Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

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Liquid Sloshing Dynamics

Parametrically forced surface waves

Recent Advances in Liquid Sloshing Dynamics

A modulated cross-wave of resonant frequencyω1, carrier frequencyω =ω1 {1 + O(e)}, slowly varying complex amplitude O(e½b), longitudinal scale b/e½ and timescale 1/eω is induced in a long channel of breadth b that contains water of depth d and is subjected to a vertical oscillation of amplitude O(eb) and frequency 2ω, where 0 0.045 for d/b [gsim ] 1. The corresponding cnoidal waves (of which the solitary wave is a limiting case) are considered in an appendix.

Parametrically excited solitary waves

This paper describes an experimental study of a stratified fluid which is flowing over a smooth two-dimensional obstacle which induces no flow separation and in which effects of viscosity and diffusion are not important. The results are restricted to fluid of finite depth. Various properties of the flow field, in particular the criterion for the onset of gravitational instability in the lee-wave field, are measured and compared with the theoretical predictions of Long's model. The agreement is found to be generally poor, and the consequent inapplicability of Long's model is explained by the failure of Long's hypothesis of no upstream influence, which is demonstrably invalid when stationary lee waves are possible. The obstacle generates upstream motions with fluid velocities which appear to be of first order in the obstacle height. These motions have some of the character of shear fronts or columnar disturbance modes and have the same vertical structure as the corresponding lee-wave modes generated downstream. They result in a reduced fluid velocity upstream below the level of the top of the obstacle, together with a jet of increased fluid velocity above this level which pours down the lee side of the obstacle. This phenomenon becomes more pronounced as the number of modes is increased.

Upstream influence and Long's model in stratified flows

In this paper experiments in which cross-waves were generated in front of a wave-maker at one end of a long channel are described. The primary field produced was a progressive wave train, but, at certain frequencies, a standing wave developed in front of the wavemaker. This wave, whose crests were at right angles to the wavemaker and which had frequency half that of the wavemaker, is known as a cross-wave.An instability mechanism for the formation of cross-waves in a long channel has been presented in Part 1 (Mahony 1972). In the present paper we describe some measurements concerning the formation of the waves. The initial growth rate of the waves has been measured and the curves of neutral stability determined for two of the possible modes. The results are in good agreement with Mahony's theory. After the early stages in the development of the waves, there is an increase in their growth rate. This new rate was found to be about twice the initial growth rate. Also reported are some measurements of the amplitude of the cross-wave field along the channel.

Cross-waves. Part 2. Experiments

Experiments are described in which a body was towed at large Richardson numbers through a uniformly stratified fluid, with particular attention being given to the parameter range in which diffusive effects are likely to play an important role. When the Richardson number is only moderately large (i.e. less than about 106) it appears that a linear viscous diffusive theory fails to model the experimental observations, but at larger Richardson numbers it seems that such a theory is a strong candidate for modelling the physical situation. At Richardson numbers of about 103 a non-diffusive viscous model (see the theories of Graebel 1969; Janowitz 1971) appears to give a fairly good description of the experimental results, except for some discrepancies between the theories and the observed flows in the wake downstream from the body. However, the conditions under which the experiments were carried out were not completely favourable for the application of these theories.A brief survey and new interpretations of some of the work on the rotatingfluid counterparts of the present experiments are also given. In view of the theoretical similarities between the two situations we have tried to assess and compare (where possible) the results of experimental investigations in each field and we feel that the observations in the rotating-fluid experiments are in good qualitative agreement with the present results. No measurements of the drag acting on the body were made in the present experiments but this has been done carefully in the rotating-fluid experiments, and the results of those measurements are in poor agreement with the theoretical predictions. Because of the close similarities between the axisymmetric model for viscous rotating fluids (cf. Moore & Saffman 1969) and the two-dimensional model for diffusive stratified fluids (Freund & Meyer 1972), an explanation for the discrepancy should be sought before the linear viscous diffusive model can be applied with any confidence to the flow of stratified fluids at very large Richardson numbers.

B. J. S. Barnard

Papers

Cross-waves. Part 2. Experiments

The motion generated by a body moving through a stratified fluid at large Richardson numbers