The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution

Abstract

An approximate theory is given for the generation of internal gravity waves in a viscous Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder. A parameter λ which is proportional to the square of the ratio of the thickness of the oscillatory boundary layer that surrounds the cylinder to a typical dimension of its cross-section is introduced. When λ 1 (or equivalently when the Reynolds number R 1), the viscous boundary condition at the surface of the cylinder may to first order in λ be replaced by the inviscid one. A viscous solution is proposed for the case λ 1 in which the Fourier representation of the stream function found in Part 1 (Hurley 1997) is modified by including in the integrands a factor to account for viscous dissipation. In the limit λ 4 0 the proposed solution becomes the inviscid one at each point in the flow field.For ease of presentation the case of a circular cylinder of radius a is considered first and we take a to be the typical dimension of its cross-section in the definition of λ above. The accuracy of the proposed approximate solution is investigated both analytically and numerically and it is concluded that it is accurate throughout the flow field if λ is sufficiently small, except in a small region near where the characteristics touch the cylinder where viscous effects dominate.Computations indicate that the velocity on the centreline on a typical beam of waves, at a distance s along the beam from the centre of the cylinder, agrees, within about 1 %, with the (constant) inviscid values provided λs\a is less than about 10 −$ . This result is interpreted as indicating that those viscous effects which originate from the characteristics that touch the cylinder (places where the inviscid velocity is singular) reach the centreline of the beam when λs\a is about 10 −$ . For larger values of s, viscous effects are significant throughout the beam and the velocity profile of the beam changes until it attains, within about 1 % when λs\a is about 2, the value given by the similarity solution obtained by Thomas & Stevenson (1972). For larger values of λs\a, their similarity solution applies.In an important paper Makarov et al. (1990) give an approximate solution for the circular cylinder that is very similar to ours. However, it does not reduce to the inviscid one when the viscosity is taken to be zero.Finally it is shown that our results for a circular cylinder apply, after small modifications, to all elliptical cylinders.

New wave generation

“…We choose to investigate the so-called Thomas-Stevenson profile (Thomas & Stevenson 1972), a viscous self-similar solution of (2.1) that can be considered as the far-field limit of the viscous solution of elliptic cylinder oscillating in a stratified fluid (Hurley & Keady 1997). It has been shown that this profile describes oceanographically relevant internal wave beams far from their generation site at the continental shelf ; after their reflection at the bottom of the ocean, such wave beams are thought to be the cause of solitons generated at the thermocline (see New & Da Silva 2002;Gerkema 2001, for instance).…”

Tangential oscillations of a circular disk in a viscous stratified fluid

“…However, in one important respect, viscosity has not yet been treated fully consistently in theoretical work, even for the simplest linear generation problem. As discussed by Voisin (2003), previous studies have not applied the correct no-slip boundary condition but instead have carried out viscous calculations with free-slip boundary conditions (Hurley & Hood 1997;Hurley & Keady 1997) or applied the no-slip condition at a fictitious interface, with an iterative procedure to correct the latter approach (e.g. Vasil'ev & Chashechkin 2006, and a number of other papers by Chashechkin and collaborators).…”

Internal tide generation by tall ocean ridges

“…These conditions must be satisfied to use the weak topography approximation (WTA), by which the bottom boundary condition is applied at z = 0. [40] for waves generated by an oscillating ellipse, which, in the limit where the width of the ellipse goes to zero, models a knife-edge.…”