Generation of beams of three-dimensional periodic internal waves by sources of various types

Abstract

The energy and force characteristics of periodic internal wave beams in a viscous exponentially stratified fluid are analyzed. The exact solutions of linearized problems of generation obtained by integral transformations describe not only three-dimensional internal waves but also the associated boundary layers of two types. The solutions not containing empirical parameters are brought to a form that allows a direct comparison with experimental data for generators of various types (friction, piston, and combined) of rectangular or elliptic shape. The stress tensor and force components acting on the generator are given in quadratures. In the limiting cases, the solutions are uniformly transformed to the corresponding expressions for the problems in a two-dimensional formulation.Introduction. Internal waves, which play an important role in the dynamics of the ocean, atmosphere, and other stratified media, have been studied analytically [1], numerically, and experimentally under laboratory and natural conditions. In analytical calculations of wave fields, the real boundary conditions on the generators are simulated by sets of singular sources, whose properties are postulated [2] or borrowed from ideal fluid theory [3]. In an analysis of the perturbations induced in a fluid by a horizontal cylinder which performs rectilinear or torsional oscillations of small amplitude, the parameters of the wave beams and boundary layers are calculated separately [4,5]. The results obtained, which are used to determine the regions of applicability of asymptotic approximations, are usually given in integral form [4,5], which prevents their practical application.Accounting for all roots of the dispersion equation allows one to simultaneously calculate the parameters of two-dimensional internal waves and associated boundary layers [6]. The results of such calculations agree with experimental data [7]. The approach proposed in [6] extends the classical Stokes method [8] to inhomogeneous media. The satisfaction of the exact boundary conditions allows one to calculate both the waves and the fine structure of the boundary layers resulting from linear oscillations of segments of a flat surface in arbitrary directions. The nonlinear problem of wave generation by the boundary layers on a disk performing torsional oscillations is considered in [9].In the case of three-dimensional motion, the boundary layer on the oscillating part of the plane is more complex and includes an analog of the Stokes layer in a homogeneous fluid and a special inner boundary layer [10]. The method of constructing the solution [9] is fairly universal and suitable for calculating flows for more complex motions of the wave-generating surfaces used in experiments to increase the wave amplitude. The goal of the present study is to estimate the dynamic characteristics of beams of three-dimensional periodic internal waves generated by sources of various shapes. Governing Equations and Boundary Conditions.We study steady-state cyclic motion in a viscous incompressible exponen...

Near-field internal wave beams in two dimensions

“…By contrast, for thin forcing, the evolution of the waves is set by the distance normal to the forcing. This can be seen in the inviscid calculations of Oser (1957), Reynolds (1962), Martin & Llewellyn Smith (2011, 2012 b ) and Davis (2012) for a horizontal disc, Hurley (1969) for an inclined plate and Llewellyn Smith & Young (2003) for a vertical plate, or in the viscous calculations of Kistovich & Chashechkin (1999 a , b ) for a two-dimensional inclined plate, Vasil'ev & Chashechkin (2003, 2006 a , b , 2012) for a three-dimensional inclined plate, Tilgner (2000), Bardakov, Vasil'ev & Chashechkin (2007), Davis & Llewellyn Smith (2010), Le Dizès (2015) and Le Dizès & Le Bars (2017) for a horizontal disc, Maurer etal. (2017) and Boury, Peacock & Odier (2019) for a horizontal wave generator and Beckebanze, Raja & Maas (2019) for a vertical wave generator.…”

Tangential oscillations of a circular disk in a viscous stratified fluid

“…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). † Email address for correspondence: sgls@ucsd.edu In this paper we provide the first consistent calculation of the linear internal waves generated by a moving object in a stratified fluid.…”

Foundations of Engineering Mathematics Applied for Fluid Flows

“…Using all solutions for the system in Equation ( 5) and the dispersion in Equation ( 6) allows for solving the linear problem of periodic internal wave generation by an oscillating body in complete 2D and 3D formulations with physically justified initial and boundary conditions [61].…”