A fully nonlinear model for sloshing in a rotating container (original) (raw)
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Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth
Journal of Fluid Mechanics, 2000
The discrete infinite-dimensional modal system describing nonlinear sloshing of an incompressible fluid with irrotational flow partially occupying a tank performing an arbitrary three-dimensional motion is derived in general form. The tank has vertical walls near the free surface and overturning waves are excluded. The derivation is based on the Bateman-Luke variational principle. The free surface motion and velocity potential are expanded in generalized Fourier series. The derived infinite-dimensional modal system couples generalized time-dependent coordinates of free surface elevation and the velocity potential. The procedure is not restricted by any order of smallness. The general multidimensional structure of the equations is approximated to analyse sloshing in a rectangular tank with finite water depth. The amplitude-frequency response is consistent with the fifth-order steady-state solutions by . The theory is validated by new experimental results. It is shown that transients and associated nonlinear beating are important. An initial variation of excitation periods is more important than initial conditions. The theory is invalid when either the water depth is small or water impacts heavily on the tank ceiling. Alternative expressions for hydrodynamic loads are presented. The procedure facilitates simulations of a coupled vehicle-fluid system.
Response regimes in equivalent mechanical model of strongly nonlinear liquid sloshing
International Journal of Non-Linear Mechanics
We consider equivalent mechanical model of liquid sloshing in partially-filled cylindrical vessel both for free vibrations case and for horizontal harmonic ground excitation. The model treats both the regime of linear sloshing, and strongly nonlinear sloshing regime; the latter is related to hydraulic impacts applied to the vessel walls. These hydraulic impacts are commonly simulated with the help of high-power potential and dissipation functions. For the sake of analytic exploration, we substitute this traditional approach by treatment of an idealized vibro-impact system with velocity-dependent restitution coefficient. Parameters of the vibro-impact model are derived from the high-power potential and dissipation functions. The obtained reduced model is similar to recently explored system of linear primary oscillator with attached vibro-impact energy sink. Analysis is based on a multiplescale approach; the ratio of modal mass of the first sloshing mode to the total mass of the liquid and the tank serves as a natural small parameter. In the case of external ground forcing, steady-state responses and chaotic strongly modulated responses are revealed. Besides, the system response to horizontal periodic excitation with additional Gaussian white noise, and corresponding dynamics on the slow invariant manifold are explored. All analytical predictions of the reduced vibro-impact model are validated against direct numerical simulations of "initial" equivalent model with high-power smooth potential and dissipation functions, and good agreement is observed.
Nonlinear modeling of liquid sloshing in a moving rectangular tank
Ocean Engineering, 2002
A nonlinear liquid sloshing inside a partially filled rectangular tank has been investigated. The fluid is assumed to be homogeneous, isotropic, viscous, Newtonian and exhibit only limited compressibility. The tank is forced to move harmonically along a vertical curve with the rolling motion to simulate the actual tank excitation. The volume of fluid technique is used to track the free surface. The model solves the complete Navier-Stokes equations in primitive variables by use of the finite difference approximations. At each time step, a donar-acceptor method is used to transport the volume of fluid function and hence the locations of the free surface. In order to assess the accuracy of the method used, computations are verified through convergence tests and compared with the theoretical solutions and experimental results.
Weakly nonlinear sloshing in a truncated circular conical tank
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.
Response Regimes in Equivalent Mechanical Model of Moderately Nonlinear Liquid Sloshing
arXiv (Cornell University), 2017
The paper considers non-stationary responses in reduced-order model (ROM) of partially liquid-filled tank under external forcing. The model involves one common degree of freedom for the tank and the non-sloshing portion of the liquid, and the other onefor the sloshing portion of the liquid. The coupling between these degrees of freedom is nonlinear, with the lowest-order potential dictated by symmetry considerations. Since the mass of the sloshing liquid in realistic conditions does not exceed 10% of the total mass of the system, the reduced-order model turns to be formally equivalent to well-studied oscillatory systems with nonlinear energy sinks (NES). Exploiting this analogy, and applying the methodology known from the studies of the systems with NES, we predict a multitude of possible nonstationary responses in the considered ROM. These responses conform, at least on the qualitative level, to the responses observed in experimental sloshing settings, multi-modal theoretical models and full-scale numeric simulations.
The semi-Lagrangian procedure is widely used for updating the fully-nonlinear free surface in the time domain. However, this procedure is only available to cases when the body surface is vertical near the waterline. Present study introduces an improved semi-Lagrangian procedure which removes this ‘vertical-wall’ limitation. Coupling with the boundary element method, the improved semi-Lagrangian procedure is applied to the simulation of fully-nonlinear sloshing waves in non-wall-sided tanks. From the result comparison with the open source CFD software OpenFOAM, it is confirmed that this numerical scheme could guarantee a sufficient accuracy. Further series studies on 2D and 3D fully-nonlinear sloshing waves in wedged tanks are performed. Featured phenomena are observed which are distinct from those in wall-sided tanks. Application of an improved semi-Lagrangian procedure to fully-nonlinear simulation of sloshing in non-wall-sided tanks. Available from: https://www.researchgate.net/publication/274193712\_Application\_of\_an\_improved\_semi-Lagrangian\_procedure\_to\_fully-nonlinear\_simulation\_of\_sloshing\_in\_non-wall-sided\_tanks [accessed May 14, 2015].
Recent advances in liquid sloshing dynamics
A liquid free surface in partially filled containers can experience a wide spectrum of motions such as planar, non-planar, rotational, quasi-periodic, chaotic, and disintegration. Civil engineers and seismologists have been studying liquid sloshing effects on large dams, oil tanks and elevated water towers under ground motion. Since the early 1960's, the problem of liquid sloshing dynamics has been of major concern to aerospace engineers studying the influence of liquid propellant sloshing on the flight performance of jet vehicles. Since then, new areas of research activities have emerged. The modern theory of nonlinear dynamics has indeed promoted further studies and uncovered complex nonlinear phenomena. These include rotary sloshing, Faraday waves, nonlinear liquid sloshing interaction with elastic structures, internal resonance effects, stochastic sloshing dynamics, hydrodynamic sloshing impact dynamics, g-jitter under microgravity field, cross-waves, and spatial resonance. The dynamic stability of liquid gas tankers and ship cargo tankers, and liquid hydrodynamic impact loading are problems of current interest to the designers of such systems. This article will address the means of passive control of liquid sloshing and the use of liquid sloshing forces to control vibratory structures. Other important contributions include the development of digital computer codes to solve complex problems that were difficult to handle in the past. The purpose of this article is to review the research work developed in different applications. It will highlight the major achievements and results reported in the literature. Some early work will be cited very briefly in order to provide an updated bibliography of liquid sloshing dynamics. This review article contains 1,319 references.
Classification of three-dimensional nonlinear sloshing in a square-base tank with finite depth
Journal of Fluids and Structures, 2005
The paper classifies steady state three-dimensional resonant waves in a square-base tank by using the asymptotic modal system proposed by the authors in 2003. The effective frequency domains of stable steady state motions are analysed versus mean fluid depths and forcing amplitude. The results are validated by experiments both qualitatively and quantitatively. r (O.M. Faltinsen). related to this study are reviewed by and . Three different approaches to theoretical sloshing modelling are distinguished. One of them focuses on low-order asymptotic mathematical theories and appropriate Hamiltonian formalism for the system of ordinary differential equations governing the dominating standing waves. Another approach is based on computational fluid dynamics (CFD) [see surveys by , Gerrits (2001), Celebi and Akyildiz ]. The third approach deals with multimodal/pseudospectral methods. Such methods are able to provide in different versions both analytical and numerical studies and 'build a bridge' between the first and second ones. All three approaches have their advantages and disadvantages from mathematical, physical and engineering points of view outlined in details by the mentioned surveys. Links, common features and differences should be demonstrated. This is a difficult problem with regard to the lower-order mathematical theories (first approach) and modal/pseudospectral methods [see some details given by ; Hill ]. Both approaches reduce the original free boundary problem to systems of ordinary differential equations with finite nonlinear kernel and often focus on nonlinear steady state waves. The difference is that the modal methods account for the full set of activated modes and their arbitrary initial perturbation, while the first approach studies the behaviour of the leading modes. Generally speaking, the multidimensional modal approach is more general, because under some additional asymptotic assumptions a corresponding low-order Hamiltonian system can be derived from the modal systems. The opposite is not true. AlthoughHill presented a version of single-dominant theory of two-dimensional sloshing, where the behaviour of some higher modes can be restored, his scheme is invalid for arbitrary initial conditions (for the higher modes) and requires the single harmonic forcing in a very small vicinity of the primary resonance.
Multimodal method for linear liquid sloshing in a rigid tapered conical tank
Engineering Computations, 2012
PurposeThe purpose of this paper is to derive linear modal equations describing the forced liquid sloshing in a rigid truncated (tapered) conical tank, as well as to show how to couple these modal equations with “global” dynamic equations of a complex mechanical system carrying this tank.Design/methodology/approachDerivation of the modal equations can be based on the Trefftz variational method developed by the authors in a previous paper. Describing the coupled dynamics utilizes Lukovsky' formulas for the resulting hydrodynamic force and moment due to liquid sloshing.FindingsThe so‐called Stokes‐Joukowski potentials can be found by using the Trefftz method from the authors' previous paper with the same polynomial‐type functional basis. Coupling the modal equations with the global dynamic equations becomes a relatively simple task facilitated by Lukovsky's formulas. Using the linear multimodal method can be an efficient alternative to traditional numerical and analytical ...