On the theory of water waves

1974 ◽  
Vol 338 (1612) ◽  
pp. 43-55 ◽  
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
De Vries ◽  
Set Up ◽  
Propagation Of Waves

The aim of this paper is to formulate a two-dimensional theory for the propagation of fairly long water waves. The approach differs from the usual in that the theory is set up via two-dimensional postulates. Subsequently, it is shown how a simple three-dimensional approxi­mation enables us to relate the two-dimensional theory to the three-dimensional theory. The resulting equations are used to discuss the unidirectional propagation of waves. lt is shown how the results obtained from the theory proposed here are related to the results of Korteweg & de Vries (1895) and to those of Benjamin, Bona & Mahony (1972).

Directed fluid sheets

1976 ◽  
Vol 347 (1651) ◽  
pp. 447-473 ◽  
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Steady Motion ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Inviscid Fluids ◽  
Two Dimensional Flows

This paper is concerned mainly with incompressible inviscid fluid sheets but the incompressible linearly viscous fluid sheet is also considered. Our development is based on a direct formulation using the two dimensional theory of directed media called Cosserat surfaces . The first part of the paper deals with the formulation of appropriate nonlinear equations (which may include the effects of gravity and surface tension) governing the two dimensional motion of incompressible inviscid media for two categories, namely those ( a ) for two dimensional flows confined to a plane perpendicular to a specified direction and ( b ) for propagation of fairly long waves in a stream of variable initial depth. The latter development is a generalization of an earlier direct formulation of a theory of water waves when the fixed bottom of the stream is level (Green, Laws & Naghdi 1974). In the second part of the paper, special attention is given to a demonstration of the relevance and applicability of the present direct formulation to a variety of two dimensional problems of inviscid fluid sheets. These include, among others, the steady motion of a class of two-dimensional flows in a stream of finite depth in which the bed of the stream may change from one constant level to another, the related problem of hydraulic jumps, and a class of exact solutions which characterize the main features of the time-dependent free surface flows in the three dimensional theory of incompressible inviscid fluids.

Membrane theory

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Three Dimensional ◽  
Thin Sheets ◽  
Two Dimensional ◽  
Leading Order ◽  
Membrane Theory ◽  
Dimensional Theory ◽  
Small Thickness

This chapter develops two-dimensional membrane theory as a leading order small-thickness approximation to the three-dimensional theory for thin sheets. Applications to axisymmetric equilibria are developed in detail, and applied to describe the phenomenon of bulge propagation in cylinders.

On the subharmonic instabilities of steady three-dimensional deep water waves

1994 ◽  
Vol 262 ◽  
pp. 265-291 ◽  
Author(s):  
Mansour Ioualalen ◽  
Christian Kharif
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Transverse Direction ◽  
Plane Waves ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Two Dimensional ◽  
Wave Steepness ◽  
Oblique Angle

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.

Contact Temperature in Dry Bearings. Three Dimensional Theory and Verification

1981 ◽  
Vol 103 (2) ◽  
pp. 243-251 ◽  
Author(s):  
A. Floquet ◽  
D. Play
Keyword(s):  
Fourier Transform ◽  
Boundary Conditions ◽  
Experimental Verification ◽  
Three Dimensional ◽  
Contact Temperature ◽  
3D Analysis ◽  
New Work ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Infra Red

Boundary conditions were arbitrarily specified in an earlier two dimensional (2D) analysis of contact temperature. In this new work a general three dimensional (3D) Fourier transform solution is obtained from which for specific cases, the boundary conditions can be estimated. Further, experimental verification of 3D analysis was performed using infra-red technique.

On Three and Two-Dimensional Disturbances of Pipe Flow

1960 ◽  
Vol 27 (3) ◽  
pp. 381-389 ◽  
Author(s):  
Kurt Spielberg ◽  
Hans Timan
Keyword(s):  
Order Differential Equation ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Main Flow ◽  
Two Dimensional ◽  
Straight Pipe ◽  
Special Cases ◽  
Set Up ◽  
The Stability ◽  
The Given

A system of ordinary, coupled differential equations is set up for three-dimensional disturbances of Poiseuille flow in a straight pipe of circular cross section. The commonly treated equations are shown to be special cases arising from particular assumptions. It is shown that in the nonviscous, and therefore also in the general case, there exists, in contrast to the analogous problem in Cartesian co-ordinates, no transformation reducing the given problem to a two-dimensional one. A fourth-order differential equation is derived for disturbances independent of the direction of the main flow. The solutions, which are obtained, show that those two-dimensional disturbances, termed cross disturbances, decay with time and do therefore not disturb the stability of the main flow. Explicit expressions for the cross disturbances are obtained and a discussion of their nature is given.

Quasi-three-dimensional simulation of crescent-shaped waves

2020 ◽  
Author(s):  
Alexander Dosaev ◽  
Yuliya Troitskaya
Keyword(s):  
Water Waves ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Rogue Waves ◽  
Boundary Integral ◽  
Main Direction ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Wave Interactions

<p>Many features of nonlinear water wave dynamics can be explained within the assumption that the motion of fluid is strictly potential. At the same time, numerically solving exact equations of motion for a three-dimensional potential flow with a free surface (by means of, for example, boundary integral method) is still often considered too computationally expensive, and further simplifications are made, usually implying limitations on wave steepness. A quasi-three-dimensional model, put forward by V. P. Ruban [1], represents another approach at reducing computational cost. It is, in its essence, a two-dimensional model, formulated using conformal mapping of the flow domain, augmented by three-dimensional corrections. The model assumes narrow directional distribution of the wave field and is exact for two-dimensional waves. It was successfully applied by its author to study a nonlinear stage of of Benjamin-Feir instability and rogue waves formation.</p><p>The main aim of the present work is to explore the behaviour of the quasi-three-dimensional model outside the formal limits of its applicability. From the practical point of view, it is important that the model operates robustly even in the presence of waves propagating at large angles to the main direction (although we do not attempt to accurately describe their dynamics). We investigate linear stability of Stokes waves to three-dimensional perturbations and suggest a modification to the original model to eliminate a spurious zone of instability in the vicinity of the perpendicular direction on the perturbation wavenumber plane. We show that the quasi-three-dimensional model yields a qualitatively correct description of the instability zone generated by resonant 5-wave interactions. The values of the increment are reasonably close to those obtained from the exact equations of motion [2], despite the fact that the corresponding modes of instability consist of harmonics that are relatively far from the main direction. Resonant 5-wave interactions are known to manifest themselves in the formation of the so-called “horse-shoe” or “crescent-shaped” wave patterns, and the quasi-three-dimensional model exhibits a plausible dynamics leading to formation of crescent-shaped waves.</p><p>This research was supported by RFBR (grant No. 20-05-00322).</p><p>[1] Ruban, V. P. (2010). Conformal variables in the numerical simulations of long-crested rogue waves. <em>The European Physical Journal Special Topics</em>, <em>185</em>(1), 17-33.</p><p>[2] McLean, J. W. (1982). Instabilities of finite-amplitude water waves. <em>Journal of Fluid Mechanics</em>, <em>114</em>, 315-330.</p>

On the Transition from Two‐Dimensional to Three‐Dimensional Water Waves

2000 ◽  
Vol 104 (2) ◽  
pp. 91-127 ◽  
Author(s):  
F. Dias ◽  
M. Hărăguş–Courcelle
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Two Dimensional

scholarly journals Can turbulent reconnection be fast in 2D?

2010 ◽  
Vol 6 (S271) ◽  
pp. 385-386
Author(s):  
K. Kulpa-Dybeł ◽  
G. Kowal ◽  
K. Otmianowska-Mazur ◽  
A. Lazarian ◽  
E. Vishniac
Keyword(s):  
Three Dimensional ◽  
Open Boundary ◽  
Initial Vector ◽  
Spectral Space ◽  
Two Dimensional ◽  
Open Boundary Conditions ◽  
Random Forcing ◽  
Potential Perturbation ◽  
Set Up ◽  
External Driving

AbstractTurbulent reconnection is studied by means of two-dimensional (2D) compressible magnetohydrodynamical numerical calculations. The process of homogeneous turbulence is set up by adding two-dimensional random forcing implemented in the spectral space at small wave numbers with no correlation between velocity and forcing. We apply the initial Harris current sheet configuration together with a density profile calculated from the numerical equilibrium of magnetic and gas pressures. We assume that there is no external driving of the reconnection. The reconnection develops as a result of the initial vector potential perturbation. We use open boundary conditions. Our main goal is to find the dependencies of reconnection rate on the uniform resistivity. We present that the reconnection speed depends on the Lindquist number in 2D in the case of low as well as high resolution. When we apply more powerful turbulence the reconnection is faster, however the speed of reconnection is smaller than in the case of our three-dimensional numerical simulations.

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