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

2010 ◽  
Vol 665 ◽  
pp. 457-479 ◽  
Author(s):  
ODD M. FALTINSEN ◽  
ALEXANDER N. TIMOKHA
Keyword(s):  
Free Surface ◽  
Steady State ◽  
Analytical Form ◽  
The Body ◽  
Liquid Sloshing ◽  
Two Dimensional ◽  
Surface Conditions ◽  
Natural Modes ◽  
Body Boundary ◽  
Multimodal Method

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 free-surface 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 body-boundary 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.

scholarly journals A two-dimensional numerical and experimental study of piston and sloshing resonance in moonpools with recess

2019 ◽  
Vol 877 ◽  
pp. 142-166 ◽  
Author(s):  
Senthuran Ravinthrakumar ◽  
Trygve Kristiansen ◽  
Bernard Molin ◽  
Babak Ommani
Keyword(s):  
Free Surface ◽  
Numerical Methods ◽  
Eigenfunction Expansion ◽  
Natural Frequencies ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Surface Conditions ◽  
Body Boundary ◽  
Sloshing Mode ◽  
The Mean

The piston and first sloshing modes of two-dimensional moonpools with recess are investigated. Dedicated forced heave experiments are carried out. Different recess lengths are tested from $1/4$ to $1/2$ of the length of the moonpool at the mean waterline. A theoretical model to calculate the natural frequencies is developed based on linearized potential flow theory and eigenfunction expansion. Two numerical methods are implemented: a boundary element method (BEM) and a Navier–Stokes solver (CFD). Both the BEM and CFD have linearized free-surface and body-boundary conditions. As expected, the BEM over-predicts the moonpool response significantly, in particular at the first sloshing mode. The CFD is in general able to predict the maximum moonpool response adequately, both at the piston and first sloshing modes. Both numerical methods fail to predict the Duffing-type behaviour at the first sloshing mode, due to the linearized free-surface conditions. The Duffing behaviour is more pronounced for the largest recess. The main source of damping in the proximity of the first sloshing mode is discussed.

Analytically approximate natural sloshing modes for a spherical tank shape

2012 ◽  
Vol 703 ◽  
pp. 391-401 ◽  
Author(s):  
Odd M. Faltinsen ◽  
Alexander N. Timokha
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Laplace Equation ◽  
Three Dimensional ◽  
Analytical Continuation ◽  
Liquid Sloshing ◽  
Two Dimensional ◽  
Spherical Tank ◽  
The Mean ◽  
Multimodal Method

AbstractThe multimodal method requires analytical (exact or approximate) natural sloshing modes that exactly satisfy the Laplace equation and boundary condition on the wetted tank surface. When dealing with the nonlinear sloshing problem, the modes should also allow for an analytical continuation throughout the mean free surface. Appropriate analytically approximate modes were constructed by Faltinsen & Timokha (J. Fluid Mech., vol. 695, 2012, pp. 467–477) for the two-dimensional circular tank. The present paper extends this result to the three-dimensional, spherical tank shape and, based on that, establishes specific properties of the linear liquid sloshing.

Nonlinear Aspects of Subcritical Shallow-Water Flow Past Two-Dimensional Obstructions

1978 ◽  
Vol 22 (04) ◽  
pp. 203-211
Author(s):  
Nils Salvesen ◽  
C. von Kerczek
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Flow Problem ◽  
Free Stream ◽  
The Body ◽  
Two Dimensional ◽  
Third Order ◽  
Submerged Body ◽  
The Mean ◽  
The Waves

Some nonlinear aspects of the two-dimensional problem of a submerged body moving with constant speed in otherwise undisturbed water of uniform depth are considered. It is shown that a theory of Benjamin which predicts a uniform rise of the free surface ahead of the body and the lowering of the mean level of the waves behind it agrees well with experimental data. The local steady-flow problem is solved by a numerical method which satisfies the exact free-surface conditions. Third-order perturbation formulas for the downstream free waves are also presented. It is found that in sufficiently shallow water, the wavelength increases with increasing disturbance strength for fixed values of the free-stream-Froude number. This is opposite to the deepwater case where the wavelength decreases with increasing disturbance strength.

Large-Amplitude Transient Motion of Two-Dimensional Floating Bodies

1979 ◽  
Vol 23 (01) ◽  
pp. 20-31
Author(s):  
R. B. Chapman
Keyword(s):  
Boundary Condition ◽  
Wave Field ◽  
Large Amplitude ◽  
The Body ◽  
Body Motion ◽  
Source Distribution ◽  
Floating Body ◽  
Two Dimensional ◽  
Body Boundary ◽  
Free Mode

A numerical method is presented for solving the transient two-dimensional flow induced by the motion of a floating body. The free-surface equations are linearized, but an exact body boundary condition permits large-amplitude motion of the body. The flow is divided into two parts: the wave field and the impulsive flow required to satisfy the instantaneous body boundary condition. The wave field is represented by a finite sum of harmonics. A nonuniform spacing of the harmonic components gives an efficient representation over specified time and space intervals. The body is represented by a source distribution over the portion of its surface under the static waterline. Two modes of body motion are discussed—a captive mode and a free mode. In the former case, the body motion is specified, and in the latter, it is calculated from the initial conditions and the inertial properties of the body. Two examples are given—water entry of a wedge in the captive mode and motion of a perturbed floating body in the free mode.

Waves and wave resistance of thin bodies moving at low speed: the free-surface nonlinear effect

1975 ◽  
Vol 69 (2) ◽  
pp. 405-416 ◽  
Author(s):  
G. Dagan
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Surface Condition ◽  
Perturbation Expansion ◽  
Wave Resistance ◽  
The Body ◽  
Thin Body ◽  
Two Dimensional ◽  
First Order ◽  
Flow Past

The linearized theory of free-surface gravity flow past submerged or floating bodies is based on a perturbation expansion of the velocity potential in the slenderness parameter e with the Froude number F kept fixed. It is shown that, although the free-wave amplitude and the associated wave resistance tend to zero as F → 0, the linearized solution is not uniform in this limit: the ratio between the second- and first-order terms becomes unbounded as F → 0 with ε fixed. This non-uniformity (called ‘the second Froude number paradox’ in previous work) is related to the nonlinearity of the free-surface condition. Criteria for uniformity of the thin-body expansion, combining ε and F, are derived for two-dimensional flows. These criteria depend on the shape of the leading (and trailing) edge: as the shape becomes finer the linearized solution becomes valid for smaller F.Uniform first-order approximations for two-dimensional flow past submerged bodies are derived with the aid of the method of co-ordinate straining. The straining leads to an apparent displacement of the most singular points of the body contour (the leading and trailing edges for a smooth shape) and, therefore, to an apparent change in the effective Froude number.

Time-Domain Simulations of Radiation and Diffraction Forces

2010 ◽  
Vol 54 (02) ◽  
pp. 79-94 ◽  
Author(s):  
Xinshu Zhang ◽  
Piotr Bandyk ◽  
Robert F. Beck
Keyword(s):  
Free Surface ◽  
Time Domain ◽  
Three Dimensional ◽  
The Body ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Forward Speed ◽  
Time Step ◽  
Surface Boundary Conditions ◽  
Strip Theory

Large-amplitude, time-domain, wave-body interactions are studied in this paper for problems with forward speed. Both two-dimensional strip theory and three-dimensional computation methods are shown and compared by a number of numerical simulations. In the present approach, an exact body boundary condition and linearized free surface boundary conditions are used. By distributing desingularized sources above the calm water surface and using constant-strength flat panels on the exact body surface, the boundary integral equations are solved numerically at each time step. The strip theory method implements Radial Basis Functions to approximate the longitudinal derivatives of the velocity potential on the body. Once the fluid velocities on the free surface are computed, the free surface elevation and potential are updated by integrating the free surface boundary conditions. After each time step, the body surface and free surface are regrided due to the instantaneous changing wetted body geometry. Extensive results are presented to validate the efficiency of the present methods. These results include the added mass and damping computations for a Wigley III hull and an S-175 hull with forward speed using both two-dimensional and three-dimensional approaches. Exciting forces acting on a Wigley III hull due to regular head seas are obtained and compared using both the fully three-dimensional method and the two-dimensional strip theory. All the computational results are compared with experiments or other numerical solutions.

Radiation and diffraction of second-order surface waves by floating bodies

1988 ◽  
Vol 196 ◽  
pp. 65-91 ◽  
Author(s):  
P. D. Sclavounos
Keyword(s):  
Free Surface ◽  
Linear Problem ◽  
Velocity Potential ◽  
Surface Condition ◽  
Second Order ◽  
The Body ◽  
Green Functions ◽  
Floating Bodies ◽  
Free Surface Condition ◽  
Body Boundary

The paper studies the radiation and diffraction by floating bodies of deep-water bichromatic and bidirectional surface waves subject to the second-order free-surface condition. A theory is developed for the evaluation of the second-order velocity potential and wave forces valid for bodies of arbitrary geometry, which does not involve the evaluation of integrals over the free surface or require an increased accuracy in the solution of the linear problem. Explicit sum- and difference-frequency ‘Green functions’ are derived for the radiation and diffraction problems, obtained from the solution of initial-value problems that ensure they satisfy the proper radiation condition at infinity. The second-order velocity potential is expressed as the sum of a particular and a homogeneous component. The former satisfies the non-homogeneous free-surface condition and is expressed explicitly in terms of the second-order Green functions. The latter is subject to the homogeneous free-surface condition and enforces the body boundary condition by the solution of a linear problem. An analysis is carried out of the singular behaviour of the second-order potential near the intersection of the body boundary with the free surface.

scholarly journals Coupling Between Flexible Ship and Liquid Sloshing Using Potential Flow Analysis

2010 ◽  
Author(s):  
Youngbum Lee ◽  
Mingyi Tan ◽  
Pandeli Temarel ◽  
Shihua Miao
Keyword(s):  
Rigid Body ◽  
Potential Flow ◽  
Coupling Effect ◽  
Flow Analysis ◽  
Hydrodynamic Forces ◽  
The Body ◽  
Liquid Sloshing ◽  
Ship Motions ◽  
Resonance Frequencies ◽  
Body Boundary

The significant increase in demand for Liquefied Natural Gas (LNG) and the economic aspects of its transportation resulted in increases in the number and size of LNG carriers. One of the design issues for LNG carriers is the sloshing phenomenon because containment systems widely used nowadays have no internal structures. Furthermore, because the weights of ship and cargo are comparable and ship operators want more flexible operations allowing partial fillings in tanks, the coupling effect between ship motions and sloshing requires further investigation, including the effect of ship distortion. The previous study on coupling between rigid body and sloshing shows good agreement between methods of prediction and measurements [1,2]. Hence, in this paper the potential flow approach adopted for the coupling effect between rigid body ship motion and sloshing is extended to flexible ship-partially filled tank system, using the desingularised Rankine source method. In this case, the global deflection of the flexible ship is used for application of the body boundary condition on the partially filled tank. The aim of this paper is to investigate the influence of hull flexibility on the hydrodynamic forces and moments associated with liquid sloshing and vice versa, as well as the dynamic characteristics (e.g. resonance frequencies) of the whole system. As there are no experimental results available, the method is validated by comparing hydrodynamic forces from sloshing obtained using rigid and flexible body approaches. The coupling effect between flexible ship and sloshing in partially filled tanks is investigated for an idealized LNG Carrier in beam regular waves, considering different partial filling scenarios.

A Localized Finite-Element Method for Two-Dimensional Steady Potential Flows with a Free Surface

1978 ◽  
Vol 22 (04) ◽  
pp. 216-230
Author(s):  
Kwang June Bai
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Boundary Conditions ◽  
Complex Geometry ◽  
The Body ◽  
Finite Domain ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Test Functions ◽  
Infinite Domain

A numerical method is presented for solving two-dimensional uniform flow problems with a linearized free-surface boundary condition. The boundary-value problem governed by Laplace's equation is replaced by a weak formulation (also known as Galerkin's method) with certain essential boundary conditions. The infinite domain of the fluid is reduced to a finite domain by utilizing known solution spaces in certain subdomains. The bases for the trial and test functions are chosen from the same subspace of the polynomial function space in the reduced subdomain. The essential boundary conditions are properly taken into account by an unconventional choice of the basis for the trial functions, which is different from that for the test functions in other subdomains. This method is applied to two-dimensional steady flow past a submerged elliptic section, a hydrofoil at an arbitrary angle of attack, and a bump on the bottom. In each example the body boundary condition is satisfied exactly. Both subcritical and supercritical flows are treated. We present the numerical results of wave resistance, lift force, moment, circulation strength, and flow blockage parameter. The computed pressure distributions on the hydrofoil and wave profiles are shown. The test results obtained by the present method agree very well with existing results. The main advantage of this method is that any complex geometry of the boundary can be easily accommodated.

Slamming Loads on a Wedge Elastically Suspended on a Marine Structure

2015 ◽  
Author(s):  
Hui Sun ◽  
Jens B. Helmers
Keyword(s):  
Free Surface ◽  
Impact Force ◽  
Water Entry ◽  
The Body ◽  
Water Impact ◽  
Surface Conditions ◽  
Marine Structure ◽  
Nonlinear Free Surface ◽  
Stiffness And Damping ◽  
Water Exit

Slamming loads on a two-dimensional wedge elastically suspended on a marine structure are analyzed by using either a combined Wagner and von Karman method (W-vK) or a boundary element method (BEM). Fully nonlinear free surface conditions are satisfied in the BEM. Hydroelasticity effects are considered in both methods. A sinusoidal free surface motion relative to the marine structure is specified for the slamming event. Both the water entry phase and the water exit phase are simulated. The numerical results by the two different methods are compared. The W-vK method can generally predict the same trend of the variation of the body motions and the water forces, although the predicted maximum forces are larger than those by the BEM. The influence of the stiffness and damping of the elastic connection on the water impact force are discussed.

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