Recent Advances in Liquid Sloshing Dynamics

Numerical simulation of sloshing waves in a 3D tank based on a finite element method

A free surface may be deformed by fluid motions; such deformation may lead to surface roughness, breakup, or disintegration. This paper describes the wide range of free-surface deformations that occur when there is turbulence at the surface, and focuses on turbulence in the denser, liquid, medium. This turbulence may be generated at the surface as in breaking water waves, or may reach the surface from other sources such as bed boundary layers or submerged jets. The discussion is structured by consideration of the stabilizing influences of gravity and surface tension against the disrupting effect of the turbulent kinetic energy. This leads to a two-parameter description of the surface behaviour which gives a framework for further experimental and theoretical studies. Much of the discussion is necessarily heuristic, and is often limited by a lack of appropriate experimental observations. It is intended that such experiments be stimulated, to test the value or otherwise of our two-parameter description.

The dynamics of strong turbulence at free surfaces. Part 1. Description

Sloshing motions in excited tanks

A liquid drop placed on a vibrating diaphragm will burst into a fine spray of smaller secondary droplets if it is driven at the proper frequency and amplitude. The process begins when capillary waves appear on the free surface of the drop and then grow in amplitude and complexity as the acceleration amplitude of the diaphragm is slowly increased from zero. When the acceleration of the diaphragm rises above a well-defined critical value, small secondary droplets begin to be ejected from the free-surface wave crests. Then, quite suddenly, the entire volume of the drop is ejected from the vibrating diaphragm in the form of a spray. This event is the result of an interaction between the fluid dynamical process of droplet ejection and the vibrational dynamics of the diaphragm. During droplet ejection, the effective mass of the drop–diaphragm system decreases and the resonance frequency of the system increases. If the initial forcing frequency is above the resonance frequency of the system, droplet ejection causes the system to move closer to resonance, which in turn causes more vigorous vibration and faster droplet ejection. This ultimately leads to drop bursting. In this paper, the basic phenomenon of vibration-induced drop atomization and drop bursting will be introduced, demonstrated, and characterized. Experimental results and a simple mathematical model of the process will be presented and used to explain the basic physics of the system.

Vibration-induced drop atomization and bursting

Mild to steep standing waves of the fundamental mode are generated in a narrow rectangular cylinder undergoing vertical oscillation with forcing frequencies of 3.15 Hz to 3.34 Hz. A precise, non-intrusive optical wave profile measurement system is used along with a wave probe to accurately quantify the spatial and temporal surface elevations. These standing waves are also simulated by a two-dimensional spectral Cauchy integral code. Experiments show that contact-line effects increase the viscous natural frequency and alter the neutral stability curves. Hence, as expected, the addition of the wetting agent Photo Flo significantly changes the stability curve and the hysteresis in the response diagram. Experimentally, we find strong modulations in the wave amplitude for some forcing frequencies higher than 3.30 Hz. Reducing contact-line effects by Photo-Flo addition suppresses these modulations. Perturbation analysis predicts that some of this modulation is caused by noise in the forcing signal through ‘sideband resonance’, i.e. the introduction of small sideband forcing can generate large modulations of the Faraday waves. The analysis is verified by our numerical simulations and physical experiments. Finally, we observe experimentally a new form of steep standing wave with a large symmetric double-peaked crest, while simulation of the same forcing condition results in a sharper crest than seen previously. Both standing wave forms appear at a finite wave steepness far smaller than the maximum steepness for the classical standing wave and a surface tension far smaller than that for a Wilton ripple. In both physical and numerical experiments, a stronger second harmonic (in time) and temporal asymmetry in the wave forms suggest a 1:2 resonance due to a non-conventional quartet interaction. Increasing wave steepness leads to a new form of breaking standing waves in physical experiments.

/pdf/moderate-and-steep-faraday-waves-instabilities-modulation-jdl3h3l2el.pdf

Moderate and steep Faraday waves: instabilities, modulation and temporal asymmetries

We examine the dynamics of two-dimensional steep and breaking standing waves generated by Faraday-wave resonance. Jiang et al. (1996) found a steep wave with a double-peaked crest in experiments and a sharp-crested steep wave in computations. Both waveforms are strongly asymmetric in time and feature large superharmonics. We show experimentally that increasing the forcing amplitude further leads to breaking waves in three recurrent modes (period tripling): sharp crest with breaking, dimpled or flat crest with breaking, and round crest without breaking. Interesting steep waveforms and period-tripled breaking are related directly to the nonlinear interaction between the fundamental mode and the second temporal harmonic. Unfortunately, these higher-amplitude phenomena cannot be numerically modelled since the computations fail for breaking or nearly breaking waves. Based on the periodicity of Faraday waves, we directly estimate the dissipation due to wave breaking by integrating the support force as a function of the container displacement. We find that the breaking events (spray, air entrainment, and plunging) approximately double the wave dissipation.

/pdf/period-tripling-and-energy-dissipation-of-breaking-standing-52avslzvc8.pdf

Period tripling and energy dissipation of breaking standing waves

Parasitic ripple generation on short gravity waves (4 cm to 10 cm wavelengths) is examined using fully nonlinear computations and laboratory experiments. Timemarching simulations show sensitivity of the ripple steepness to initial conditions, in particular to the crest asymmetry. Signicant crest fore{aft asymmetry and its unsteadiness enhance ripple generation at moderate wave steepness, e.g. ka between 0.15 and 0.20, a mechanism not discussed in previous studies. The maximum ripple steepness (in time) is found to increase monotonically with the underlying (lowfrequency bandpass) wave steepness in our simulations. This is dierent from the subor super-critical ripple generation predicted by Longuet-Higgins (1995). Unsteadiness in the underlying gravity{capillary waves is shown to cause ripple modulation and an interesting ‘crest-shifting’ phenomenon { the gravity{capillary wave crest and the rst ripple on the forward slope merge to form a new crest. Including boundary layer eects in the free-surface conditions extends some of the simulations at large wave amplitudes. However, the essential process of parasitic ripple generation is nonlinear interaction in an inviscid flow. Mechanically generated gravity{capillary waves demonstrate similar characteristic features of ripple generation and a strong correlation between ripple steepness and crest asymmetry.

Unsteady ripple generation on steep gravity–capillary waves

Parasitic capillary waves and Wilton's ripples under Faraday resonance

New experimental data on the frequency and damping of Faraday water waves in glass tanks are presented to demonstrate the contact-line effect on free surface flows. We find a complicated nonlinear relationship between wave frequency and amplitude near contact lines: The amplitude dispersion for decaying standing waves directly progresses from a nonlinear regime due to large amplitude to a regime due to contact-line nonlinearity. The relative damping rate is also a function of the wave amplitude, increasing significantly at smaller wave amplitude. These results are discussed in relation to different formulations of contact-line conditions for oscillatory motions and free surface flows. A new model is proposed to explain the observed amplitude scaling in the frequency and damping rate, and to relate these behaviors to slip-length and other contact-line measurements by Ting and Perlin [J. Fluid Mech. 295, 263 (1995)].

Contact-line dynamics and damping for oscillating free surface flows

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