Second-order sloshing over an arbitrary bed
G.J.D. Chapman & R. Porter, 2005, J. Fluid Mech. 524, 331--355.
The two-dimensional problem of the free sloshing of an inviscid fluid
in a vertically- walled tank with an arbitrary bed shape is solved at
both first and second-order in the Stokes expansion of the velocity
potential. The approach employed at both orders uses Green¿s functions
for a flat-bed in conjunction with the Cauchy-Riemann equations to derive
integral equations for the tangential flux along the varying bed. The
first and second-order potentials everywhere in the fluid may then be
related to these fluxes. Significant analytic progress is made with the
calculation of various contributions to the integral equations at second
order. The equations at first and second-order are ultimately solved using
a variational principle equivalent to the Galerkin method giving efficient
and accurate results. In particular, the work involved in determining the
second-order solution is no more intensive than solving the first-order
problem. The first-order solution is shown to reproduce known results for
specific bed shapes. The method is applied to a range of bed shapes and
the second-order correction to the free-surface elevation is illustrated.
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