Sinuous waves

First Wind-Induced Breakup (Sinuous Wave Breakup) Surface Tension Force, Dynamic Pressure of Ambient Air 1.2 + 3.41Oh09 < We <13... 

Conical Sheets. Conical sheets are deemed to be shorter than flat sheets because the radius of the curvature may have a destabilizing effect on fluctuations.Fraser et al.[116] indicated that dilational waves in a sheet may be neglected because the degree of instability of these waves is always less than that of sinuous waves. 

Linear stability theories have also been applied to analyses of liquid sheet breakup processes. The capillary instability of thin liquid sheets was first studied by Squire[258] who showed that instability and breakup of a liquid sheet are caused by the growth of sinuous waves, i.e., sideways deflections of the sheet centerline. For a low viscosity liquid sheet, Fraser et al.[73] derived an expression for the wavelength of the dominant unstable wave. A similar formulation was derived by Li[539] who considered both sinuous and varicose instabilities. Clark and DombrowskF540 and Reitz and Diwakar13161 formulated equations for liquid sheet breakup length. 

Generally, two modes of oscillations are considered symmetric and antisymmetric. In the symmetric mode, also referred to as the dilational mode or varicose waves, the middle plane is undisturbed. In the antisymmetric mode, also referred to as sinuous waves, the free surfaces move in the same direction and with the same magnitude. Squire [3] and Hagerty and Shea [4] showed that for the case of inviscid sheets, the antisymmetric mode is the dominant mode of disturbance. However, later studies have revealed that this is not generally the case [22]. 

In order to analyze the atomization mechanism of the air-shrouded injector, the atomization characteristics of the fabricated atomizer was investigated using a phase Doppler particle analyzer (PDPA). The Sauter mean diameter (SMD) and mean velocity distribution at 5 ms ASI are shown in Fig. 34.8. As the air pressure increases, the air velocity increases and the air dispersion area is enlarged proportionally. The maximum velocity achieved is 55 m/s when the air pressure is 50 kPa. The degree of atomization is greater at the center flow because the air velocity at the center flow is greater. Spray patterns for various air pressures are shown in Fig. 34.9. It can be seen that as the air pressure increases, the atomization process transitions from varicose wave to sinuous wave mode. Atomization at low air pressure and low fuel pressure can be seen to be affected by a twisted or sinuous mode. The spray angle... 

Equation 3.2 for the sinuous and varicose growth rates are shown in Figs. 3.3 and 3.4 for gas Weber numbers We = p Uo a/a of 0.5 and 5.0, respectively. Each figure also shows the results for the long wave (Equation 3.4), that tanh( ) ka and short wave (equation 3.6) approximation. Long wave approximation is similar to that of Hagerty and Shea [4]. For a Weg = 5.0, the dimensionless growth rate curves are very similar, except at low values of the dimensionless wave number ka. 

For a viscous liquid sheet, assuming to have the same pressure distribution as the inviscid liquid, the liquid velocity components can be described as u = ui + uy and vi = Vi + vy. The potential and stream functions that satisfy the continuity equations may have the following forms (p = (p y) exp ikx + (ot), =Oi(y) xp ikx + cot), and i/r = (y) exp ikx + cot). Similar analysis is completed on the gas phase. After substitution, a relation between the complex growth rate and the disturbance wave number k is obtained. Senecal et al. [20] provided the following relation for the growth rate for the sinuous mode ... 

Jazayeri and Li [41] developed up to the third order nonlinear analysis of a liquid sheet to determine the breakup length of the sheet A typical result of their solution for the surface deformation as a function of distance is shown in Fig. 3.9. This case is for the initial disturbance amplitude of 0.1, the Weber number of 40 and the gas-to-liquid density ratio of 10, which approximates the situation of liquid water in ambient air. The wave number of 0.02 is almost equal to the dominant wave number for the sinuous disturbance of the linear theory. It is seen that the surface wave grows in time, and maintains its sinuous character for the majority of its growth... 

It is seen that although the wave remains sinuous for most of the sheet length, nonlinear effects cause the sheet thinning and pinching that lead to the eventual breakup of the sheet. As observed earlier, the breakup time decreases for each initial amplitude Co until it reaches a minimum value and then approaches infinity when the wave number approaches the cut-off wave number k. ... 

Directional wavemakers are vertically segmented wavemakers that undulate sinuously and, consequently, are also called snake wavemakers. Segmented directional wavemakers may be driven either in the middle of each vertical segment or at the joint between vertical segments. Because of these two methods of wave generation, parasitic waves are formed along the wavemaker due to either the discontinuity of... 

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