Analysis of Disintegration of Planar Liquid Sheet Sandwiched between Gas Streams with Unequal Velocities and Resulting Spray Formation

A temporal stability analysis was carried out for a planar liquid sheet sandwiched between two gas streams moving with unequal velocities. Perturbation analysis was carried out up to second order using the initial amplitude of disturbance of the liquid sheet as the perturbation parameter. Both para-sinuous and para-varicose modes were investigated to determine the dominant break-up mechanism. The nonlinear stability analysis gives the ligament area and the breakup time corresponding to the most unstable wave numbers .Gaster transformation is used to obtain the breakup length from the breakup time. Secondary breakup of the ligament into droplets was modeled following Rayleigh mechanism. Droplet size and velocity distributions were predicted using Maximum Entropy Formulation (MEF). The results show that the asymmetry in the gas velocity significantly affects both the first order and the second order results. The range of unstable wavelengths, the maximum growth rate, the breakup length and the mean droplet diameters were found to be determined through a complex interaction of the gas phase velocities and their differences. It is found that increase in difference the gas velocities on two sides of the liquid sheet leads to a significantly narrower droplet size distribution though the velocity distribution is less affected. Introduction Disintegration of liquid sheets has been investigated for the last several decades owing to its importance in spray formation. Excellent reviews on the subject have been provided by Lin and Reitz [1], Lasheras and Hopfinger [2], Sirignano and Mehring [3] and Lin [4]. Although a majority of the studies are limited to linear stability analysis (e.g., [5, 6, 7, 8]), linear analysis cannot provide information on parameters like breakup length. Consequently, in the last few years, a number of researchers have addressed the nonlinear stability problem. Clark and Dombrowski [9] investigated the instability of a planar liquid sheet in a quiescent gas. They used regular perturbation analysis and retained terms up to the second order. Jazayeri and Li [10] extended the work of Clark and Dombrowski [9] by including terms up to third order. They showed that the inclusion of the third order term can significantly alter the sheet profile at breakup. Unlike these works which considered breakup of liquid sheets in stationary gas, Mitra [11] examined the disintegration of planar liquid sheets in a moving gaseous medium. In addition, he considered both sinuous and varicose modes of perturbation unlike the earlier works, which considred only the sinuous mode. Very recently, Nath et al. [12] analysed the breakup of planar liquid sheets in moving gaseous medium, considering gas phase velocity both higher and lower than the liquid sheet velocity. Their analysis identified certain shortcomings in the formulation of both Clark and Dombrowski [9] and Jazayeri and Li [10]. Their prediction of breakup length using second order analysis closely matched that of the third order results of Jazayeri and Li [10] and Mitra [11] in spite of significant differences in the breakup pattern. Their result also showed that within the range of parameters normally encountered in atomization, both sinuous and varicose modes can dominate. In a follow-up work, Nath et al. [13] combined the nonlinear stability analysis with a maximum entropy based model for predicting droplet size and velocity distribution. This approach enables determination of the spray characteristics in terms of atomizer parameters. All the above works considered equal velocities of the gas layers on either side of the liquid sheet. Very often annular liquid sheets along with high speed inner gas streams are injected into a nearly quiescent gas pool. This configuration is markedly different from that of the studies reported above where the gas stream velocities were equal on each side. Gretzinger and Marshall [14] present photographic images of the breakup of a flat liquid sheet, exposed to high velocity gas on one side and quiescent gas on the other side. It was reported that asymmerty in the gas velocities stabilize the spray on the lower velocity side. Consequently, larger droplets are generally formed on that side. A temporal instability analysis of an inviscid liquid sheet sandwiched between two gas streams of unequal velocity was first studied by Li [15]. He investigated that for unequal gas velocities, * Corresponding author: [email protected] 12 ICLASS 2012 Spray Formation by Disintegration of Planar Liquid Sheets with Unequal Gas Velocities 2 two independent modes of instabilities exist, which closely resemble the sinuous and varicose modes observed in the case of equal gas velocities. These modes were termed para-sinuous (when the phase difference between two interfaces is close to zero) and para-varicose (when the phase difference between two interfaces is close to π). According to his linear stability analysis, para-sinuous mode was found to be more unstable than para-varicose mode at large Weber numbers and at lower Weber numbers; the para-varicose mode dominates the breakup process. It was reported that surface tension always has a stabilizing effect whereas viscosity has dual effect depending on the conditions. By analyzing the disturbance energy equation he demonstrated that interfacial pressure fluctuations are responsible for instability of the liquid sheet. He also investigated that for Weber number much larger than unity the growth rate curve of para-sinuous mode has only one local maximum, but for Weber number slightly larger and less than unity there exists two maximum values in the growth rate curve. Among these two, growth rate at the longer wavelength is independent of the viscosity, whereas growth rate at the shorter wave length depends on the viscosity of the liquid. But for the para-varicose mode, there is only one maximum at all Weber numbers. Tharakan and Ramamurthi [16] investigated the breakup of a thin viscous liquid sheet coflowing with the gas streams with different relative velocity and in presence of both longitudinal and lateral waves over liquid-gas interfaces. They reported that for larger Weber mumber and in presence of lateral wave, para-sinuous made have larger growth rate. But the influence of lateral wave on growth rate in not significant when Weber number is more than 10. Witherspoon and Parthasarathy [17] also investigated the breakup of a viscous liquid sheet co flowing with gas phase in both side having both equal and unequal velocity. They used a linear spatial stability analysis to determine the most unstable wave number corresponding to maximum growth rate. The above studies were all limited to linear analysis. Only few nonlinear analyses of liquid film subjected to unequal gas velocities have been reported in the literature. Ibrahim and Jog [18] carried out a nonlinear stability analysis for annular liquid fuel sheet subjected to unequal inner and outer gas velocities by a perturbation expansion technique with the initial amplitude of the disturbance as the perturbation parameter. However, they investigated mainly configurations with one stream quiescent. Their studies with both gas streams moving were mostly limited to equal gas velocities. Although in most practical injectors, the fuel emerges as an annular liquid sheet, a planar sheet can be considered as an idealization of the annular liquid sheet owing to the high ratio of liquid sheet radius to its thickness. The present work considers nonlinear temporal stability analysis of planar liquid sheet sandwiched between two gas streams moving with unequal velocities. The instability analysis predicting the breakup of the liquid sheet is coupled with a stochastic model based on maximum entropy formulation for predicting droplet size and velocity distribution. Mathematical Formulation