A. N. Timokha

Resonant three-dimensional nonlinear sloshing in a square-base basin

An asymptotic modal system is derived for modelling nonlinear sloshing in a rectangular tank with similar width and breadth. The system couples nonlinearly nine modal functions describing the time evolution of the natural modes. Two primary modes are assumed to be dominant. The system is equivalent to the model by Faltinsen et al. (2000) for the two-dimensional case. It is validated for resonant sloshing in a square-base basin. Emphasis is on finite fluid depth but the behaviour with decreasing depth to intermediate depths is also discussed. The tank is forced in surge/sway/roll/pitch with frequency close to the lowest degenerate natural frequency. The theoretical part concentrates on periodic solutions of the modal system (steady-state wave motions) for longitudinal (along the walls) and diagonal (in the vertical diagonal plane) excitations. Three types of solutions are established for each case: (i) ‘planar’/‘diagonal’ resonant standing waves for longitudinal/diagonal forcing, (ii) ‘swirling’ waves moving along tank walls clockwise or counterclockwise and (iii) ‘square’-like resonant standing wave coupling in-phase oscillations of both the lowest modes. The frequency domains for stable and unstable waves (i)–(iii), the contribution of higher modes and the influence of decreasing fluid depth are studied in detail. The zones where either unstable steady regimes exist or there are two or more stable periodic solutions with similar amplitudes are found. New experimental results are presented and show generally good agreement with theoretical data on effective domains of steady-state sloshing. Three-dimensional sloshing regimes demonstrate a significant contribution of higher modes in steady-state and transient flows.

Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth

The modal system describing nonlinear sloshing with inviscid flows in a rectangular rigid tank is revised to match both shallow fluid and secondary (internal) resonance asymptotics. The main goal is to examine nonlinear resonant waves for intermediate depth/breadth ratio 0.1 [lsim ] h / l [lsim ] 0.24 forced by surge/pitch excitation with frequency in the vicinity of the lowest natural frequency. The revised modal equations take full account of nonlinearities up to fourth-order polynomial terms in generalized coordinates and h / l and may be treated as a modal Boussinesq-type theory. The system is truncated with a high number of modes and shows good agreement with experimental data by Rognebakke (1998) for transient motions, where previous finite depth modal theories failed. However, difficulties may occur when experiments show significant energy dissipation associated with run-up at the walls and wave breaking. After reviewing published results on damping rates for lower and higher modes, the linear damping terms due to the linear laminar boundary layer near the tanks surface and viscosity in the fluid bulk are incorporated. This improves the simulation of transient motions. The steady-state response agrees well with experiments by Chester & Bones (1968) for shallow water, and Abramson et al. (1974), Olsen & Johnsen (1975) for intermediate fluid depths. When h / l [lsim ] 0.05, convergence problems associated with increasing the dimension of the modal system are reported.

Two-dimensional resonant piston-like sloshing in a moonpool

This paper presents combined theoretical and experimental studies of the two-dimensional piston-like steady-state motions of a fluid in a moonpool formed by two rectangular hulls (e.g. a dual pontoon or catamaran). Vertical harmonic excitation of the partly submerged structure in calm water is assumed. A high-precision analytically oriented linear-potential-flow method, which captures the singular behaviour of the velocity potential at the corner points of the rectangular structure, is developed. The linear steady-state results are compared with new experimental data and show generally satisfactory agreement. The influence of vortex shedding has been evaluated by using the local discrete-vortex method of Graham (1980). It was shown to be small. Thus, the discrepancy between the theory and experiment may be related to the free-surface nonlinearity.

Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 2. Effect of higher modes

The paper continues our investigations of three-dimensional nonlinear resonant fluid sloshing in a square-base basin with finite depth (mean depth/tank length ratio

A multimodal method for liquid sloshing in a two-dimensional circular tank

h\,{\ge}\, 0.3

Transient and steady-state amplitudes of resonant three-dimensional sloshing in a square base tank with a finite fluid depth

). The sloshing is forced by a combined sway/surge resonant harmonic excitation of the two lowest natural modes. The new studies are strongly motivated by a discrepancy between previous quantitative theoretical results and experimental measurements and consist of a more precise description of the strong nonlinear amplification of higher modes. The latter is justified here by secondary resonance. Effective frequency domains of the secondary resonance are quantified. An adaptive asymptotic modal theory improves agreement with earlier and new experimental data both in the transient and steady-state conditions. Local breaking and overturning near the walls, that may lead to a ‘switch’ between distinct steady regimes, increase both global damping and generate random-like excitation of higher modes, are extensively discussed.

Steady-state liquid sloshing in a rectangular tank with a slat-type screen in the middle: Quasilinear modal analysis and experiments

Two-dimensional forced liquid sloshing in a circular tank is studied by the multimodal method which uses an expansion in terms of the natural modes of free oscillations in the unforced tank. Incompressible inviscid liquid, irrotational flow and linear freesurface conditions are assumed. Accurate natural sloshing modes are constructed in an analytical form. Based on these modes, the ‘multimodal’ velocity potential of both steady-state and transient forced liquid motions exactly satisfies the bodyboundary condition, captures the corner-point behaviour between the mean free surface and the tank wall and accurately approximates the free-surface conditions. The constructed multimodal solution provides an accurate description of the linear forced liquid sloshing. Surface wave elevations and hydrodynamic loads are compared with known experimental and nonlinear computational fluid dynamics results. The linear multimodal sloshing solution demonstrates good agreement in transient conditions of small duration, but fails in steady-state nearly-resonant conditions. Importance of the free-surface nonlinearity with increasing tank filling is explained. Sloshing must be considered for almost any moving vehicle or structure containing a liquid with a free surface and can be the result of both transient and resonant excitations of the tank. The hydrodynamics of sloshing is complicated, depending on the tank shape, liquid depth and the forcing conditions. Its understanding requires a combination of theory, computational fluid dynamics (CFD) and experiments. We must distinguish from a physical point of view between global flow and local flow associated with impact between the free surface and the tank structure. The present paper concentrates on the global flow and the resulting hydrodynamic loads due to forced two-dimensional transverse liquid sloshing in a circular-shaped tank. The latter is needed in predicting the dynamics of vehicles and structures relevant, for instance for wave liquid motions in lorry tanks, horizontal cylindrical ship tanks, railway cisterns and storage containers exposed to seismic excitations. Experiments and CFD simulations of two-dimensional forced liquid sloshing in circular tanks are, for instance, reported by Strandberg (1978), Kobayashi et al. (1989), Aliabadi, Johnson & Abedi (2003), Djavareshkian & Khalili (2006), Moderassi-Tehrani, Rakheja & Sedaghati (2006), Karamanos, Papaprokopiou &

Natural sloshing frequencies in rigid truncated conical tanks

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 ⩾0.4. 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.

MODAL MODELLING OF THE NONLINEAR RESONANT FLUID SLOSHING IN A RECTANGULAR TANK I: A SINGLE-DOMINANT MODEL

Two-dimensional resonant liquid sloshing in a rectangular tank equipped with a central slat-type screen is studied theoretically and experimentally with focus on nonsmall solidity ratios of the screen (0.5≲Sn≲0.95), nonlarge number of slots (N≲50), and steady-state conditions. The tank is horizontally and harmonically excited with frequencies in a range covering the two lowest primary-excited natural sloshing resonance frequencies in the corresponding clean tank. The liquid depth is finite. Theoretical analysis is based on the multimodal method with linear free-surface conditions and a quadratic pressure drop condition at the screen expressing an “integral” effect of the screen-induced cross-flow separation (or jet flow). New experimental data on the maximum wave elevations at the wall are compared with the theoretical predictions. Very good agreement is shown for the smallest forcing amplitudes (the forcing amplitude-to-tank width ratio is ≈0.001). Increasing the nondimensional forcing amplitude to ≈0.01...

Effect of central slotted screen with a high solidity ratio on the secondary resonance phenomenon for liquid sloshing in a rectangular tank

Purpose – The main purpose of this paper is to develop two efficient and accurate numerical analytical methods for engineering computation of natural sloshing frequencies and modes i the case of truncated circular conical tanks.Design/methodology/approach – The numerical‐analytical methods are based on a Ritz Treftz variational scheme with two distinct analytical harmonic functional bases.Findings – Comparative numerical analysis detects the limit of applicability of variational methods in terms of the semi‐apex angle and the ratio between radii of the mean free surface and the circular bottom. The limits are caused by different analytical properties of the employed functional bases. However, parallel use of two or more bases makes it possible to give an accurate approximation of the lower natural frequencies for relevant tanks. For V‐shaped tanks, dependencies of the lowest natural frequency versus the semi‐apex angle and the liquid depth are described.Practical implications – The methods provide the nat...