THE LONGITUDINAL INSTABILITY IN ANAIR-BLASTED LIQUID SHEE T

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1 ILASS-Europe'99 Toulouse 5-7 July 1999 THE LONGITUDINAL INSTABILITY IN ANAIR-BLASTED LIQUID SHEE T Antonio Lozano*, Félix Barreras, César Dopazo * o * LITEC/CSIC, Maria de Luna, 3, Zaragoza, Spai n Centro Politécnico Superior de Ingenieros, Area de Mecánica de Fluido s Universidad de Zaragoza, Maria de Luna, 3, Zaragoza, Spain TeL Fax: alozano@ideafix.litec.csic.es ABSTRACT Continuation of earlier studies on the sinusoidal longitudinal waves that are formed in an air-blasted liqui d sheet has provided new data that contribute to the consolidation of some hypothesis that had not been prove d experimentally before. Oscillation frequency measurements obtained by laser diffractometry have bee n analyzed as a function of the momentum flux ratio, obtaining a satisfactory collapse of the experimental curves. Wavelengths and wave growth rates have been measured using planar laser-induced fluorescence to visualiz e the liquid sheet and the surrounding air streams. These results are compared with those predicted by a simplified linear stability analysis. Some observed trends correspond to the predictions. INTRODUCTION Over the last decade, studies on large aspect ratio liquid sheets have experienced a renewed interest, almos t becoming a canonical flow to study primary atomization. The first reports on large aspect ratio liquid sheet s date back to the fifties [1], but with the liquid flow exiting in quiescent ambient air. The addition of high speed air coflows is described much more recently [2-6] and its effects on the atomization have been demonstrated t o be decisive. Most of the initial studies, based on visual observations, were phenomenological with a main interest in predicting the droplet cloud characteristics. Recent works, however, have emphasized the analysis of the near field region, close to the exit slit, to explain the development and growth of the instabilities tha t ultimately cause the liquid sheet break-up [7-10]. Simultaneous to the experimental work, efforts have been dedicated to analyze the problem theoretically an d nurnerically. Modeling the liquid sheet atomization is still a research in progress. A majority of the studie s have been 2-D, based on linear instability analysis, and in analogous way to the development of the experimental research, quite often the situation described is a liquid sheet moving in an atmosphere at rest [11-13]. In general flowing air has only been considered when treating a single mixing layer of a liquid film flowing over a solid wall [14,15]. Rangel and Sirignano [16], Yang [17] and Ibrahim [18] have recently analyzed the sheet problem in the context of an air-blasted configuration, with the air exit velocity higher tha n that of the liquid. Longitudinal and transverse waves are involved in the atomization process. To provide a complete explanation, the three-dimensional character cannot be ignored, and it is required, for example to correctly predict the mean droplet size produced by primary break-up. If a simplified model is to be developed to describe only the longitudinal oscillations, some reasonable predictions can be obtained from 2-D linea r stability analysis. Using this method, results have been obtained in conditions similar to those in th e experiments, so that they can be compared with the measurements. EXPERIMENTAL FACILITY The experimental facility where the present study has been developed is similar in its design to the ones in Mansour and Chigier [6] and Lozano et al.[7]. The main difference is that in the present case both water and air nozzle profiles have been contoured to provide at the exit parallel air and water uniform flows. The set up is described in detail in Barreras [9] and Lozano et al. [10]. Water injected at the top of the nozzle head exits vertically through a 0.35 mm wide slit. The span of the sheet is 80 mm, yielding an aspect ratio of 230. Air is

also introduced from the top following a settling chamber with two honeycombs and a wire mesh screen t o smooth the flow. The exit widths of the air channels is 3.45 mm.

2 also introduced from the top following a settling chamber with two honeycombs and a wire mesh screen t o smooth the flow. The exit widths of the air channels is 3.45 mm. The simplicity introduced by the parallel exi t velocity profiles (as opposed to air/water impingement at an angle), is an advantage in order to compare th e measurements with numerical simulations and to identify basic break up mechanisms. For the conditions under study, water velocities have ranged from 0.6 to 6 m/s, while air velocities have varied between 15 and 75 m/s. To measure the sheet oscillation frequency, a laser diffraction technique has been used, detecting the passage o f the oscillating liquid sheet by the obscuration of a diode laser beam shining onto a photodiode. This procedure is also described in previous papers [6,7,10]. To visualize the sheet wavelength, Planar Laser-Induce d Fluorescence has been used, seeding the liquid with Sulforhodamine B (Kiton Red), and illuminating the flo w with a 532 nm, 2 cm high laser plane originated at a double cavity Quantel YG781C-10 pulsed Nd:YAG laser. The fluorescence peak of this dye is located at 575 nm, which allows for an efficient discrimination betwee n the emitted signal and scattering of the excitation light, for example from the liquid droplets. To image the fluorescence emission, a Princeton Instruments slow scan CCD camera has been used, with a 50 mm F1.2 Nikon lens. It should be noted that the short lifetime of the fluorescence enables high temporal resolutio n imaging without the need of gated cameras. Images have been acquired for longitudinal sections slicing th e water sheet perpendicularly through the middle of the exit slit. Data sets have been recorded for two different fields of view, 26x35 mm with a resolution of 90 µm/pixel and 13x 17.5 mm with a resolution of 45.tm/pixel. To avoid imaging through the sheet edge, thickened by the surface tension, the camera was not locate d perpendicular to the liquid sheet, but with an angle of 25.Results have been corrected for this image artifact. To measure the oscillation amplitude growth ratio, mean images have been obtained averaging on-chip on e thousand single shot images. To avoid saturation, the lens aperture was reduced to f#16, as well as the lase r energy. To determine the interface location the images were thresholded and binarized. Once determined th e coordinates of the interface, the points were plotted and fitted to an exponential curve of the form 11 = rl oek'x, where k,. is the growth factor of a sinusoidal wave described by '11 =,goe tk, +rk; tx+wr Air Velocity (m/s) o _p / 0.01 b Ay Ñ o 7 5 LL Water Velocity (m/s) MFR Fig.1 a) Frequency measurements as a function of water exit velocity for different values of the air velocity. b ) Same plot for nondimensional variables as explained in the text. RESULTS The frequency measurements obtained in this work are presented in Fig. 1. They are in very good agreemen t with those previously reported in [7] and [9]. It is confirmed that the oscillation frequency depends linearly o n air exit velocity and scales with the sheet thickness. If the air velocity is kept constant, for a certain range of water velocities, the sheet oscillates in a dominant sinusoidal mode and under this situation the frequency vs. water velocity plots can be fitted to parabolas presenting a point of maximum value. It is interesting to observe that if the frequency measurements are plotted against the Momentum Flux Ratio (MFR) defined as MFR = (PaU 2)/(P1Ur2)

3 where the subscripta a and l stand for air and liquid respectively, all the frequency maxima in the curves fo r different air velocities align for a fixed 0.5 value. The frequency axis can also be collapsed with the followin g nondimensional definition f*= Ua which corresponds to a Strouhal number, where d is the sheet thickness, and U, corrects for the fact that th e linear relation between frequency and air velocity doesn't pass trough the origin. Figure 1 b) displays th e nondimensional plot. Wavelengths have been measured directly from the fluorescence images. There is some ambiguity in defining a perturbation wavelength because as the wave propagates with the accelerating water sheet, its velocity increases while moving downstream. As the frequency remains constant, the wavelength grows according t o the fundamentalf 2k, = c relation, wheref is the oscillation frequency, ñ is the wavelength and c is the phas e velocity. This is aggravated by the fact that the sinusoidal waves become highly distorted prior to break-up, which takes place after only one or two wavelengths. Having this in mind, and accepting some errors in th e wavelength measurements, the general behavior is summarized in Fig. 4, where the wavelength in mm is plotted vs. water velocity, for different air velocities. The sudden increase in wavelengths for increasing wate r velocity values appears to coincide with the transition from a regime with dominant sinusoidal oscillation mod e to another one with a mixture of sinusoidal and dilatational waves. Figure 4 shows also the wavelength values that are calculated dividing the propagation velocity by the frequency measurements. As this velocity has not been measured, the water exit velocity has been used. The good agreement for the range of the obtained result s indicates that the wave appears to propagate with the local (and accelerating) water velocity. This would als o explain, at least partially, why the calculated values are lower than the measured ones. f d -U, Air Velocity (m/s) -* 1 5 E s d o --e 25 *- 3 0 * 40 e e Water Velocity (m/s ) Fig. 2. Wavelength as a function of water velocity for different air velocities From averaged images, the spatial growth rate has been calculated as explained in the previous section. Thi s rate determines how the wave amplitude increases while moving downstream. Fig. 3 is an example of the binarized images, and of the exponential fits to the location of the air/water interface. It can be seen how initially the wave growth adjusts accurately to the exponential curve, as predicted by linear stability theory. As the wave evolves, the growth departs from that predicted by linear theory. It is no longer exponential, and the wave profile deforms from a sinusoidal shape. LINEAR STABILITY ANALYSIS

4 As it is well known, linear perturbation analysis is based on perturbing a steady state solution of the flow with a small amplitude wave in normal modes, analyzing its growth rate with the linearized Navier-Stokes equations E 0*--*, i Downstream Distance (mm ) Fig. 3. Left: average image of 1,000 single-shot images of the longitudinal section of the sheet, for air and water velocities of 15 m/s and 0.6 m/s respectively. Right: location of the right hand air/water interface of the average image on the left, fitted to an exponential function. The steady state solution is usually assumed, according to certain simplifications, e.g. assuming quiescent air, neglecting its shearing effects (zero viscosity), constant water velocity profile, etc. In the present case, a very simple steady state basic flow has been considered, in which both air and water have non-zero velocities, bu t both have been supposed to be inviscid. In this situation the velocity profiles are flat with a constant value. When analyzing the perturbations, however, viscosity has been included in both fluids. The instability growth has been analyzed temporally in an infinite sheet, i.e. the angular frequency term in the normal modes, co has been assumed to be complex, while the wavenumber k has been forced to be real. The stability analysis provides a dispersion relation that, for fixed velocity conditions, gives the temporal growt h rate for a perturbation of wavenumber k. Such relations are plotted in Fig. 4 for a liquid Re number of 850 an d different Ug values, where Ug is defmed as the ratio Ua/Ul. Re i = k R (non-dim. ) Fig. 4. Temporal non-dimensional wave growth rate as a function of the non-dimensional wavenumber for a fixed water Re number and different velocity ratios Ug = U* /U1 If the wavenumber for which the growth ratio is maximum is stored for the different conditions of air an d water velocities, and then it is transformed into a frequency according to the relation ao/k = c, where c is the

5 wave phase velocity and the medium is assumed to be non-dispersive, a piot similar to the one presented in Fig. 1 for the experimental measurements can be generated. In accordance with the experimental results, the wate r exit velocity has been taken as the wave speed. The result is shown in Fig. 5, where the non-dimensional variables have been transformed into dimensional ones to be compared with the measurements. It is interesting to observe the quadratic dependence of the frequency on water velocity, with the parabola maxima displacing t o higher values as the air velocity is increased. The behavior is qualitatively similar to the experimental curves but the predicted frequency values are higher than the measured ones. u a (m/s) U I (m/s) Fig. 5. Predicted sheet oscillation frequency values for maximum growth rate as a function of water velocit y and different air exit velocities. The dotted line joins the parabola maxima. The predicted growth rate can also be compared with the values obtained from the imagen. Again, the temporal values has to be transformed into spatial ones by means of the wave propagation velocity. Predicted an d measured results are presented in Fig. 6 for a fixed air/water velocity ratio of 7.5. The agreement is reasonable. U g =7. 5 k (Iin.stab.) k 1 (exper.) U I (m/s) Fig. 6. Comparison of the predicted and measured spatial growth rates for a fixed air/water velocity ratio. CONCLUSIONS Some measurements have been obtained in an air-blasted liquid sheet to investigate the longitudinal instabilities. It has been observed how the maxima of the oscillation frequency values when varying the wate r exit velocity while maintaining the air velocity fixed align if they are plotted as a function of the momentu m flux ratio MFR. Wavelengths and growth ratios have also been measured. From them it can be deduced that th e waves propagate with the local water velocity. Results predicted by temporal linear stability analysis have a

6 reasonable agreement, at least qualitatively, with the measurements, even when using a very simplisti c approximation of the steady state flow. ACKNOWLEDGMENT S This work has been partially funded by the Dirección General de Estudios Superiores of the Spanis h Government under proposal PB O3-03. REFERENCE S [1] Hagerty, W.W., and Shea, J.F., A study ofthe stability ofplane fluid sheets, J. Appl. Mech. December, , [2] Rizk, N.K., and Lefebvre, A.H., The Influence ofliquid Film Thickness on Airblast Atomization, J. Eng. Power 102, , [3] Arai, T. and Hashimoto, H., Disintegration of a thin liquid sheet in a concurrent gas stream. Proc. ICLASS-85, London, [4] Mansour, A. and Chigier, N., Disintegration ofliquid sheets. Phys. Fluids A, 2,(5), , [5] Stapper, B. E., and Samuelsen, G. S., An experimental study of the breakup of a two-dimensional liquid sheet in the presence of co flow air shear. AIAA paper # , [6] Mansour, A. and Chigier, N., (1991), Dynamic Behavior of Liquid Sheets. Phys. Fluids A, vol. 3,(12), [7] Lozano, A., Call, C.J., Dopazo, C. García-Olivares, A., (1996), An Experimental and Numerical Study of the Atomization of a Planar Liquid Sheet, Atomization and Sprays, vol. 6, [8] Vich, G., Dumouchel, C., Ledoux, M., (1996), Mechanisms of Disintegration of Flat Liquid Sheets, 12th Ann. Conf. ILASS-Europe, Lund, Sweden, June [9] Barreras, F., (1998), Experimental Study of the Break-up and Atomization of a Liquid Sheet, Ph. D. Dissertation University of Zaragoza (in Spanish). [10] Lozano, A :, Barreras, F., Dopazo, C., "Experimental Study ofthe Near Field ofan Air-Blaste d Liquid Sheet", submitted to J. Fluid Mech. [11] Squire, H.B., Investigation of the instability of a moving liquid film, Brit. J. of Appl. Phys., 4, 167, [12] Li, X., Tankin, R.S., On the temporal instability ofa two-dimensional viscous liquid sheet, J. Fluid Mech. 226, , [13] Ibrahim, E.A., Spatial Instability of a Viscous Liquid Sheet, AIAA paper , 1994 [14] Yih, C., Wave formation on a liquid layer for de-icing airplane wings, J. Fluid Mech., 212, 41-53, 1990 [15] Miesen R., Boersma B.J., Hydrodynamic stability of a sheared liquid film, J. Fluid Mech. 301, , [16] Rangel, R.H., Sirignano, W.A., The linear and non-linear shear instability of a fluid sheet, Phys. Fluids A, 3, (10), , [17] Yang, H.Q., Interfacial Instability Between a Liquid Film and the Surrounding Compressible Gas, AIAA paper , [18] Ibrahim, E.A., Effectsof compressibility on the instability of liquid sheets, AIAA paper , 1995.

Wavelet analysis of an unsteady air-blasted liquid sheet. University of Bergamo, Faculty of Engineering, viale Marconi 5, Dalmine, Italy

Wavelet analysis of an unsteady air-blasted liquid sheet. University of Bergamo, Faculty of Engineering, viale Marconi 5, Dalmine, Italy , 23rd Annual Conference on Liquid Atomization and Spray Systems, Brno, Czech Republic, September 2010 E. Calvo*, A. Lozano*, O. Nicolis, M. Marengo * LITEC, CSIC-Univ. Zaragoza, María de Luna, 10, 50018