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.

scholarly journals On bounds for steady waves with negative vorticity

2021 ◽  
Vol 23 (2) ◽  
Author(s):  
Evgeniy Lokharu
Keyword(s):  
Froude Number ◽  
Solitary Waves ◽  
Upper Bound ◽  
Critical Value ◽  
Two Dimensional ◽  
Absolute Value ◽  
Constant Vorticity ◽  
Negative Constant

AbstractWe prove that no two-dimensional Stokes and solitary waves exist when the vorticity function is negative and the Bernoulli constant is greater than a certain critical value given explicitly. In particular, we obtain an upper bound $$F \le \sqrt{2} + \epsilon $$ F ≤ 2 + ϵ for the Froude number of solitary waves with a negative constant vorticity, sufficiently large in absolute value.

scholarly journals On periodic geophysical water flows with discontinuous vorticity in the equatorial f -plane approximation

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170096 ◽  
Author(s):  
Calin Iulian Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Small Amplitude ◽  
Wave Speed ◽  
Laminar Flows ◽  
Local Bifurcation ◽  
Two Dimensional ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Water Flows

We are concerned here with geophysical water waves arising as the free surface of water flows governed by the f -plane approximation. Allowing for an arbitrary bounded discontinuous vorticity, we prove the existence of steady periodic two-dimensional waves of small amplitude. We illustrate the local bifurcation result by means of an analysis of the dispersion relation for a two-layered fluid consisting of a layer of constant non-zero vorticity γ 1 adjacent to the surface situated above another layer of constant non-zero vorticity γ 2 ≠ γ 1 adjacent to the bed. For certain vorticities γ 1 , γ 2 , we also provide estimates for the wave speed c in terms of the speed at the surface of the bifurcation inducing laminar flows. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals No steady water waves of small amplitude are supported by a shear flow with a still free surface

2013 ◽  
Vol 717 ◽  
pp. 523-534 ◽  
Author(s):  
Vladimir Kozlov ◽  
Nikolay Kuznetsov
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Gravity Waves ◽  
Water Waves ◽  
Small Amplitude ◽  
Finite Depth ◽  
Coordinate Frame ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Positive Coefficients

AbstractThe two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.

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’.

A note on the instabilities of a horizontal shear flow with a free surface

2000 ◽  
Vol 406 ◽  
pp. 337-346 ◽  
Author(s):  
L. ENGEVIK
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Shear Flow ◽  
Velocity Profile ◽  
Froude Number ◽  
The Other ◽  
Algebraic Equations ◽  
Horizontal Shear ◽  
Analytical Treatment ◽  
Unstable Region

The instabilities of a free surface shear flow are considered, with special emphasis on the shear flow with the velocity profile U* = U*0sech2 (by*). This velocity profile, which is found to model very well the shear flow in the wake of a hydrofoil, has been focused on in previous studies, for instance by Dimas & Triantyfallou who made a purely numerical investigation of this problem, and by Longuet-Higgins who simplified the problem by approximating the velocity profile with a piecewise-linear profile to make it amenable to an analytical treatment. However, none has so far recognized that this problem in fact has a very simple solution which can be found analytically; that is, the stability boundaries, i.e. the boundaries between the stable and the unstable regions in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic equations in k and F. This applies also when surface tension is included. With no surface tension present there exist two distinct regimes of unstable waves for all values of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by 0 < k < (1 − F2/6), the other by F−2 < k < 9F−2, which is a very extended region on the k-axis. When F [Gt ] 1 there is one small unstable region close to k = 0, i.e. 0 < k < 9/(4F2), the other unstable region being (3/2)1/2F−1 < k < 2 + 27/(8F2). When surface tension is included there may be one, two or even three distinct regimes of unstable modes depending on the value of the Froude number. For small F there is only one instability region, for intermediate values of F there are two regimes of unstable modes, and when F is large enough there are three distinct instability regions.

Waves and wave resistance of thin bodies moving at low speed: the free-surface nonlinear effect

1975 ◽  
Vol 69 (2) ◽  
pp. 405-416 ◽  
Author(s):  
G. Dagan
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Surface Condition ◽  
Perturbation Expansion ◽  
Wave Resistance ◽  
The Body ◽  
Thin Body ◽  
Two Dimensional ◽  
First Order ◽  
Flow Past

The linearized theory of free-surface gravity flow past submerged or floating bodies is based on a perturbation expansion of the velocity potential in the slenderness parameter e with the Froude number F kept fixed. It is shown that, although the free-wave amplitude and the associated wave resistance tend to zero as F → 0, the linearized solution is not uniform in this limit: the ratio between the second- and first-order terms becomes unbounded as F → 0 with ε fixed. This non-uniformity (called ‘the second Froude number paradox’ in previous work) is related to the nonlinearity of the free-surface condition. Criteria for uniformity of the thin-body expansion, combining ε and F, are derived for two-dimensional flows. These criteria depend on the shape of the leading (and trailing) edge: as the shape becomes finer the linearized solution becomes valid for smaller F.Uniform first-order approximations for two-dimensional flow past submerged bodies are derived with the aid of the method of co-ordinate straining. The straining leads to an apparent displacement of the most singular points of the body contour (the leading and trailing edges for a smooth shape) and, therefore, to an apparent change in the effective Froude number.

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)).

scholarly journals A note on the flow induced by a line sink beneath a free surface

1991 ◽  
Vol 32 (3) ◽  
pp. 251-260 ◽  
Author(s):  
G. C. Hocking ◽  
L. K. Forbes
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Ideal Fluid ◽  
Critical Value ◽  
Homogeneous Fluid ◽  
Substitution Method ◽  
A Value ◽  
Line Sink ◽  
Low Froude Number

AbstractThe problem of withdrawing water through a line sink from a region containing an homogeneous fluid beneath a free surface is considered. Assuming steady, irrotational flow of an ideal fluid, solutions with low Froude number containing a stagnation point on the free surface above the sink are sought using a series substitution method. The solutions are shown to exist for a value of the Froude number up to a critical value of about 1.4. No solutions of this type are found for Froude numbers greater than this value.

Withdrawal of a stratified fluid from a vertical two-dimensional duct

1975 ◽  
Vol 70 (2) ◽  
pp. 321-332 ◽  
Author(s):  
Jorg Imberger ◽  
Chris Fandry
Keyword(s):  
Number Theory ◽  
Free Surface ◽  
Froude Number ◽  
Stratified Fluid ◽  
Two Dimensional ◽  
Tank Bottom ◽  
Vertical Duct

On the basis of a small Froude number theory it is shown that previous horizontalduct models of the withdrawal of fluid from a stratified tank with a free surface and a fixed bottom are inappropriate at large times and that the flow should be modelled by that from an infinite vertical duct. The falling horizontal free surface in the tank is replaced by a vertically moving column of stratified fluid and the tank bottom is modelled by a stagnant pool of fluid below. This model is analysed and a solution uniformly valid in time is presented. The properties of the solution are then compared with existing theories and experiments.

scholarly journals Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170102 ◽  
Author(s):  
D. Henry ◽  
G. P. Thomas
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Surface Profile ◽  
Two Dimensions ◽  
Arbitrary Distribution ◽  
Nonlinear Water Waves ◽  
Steady Current ◽  
Linear Waves ◽  
Free Surface Profile ◽  
Near Shore

This paper considers the pressure–streamfunction relationship for a train of regular water waves propagating on a steady current, which may possess an arbitrary distribution of vorticity, in two dimensions. The application of such work is to both near shore and offshore environments, and in particular, for linear waves we provide a description of the role which the pressure function on the seabed plays in determining the free-surface profile elevation. Our approach is shown to provide a good approximation for a range of current conditions. This article is part of the theme issue ‘Nonlinear water waves’.

Sign in / Sign up

Export Citation Format

Share Document