Wave Emission From Bottom Vibrations in Subsurface Open-channel Shear Flow
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Wave Emission From Bottom Vibrations in Subsurface Open-channel Shear Flow Peder A. Tyvand1 · Eivind B. Sveen1,2 Received: 26 February 2020 / Accepted: 25 April 2020 © The Author(s) 2020
Abstract The linearized water-wave radiation problem for a 2D oscillating bottom source in an inviscid shear flow with a free surface is investigated analytically. The fluid depth is constant. The velocity of the basic flow varies linearly with depth (uniform vorticity), with zero surface velocity. The far-field surface waves radiated out from the 2D source are calculated, based on Euler’s equation of motion with the application of radiation conditions. There are always two waves, one emitted in the upstream direction and the other in the downstream direction. The energy fluxes of these two waves are calculated. The hydrostatic limit of zero wave number is related to the theory of undular bores. Keywords Green function · Oscillating bottom source · Radiation · Shear flow · Water waves
1 Introduction The submerged oscillatory source is recognized as an elementary solution for linearized water waves governed by Laplace’s equation. Oscillatory sources are Green functions that satisfy the linearized free-surface condition and radiation conditions at infinity. The first mathematical solutions were given by Kochin [12], summarized in [21]. These singular solutions are important building blocks for water waves interacting with submerged and floating bodies [9,14]. We will derive a fundamental solution for two-dimensional (2D) bottom vibrations for a fluid layer with a free surface in the presence of a subsurface shear flow. This
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Eivind B. Sveen [email protected] Peder A. Tyvand [email protected]
1
Faculty of Mathematical Sciences and Technology, Norwegian University of Life Science, 1432 Ås, Norway
2
Present Address: Institute for Energy Technology, 2007 Kjeller, Norway
P. A. Tyvand, E. B. Sveen
is a Green function formalism which considers one singular point of bottom oscillations, representing an oscillatory Dirac singularity for the normal velocity. The exact dispersive theory of linearized water waves is applied, while the conventional theories of open-channel hydraulics apply shallow-water approximations. Our assumption of 2D waves is representative also for 3D waves in open channels when the ratio of wavelength to channel width is of order one or greater. The solution will be based on the Euler equation of motion for an incompressible inviscid fluid. The amplitudes of the radiated far-field waves will be calculated analytically. We will consider only the case of zero surface velocity where there are no Doppler effects. With zero surface velocity, there are always two waves emitted from the source. One upstream wave and one downstream wave. The established research on fully dispersive water waves on shear flows [16] has focus on ocean wave drift and weakly nonlinear interactions. The possibilities of stopping a surface wave by generating a shear flow has been studied [4]. After a detailed discussion of the dispersio
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