liquid jet

P1: ARS/kja P2: HCS/plb QC: MBL/agr T1: MBL

November 25, 1997 11:20 Annual Reviews AR049-04

Annu. Rev. Fluid Mech. 1998. 30:85–105Copyright c© 1998 by Annual Reviews Inc. All rights reserved

DROP AND SPRAY FORMATIONFROM A LIQUID JET

S. P. LinMechanical and Aeronautical Engineering Department, Clarkson University, Potsdam,New York 13699; e-mail: gw02@ sun.soe.clarkson.edu

R. D. ReitzMechanical Engineering Department, University of Wisconsin, Madison, Wisconsin,53706; e-mail: [email protected]

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 intodroplets of diameters several orders of magnitude smaller. The sizes of the dropsformed from a liquid jet without external control are in general not uniform. Thesizes as well as the size distribution depend on the range of flow parameters inwhich the jet is produced. The jet breakup exhibits different characteristics indifferent regimes of the relevant flow parameters because of the different physicalmechanisms involved. Some recent works based on linear stability theories aimedat the delineation of the different regimes and elucidation of the associated phys-ical mechanisms are reviewed, with the intention of presenting current scientificknowledge on the subject. The unresolved scientific issues are pointed out.

1. INTRODUCTION

The breakup of a liquid jet emanating into another fluid has been quantitativelystudied for more than a century. Plateau (1873) observed that the surface energyof a uniform circular cylindrical jet is not the minimum attainable for a givenjet volume. He argued that the jet tends to break into segments of equal length,each of which is 2π times longer than the jet radius, such that the sphericaldrops formed from these segments give the minimum surface energy if a dropis formed from each segment. Rayleigh (1879a, b) showed that the jet breakup

850066-4189/98/0115-0085$08.00

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86 LIN & REITZ

is the consequence of hydrodynamic instability. Neglecting the ambient fluid,the viscosity of the jet liquid, and gravity, he demonstrated that a circularcylindrical liquid jet is unstable with respect to disturbances of wavelengthslarger than the jet circumference. Among all unstable disturbances, the jet ismost susceptible to disturbances with wavelengths 143.7% of its circumference.Rayleigh also considered the cases of a viscous jet in an inviscid gas (1892a)and an inviscid gas jet in an inviscid liquid (1892b). He showed that if the massof the gas is neglected, the most amplified disturbance in the first case possessesan infinitely long wave length, and that for the second case it is 206.5% of thejet circumference.

Tomotika (1935) showed that an optimal ratio of viscosities of the jet andthe ambient fluid exists for which a disturbance of finite wavelength attainsthe maximum growth rate. Chandrasekhar (1961) took into account the liquidviscosity and the liquid density, which was neglected by Rayleigh, and showedmathematically that the viscosity tends to reduce the breakup rate and increasethe drop size. He also showed that the physical mechanism of the breakup of aviscous liquid jet in a vacuum is capillary pinching. The theoretical results ofRayleigh and Chandrasekhar appear to be in agreement with the experimentsof Donnelly & Glaberson (1966) and Goedde & Yuen (1970).

Weber (1931) considered the effects of the liquid viscosity as well as thedensity of the ambient fluid. His theoretical prediction did not agree well withexperimental data, as pointed out by Sterling & Sleicher (1975), who improvedWeber’s theory with partial success. Taylor (1962) showed that the density ofthe ambient gas has a profound effect on the form of the jet breakup. For asufficiently large gas inertia force (which is proportional to the gas density)relative to the surface tension force per unit of interfacial area, the jet maygenerate at the liquid-gas interface droplets with diameters much smaller thanits 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 indeedvery large owing to the increasingly wide applications of the jet breakup pro-cesses. There have been several review articles in this area (e.g. Sirignano1993). The latest ones are by Chigier & Reitz (1996) and Lin (1996). In thisreview, we focus on the physical mechanisms that cause the onset of the jetbreakup at the liquid-gas interface. The nonlinear evolution after the onset ofjet breakup is not considered. The physical mechanism of breakup frequentlyremains the same during the nonlinear evolution, although the nonlinear theorymay produce additional quantitative results. For example, the satellite dropletsformed from the ligament between two main drops are not predicted by lin-ear theories, but the mechanism of the satellite formation remains capillarypinching of the Rayleigh mode.

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JET BREAKUP 87

In addition, previous works on the breakup processes of a liquid jet in anotherliquid (which are relevant to the emulsification process) are not reviewed. Nordo we review works that focus on the application of the jet breakup process,although they are interesting and are of considerable importance in technologyand science. Even within the area of the fluid dynamics of the onset of liquidjet 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 withcorrelations in terms of relevant flow parameters are reviewed. These correla-tions are based mainly on temporal linear stability theory. Section 2 provides aframework for the discussion in Section 3 of the physical mechanisms at workin the different regimes. The elucidations of the physical mechanisms are basedon absolute and convective instability theories. A general critical discussionof the recent work in the field is given in Section 4. Section 5 summarizes thescientific issues that remain to be investigated.

2. BREAKUP REGIMES

The breakup of a liquid jet injected through a circular nozzle hole into a stagnantgas has been studied most frequently. Previous studies have established thatthe spray properties are influenced by an unusually large number of parameters,including nozzle internal flow effects resulting from cavitation, the jet velocityprofile and turbulence at the nozzle exit, and the physical and thermodynamicstates of both liquid and gas (e.g. Wu et al 1992, Eroglu et al 1991, and Reitz& Bracco 1979). The precise mechanisms of breakup are still being researched(e.g. Lin 1996, Chigier & Reitz 1996). However, linear stability theory canprovide qualitative descriptions of breakup phenomena and predict the existenceof various breakup regimes. It is noteworthy that the influence of nozzle internalflow effects is included only empirically in most jet breakup theories. Theseeffects are known to be important, particularly for high-speed jet breakup.

Jet breakup phenomena have been divided into regimes that reflect differ-ences in the appearance of jets as the operating conditions are changed. Theregimes are due to the action of dominant forces on the jet, leading to its breakup,and it is important that these forces be identified in order to explain the breakupmechanism in each regime (Reitz & Bracco 1986). The case of a round liq-uid jet injected into a stagnant gas is shown in Figure 1. Four main breakupregimes have been identified that correspond to different combinations of liquidinertia, surface tension, and aerodynamic forces acting on the jet. These havebeen named the Rayleigh regime, the first wind-induced regime, the secondwind-induced regime, and the atomization regime (Figure 1) (Reitz & Bracco1986).

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88 LIN & REITZ

Figure 1 (a) Rayleigh breakup. Drop diameters larger than the jet diameter. Breakup occurs manynozzle diameters downstream of nozzle. (b) First wind-induced regime. Drops with diameters ofthe 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 somedistance downstream of nozzle. (d) Atomization regime. Drop sizes much smaller than the jetdiameter. Breakup starts at nozzle exit.

At low jet velocities, the growth of long-wavelength, small-amplitude distur-bances on the liquid surface promoted by the interaction between the liquid andambient gas is believed to initiate the liquid breakup process. The existenceof these waves is clearly demonstrated in Figure 1a and b. For high-speedliquid jets, the breakup is thought to result from the unstable growth of short-wavelength waves (Figure 1c andd) (Reitz & Bracco 1982). The breakup dropsizes are on the order of the jet diameter in the Rayleigh and first wind-induced

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JET BREAKUP 89

ΛL

Λ

L

U

L

AB

CD

Figure 2 Schematic diagram of the jet breakup length curve.

breakup regimes. The drop sizes are very much less than the jet diameter in thesecond wind-induced and atomization regimes.

A convenient method for categorizing jet breakup regimes is to considerthe 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 linearlywith increasing jet velocity, reaches a maximum, and then decreases (regions Aand B). Drops are pinched off from the end of the jet, with diameters comparable

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90 LIN & REITZ

to that of the jet (Figure 1a andb). These first two breakup regimes, which arereasonably well understood, correspond to the Rayleigh and first wind-inducedbreakup regimes.

The form of the breakup curve in these two regimes is well predicted by linearstability theories such as that of Sterling & Sleicher (1975). In this temporalstability 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

η = η0 exp(ωt + ikx), (1)

whereη = η(x, t) is the displacement of the liquid surface. A cylindricalcoordinate system is used that moves in the axial direction, x, at the jet velocity,U. The fluid is located at the origin, the nozzle exit x= 0, when t= 0. η0 isan initial disturbance level (the initial amplitude of the perturbation), k is thewave number of the disturbance, andω is the complex frequency, the real partof which, ωr, is the growth rate. The stability of the liquid surface to linearperturbations is examined and a dispersion equation is derived that relates thecomplex frequency,ω, of an initial perturbation of infinitesimal amplitude, toits wavelengthλ (or wavenumber k= 2π/λ). The relationship also includesthe 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 mostunstable wave. Further discussion of the characteristics of the linear solutionis given in the next section.

Reitz (1987) generated curve-fits of numerical solutions to the dispersionequation for the maximum growth rate (ωr = �) and for the correspondingwavelength (λ = 3) of the form

3

a= 9.02

(1+ 0.45 Z0.5)(1+ 0.4 T0.7)

(1+ 0.87 We1.672 )0.6

(2a)

�ρ1a3

σ 0.5= 0.34+ 0.38 We1.5

2

(1+ Z)(1+ 1.4 T0.6)(2b)

whereZ = We0.51 /Re1, T = ZWe0.52 , We1 = ρ1U

2a/σ , We2 = ρ2U2a/σ ,

and Re1 = Ua/ν1. U is the relative velocity between the jet and the gas,and the subscripts 1 and 2 identify properties based on the liquid and the gas,respectively. As can be seen from Equations 2a and 2b, the maximum wavegrowth rate increases and the corresponding wavelength decreases rapidly withincreasing Weber number, which is the ratio of the inertia force to surfacetension force acting on the jet. The effect of the liquid viscosity (which appearsin the Reynolds number,Re, and the Ohnesorge number,Z) is seen to reducethe wave growth rate and to increase the wave length significantly as the liquidviscosity increases.

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JET BREAKUP 91

At low jet velocities (small Weber numbers) it is reasonable to assume thatdisruption of the jet occurs when the dominant wave’s amplitude is equal to thejet radius. In this case, the jet breakup time,τ , is given bya = η0 exp (�τ ),and the breakup length is predicted to be

L = Uτ = U/� ln(a/η0) (3)

For low-speed jets in the Rayleigh breakup regime, the parameter ln(a/η0)has been determined from experiments to be roughly equal to 12. For aninviscid liquid it is readily seen that Equation 3, when combined with Equation2b, predicts the linear increase in jet breakup length with jet velocity at lowgas densities, sinceτ is independent of the jet velocity. A maximum in thebreakup length curve is predicted if the gas density is non-zero, i.e. the theorypredicts that aerodynamic effects are responsible for the decrease in the breakuplength as the Weber number is increased beyond the maximum point. However,discrepancies have been found between the predicted location of the maximumpoint and experimental data.

The shape of the breakup curve has been reviewed by many researchers,including Grant & Middleman (1966) and McCarthy & Malloy (1974) whodiscussed the effects of the ambient gas, fluid properties, and nozzle design.Leroux et al (1996) pointed out that the location of the maximum point de-pends on nozzle parameters [presumably through the influence of the initialdisturbance term, ln(a/η0)], and also on the magnitude of the gas density itself.Indeed, Leroux et al (1996) proposed empirical modifications to account forthese effects, which extend the theory of Sterling & Sleicher (1975).

Beyond the first wind-induced breakup regime (region B, Figure 2) there iseven more confusion about the breakup-length trends. For example, Haenlein(1932) reported that the jet breakup length increases again with increasing jetvelocity (region C), and then abruptly reduces to zero (region D). McCarthy& Malloy (1974) reported that the breakup length continually increases. Morerecently, Hiroyasu et al (1991) discovered discontinuous elongations and short-enings of the jet with changes in the jet velocity. These apparent anomalies areassociated with changes in the nozzle internal flow patterns caused by separationand cavitation phenomena, which also exhibited hysteresis effects. Jets fromcavitating nozzles were found to have very short breakup lengths. Detachedflow jets have long breakup lengths. These phenomena may help explain theprevious discrepancies in measurements of breakup lengths in the spray liter-ature, since only recently have investigators paid attention to nozzle flow andgeometry effects.

Equation 3 predicts that the breakup length decreases continuously as the jetvelocity is increased when the effect of the gas density is significant. How-ever, the validity of the assumption thatL = Uτ becomes questionable for

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92 LIN & REITZ

high-speed jets, because the breakup mechanism is no longer due to capillarypinching, but is now due to the unstable growth of short-wavelength surfacewaves (Figure 2). In fact, as the jet velocity is increased, it becomes difficultto define a precise breakup length, and probability density functions are founduseful to quantify the breakup length (e.g. Leroux et al 1996).

The details of the unstable growth of short-wavelength waves on the surface ofthe liquid jet near the nozzle exit are obscured by the dense spray that surroundsthe jet. However, it is generally believed that the jet consists of an unbrokeninner liquid core in the vicinity of the nozzle exit, and droplets are strippedfrom the core by the action of aerodynamic forces at the liquid-gas interface(Reitz & Bracco 1982). Attempts have been made to measure the length ofthe core region by using intrusive techniques such as electrical conductivitymeasurements (e.g. Chehroudi et al 1985, Hiroyasu et al 1991), and laser sheetvisualization (e.g. Gulder et al 1994, Dan et al 1997). The core length dependson the liquid/gas density ratio and only weakly on the fluid properties and thejet velocity.

These trends can be demonstrated by using Taylor’s (1940) analysis of high-speed liquid jet breakup. Taylor considered the rate of mass loss per unit lengthof the jet caused by droplet erosion from the liquid surface resulting from theunstable growth of short-wavelength surface waves and showed that the breakuplength of a high-speed jet is given by

L/a = B(ρ1/ρ2)1/2/ f (T) (4)

whereT is Taylor’s parameter,T = ρ1/ρ2(Re1/We1)2, and the functionf (T )

has been approximated from Taylor’s numerical results asf (T ) = √3/6 [1−exp(−10T )] by Dan et al (1997).

The constantB in Equation 4 has a recommended value of 4.04 for typicaldiesel spray nozzles (Chehroudi & Bracco 1985). However, nozzle internaldesign effects are clearly important for high-speed jet breakup. It is known thathigh-speed liquid jets in jet cutting applications remain intact for a distanceof many diameters away from the nozzle. On the other hand, modern dieselinjectors employ very similar injection pressures, but diesel spray breakup startsat the nozzle exit (e.g. Figure 1d). The significant differences in the interiornozzle design features of these two applications account for their differentperformances. Diesel nozzles are typically short-length holes with sharp-edgedinlets, whereas jet cutting nozzles consist of contoured nozzles to minimizeinitial disturbance levels to the liquid flow.

Criteria for predicting the onset of the breakup regimes have been reviewedby Chigier & Reitz (1996). Consideration of the balance between the liquidinertia 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

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JET BREAKUP 93

(i.e. a jet is formed) ifWeL > 8, whereWeL = ρ1U2(2a)/σ . The criterion

Weg ≡ ρ2U2(2a)/σ < 0.4 corresponds to the point where the inertia force of

the surrounding gas reaches about 10% of the surface tension force. Ranz(1956) suggested that this would mark the beginning of the first wind-inducedbreakup regime, where the effects of the ambient gas are no longer negligible.Numerical results of Sterling & Sleicher (1975) indicate that the maximum inthe jet breakup length (see Figure 2) occurs whenWeL = 1.2+ 3.41Z0.9

1 , whereZ1 ≡ We0.5

L /ReL, ReL ≡ U(2a)/ν1. This could also indicate the importance ofaerodynamic effects, so that the criteria for Rayleigh breakup (see Figure 1a)would be

WeL > 8 and Weg < 0.4 or 1.2+ 3.41Z0.91 . (5)

Note, however, that nozzle turbulence and other flow effects are not includedin Equation 5. Ranz (1956) argued that the gas inertia force is of the same orderas the surface tension force whenWeg = 13. This could serve as a definitionof the end of the first wind-induced regime (see Figure 1b), which then occurswhen

1.2+ 3.41Z0.91 < 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 criterionWeg > 40.3 to predict the onset of theatomization regime, the point at which breakup appears to start at the nozzleexit (see Figure 1d). Thus, the criteria for breakup in the second wind-inducedregime are

13< Weg < 40.3 (7)

In the second wind-induced regime, the breakup starts some distance down-stream of the nozzle exit, and a smooth unbroken section of the jet is visibledownstream of the nozzle exit (Figure 1c).

As mentioned previously, no account is made of nozzle internal flow effectsin the above correlations. To address this shortcoming, Reitz (1978) assumedthat atomization corresponds to a critical value of the breakup length/nozzlediameter ratio. With this assumption, the onset of atomization is predicted tooccur when

ρ2/ρ1 > K f (T)−2 (8)

In this case, the parameterK was obtained from experiments on atomizing jetsand was found to be a function of the nozzle geometry, where

K = (0.53[3.0+ (`/2a)]1/2− 1.15)/744 (9)

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94 LIN & REITZ

and`/2a is the nozzle length-to-diameter ratio.K empirically accounts for theeffect of initial disturbances in the flow caused by nozzle internal flow phenom-ena such as turbulence, cavitation, and flow separation. Equation 9 includesthe effect of liquid viscosity and nozzle internal flows, and it predicts that at-omization is favored at high gas densities and for sharp inlet edge nozzles, withsmall length-to-diameter ratio. These trends agree with experiments reportedin the literature (Hiroyasu et al 1991, Reitz 1978, Reitz & Bracco 1979).

When injection takes place into a coflowing gas, additional breakup regimesare observed, as described by Chigier & Reitz (1996). This situation is ofmuch practical interest and it is frequently used in air-blast coaxial atomizersto improve the quality of atomization and to maintain it over a wide rangeof liquid flow rates. High gas velocities (up to sonic) are generated by high-pressure 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, causingstretching, destabilization, and flapping of the liquid jet. Eroglu et al (1991)measured breakup lengths of round liquid jets in annular coaxial air streamsand found that the breakup length decreases with increasing Weber number andincreases with increasing liquid jet Reynolds number according to the relation

L/2a = 0.5We−0.4L Re0.6

g (10)

whereL is the liquid intact length,a is the central tube inner radius, and theWeber and Reynolds numbers are based on the relative velocity between thegas and the liquid.

Jet breakup in coaxial flows is highly unsteady, and unstable liquid structuresare 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 flappingmotion and can be transformed into a curling liquid sheet. The sheet becomesstretched into a membrane bounded by thicker rims, which finally burst intoligaments and drops of various sizes.

Farago & Chigier (1992) classified coaxial jet disintegration into three maincategories: (a) Rayleigh-type breakup where the mean drop diameter is of theorder of the jet diameter—both axisymmetric breakup (forWeg< 15) and non-axisymmetric breakup patterns (for 15< Weg < 25) were observed; (b) jetdisintegration via the stretched-sheet mechanism, which produces membrane-type ligaments (25<Weg< 70)—in this case, the diameter of the drops formedis considerably smaller than the diameter of the jet; and (c) jet disintegrationvia fiber-type ligaments (100< Weg < 500)—at even higher air-flow rates,fibers are formed that peel off the liquid-gas interface. This breakup mecha-nism resembles the short-wavelength breakup mechanism of jets in the second

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JET BREAKUP 95

wind-induced and atomization regimes mentioned above. The atomization be-gins with the unstable growth of short-wavelength waves, the formation offibers, and their peeling off from the main liquid core. The fibers break intodroplets by the nonaxisymmetric Rayleigh-type jet disintegration mode. Again,the drop diameter is much smaller than the jet diameter.

3. BREAKUP MECHANISMS

Jet InstabilityThe regimes of jet breakup have been delineated above with correlations interms of relevant parameters. The results of recent works based on the the-ory of absolute and convective instability of liquid jets enable us to elucidatethe different physical mechanisms responsible for the jet breakup in the variousregimes. In the aerodynamic theory of spontaneous jet breakup without externalexcitation, it is assumed that the onset of breakup is caused by the amplificationof natural disturbances in a jet. Any arbitrary form of disturbance can be con-structed by superposition of all Fourier components. Each Fourier componenthas the form A exp[ikx+ ωt], where A is the wave amplitude, k= kr + iki isthe complex wave number whose real and imaginary parts give, respectively,the number of waves over a distance 2π and the exponential spatial growth rateper unit distance in the axial x-direction, andω = ωr + iωi is the complexwave frequency, the real and imaginary part of which give, respectively, theexponential temporal growth rate and the frequency of the Fourier wave.

Not all the Fourier components are capable of extracting energy from the jetsystem and amplifying, however. The Fourier components must have specialvalues of k andω, which depend on specific characteristics of the jet system,in order to grow from initially infinitesimal amplitudes. Mathematically, (k,ω)is determined as the eigenvalue of a linear system containing relevant flowparameters. The eigenvalues or the characteristic values are so determined thatthe condition of the existence of a nontrivial solution of the system is satisfied.This condition is the so-called characteristic equation or dispersion relation. Inthe linear aerodynamic theory of jet instability, the finite amplitude disturbancesintroduced outside or inside the nozzle by the various means mentioned aboveare excluded from consideration.

Absolute Instability and Formability of a JetIn the pioneering works cited above, the liquid jet is considered to be infinitelylong and k is assumed to be real. Thus the disturbance must grow or decayeverywhere in space at the same time rateωr. However, Keller et al (1972)noted that the disturbance initiating from the nozzle tip actually grows in spaceas it is swept downstream to break up the jet into drops, leaving a section of jet

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96 LIN & REITZ

intact near the nozzle tip. They set k to be complex and allow the disturbanceto grow in space as well as in time in a semi-infinite weightless inviscid jet ina vacuum. They found that Rayleigh’s results are relevant only in the case oflarge Weber number,WeL (WeL = ρLU

22a/σ is based on the liquid density).They also showed that in the limit ofWeL→∞, the spatial growth rate kI canbe inferred from the temporal growth rateωr by the relation ki = ±ωr + O(1/WeL), while the disturbance travels at the jet velocity. For Weber number lessthan the order of one, they found a new mode of faster-growing disturbanceswhose wavelengths are so long that they may not be actually observable.

Using the theory of absolute and convective instability (Briggs 1964, Bers1983), Leib & Goldstein (1986b) showed that the new mode actually corre-sponds to absolute instability arising from a saddle-point singularity in thecharacteristic equation. The unstable disturbances in an absolutely unstable jetmust propagate in both upstream and downstream directions. Thus, the unsta-ble disturbances expand in space over the course of time. This contrasts withwhat is observed in a Rayleigh jet, wherein unstable disturbances grow overtime as they are convected in a wave packet in the downstream direction withthe group velocity dωi/dkr (Lighthill 1987, Mei 1989). For a brief introductionto the theory of absolute and convective instability in the context of jet breakup,see the recent work of Lin (1996).

The critical Weber numberWeLc, below which an inviscid jet under weight-less condition in vacuum is absolutely unstable, and above which the jet isconvectively unstable, was found by Leib & Goldstein (1986b) to beπ . Whenthe viscosity of the jet is taken into account, the critical Weber number dependson the Reynolds numberReL = U(2a)/ν1, whereν1 is the liquid viscosityas shown in Figure 3 with the curve Q= 0, which was obtained by Leib &Goldstein (1986a). The other two curves for nonvanishing values of Q≡ ρ2/ρ1were obtained by Lin & Lian (1989), who were motivated to find out whetherthe absolute instability discovered by Leib & Goldstein is physical or mathe-matical, arising from the neglected ambient gas effect. It turns out that the effectof the gas density is to increaseWeLc (Figure 1). Thus, the gas density promotesabsolute instability in the sense that the given jet that is convectively unstablemay be made absolutely unstable by increasing the ambient gas density.

The absolute instability cannot be suppressed by either the gas compress-ibility (Zhou & Lin 1992a,b, Li & Kelly 1992) or by the gas viscosity (Lin& Lian 1993). Absolute instability is a real physical phenomenon, at least inthe absence of gravity, which is neglected in the theories. However, jet ab-solute instability under weightless conditions has not yet been reported in theliterature. The delineation of transition from convective to absolute instabil-ity in the absence of gravity is yet to be completed (Honohan 1995, Vihinen1996). In such a delineation, the critical Weber number is a function ofReL,

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98 LIN & REITZ

Q, and N≡ ν2/ν1, whereν2 is the viscosity of the surrounding gas. Exist-ing theoretical results show that for a given set ofReL, Q, and N, a liquid jetmay be made absolutely unstable by reducing the Weber number to be belowWeLc. The theoretical prediction that the unstable disturbance must propagatein both downstream and upstream directions when the jet velocity is smallerthan that corresponding toWeLc signifies that absolute instability occurs whenthe jet inertia is not sufficiently large to carry downstream all of the unstabledisturbances that derive their energy from the surface tension. Thus, surfacetension remains the source of instability. Part of the unstable disturbanceswill propagate back to the nozzle tip to interrupt the formation of a jet of anylength. It is likely that the phenomenon of absolute instability also exists in ajet in the presence of gravity, because the physical mechanism is unlikely tobe altered significantly by the gravity-induced variation in the thickness andthe velocity of the jet along its axis. Thus, the transition from absolute toconvective 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 theabsolutely unstable regime. The origin of nonformability of a jet is the surfacetension.

Capillary Pinching with Wind AssistanceThe jet breakup in the Rayleigh, the first wind-induced, the second wind-induced, and the atomization regimes defined in the previous section are allthe manifestations of convective instability. As mentioned earlier, the unstabledisturbances amplify in time as they are convected in a group in the down-stream direction with the group velocity. However, the physical mechanism ofbreakup in the second wind-induced and atomization regimes is fundamentallydifferent from that in the other regimes. Neglecting the viscosity of gas, Lin &Creighton (1990) calculated from the Navier-Stokes equations the mechanicalenergy budget of a liquid jet in a parameter range of convective instability. Thestability analysis in this range was completed earlier by Lin & Lian (1990).They expressed the total time rate of changes of the disturbance kinetic en-ergy in a controlled volume of the jet, over a wavelength of the most amplifieddisturbance, as the sum of the rate of work done by various relevant forces, i.e.

E = Pg + P̀ + S+ V + D, (11)

wherePg is the rate of work done by the gas pressure fluctuation at the liquid-gasinterface,P̀ is the rate of work done by the liquid pressure fluctuation at theinlet and outlet of the control volume,S is the rate of work done by the surfacetension,V is the rate of work done by the liquid viscous stress, andD is the rateof viscous dissipation of mechanical energy.

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Some typical values of (pg, p`, s, v, d) ≡ (Pg, P̀ , S, V, D)/E for variousbreakup regimes are shown in Table 1. The first four rows in this table belongto the Rayleigh and the first wind-induced breakup regimes. The last four rowsbelong to the second wind-induced and the atomization regimes. The valueskr = 2πa/λm in the fourth column are the wave numbers corresponding tothe wavelength,λm, of the most amplified disturbance predicted by the lineartheory for the flow parameters specified in the first three columns. Q= 0.0013corresponds to the case of a water jet in air under one atmosphere. The highReynolds number jet at low gas density depicted in the fourth row is closestto the idealized Rayleigh jet. The presence of the low density gas increasesonly slightly the most amplified wave number, 0.696, predicted by Rayleigh.Moreover,sdominates all other work terms. This is a classical case of Rayleighbreakup by capillary pinching, which produces drops of a diameter comparableto the jet diameter.

Capillary pinching remains the mechanism of breakup for the low Reynoldsnumber jet depicted in the first row (Table 1). Note that the value of kr isalmost four times smaller than that predicted by Rayleigh. As the drop sizeis inversely proportional to kr, the liquid viscosity tends to increase the dropsize considerably. This regime, which is not shown in Figure 1, may be termedthe Weber-Chandrasekhar regime to emphasize the important role of liquidviscosity.

With Q kept at 0.0013 in rows 2 and 3,pg increases with increasingReL

almost to the same order of magnitude ass. All other work terms remaininsignificant. Thus, as the relative speed of gas-to-liquid (wind speed) increases,the gas pressure fluctuation assists significantly the capillary force to break upthe liquid jet. Nevertheless the capillary force remains dominant over thewind force. Comparing the values of kr in these two rows with that of theRayleigh jet, it is seen that the drop size in this first wind-induced breakupregime may be slightly larger or smaller than that in the Rayleigh regime,

Table 1 Energy budget in jet breakup

ReL 103/WeL Q × 103 kr s pg p` v d

2 1.25 1.3 0.1669 117.5 3.5 −0.2 0.2 −21.6102 1.25 1.3 0.5725 97.7 26.2 −4.9 1.6 −20.64 × 102 2.50 1.3 0.7701 65.5 41.8 0.0 0.1 −7.44 × 104 1.25 0.1 0.7088 96.2 3.5 0.4 0.0 −0.111016 0.882(−2) 1.3 35.417 −296.0 547.7 −2.1 −0.1 −149.536720 0.882(−2) 1.3 40.051 −215.1 351.7 −2.0 0.1 −34.767411 0.882(−2) 1.3 41.580 −210.4 331.6 −1.9 0.0 −19.3116122 0.802(−2) 1.3 42.368 −214.7 332.1 −1.0 −0.2 −16.2

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depending on the flow parameters (see Figure 1b). Since the surface tension ismainly responsible for the breakup in this regime, and the gas inertia force onlyassists in the breakup, rather than inducing the breakup, it is probably moreappropriate to call this regime the wind assisted breakup regime instead of thefirst wind-induced breakup regime. The second wind-induced breakup regimeis genuinely wind-induced, as is explained below.

Interfacial Stress Against Surface TensionFor the parameter range specified in the last four rows (Table 1), kr is foundto 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 thanthe jet radius (see Figure 1c andd ). In contrast to the previous four cases, thepressure work-term dominates the surface tension work-term in the last fourrows. In fact the surface tension term is negative. That is to say, the surfacetension acts against the formation of small droplets generated by the interfacialpressure fluctuation in the second wind-induced and atomization regimes. Partof the kinetic energy in a jet is converted through the pressure work-term to thesurface energy in the droplets. Thus the second wind-induced and atomizationregimes are genuinely wind-induced.

It is seen (Table 1) that while the atomization and second wind-inducedregimes exist in the parameter rangeWeL� Q, the rest of the regimes exist forWeL ≤ 1. It has been shown (Kang & Lin 1987) that the unstable disturbancesin the atomization regime scale with the gas capillary length c= σ/ρ2U

2. TheconditionWeL�Q implies that c is much smaller than the jet diameter (Lin &Lian 1990). The interfacial shear stress fluctuation will augment the pressurefluctuation in the breakup process, if the gas viscosity is considered.

It should be remembered that the linear stability theory is not capable ofdifferentiating between the second wind-induced and the atomization regimes.The linear theory can only predict the onset of instability, which produces in-terfacial waves of different length scales depending on the parameters. Thenonlinear processes of pinching off a small droplet from the interface, subse-quent to the onset, and the continuous generation of droplets from the recedinginterface toward the core of a jet are involved in reaching the atomization state(see Figure 1d). However, the physical mechanism involved in the initial stageof the second wind-induced and the atomization regimes may be the same.

4. DISCUSSION

Interfacial Shear LayerA serious defect common to all of the above reviewed works is the lack ofa rigorous treatment of the effect of the gas viscosity. Sterling & Sleicher

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(1975) assumed with Benjamin (1959) that the Kelvin-Helmholtz model canbe applied locally along the interfacial wave with an arbitrary correction factorthat—because of the viscous effect—is used to reduce the pressure distributionpredicted by the Kelvin-Helmholtz model. Thus the possibility of generationof droplets by shear waves is missed. Lin & Lian (1990) modeled the liquid-gasinterfacial boundary layer with the boundary layer over a wavy solid surface inorder to estimate the interfacial shear effect. This is not satisfactory because afluid-fluid interfacial shear layer is fundamentally different from that of a solid-fluid boundary layer. The effect of gas viscosity was rigorously analyzed byLin & Ibrahim (1990) with temporal theory, and by Lin & Lian (1993) with thetheory of absolute and convective instability for a viscous liquid jet in a viscousgas 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 stronginterfacial shear were not sufficiently accurate to allow the drawing of definitiveconclusions on the effect of the gas viscosity on the atomization process. Therelative importance of the shear stress fluctuation to the pressure fluctuation inthe atomization process in various parameter ranges remains unknown.

Nonaxisymmetric DisturbancesThe above description is based on works that assume that axisymmetric distur-bances are more unstable than asymmetric ones. Rayleigh was able to provethat asymmetric temporal disturbances in an inviscid jet are all stable. Tem-porally stable waves are also convectively stable evanescent waves (Huerre &Monkewitz 1990). ForReL = O (10), Lin & Webb (1994) showed that theasymmetric disturbances are evanescent waves in the parameter range 10−4≤Q≤ 10−2, 10≤We≤ 103. Yang (1992) demonstrated that temporally grow-ing asymmetric long-wavelength disturbances may become dominant whenthe Weber number of an inviscid jet in an inviscid gas is in the atomizationregime. However, the viscosity of the liquid tends to bring down the temporalgrowth rate of nonaxisymmetric disturbances (Avital 1995), and the maximumtemporal growth rates of axisymmetric disturbances remain higher than thoseof asymmetric ones except when the jet is almost inviscid, for example whenReL = 105 andWeL = 104 (Li 1995).

A similar conclusion was reached by EA Ibrahim (personal communication),who investigated convectively unstable asymmetric disturbances in a viscousjet emanating into an inviscid gas, whenWeL and ReL are of order 103 andhigher. There is also experimental evidence of asymmetric disturbances inthe atomization regime mentioned above (Meister & Scheele 1969, Taylor &Hoyt 1983, Eroglu et al 1991). However, the appearance of non-axisymmetricdisturbances may also be caused by the secondary instability after the onset ofinstability from axisymmetric disturbances. In the Rayleigh, the wind-induced,

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and the wind-assisted regimes, the asymmetric disturbances may be brought outprominently by the swirl in the liquid jet (Ponstein 1959, Kang & Lin 1989) orless prominently by the swirl in the gas (Lin & Lian 1990).

Success and Shortcoming of Linear TheoryDespite these shortcomings, the linear stability analysis started by Rayleigh pro-vides a qualitative description of the physical mechanisms involved in variousregimes of jet breakup. The linear theories have even enjoyed some reason-able semiquantitative comparisons with experiments in the atomization regime(Kang & Lin 1987, Reitz & Bracco 1982). These comparisons include theintact length, spray angle, and droplet size, which scales with the gas capillarylengthσ/ρ2U

2. The quantities that have just been mentioned are the productsof highly nonlinear processes. The reason such a reasonable comparison basedon linear theory is possible is probably due to the fact that the physical mech-anisms at work in various breakup regimes are already basically determined atthe onset of instability. The nonlinear evolution subsequent to the onset onlymodifies quantitatively the physical mechanisms.

On the other hand, the excellent quantitative comparison between the theoryof Rayleigh and the experiments of Goedde & Yuen (1970) and Donnelly &Glaberson (1966) is probably fortuitous. While Rayleigh neglected the exis-tence of the ambient gas, the liquid viscosity, and gravity, none of these areneglected in experiments. Moreover, the Rayleigh jet breakup is due to capil-lary pinching, and yet the experiments do not seem to be sensitive to the Webernumber. A complete delineation of where each regime should start and endin the parameter space (Q, N, Re, We, Fr) may be made within the frameworkof linear theory only if the various effects of nozzle flows on jet instabilityare known or assumed known. Here, the Froude number, Fr= U 2/g(2a),represents the ratio of inertial to gravitational force.

5. SUMMARY AND UNRESOLVEDSCIENTIFIC ISSUES

This study discusses various regimes of breakup of liquid jets injected intoboth stagnant and coflowing gases. Available criteria for the transition betweenthe regimes are reviewed. The physical mechanisms at work in the differentbreakup regimes are described. The influence of nozzle internal flow effects isshown to be important, but these effects are only included empirically in currentwave-stability theories.

A useful area for future research would be the development of fundamentallybased models that account for the effect of nozzle internal flows on the liquidbreakup process. In addition, current breakup models need to be extended

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to account for the nonlinear effects of liquid distortion, membrane formation,and stretching on the atomization process. These phenomena are especiallyimportant 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 rigorouslyanalyzed and tested. This knowledge is not only important for applicationsinvolving jet breakup, it will also advance our scientific understanding of var-ious processes in nature as well as in industries. The impact of the stick-slipcondition experienced by the jet liquid exiting the nozzle tip and the spatialdevelopment of the liquid-gas interfacial shear layer (before the jet flow is fullydeveloped) on the receptivity of the jet to instability need to be investigated. Theeffect of gravity in the absolute instability regime and the Weber-Chandrasekharregimes, where the Froude number is small, remains to be elucidated. In theatomization regime, Fr is so large that the gravitational effect may not be signif-icant. A complete delineation of the jet breakup regimes in the entire parameterspace (Q, N, Re, We, Fr), even in the framework of linear theory, has not yetappeared. The nonlinear studies, which take into account the finite amplitudedisturbances originated in the nozzle, and which elucidate the nonlinear processsubsequent to the onset of unstable waves of various-length scales, will con-tribute to the understanding of the formation of sprays and drops from a liquidjet.

ACKNOWLEDGMENTS

Support for SP Lin was provided in part by Army Research Office GrantDAAL03-89-K-0179 and NASA Grant NAG3-1891. Support for R Reitz wasprovided by the Army Research Office Contract DAAL03-86-K-0174.

Visit the Annual Reviews home pageathttp://www.AnnualReviews.org.

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Annual Review of Fluid Mechanics Volume 30, 1998

CONTENTSLewis Fry Richardson and His Contributions to Mathematics, Meteorology, and Models of Conflict, J.C.R. Hunt 0

Aircraft Laminar Flow Control , Ronald D. Joslin 1

Vortex Dynamics in Turbulence, D. I. Pullin, P. G. Saffman 31

Interaction Between Porous Media and Wave Motion, A. T. Chwang, A. T. Chan 53

Drop and Spray Formation from a Liquid Jet, S. P. Lin, R. D. Reitz 85

Airplane Trailing Vortices, Philippe R. Spalart 107

Diffuse-Interface Methods in Fluid Mechanics, D. M. Anderson, G. B. McFadden, A. A. Wheeler 139

Turbulence in Astrophysics: Stars, V. M. Canuto, J. Christensen-Dalsgaard 167

Vortex-Body Interactions, Donald Rockwell 199

Nonintrusive Measurements for High-Speed, Supersonic, and Hypersonic Flows, J. P. Bonnet, D. Grésillon, J. P. Taran 231

Renormalization-Group Analysis of Turbulence, Leslie M. Smith, Stephen L. Woodruff 275

Control of Turbulence, John Lumley, Peter Blossey 311

Lattice Boltzmann Method for Fluid Flows, Shiyi Chen, Gary D. Doolen 329

Boiling Heat Transfer, V. K. Dhir 365

Direct Simulation Monte Carlo--Recent Advances and Applications, E.S. Oran, C.K. Oh, B.Z. Cybyk 403

Air-Water Gas Exchange, B. Jähne, H. Haußecker 443

Computational Hypersonic Rarefied Flows, M. S. Ivanov, S. F. Gimelshein 469

Turbulent Flow Over Hills and Waves, S. E. Belcher, J. C. R. Hunt 507

Direct Numerical Simulation: A Tool in Turbulence Research, Parviz Moin, Krishnan Mahesh 539

Micro-Electro-Mechanical-Systems (MEMS) and Fluid Flows, Chih-Ming Ho, Yu-Chong Tai 579

Fluid Mechanics for Sailing Vessel Design, Jerome H. Milgram 613

Direct Numerical Simulation of Non-Premixed Turbulent Flames, Luc Vervisch, Thierry Poinsot 655

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