scholarly journals Hamiltonian long–wave expansions for water waves over a rough bottom

2005 ◽  
Vol 461 (2055) ◽  
pp. 839-873 ◽  
Author(s):  
Walter Craig ◽  
Philippe Guyenne ◽  
David P. Nicholls ◽  
Catherine Sulem
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Kdv Equation ◽  
Bottom Topography ◽  
Multiple Scale ◽  
Three Dimensions ◽  
Length Scale ◽  
Two Dimensional ◽  
Long Wave ◽  
Starting Point

This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long–wave asymptotic regime, extending the results of R. Rosales & G. Papanicolaou (1983 Stud. Appl. Math. 68 , 89–102) on periodic bottoms for two–dimensional flows.We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being Zakharov's Hamiltonian (V. E. Zakharov 1968 J. Appl. Mech. Tech. Phys. 9, 1990–1994) for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length–scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length–scale as, or longer than, the order of the wavelength of the surface waves. We do not take up the question of random bottom variations, a topic which is considered in Rosales & Papanicolaou (1983). In the two–dimensional case of waves in a channel, we give an alternate derivation of the effective Korteweg–de Vries (KdV) equation that is obtained in Rosales & Papanicolaou (1983). In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long–scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three–dimensional long–wave equations in a Boussinesq scaling regime, and again in certain cases an effective Kadomtsev–Petviashvili (KP) system in the appropriate unidirectional limit. The computations for these results are performed in the framework of an asymptotic analysis of multiple–scale operators. In the present case this involves the Dirichlet–Neumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.

scholarly journals Steady Two-Dimensional Free-Surface Flow Past Disturbances in an Open Channel: Solutions of the Korteweg–De Vries Equation and Analysis of the Weakly Nonlinear Phase Space

Fluids ◽  
2019 ◽  
Vol 4 (1) ◽  
pp. 24 ◽  
Author(s):  
Benjamin Binder
Keyword(s):  
Free Surface ◽  
Open Channel ◽  
Kdv Equation ◽  
Bottom Topography ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Two Dimensional ◽  
Model Approximation ◽  
Flow Past ◽  
Channel Bottom

Two-dimensional free-surface flow past disturbances in an open channel is a classical problem in hydrodynamics—a problem that has received considerable attention over the last two centuries (e.g., see Lamb’s Treatise, 1879). With traces back to Russell’s experimental observations of the Great Wave of Translation in 1834, Korteweg and de Vries (1895), and others, derived an unforced equation to describe the balance between nonlinearity and dispersion required to model the solitary wave. More recently, Akylas (1984) derived a forced KdV equation to model a pressure distribution on the free-surface (e.g., a ship). Since then, the forced KdV equation has been shown to be a useful model approximation for two-dimensional flow past disturbances in an open channel. In this paper, we review the stationary solutions of the forced KdV equation for four types of localised disturbances: (i) a flat plate separating two free surfaces; (ii) a compact bump, or dip in the channel bottom topography; (iii) a compact distribution of pressure on the free surface and (iv) a step-wise change in the otherwise constant horizontal level of the channel bottom topography. Moreover, Dias and Vanden-Broeck (2002) developed a phase plane method to analyse flow over a bump, and this general approach has also been applied to the three other types of forcing (see Binder et al., 2005–2015, and others). In this study, we use eleven basic flow types to classify the steady solutions of the forced KdV equation using the phase plane method. Additionally, considering solutions that are wave-free both far upstream and far downstream, we compare KdV model approximations of the uniform flow conditions in the far-field with exact solutions of the full problem. In particular, we derive a new KdV model approximation for the upstream dimensionless flow-rate which is conveniently given in terms of the known downstream dimensionless flow-rate.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

Vorticity and curvature at a free surface

1998 ◽  
Vol 356 ◽  
pp. 149-153 ◽  
Author(s):  
MICHAEL S. LONGUET-HIGGINS
Keyword(s):  
Free Surface ◽  
Simple Relation ◽  
Surface Curvature ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Umbilical Point ◽  
Two Dimensional Flow ◽  
Principal Directions ◽  
Component Parallel

For two-dimensional flow there is a simple relation between the vorticity at a stress-free surface and the surface curvature. In this note the relation is generalized to flow in three dimensions. It is shown that in addition to a component of vorticity perpendicular to the flow there is also a component parallel to the direction of flow. The latter vanishes only at an umbilical point or when the flow is in one of the two principal directions of curvature.

scholarly journals Modeling the Contact Mechanics of Hydrogels

Lubricants ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 35
Author(s):  
Martin H. Müser ◽  
Han Li ◽  
Roland Bennewitz
Keyword(s):  
Contact Mechanics ◽  
Feasibility Study ◽  
Three Dimensions ◽  
Coarse Grained ◽  
Length Scale ◽  
Two Dimensional ◽  
Spring Model ◽  
Mechanical Pressure ◽  
Solvent Quality ◽  
A Chain

A computationally lean model for the coarse-grained description of contact mechanics of hydrogels is proposed and characterized. It consists of a simple bead-spring model for the interaction within a chain, potentials describing the interaction between monomers and mold or confining walls, and a coarse-grained potential reflecting the solvent-mediated effective repulsion between non-bonded monomers. Moreover, crosslinking only takes place after the polymers have equilibrated in their mold. As such, the model is able to reflect the density, solvent quality, and the mold hydrophobicity that existed during the crosslinking of the polymers. Finally, such produced hydrogels are exposed to sinusoidal indenters. The simulations reveal a wavevector-dependent effective modulus E * ( q ) with the following properties: (i) stiffening under mechanical pressure, and a sensitivity of E * ( q ) on (ii) the degree of crosslinking at large wavelengths, (iii) the solvent quality, and (iv) the hydrophobicity of the mold in which the polymers were crosslinked. Finally, the simulations provide evidence that the elastic heterogeneity inherent to hydrogels can suffice to pin a compressed hydrogel to a microscopically frictionless wall that is undulated at a mesoscopic length scale. Although the model and simulations of this feasibility study are only two-dimensional, its generalization to three dimensions can be achieved in a straightforward fashion.

Generalized vortex methods for free-surface flow problems

1982 ◽  
Vol 123 ◽  
pp. 477-501 ◽  
Author(s):  
Gregory R. Baker ◽  
Daniel I. Meiron ◽  
Steven A. Orszag
Keyword(s):  
Water Waves ◽  
Evolution Equations ◽  
Bottom Topography ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Three Dimensions ◽  
Fredholm Integral Equations ◽  
Free Surfaces ◽  
Flow Problems ◽  
Dipole Vortex

The motion of free surfaces in incompressible, irrotational, inviscid layered flows is studied by evolution equations for the position of the free surfaces and appropriate dipole (vortex) and source strengths. The resulting Fredholm integral equations of the second kind may be solved efficiently in both storage and work by iteration in both two and three dimensions. Applications to breaking water waves over finite-bottom topography and interacting triads of surface and interfacial waves are given.

Note on the scattering of water waves by a vertical barrier

1966 ◽  
Vol 62 (3) ◽  
pp. 507-509 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Vertical Barrier ◽  
Alternative Approach

Introduction. In this note an alternative approach is presented to the problem of the scattering of small amplitude two-dimensional water waves by a fixed barrier, one edge of the barrier lying in the free surface of the water. This problem was first solved by Ursell ((1)) and generalizations of the problem have been considered by John ((2)) and Lewin ((3)).

Water waves over a random bottom

2009 ◽  
Vol 640 ◽  
pp. 79-107 ◽  
Author(s):  
W. CRAIG ◽  
P. GUYENNE ◽  
C. SULEM
Keyword(s):  
Free Surface ◽  
Random Process ◽  
Water Waves ◽  
Large Scale ◽  
Principal Component ◽  
Random Coefficients ◽  
Homogenization Theory ◽  
Dynamical Regime ◽  
Long Wave ◽  
Imperfect Knowledge

This paper gives a new derivation and an analysis of long-wave model equations for the dynamics of the free surface of a body of water which has random bathymetry. This is a problem of hydrodynamical significance to coastal regions and to global-scale propagation of tsunamis, for which there may be imperfect knowledge of the detailed topography of the bottom. The surface motion is assumed to be in a long-wavelength dynamical regime, while the bottom of the fluid region is given by a stationary random process whose realizations vary over short length scales and are decorrelated on the longer principal length scale of the surface waves. Our basic conclusions are that coherent solutions propagating over a random bottom maintain basic properties of their structure over long distances, but however, the effect of the random bottom introduces uncertainty in the location of the solution profile and modifies the amplitude by random factors. It also gives rise to a random scattered component of the solution, but this does not result in the dispersion of the principal component of the solution, at least over length and time scales considered in this regime. We illustrate these results with numerical simulations.The mathematical question is one of homogenization theory in the long-wave scaling regime, for which our work is a reappraisal of the paper of Rosales & Papanicolaou (Stud. Appl. Math., vol. 68, 1983, pp. 89–102). In particular, we derive appropriate Boussinesq and Korteweg–deVries type equations with random coefficients which describe the free-surface evolution in this regime. The derivation is performed from the point of view of perturbation theory for Hamiltonian partial differential equations with a small parameter, with a subsequent analysis of the random effects in the resulting solutions. In the analysis, we highlight the distinction between the effective equations for a fixed typical realization, for which there are coherent solitary-wave solutions, and their ensemble average, which may exhibit diffusive effects. Our results extend the prior analysis to the case of non-zero variance σ2β > 0, and furthermore the analysis identifies the canonical limit random process as a white noise with covariance σβ2δ(X − X′) and quantifies the variations in phase and amplitude of the principal and scattered components of solutions. We find that the random topography can give rise to an additional linear term in the KdV limit equations, which depends upon a skew property of the random process and whose sign affects the stability of solutions. Finally we generalize this analysis to the case in which the bottom has large-scale deterministic variations on which are superposed random fluctuations with slowly varying statistical properties.

On the Disturbance of a Thin Layer of Liquid by a Moving Obstruction

1990 ◽  
Vol 57 (4) ◽  
pp. 1066-1072
Author(s):  
Roger F. Gans ◽  
Chung-Hai Wang
Keyword(s):  
Surface Tension ◽  
Reynolds Number ◽  
Free Surface ◽  
Thin Layer ◽  
Viscous Flow ◽  
Length Scale ◽  
Two Dimensional ◽  
Thin Liquid Layer ◽  
Horizontal Length ◽  
Small Ratio

We calculate the free surface shapes upstream and downstream of an obstacle obstructing a thin liquid layer on a moving surface, taking into account gravity and surface tension. We assume low Reynolds number viscous flow, a two-dimensional layer, and small ratio of vertical to horizontal length scale. The upstream and downstream shapes are very different. The upstream liquid piles up against the obstacle to provide an overpressure sufficient to drive the Poiseuille component of the lubrication flow under the obstacle. The downstream liquid is disturbed only by surface tension.

A Two-dimensional Boussinesq equation for water waves and some of its solutions

1996 ◽  
Vol 323 ◽  
pp. 65-78 ◽  
Author(s):  
R. S. Johnson
Keyword(s):  
Soliton Solution ◽  
Water Waves ◽  
Boussinesq Equation ◽  
Kdv Equation ◽  
Boussinesq Equations ◽  
Parametric Representation ◽  
Two Dimensional ◽  
Completely Integrable ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

A two-dimensional Boussinesq equation, \[u_{tt} - u_{xx} + 3(u^2)_{xx} - u_{xxxx} - u_{yy} = 0,\] is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation.A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

scholarly journals The Froude number for solitary water waves with vorticity

2015 ◽  
Vol 768 ◽  
pp. 91-112 ◽  
Author(s):  
Miles H. Wheeler
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Froude Number ◽  
Simple Proof ◽  
Upper Bound ◽  
Water Waves ◽  
Wave Speed ◽  
Critical Value ◽  
Arbitrary Distribution ◽  
Two Dimensional

We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.

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