On Three-Dimensional Nonlinear Subharmonic Resonant Surface Waves in a Fluid: Part I—Theory

Abstract

Three-dimensional surface waves in a rectangular container subjected to vertical excitation are studied. The analysis includes the effects of surface tension, energy dissipation, and critical depth. Both steady state and transient phenomena are discussed.

Sloshing motions in excited tanks

“…At the critical depth the response changes from a ''hard-spring'' to a ''soft-spring''. Gu et al [20], Faltinsen [11], Waterhouse [41] found h c =k ¼ 0:583 m or in general h c ¼ 0:337 Â b for the first mode, respectively. Since h s is 0.6 m, this particular study is a near critical depth case.…”

“…It has been established that there is a critical liquid depth that delineates two non-linear regimes of the liquid free surface referred to as soft and hard spring characteristics. Gu and Sethna [19], Gu et al [20], Virnig et al [40] have examined the role of the liquid critical depth in rectangular tanks subjected to vertical sinusoidal excitation. Another important feature is that there is an excitation frequency range over which the free surface exhibits chaotic motion [23].…”

Simulation of sloshing motions in fixed and vertically excited containers using a 2-D inviscid σ-transformed finite difference solver

“…It has been established that there is a critical liquid depth that delineates two nonlinear regimes of the liquid free surface referred to as soft and hard spring characteristics. Gu and Sethna (1987), Gu et al (1988) and Virnig et al (1988) have examined the role of the liquid critical depth in rectangular tanks subjected to vertical sinusoidal excitation. Another important feature is that there is an excitation frequency range over which the free surface exhibits chaotic motion (Ibrahim et al, 2001).…”

Numerical predictions of tuned liquid tank structural systems

“…Further, the wave steepness, defined as S ¼ k nẑ ¼ 1:2; 0:64 (where k n ¼ np=b is the wavenumber), respectively, versus the wave steepness of 0.0004 of the linear case. Also, this particular tank aspect ratio (b=h ¼ 2) has a water depth above the critical water depth, defined asĥ c =l w ¼ 0:162 (Gu et al, 1988) wherê l w ¼ 2b=n denotes the wavelength. The b=h ¼ 2 is a deep water case as k 1ĥ 4p=2: Note in an uncoupled system the liquid is known to exhibit softening spring behavior whereas the coupled system shows increasing amplitudes with respect to linearly predicted frequencies (as discussed later, e.g., Fig.…”