ILASS-Americas 29th Annual Conference on Liquid Atomization and Spray Systems, Atlanta, GA, May 2017

Planar Liquid Sheet Breakup Mechanisms, Time Scales, and Length ScaleCascade

A. Zandian1∗, W.A. Sirignano1, and F. Hussain2

1Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaIrvine, CA 92697 USA

2Department of Mechanical EngineeringTexas Tech University

Lubbock, TX 79409, USA

AbstractThe temporal, three-dimensional instabilities on a planar liquid sheet are studied using Direct NumericalSimulation and the level-set and volume-of-fluid interface tracking. Three breakup mechanisms are distin-guished at early breakup, which are well categorized on a gas Weber number (Weg) versus liquid Reynoldsnumber (Rel) diagram. At low Rel and low Weg, liquid lobes stretch directly into ligaments. Thick lig-aments and large droplets occur in this zone. With increasing Rel at high Weg, the breakup mechanismmanifests hole formation. Breakup is initiated with lobe thinning and perforation, leading to formation ofbridges and then ligaments. At lower Ohnesorge number (Oh) and higher Rel, hole formation is prohib-ited at early breakup. The lobes are corrugated; thin ligaments result from corrugation stretching. Thesemechanisms are relatively independent of the jet configuration - seen in both planar and circular liquid jets.The characteristic times for the hole formation and the ligament stretching differ - the former dependingon surface tension and the latter on liquid viscosity. In the transitional region, both characteristic timesare of the same order. Using the local radius of curvature of the surface and the local cross-flow coordinateof the spray surface, two series of PDFs are obtained over a wide range of length scales. The radius PDFshows that, with increasing Weg, the average curvature increases, the number of small droplets increases,and cascade occurs at a faster rate. The other PDF shows the spray expansion, with the spray angle beinglarger at higher Weg, higher density ratios, and lower Rel.

∗Corresponding Author: azandian@uci.edu

Introduction

Atomization is of importance in numerous in-dustrial, automotive, and aerospace fields, and isencountered in applications such as fuel injection,propulsion and combustion systems, agriculturalsprays and chemical reactors. The initial breakup ofa liquid jet is often referred to as primary breakupor primary atomization. A population of largerdroplets produced in the primary atomization maybe unstable at some critical flow conditions and thusmay undergo further instability leading to smallerdroplets. This process is usually termed secondarybreakup. Controlling the droplet size distributionproduced during primary atomization is the key todesigning an efficient apparatus, as it leads to highervolumetric heat release rates, wider burning ranges,and lower exhaust concentrations of pollutant emis-sions.

While increasing relative gas/liquid velocity hasbeen shown to produce smaller droplets, the mech-anism for breakup has not been clearly established.The common notion is that the shear causes span-wise waves to form on the liquid surface, and thewaves grow and separate from the sheet in the formof ligaments, which then fragment into droplets [1].There have been several numerical and experimen-tal studies on liquid sheet breakup; however, untilrecently, none of them have been able to explain thebreakup mechanisms and delineate the flow parame-ters affecting those mechanisms. Understanding thebreakup mechanisms during primary atomization isthe first step towards controlling the size distribu-tion of the droplets and the spray angle. The pur-pose of this study is to explain these mechanismsand to analyze their effects on the ultimate dropletsize or spray expansion.

Analytical studies

Linear analytical studies have shown that therecan only exist two modes of unstable waves on thetwo gas/liquid interfaces, corresponding to the twosurface waves oscillating in and out of phase, com-monly referred to as the sinuous (anti-symmetric)and varicose (symmetric) modes [2, 3]. The nonlin-ear calculations of Rangel and Sirignano [4] indicatethe dominance of varicose modes when the densityratio is of the order of one, while sinuous waves pre-vail at lower gas densities.

Reitz and Bracco [5] identified four distinctregimes of liquid jet breakup. The regimes were sep-arated by straight lines with negative slope on thelog-log plot of Weber number (We) versus Reynoldsnumber (Re). The first domain at lower values ofWe and Re is the Rayleigh capillary mechanism re-

gion, where aerodynamic interaction with the gasis not significant, and symmetric distortion occurswith the formation of droplets that have diametersof the same order as the jet thickness. Next is thefirst wind-induced region where sinuous oscillationsoccur resulting in droplet sizes still comparable tothe jet thickness. The second wind-induced regioninvolves smaller droplets, while the atomization re-gion at the highest values of We and Re producesthe smallest droplets. As We or Ohnesorge num-ber (Oh) increases, the breakup length tends to de-crease. The atomization regime is the domain wherethe breakup process begins essentially at the orificeexit and is of much broader practical interest but yetfar less studied than the other three domains. Thatatomization domain is the focus of our study.

Experimental studies

The transient behavior of liquid injection hasbeen the subject of numerous experimental studiesfor planar liquid sheets [1, 6–9]. Typically, theseexperiments are conducted at atmospheric pressurewith very low gas-to-liquid density ratios. Fraserand Eisenklam [6] defined three modes of sheet dis-integration, described as rim, wave, and perforated-sheet disintegration. Stapper and Samuelsen [1] sug-gested two regimes of liquid sheet breakup: cellularbreakup at high gas-to-liquid relative velocities, andstretched ligament breakup at lower relative veloc-ities. The relative strengths of the spanwise andstreamwise vortical waves in the sheet was a control-ling factor in the breakup mechanism. Their studyrevealed that the relative gas/liquid velocity is a pri-mary factor in the ligament formation and breakup.Breakup is the result of a complex three-dimensionalinteraction of spanwise and streamwise waves, whichstretch thin film membranes; an increase in the rel-ative velocity leads to further stretching of both theliquid membranes and streamwise ligaments, bothleading to smaller droplets. They also showed thatthe variation in the liquid properties did not alterthe general character of the two breakup mecha-nisms; however, it had a pronounced effect on thetime and length scales. Most of their study was fo-cused on relatively low Re (Re < 3000) and low We(We < 103), and did not clearly specify the transi-tion between their breakup regimes.

Planar laser-induced fluorescence (PLIF) wasused by Lozano and Barreras [7] to visualize the airflow in the near field of an air-blasted breaking wa-ter sheet. They used the air-water momentum ratioto characterize different breakup regimes. They re-vealed detachment of the air boundary layer over theair-water interface behind the zones of strong cur-

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vature, and identified the pressure field induced bythese vortices as a cause of the enhanced sheet flap-ping and instability growth. Their experiments wereconducted at a fairly high Re of O(104) and a lowWe of O(103). Later, Wahono et al. [9] used high-speed imaging to qualitatively visualize the structureof the annular spray at low We and relatively lowRe. They showed that thicker and longer ligamentsare formed at higher Re’s, which is attributed tohigher liquid momentum.

Even though several types of breakup mecha-nisms and the corresponding ligament and dropletsizes have been introduced by several experimen-talists, they have not clearly identified the domainin which these breakups occur. Moreover, many ofthe studies focused on very low Re and We, whichis far from the domain of interest in most applica-tions. Generally, experiments have not been ableto reveal the details of the smaller structures thatdevelop during cascade of atomization and only thequality of the developed ligaments and droplets atthe end of the breakup has been discussed. In thisregard, the numerical simulations have proven to bea much better tool to investigate the mechanisms ofthe primary atomization.

Computational studies

Scardovelli and Zaleski [10] distinguished twoscenarios by which three-dimensional instabilitiesset in in response to spanwise perturbations of thetwo-dimensional base flow in their three-dimensionalsimulations. For weak three-dimensional perturba-tions, the simulation remains two-dimensional untilthe sheet breaks up and a cylinder with spanwiseaxis pinches off. Subsequently, the capillary insta-bility gives a three-dimensional structure to the flow.In contrast, for higher three-dimensional perturba-tions, the rim-like edge concentrates in protrudingligaments that are subject to the capillary instabil-ity. However, the unstable cylinder is now stream-wise oriented. The formation of streamwise liga-ments is also observed in real experiments on shearlayers [11].

A detailed numerical investigation of turbu-lent liquid jets in quiescent air at 0.025 gas-to-liquid density ratio was conducted by Desjardins andPitsch [12] to identify the key atomization mecha-nisms, and to analyze the effects of the jet Re andWe. Their work was limited to We of O(102–103),which were low compared to our range of interest forcommon liquid fuels and high-pressure operations.They found that liquid turbulence plays an impor-tant role in the generation of the first droplets, whilethe Kelvin-Helmholtz (KH) instabilities are not vis-

ible at early times. Several detailed atomizationmechanisms have also been identified by them, suchas bubble formation through sweep-ejection eventsand ligament generation due to bubble bursting ordroplet collision. The instantaneous liquid-gas in-terface in their computations showed some hole for-mation near the tips of a few of the liquid sheetsexpelled from the planar jet core, tearing of thosesheets and formation of the ligaments due to thetearing. However, they did not explain the mecha-nisms of hole formation.

Computational studies of Shinjo and Umemura[13] on round liquid jets at higher We, e.g. O(104–105), and lower Re, e.g. O(103–104), than the simu-lations of Desjardins and Pitsch [12], indicated liga-ment formation from the development of holes in theliquid crests. They showed that disturbances prop-agate upstream from the liquid jet tip through vor-tices and droplet re-collision. They claimed that col-lision of the droplets, that were broken from the lig-aments formed at the back of the mushroom-shapedcap, with the liquid jet core formed holes on the liq-uid lobes. When the lobe surface area increased andits rims became thicker, the lobe surface puncturedto form two or three ligaments as the hole extendedto the tip of the lobe.

Lobe perforation leading to ligament formationsimilar to the mechanism observed by Shinjo andUmemura [13] was also observed by Jarrahbashi andSirignano [14] for larger gas-to-liquid density ratios.However, they observed holes forming on the lobeseven before the ligaments form and break up intodroplets. Therefore, most probably, droplet colli-sion with the liquid jet core is not the only factor,nor even the major factor, promoting hole formation[15]. Herrmann [16] also showed the disintegrationof the liquid core into ligaments and droplets, andalso predicted the droplet sizes. However, ligamentformation mechanisms were not addressed.

Jarrahbashi and Sirignano [14] and Jarrahbashiet al. [15] covered a wide range of density ratio (0.05–0.9), Re (320–16, 000) and We (2000–230, 000) intheir computational studies of round liquid jets. Therole of vorticity dynamics was examined via post-processing. They showed that the three-dimensionalinstability starts as a result of the appearance ofcounter-rotating pairs of streamwise vortices in be-tween two consecutive vortex rings, which are ac-tually hairpin vortices wrapped around the vortexrings, and produce the first lobes. They also foundthat, for low gas-to-liquid density ratios, ligamentsmostly form due to elongation of the lobes them-selves; however, most of the ligaments form as a re-sult of hole tearing at higher density ratios.

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In the latest numerical analysis by Zandian etal. [17], various breakup patterns are identified atdifferent flow regimes for a wide range of viscosityratio, Re, and We similar to the ranges covered byJarrahbashi and Sirignano [14], and high density ra-tios (0.5–0.9) of a thin liquid sheet. Two distinctmechanisms were proposed regarding breakup of theliquid lobes into droplets: hole formation and ex-pansion, and ligaments formation due to stretchingof the lobes, which happen at low/medium and highRe’s, respectively. The cascade of structures andtheir causes were explained from a vorticity dynam-ics perspective.

Droplet size distribution and spray angle

In the second part of this paper, we analyze theeffects of different flow parameters, e.g. Re, We, anddensity ratio, on the temporal variation of the lengthscales of the jet and the spray expansion rate in thevarious breakup domains that we introduce in thefirst part. The spray angle measurement is a formof evaluating the amplitude of the liquid sheet in-stability, as the former is closely dependent on thelatter. The focus of most of the previous studies hasbeen to assess the manner by which the final dropletsize distribution is affected by gas and liquid proper-ties, and the geometry of the nozzle [1, 7, 11, 18–23].The general conclusions are that the Sauter MeanDiameter (SMD) decreases with increasing relativeair/liquid velocity, increasing liquid density, and de-creasing surface tension, while viscosity is found tohave little or no effect [1]. While the main focus hasbeen on the final droplet size distribution and thespatial growth of the spray (spray angle), little em-phasis has been placed on the temporal cascade ofthe droplet size and the expansion rate of the sprayin primary atomization. This is the main focus ofour research.

Dombrowski and Hooper [18] developed theo-retical expressions for the size of drops producedfrom fan spray sheets, and demonstrated that thedrop size increases with ambient density. They alsoshowed that, for relatively thin sheets, the drop sizeis a direct function of the surface tension and theliquid density, and an inverse function of the liq-uid injection pressure and the gas density. However,for relatively thick sheets, drop size is independentof surface tension or injection pressure and increaseswith increasing gas density. Later, Senecal et al. [19]showed that the ligament diameter is directly pro-portional to the sheet thickness, and inversely pro-portional to the square root of the gas Weber (Weg)number.

Detailed measurements of mean drop size and

velocity on liquid sheets have been made by Mansourand Chigier [20], showing that substantial decreasein the spray angle occurs as a result of increasing theliquid flow rate while maintaining the same air pres-sure. They related this behavior to the reduction inthe specific energy of air per unit volume of liquidleaving the nozzle. It has also been shown that in-creasing the air pressure for a fixed liquid flow rateresults in the increase of the spray angle. They ob-served a substantial increases in SMD at all axiallocations, from the central part toward the edge ofthe spray, which were related to the lower air-liquidrelative velocity on the sheet edges [20]. Lozano etal. [7] obtained correlations relating physical fluidproperties, nozzle geometry and flow conditions withmean droplet diameter. They found that the meandiameter decreases with increasing gas velocity. Fora fixed gas velocity, and varying liquid velocities, themean diameter presents a minimum roughly coinci-dent with the frequency maximum. When measur-ing at larger downstream distances for a fixed gasvelocity the diameter always tends to decrease withincreasing momentum flux ratio due to secondarybreakup.

Carvalho et al. [21] performed detailed measure-ments of the spray angle versus gas and liquid ve-locities in a flat liquid film surrounded by two airstreams. They showed that, for low gas-to-liquidmomentum ratio, the atomization quality is ratherpoor and a narrow spray angle is obtained. Forhigher gas-to-liquid momentum ratios, the atomiza-tion quality is considerably improved, as the sprayangle increases significantly.

At present, it is not possible to predict the vari-ation of droplet diameters and their distribution as afunction of injection conditions. For combustion ap-plications, many empirical correlations are availablefor the droplet size as a function of injection parame-ters as compiled by Lefebvre [24]; however, more de-tailed studies of fundamental breakup mechanismsare clearly needed in order to construct predictivemodels. The dependence of the primary droplet size,d, on the atomizing gas velocity, Ug, is most often ex-pressed in the form of a power law, d ∝ U−ng , where0.7 ≤ n ≤ 1.5. Physical explanations for particu-lar values of the exponent n are generally lacking.Varga et al. [22] found that the mean droplet sizeis not very sensitive to the liquid jet diameter. Thetrend is actually opposite to intuition; the dropletsize is observed to be slightly larger for a smallerliquid nozzle diameter. This effect was attributed tothe slightly longer gas boundary-layer attachmentlength.

Marmottant and Villermaux [11] presented

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probability density functions (PDF) of the ligamentsize and the droplet size after breakup. The lig-ament size d0 was found to be distributed aroundthe mean in a nearly symmetric bell-shaped distribu-tion. The droplet diameter d was more broadly dis-tributed with a skewed distribution. When the sizeswere rescaled by d0, which depends on the gas ve-locity ug, their distribution keeps roughly the sameshape for various gas flows. The droplet size distri-bution P (d) displays an exponential fall-off at largediameters solely parameterized by the average liga-ment size 〈d0〉, P (d) ∼ exp(−nd/ 〈d0〉). The param-eter n ≈ 3.5 slowly increased with the gas velocity.The mean droplet size in the spray decreases likeu−1g .

Most of the numerical studies on the liquid-jetbreakup in the past few years have focused on qual-itative investigation of the effects of fluid propertiesand flow parameters on the final droplet size distri-bution or the spray size. In only a few cases, quan-titative results have been presented for the spatialvariation of the droplet/ligament size along or acrossthe spray axis. However, there has not been any de-tailed study on the liquid-structure size distributionand the spray-size (spray-angle) variation throughtime.

Desjardins and Pitsch [12] defined the half-widthof the planar jets as the distance from the jet center-line to the point at which the mean streamwise ve-locity excess is half of the centerline velocity. Theyshowed that high We jets grow faster, which sug-gests that surface tension forces tend to stabilizethe jets. They also demonstrated that ligaments arelonger, thinner, and more numerous as We is in-creased. The corrugation length scales appear largerfor lower Re. This is attributed to less energy that iscontained in small eddies for low Re case comparedto higher Re. Consequently, the early deformationon the smaller scales of the interface is more likelyto take place on relatively larger length scales as Reis reduced.

Jarrahbashi et al. [15] defined the radial scale ofthe two-phase mixture as the outermost radial po-sition of the continuous liquid, and showed that theradial spray growth increases with increasing gas-to-liquid density ratio. They qualitatively showedthat larger droplets are formed for higher gas den-sities and lower Weber numbers. Zandian et al. [17]in a study similar to Jarrahabshi et al. [15], butfor planar liquid jets, attempted to show the ef-fects of We on the droplet size as well as the liq-uid sheet expansion rate. They showed the vari-ation in droplet size and the number of dropletsfor the range of 3000 < We < 72, 000; however,

both the size and the distribution of droplets wereobtained from visual post-processing, which incurnon-negligible errors. Their qualitative comparisonshowed that droplet and ligament sizes decrease withincreasing We, while the number of droplets in-creases. They did not report the temporal cascadeof the length scales though. Zandian et al. [17] main-tained the same spray size definition used by Jarrah-bashi et al. [15] This definition lacks any statisticalinformation about the whole liquid jet cross-section;i.e. it does not show the number density of the liquiddroplets or ligaments at any cross-flow distance fromthe centerline. Thus, this definition is not a properrepresentative of the liquid spray size, and fails toshow the effects of We on spray expansion rate.

Objectives

Our objectives are to detail and explain (i) cas-cade of structures on the liquid surface with time,including lobe, ligament, and droplet formations; (ii)the breakup mechanisms at different flow conditions;(iii) the effects of density ratio, viscosity ratio, andsheet thickness on the liquid sheet breakup; and (iv)proper definition of the time scale of each of thebreakup mechanisms, which would help predict thedominant mechanism at different flow conditions. Inthe second part of this paper, we study the effects ofdifferent flow parameters and fluid properties on thetemporal variation of the spray size and the liquid-structure length scales. The intention is to establisha new model and definition for measurement of thedroplet size distribution and spray size, and to ex-plain the role different breakup regimes play in thisrespect.

Numerical Modeling

The three-dimensional Navier-Stokes with level-set and volume-of-fluid surface tracking methodsyield computational results for the liquid-segmentwhich captures the liquid-gas interface deformationswith time.

The incompressible continuity and Navier-Stokes equations, including the viscous diffusion andsurface tension forces and neglecting the gravita-tional force, are as follows;

∇ · u = 0, (1)

∂(ρu)

∂t+∇·(ρuu) = −∇p+∇·(2µD)−σκδ(d)~n, (2)

where D is the rate of deformation tensor,

D =1

2

[(∇u) + (∇u)T

]. (3)

u is the velocity field; p, ρ, and µ are the pres-sure, density and dynamic viscosity of the fluid, re-

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spectively. The last term in Eq. (2) is the surfacetension force per unit volume, where σ is the surfacetension coefficient, κ is the surface curvature, δ(d)is the Dirac delta function and ~n is the unit vectornormal to the liquid/gas interface.

Direct numerical simulation is done by usingan unsteady three-dimensional finite-volume solverto solve the Navier-Stokes equations for the planarincompressible liquid sheet segment (initially stag-nant), which is subject to instabilities due to a gasstream that flows past it on both sides. A uni-form staggered grid is used with the mesh size of2.5 µm and a time step of 5 ns. Third-order accu-rate QUICK scheme is used for spatial discretizationand the Crank-Nicolson scheme for time marching.The continuity and momentum equations are cou-pled through the SIMPLE algorithm.

The level-set method developed by Osher andhis coworkers [25–27] tracks the liquid-gas interface.The level set φ is a distance function with zero valueat the liquid-gas interface; positive values in the gasphase and negative values in the liquid phase. Allthe fluid properties for both phases in the Navier-Stokes equations are defined based on the φ valueand the equations are solved for both phases simul-taneously. Properties such as density and viscosityvary continuously but with a very large gradient nearthe liquid-gas interface. The level-set function φ isalso advected by the velocity field:

∂φ

∂t+ u · ∇φ = 0. (4)

For detailed descriptions for this interface track-ing see [26].

At low density ratios, a transport equation sim-ilar to Eq. (4) is used for the volume fraction f ,also called the VoF-variable, in order to describe thetemporal and spatial evolution of the two phase flow[28]. The VoF-variable f , introduced by Hirt andNichols [28], represents the volume of (liquid phase)fluid fraction in each cell.

The formulation for the fully conservative mo-mentum convection and volume fraction transport,the momentum diffusion, and the surface tensionare treated explicitly. To ensure a sharp interfaceof all flow discontinuities and to suppress numeri-cal dissipation of the liquid phase, the interface isreconstructed at each time step by the PLIC (piece-wise linear interface calculation) method of Riderand Kotche [29]. The liquid phase is transportedon the basis of its reconstructed distribution. Thecapillary effects in the momentum equations are rep-resented by a capillary tensor [10].

Flow Configuration

The computational domain, shown in Fig. 1,consists of a cube, which is discretized into uniform-sized cells. The liquid segment, which is a sheetof thickness h (h = 50 µm for the thin sheet and200 µm for the thick sheet cases in this study), islocated at the center of the box and is stationaryin the beginning. The domain size in terms of thesheet thickness is 16h × 10h × 10h, in the x, y andz directions, respectively, for the thin sheet, and4h×4h×8h for the thick sheet. The liquid segmentis surrounded by the gas zones on top and bottom.The gas moves in the positive x-direction (stream-wise direction), with a constant velocity at the topand bottom boundaries, and its velocity diminishesto the interface velocity with a hyperbolic tangentprofile. The velocity decays exponentially to zeroat the center of the liquid sheet. For more detaileddescription of the initial conditions see Zandian etal. [17]

Figure 1. The computational domain with the ini-tial liquid and gas zones.

The liquid/gas interface is initially perturbedsymmetrically on both sides with a sinusoidal pro-file and predefined wavelength and amplitude. Bothstreamwise (x-direction) and spanwise (y-direction)perturbations are considered in this study. Periodicboundary condition for all components of velocity aswell as the level-set/VoF variable is imposed on thefour sides of the computational domain.

The most important dimensionless groupings inthis study are the Reynolds number (Re), the Webernumber (We), and the gas-to-liquid density ratio (ρ̂)and viscosity ratio (µ̂), as defined below. The ini-

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tial wavelength-to-sheet-thickness ratio (Λ) is alsoan important parameter.

Re =ρlUh

µl, We =

ρlU2h

σ, (5a)

ρ̂ =ρgρl, µ̂ =

µg

µl, Λ =

λ

h. (5b)

The sheet thickness h and the velocity of thefar field gas U are considered as the characteristiclength and velocity. The subscripts l and g refer tothe liquid and gas, respectively.

Post-processing methods

Twice the inverse of the liquid surface curvature(κ = 1

R1+ 1

R2, where R1 and R2 are the two radii of

curvature of the surface in a 3D domain) has beendefined as the local length scale in this study. Thislength scale represents the radius of curvature of theinterface, which would eventually asymptote to thedroplet radius. Based on this definition, a lengthscale is obtained at each computational cell in thefuzzy zone at the interface.

Lijk =2

|κijk|, (6)

where κ is the curvature, and the ijk indices showthe coordinates of the cell in a three-dimensional do-main. The length scale Lijk of each cell is used tocreate a PDF of the length scales. The κijk valueis measured in the computational cells containingthe interface to obtain the PDFs. The bin size con-sidered in this study for the length-scale analysis isdL = 2dx = 5µm. The probability of the lengthscale in the interval (L,L+ dL) is obtained by mul-tiplying the PDF value at that length scale f(L) bythe bin size:

prob(L ≤ L′ ≤ L+ dL) ≡ P (L) = f(L)dL. (7)

This is an operational definition of the PDFf(L). Since the probability is unit-less, then f(L)has units of the inverse of the length scale; i.e. 1/m.However, in our study, the length scale is non-dimensionalized by the initial wavelength. Thus,f(L/λ0) is dimensionless. The relation between thelength-scale PDF and its probability can be derivedfrom equation (7);

f(L/λ0) =P (L/λ0)

dL/λ0= 20P (L/λ0), (8)

where dL is the bin size, and λ0 = 100 µm is theinitial perturbation wavelength.

In order to obtain the average length scale ateach time step, the length scales should be inte-grated along the liquid/gas interface and divided bythe total interface area. The average length scale δis non-dimensionalized using the initial perturbationwavelength. The average length scale is defined as

δ =1

λ0

∫Lds

S≈ 1

λ0

∫Ldn

N=

1

λ0

∑Li

N, (9)

where S is the total surface area of the interface,and N is the total number of cells in the fuzzy zonealong the interface. As is described in this equation,instead of integrating the surface area of the inter-face, we count the number of cells that the interfacecrosses as an approximation. Using the number ofcells along the interface for computing the averagelength scale is satisfactory for our purpose.

Since the length scale L has a wide range from afew microns to infinity (if the curvature is zero at acell), the length scales that are very large (L > 10λ0)are neglected, so that the average length scale wouldnot be biased towards high scales due to those offvalues.

Similar to the length scale, a PDF is obtainedfor the normal (cross-flow) distance of the cells nearthe liquid surface from the center plane. This PDFalso has units of 1/m, as discussed before; however,we normalize it by the initial thickness h0 to makeit dimensionless. This gives a better statistical dataabout the distribution of the normal coordinate ofthe liquid surface along the interface. In order totake into account both sides of the liquid sheet inthis analysis, we define the cross-flow distance hijk ofeach local point on the liquid surface as the absolutevalue of its z-coordinate;

hijk = |zijk|. (10)

The probability of the spray size is obtainedfrom an equation similar to equation (7), where Lis replaced by h. The bin size in the computation ofthe spray size PDF is taken to be equal to the meshsize; i.e. dh = dx = 2.5 µm.

The average spray size ζ is also obtained by inte-grating h along the liquid surface and dividing it bythe total liquid surface area. The spray size is non-dimensionalized by the initial sheet thickness h0;

ζ =1

h0

∫hds

S≈ 1

h0

∑hiN

, (11)

where the cross-flow distance of the liquid surfaceis measured per computational cell. This definitiongives a more realistic representation of the spraygrowth rate.

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Figure 2. Sketch showing the cascade of structures on a liquid lobe for the LoLiD (top), LoCLiD (center),and LoHBrLiD (bottom) mechanisms. The gas flows from left to right and time increases to the right.

Results and discussion

Breakup classification

Three different mechanisms for liquid surface de-formation and breakup are found, each applying ina different domain of the flow Re and We. At highRe these breakup characteristics change based onthe Ohnesorge number (Oh =

√We/Re). At rela-

tively high Re (> 2500), two of these mechanismswere observed: (i) at high Oh, the lobes becomethin and punctured, creating holes and bridges. Thebridges break as the perforation expands and createligaments. The ligaments then stretch and breakup into droplets by capillary action. This mecha-nism is hereafter called LoHBrLiD, based on thecascade or sequence of the structures seen in thisdomain; e.g. Lo ≡ Lobe, H ≡ Hole, Br ≡ Bridge,Li ≡ Ligament, D = Droplet. (ii) at low Oh, lobeperforations are not seen at early times; instead, thelobes stretch directly into ligaments. At high Re,many corrugations form on the lobe edges, and arethen stretched into ligaments. This LoCLiD mech-anism (C ≡ Corrugation) results in ligaments anddroplets without having the hole and bridge forma-tion steps. The ligaments formed in this mechanismare typically shorter and thinner, compared to thelong and thick ligaments in the former mechanism.

The third mechanism follows a LoLiD processand occurs at low Re and low We. The main dif-ference between the two ligament formation mech-anisms at high and low Re’s is that, at higher Re

the lobes become corrugated and then stretchedinto multiple ligaments which are much thinner andshorter than the ligaments in the low Re case. Atlow Re, on the other hand, because of the higherviscosity, the entire lobe stretches into one thick andusually long ligament. The droplets produced fromthe ligament breakup at lowRe are fairly larger. Thestructures in these breakup mechanisms are sketchedin Fig. 2.

More cases were computed near the borderof the LoLiD/LoCLiD and LoHBrLiD domains,showing that there is a transitional region in whichboth lobe/ligament stretching and hole formationmechanisms occur. Based on the cases run and alsofrom the flow physics, it seems that the liquid viscos-ity, and thus the liquid Re, has the most significanteffect on the stretching characteristic, while the sur-face tension and the gas inertia, thus the gas We,has an important role in the hole formation. Pur-suant to this notion, different breakup characteristicdomains are defined in the parameter space of theliquid Reynolds number (Rel) and gas Weber num-ber (Weg). Using Weg allows embedding the effectsof density ratio in the classification. This brings theresults of the lower density ratio closer to the Relaxis than using Wel instead, since Weg = ρ̂×Wel.Thus, a generic map is produced that represents alldensity ratios on a single diagram, shown in Fig. 3.A very similar mapping is also obtained for roundliquid jets (not shown here).

8

Figure 3. The breakup characteristics based onWeg and Rel, showing the LoLiD mechanism de-noted by diamonds (atomization domain I), theLoHBrLiD mechanism denoted by circles (atom-ization domain II), the LoCLiD mechanism denotedby squares (atomization domain III), and the transi-tional region denoted by triangles. The ρ̂ = 0.1 casesare shaded. The low and high density ratio casesthat overlap have been marked. – · – · –, transitionalboundary at low Rel; – – –, transitional boundaryat high Rel.

The breakup domains having the LoLiD,LoHBrLiD, and LoCLiD cascade processes are in-dicated as the atomization domains I, II, and III,respectively, in Fig. 3. The three other regimes pre-sented by Reitz and Bracco [5], that are at lower Reland lower Weg compared to the atomization regime,occupy a small region at the lower left corner of ourdiagram. The atomization domain dwarfs the otherdomains in both parameter range and importancein practical spray formation. Yet, surprisingly thisdomain is the least studied of all. The atomizationregime is the main focus of our study.

The transitional region at low Rel follows a hy-perbolic relation, i.e. Weg = A/Rel, shown in dash-dotted line, while at high Rel limit it follows aparabolic curve, i.e. Weg = B2Re2l , shown in dashedline. The constant B is a critical value of the prod-uct of Oh and the square root of density ratio, andhereafter is defined as a modified Oh, i.e. Ohm.Ohm is defined using the gas We and liquid Re,i.e. Ohm =

√Weg/Rel =

√ρ̂ × Oh. The critical

Ohm at the boundary for high Rel then becomesB ≈ 0.021.

Two parameters, A and B, are used to define

the domain boundary. At high Rel, the parameterB involves four forces and is the ratio of the productof liquid viscous force and square root of gas iner-tia to the product of liquid inertia and square rootof surface tension force (B ∝

√Weg/Rel). Asymp-

totically, the domain boundary goes to constant B(constant Ohm) at high Rel. The second parameterA = Weg × Rel is the ratio of the product of gasinertia and liquid inertia to the product of surfacetension force and liquid viscous force. Again, fourforces are involved. At low Rel, the domain bound-ary approaches A = const. as an asymptote.

Clearly, four forces are involved in different waysat the two (left and right) boundaries. The two bor-derlines can be combined into a single function withsome extra constants for a better fit as follows

Weg =A

Rel + ε+B2Re2l + C , (12)

where A, B and C are empirical constants, and εis a small parameter for better curve fitting. AsRel gets very large, we would retain the parabolicfunction (second term) with the constant B beingthe product of criticalOh and

√ρ̂. In the limit where

Rel gets very small, the hyperbolic function (firstterm) dominates and gives the asymptote.

Based on these results, there are different char-acteristic times for formation of holes and stretchingof lobes/ligaments. At the same Rel, as the surfacetension increases (decreasing Oh), the characteristictime for hole formation becomes larger, hence delay-ing the hole formation. Thus, most of the earlier lig-aments are formed via direct stretching of the lobesand corrugations, while hole formation is hindered.On the other hand, as the liquid viscosity increases(decreasing Rel and increasing Oh), while keepingWeg the same, the characteristic time for ligamentstretching gets larger if Rel is high. In this case, thehole formation prevails over the ligament stretching,resulting in more holes on the lobes surface. At verylow Rel, the characteristic time of the hole forma-tion is also influenced by the liquid viscosity andgets larger with decreasing Rel; thus, the breakupmechanism switches back to direct stretching as Relis sufficiently lowered.

In conclusion, the transitional region is notmonotonic on a Weg vs. Rel diagram, and has aminimum value around Rel ≈ 2500. Thus, if Wegis less than a certain value (Weg ≈ 5, 000), the lig-ament stretching mechanism will always prevail re-gardless of Rel. At higher Weg > 5, 000, the LoLiDand LoCLiD mechanisms are dominant at very lowand very high Rel, respectively, but there will be ahole formation region between them.

9

Figure 4. Liquid surface deformation in the LoHBrLiD mechanism (domain II); Rel = 320, Weg = 115, 000(Ohm = 1.06), ρ̂ = 0.5, and µ̂ = 0.0022, at t = 18 µs (a), 22 µs (b), 26 µs (c), 28 µs (d), 30 µs (e), and32 µs (f). Gas flows from left to right.

The breakup mechanisms are functions of Reland Weg only. The qualitative behavior is not muchaffected by the viscosity ratio (thus the gas Re); theinfluence of density ratio appears only through Weg,and the effect of the sheet thickness appears onlythrough Rel and Weg.

The breakup domains could be divided intothree classes only based on their Rel and Weg, asshown in Fig. 3: (I) LoLiD at low Rel and low Weg;(II) LoHBrLiD, which happens at high Weg andmoderate Rel; and (III) LoCLiD at high Rel andlow Weg, which involves corrugation formation. Thetransitional region between the LoLiD/LoCLiDand LoHBrLiD zones involves overlaps of the mech-anisms.

The LoHBrLiD cascade is shown in Fig. 4. Thelobes form and thin on the primary KH waves. Themiddle section of the lobes (the braid), where thehighest strain occurs, thins faster and thus perfo-rates, creating a hole and a bridge on the lobe rim.Bridges become thinner as the holes expand. Fi-nally, the bridges break and create one or two lig-aments (depending on the breakup location). Theligaments stretch and eventually break into dropletsunder capillary action. The final ligaments are com-paratively long and thick in this zone.

The LoCLiD process in the atomization domainIII is shown in Fig. 5. The lobes form similar tothe previous case, but do not stretch as much. Thelobe edges are not as smooth as in the previous case,and corrugations form on the lobes’ front edge and

stretch to create ligaments. Multiple ligaments areformed per lobe, typically shorter and thinner com-pared to the ligaments seen in the LoHBrLiD do-main. Eventually, the ligaments detach from theliquid jet and break up into droplets by capillary ac-tion. These droplets are consequently smaller thanthe ones formed in the LoHBrLiD mechanism.

For the LoLiD mechanism which prevails at lowRel and low Weg, as shown in Fig. 3, the surface ten-sion force does not allow the lobes to perforate easily.The liquid viscosity is also fairly high and preventsany corrugation formation on the lobe front edge.Consequently, the entire lobe stretches slowly intoa thick and long ligament, which eventually breaksinto large droplets. The LoLiD breakup process isshown step-by-step in Fig. 6.

Structure cascade time scales

As mentioned earlier, there are two differentcharacteristic times for the formation of holes andthe stretching of lobes and ligaments. At the sameRel, as the surface tension increases (decreasing Ohand We), the characteristic time for hole forma-tion increases, hence delaying the hole formation.Thus, for lower Oh (or We) most of the earlier lig-aments are formed due to direct stretching of thelobes and/or corrugations, while the hole formationis inhibited. On the other hand, at relatively largeRel (Rel > 3000), as the liquid viscosity is increased(decreasing Rel and increasing Oh), at the sameWe, the ligament-stretching time gets larger. In this

10

Figure 5. Liquid surface deformation in the LoCLiD mechanism (domain III); Rel = 5000, Weg = 7200(Ohm = 0.017), ρ̂ = 0.5, and µ̂ = 0.0066, at t = 44 µ (a), 48 µs (b), 50 µs (c), 52 µs (d), 56 µs (e), and60 µs (f). Gas flows from left to right.

case, hole formation prevails compared to the liga-ment stretching mechanism, resulting in more holeson the liquid lobes.

Our results show that as We increases, the firstholes form earlier. This indicates that the hole for-mation characteristic time should be inversely pro-portional to We. The holes expand more rapidly athigher We. At high Rel, the hole formation is de-layed as ρ̂ decreases. Thus, the hole-formation timescale should be inversely proportional to ρ̂. Thiseffect may be combined with the effect of We byusing the gas Weg instead of the liquid We. Twodifferent characteristic times are formed at high Rel- one for each mechanism - involving surface ten-sion and viscosity. The hole-formation characteris-tic time is directly proportional to the surface ten-sion and inversely proportional to the density ratio,while the stretching characteristic time is propor-tional to the liquid viscosity. Thus, the followingtwo non-dimensional characteristic times are pro-posed for these mechanisms:

Uτhh

∝ σ

ρgU2h=

1

Weg, (13)

Uτsh

∝ µl

ρlUh=

1

Rel, (14)

where τh and τs are the dimensional characteristictimes for hole formation and ligament stretching, re-spectively.

As indicated in the equations above, the char-acteristic times can be written in terms of the sheetthickness and jet velocity. In this form, the hole-formation characteristic time becomes inversely pro-

portional to Weg, while the stretching characteris-tic time is inversely related to Rel. However, notethat real times are independent of the thickness andonly depend on the fluid properties. Clearly, h/Uis a convenient normalizing time. Aside from its ac-cord with our results, these time scales are consis-tent with intuition. The surface tension hinders holeformation, thus increasing its time scale, while liq-uid viscosity resists stretching, hence increasing thestretching characteristic time. Also, the time scalesfor both mechanisms become larger for the same di-mensionless parameters as the jet becomes thicker.

Combination of the two Eqs. (13) and (14) yieldsa relation between the two time scales at high Relregions of interest, involving Ohm.

Uτhh∝(Uτs/h

Ohm

)2

. (15)

In the transitional region, near the boundary,where the two characteristic times are of the sameorder, both hole formation and corrugation stretch-ing appear at different parts of the liquid sheet.

At the low Rel < 3000, the liquid viscosity hasopposite effects on the hole formation and ligamentstretching. As shown in Fig. 3, near the left bound-ary, the time scale of the stretching becomes rela-tively smaller than the hole-formation time scale asRel is reduced at a constant Weg. Hence, there isa move back to ligament stretching from hole for-mation with decreasing Rel at a fixed Weg. Weconclude that a term should be added in the hole-formation time scale, which becomes larger at lowerRel. At higher Rel, however, this term would van-

11

Figure 6. Liquid surface showing the LoLiD mechanism (domain I) at low Rel; Rel = 320, Weg = 23, 000(Ohm = 0.47), ρ̂ = 0.1 and µ̂ = 0.0022, at t = 26 µs (a), 36 µs (b), 40 µs (c), 44 µs (d), 46 µs (e), 48 µs (f),and 52 µs (g). Gas flows from left to right.

ish, with a return to the time scale in Eq. (13). Inorder to generalize the hole-formation time scale, aterm is added with the reciprocal of Rel as its coeffi-cient to fix the opposite behavior seen at lower Rel.The new hole formation time scale then becomes:

Uτhh∝ σ

ρgU2h(1+

k

Rel) =

1

Weg

(1 +

k

Rel

), (16)

where k is a non-dimensional constant. The stretch-ing time scale does not need any modification forlow Rel; the relation between the two time scalesbecomes

Uτhh∝(Uτs/h

Ohm

)2(1 +

k

Rel

). (17)

At high Rel, the earlier derived relation Eq. (15)is retained. At the low Rel limit, the asymptoticrelation between the two time scales becomes

Uτhh∝ 1

Rel

(Uτs/h

Ohm

)2

=RelWeg

(Uτsh

)2

, (18)

so, at low Rel, the hole formation and the ligamentstretching time scales are not related only via themodified Ohnesorge number; the flow Rel also has asignificant role in the breakup mechanism at lowerRel.

Structure cascade length scale

The temporal variation of the average lengthscale for low, medium, and high Weg and moder-ate Rel are illustrated in Fig. 7. The density ra-tio is kept the same among all these cases; hence

Figure 7. Effect of Weg on the temporal varia-tion of the average dimensionless length scale; Rel =2500, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

Weg is only changed through the surface tension.The effects of density ratio will be shown later.The lowest Weg case falls in the ligament stretch-ing (LoLiD) category (domain I), while the twohigher Weg cases follow the hole-formation mecha-nism (domain II), i.e. LoHBrLiD, in their breakup.The average length scale decreases with time forall cases, which is consistent with the cascade ofstructures that were presented in the earlier stud-ies [15, 17] (lobes to holes and bridges, to ligaments

12

Figure 8. Probability of the normalized length scales at different times for Weg = 1500 (a),Weg = 7, 250 (b), and Weg = 36, 000 (c); Rel = 2500, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

and then to droplets). By increasing Weg, the rateof length-scale reduction is increased as expected.Moreover, the average length scale becomes smalleras Weg is increased. At 40 µs, the average lengthscale for a liquid jet with Weg = 36, 000 is almost0.3λ0 = 30 µm, while for Weg = 7, 250 this in-creases to 0.4λ0; at the lowest Weg it increases to0.7λ0. This shows the clear influence of the surfacetension on the length-scale cascade and the size ofthe droplets. Surface tension stabilizes the instabil-ities and increases the structures’ sizes.

The length scale starts from 1.2λ0 in all threecases, due to the initial perturbations, and the aver-age scale increases until about 10 µs, when it reachesa maximum. This initial increase in the length scale

is caused by the initial distortion and stretching ofthe waves, which create a flat region near the braids.These flat regions have very small curvatures, hencevery large length scales, contributing to an increasein the average length scale. Later, when lobes, lig-aments and droplets are formed, the average lengthscale decreases because (i) the radius of curvatureof the ligaments and droplets are much smaller thanthe initial waves, and (ii) the total interface areaincreases by the formation of holes, ligaments anddroplets. Hence, the large scale loses its significanceon the average length-scale calculation.

In order to gain a better insight into the distri-bution of the length scales, the probability distribu-tion of different length scales of the three Weg’s are

13

Figure 9. Effect of Rel on the temporal variationof the average dimensionless length scale; Weg =7, 250, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

compared at different times on a log-scale in Fig. 8.All of the cases start from a slightly skewed bell-shaped length-scale distribution indicated by the redline. Initially, the length scales with a range of0.5λ0–0.6λ0 have the maximum probability of ap-proximately 10%. As time passes, the most proba-ble length scale becomes smaller, while the proba-bility of the dominant structure size increases. Thistransition towards smaller scales is faster as Wegincreases, since lowering surface tension reduces theresistance of the liquid surface against deformations.At 50 µs, the most probable length scale becomes0.1λ0 for Weg = 1500 (the dashed-line in Fig. 8a).The higher Weg cases reach the same most proba-ble length scale at 40 µs (dash-dotted line in Fig. 8b)for Weg = 7, 250, and at less than 10 µs (not shownhere) for Weg = 36, 000. Clearly, the rate of length-scale reduction increases with increasing Weg.

The PDFs also show an increase in the prob-ability of small scales at higher Weg. Consider-ing the probabilities at t = 50 µs for example, theWeg = 1500 case has a probability of about 8% forthe smallest computed length scale, i.e. 2.5 µm. Theprobability of the same length scale at the same timeincreases to 38% as Weg increases to 7, 250. At evenhigher Weg = 36, 000, the probability of the surfaceshaving length scale of 2.5 µm or lower is slightlymore than 45% at 50 µs. This means that the num-ber density of the smaller droplets increases withWeg; i.e. more droplets with smaller sizes occur athigher Weg.

Figure 9 demonstrates the effects of Rel on the

average liquid jet length scale through time. ThreedifferentRel’s are compared in this plot, Rel = 1000,2500, and 5000, each representing one of the zonesin the Weg–Rel plot with a particular breakup char-acteristic; see Fig. 3. As expected, the liquid jetstretches more in the streamwise direction by in-creasing Rel. This creates flatter stretched surfacesat the early stages of the spray formation. Thus,in average, relatively larger length scales occur atearly times as Rel increases. The higher Rel alsohas its maximum at a later time, meaning that thelobes stretch for a longer period before the actualbreakup occurs and the length scales decrease.

After the early injection period, the rate of cas-cade of the large liquid structures, e.g. lobes andbridges, into smaller structures, e.g. ligaments anddroplets, is greater at higher Rel. This can be ob-served from the mean slope of the curves in Fig. 9.By decreasing the liquid viscosity, i.e. increasing Rel,and keeping every other fluid properties and flowparameters unchanged, the breakup happens fasterand the instantaneous rate of cascade of length scalesgrows. Figure 9 also shows that Rel does not signif-icantly affect the ultimate mean length scale, aftera long enough time. All three cases asymptoticallyreach a mean length scale of about 0.12λ0, althoughat different times and rates. However, the actualfinest length scale gets smaller as Rel increases be-cause the higher Rel case also has the largest lengthscales, as described before. Thus, to compensate forthe many cells with larger length scales, i.e. stretchedflat surfaces that occur at high Rel, more small scalestructures are required.

We can conclude that there are two portions inthe liquid sheet distortion: (i) the initial stage ofdistortion where the length scales grow, and (ii) thefinal asymptote in time. Viscosity but not surfacetension is effective in the first stage, which is iner-tially driven, while surface tension but not viscosityaffects the final stage and the ultimate mean scale.

The effect of density ratio on the average liquidjet length scale is shown in Fig. 10. Three density ra-tios have been studied here in a range of 0.1 (low gasdensity) to 0.9 (high gas density). The liquid We,hence the surface tension coefficient, has been keptthe same for all three cases; thus, Weg is also differ-ent for these three cases through gas density. Thelowest density-ratio case falls in the atomization do-main I, while the other two high density-ratio casesfollow the LoHBrLiD mechanism in domain II.

Density ratio does not change the ultimatelength scale significantly. All cases asymptoticallyreach the same average length scale of about 0.12λ0.However, density ratio slightly affects the rate at

14

Figure 10. Effect of density ratio on the temporalvariation of the average dimensionless length scale;Rel = 2500, Wel = 14, 500, µ̂ = 0.0066, and Λ =2.0.

which the ultimate length scale is achieved. As den-sity ratio is increased, cascade of length scales be-comes slower. The ultimate length scale is achievedat about 45 µs for ρ̂ = 0.1, but at about 50 µs forρ̂ = 0.5, and at 60 µs for ρ̂ = 0.9. The densityratio does not alter the initial stage of the lengthscale growth, which proves it to be correlated withthe liquid inertia and not the gas inertia. The finalasymptotic stage is also driven by the liquid inertiaand is independent of the gas density. Earlier, wesaw that this stage is also correlated with surfacetension. Thus, we conclude that the liquid Wel andnot the gas Weg is the key parameter in determiningthe asymptotic droplet size.

Even though two jets at the same Rel and Wegexhibit the same breakup mechanism, the lengthscales of the resulting liquid structures depend onthe density ratio. The higher liquid density results infiner structures. In other words, the breakup regimeonly determines the breakup quality (the type ofprocess that takes place), but other factors need tobe considered to control the quantitative character-istics of the breakup; e.g. droplet size and structurelength scales.

Our computations also show that viscosity ratiohas negligible effects on both the cascade rate andthe ultimate length scale. The sheet thickness on theother hand affects the cascade of length scales sig-nificantly. As the sheet becomes thicker, the cascadeoccurs much slower and asymptotes to a much largerscale. For example, as we make the sheet 4 times

Figure 11. Average dimensionless spray size ζ vari-ation with time for different Weg; Rel = 2500,ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

thicker, the asymptotic mean length scale becomesabout 4.16 times larger, which is in good agreementwith the analytical findings of Senecal et al. [19].They found that ligament diameter is directly pro-portional to the sheet thickness.

Spray growth rate

The effect of Weg on the temporal evolution ofthe average spray size is shown in Fig. 11. Thesprays expand faster as Weg increases since thesmaller structures produced at higher Weg are cat-apulted in the normal direction and fly away fromthe jet core. The average spray size remains closeto the initial sheet thickness for the first 50 µs forWeg = 1500, since the high surface tension sup-presses the instability wave, the lobe stretching, andthe ligament breakup. The jet starts to expandmuch sooner at higher Weg, for example, at about20 µs for Weg = 7, 250, and 7 µs for Weg = 36, 000.The liquid structures stretch much quicker at higherWeg and are less suppressed by the surface tensionforces; hence, they can expand in the normal direc-tion more freely and fly away from the interface afterbreakup.

The spikes and oscillations that are seen in theaverage spray size ζ are caused by the detachmentof a liquid blob, e.g. bridges, ligaments or droplets,from the jet core. The spray size PDFs, given inFig. 12, support this reason.

All three cases in Fig. 12 start from a bell-shaped distribution around h = h0, shown by thered solid lines. The two peaks on the two sides of

15

Figure 12. Non-dimensional spray size probability distribution at different times for Weg = 1500 (a),Weg = 7, 250 (b), and Weg = 36, 000 (c); Rel = 2500, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

h/h0 = 1 are due to the initial perturbations ampli-tude of 5 µm. Since there are more computationalcells near the peak and trough of the perturbationscompared to the neutral plane, i.e. h/h0 = 1, theprobability of those sizes are slightly higher. Theprobability of the initial thickness value increases inall cases during the first 20 µs, since the wave am-plitude decreases as the waves get distorted in theinitial stage.

Later, when the waves start to grow due to theKH instability and roll-up or stretching of the lobesover the initial KH vortex, the jet size increases andthe distribution becomes wider and skews towardsthe larger values on the right. Meanwhile, smallersizes are also observed. With time, the peak of the

initial distribution curve is diminished while the dis-tribution broadens. This decrease occurs faster athigher Weg, showing that the spray grows faster forlower surface tension. Considering the t = 50 µslines, only 8% of the liquid surface lies near the ini-tial thickness for Weg = 36, 000 (see the black solidline in Fig. 12c), and the spray has grown upto threetimes the initial sheet thickness; i.e. h/h0 ≈ 3. Atthe same time, the farthest normal distance reachedby the liquid surface is about 2.8h0 from the centerplane for Weg = 7, 250; see where the dashed-linein Fig. 12b meets the horizontal axis. At still lowerWeg, the maximum spray size is just slightly morethan 2h0 at 50 µs, and still more than 16% of theliquid surface lies around the initial sheet thickness;

16

see Fig. 12(a).Figure 12(b) shows that there is a miss-

ing section (having zero probability) around1.4 < h/h0 < 1.8 at t = 30 µs. The same missingsection moves to the right (to 1.5 < h/h0 < 2.0)at t = 40 µs, and finally vanishes at 50 µs. Thesesections coincide with the places where the suddendecrease in the average jet size were seen in Fig. 11for Weg = 7, 250; hence, explaining the oscillationsin the average spray size. The missing section ap-pears since some part of the liquid jet (ligament orbridge) detaches from the jet core and moves away inthe normal direction, leaving behind a region emptyof any liquid surface at the breakup location. Whilethe detached liquid blob moves away from the inter-face, the jet surface starts to stretch outward againdue to KH instability; the missing zone moves out-ward following it. The average spray size starts togrow again when the new lobes and ligaments stretchenough to compensate for the broken (missing) sec-tion. This is when the lobes and ligaments fill in themissing gap and the broken liquid structures advectfarther from the interface.

The above discussion shows that the PDFs andthe average spray size plots can indicate the first in-stance of ligament/lobe breakup. This correspondsto the first instant where the average spray size ζstarts to decline. Since the lower Weg case hasless stretching and fewer ligament detachments atearly times (due to the high surface tension), itsspike is less intense and also appears much later(around 55 µs). The higher Weg case, however,breaks sooner, at around 25 µs. The mean-spray-size plots thus provide this useful information.

Figure 13 shows the effects of the liquid viscosityon the jet expansion. Since the inertial effects domi-nate over the viscous effects at higher Rel, the sprayis more oriented in the streamwise direction, and themean spray size and accordingly the spray angle aresmaller at higher Rel. This is consistent with bothnumerical computations [15] and experimental re-sults [20, 21]. Our results show that the growth rateof the spray angle is also lower at higher Rel. Eventhough both the lowest and the highest Rel casesfollow the same lobe stretching mechanism, there isa significant difference in their spray expansion. Athigh Rel, the corrugations on the lobes stretch fasterinto ligaments in the streamwise direction, resultingin thinner and hence shorter ligaments. At low Rel,on the other hand, the lobes directly stretch into lig-aments, and the stretching is oriented in the normaldirection as the viscous forces resist against stream-wise stretching caused by the inertial effects. Theresulting ligaments are thicker and longer, as shown

Figure 13. Average dimensionless spray size ζ vari-ation with time for different Rel; Weg = 7, 250,ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

by Marmottant and Villermaux [11]. This differenceoriginates from the differences in the vortex struc-tures of these two regimes, and shows that these twomechanisms have different causes from vorticity dy-namics perspective. For example, the streamwisevortex stretching is stronger at higher Rel, causingstreamwise oriented ligaments.

Figure 14 compares the average spray sizegrowth for low, medium, and high density ratios.The average expansion rate increases with increas-ing density ratio, consistent with the findings of Jar-rahbashi et al. [15] for round jets. The jet with thehighest density ratio (the dash-dotted line in Fig. 14)expands more rapidly than the case with smallerdensity ratio (the solid line). The spray size startsto grow from its initial thickness at around 10 µs forthe highest density ratio, while the initiation of theexpansion is postponed to 45 µs for the lowest den-sity ratio. Some researchers have reported growth ofthe spray angle with increasing the density ratio orgas-to-liquid momentum ratio [15, 21]; their reportshave been based on the final stage of the expandedjet, but none has shown the rate of the spray growthwith the density ratio. Our results show that the jetexpansion rate is higher at higher density ratios.

Even though the average spray expansion rateis lower for lower gas densities, the asymptotic ex-pansion rate is approximately the same after a longenough time from the start of the injection, regard-less of the density ratio. This can be seen from theslopes of the curves in Fig. 14, which become ap-proximately equal near the end of the computations;

17

Figure 14. Effect of density ratio on the temporalvariation of the average dimensionless spray size ζ;Rel = 2500, Wel = 14, 500, µ̂ = 0.0066, and Λ =2.0.

i.e. the asymptotic slopes appear to be independentof the gas density. Jarrahbashi et al. [15] also foundthat the spray expansion rate is higher at higherdensity ratios for circular liquid jets. However, theyused the traditional definition for the jet size, i.e. dis-tance of the farthest continuous liquid structure fromthe centerline, which would be troublesome in somecases, as discussed.

Since an increase in density ratio increases thespray size and expansion rate, and the sheet expan-sion is also directly proportional to the liquid We,bothWe and ρ̂move in the same direction in relationwith the jet expansion. It is interesting now to ex-amine the effects of We and density ratio combinedthrough the gas Weber number Weg. This is demon-strated in Fig. 15, where the temporal evolution ofthe spray size of the two cases that overlapped at thesame data point on the Weg–Rel map of Fig. 3 arecompared. Both cases have the same Weg = 1500,but different ρ̂ and Wel. Since Weg = ρ̂Wel, thedensity ratio and Wel should change in opposite di-rections to keep Weg constant; i.e. as density ratioincreases (increasing the jet expansion), Wel shoulddecrease (decreasing in the jet expansion). Figure 15shows that the two cases behave very similarly intemporal expansion; both sprays start to expand atalmost the same time and at the same rate. Hence,the two parameters, Wel and ρ̂, could be combinedinto a single parameter Weg for the matter of jetexpansion analysis. The conclusion here is that gasinertia (and not the liquid inertia) and liquid sur-

Figure 15. Effect of density ratio and Wel onthe temporal variation of the average dimensionlessspray size; Rel = 2500, Weg = 1500, µ̂ = 0.0066,and Λ = 2.0.

face tension are the key parameters in determiningthe spray size. This confirms Weg to be the properchoice for categorizing the liquid jet breakup char-acteristics.

Our computations show that viscosity ratio doesnot have a considerable influence on the expansionrate of the sheet. The sheet thickness though, affectsthe spray expansion rate. Spray expansion gets de-layed significantly as the sheet thickness increases.Even though a thin sheet has higher growth rate atthe early stages of spray formation, the thicker sheetachieves the highest growth rate at the final stage.The reason for this major difference is related toslower transition towards antisymmetry as the sheetbecomes thicker. The wavelength of the antisym-metric waves are much larger for thicker sheets [17].Hence, the growth rate of those larger waves areclearly higher than the short waves that exist in thin-ner sheets. Further studies should explain this.

Conclusions

The temporal development of surface waves andtheir breakup into droplets are studied numerically.Three main breakup mechanisms are identified. Thebreakup characteristics are well categorized on a pa-rameter space of gas Weber number (Weg) versusliquid Reynolds number (Rel). The atomizationregime is now separated as three sub-domains.

At high Rel, the breakup characteristics changebased on the Ohnersorge number (Oh). At high Oh,the lobes thin and perforate to form bridges, which

18

eventually break into one or two ligaments. At lowerOh, the hole formation is hindered and instead, thelobe rims corrugate and stretch into small ligaments.There is also a transition region where both mech-anisms co-exist. The transition region at high Relfollows a constant Oh line for a given density ratio.At low Rel, the transitional region follows a hyper-bolic relation in the Weg–Rel plot. At low Weg, thelobes stretch directly into ligaments. The ligamentscreated in this domain are fairly thick and long, andresult in larger droplets. As Weg is increased ata constant Rel, the hole formation mechanism pre-vails.

Different characteristic time scales are intro-duced for the hole formation and lobe stretching- mainly related to the surface tension and liquidviscosity, respectively. At any flow condition, themechanism having a smaller characteristic time isthe dominant mechanism. In the transitional region,both characteristic times (hole formation and lobestretching) are of the same order; thus, both mecha-nisms occur simultaneously and the breakup mecha-nism varies locally. The two characteristic times arerelated to each other by the modified Ohnersorgenumber, which involves the gas We and the liquidRe.

Two PDFs were formed for the liquid structuresize and the spray size from our numerical data. ThePDFs of the structure size provided statistical infor-mation about the droplet size distribution and thequalitative number-density of droplets in a liquid jetbreakup. The temporal variation of the mean of thePDFs gave useful information about the rate of cas-cade of liquid structures (from lobes to ligamentsand to droplets) in different flow conditions. Themean and PDF of the spray size also showed thefirst instance of breakup of ligaments/droplets andalso qualitatively showed the transition of the liquidjet from symmetric mode towards anti-symmetricmode.

Our method was able to accurately predict theeffects of Weber (We) and Reynolds (Re) number,density ratio, viscosity ratio, and the sheet thick-ness on the droplet size and spray size. The resultsshowed that the ultimate size of the droplets de-creases with increasing We, and the reduction in thestructure (droplet) size happens at a higher rate forhigher We. The spray size (spray angle) also growsat a higher rate for higher We. Re does not affectthe final structure size significantly, but has a clearrole in the rate of structure size reduction. The cas-cade of structures happen much faster at higher Re.The spray size is larger at lower Re, and the sprayangle and the rate of spray growth decreases as Re

increases. The density ratio does not alter the finaldroplet size significantly, but has a direct impact onthe spray size. The spray size grows significantlywith increasing gas density.

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