Peaked sloshing in a wedge container

Finite-amplitude free-surface flow in a wedge container is investigated analytically. We study a motionless standing wave of pure potential-flow acceleration with maximal amplitude where its right-angle surface peak falls from rest. The nonlinear free-surface conditions are satisfied by a family of flows where the chosen initial acceleration field is governed by one single dipole plus its three image dipoles. Streamlines and isobars are plotted, with the free surface as the zero-pressure isobar. The key geometric parameters are tabulated for each case, supplied with force calculations for an upright wedge container. The present approach is assessed against established eigenfunctions for linearized standing waves in a wedge container. The present dipole flows constitute a much richer family of peaked free sloshing shapes than the classical Fourier modes of free oscillation.acceptedVersio

Stagnant peaked free surface released at a sloping beach

A stagnant free-surface flow is an instantaneous flow field of pure acceleration with zero velocity and a deformed surface. There exists a potential-flow acceleration field. With zero velocity and the acceleration field given, there is a limiting free-surface position which possesses one peak at its point of highest elevation. By complex analysis, it can be shown that the surface peak has a right angle. We elaborate on an elementary model of two-dimensional stagnant free-surface flow with a peak. Our model may serve to describe a situation of maximal single-wave run-up with a given energy at a uniformly sloping beach. The highest possible run-up of an incoming solitary wave corresponds to zero kinetic energy. It encompasses an idealized situation where the kinetic wave energy is converted into potential energy in a water mass piling up along the slope to become stagnant at one single moment. Multipoles with singularities outside the fluid domain may give rise to a smooth and gradual deceleration needed for a non-breaking run-up process. A pair of dipoles with an orientation perpendicular to a given slope represents the stagnant acceleration fields with the highest surface peak spatially concentrated along the slope. Thereby, a one-parameter family of surface shapes is constituted, only dependent on the slope angle. The initial flow field, the initial free surface, the initial isobars and the geometric parameters are all calculated for different slope angles.publishedVersio

Laterally Penetrative Onset of Convection in a Horizontal Porous Layer

The onset of Darcy–Bénard convection in an unlimited horizontal porous layer is studied theoretically. The thermomechanical boundary conditions of Dirichlet or Neumann type at the lower and upper plane are switched from one type to another, at certain values of the horizontal x-coordinate. A semi-infinite portion of the lower boundary is defined as thermally conducting and impermeable, while the remaining boundary is open and with given heat flux. At the upper boundary, the same thermomechanical conditions are applied, but with a relative spatial displacement L and in the opposite spatial order. A domain of local destabilization around the origin is generated between the lines of discontinuity =±/2 x = ± L / 2 . The marginal state of convection is triggered centrally, while it is penetrative in the domains exterior to the central domain. The onset problem is solved numerically, with a general 3D mode of disturbance, but 2D disturbances are preferred in most cases. The critical Rayleigh number is given as a function of the dimensionless gap width L and the wavenumber k in the y direction along the lines of discontinuity in the boundary conditions. An asymptotic formula for 2D penetrative eigenfunctions is shown to be in agreement with the numerical results.publishedVersio