Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth

doi:10.1017/S0022112099007569

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Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth

Journal Article•DOI•

Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth

Odd M. Faltinsen¹, Olav F. Rognebakke¹, I. A. Lukovsky², Alexander Timokha²•Institutions (2)

Norwegian University of Science and Technology¹, National Academy of Sciences of Ukraine²

25 Mar 2000-Journal of Fluid Mechanics (Cambridge University Press)-Vol. 407, Iss: 1, pp 201-234

TL;DR: In this paper, a multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth is presented, where the modality is modelled as a set of modalities.

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Abstract: Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth

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Liquid Sloshing Dynamics

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Raouf A. Ibrahim¹•Institutions (1)

Wayne State University¹

19 May 2005

TL;DR: In this article, the authors present a detailed review of liquid sloshing dynamics in rigid containers, including linear forced and non-linear interaction under external and parametric excitations.

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Abstract: Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

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920 citations

Journal Article•DOI•

Recent Advances in Liquid Sloshing Dynamics

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Raouf A. Ibrahim¹, Valery N. Pilipchuk¹, T. Ikeda²•Institutions (2)

Wayne State University¹, Shimane University²

01 Mar 2001-Applied Mechanics Reviews

319 citations

Journal Article•DOI•

A numerical study of three-dimensional liquid sloshing in tanks

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Dongming Liu¹, Pengzhi Lin²•Institutions (2)

National University of Singapore¹, Sichuan University²

01 Apr 2008-Journal of Computational Physics

TL;DR: A numerical model NEWTANK (Numerical Wave TANK) has been developed to study three-dimensional (3-D) non-linear liquid sloshing with broken free surfaces to solve the spatially averaged Navier-Stokes equations for two-phase flows.

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307 citations

Journal Article•DOI•

Sloshing motions in excited tanks

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Jannette Frandsen¹•Institutions (1)

Louisiana State University¹

01 May 2004-Journal of Computational Physics

TL;DR: In this paper, a fully non-linear finite difference model was developed based on inviscid flow equations for liquid sloshing induced by harmonic base excitations, which is valid for any water depth except for small depth when viscous effects would become important.

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218 citations

Cites background or methods from "Multidimensional modal analysis of ..."

...A multimodal algorithm should be used [11]....
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...16 represents an off-resonance case [11,21] with a forcing frequency higher than the first natural sloshing frequency (xh=x1 1⁄4 1:283), similar to...
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...[11], Ockendon and Ockendon [31], Hill [21]....
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...The role of critical depth in tanks subjected to horizontal motions have also been studied [10–13,21]....
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...[20], Faltinsen [11], Waterhouse [41] found hc=k 1⁄4 0:583 m or in general hc 1⁄4 0:337 b for the first mode, respectively....
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Journal Article•DOI•

An adaptive multimodal approach to nonlinear sloshing in a rectangular tank

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Odd M. Faltinsen¹, Alexander Timokha²•Institutions (2)

Norwegian University of Science and Technology¹, National Academy of Sciences of Ukraine²

10 Apr 2001-Journal of Fluid Mechanics

TL;DR: In this paper, a modal theory based on an innite-dimensional system of nonlinear ordinary dierential equations coupling generalized coordinates of the free surface and fluid motion associated with the amplitude response of natural modes is presented.

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Abstract: Two-dimensional nonlinear sloshing of an incompressible fluid with irrotational flow in a rectangular tank is analysed by a modal theory. Innite tank roof height and no overturning waves are assumed. The modal theory is based on an innite-dimensional system of nonlinear ordinary dierential equations coupling generalized coordinates of the free surface and fluid motion associated with the amplitude response of natural modes. This modal system is asymptotically reduced to an innite-dimensional system of ordinary dierential equations with fth-order polynomial nonlinearity by assuming suciently small fluid motion relative to fluid depth and tank breadth. When introducing inter-modal ordering, the system can be detuned and truncated to describe resonant sloshing in dierent domains of the excitation period. Resonant sloshing due to surge and pitch sinusoidal excitation of the primary mode is considered. By assuming that each mode has only one main harmonic an adaptive procedure is proposed to describe direct and secondary resonant responses when Moiseyev-like relations do not agree with experiments, i.e. when the excitation amplitude is not very small, and the fluid depth is close to the critical depth or small. Adaptive procedures have been established for a wide range of excitation periods as long as the mean fluid depth h is larger than 0.24 times the tank breadth l. Steady-state results for wave elevation, horizontal force and pitch moment are experimentally validated except when heavy roof impact occurs. The analysis of small depth requires that many modes have primary order and that each mode may have more than one main harmonic. This is illustrated by an example for h=l =0 :173, where the previous model by Faltinsen et al. (2000) failed. The new model agrees well with experiments.

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216 citations

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Journal Article•DOI•

Über Stoß- und Gleitvorgänge an der Oberfläche von Flüssigkeiten

[...]

Herbrt Wagner

01 Jan 1932-Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik

925 citations

Journal Article•DOI•

A variational principle for a fluid with a free surface

[...]

J. C. Luke¹•Institutions (1)

California Institute of Technology¹

02 Feb 1967-Journal of Fluid Mechanics

TL;DR: In this paper, the full set of equations of motion for the classical water wave problem in Eulerian co-ordinates is obtained from a Lagrangian function which equals the pressure.

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Abstract: The full set of equations of motion for the classical water wave problem in Eulerian co-ordinates is obtained from a Lagrangian function which equals the pressure. This Lagrangian is compared with the more usual expression formed from kinetic minus potential energy.

...read moreread less

520 citations

"Multidimensional modal analysis of ..." refers background or methods in this paper

...Multidimensional modal analysis of nonlinear sloshing 203 (see Shemer 1990 and Tsai, Yue & Yip 1990) or a system of four first-order ordinary differential equations for a vertical circular cylindrical tank (see Miles (1984a, b)). Funakoshi & Inoue (1991) used Miles’ model in their detailed simulations....
[...]
...The canonical formulation of this principle is given by Bateman (1944) and Luke (1967) (for gravity surface waves in infinite basins)....
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...Multidimensional modal analysis of nonlinear sloshing 203 (see Shemer 1990 and Tsai, Yue & Yip 1990) or a system of four first-order ordinary differential equations for a vertical circular cylindrical tank (see Miles (1984a, b)). Funakoshi & Inoue (1991) used Miles’ model in their detailed simulations. The averaging technique and small-dimensional modal modelling complement each another in the analysis of the steady-state free surface response due to periodic tank excitations. But these methods are questionable in modelling coupled fluid–structure interaction with complicated non-periodic tank motions when transient effects matter. These complex motions are simulated in engineering applications either numerically or by phenomenological (usually pendulum) models (see Chapter 5 of Narimanov et al. 1977 or Pilipchuk & Ibrahim 1997). An alternative is to use Narimanov’s original technique with the modal representation in the form (1.1) and more general asymptotic assumptions of βi and Rn in order to reach reasonable dimensions of the modal systems. The successful use of this approach is reported by Limarchenko & Yasinsky (1997) and Lukovsky & Timokha (1995) for simplified models of spacecraft....
[...]

Journal Article•DOI•

Partial Differential Equations of Mathematical Physics

[...]

Edmund Taylor Whittaker

01 Jun 1932-Nature

TL;DR: Bateman as mentioned in this paper argued that the main work of mathematical physicists is to represent the sequence of phenomena in time and space by means of differential equations, and to solve these equations. But the discovery of wave mechanics restored the status quo ante, and today differential equations are more important than ever before.

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Abstract: THE main work of mathematical physicists is to represent the sequence of phenomena in time and space by means of differential equations, and to solve these equations. Even the revolution effected by relativity and quantum theory has not changed this fundamental fact. A few years ago, indeed, when the discrete nature of quantum phenomena was being emphasised, it seemed as if differential equations might be replaced in ultimate physics by difference equations: but the discovery of wave mechanics restored the status quo ante, and today differential equations are more important than ever before. Partial Differential Equations of Mathematical Physics. By Prof. H. Bateman. Pp. xxii + 522. (Cambridge: At the University Press, 1932.) 42s. net.

...read moreread less

336 citations

"Multidimensional modal analysis of ..." refers background or methods or result in this paper

...The general form of a discrete infinite-dimensional modal system is derived in the first part of this paper by the Bateman–Luke (pressure-integral Lagrangian) variational principle. This idea was proposed independently by Miles (1976) and Lukovsky (1976). They studied forced small-amplitude translatory motions of a vertical circular cylindrical tank. The surfacemodes and domainmodes were obtained by linear theory and related by (1.2). Our derivation of a discrete infinite-dimensional modal system is not restricted to a particular type of body motion. The surface and domain modes are not associated with natural modes and no asymptotic assumptions are introduced in the first stage of the derivation. The infinite-dimensional modal system can be reduced to a finite-dimensional form by assuming small-amplitude forced oscillations and associate order of magnitudes of the different modes. This is done in the second part of the paper to analyse nonlinear sloshing in a two-dimensional rectangular smooth tank with finite water depth. Both forced translatory and rotational body motions are considered. The lowest natural mode is assumed to dominate and the three lowest modes interact nonlinearly with each other. Several modes having higher order are considered by linear theory. The asymptotic theory constructed is a special multidimensional analogue of the model by Ikeda & Nakagawa (1997) and the direct generalization of the third-order hydrodynamic theory by Faltinsen (1974). Experiments on nonlinear sloshing caused by primary mode resonant excitation have been conducted....
[...]
...The general form of a discrete infinite-dimensional modal system is derived in the first part of this paper by the Bateman–Luke (pressure-integral Lagrangian) variational principle. This idea was proposed independently by Miles (1976) and Lukovsky (1976). They studied forced small-amplitude translatory motions of a vertical circular cylindrical tank. The surfacemodes and domainmodes were obtained by linear theory and related by (1.2). Our derivation of a discrete infinite-dimensional modal system is not restricted to a particular type of body motion. The surface and domain modes are not associated with natural modes and no asymptotic assumptions are introduced in the first stage of the derivation. The infinite-dimensional modal system can be reduced to a finite-dimensional form by assuming small-amplitude forced oscillations and associate order of magnitudes of the different modes. This is done in the second part of the paper to analyse nonlinear sloshing in a two-dimensional rectangular smooth tank with finite water depth. Both forced translatory and rotational body motions are considered. The lowest natural mode is assumed to dominate and the three lowest modes interact nonlinearly with each other. Several modes having higher order are considered by linear theory. The asymptotic theory constructed is a special multidimensional analogue of the model by Ikeda & Nakagawa (1997) and the direct generalization of the third-order hydrodynamic theory by Faltinsen (1974)....
[...]
...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 Waterhouse (1994). The theory is validated by new experimental results....
[...]
...The canonical formulation of this principle is given by Bateman (1944) and Luke (1967) (for gravity surface waves in infinite basins)....
[...]
...We will use a Bateman–Luke variational principle and introduce the pressure in the Lagrangian of the Hamilton principle. The idea of the pressure integral as the Lagrangian in hydrodynamic problems was first proposed by Hargneaves (1908). The canonical formulation of this principle is given by Bateman (1944) and Luke (1967) (for gravity surface waves in infinite basins)....
[...]

Journal Article•DOI•

A Nonlinear Theory of Sloshing in Rectangular Tanks

[...]

Odd M. Faltinsen¹•Institutions (1)

DNV GL¹

01 Dec 1974-Journal of Ship Research

TL;DR: In this paper, a nonlinear, inviscid boundary-value problem of potential flow is formulated and the steady-state solution is found as a power series in epsilon to the one-third correctly to the order Epsilon.

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Abstract: A two-dimensional rigid, rectangular, open tank without baffles is forced to oscillate harmonically with small amplitudes of sway or roll oscillation in the vicinity of the lowest natural frequency for the fluid inside the tank. The breadth of the tank is 0 (1) and the depth of the fluid is either (1) or infinite. The excitation is of the order epsilon and the response is of the order epsilon to the one-third. A nonlinear, inviscid boundary-value problem of potential flow is formulated and the steady-state solution is found as a power series in epsilon to the one-third correctly to the order epsilon. Comparison between theory and experiment shows reasonable agreement. The stability of the steady-state solution has been studied.

...read moreread less

218 citations

"Multidimensional modal analysis of ..." refers background or methods or result in this paper

...It is also necessary that the water depth is not shallow and the fluid does not hit the tank ceiling (see, also, physical arguments presented in Faltinsen 1974 and the book by Mikishev 1978)....
[...]
...The asymptotic theory constructed is a special multidimensional analogue of the model by Ikeda & Nakagawa (1997) and the direct generalization of the third-order hydrodynamic theory by Faltinsen (1974)....
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...This was demonstrated by Solaas & Faltinsen (1997), where Moiseev’s theory was applied to two-dimensional sloshing....
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...It is assumed, as in the theory by Faltinsen (1974), that O(β31 ) = O(H) = O(ψ0) = ....
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...The branches presented differ from diagrams obtained by Faltinsen’s theory only for large values of |A|/l and far away from the main resonance σ̄1 = 1. The last difference is due to the change of m1 when varying σ. The results agree with the fifth-order theory by Waterhouse (1994) for sufficiently small amplitudes....
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Journal Article•DOI•

Nonlinear surface waves in closed basins

[...]

John W. Miles¹•Institutions (1)

University of California, Los Angeles¹

11 Jun 1976-Journal of Fluid Mechanics

TL;DR: The Lagrangian and Hamiltonian for nonlinear gravity waves in a cylindrical basin are constructed in terms of the generalized co-ordinates of the free-surface displacement, {qn(t)} ≡ q, thereby reducing the continuum-mechanics problem to one in classical mechanics as discussed by the authors.

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Abstract: The Lagrangian and Hamiltonian for nonlinear gravity waves in a cylindrical basin are constructed in terms of the generalized co-ordinates of the free-surface displacement, {qn(t)} ≡ q, thereby reducing the continuum-mechanics problem to one in classical mechanics. This requires a preliminary description, in terms of q, of the fluid motion beneath the free surface, which kinematical boundary-value problem is solved through a variational formulation and the truncation and inversion of an infinite matrix. The results are applied to weakly coupled oscillations, using the time-averaged Lagrangian, and to resonantly coupled oscillations, using Poincare's action—angle formulation. The general formulation provides for excitation through either horizontal or vertical translation of the basin and for dissipation. Detailed results are given for free and forced oscillations of two, resonantly coupled degrees of freedom.

...read moreread less

189 citations

"Multidimensional modal analysis of ..." refers methods in this paper

...The derivation of the modal system from the original free boundary problem was first proposed by Narimanov (1957) based on a perturbation technique. It has been further developed by Dodge, Kana & Abramson (1965), Narimanov, Dokuchaev & Lukovsky (1977) and Lukovsky (1990)....
[...]
...The derivation of the modal system from the original free boundary problem was first proposed by Narimanov (1957) based on a perturbation technique....
[...]
...This idea was proposed independently by Miles (1976) and Lukovsky (1976)....
[...]
...Using the Bateman–Luke variational principle, we generalize the procedure proposed by Miles (1976) and Lukovsky (1976) to derive a modal system describing nonlinear sloshing of an incompressible perfect fluid with irrotational flow partly occupying a tank performing an arbitrary three-dimensional…...
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