Analytically approximate natural sloshing modes for a spherical tank shape

Faltinsen, Odd M.; Timokha, Alexander

doi:10.1017/jfm.2012.237

Cited by 27 publications

(23 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This effect was proved rigorously by Kulczycki and Kwaśnicki in [30]. It has also been studied for spherical tanks by Faltinsen and Timokha [13], using analytic approximations.…”

Section: Introductionmentioning

confidence: 79%

“…As such, the model is well suited for infinitesimal sloshing of a perfect liquid (cf. [17,26,13,14]). In other words, our approximations only apply to a nearly still water surface, as opposed to turbulent phenomena in strong sloshing (as in [42]).…”

Section: Mathematical Modelmentioning

confidence: 99%

“…We consider a sphere of radius 1, filled to a depth d < 2. Note that the same parametrization was already considered by McIver [39] and Faltinsen-Timokha [13]. In both papers authors use the geometry of the sphere to either numerically or analytically approximate the sloshing eigenvalues of the spherical tank.…”

Section: 23mentioning

confidence: 99%

“…We also provide sloshing profiles for the fundamental modes for various depths on Figure 11, where we also mark the high-spots (cf. Figure 2 in Faltinsen-Timokha [13]). We include the ice-fishing problem (discussed and solved numerically in Section 4.1.1), since it can be understood as the limiting case d = 2 (full tank), see McIver [39].…”

Section: 23mentioning

confidence: 99%

See 3 more Smart Citations

The shape of the fundamental sloshing mode in axisymmetric containers

2015

View full text Add to dashboard Cite

ABSTRACT. In the paper we numerically study positions of high spots (extrema) of the fundamental sloshing mode of a liquid in an axisymmetric tank. Our approach is based on a linear model reducing the problem to appropriate Steklov eigenvalue problem. We propose a numerical scheme for calculating sloshing modes and a novel method of making images of oscillating fluid. We also describe the relation of the high spot problem to the celebrated hot spots conjecture.

show abstract

“…This effect was proved rigorously by Kulczycki and Kwaśnicki in [30]. It has also been studied for spherical tanks by Faltinsen and Timokha [13], using analytic approximations.…”

Section: Introductionmentioning

confidence: 79%

Section: Mathematical Modelmentioning

confidence: 99%

Section: 23mentioning

confidence: 99%

Section: 23mentioning

confidence: 99%

See 2 more Smart Citations

The shape of the fundamental sloshing mode in axisymmetric containers

2015

View full text Add to dashboard Cite

show abstract

“…Уровень заполнения h, м Рассмотрены различные уровни заполнения жидкостью, включая h 1 =1.99, что соответствует «ice-fishing problem», [11]. Результаты расчетов близки, в некоторых случаях МГЭ дает более высокую точность.…”

Section: методunclassified

Method of singular integral equations in liquid vibration problems for coaxial shells

2019

Bulletin of V.N. Karazin Kharkiv National University, Series «Mathematical Modeling. Information Technology. Automated Control

View full text Add to dashboard Cite

The paper deals with the problem of free vibrations of an ideal incompressible fluid in coaxial shells of revolution. It is assumed that the motion of the fluid is irrotational that allows us to introduce the velocity potential. In these suppositions the potential is satisfied to Laplace equation. The boundary conditions are formulated on the wetted surfaces of the shells and on the free liquid surface. The non-penetration conditions are applied to the wetted surfaces. On the free surface we consider dynamical and kinematical boundary conditions. The dynamical condition consists in equality of the liquid pressure on the free surface to the atmospheric one. The kinematic condition requires that total time derivative of the free surface elevation will be equal to zero at any instant. Regarding the potential of velocities, a boundary value problem is formulated that is further reduced to the eigenvalue problem. To solve the boundary value problem for the Laplace equation, the boundary element method is used in a direct formulation. The axial symmetric form of the shells allows us to reduce the obtained system of singular equations to one-dimensional equations. The kernels in singular operators of obtained integral equations are expressed on terms of elliptical integrals of the first and second kinds, and have the logarithmic singularities. The special numerical technique is elaborated to treat with such kind integral equations. The resulting one-dimensional singular equation is solved by the method of discrete singularities. The integration region contains the free surface of the fluid that in the case of coaxial shells is a ring. So, the possibility of using the boundary integral equation approach coupled with application of the discrete singularities method is established to solution of the singular integral equation with incoherent boundaries. A numerical study has been carried out that made it possible to determine the frequencies and modes of the liquid sloshing in the shells for different ratios of the inner and outer radii of cylindrical coaxial shells. The obtained modes of natural vibrations will be used for numerical simulation of forced liquid vibrations in the tanks and reservoirs.

show abstract

The Narimanov–Moiseev Multimodal Analysis of Nonlinear Sloshing in Circular Conical Tanks

Solodun

Timokha

2019

Applied Mathematical Analysis: Theory, Methods, and Applications

View full text Add to dashboard Cite

The chapter reports mathematical aspects of the Narimanov-Moiseev multimodal modelling for the liquid sloshing in rigid circular conical tanks, which perform small-magnitude oscillatory motions with the forcing frequency close to the lowest natural sloshing frequency. To derive the corresponding nonlinear modal system (of ordinary differential equations), we introduce an infinite set of the sloshing-related generalised coordinates governing the free-surface elevation but the velocity potential is posed as a Fourier series by the natural sloshing modes where the time-depending coefficients are treated as the generalised velocities. The employed approximate natural sloshing modes exactly satisfy both the Laplace equation and the zero-Neumann boundary condition on the wetted tank walls. The Lukovsky non-conformal mapping technique transforms the inner (conical) tank (physical) domain to an artificial upright circular cylinder, for which the single-valued representation of the free surface is possible. Occurrence of secondary resonances for the V-shaped truncated conical tanks is evaluated. The Narimanov-Moiseev modal equations allow for deriving an analytical steady-state (periodic) solution, whose stability is studied. The latter procedure is illustrated for the case of longitudinal harmonic excitations. Standing (planar) waves and swirling as well as irregular sloshing (chaos) are established in certain frequency ranges. The corresponding amplitude response curves are drawn and extensively discussed.

show abstract

Analytically approximate natural sloshing modes for a spherical tank shape

Cited by 27 publications

References 18 publications

The shape of the fundamental sloshing mode in axisymmetric containers

The shape of the fundamental sloshing mode in axisymmetric containers

Method of singular integral equations in liquid vibration problems for coaxial shells

The Narimanov–Moiseev Multimodal Analysis of Nonlinear Sloshing in Circular Conical Tanks

Contact Info

Product

Resources

About