Wave growth

Fa.n Spra.ys, It was demonstrated around the 1950s that iastabiHty theory can be used to analyze the wave growth on a thin Hquid sheet (18). This analysis predicted the existence of an optimum wavelength at which a wave would grow rapidly. This optimum wavelength, X corresponds to a condition that leads to Hquid sheet disiategration. It can be expressed as ia equatioa 2 ... 

Wu, Ruff and Faeth12491 studied the breakup of liquid jets with holography. Their measurements showed that the liquid volume fraction on the spray centerline starts to decrease from unit atZ/<70=150 for non-turbulent flows, whereas the decrease starts at aboutZ/<70=10 for fully developed turbulent flows. Their measurements of the primary breakup also showed that the classical linear wave growth theories were not effective, plausibly due to the non-linear nature of liquid breakup processes. 

As described previously, in the atomization sub-model, 232 droplet parcels are injected with a size equal to the nozzle exit diameter. The subsequent breakups of the parcels and the resultant droplets are calculated with a breakup model that assumes that droplet breakup times and sizes are proportional to wave growth rates and wavelengths obtained from the liquid jet stability analysis. Other breakup mechanisms considered in the sub-model include the Kelvin-Helmholtz instability, Rayleigh-Taylor instability, 206 and boundary layer stripping mechanisms. The TAB model 310 is also included for modeling liquid breakup. 

These limiting factors of the wave growth in an enclosed sea with the dimensions as the Baltic cause the average wave height to be smaller in particular during high wind speeds compared with the ocean where the fetch is limited only by the dimension of the wind field and the wave motion does not come into contact with the ocean bottom. 

In sea areas similar to the Baltic Sea, the wind direction determines the fetch and thereby the wave growth. 

Fetch The fetch is the length of a wind field, in which the direction and speed of the wind are approximately constant (see Fig. 7.6). This is the part of the sea surface on which the wind transmits its energy and thus causes wave growth. 

Depending on wind speed and fetch, wave growth is limited by the water depth (Fig. 7.11, top right). In areas with decreasing water depth ingoing waves will be deformed, and their transformation starts at a depth lower than half the wavelength (see Fig. 7.2). The life cycles of waves are finally ended when they break. 

The wind caused the significant wave height to increase from 1.5 m to almost 5 m within 3 h foUowing the increase in wind speed with a delay of approximately 2-3 h. There was a systematic difference of 20° to 30° degrees between mean wind direction (WSW) and sea direction (nearly west). A possible cause for this is that due to topography, the most effective direction for wave growth differs from the prevailing wind direction. The peak period increased to 8-9 s. 

Assuming that atomization is caused by aerodynamic surface wave growth, Ranz (37) predicted the spreading angle, 0, of the atomizing jet by relating the radial velocity of the unstable surface waves to the axial injection velocity... 

Problem 12-2. Capillary Instability of a Varicose Sheet. We have seen that a cylinder of fluid surrounded by air is unstable because of growing capillary waves. An equivalent problem is the stability of capillary waves on a round laminar jet when the velocity profile across the jet is uniform. We now wish to consider whether the same type of instability is relevant to a fluid sheet of finite width <7 and infinite lateral extent (the sheet has an interface above and below). Equivalently, we could also ask whether a planar (2D) laminar jet is unstable to capillary wave growth. Assume that the sheet is subject to an infinitesimal ID wavelike disturbance that is symmetric about the center plane and corresponds to an initial sheet thickness h = ho( + e sin foe). Prove whether the sheet is linearly stable or unstable to this type of disturbance. 

A commonly used primary atomization model for liquid jets has been developed by Huh et al. [1], The model considers the effects of both infinitesimal wave growth on the jet surface and jet turbulence including cavitation dynamics. Initial perturbations on the jet surface are induced by the turbulent fluctuations in the jet, originating from the shear stress along the nozzle wall and possible cavitation effects. This approach overcomes the inherent difficulty of wave growth models, where the exponential wave growth rate becomes zero at zero perturbation amplitude. 

The time scale of atomization is the linear sum of the turbulence and wave growth time scales ... 

The wave growth timescale is approximated by neglecting the surface tension and viscous effects and maintaining only the aerodynamic destabilizing term ... 

G. D. Crapper, N. Dranhrowski, and W. P. JepsOTi Wave growth on thin sheets of non-newtonian liquids. Proeeedings of the Royal Society of London, A. 342, 225—236 (1975). 

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