The pendulum-slosh problem: Simulation using a time-dependent conformal mapping

Abstract

-AbstractSuspending a rectangular vessel which is partially filled with fluid from a single rigid pivoting pole produces an interesting theoretical model with which to investigate the dynamic coupling between fluid motion and vessel rotation. The exact equations for this coupled system are derived with the fluid motion governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion governed by a modified forced pendulum equation. The nonlinear equations of motion for the fluid are solved numerically via a timedependent conformal mapping, which maps the physical domain to a rectangle in the computational domain with a time dependent conformal modulus. The numerical scheme expresses the implicit free-surface boundary conditions as two explicit partial differential equations which are then solved via a pseudospectral method in space. The coupled system is integrated in time with a fourth-order Runge-Kutta method. The starting point for the simulations is the linear neutral stability contour discovered by Turner, Alemi Ardakani & Bridges (2014, J. Fluid Struct. 52, 166-180). Near the contour the nonlinear results confirm the instability boundary, and far from the neutral curve (parameterised by longer pole lengths) nonlinearity is found to significantly alter the vessel response. Results are also presented for an initial condition given by a superposition of two sloshing modes with approximately the same frequency from the linear characteristic equation. In this case the fluid initial conditions generate large nonlinear vessel motions, which may have implications for systems designed to oscillate in a confined space or on the slosh-induced-rolling of a ship.

Time-dependent conformal mapping of doubly-connected regions

“…To keep it realistic, the amount of fluid in the vessel is conserved for all time. Although this profile is a simplification to a true sloshing profile, near straight free-surface profiles have been observed in the pendulum vessel simulations of Turner et al [28].…”

“…Numerical results obtained with this approach are compared to those of Frandsen [14] and they show excellent agreement-when the conformal modulus is properly included in the calculation. This time-dependent conformal mapping strategy is extended to simulate sloshing in a vessel undergoing pendular rotation in Turner et al [28].…”

Coupled shallow-water fluid sloshing in an upright annular vessel

“…In this case, the vessel's motion is not given a priori, instead the fluid and vessel's motions are solved simultaneously, and are intrinsically coupled through the pressure force due to the fluid impacting on the vessel walls. The first investigation of a dynamically coupled sloshing problem was by [6], who considered a vessel partially filled with fluid, attached to a rigid pole anchored at one end about which it rotates, the so called 'pendulum slosh problem' [3,7,8]. The pendulum provides a restoring force on the vessel, forcing the fluid into motion, which in turn modifies the vessel's motion via the hydrodynamic force it generates on the vessel walls.…”

On the Harmonic Oscillations for the Motion of a Dynamical System

“…The motion of a pendulum whether it has a rigid arm or elastic one has shed the interest of many researchers during the last century. The study of this motion has been widely spread in the last three decades due to its great applications in different fields like, clinical studies [1][2], physics [3][4], military [5] and engineering applications [6].…”