Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-14T22:39:35.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Sloshing frequencies of longitudinal modes for a liquid contained in a trough

Published online by Cambridge University Press:  26 April 2006

P. McIver  and
M. McIver
P. McIver
Affiliation:
Department of Mathematical Sciences, Loughborough University of Technology, Loughborough, Leicestershire, LE11 3TU, UK
M. McIver
Affiliation:
Department of Mathematical Sciences, Loughborough University of Technology, Loughborough, Leicestershire, LE11 3TU, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The sloshing under gravity is considered for a liquid contained in a horizontal cylinder of uniform cross-section and symmetric about a vertical plane parallel to its generators. Much of the published work on this problem has been concerned with twodimensional, transverse oscillations of the fluid. Here, attention is paid to longitudinal modes with variation of the fluid motion along the cylinder. There are two known exact solutions for all modes; these are for cylinders whose cross-sections are either rectangular or triangular with a vertex semi-angle of ¼π. Numerical solutions are possible for an arbitrary geometry but few calculations are reported in the open literature. In the present work, some general aspects of the solutions for arbitrary geometries are investigated including the behaviour at low and high frequency of longitudinal modes. Further, simple methods are described for obtaining upper and lower bounds to the frequencies of both the lowest symmetric and lowest antisymmetric modes. Comparisons are made with numerical calculations from a boundary element method.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 525 - 541
Copyright
© 1993 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics, vol. 1. Interscience.
Davis, A. M. J. 1965 Two-dimensional oscillations in a canal of arbitrary cross-section. Proc. Camb. Phil. Soc. 61, 827–846.Google Scholar
Davis, A. M. J. 1992 Discussion related to Watson and Evan: ‘Resonant frequencies of a fluid in containers with internal bodies’. J. Engng Math. 26, 445–454.Google Scholar
Evans, D. V. & Linton, C. M. 1993 Sloshing frequencies. Q. J. Mech. Appl. Maths (to appear).Google Scholar
Evans, D. V. & Mciver, P. 1991 Trapped waves over symmetric thin bodies. J. Fluid Mech. 223, 509–519.Google Scholar
Fox, D. W. & Kuttler, J. R. 1983 Sloshing frequencies. Z. Angew Math. Phys. 34, 668–696Google Scholar
Isaacson, E. 1950 Water waves over a sloping beach. Commun. Pure Appl. Math. 3, 1–32.Google Scholar
Kobayashi, N., Mieda, T., Shibata, H. & Shinozaki, Y. 1989 A study of the liquid slosh response in horizontal cylindrical tanks. J. Pressure Vessel Technol. 111, 32–38.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th Edn. Cambridge University Press.
McIver, P. 1989 Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth. J. Fluid Mech. 201, 243–257.Google Scholar
Mciver, P. & Smith, S. R. 1987 Free-surface oscillations of fluid in closed basins. J. Engng Math. 21, 139–148.Google Scholar
Moiseev, N. N. & Petrov, A. A. 1966 The calculation of free oscillations of a liquid in a motionless container. Adv. Appl. Mech. 9, 91–154.Google Scholar
Packham, B. A. 1980 Small-amplitude waves in a straight channel of uniform triangular crosssection. Q. J. Mech. Appl. Math. 33, 179–187.Google Scholar
Peters, A. S. 1952 Water waves over sloping beaches and the solution of a mixed boundary value problem for ∇2ϕ-k2ϕ = 0 in a sector. Commun. Pure Appl. Maths 5, 87–108.Google Scholar
Protter, M. H. & Weinberger, H. F. 1984 Maximum Principles In Differential Equations. Springer.
Shi, J. & Yih, C.-S. 1984 Waves in open channels. J. Engng Mech. 110, 847–870.Google Scholar
Ursell, F. 1952 Edge waves on a sloping beach. Proc. R. Soc. Lond. A 214, 79–97.Google Scholar