Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth

Abstract

The two-dimensional sloshing of a fluid in a horizontal circular cylindrical container and the three-dimensional sloshing of a fluid in a spherical container are considered. The linearized theory of water waves is used to determine the frequencies of free oscillations under gravity of an arbitrary amount of fluid in such tanks. Special coordinate systems are used and the problems are formulated in terms of integral equations which are solved numerically for the eigenvalues. Detailed tables of the sloshing frequencies are presented for a range of fill-depths of the containers.

Finite energy solutions of self-adjoint elliptic mixed boundary value problems

“…Eigenfunctions of this type for the Laplacian with n = 2 or 3 have been studied as modes in the theory of sloshing of a fluid and some analyses of these problems are described in [20,21]. When = * , solutions of (58) are called Steklov eigenfunctions and were studied by the author in [2,19].…”

On the oblique water-entry problem of a rigid sphere

“…In order to numerically determine the variation of the sway added-mass with submergence, one has to solve the integral equation (18) and substitute the results in (21). This integral equation has been solved numerically by using McIver's [13] procedure for evaluating the kernel function I(q, p; 00) for 7r > 00 >0, which covers the whole range of depths starting from first contact ( r = 0 ) to full submergence ( r = 2 ) . Once Ci(0o) has been calculated, the horizontal slamming coefficient dCl (7-) dC,(Oo) -sin 00 -- . '…”

“…It is known that a general solution of the Laplace equation may be expressed in terms of exterior toroidal harmonics in a form which is only partially separable. Hence, following Sneddon [26], an arbitrary exterior potential function which decays at infinity, may be represented as Kp(COsh rt) = Pm I/e+ip(COsh n)" (12) Thus, since the surface z = 0 is being represented now by 0 = 0, equations (3) and (11) suggest that 4~01)(T/, 8, ~; 00) = a cos 0(cosh n -cos 8) 1/2 f0 ~ sinh pO A(p; 8o)Klp(cosh~l) cosh P~---~0 dp (13) represents the horizontal velocity potential with Klp(cosh ~/)= (d/dT1)Kp(COsh ~/), and that the vertical potential is similarly given by ~b~2)(n, O, qJ; 8o) = a(cosh n -cos 0) 1/2 fo ~ sinh pO B(p; Oo)Kp(cosh ~?) cosh p~ d p , (14) where A(p) and B(p) are some unknown real coefficients.…”

“…A general procedure for solving such an integral equation has been recently outlined by Mclver [13] in his study of the sloshing frequencies of a partially-filled spherical container. The proposed numerical scheme is rather involved and is based on the representation of the toroidal functions in terms of the Gauss hypergeometric functions (Gradshteyn and Ryzhik [2], p. 1039).…”

Sloshing of Liquid in Rigid Cylindrical Container with a Rigid Annular Baffle. Part I: Free Vibration

“…The weighted problem could be treated very effectively by using the powerful methods of intermediate problems. Mciver [10] considered the two-dimensional sloshing of a fluid in a horizontal circular cylindrical container and the three-dimensional sloshing of a fluid in a spherical container. The special coordinate systems are used to formulate the sloshing problem as an integral equation which was solved numerically for eigenvalues.…”