Resonance in a model for Cooker's sloshing experiment

Euro. J. Mech. B/Fluids, 2022

The problem of dynamic coupling between a rectangular container undergoing planar pendular oscillations and its interior potential fluid sloshing is studied. The Neumann boundaryvalue problem for the fluid motion inside the container is deduced from the Bateman-Luke variational principle. The governing integro-differential equation for the motion of the suspended container, from a single rigid pivoting rod, is the Euler-Lagrange equation for a forced pendulum. The fluid and rigid-body partial differential equations are linearised, and the characteristic equation for the natural and resonant frequencies of the coupled dynamical system are presented. It is found that internal 1 : 1 resonances exist for an experimentally realistic setup, which has important physical implications. In addition, a new instability is found in the linearised coupled problem whereby instability occurs when the rod length is shorter than a critical length, and an explicit formula is given.

Journal of Hydrodynamics, Ser. B, 2014

Surface tension effects on fluid sloshing in a tank subjected to external excitation has been less studied. This work aims at understanding this phenomenon in order to derive practical solutions to problems faced in several engineering. A tank containing a fluid with a free surface is submitted to gravity and capillary forces and subject to external dynamic excitation. Introduction of vertical sinusoidal dynamical excitation leads to a problem of paramtric oscillations governed by the Mathieu equation. Analysis of the Mathieu equation shows the existence of stable and unstable regions in the stability diagram. Some results induced by harmonic excitations on the fluid sloshing are presented. When the external dynamical excitation amplitude  is small, periodic solutions appear in stable regions and when  increases, the fluid behavior is not perfectly periodic and the amplitudes are not regular.

Fluid Dynamics Research, 2013

Sloshing of an ideal incompressible liquid in a rigid truncated (tapered) conical tank is considered when the tank performs small-magnitude oscillatory motions with the forcing frequency close to the lowest natural sloshing frequency. The multimodal method, the non-conformal mapping technique and the Moiseev type asymptotics are employed to derive a finite-dimensional system of weakly nonlinear ordinary differential (modal) equations. This modal system is a generalization of that by Gavrilyuk et al 2005 Fluid Dyn. Res. 37 399-429. Using the derived modal equations, we classify the resonant steady-state wave regimes occurring due to horizontal harmonic tank excitations. The frequency ranges are detected where the 'planar' and/or 'swirling' steady-state sloshing are stable as well as a range in which all steadystate wave regimes are not stable and irregular (chaotic) liquid motions occur is established. The results on the frequency ranges are qualitatively supported by experiments by Matta E 2002 PhD Thesis Politecnico di Torino, Torino.

Fluid Dynamics Research, 2005

Linear and nonlinear fluid sloshing problems in a circular conical tank are studied in a curvilinear coordinate system. The linear sloshing modes are approximated by a series of the solid spheric harmonics. These modes are used to derive a new nonlinear modal theory based on the Moiseyev asymptotics. The theory makes it possible to both classify steady-state waves occurring due to horizontal resonant excitation and visualise nonlinear wave patterns. Secondary (internal) resonances and shallow fluid sloshing (predicted for the semi-apex angles > 60 • ) are extensively discussed.

J. Fluids Struct., 2016

The synchronous interaction between a sloshing fluid and its container may take place when the motion of the container is not prescribed a priori. When a partially-filled container is suspended as a bifilar pendulum, the fluid contributes to the restoring force by its weight through the wire suspensions and may either augment or diminish the restoring force through hydrodynamic interaction with the container. Here we report on linearized shallow-water theory and experiments for multi-chamber box containers, a cylinder and an upright 90 • wedge. The nature of system response at low fluid mass is analyzed in detail using linear theory and shallow-water simulations. These results show that the system frequency smoothly transitions through successive higher frequency eigenmodes so that, in the limit where the fluid mass tends to zero, the system frequency is that of the dry container.

Physics of Fluids, 2006

An adaptive asymptotic nonlinear modal system is used for systematic quantification of three-dimensional steady-state resonant sloshing in a square base tank with a finite fill depth. The depth/breadth ratios are ജ0.4. The tank is laterally excited with frequency close to the lowest natural frequency. The main emphasis is on the "swirling" wave regime and its special features, e.g., stability, feedback of higher modes, and regular and irregular switch of the apparent direction of rotation. Theoretical results are validated for both steady-state solutions and "beating" that does not die out in experimental investigations. Frequency domains with no stable steady-state waves and occurrence of "chaotic" waves are discussed.

5th International Symposium on Fluid Structure International, Aeroeslasticity, and Flow Induced Vibration and Noise, 2002

ABSTRACT Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

Journal of Fluids and Structures, 2011

Internal bodies (baffles) are used as damping devices to suppress the fluid sloshing motion in fluid-structure interaction systems. An analytical method is developed in the present article to investigate the effects of a rigid internal body on bulging and sloshing frequencies and modes of a cylindrical container partially filled with a fluid. The internal body is a thin-walled and open-ended cylindrical shell that is coaxially and partially submerged inside the container. The interaction between the fluid and the structure is taken into account to calculate the sloshing and bulging frequencies and modes of the coupled system using the Rayleigh quotient, Ritz expansion and Galerkin method. It is shown that the present formulation is an appropriate and new approach to tackle the problem with good accuracy. The effects of fluid level, number of nodal diameters, internal body radius and submergence ratio on the dynamic characteristics of the coupled system are also investigated.

Phys. Rev. Fluids, 2017

This paper investigates the coupled motion between the dynamics of N vessels coupled together in a one-dimensional array by springs, and the motion of the inviscid fluid sloshing within each vessel. We develop a fully-nonlinear model for the system relative to a moving frame such that the fluid in each vessel is governed by the Euler equations and the motion of each vessel is modelled by a forced spring equation. By considering a linearization of the model, the characteristic equation for the natural frequencies of the system is derived, and analysed for a variety of non-dimensional parameter regimes.

In this paper a theoretical and experimental analysis of sloshing in 2D and 3D free-surface con gurations is performed. In particular, the case of a tank rotating around a horizontal axis has been considered. The uid is assumed to be incompressible and inviscid. A fully nonlinear mathematical model is de ned by applying the variational method to the sloshing. The damping of gravity waves has been accounted by introducing a suitable dissipation function from which generalized dissipative forces are derived. A modal decomposition is then adopted for the unknowns and a dynamical system is derived to describe the evolution of the physical system. An experimental technique has been applied to select the leading modes, whose evolution characterizes the physical process, i.e. captures the most of the kinetic energy of the process. A very good agreement between experimental and numerical results con rms the validity of the methodological approach followed.