scholarly journals Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current

2021 ◽  
Vol 927 ◽  
Author(s):  
Curtis Hooper ◽  
Karima Khusnutdinova ◽  
Roger Grimshaw
Keyword(s):  
Plane Waves ◽  
Large Family ◽  
Tangent Plane ◽  
Stratified Fluid ◽  
Internal Ring ◽  
Interfacial Waves ◽  
Homogeneous Fluid ◽  
Modal Structure ◽  
Shear Current ◽  
Linear Shear

We study long surface and internal ring waves propagating in a stratified fluid over a parallel shear current. The far-field modal and amplitude equations for the ring waves are presented in dimensional form. We re-derive the modal equations from the formulation for plane waves tangent to the ring wave, which opens a way to obtaining important characteristics of the ring waves (group speed, wave action conservation law) and to constructing more general ‘hybrid solutions’ consisting of a part of a ring wave and two tangent plane waves. The modal equations constitute a new spectral problem, and are analysed for a number of examples of surface ring waves in a homogeneous fluid and internal ring waves in a stratified fluid. Detailed analysis is developed for the case of a two-layered fluid with a linear shear current where we study their wavefronts and two-dimensional modal structure. Comparisons are made between the modal functions (i.e. eigenfunctions of the relevant spectral problems) for the surface waves in homogeneous and two-layered fluids, as well as the interfacial waves described exactly and in the rigid-lid approximation. We also analyse the wavefronts of surface and interfacial waves for a large family of power-law upper-layer currents, which can be used to model wind generated currents, river inflows and exchange flows in straits. Global and local measures of the deformation of wavefronts are introduced and evaluated.

scholarly journals On the Dynamics of Two-Dimensional Capillary-Gravity Solitary Waves with a Linear Shear Current

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Dali Guo ◽  
Bo Tao ◽  
Xiaohui Zeng
Keyword(s):  
Solitary Waves ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Numerical Study ◽  
Two Dimensional ◽  
Shear Current ◽  
Linear Shear ◽  
The Stability ◽  
Depression Wave ◽  
Elevation Wave

The numerical study of the dynamics of two-dimensional capillary-gravity solitary waves on a linear shear current is presented in this paper. The numerical method is based on the time-dependent conformal mapping. The stability of different kinds of solitary waves is considered. Both depression wave and large amplitude elevation wave are found to be stable, while small amplitude elevation wave is unstable to the small perturbation, and it finally evolves to be a depression wave with tails, which is similar to the irrotational capillary-gravity waves.

The stability of a stratified fluid

1967 ◽  
Vol 29 (2) ◽  
pp. 233-240
Author(s):  
J. B. Hinwood
Keyword(s):  
Turbulent Flow ◽  
Froude Number ◽  
Reynolds Numbers ◽  
Stratified Fluid ◽  
Stratified Flows ◽  
Rectangular Duct ◽  
Interfacial Waves ◽  
Homogeneous Flow ◽  
Critical Reynolds Numbers ◽  
The Stability

For the flow of a stably-stratified fluid in the inlet region of a rectangular duct, it is shown experimentally that the upper and lower critical Reynolds numbers are functions of the interfacial Froude number F, and that if F is large they are lower than for a homogeneous flow. In stratified flows the disturbances leading to turbulent flow sometimes arise at the interface and lead to interfacial waves, whose wavelength at breaking is equal to the conduit depth.

scholarly journals Gravity–capillary waves in finite depth on flows of constant vorticity

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160363 ◽  
Author(s):  
Hung-Chu Hsu ◽  
Marc Francius ◽  
Pablo Montalvo ◽  
Christian Kharif
Keyword(s):  
Surface Profile ◽  
Finite Depth ◽  
Bond Number ◽  
Perturbation Series ◽  
Capillary Waves ◽  
Constant Vorticity ◽  
Expansion Procedure ◽  
Shear Current ◽  
Linear Shear

This paper considers two-dimensional periodic gravity–capillary waves propagating steadily in finite depth on a linear shear current (constant vorticity). A perturbation series solution for steady periodic waves, accurate up to the third order, is derived using a classical Stokes expansion procedure, which allows us to include surface tension effects in the analysis of wave–current interactions in the presence of constant vorticity. The analytical results are then compared with numerical computations with the full equations. The main results are (i) the phase velocity is strongly dependent on the value of the vorticity; (ii) the singularities (Wilton singularities) in the Stokes expansion in powers of wave amplitude that correspond to a Bond number of 1/2 and 1/3, which are the consequences of the non-uniformity in the ordering of the Fourier coefficients, are found to be influenced by vorticity; (iii) different surface profiles of capillary–gravity waves are computed and the effect of vorticity on those profiles is shown to be important, in particular that the solutions exhibit type-2-like wave features, characterized by a secondary maximum on the surface profile with a trough between the two maxima.

scholarly journals Steady boundary-layer solutions for a swirling stratified fluid in a rotating cone

1999 ◽  
Vol 384 ◽  
pp. 339-374 ◽  
Author(s):  
R. E. HEWITT ◽  
P. W. DUCK ◽  
M. R. FOSTER
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Swirling Flow ◽  
Rotating Disk ◽  
Stratified Fluid ◽  
Steady States ◽  
General Description ◽  
Homogeneous Fluid ◽  
Axisymmetric Solution ◽  
Steady Solutions

We consider a set of nonlinear boundary-layer equations that have been derived by Duck, Foster & Hewitt (1997a, DFH), for the swirling flow of a linearly stratified fluid in a conical container. In contrast to the unsteady analysis of DFH, we restrict attention to steady solutions and extend the previous discussion further by allowing the container to both co-rotate and counter-rotate relative to the contained swirling fluid. The system is governed by three parameters, which are essentially non-dimensional measures of the rotation, stratification and a Schmidt number. Some of the properties of this system are related (in some cases rather subtly) to those found in the swirling flow of a homogeneous fluid above an infinite rotating disk; however, the introduction of buoyancy effects with a sloping boundary leads to other (new) behaviours. A general description of the steady solutions to this system proves to be rather complicated and shows many interesting features, including non-uniqueness, singular solutions and bifurcation phenomena.We present a broad description of the steady states with particular emphasis on boundaries in parameter space beyond which steady states cannot be continued.A natural extension of this work (motivated by recent experimental results) is to investigate the possibility of solution branches corresponding to non-axisymmetric boundary-layer states appearing as bifurcations of the axisymmetric solutions. In an Appendix we give details of an exact, non-axisymmetric solution to the Navier–Stokes equations (with axisymmetric boundary conditions) corresponding to the flow of homogeneous fluid above a rotating disk.

Motion of an axisymmetric body in a rotating stratified fluid confined between two parallel planes

1969 ◽  
Vol 38 (4) ◽  
pp. 833-842 ◽  
Author(s):  
D. V. Krishna ◽  
L. V. Sarma
Keyword(s):  
Steady State ◽  
Oblate Spheroid ◽  
Vertical Axis ◽  
Vertical Gradient ◽  
Steady State Solution ◽  
Stratified Fluid ◽  
Axisymmetric Body ◽  
The Body ◽  
Homogeneous Fluid ◽  
Axis Of Rotation

We consider here the flow due to the oscillation of a slender oblate spheroid in a non-homogeneous, rotating fluid confined between two parallel planes which are perpendicular to the (vertical) axis of rotation. The direction of oscillation of the spheroid is perpendicular to the axis of rotation. By solving a set of dual integrals the steady-state solution is obtained in the two cases when the plates are at an infinite distance from the body and when they are at a large but finite distance. The singular or discontinuous surfaces observed in the case of homogeneous fluid are absent here. Also, the steady-state velocity is no longer independent of the distance along the axis of rotation. The velocity has now a vertical gradient, an important feature in the case of stratified fluid. It is also found that the presence of the plane boundaries increases the force on the body.

Large-amplitude interfacial waves on a linear shear flow in the presence of a current

1993 ◽  
Vol 249 (-1) ◽  
pp. 499 ◽  
Author(s):  
George Breyiannis ◽  
Vasilis Bontozoglou ◽  
Dimitris Valougeorgis ◽  
Apostolos Goulas
Keyword(s):  
Shear Flow ◽  
Large Amplitude ◽  
Interfacial Waves ◽  
Linear Shear ◽  
A Current

Inviscid waves on a Lamb–Oseen vortex in a rotating stratified fluid: consequences for the elliptic instability

2008 ◽  
Vol 597 ◽  
pp. 283-303 ◽  
Author(s):  
STÉPHANE LE DIZÈS
Keyword(s):  
Plane Waves ◽  
Stratified Fluid ◽  
Lagrangian Description ◽  
Small Range ◽  
Local Approach ◽  
Coriolis Parameter ◽  
Centrifugal Instability ◽  
Local Plane ◽  
Core Modes ◽  
Oseen Vortex

The inviscid waves propagating on a Lamb–Oseen vortex in a rotating medium for an unstratified fluid and for a strongly stratified fluid are analysed using numerical and asymptotic approaches. By a local Lagrangian description, we first provide the characteristics of the local plane waves (inertia–gravity waves) as well as the local growth rate associated with the centrifugal instability when the vortex is unstable. A global WKBJ approach is then used to determine the frequencies of neutral core modes and neutral ring modes. We show that these global Kelvin modes only exist in restricted domains of the parameters. The consequences of these domain limitations for the occurrence of the elliptic instability are discussed. We argue that in an unstratified fluid the elliptic instability should be active in a small range of the Coriolis parameter which could not have been predicted from a local approach. The wavenumbers of the sinuous modes of the elliptic instability are provided as a function of the Coriolis parameter for both an unstratified fluid and a strongly stratified fluid.

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