Boundary integrals for oscillating bodies in stratified fluids

Abstract

The theoretical foundations of the boundary integral method are considered for inviscid monochromatic internal waves, and an analytical approach is presented for the solution of the boundary integral equation for oscillating bodies of simple shape: an elliptic cylinder in two dimensions, and a spheroid in three dimensions. The method combines the coordinate stretching introduced by Bryan and Hurley in the frequency range of evanescent waves, with analytic continuation to the range of propagating waves by Lighthill's radiation condition. Not only are the waves obtained for arbitrary oscillations of the body, with application to radial pulsations and rigid vibrations, but also the distribution of singularities equivalent to the body, allowing later inclusion of viscosity in the theory. Both a direct representation of the body as a Kirchhoff–Helmholtz integral involving single and double layers together, and an indirect representation involving a single layer alone, are considered. The indirect representation is seen to require a certain degree of symmetry of the body with respect to the horizontal and the vertical. As the surface of the body is approached the single- and double-layer potentials exhibit the same discontinuities as in classical potential theory.

Buoyancy oscillations

“…We consider an elliptic cylinder, typical of two-dimensional bodies, and a spheroid, typical of three-dimensional bodies. The difference from the same investigations for surface gravity waves, discussed in § 1, is that the boundary condition has been solved in analytical form for these bodies by Voisin (2021), and deduced from the solution in Part 1. Accordingly, all that is left now is the inversion of (2.11).…”

“…The first is wave damping, acting through the hydrodynamic pressure force F . Introducing the added mass coefficients C ij (ω) of the body (Ermanyuk 2002;Voisin 2024), we have in the frequency domain, for time variation as exp(−iωt),…”

“…The first is wave damping, acting through the hydrodynamic pressure force . Introducing the added mass coefficients of the body (Ermanyuk 2002; Voisin 2024), we have in the frequency domain, for time variation as , where suffix notation is used, with ranging over , and the vertical direction , and is the velocity of the body. Taking inverse Fourier transforms and introducing the impulse response function where , the hydrodynamic force appears as namely a combination of two terms: the same acceleration reaction as in a homogeneous fluid (Batchelor 1967, § 6.4; Landau & Lifshitz 1987, § 11), with coefficients equal to for a sphere, where is the Kronecker delta symbol; and a memory integral representing the effect of internal wave radiation, with a kernel given at large time for the vertical motion of a sphere by exhibiting an aperiodic decay as , together with oscillations of frequency and amplitude decay as .…”

“…For each system, the equation of motion is written for an arbitrary body, and the Fourier transform of its solution is expressed in terms of the added mass of the body. Then, depending on what is relevant in each case, the added masses derived for an elliptic cylinder of horizontal axis and a spheroid of vertical axis in Voisin (2024), hereafter referred to as Part 1, from the boundary integral calculations in Voisin (2021), are used. The Fourier transform is inverted exactly, in analytical or numerical form, and its expansion for large time is calculated.…”

“…The outcome is consistent with Brennen (1982) and Korotkin (2009, §§ 2.2.1 and 3.2). It involves for the spheroid becoming for the sphere. For the Basset–Boussinesq tensor , the irrotational flow was obtained as the limit of the results of Voisin (2021), then the dissipation rate was calculated using (C4). The outcome is consistent with Lawrence & Weinbaum (1988), Pozrikidis (1989) and Loewenberg (1993 a ) for the spheroid, Nuriev, Egorov & Kamalutdinov (2021) for the elliptic cylinder – it involves the complete elliptic integrals and of modulus – and Batchelor (1967, § 5.13) and Landau & Lifshitz (1987, § 24) for the sphere and the circular cylinder. The Stokes resistance tensor was obtained by applying the procedure of Lamb (1932, § 339) and Happel & Brenner (1983, § 5.11).…”