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Research Papers: Fluid-Structure Interaction

# Theoretical Determination of Modal Damping Ratio of Sloshing Using a Variational Method

[+] Author and Article Information
M. Utsumi

Department of Machine Element, Technical Research Laboratory, IHI Corporation, 1 Shinnakaharacho, Isogo-ku, Yokohama, Kanagawa Prefecture 235-8501, Japanmasahiko_utsumi@ihi.co.jp

J. Pressure Vessel Technol 133(1), 011301 (Jan 21, 2011) (10 pages) doi:10.1115/1.4002057 History: Received February 15, 2010; Revised June 17, 2010; Published January 21, 2011; Online January 21, 2011

## Abstract

This paper investigates a variational method for theoretically determining the first damping ratio of sloshing in cylindrical and arbitrary axisymmetric tanks. In this method, a virtual work expression for the viscous terms in the Navier–Stokes equations is transformed into the first-mode damping term, thereby extending Hamilton’s principle for nonviscous sloshing to a variational principle of viscous sloshing. By applying the Galerkin method to the variational principle, a computationally efficient analysis is conducted. For arbitrary axisymmetric tanks, a method for reducing the influence of the change in the fluid velocity boundary condition on the damping estimation is investigated. The proposed method provides theoretical foundations for past empirical results for cylindrical and spherical tanks.

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## Figures

Figure 8

Sloshing mass normalized by mass of the liquid, which completely fills the tank

Figure 9

Comparison with earlier experimental results given in Ref. 2; half-full spherical tank; chain line, present analysis; solid line, experimental equation given by Ref. 4

Figure 1

Cylindrical tank and coordinate systems

Figure 2

Logarithmic decrement δd=2πζd of sloshing in cylindrical tank; ρf=1000 kg/m3, μ=0.0011 N s/m2, g=9.8 m/s2; , present analysis; ●, calculation in Ref. 8; ×, experiment in Ref. 8 (polished tank); ▲, experiment in Ref. 8 (unpolished tank)

Figure 3

Logarithmic decrement δd=2πζd of sloshing in cylindrical tank; h/a=1, ρf=1000 kg/m3; , the present analysis; ●, calculation in Ref. 8; (a) δd versus μ (a=0.076 m, g=9.8 m/s2); (b) δd versus a (μ=0.0011 N s/m2, g=9.8 m/s2); (c) δd versus g (a=0.076 m, μ=0.0011 N s/m2)

Figure 4

Spherical coordinates for arbitrary axisymmetric tank

Figure 5

Numerical results for nonviscous sloshing in spherical tank; , present analysis; ●, Refs. 12-14; (a) dimensionless eigenfrequency ω/(g/a)1/2; (b) sloshing mass normalized by liquid mass

Figure 6

Logarithmic decrement δd=2πζd of sloshing in spherical tank; ρf=1000 kg/m3, a=0.076 m, μ=0.0011 N s/m2, g=9.8 m/s2; thin solid line, present analysis without correction; thick solid line, present analysis with correction; dotted line, Eq. 55 modified by multiplying by 0.64

Figure 7

Logarithmic decrement δd=2πζd of sloshing in spherical tank; filling level 0.64, ρf=1000 kg/m3; , present analysis; ●, Eq. 55 modified by multiplying by 0.64; (a) δd versus μ (a=0.076 m, g=9.8 m/s2); (b) δd versus a (μ=0.0011 N s/m2, g=9.8 m/s2); (c) δd versus g (a=0.076 m, μ=0.0011 N s/m2)

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