas Dynamics ◽  
Numerical Experiments ◽  
Method Of Averaging ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Resonance Case ◽  
Gas Dynamics Equations ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Asymptotical Analysis

A method of averaging is developed for constructing a uniformly valid asymptotic solution for weakly nonlinear one dimensional gas dynamics systems. Using this method we give the averaged system, which disintegrates into independent equations for the non‐resonance systems. Conditions of the resonance for periodic and almost periodic solutions are presented. In the resonance case the averaged system is solved numerically. Some results of numerical experiments are given.

Weakly nonlinear dispersive waves: group velocity

1982 ◽  
Vol 27 (3) ◽  
pp. 507-514
Author(s):  
Bhimsen K. Shivamoggi
Keyword(s):  
Group Velocity ◽  
Acoustic Waves ◽  
Specific Problem ◽  
Electron Plasma ◽  
Hot Electron ◽  
Ion Acoustic Waves ◽  
Dispersive Waves ◽  
Weakly Nonlinear ◽  
Longitudinal Waves ◽  
Wave Trains

For slowly varying wave trains in a linear system, it is known that a quantity proportional to the square of the amplitude propagates with the group velocity. It is shown here, by considering a specific problem of longitudinal waves in a hot electron-plasma and using an asymptotic analysis, that this result continues to be valid even when weak nonlinearities are introduced into the system provided they produce slowly varying wave trains. The method of analysis fails, however, for weakly nonlinear ion-acoustic waves.

Experimental investigation on the secondary instability of liquid-fluidized beds and the formation of bubbles

2002 ◽  
Vol 470 ◽  
pp. 359-382 ◽  
Author(s):  
PAUL DURU ◽  
ÉLISABETH GUAZZELLI
Keyword(s):  
Fluidized Beds ◽  
Secondary Instability ◽  
Analytical Models ◽  
Physical Origin ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Numerical Studies ◽  
Wavy Structure ◽  
Nonlinear Simulations ◽  
The One

The objective of the present work is to investigate experimentally the secondary instability of the one-dimensional voidage waves occurring in two-dimensional liquid- fluidized beds and to examine the physical origin of bubbles, i.e. regions devoid of particles, which arise in fluidization. In the case of moderate-density glass particles, we observe the formation of transient buoyant blobs clearly resulting from the destabilization of the one-dimensional wavy structure. With metallic beads of the same size but larger density, the same destabilization occurs but it leads to the formation of real bubbles. Comparison with previous analytical and numerical studies is attempted. Whereas the linear and weakly nonlinear analytical models are not appropriate, the direct nonlinear simulations provide a qualitative agreement with the observed destabilization mechanism.

Chaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves

2000 ◽  
Vol 41 (6) ◽  
pp. 4125-4153 ◽  
Author(s):  
David Cai ◽  
David W. McLaughlin
Keyword(s):  
Dispersive Waves ◽  
One Dimensional

A quasi-steady approach to the instability of time-dependent flows in pipes

2002 ◽  
Vol 465 ◽  
pp. 301-330 ◽  
Author(s):  
M. S. GHIDAOUI ◽  
A. A. KOLYSHKIN
Keyword(s):  
Experimental Data ◽  
Laminar Flow ◽  
Asymptotic Expansions ◽  
Unstable Mode ◽  
Landau Equation ◽  
Base Flow ◽  
Neutral Stability ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Weakly Nonlinear Analysis

Asymptotic solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe subject to rapid deceleration and/or acceleration are derived and their stability investigated using linear and weakly nonlinear analysis. In particular, base flow solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe are derived by the method of matched asymptotic expansions. The solutions are valid for short times and can be successfully applied to the case of an arbitrary (but unidirectional) axisymmetric initial velocity distribution. Excellent agreement between asymptotic and analytical solutions for the case of an instantaneous pipe blockage is found for small time intervals. Linear stability of the base flow solutions obtained from the asymptotic expansions to a three-dimensional perturbation is investigated and the results are used to re-interpret the experimental results of Das & Arakeri (1998). Comparison of the neutral stability curves computed with and without the planar channel assumption shows that this assumption is accurate when the ratio of the unsteady boundary layer thickness to radius (i.e. δ1/R) is small but becomes unacceptable when this ratio exceeds 0.3. Both the current analysis and the experiments show that the flow instability is non-axisymmetric for δ1/R = 0.55 and 0.85. In addition, when δ1/R = 0.18 and 0.39, the neutral stability curves for n = 0 and n = 1 are found to be close to one another at all times but the most unstable mode in these two cases is the axisymmetric mode. The accuracy of the quasi-steady assumption, employed both in this research and in that of Das & Arakeri (1998), is supported by the fact that the results obtained under this assumption show satisfactory agreement with the experimental features such as type of instability and spacing between vortices. In addition, the computations show that the ratio of the rate of growth of perturbations to the rate of change of the base flow is much larger than 1 for all cases considered, providing further support for the quasi-steady assumption. The neutral stability curves obtained from linear stability analysis suggest that a weakly nonlinear approach can be used in order to study further development of instability. Weakly nonlinear analysis shows that the amplitude of the most unstable mode is governed by the complex Ginzburg–Landau equation which reduces to the Landau equation if the amplitude is a function of time only. The coefficients of the Landau equation are calculated for two cases of the experimental data given by Das & Arakeri (1998). It is shown that the real part of the Landau constant is positive in both cases. Therefore, finite-amplitude equilibrium is possible. These results are in qualitative agreement with experimental data of Das & Arakeri (1998).

Solitons, solitary waves, and voidage disturbances in gas-fluidized beds

1994 ◽  
Vol 266 ◽  
pp. 243-276 ◽  
Author(s):  
S. E. Harris ◽  
D. G. Crighton
Keyword(s):  
Vries Equation ◽  
Solitary Wave Solution ◽  
Finite Amplitude ◽  
Order Equation ◽  
Leading Order ◽  
Dimensional Model ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Weakly Nonlinear Waves ◽  
Gas Fluidized Beds

In this paper, we consider the evolution of an initially small voidage disturbance in a gas-fluidized bed. Using a one-dimensional model proposed by Needham & Merkin (1983), Crighton (1991) has shown that weakly nonlinear waves of voidage propagate according to the Korteweg–de Vries equation with perturbation terms which can be either amplifying or dissipative, depending on the sign of a coefficient. Here, we investigate the unstable side of the threshold and examine the growth of a single KdV voidage soliton, following its development through several different regimes. As the size of the soliton increases, KdV remains the leading-order equation for some time, but the perturbation terms change, thereby altering the dependence of the amplitude on time. Eventually the disturbance attains a finite amplitude and corresponds to a fully nonlinear solitary wave solution. This matches back directly onto the KdV soliton and tends exponentially to a limiting size. We interpret the series of large-amplitude localized pulses of voidage formed in this way from initial disturbances as corresponding to the ‘voidage slugs’ observed in gas fluidization in narrow tubes.

Implementation of fictitious absorbing layers with deceleration effects for one-dimensional Schrödinger equations

2020 ◽  
Vol 86 (5) ◽  
Author(s):  
Hitoshi Kanai ◽  
Tomo Tatsuno
Keyword(s):  
Absorbing Boundary Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Wave Reflections ◽  
Absorbing Boundary ◽  
Dispersive Waves ◽  
One Dimensional ◽  
Phase Velocities ◽  
Absorbing Layer ◽  
And Performance

Absorbing boundary conditions or layers are used in simulations to reduce or eliminate wave reflections from the boundary; one of the most widely used absorbing layers is Berenger's perfectly matched layer (PML). In this paper, PML is extended to a compound absorbing layer which has multiple effects of damping and deceleration, and is applied to linear and nonlinear Schrödinger equations. The deceleration extends the time to damp out the modes with higher phase velocities, leading to remarkably reduced total reflection for dispersive waves. By invoking the two effects independently, the flexibility and performance are enhanced. Since this method is based on the WKB formalism, it requires an absorbing layer of a moderate size.

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