scholarly journals Second Summary-Introduction: Steady One-Dimensional Fluid-Magnetic Collisionless Shock Theory

1960 ◽  
Vol 12 ◽  
pp. 459-467
Author(s):  
A. A. Blank ◽  
H. Grad
Keyword(s):  
Wave Front ◽  
Mean Free Path ◽  
Finite Amplitude ◽  
Wave Profile ◽  
The State ◽  
Compressive Wave ◽  
Dissipative Effects ◽  
Dissipative Mechanism ◽  
Weak Shocks ◽  
Shock Profile

Shock-waves represent one of the most important mechanisms for creating and heating a plasma. In classical non-dissipative gas dynamics, the formation of a shock is indicated by the progressive steepening of a finite-amplitude compressive wave front to the point where it becomes multivalued and consequently without physical meaning. This difficulty is avoided by the inclusion of dissipative effects, usually in the form of heat flow and viscosity. The dissipative mechanisms become more effective as the wave front steepens, and the result is a steady wave profile for which the non-linear and dissipative effects are counterbalanced. The scale length for the dissipative transition zone or wave profile is the mean-free-path; the actual thickness may range from one to several mean-free-paths, or even more for very weak shocks. Given the strength of the shock, the state on one side of the shock may be computed from the state on the other side directly from the laws of conservation of mass, momentum and energy (Hugoniot relations). Accordingly, the nature of the particular dissipative mechanism affects only the shape of the shock profile but not the end states.

Steep gravity waves in water of arbitrary uniform depth

1977 ◽  
Vol 286 (1335) ◽  
pp. 183-230 ◽  
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Perturbation Parameter ◽  
Intermediate Energy ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Accurate Calculation ◽  
Theoretical Developments ◽  
Deep Water Waves

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

A Virtual Piston-Type Wave Maker Based on OpenFOAM

2017 ◽  
Vol Vol 159 (A2) ◽  
Author(s):  
J Yao
Keyword(s):  
Fluid Dynamics ◽  
Experimental Data ◽  
Computational Procedure ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Piston Motion ◽  
Time Step ◽  
Simulation Accuracy ◽  
Wave Maker ◽  
Permanent Form

OpenFOAM is an open source CFD (Computational Fluid Dynamics) toolbox and recently attracts many researchers to develop codes based on it for their own applications. In order to numerically generate waves based on the wave-maker theory for a piston motion, numerical improvements have been done on the base of OpenFOAM by the author. In gen-eral, the present new tool can be employed to simulate wave generation as long as the piston motion is given. This paper presents the related computational procedure and simulations for generating relatively long finite-amplitude waves ac-cording to Madsen’s second-order wave-maker theory. The sensitivities of the computed incident wave profile to grid density and time step are investigated for the case of generating a wave with permanent form. The simulation accuracy is validated by comparison with the analytical solution and available experimental data.

A VIRTUAL PISTON-TYPE WAVE MAKER BASED ON OPENFOAM

2021 ◽  
Vol 159 (A2) ◽  
Author(s):  
J Yao
Keyword(s):  
Fluid Dynamics ◽  
Experimental Data ◽  
Computational Procedure ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Piston Motion ◽  
Time Step ◽  
Simulation Accuracy ◽  
Wave Maker ◽  
Permanent Form

OpenFOAM is an open source CFD (Computational Fluid Dynamics) toolbox and recently attracts many researchers to develop codes based on it for their own applications. In order to numerically generate waves based on the wave-maker theory for a piston motion, numerical improvements have been done on the base of OpenFOAM by the author. In gen- eral, the present new tool can be employed to simulate wave generation as long as the piston motion is given. This paper presents the related computational procedure and simulations for generating relatively long finite-amplitude waves ac- cording to Madsen’s second-order wave-maker theory. The sensitivities of the computed incident wave profile to grid density and time step are investigated for the case of generating a wave with permanent form. The simulation accuracy is validated by comparison with the analytical solution and available experimental data.

scholarly journals On the breaking of water waves of finite amplitude on a sloping beach

1958 ◽  
Vol 4 (3) ◽  
pp. 330-334 ◽  
Author(s):  
H. P. Greenspan
Keyword(s):  
Particle Velocity ◽  
Water Waves ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Shallow Water Theory ◽  
Wave Shape ◽  
Compressive Wave ◽  
Sloping Beach ◽  
Explicit Relation ◽  
Linear Shallow Water Theory

In a recent paper Carrier & Greenspan (1958) showed that, within the framework of the non-linear shallow-water theory, there exist waves which do not break as they climb a sloping beach. The formation of a shock or bore is dependent on a variety of factors (wave shape, particle velocity, etc.) and, as yet, no general criteria for breaking have been found. In this paper, we consider waves which propagate shoreward into quiescent water; it is shown that any compressive wave (a wave of positive amplitude) which has a non-zero slope at the wave-front eventually breaks before reaching the coastline. In fact, an explicit relation is obtained between the initial conditions and the position where breaking occurs.

scholarly journals Short Note: Steady Profile of a Finite-Amplitude Kinematic Wave on a Glacier

1968 ◽  
Vol 7 (49) ◽  
pp. 117-119 ◽  
Author(s):  
James N. Johnson
Keyword(s):  
Wave Front ◽  
Wave Velocity ◽  
Short Note ◽  
Finite Amplitude ◽  
Ice Thickness ◽  
Kinematic Wave ◽  
Order Of Magnitude

Abstract The steady profile of a finite-amplitude kinematic wave on a glacier is calculated with the assumption that the wave velocity varies linearly with h 1, the departure of the ice thickness from the datum state. If the flow of ice is due mainly to sliding of the glacier on its bed, the width of the calculated steady profile is several hundred times the datum state ice thickness. The width of an observed kinematic wave front on Nisqually Glacier, Mt. Rainier, Washington is at least an order of magnitude smaller than the calculated steady profile. This indicates that the observed steepening of the wave may be due to effects other than variation of wave velocity with ice thickness.

Spectral Analysis of Nonlinear Random Wave Loadings

2005 ◽  
Author(s):  
Rujian Ma ◽  
Guixi Li ◽  
Dong Zhao
Keyword(s):  
Spectral Analysis ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Circular Cylinders ◽  
Random Wave ◽  
Wave Spectra ◽  
Random Waves ◽  
First Order ◽  
Nonlinear Random Wave ◽  
Nonlinear Spectral Analysis

The spectral analysis of nonlinear random wave loadings on circular cylinders is performed in this paper by means of nonlinear spectral analysis. The study is carried out by expressing the wave profile and velocities of water particles as a nonlinear composition of the first order wave profile. Under the assumption of the first order wave profile being a zero-mean Gaussian process, the random wave spectra of finite amplitude waves are given. In order to solve the loading spectra of the finite amplitude random waves, the drag force is extended into power series of velocity. The loadings of the finite amplitude random waves are then expressed as nonlinear compositions of the first order wave profile and its derivatives. These techniques made it easier to compute the spectral densities of the finite amplitude random wave loadings.

scholarly journals Short Note: Steady Profile of a Finite-Amplitude Kinematic Wave on a Glacier

1968 ◽  
Vol 7 (49) ◽  
pp. 117-119 ◽  
Author(s):  
James N. Johnson
Keyword(s):  
Wave Front ◽  
Wave Velocity ◽  
Short Note ◽  
Finite Amplitude ◽  
Ice Thickness ◽  
Kinematic Wave ◽  
Order Of Magnitude

AbstractThe steady profile of a finite-amplitude kinematic wave on a glacier is calculated with the assumption that the wave velocity varies linearly with h1, the departure of the ice thickness from the datum state. If the flow of ice is due mainly to sliding of the glacier on its bed, the width of the calculated steady profile is several hundred times the datum state ice thickness. The width of an observed kinematic wave front on Nisqually Glacier, Mt. Rainier, Washington is at least an order of magnitude smaller than the calculated steady profile. This indicates that the observed steepening of the wave may be due to effects other than variation of wave velocity with ice thickness.

Nonlinear Breaking of Wave Fronts in a Vibrationally Relaxing Gas

1982 ◽  
Vol 49 (1) ◽  
pp. 10-12
Author(s):  
R. Shyam ◽  
V. D. Sharma ◽  
V. V. Menon
Keyword(s):  
Wave Front ◽  
Vibrational Relaxation ◽  
Method Of Characteristics ◽  
Spherically Symmetric ◽  
Wave Fronts ◽  
Compressive Wave ◽  
Initial Discontinuity ◽  
Explicit Criteria ◽  
Front Curvature ◽  
Relaxing Gas

Using the method of characteristics, some explicit criteria for the nonlinear breaking of wave fronts in a plane, cylindrically or spherically symmetric flows of a vibrationally relaxing gas are obtained. It is shown that except a spherically converging compressive wave, all compressive waves break only when the initial discontinuity at the wave-head exceeds a critical value, and the effects of vibrational relaxation and that of the wave front curvature are to increase the time of their breaking. It is found that the stabilizing influences of vibrational relaxation and that of the wave front curvature are not strong enough to overcome the breaking tendency of a spherically converging compressive wave; in fact, in contrast to a cylindrically converging wave, it always breaks down before the formation of a focus, no matter how small be the initial discontinuity at the wave-head.

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