ILASS Americas 2th Annual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 27 Predicting Breakup Characteristics of Liquid Jets Disturbed by Practical Piezoelectric Devices M. Rohani, D. Dunn-Rankin, and F. Jabbari Department of Mechanical and Aerospace Engineering University of California Irvine, Irvine, CA 92697 Abstract In this paper, we study the breakup characteristics of a jet of liquid. To break a capillary jet into droplets, a piezoelectric is often used to generate disturbances growing along the jet. Rayleigh s linear theory predicts that uniform droplets are produced when the jet is perturbed by a single wavenumber disturbance. However, studies limited to linear behavior are unable to predict how the jet behaves if it is subjected to a multiplefrequency input. Measurements of an actual piezoelectric s dynamics show that driven with single frequency harmonic signal, it disturbs the jet with three output sine waves: one steady state response and two very lightly damped modes corresponding to the structural resonances of the device. In this paper, we study the interaction of these output waves to estimate the range of frequencies where irregularity effects are likely to occur on the surface of the jet that might lead to nonuniform droplet formation. Control strategies designed to eliminate these unwanted dynamics can then retain uniform breakup over a wide range of input frequencies.
Introduction Spray and atomization has been an important topic of research for a long time because of its broad applications in combustion, inkjet printing, surface coating, and many others. In particular, hydrodynamic instability of capillary jets has been studied to control the resulting droplets sizes and spacings. Theories on the instability of the jets are mainly inspired by the pioneering work of Lord Rayleigh in 878 []. Rayleigh studied the capillary instability that grows temporally in an infinitely long jet. Rayleigh s work was modified in 93 by Weber, [8], who took ambient fluid and liquid viscosity effects into account. In 973, Keller et al. studied the spatial instability of a capillary jet. They introduced an instability governed by a spatially growing disturbance that is initiated at the base of the jet. This spatially oriented configuration is more applicable to our experiment compared to the temporal instability configuration which would require a jet of infinite length simultaneously excited along its entire length. They also showed that temporal and spatial theories coincide at high Weber numbers [2]. Later in 986, Leib and Goldstein argued that the Keller theory is not valid for Weber numbers below a critical value and that the jet is absolutely unstable in the low Weber number range [3, 4]. Since studies limited to linear theories are unable to explain some phenomena such as satellite droplet formation, much emphasis has been placed on studying nonlinearities. In 974, H. C. Lee used nonlinear theories to study satellite formation [5], and Pimbley et al. used spatial instability analysis to explain nonuniform droplet formation, both experimentally and theoretically [6]. Explaining satellite formation to a high degree of accuracy requires nonlinear studies. However, in this paper, the goal is to avoid nonlinear studies by determining the interval of wavenumbers in which irregularities on the surface of the jet tend to occur. Subsequently, we introduce certain control strategies to eliminate satellites in that interval. In the first section, we discuss theories that explain the disturbance growth on the surface of the jet stream. Then, we consider the breakup characteristics of a jet perturbed by a composition of disturbances. Finally, we suggest a control method to eliminate the undesirable droplets that would be formed due to nonlinear effects. Theory Disintegration of a jet of fluid is a result of the Kelvin-Helmholtz instability and capillary instability [7]. The Kelvin-Helmholtz instability results from differences in velocities of the jet and its surrounding fluid. It can be shown that the effect of the Kelvin-Helmholtz instability is absent in a jet ejected into the vacuum due to the lack of hydrodynamic interaction between the jet of fluid and its ambient [7]. In this case, only the capillary instability acts on the jet. Capillary force, which is the surface tension over the radius of the jet, increases by narrowing the stream and results in a capillary collapse. Therefore, in the presence of both an ambient and surface tension, a combination of Kelvin- Helmholtz and capillary instability leads the jet to breakup. Rayleigh considered capillary instability in an infinitely long incompressible inviscid fluid cylinder, neglecting the ambient. He showed that any disturbance with a wavelength larger than the circumference of the cylinder jet grows exponentially until it pinches into droplets. For small disturbances, the problem can be solved linearly and the solution for the radius of the jet in time and space is: r(x,t) r s = Re[ε e i(kx ωt) ], () where x is the axis along the jet, r s is the unperturbed jet stream radius, and ε is the initial amplitude of an axisymmetric disturbance applied to the stream. Rayleigh studied the disturbances that grow with time while having a sinusoidal oscillation in space. The assumption that Rayleigh made represents a temporal instability. This requires that k, the wave number, be real and ω be a complex number consisting of the disturbance frequency and the temporal growth rate as real and imaginary parts, respectively, ω = ω R + iω I. In the case of an unstable perturbation, ω I, the growth rate, is greater than zero. For practical purposes, spatial instability is taken into account by assuming a complex k (k = k R + ik I ) and real ω in Equation () [2]. The imaginary part of k brings about the disturbance growth in space. Since ω is real, the disturbance oscillates without any amplification in time. Equation (2) can be rewritten for the spatial instability as: r(x,t) r s = Re[ε e i(ωt) e x(β ik) ], (2) where k R is the wave-number and k I is assumed to be the temporal growth rate when it is large relative to the Weber number [2, 7]. To study breakup characteristics of the jet, it is of interest to understand how the growth rate varies with the disturbance wavenumber. Considering the
stabilizing effect of viscosity and the surrounding fluid, Weber derived an expression for the growth rate in terms of the wavenumber and the fluid characteristics [8]: β 2 + 3µ(k ) 2 ρ s rs 2 β σ(k ) 2 2ρ s rs 3 ( (k ) 2 ) V s 2 ρ a (k ) 2 K (k ) 2ρ s rsk 2 (k, ) (3) where µ is the stream dynamic viscosity, ρ s is the density of the liquid in the stream, k = 2πrs λ is the non-dimensional wave number of the disturbance, r s is the unperturbed jet radius, σ is the surface tension of the stream fluid, V s is the stream speed, ρ s is the ambient fluid density, and K n is the n th order Bessel function of the second kind. Fig. shows the plot of the growth rate expression derived by Weber as a function of input frequency. Growth rate (s ) 2.5 x 4 2.5.5.5.5 2 2.5 Input Frequency (Hz) x 4 Figure. Variation of growth rate as a function of input frequency. For the disturbance to be unstable, the nondimensional wavenumber has to be less than one. The peak represents the fastest growth rate that causes the shortest pinch-off time when the amplitude of the disturbance is uniform for all wavenumbers. This is the desirable range of operation for practical systems, since it is the region most likely to produce uniform droplets. Breakup Analysis As discussed earlier, linear theory can be used to investigate breakup characteristics of a jet, subjected to a single frequency disturbance. In this section, we explain how the jet behaves in the presence of a combination of distortions applied to the surface of the jet, based on actual piezoelectric forcing behavior. In general, practical oscillating devices are such that, induced with a single frequency signal, their output consists of other modes with different frequencies representing structural responses of the device. As shown in previous work [9], it is observed experimentally that for the device used in our study, there always exist two very lightly damped modes corresponding to the structural resonances of the device. A 4 th order function that represents the observed response is determined based on experimental frequency response data. Therefore, the output of the transfer function driven with a harmonic signal oscillating at driving frequency, f d, consists of three components as shown in Fig. 2. The first component, called the input mode, corresponds to the permanent response of the transfer function and oscillates with the driving frequency. Two other output components, known as the structural modes, correspond to very lightly damped resonant modes that oscillate at natural frequencies of the actuator, (6kHz and 6kHz, for the device we used, [9]). Fig. 3 shows the amplitudes of A(f d ), B(f d ), and C(f d ) of Fig. 2 as the frequency of input is swept from 4kHz to 8.5kHz. We have scaled the amplitudes of the output components by dividing them by a constant value equal to the maximum amplitude occurring at 6kHz. According to the linear theory, all of the perturbation components should behave independently of the effects of other components. Hence, we can use superposition of a total of N growing disturbances to express the radius of the jet. This operation is formulated as: r(x,t) r s = N Re[ε j e i(ωjt).e x(βj ikj) ] (4) j= Unfortunately, although linear theory agrees well with experimental results for a single frequency perturbation, it does not explain the jet behavior in the case of multiple components over the entire frequency range. The interaction of disturbance components gives rise to nonlinearities, especially close to the pinch-off, and the result is no longer predictable with linear theory. Superposition can provide a good estimation result only when there is one dominant wave which dominates the effect of other components. Having a dominant disturbance causes the jet to act as if it is disturbed by a single frequency disturbance. Without any interference from the other modes, linear theory predicts a uniform breakup for the jet. Here, we investigate the interaction of waves and describe how they compete with each other for growth. We expect to observe nonuniform droplets sizes and spacings in most of the regions where most competitions between modes occur.
Figure 2. Three harmonic components are produced at the output of the actuator induced by a single frequency input. (a) Scaled amplitude (b) Scaled amplitude (c) Scaled amplitude.6.4.2.4.6.2.4.6.8 x 4.6.4.2.4.6.2.4.6.8 x 4.6.4.2 Input mode oscillating with driving frequency First structural mode oscillating at 6Hz Second structural mode oscillating at 6Hz.4.6.2.4.6.8 x 4 Figure 3. Amplitude of the disturbance components as a function of input frequency. (a), (b), and (c) respectively show A(f d ), B(f d ), and C(f d ), which are the amplitudes of the output components shown in Fig. 2. Interaction among disturbances As mentioned earlier, the actual disturbance consists of three harmonics with initial amplitudes varying with input frequency. According to Equation (3), the growth rate is a function of the wavenumber. Therefore, for the input mode, the growth rate varies with driving frequency variation, whereas structural modes have constant growth rates as depicted in Fig. 4. Considering that growth rate has a more significant effect on breakup compared to the initial disturbance amplitude, one can conservatively say that the mode with growth rate near the peak, i.e., the second structural mode, tends to grow much faster than the disturbance at the first structural mode. Therefore, we only need to study the interaction of the second structural mode and the driving mode, except at frequencies very close to the first natural frequency, where the first structural mode has an effect on the pinch-off due to its large amplitude. It is only near this resonant frequency that the amplitude has a dominant effect on breakup. In this case, the resonant mode and the driving mode have equal frequencies, amplitudes, and growth rates. This is thus similar to disturbing the jet with a single frequency perturbation that has a very large amplitude. Hence, the pinch-off times due to both modes coincide and a uniform breakup is likely to occur at the first natural frequency. Fig. 4 shows a superimposition of the growth rate (Fig. ) over the amplitude (Fig. 3(a)), for the input mode. The purpose of this curve is to display both the amplitude and the growth rate of the input mode at each frequency. The combination of the disturbance amplitude and the growth rate produces a pinch-off or breakup time. As mentioned earlier, the strongest resonance of the device occurs around 6Hz, but there is a second structural resonance around 6Hz. The latter is in the region of very high growth rate while the former is in a region of more modest growth rate. We now increase the driving frequency up to 8kHz. The amplitude of the second structural mode grows much faster compared to the other two modes, even though its absolute value is initially much smaller than the other two. Fast growth results in
.5.5 2 2.5 Growth rate (s ).4.2.6.4.2 Input mode growth rate Input mode amplitude Input Frequency [Hz] Figure 4. Superimposition of scaled amplitude over scaled growth rate for the input mode, as a function of driving frequency. a fast pinch-off. It is possible for the short pinch-off time of the structural mode to dominate the effects of the other components. With this competition in mind, Fig. 5 displays pinch-off times versus frequency. In this figure, the input mode and both structural modes are considered independently to determine the pinch-off time of the same jet. Combining the growth rates and the amplitudes discussed in the previous section, pinch-off time at each frequency can be derived using Equation 2. Short pinch-off time of the second structural mode is clear from the figure at 7-8kHz. x 4 than that of the second structural mode. However, since the growth rate of input mode is somewhat smaller in this range (see Fig. 3), the combination of the amplitude and growth rate leads to close pinchoff times as shown in Fig. 5. As a result, competition can occur between the input mode and the second structural mode, which can lead to pronounced nonlinear effects and eventual satellite formation. At high frequencies around the peak of the growth rate dome, approximately from khz to 5kHz, we can see that the input mode has the shortest pinch-off time. In this region we would predict that the input mode is the dominant disturbance component. This can be explained by the very high growth rate of the input mode as well as its amplitude. Therefore, the combination of the amplitude and the growth rate results in a short pinch-off time for the input mode as shown in Fig. 5. The input mode dominates the effect of the structural mode and the jet can be assumed to be disturbed by a single frequency disturbance that can lead to generation of uniform droplets in the downstream region. The existence of a dominant disturbance wave allows us to use linear theory for simulating the breakup in this region of the frequency spectrum. Fig. 6 shows the simulation results of the jet breakup at the frequency corresponding to the peak of the growth rate. The jet is disturbed at 4.5kHz, which is the frequency where the growth rate has its maximum value. Substituting the growth rate, the wave number, and the frequency in Equation 2, the radius of the jet is determined along the jet, at time instance. 2.6 x 3 2.4 Pinch off times Input mode First structural mode Second structural mode Breakup simulation 2.2 First natural frequency: 6kHz Second natural frequency: 6kHz 2 Pinch off time.8.6.4.2.4.6.2.4.6.8 2 4 x Figure 5. Pinch-off time as a function of driving frequency. Each mode is considered independently. At slightly higher frequencies, but still prior to the peak of the growth rate dome (i.e., 8-kHz), the amplitude of the input mode is considerably larger Figure 6. Breakup simulation results derived from applying linear theory. Uniform circles in the picture demonstrate uniform pinch-off at the frequency where the growth rate has its maximum value. An experimental image of the jet breakup at the same frequency is shown in Fig. 7. Uniform breakup shown in this picture confirms that the linear assumptions made in the simulation are valid for this frequency range.
Figure 7. Experimental image of uniform droplets generation at 4.5kHz, the frequency of the peak of the growth rate. What happens at inputs near the second natural frequency, is similar to what was described at the first natural frequency. In this case the second structural mode has the same frequency, amplitude, and growth rate as the input mode. This results in equal pinch-off times, as shown in Fig. 5. That is, the input mode and the second structural mode coincide strongly in both pinch-off time and in the expected frequency of the resulting droplets. Going even higher in frequency and passing the second natural frequency, the growth rate of the driving mode decreases quickly and the second structural mode again becomes the dominant one. This is represented in Fig. 5 where the pinch-off time for the second resonant mode is shorter than for the input mode. Considering the initial disturbance of the piezoelectric as a function of time confirms to some extent what we concluded from the analysis of the interaction of disturbance components. The initial disturbance is the superposition of the output components of the piezoelectric device (Fig. 2) before they start growing. Studying the initial amplitude profile provides us with an approximation of how the disturbance is distributed on the surface of the jet. In Fig. 8, variation of the initial disturbances with time is demonstrated at the first natural frequency (6kHz) and at 9kHz. The beat frequency in the first subplot in Fig. 8 is 43Hz. the piezoelectric (even though the amplitude varies somewhat) suggests a uniform dispersion of pinchoffs. The frequency is uniform so that the instability growth will still pinch off the same size fluid parcel each time. This situation should be compared with Fig., which shows the initial disturbance variation at 9Hz where competition between modes was predicted based on the breakup time analysis shown in Fig. 5. The effect of a high frequency disturbance is obvious from Fig. and it is plausible that the driving mode competes with the highfrequency structural resonance mode. Initial amplitude 6 Hz.2.4.6.2.4.6.8 x 3 Figure 9. Initial disturbance amplitude variation at 6kHz. Initial amplitude 9 Hz Initial amplitude.2.4.6.2.4.6.8 x 3 Amplitude Amplitude.5..5.2.25.3.35.4.5..5.2.25.3 Figure. Initial disturbance amplitude variation at 9kHz. An experimental image from the jet stream at 9Hz, (Fig. ), clearly shows a satellite formation that can result from competing breakup modes. Figure 8. Initial disturbance amplitudes at 6Hz and 9Hz. Fig. 9 shows a close up view of the initial disturbance variation at 6Hz through the window shown in left side of the figures. Uniform motion of Figure. Experimental image of satellite formation at 9Hz.
Control Approach We are interested in avoiding nonuniform droplet formation by suppressing the extraneous disturbances. For this purpose, control methods are proposed that manipulate the piezoelectric input signal to provide us with a single wavenumber perturbation over the entire range of driving frequencies. This system can be assumed to be a droplet generator followed by a capillary stream function. Figure 3. The output of the actuator is tracked and the controller is designed to reduce the error in the closed loop system. Figure 2. Open loop block diagram of the system. Fig. 2, is an open loop model of the system. The input of the droplet generator is the voltage oscillating at the driving frequency and the output is the summation of structural and driving harmonic functions of the input frequency. One simple approach to control the droplets to breakup uniformly is to find the input needed to get a single frequency at the output of the actuator. The input can easily be calculated from an inverse Laplace transform of, I(s) = R(s) H(s), where R(s) is different at each frequency since it is the Laplace transform of the desired single frequency disturbance. H(s), the transfer function of the actuator, is experimentally determined as discussed earlier, using the procedure described in [9]. Open loop control, however, is not likely to be a good solution, since it is sensitive to parameter variation. In other words, if characteristics of the system change due to temperature or pressure variations, the results might not be valid. To have a less sensitive system to variation of parameters, a closed loop feedback control can be used. It can provide us with uniform breakup even if the model does not match exactly the real system. Droplet size, average droplet frequency, or output of the piezoelectric can be used as output feedback of the system. The closed loop system is shown in Fig. 3, schematically. This is the subject for our future activity. Summary and Conclusion According to Rayleigh s linear theory, any disturbance imposed on a jet of fluid, grows in an exponential manner if its wavelength exceeds the circumference of the jet stream. One common practical device that exerts this distortion is a piezoelectric actuator that responds to an input voltage by applying a distortion onto the surface of the jet. In this paper we show that in a practical device, typical piezoelectric actuators produce multiple wavenumber disturbances which compete with each other in some frequencies. Disturbance components competition results in nonuniform droplet formation. Analyzing the interaction of waves allows us to recognize the range of frequencies prone to satellite formation. Also, an output feedback control model is proposed whose job would be to eliminate satellites by tracking the output of the actuator and preventing disturbances away from the desired input frequency. References [] Lord Rayleigh, On the instability of jets, Proc. London Math. Soc., :4-3, 878. [2] J. S. Keller, S. I. Rubinow, and Y. O. Tu, Spatial instability of a jet, Phys. Fluids, 6:252-255, 973. [3] S. J. Leib and M. E. Goldstein, The generation of capillary instabilities on a liquid jets, J. Fluid Mech., 68:479-5, 985. [4] S. J. Leib and M. E. Goldstein, Convective and absolute instability of a viscous liquid jet, Phys. Fluids, 29:952-954, 986. [5] H. C. Lee Drop formation in a liquid jet, IBM J. Res. Develop., 8:364-369, 974. [6] W. T. Pimbley and H. C. Lee, Satellite droplet formation in a liquid jet, IBM J. Res. Develop., 2:2-3, 977. [7] T. Funada, D. D. Joseph, and S. Yamashita, Stability of a liquid jet into incompressible gases and liquids, Int. J. Multiphase Flow, 3:279-3, 24. [8] C. Weber, Zum Zerfall eines Flüssigkeitsstrahles, Ztschr. f. angew. Math und Mech, :36-54, 93. [9] D. K. Iobbi, Controlling piezoelectric generated droplets, Master thesis, University of california, Irvine,, 24.