DROP AND SPRAY FORMATION FROM A LIQUID JET

Transcription

1 Annu. Rev. Fluid Mech : Copyright 1998 by Annual Review., Inc. All rights reserved DROP AND SPRAY FORMATION FROM A LIQUID JET S. P. Lin Mechanical and Aeronautical Engineering Department, Clarkson University, Potsdam, New York 13699; gw02@ sun.soe.clarkson.edu R. D. Reitz Mechanical Engineering Department, University of Wisconsin, Madison, Wisconsin, 53706; reitz@engr.wisc.edu KEY WORDS: jet instability, breakup regimes, atomization, sprays, surface tension ABSTRACT A liquid jet emanating from a nozzle into an ambient gas is inherently unstable. It may break up into drops of diameters comparable to the jet diameter or into droplets of diameters several orders of magnitude smaller. The sizes of the drops formed from a liquid jet without external control are in general not uniform. The sizes as well as the size distribution depend on the range of flow parameters in which the jet is produced. The jet breakup exhibits different characteristics in different regimes of the relevant flow parameters because of the different physical mechanisms involved. Some recent works based on linear stability theories aimed at the delineation of the different regimes and elucidation of the associated physical mechanisms are reviewed, with the intention of presenting current scientific knowledge on the subject. The unresolved scientific issues are pointed out. 1. INTRODUCTION The breakup of a liquid jet emanating into another fluid has been quantitatively tudied for more than a century. Plateau (1873) observed that the surface energy f a uniform circular cylindrical jet is not the minimum attainable for a given jet volume. He argued that the jet tends to break into segments of equal length, each of which is 2][ times longer than the jet radius, such that the spherical drops formed from these segments give the minimum surface energy if a drop is formed from each segment. Rayleigh (l879a, b) showed that the jet breakup /98/ $

2 86 UN & REITZ is the consequence of hydrodynamic instability. Neglecting the ambient fluid, the viscosity of the jet liquid, and gravity, he demonstrated that a circular cylindrical liquid jet is unstable with respect to disturbances of wavelengths larger than the jet circumference. Among all unstable disturbances, the jet is most susceptible to disturbances with wavelengths 143.7% of its circumference. Rayleigh also considered the cases of a viscous jet in an inviscid gas (I 892a) and an inviscid gas jet in an inviscid liquid (1892b). He showed that if the mass of the gas is neglected, the most amplified disturbance in the first case possesses an infinitely long wave length, and that for the second case it is 206.5% of the jet circumference. Tomotika (1935) showed that an optimal ratio of viscosities of the jet and the ambient fluid exists for which a disturbance of finite wavelength attains the maximum growth rate. Chandrasekhar (1961) took into account the liquid viscosity and the liquid density, which was neglected by Rayleigh, and showed mathematically that the viscosity tends to reduce the breakup rate and increase the drop size. He also showed that the physical mechanism of the breakup of a viscous liquid jet in a vacuum is capillary pinching. The theoretical results of Rayleigh and Chandrasekhar appear to be in agreement with the experiments of Donnelly & Glaberson (1966) and Goedde & Yuen (1970). Weber (1931) considered the effects of the liquid viscosity as well as the density of the ambient fluid. His theoretical prediction did not agree well with experimental data, as pointed out by Sterling & Sieicher (1975), who improved Weber's theory with partial success. Taylor (1962) showed that the density of the ambient gas has a profound effect on the form of the jet breakup. For a sufficiently large gas inertia force (which is proportional to the gas density) relative to the surface tension force per unit of interfacial area, the jet may generate at the liquid-gas interface droplets with diameters much smaller than its own diameter. This Taylor mode of jet breakup is the so-called "atomization" that leads to fine spray formation. The number of publications following the above pioneering works is indeed very large owing to the increasingly wide applications of the jet breakup processes. There have been several review articles in this area (e.g. Sirignano 1993). The latest ones are by Chigier & Reitz (1996) and Lin (1996). In this review, we focus on the physical mechanisms that cause the onset of the jet breakup at the liquid-gas interface. The nonlinear evolution after the onset of jet breakup is not considered. The physical mechanism of breakup frequently remains the same during the nonlinear evolution, although the nonlinear theory may produce additional quantitative results. For example, the satellite droplets formed from the ligament between two main drops are not predicted by linear theories, but the mechanism of the satellite formation remains capillary pinching of the Rayleigh mode.

3 JETBREAKUP 87 In addition, previous works on the breakup processes of a liquid jet in another liquid (which are relevant to the emulsification process) are not reviewed. Nor do we review works that focus on the application of the jet breakup process, although they are interesting and are of considerable importance in technology and science. Even within the area of the fluid dynamics of the onset of liquid et breakup in a gas, many important works are not commented on explicitly. Most of these important works are cited in the papers discussed in this chapter. In Section 2, works on the delineation of different regimes of jet breakup with correlations in terms of relevant flow parameters are reviewed. These correlations are based mainly on temporal linear stability theory. Section 2 provides a framework for the discussion in Section 3 of the physical mechanisms at work in the different regimes. The elucidations of the physical mechanisms are based on absolute and convective instability theories. A general critical discussion of the recent work in the field is given in Section 4. Section 5 summarizes the scientific issues that remain to be investigated. 2. BREAKUP REGIMES The breakup of a liquid jet injected through a circular nozzle hole into a stagnant gas has been studied most frequently. Previous studies have established that the spray properties are influenced by an unusually large number of parameters, including nozzle internal flow effects resulting from cavitation, the jet velocity rofile and turbulence at the nozzle exit, and the physical and thermodynamic tates of both liquid and gas (e.g. Wu et al 1992, Eroglu et a11991, and Reitz Bracco 1979). The precise mechanisms of breakup are still being researched e.g. Lin 1996, Chigier & Reitz 1996). However, linear stability theory can provide qualitative descriptions of breakup phenomena and predict the existence of various breakup regimes. It is noteworthy that the influence of nozzle internal flow effects is included only empirically in most jet breakup theories. These effects are known to be important, particularly for high-speed jet breakup. Jet breakup phenomena have been divided into regimes that reflect differences in the appearance of jets as the operating conditions are changed. The regimes are due to the action of dominant forces on thejet, leading to its breakup, and it is important that these forces be identified in order to explain the breakup mechanism in each regime (Reitz & Bracco 1986). The case of a round liquid jet injected into a stagnant gas is shown in Figure 1. Four main breakup regimes have been identified that correspond to different combinations of liquid inertia, surface tension, and aerodynamic rorces acting on the jet. These have been named the Rayleigh regime, the first wind-induced regime, the second wind-induced regime, and the atomization regime (Figure 1) (Reitz & Bracco 1986).

4 88 UN & REITZ I"..~~,.I;t ( bj (c) (d) (0 ) Figure J (a) Rayleigh breakup. Drop diameters larger than the jet diameter. Breakup occurs many nozzle diameters downstream of nozzle. (b) First wind-induced regime. Drops with diameters of the order of jet diameter. Breakup occurs many nozzle diameters downstream of nozzle. (c) Second wind-induced regime. Drop sizes smaller than the jet diameter. Breakup starts some distance downstream of nozzle. (d) Atomization regime. Drop sizes much smaller than the jet diameter. Breakup starts at nozzle exit. At low jet velocities, the growth of long-wavelength, small-amplitude disturbances on the liquid surface promoted by the interaction between the liquid and ambient gas is believed to initiate the liquid breakup process. The existence of these waves is clearly demonstrated in Figure la and b. For high-speed liquid jets, the breakup is thought to result from the unstable growth of shortwavelength waves (Figure Ie and d) (Reitz & Bracco 1982). The breakup drop sizes are on the order of the jet diameter in the Rayleigh and first wind-induced

5 JET BREAKUP 89 L + T ia JIL 1 i ' la L~ jo. o 000 vcfj,t\) ~ eto o A o u Figure 2 Schematic diagram of the jet breakup length curve. breakup regimes. The drop sizes are very much less than the jet diameter in the second wind-induced and atomization regimes. A convenient method for categorizing jet breakup regimes is to consider the length of the coherent portion of the liquid jet or its unbroken length, L, as a function of the jet exit velocity, U (Figure 2) (e.g. Leroux et al 1996). Beyond the dripping flow regime, the breakup length at first increases linearly with increasing jet velocity, reaches a maximum, and then decreases (regions A and B). Drops are pinched off from the end ofthejet, with diameters comparable

6 90 UN & REITZ to that of the jet (Figure la and b). These first two breakup regimes, which are reasonably well understood, correspond to the Rayleigh and first wind-induced breakup regimes. The form of the breakup curve in these two regimes is well predicted by linear stability theories such as that of Sterling & Sleicher (1975). In this temporal stability theory, it is assumed that the interface, r = a, of a circular jet of radius, a, is perturbed by an axisymmetric wave with a Fourier component of the form 71 = 710 exp(wt + ikx), where 71 = TJ(x, t) is the displacement of the liquid surface. A cylindrical coordinate system is used that moves in the axial direction, x, at the jet velocity, V. The fluid is located at the origin, the nozzle exit x = 0, when t = O. 710 is an initial disturbance level (the initial amplitude of the perturbation), k is the wave number of the disturbance, and w is the complex frequency, the real part of which, w ' is the growth rate. The stability of the liquid surface to linear perturbations is examined and a dispersion equation is derived that relates the complex frequency, w, of an initial perturbation of infinitesimal amplitude, to its wavelength)" (or wavenumber k = 2n / )..). The relationship also includes the physical and dynamical parameters of the liquid jet and the surrounding gas (e.g. Reitz & Bracco 1986), and there exists a maximum growth rate or most unstable wave. Further discussion of the characteristics of the linear solution is given in the next section. Reitz (1987) generated curve-fits of numerical solutions to the dispersion equation for the maximum growth rate (w, = Q) and for the corresponding wavelength ().. = 1\) of the form a -. ( Wei 67)0.6 1\ ( ZO.5)( TO.7) Q- Pla3 = Wei.5 ao.5 (I + Z)(I TO.6) where Z = We? 5/Re\, T = ZWe~ 5, We\ = PIV2a/a, We2 = p2v2a/a, (1) (2a) and ReI = Va/vI' V is the relative velocity between the jet and the gas, and the subscripts I and 2 identify properties based on the liquid and the gas, respectively. As can be seen from Equations 2a and 2b, the maximum wave growth rate increases and the corresponding wavelength decreases rapidly with increasing Weber number, which is the ratio of the inertia force to surface tension force acting on the jet. The effect of the liquid viscosity (which appears in the Reynolds number, Re, and the Ohnesorge number, Z) is seen to reduce the wave growth rate and to increase the wave length significantly as the liquid viscosity increases. (2b)

7 JET BREAKUP 91 At low jet velocities (small Weber numbers) it is reasonable to assume that disruption of the jet occurs when the dominant wave's amplitude is equal to the jet radius. In this case, the jet breakup time,., is given by a = 170exp (!;h), and the breakup length is predicted to be L = U. = U/fJ.ln(a/17o) For low-speed jets in the Rayleigh breakup regime, the parameter In(a/17o) has been determined 'from experiments to be roughly equal to 12. For an inviscid liquid it is readily seen that Equation 3, when combined with Equation 2b, predicts the linear increase in jet breakup length with jet velocity at low gas densities, since. is independent of the jet velocity. A maximum in the breakup length curve is predicted if the gas density is non-zero, i.e. the theory predicts that aerodynamic effects are responsible for the decrease in the breakup length as the Weber number is increased beyond the maximum point. However, discrepancies have been found between the predicted location of the maximum point and experimental data. The shape of the breakup curve has been reviewed by many researchers, including Grant & Middleman (1966) and McCarthy & Malloy (1974) who discussed the effects of the ambient gas, fluid properties, and nozzle design. Leroux et al (1996) pointed out that the location of the maximum point depends on nozzle parameters [presumably through the influence of the initial disturbance term, In(a/17o)], and also on the magnitude of the gas density itself. Indeed, Leroux et al (1996) proposed empirical modifications to account for these effects, which extend the theory of Sterling & Sleicher (1975). Beyond the first wind-induced breakup regime (region B, Figure 2) there is even more confusion about the breakup-length trends. For example, Haenlein (1932) reported that the jet breakup length increases again with increasing jet velocity (region C), and then abruptly reduces to zero (region D). McCarthy & Malloy (1974) reported that the breakup length continually increases. More recently, Hiroyasu et al (1991) discovered discontinuous elongations and shortenings of the jet with changes in the jet velocity. These apparent anomalies are associated with changes in the nozzle internal flow patterns caused by separation and cavitation phenomena, which also exhibited hysteresis effects. Jets from cavitating nozzles were found to have very short breakup lengths. Detached flow jets have long breakup lengths. These phenomena may help explain the previous discrepancies in measurements of breakup lengths in the spray literature, since only recently have investigators paid attention to nozzle flow and geometry effects. Equation 3 predicts that the breakup length decreases continuously as the jet velocity is increased when the effect of the gas density is significant. However, the validity of the assumption that L = U. becomes questionable for (3)

8 92 UN & REITZ high-speed jets, because the breakup mechanism is no longer due to capillary pinching, but is now due to the unstable growth of short-wavelength surface waves (Figure 2). In fact, as the jet velocity is increased, it becomes difficult to define a precise breakup length, and probability density functions are found useful to quantify the breakup length (e.g. Leroux et al 1996). The details ofthe unstable growth of short-wavelength waves on the surface of the liquid jet near the nozzle exit are obscured by the dense spray that surrounds the jet. However, it is generally believed that the jet consists of an unbroken inner liquid core in the vicinity of the nozzle exit, and droplets are stripped from the core by the action of aerodynamic forces at the liquid-gas interface (Reitz & Bracco 1982). Attempts have been made to measure the length of the core region by using intrusive techniques such as electrical conductivity measurements (e.g. Chehroudi et a11985, Hiroyasu et ai1991), and laser sheet visualization (e.g. Guider et a11994, Dan et ai1997). The core length depends on the liquid/gas density ratio and only weakly on the fluid properties and the jet velocity. These trends can be demonstrated by using Taylor's (1940) analysis of highspeed liquid jet breakup. Taylor considered the rate of mass loss per unit length of the jet caused by droplet erosion from the liquid surface resulting from the unstable growth of short-wavelength surface waves and showed that the breakup length of a high-speed jet is given by Lla = B(pt! P2)1/2 1f(T) where T is Taylor's parameter, T = pif P2(ReIfWe()2, and the functionf(t) has been approximated from Taylor's numerical results asf(t) =./3/6 [1 exp(-iot)] by Dan et al (1997). The constant B in Equation 4 has a recommended value of 4.04 for typical diesel spray nozzles (Chehroudi & Bracco 1985). However, nozzle internal design effects are clearly important for high-speed jet breakup. It is known that high-speed liquid jets in jet cutting applications remain intact for a distance of many diameters away from the nozzle. On the other hand, modern diesel injectors employ very similar injection pressures, but diesel spray breakup starts at the nozzle exit (e.g. Figure Id). The significant differences in the interior nozzle design features of these two applications account for their different performances. Diesel nozzles are typically short-length holes with sharp-edged inlets, whereas jet cutting nozzles consist of contoured nozzles to minimize initial disturbance levels to the liquid flow. Criteria for predicting the onset of the breakup regimes have been reviewed by Chigier & Reitz (1996). Consideration of the balance between the liquid inertia force and the surface tension force of a free column of liquid led Ranz (1956) to the criterion that dripping no longer occurs from the nozzle exit (4)

9 JETBREAKUP 93 ).e. a jet is formed) if WeL > 8, where WeL = PI cj2(2a)ja. The criterion Weg = p2u2(2a)/a < 0.4 corresponds to the point where the inertia force of :he surrounding gas reaches about 10% of the surface tension force. Ranz (1956) suggested that this would mark the beginning of the first wind-induced breakup regime, where the effects of the ambient gas are no longer negligible. Numerical results of Sterling & Sieicher (1975) indicate that the maximum in the jet breakup length (see Figure 2) occurs when WeL = Z? 9, where ZI = We15/Rev ReL = U(2a)jvI' This could also indicate the importance of aerodynamic would be effects, so that the criteria for Rayleigh breakup (see Figure la) WeL > 8 and Weg < 0.4 or Z? 9. (5) Note, however, that nozzle turbulence and other flow effects are not included in Equation 5. Ranz (1956) argued that the gas inertia force is of the same order as the surface tension force when Weg = 13. This could serve as a definition of the end of the first wind-induced regime (see Figure Ib), which then occurs when Z? 9 < Weg < 13 (6) In this case, Weg > 13 marks the onset of the second wind-induced regime, where the interaction with the surrounding gas starts to become dominant. Miesse (1955) suggested the criterion Weg > 40.3 to predict the onset of the atomization regime, the point at which breakup appears to start at the nozzle exit (see Figure Id). Thus, the criteria for breakup in the second wind-induced regime are 13 < Weg < 40.3 (7) In the second wind-induced regime, the breakup starts some distance downstream of the nozzle exit, and a smooth unbroken section of the jet is visible downstream of the nozzle exit (Figure Ie). As mentioned previously, no account is made of nozzle internal flow effects in the above correlations. To address this shortcoming, Reitz (1978) assumed that atomization corresponds to a critical value of the breakup length/nozzle diameter ratio. With this assumption, the onset of atomization is predicted to occur when P2/ PI > Kf(T)-2 (8) In this case, the parameter K was obtained from experiments on atomizing jets and was found to be a function of the nozzle geometry, where K = (0.53[3.0 + (fj2a)]1/2-1.15)/744 (9)

10 94 UN & REITZ and f./2a is the nozzle length-to-diameter ratio. K empirically accounts for the effect of initial disturbances in the flow caused by nozzle internal flow phenomena such as turbulence, cavitation, and flow separation. Equation 9 includes the effect of liquid viscosity and nozzle internal flows, and it predicts that atomization is favored at high gas densities and for sharp inlet edge nozzles, with smalliength-to-diameter ratio. These trends agree with experiments reported in the literature (Hiroyasu et al 1991, Reitz 1978, Reitz & Bracco 1979). When injection takes place into a coflowing gas, additional breakup regimes are observed, as described by Chigier & Reitz (1996). This situation is of much practical interest and it is frequently used in air-blast coaxial atomizers to improve the quality of atomization and to maintain it over a wide range of liquid flow rates. High gas velocities (up to sonic) are generated by highpressure gas flows passing through annular orifices surrounding the liquid jet. The high coflowing gas velocity transmits momentum to the liquid interface. Large-scale eddy structures in the gas flow impact upon the liquid jet, causing stretching, destabilization, and flapping of the liquid jet. Eroglu et al (1991) measured breakup lengths of round liquid jets in annular coaxial air streams and found that the breakup length decreases with increasing Weber number and increases with increasing liquid jet Reynolds number according to the relation L/2a = O.5WeL".4 Re~ 6 (10) where L is the liquid intact length, a is the central tube inner radius, and the Weber and Reynolds numbers are based on the relative velocity between the gas and the liquid. Jet breakup in coaxial flows is highly unsteady, and unstable liquid structures are observed to disintegrate in a time-varying, bursting manner. Farago & Chigier (1992) refer to these as pulsating and super-pulsating breakup processes. At high air-flow rates, the unstable liquid cylindrical jet undergoes a flapping motion and can be transformed into a curling liquid sheet. The sheet becomes stretched into a membrane bounded by thicker rims, which finally burst into ligaments and drops of various sizes. Farago & Chigier (1992) classified coaxial jet disintegration into three main categories: (a) Rayleigh-type breakup where the mean drop diameter is of the order of the jet diameter-both axisymmetric breakup (for Weg < 15) and nonaxisymmetric breakup patterns (for 15 < Weg < 25) were observed; (b) jet disintegration via the stretched-sheet mechanism, which produces membranetype ligaments (25 < Weg < 70)-in this case, the diameter of the drops formed is considerably smaller than the diameter of the jet; and (c) jet disintegration via fiber-type ligaments (100 < Weg < 500)-at even higher air-flow rates, fibers are formed that peel off the liquid-gas interface. This breakup mechanism resembles the short-wavelength breakup mechanism of jets in the second

11 JETBREAKUP 95 wind-induced and atomization regimes mentioned above. The atomization begins with the unstable growth of short-wavelength waves, the formation of fibers, and their peeling off from the main liquid core. The fibers break into droplets by the nonaxisymmetric Rayleigh-type jet disintegration mode. Again, the drop diameter is much smaller than the jet diameter. 3. BREAKUP MECHANISMS Jet Instability The regimes of jet breakup have been delineated above with correlations in terms of relevant parameters. The results of recent works based on the theory of absolute and convective instability of liquid jets enable us to elucidate the different physical mechanisms responsible for the jet breakup in the various regimes. In the aerodynamic theory of spontaneous jet breakup without external excitation, it is assumed that the onset of breakup is caused by the amplification of natural disturbances in a jet. Any arbitrary form of disturbance can be constructed by superposition of all Fourier components. Each Fourier component has the form A exp[ikx + wt), where A is the wave amplitude, k = kr + ikj is the complex wave number whose real and imaginary parts give, respectively, the number of waves over a distance 21l' and the exponential spatial growth rate per unit distance in the axial x-direction, and w = wr + iwj is the complex wave frequency, the real and imaginary part of which give, respectively, the exponential temporal growth rate and the frequency of the Fourier wave. Not all the Fourier components are capable of extracting energy from the jet system and amplifying, however. The Fourier components must have special values of k and w, which depend on specific characteristics of the jet system, in order to grow from initially infinitesimal amplitudes. Mathematically, (k, w) is determined as the eigenvalue of a linear system containing relevant flow parameters. The eigenvalues or the characteristic values are so determined that the condition of the existence of a nontrivial solution of the system is satisfied. This condition is the so-called characteristic equation or dispersion relation. In the linear aerodynamic theory of jet instability, the finite amplitude disturbances introduced outside or inside the nozzle by the various means mentioned above are excluded from consideration. Absolute Instability and Formability of a Jet In the pioneering works cited above, the liquid jet is considered to be infinitely long and k is assumed to be real. Thus the disturbance must grow or decay everywhere in space at the same time rate wr. However, Keller et al (1972) noted that the disturbance initiating from the nozzle tip actually grows in space as it is swept downstream to break up the jet into drops, leaving a section of jet

12 96 UN & REITZ intact near the nozzle tip. They set k to be complex and allow the disturbance to grow in space as well as in time in a semi-infinite weightless inviscid jet in a vacuum. They found that Rayleigh's results are relevant only in the case of large Weber number, WeL (WeL = PLU22a/a is based on the liquid density). They also showed that in the limit of WeL -+ 00, the spatial growth rate kl can be inferred from the temporal growth rate Wr by the relation kj = ± Wr + 0 (1 /We J, while the disturbance travels at the jet velocity. For Weber number less than the order of one, they found a new mode of faster-growing disturbances whose wavelengths are so long that they may not be actually observable. Using the theory of absolute and convective instability (Briggs 1964, Bers 1983), Leib & Goldstein (l986b) showed that the new mode actually corresponds to absolute instability arising from a saddle-point singularity in the characteristic equation. The unstable disturbances in an absolutely unstable jet must propagate in both upstream and downstream directions. Thus, the unstable disturbances expand in space over the course of time. This contrasts with what is observed in a Rayleigh jet, wherein unstable disturbances grow over time as they are convected in a wave packet in the downstream direction with the group velocity dw;/dk. (LighthillI987, Mei 1989). For a brief introduction to the theory of absolute and convective instability in the context of jet breakup, see the recent work of Lin (1996). The critical Weber number WeLc' below which an inviscid jet under weightless condition in vacuum is absolutely unstable, and above which the jet is convectively unstable, was found by Leib & Goldstein (1986b) to be 7f. When the viscosity of the jet is taken into account, the critical Weber number depends on the Reynolds number ReL = U(2a)/vl' where VI is the liquid viscosity as shown in Figure 3 with the curve Q = 0, which was obtained by Leib & Goldstein (1986a). The other two curves for nonvanishing values ofq == P2/ PI were obtained by Lin & Lian (1989), who were motivated to find out whether the absolute instability discovered by Leib & Goldstein is physical or mathematical, arising from the neglected ambient gas effect. It turns out that the effect of the gas density is to increase WeLc (Figure 1). Thus, the gas density promotes absolute instability in the sense that the given jet that is convectively unstable may be made absolutely unstable by increasing the ambient gas density. The absolute instability cannot be suppressed by either the gas compressibility (Zhou & Lin 1992a,b, Li & Kelly 1992) or by the gas viscosity (Lin & Lian 1993). Absolute instability is a real physical phenomenon, at least in the absence of gravity, which is neglected in the theories. However, jet ab ;olute instability under weightless conditions has not yet been reported in the iterature. The delineation of transition from convective to absolute instabilty in the absence of gravity is yet to be completed (Honohan 1995, Vihinen [996). In such a delineation, the critical Weber number is a function of Rev

13 In 2.5 Q = WeLC () ~ tc iti CI1 ;I> ;;0:: ~ "0 Figure 3 Critical Weber number as a function of the Reynolds number. The jet is absolutely unstable below each curve of constant Q. The jet is convectively unstable in the rest of the parameter space. ReL \0 -.1

14 98 UN & REITZ Q, and N = \i21\i1' where \i2 is the viscosity of the surrounding gas. Existing theoretical results show that for a given set of Rev Q, and N, a liquid jet may be made absolutely unstable by reducing the Weber number to be below WeLc- The theoretical prediction that the unstable disturbance must propagate in both downstream and upstream directions when the jet velocity is smaller than that corresponding to Wel.c signifies that absolute instability occurs when the jet inertia is not sufficiently large to carry downstream all of the unstable disturbances that derive their energy from the surface tension. Thus, surface tension remains the source of instability. Part of the unstable disturbances will propagate back to the nozzle tip to interrupt the formation of a jet of any length. It is likely that the phenomenon of absolute instability also exists in a jet in the presence of gravity, because the physical mechanism is unlikely to be altered significantly by the gravity-induced variation in the thickness and the velocity of the jet along its axis. Thus, the transition from absolute to convective instability signifies the beginning of the formability of a liquid jet. The parameter range in which a liquid jet cannot be formed may be termed the absolutely unstable regime. The origin of nonformability of a jet is the surface tension. Capillary Pinching with Wind Assistance The jet breakup in the Rayleigh, the first wind-induced, the second windinduced, and the atomization regimes defined in the previous section are all the manifestations of convective instability. As mentioned earlier, the unstable disturbances amplify in time as they are convected in a group in the downstream direction with the group velocity. However, the physical mechanism of breakup in the second wind-induced and atomization regimes is fundamentally different from that in the other regimes. Neglecting the viscosity of gas, Lin & Creighton (1990) calculated from the Navier-Stokes equations the mechanical energy budget of a liquid jet in a parameter range of convective instability. The stability analysis in this range was completed earlier by Lin & Lian (1990). They expressed the total time rate of changes of the disturbance kinetic energy in a controlled volume of the jet, over a wavelength of the most amplified disturbance, as the sum of the rate of work done by various relevant forces, i.e. E = Pg + Pi + S + V + D, (II) where Pg is the rate of work done by the gas pressure fluctuation at the liquid-gas interface, Pi is the rate of work done by the liquid pressure fluctuation at the inlet and outlet of the control volume, S is the rate of work done by the surface ~ension, V is the rate of work done by the liquid viscous stress, and D is the rate )f viscous dissipation of mechanical energy.

15 JETBREAKUP 99 Some typical values of (Pg, Pt, s, v, G) == (Pg, Pt, S, V, D)j E for various Jfeakup regimes are shown in Table 1. The first four rows in this table belong :0 the Rayleigh and the first wind-induced breakup regimes. The last four rows Jelong to the second wind-induced and the atomization regimes. The values (r = 2najAm in the fourth column are the wave numbers corresponding to he wavelength, Am' of the most amplified disturbance predicted by the linear :heory for the flow parameters specified in the first three columns. Q = orresponds to the case of a water jet in air under one atmosphere. The high Reynolds number jet at low gas density depicted in the fourth row is closest :0 the idealized Rayleigh jet. The presence of the low density gas increases Jnly slightly the most amplified wave number, 0.696, predicted by Rayleigh. oreover, s dominates all other work terms. This is a classical case of Rayleigh Jfeakup by capillary pinching, which produces drops of a diameter comparable :0 the jet diameter. Capillary pinching remains the mechanism of breakup for the low Reynolds mmber jet depicted in the first row (Table 1). Note that the value of kr is lmost four times smaller than that predicted by Rayleigh. As the drop size s inversely proportional to kp the liquid viscosity tends to increase the drop ize considerably. This regime, which is not shown in Figure 1, may be termed he Weber-Chandrasekhar regime to emphasize the important role of liquid viscosity. With Q kept at in rows 2 and 3, Pg increases with increasing ReL ilmost to the same order of magnitude as s. All other work terms remain significant. Thus, as the relative speed of gas-to-liquid (wind speed) increases, he gas pressure fluctuation assists significantly the capillary force to break up he liquid jet. Nevertheless the capillary force remains dominant over the ind force. Comparing the values of kr in these two rows with that of the Rayleigh jet, it is seen that the drop size in this first wind-induced breakup egime may be slightly larger or smaller than that in the Rayleigh regime, [able ( ( Pi Pgs vd ( Energy /WeL Q xkr 1.3 budget -2) 103 in jet breakup

16 100 UN & REITZ depending on the flow parameters (see Figure Ib). Since the surface tension is mainly responsible for the breakup in this regime, and the gas inertia force only asslsts in the breakup, rather than inducing the breakup, it is probably more appropriate to call this regime the wind assisted breakup regime instead of the first wind-induced breakup regime. The second wind-induced breakup regime is genuinely wind-induced, as is explained below. Interfacial Stress Against Surface Tension For the parameter range specified in the last four rows (Table I), kr is found to be more than one order of magnitude larger than in the previous four cases. Thus, the drop radii produced in these parameter ranges are much smaller than the jet radius (see Figure 1c and d). In contrast to the previous four cases, the pressure work-term dominates the surface tension work-term in the last four rows. In fact the surface tension term is negative. That is to say, the surface tension acts against the formation of small droplets generated by the interfacial pressure fluctuation in the second wind-induced and atomization regimes. Part of the kinetic energy in ajet is converted through the pressure work-term to the surface energy in the droplets. Thus the second wind-induced and atomization regimes are genuinely wind-induced. It is seen (Table 1) that while the atomization and second wind-induced regimes exist in the parameter range WeL» Q, the rest of the regimes exist for WeL':::: 1. It has been shown (Kang & Lin 1987) that the unstable disturbances in the atomization regime scale with the gas capillary length c = a/ P2l.f2. The condition WeL» Q implies that c is much smaller than the jet diameter (Lin & Lian 1990). The interfacial shear stress fluctuation will augment the pressure fluctuation in the breakup process, if the gas viscosity is considered. It should be remembered that the linear stability theory is not capable of differentiating between the second wind-induced and the atomization regimes. The linear theory can only predict the onset of instability, which produces interfacial waves of different length scales depending on the parameters. The nonlinear processes of pinching off a small droplet from the interface, subsequent to the onset, and the continuous generation of droplets from the receding interface toward the core of a jet are involved in reaching the atomization state (see Figure Id). However, the physical mechanism involved in the initial stage of the second wind-induced and the atomization regimes may be the same. 4. DISCUSSION lnterfacial Shear Layer A serious defect common to all of the above reviewed works is the lack of a rigorous treatment of the effect of the gas viscosity. Sterling & Sleicher

17 JET BREAKUP 101 (1975) assumed with Benjamin (1959) that the Kelvin-Helmholtz model can be applied locally along the interfacial wave with an arbitrary correction factor that-because of the viscous effect-is used to reduce the pressure distribution predicted by the Kelvin-Helmholtz model. Thus the possibility of generation of droplets by shear waves is missed. Lin & Lian (1990) modeled the liquid-gas interfacial boundary layer with the boundary layer over a wavy solid surface in order to estimate the interfacial shear effect. This is not satisfactory because a fluid-fluid interfacial shear layer is fundamentally different from that of a solidfluid boundary layer. The effect of gas viscosity was rigorously analyzed by Lin & Ibrahim (1990) with temporal theory, and by Lin & Lian (1993) with the theory of absolute and convective instability for a viscous liquid jet in a viscous gas in a vertical circular pipe. The basic flow is an exact solution of the Navier Stokes equation. Unfortunately, the numerical results for the case of very strong interfacial shear were not sufficiently accurate to allow the drawing of definitive conclusions on the effect of the gas viscosity on the atomization process. The relative importance of the shear stress fluctuation to the pressure fluctuation in the atomization process in various parameter ranges remains unknown. Nonaxisymmetric Disturbances The above description is based on works that assume that axisymmetric disturbances are more unstable than asymmetric ones. Rayleigh was able to prove that asymmetric temporal disturbances in an inviscid jet are all stable. Temporally stable waves are also convectively stable evanescent waves (Huerre & Monkewitz 1990). For ReL = 0 (10), Lin & Webb (1994) showed that the asymmetric disturbances are evanescent waves in the parameter range 10-4 :S Q :S 10-2, 10 :S We :S 103. Yang (1992) demonstrated that temporally growing asymmetric long-wavelength disturbances may become dominant when the Weber number of an inviscid jet in an inviscid gas is in the atomization regime. However, the viscosity of the liquid tends to bring down the temporal growth rate of nonaxisymmetric disturbances (Avital 1995), and the maximum temporal growth rates of axisymmetric disturbances remain higher than those of asymmetric ones except when the jet is almost inviscid, for example when ReL = 105 and WeL = 104 (Li 1995). A similar conclusion was reached by EA Ibrahim (personal communication), who investigated convectively unstable asymmetric disturbances in a viscous jet emanating into an inviscid gas, when WeL and ReL are of order 103 and higher. There is also experimental evidence of asymmetric disturbances in the atomization regime mentioned above (Meister & Scheele 1969, Taylor & Hoyt 1983, Eroglu et al 1991). However, the appearance of non-axisymmetric disturbances may also be caused by the secondary instability after the onset of instability from axisymmetric disturbances. In the Rayleigh, the wind-induced,

18 102 UN & REITZ and the wind-assisted regimes, the asymmetric disturbances may be brought out prominently by the swirl in the liquid jet (Ponstein 1959, Kang & Lin 1989) or less prominently by the swirl in the gas (Lin & Lian 1990). Success and Shortcoming of Linear Theory Despite these shortcomings, the linear stability analysis started by Rayleigh provides a qualitative description of the physical mechanisms involved in various regimes of jet breakup. The linear theories have even enjoyed some reasonable semiquantitative comparisons with experiments in the atomization regime (Kang & Lin 1987, Reitz & Bracco 1982). These comparisons include the intact length, spray angle, and droplet size, which scales with the gas capillary length a / pzlf2. The quantities that have just been mentioned are the products of highly nonlinear processes. The reason such a reasonable comparison based on linear theory is possible is probably due to the fact that the physical mechanisms at work in various breakup regimes are already basically determined at the onset of instability. The nonlinear evolution subsequent to the onset only modifies quantitatively the physical mechanisms. On the other hand, the excellent quantitative comparison between the theory of Rayleigh and the experiments of Goedde & Yuen (1970) and Donnelly & Glaberson (1966) is probably fortuitous. While Rayleigh neglected the existence of the ambient gas, the liquid viscosity, and gravity, none of these are neglected in experiments. Moreover, the Rayleigh jet breakup is due to capillary pinching, and yet the experiments do not seem to be sensitive to the Weber number. A complete delineation of where each regime should start and end in the parameter space (Q, N, Re, We, Fr) may be made within the framework of linear theory only if the various effects of nozzle flows on jet instability are known or assumed known. Here, the Froude number, Fr = UZ/g(2a), represents the ratio of inertial to gravitational force. 5. SUMMARY AND UNRESOLVED SCIENTIFIC ISSUES This study discusses various regimes of breakup of liquid jets injected into both stagnant and coflowing gases. Available criteria for the transition between the regimes are reviewed. The physical mechanisms at work in the different breakup regimes are described. The influence of nozzle internal flow effects is shown to be important, but these effects are only included empirically in current wave-stability theories. A useful area for future research would be the development of fundamentally based models that account for the effect of nozzle internal flows on the liquid breakup process. In addition, current breakup models need to be extended

19 JET BREAKUP 103 to account for the nonlinear effects of liquid distortion, membrane formation, and stretching on the atomization process. These phenomena are especially important in liquid injections in a high-momentum coflowing gas. The effect of liquid-gas viscous shear layers on the onset of instabilities, which leads to various regimes of the jet breakup, remains to be rigorously analyzed and tested. This knowledge is not only important for applications involving jet breakup, it will also advance our scientific understanding of various processes in nature as well as in industries. The impact of the stick-slip condition experienced by the jet liquid exiting the nozzle tip and the spatial development of the liquid-gas interfacial shear layer (before the jet flow is fully developed) on the receptivity of the jet to instability need to be investigated. The effect of gravity in the absolute instability regime and the Weber-Chandrasekhar regimes, where the Froude number is small, remains to be elucidated. In the atomization regime, Fr is so large that the gravitational effect may not be significant. A complete delineation of the jet breakup regimes in the entire parameter space (Q, N, Re, We, Fr), even in the framework of linear theory, has not yet appeared. The nonlinear studies, which take into account the finite amplitude disturbances originated in the nozzle, and wh'ich elucidate the nonlinear process subsequent to the onset of unstable waves of various-length scales, will contribute to the understanding of the formation of sprays and drops from a liquid et. ACKNOWLEDGMENTS Support for SP Lin was provided in part by Army Research Office Grant DAAL03-89-K-0179 and NASA Grant NAG Support for R Reitz was provided by the Army Research Office Contract DAAL03-86-K Visit the Annual Reviews home page at Literature Cited Avital E Asymmetric instability of a viscid capillary jet in an inviscid Fluids.7: media. Phys. enjamin TB Shearing ftow over a wavy boundary. J. Fluid Mech. 6: ers A Space-time evolution of plasma instabilities-absolute and convective. In Handbook of Plasma Physics, ed. M. Rosenbluth, I : Amsterdam: North HoIland riggs RJ Electron Stream Interaction with Pla.rma.r. Cambridge: MIT handrasekhar S The capillary instability of a liquid jet. In Hydrodynamic and Hydromagnetic Stability, pp Oxford: Oxford Univ. Press. 652 pp. Chehroudi B, Bracco FY On the intact core of full cone sprays. Soc. Automat. Eng. Tech. Pap Chigier N, Reitz RD Regimes of jet breakup and breakup mechanisms (physical aspects). In Recent Advances in Spray Combustion: Spray Atomization and Drop Burning Phenomena, ed. KK Kuo, I: Reston: AIAA Dan T, Yamamoto T. Senda J. Fujimoto H

20 104 UN & REITZ Effect of nozzle configurations for characteristics of non-reacting diesel fuel sprays. Soc. Automot. Eng. Tech. Pap Donnelly Rl, Glaberson W Experiments on the capillary instability of a liquid jet. Proc. R. Soc. London Ser. A. 290: Eroglu H, Chigier N, Farago Z Coaxial atomizer liquid intact lengths. Phys. Fluid. A. 3:303-8 Farago Z, Chigier N Morphological classification of disintegration of round liquid jets in a coaxial air stream. At. Spray.. 2: Goedde EF, Yuen Me Experiments on liquid jet instability. J. Fluid Mech. 40: Grant RP, Middleman S Newtonian jet stability. A.I.Ch.E. J. 12: Guider OL, Smallwood Gl, Snelling DR Internal structure of transient full-one dense diesel sprays. Int. Symp. COMODIA. pp Haenlein A Uber den Zerfall eines Flussigkeitsstrahls (On the disruption of a liquid jet). NACA TM Report. 659 Hiroyasu H, Arai M, Shimizu M Breakup length of a liquid jet and internal flow in a nozzle. Proc. ICLASS-91, Pap. 26 Honohan A Experimental measurement of the spatial instability of a viscous liquidjet at microgravity. MS thesis. Clarkson Univ., Potsdam, NY. 95 pp. Huerre P, Monkewitz PA Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22: Kang 01, Lin SP Breakup of swirling liquidjets.lnt. J. Eng. Fluid Mech. 2:47-62 Keller 18, Rubinow SI, Tu YO Spatial instability of a jet. Phys. Fluids. 16: Leib SI, Goldstein ME. 1986a. Convective and absolute instability of a viscous liquid jet. Phys. Fluids. 29: Leib SI, Goldstein ME. 1986b. The generation of capillary instability on a liquid jet. J. Fluid Mech. 168: Leroux S, Dumouchel C, Ledoux M The stability curve of Newtonian liquid jets. At. Spray. 6: Li HS, Kelly RE The instability of a liquid jet in a compressible air stream. Phy.. Fluids A. 4 : Li HS, KelIy RE On the transfer of energy to an unstable liquid jet in a coflowing compressible airstream. Phys. Fluids A. 5: ~i X Mechanism of atomization of a liquid jet. At. Spray. 5 : ~ighthill Waves in Fluids, pp Cambridge: Cambridge Univ. Press. 504 pp. _in SP Regimes of jet breakup and breakup mechanisms (Mathematical aspeets). In Recent Advance.fin Spray Combu. - tion: Spray Atomization and Drop Burning Phenomena. VA: AIAA ed. KK Kuo, 1: Reston, Lin SP, Creighton B Energy budget in atomization. J. Aerosol. Sci. and Technol. 12: Lin SP, Ibrahim EA Instability of a viscous liquid jet surrounded by a viscous gas in a pipe. J. Fluid Mech. 218: Lin SP, Kang OJ Atomization ofa liquid jet. Phy.. Fluids. 30: Lin SP, Lian Zw Absolute instability in a gas. Phys. Fluids A. 1: Lin SP, Lian ZW Mechanism of atomization. AlAA J. 28: Lin SP, Lian Zw Absolute and convective instability of a viscous liquid jet surrounded by a viscous gas in a vertical pipe. Phys. Fluids A. 5: Lin SP, Webb RD Nonaxisymmetric evanescent waves in a viscous liquid jet. Phys. Fluids. 6: McCarthy Ml, Malloy NA Review of stability of liquid jets and the influence of nozzle design Chem. Eng. J. 7: 1-20 Mei Ce Applied Dynamic. of Ocean Surface Waves. Hong Kong: World Sci. 740 pp. Meister Bl, Scheele GF. I 969a. Drop formation from cylindrical jets in immiscible liquid systems. AlChE J. 15:700-6 Meister Bl, Scheele GF. I969b. Prediction of jet length in immiscible liquid system. AlChE J. 15: Miesse Ce Correlation of experimental data on the disintegration of liquid jets. Ind. Eng. Chem. 47: Plateau Statique Experimentale et Theorique de. Liquids Soumie aux Seules Forces Moleculaire, vols. I, 2. Paris: Cauthier Villars. 450 pp. 495 pp. Ponstein P Instability of rotating cylindrical jets. Appl. Sci. Re.. 8: Ranz WE On sprays and spraying. Dep. Eng. Res., Penn State Univ. Bull pp. Rayleigh L. 1879a. On the capillary phenomenon of jets. Proc. R. Soc. London. 29:71-97 Rayleigh L. 1879b. On the instability of jets. Proc. London Math. Soc. 10:4-13 Rayleigh L. I892a. On the instability of a cylinder of viscous liquid under capillary force. Phil. Mag. 34: Rayleigh L. I 892b. On the instability of cylindrical fluid surfaces. Phil. Mag. 34: Reitz RD Atomization and other breakup regimes of a liquid jet. PhD thesis. Princeton Univ., Princeton, Nl. 231 pp. Reitz RD Modeling atomization processes in high-pressure vaporizing sprays. Atom. Spray Technol. 3:309-37

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