On the nonlinear stability of a swirling liquid jet

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Abstract

The nonlinear deformation and atomization of a rotating column is considered using an axisymmetric boundary element formulation. Swirl has been considered by superposing a potential vortex to the bulk flow of the jet. The resulting model has been shown to reproduce the classical linear result due to Ponstein and parametric studies are conducted in the nonlinear regime to determine wave shapes and droplet sizes. As with prior nonlinear column breakup studies, results indicate that satellite drops are formed from the main wave under virtually all conditions. The ratio of the main drop to satellite drop diameter is shown to be remarkably constant over a variety of wave numbers/column lengths thereby providing a potential approach to produce tightly controlled bimodal sprays.

Introduction

The instability of a jet is of significant interest in both the academic community and within industries due to the wealth of practical industrial applications including: ink-jet printing, agricultural sprays, IC-engines, and numerous manufacturing and painting processes. If drop size is to be predicted using the instability theory, the predicted value can be of great value as an initial condition for a spray simulation and as validation data when designing a spray nozzle for a specific application.

The origins of the instability theory can date back to the late eighteenth century by the Lord Rayleigh (1878). The Rayleigh’s inviscid incompressible column jet analysis is probably the most classical and most frequently cited linear instability theory in the relevant literature. Weber (1931) extended Rayleigh’s analysis by adding the effect of viscosity of the jet. It is note worthy that both Rayleigh, 1878, Weber, 1931 considered the limiting case of ka  0 (where k is the wavenumber and a is the column jet radius), which indicates that the both analyses are applicable for the relatively large wavelength only; the wavelength is greater or comparable to the jet radius. The linear theories were later revisited by Levich, 1962, Sterling and Sleicher, 1975, Reitz and Bracco, 1982. Their linear theories are applicable for the viscous jet and for a wide range of wavenumber. However, all the linear theories are limited to the case of infinitesimal deformation and therefore cannot strictly be used to predict droplet sizes.

It is now well known that the nonlinearity in the surface deformation process produces several satellite drops as multiple crests are formed over a given wave on the free surface. In this case, the linear theory cannot predict the satellite drop formation. Yuen (1968) extended the Rayleigh’s linear theory to the nonlinear case, where Yuen derived the lengthy 2nd and the 3rd order terms for the dispersion equation. Yuen’s theory was later validated with the experimental data of Rutland and Jameson (1970). However, it was later pointed out, by Nayfeh (1970), that Yuen’s 3rd order terms was incorrect for the wavenumbers close to the cutoff wavenumber. Lafrance (1975) provided the correct 3rd order term in analytic form, which was free from the secular terms contained in the Yuen’s analysis. More recently, a nonlinear unsteady calculation was achieved numerically using the boundary element method (BEM) for the column jet by Hilbing and Heister (1996). They validated their predicted satellite drop size with the experimental data of Rutland and Jameson and then showed the controlling mechanism for drop size. Surely, both theoretical and numerical analyses for both linear and nonlinear cases are well archived for a column jet in the literature. However, the instability study on the effect of swirl (or rotation) on a column jet has been studied less.

It appears that Ponstein (1959) is the first author who considered the effect of swirl on the stability of a classical liquid jet/column. The Ponstein’s analysis was so complete and original that Ponstein’s dispersion equation is capable of recovering the dispersion equations of the aforementioned authors (Rayleigh, 1878, Weber, 1931, Levich, 1962, Reitz and Bracco, 1982) for their specific cases when considering the non-swirling case. The Ponstein’s equation can also recover the Kelvin–Helmoltz and the Taylor’s equation for the limiting case of ka  ∞, where the relevant wavelengths are much smaller than the jet diameter. Ibrahim (1993) solved one dimensional unsteady Navier–Stokes equations applied for the swirling jet and showed his numerical solutions were in agreement with the Ponstein’s linear theory. Ibrahim later included the convective term and presented the nonlinear results. The nonlinear result indicated that the growth rate was a bit smaller than that of the linear result. No shift in the wavenumber of the maximum growth rate was clearly observed and, thus, the information on the reduction of the main drop size due to the mass loss to the satellite drops could not be obtained.

It is our objective to extend the linear theory and prior nonlinear results in order to identify the presence of, and size of, any satellite droplets formed during the nonlinear portion of the jet deformation. The presence of satellite droplets has long been noted in the Rayleigh jet (Lundgren and Mansour, 1988, Hilbing and Heister, 1996), and today nonlinear simulations can reproduce the measured droplet sizes quite accurately. While electrostatics (Setiawan and Heister, 1997) and jet excitation (Orme and Muntz, 1990) have been used in the past to control droplet sizes in the Rayleigh regime, the use of swirl has not been studied in any detail. Here we employ an axisymmetric boundary element method (BEM) by superposing a potential vortex with the bulk mean flow. The computations are validated against Ponstein’s linear theory and parametric studies are reported for the nonlinear case.

Section snippets

Model development

Since the early 1990s, boundary element method (BEM) solutions have appeared in the literature for atomization problems of various types. Solutions for the classical liquid jet (Lundgren and Mansour, 1988), the finite-length liquid jet (Hilbing and Heister, 1996), electrostatic jets (Setiawan and Heister, 1997), and jets in wind-induced regimes (Spanger et al., 1995) have provided nonlinear companions to classical linear results dating back to Rayleigh’s classic work in the late 1800s. In case

Results and discussion

Fig. 1 shows the computational domain for the rotating infinite liquid jet in this simulation. A constant grid spacing of ds = 0.005 along the liquid surface was employed for this calculation. Thus total number of grid points is dependent on the investigated wave length; typical grids employed vary from 100 to 400 nodes in the simulations. The calculated result was compared against the linear result given by Eq. (10) by running a series of calculations at different k and Ro values. Assuming a

Conclusions

A fully nonlinear model has been developed for simulating the swirling jet. An axisymmetric boundary element formulation has been utilized wherein a potential vortex is superposed to the bulk flow to simulate the swirl in the jet/column. The corresponding form for the nonlinear Bernoulli equation free surface boundary condition has been developed.

A linear instability analysis due to Ponstein has been used to validate the newly developed model. The difference between computed growth rates and

Acknowledgements

The authors greatly acknowledge the support of the Air Force Office of Scientific Research (Grant No. F49620-03-1-0025) with program manager Dr. Mitat Birkan. The second author wishes to acknowledge the partial support of this research by Carbon Dioxide Reduction & Sequestration R&D Center and by New Faculty Research Grant, funded by Korea University.

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