ILASS Americas 27th Annual Conference on Liquid Atomization and Spray Systems, Raleigh, NC, May 2015

Early Spray Development at High Pressure:Hole, Ligament, and Bridge Formations

D. Jarrahbashi1∗, W. A. Sirignano1, P. P. Popov 1, and F. Hussain2

1Department of Mechanical & Aerospace EngineeringUniversity of California, Irvine

Irvine, CA 92697 USA2Department of Mechanical Engineering

Texas Tech UniversityLubbock, TX 79409 USA

AbstractThree-dimensional temporal instabilities, leading to spray formation, of a round liquid jet segment withco-axial gas flow at high pressure is studied by Navier-Stokes and level-set computations. Liquid-surfaceshape shows the development of smaller structures on the conical wave crests, i.e., lobes, holes, bridges, andligaments. The gas-to-liquid density ratio, liquid Reynolds number (Re), and liquid Weber number rangebetween 0.05 to 0.9, 320 to 5000, and 2000 to 230,000, respectively. At higher Re, lobes are longer andcurve more at the crest edge with a regular formation of holes. The crest rims eventually tear, transformingthe crest rims to ligaments. At higher gas densities throughout the Re range, the lobes are regular butshorter. The holes merge before the rims break to form ligaments. Consequently, liquid formations withboth rim bridges and middle bridges are more common in this domain. In cases where both gas densityand Re are lower, the well-ordered lobes are replaced by a more irregular corrugation with more wrinklesalong the conical wave crest edge. Ligaments stretch from the lobes before holes form. The more viscouscrests are thicker here explaining a delay in hole formation; still, the ligament extension is driven by pressuregradient rather than shear at the gas-liquid interface. In all cases, hole formation is correlated with hairpinand helical vortices; the perforations correlate with resulting fluid motion. Agreements with experiments arevery good.

∗Corresponding Author: djarrahb@uci.edu

Introduction

The small structures in the early spray develop-ment, i.e., holes, ligaments, and bridges, are cru-cial in determining the sizes of the droplets thatbreak from these structures due to capillary action.While the literature has identified some interest-ing phenomena at the small structure level, the im-portance of these structures to the final spray, thefluid mechanical causal relations, and the limitationswithin the parameter domain to their appearanceshave not been fully explored. The goal in atom-ization research is to resolve the size and velocitydistributions of the droplets because they stronglyaffect the performances of various operations andmachines, including especially combustion engines.Incompressible round jets or planar mixing layerswithout density gradients or interfacial tension havebeen studied, with attention to the vorticity and ve-locity fields but little attention to interface trackingand the Lagrangian flow description. Meanwhile,liquid streams flowing into a gaseous environmenthave been researched with great attention to devel-oping surface deformation and eventual breaking ofthe liquid into smaller elements; however, little or noattention is paid to velocity and vorticity. This pa-per relates the flow field variables, interface tracking,and Lagrangian portrait, thereby obtaining physicalinsights and explanations.

Liquid-jet experiments of Marmottant & Viller-maux [1] at low pressure with co-axial air flowshowed that ligaments formed by elongation of thecorrugations on the liquid cone crest rims of theprimary Kelvin-Helmholtz (KH) waves. They sug-gested that the cone-crest instability was caused bythe Rayleigh-Taylor (RT) instability due to the highflow acceleration normal to the crests. On the con-trary, the side-jets, expelled as counter-rotating vor-tices for jet flow into a like-density fluid, show thatdensity difference cannot explain fully the three-dimensional instability.

Previous simulations by Jarrahbashi & Sirig-nano [2], hereafter described as JS, of spatial in-stability of round liquid jets in high-pressure air,i.e., high gas-to-liquid density ratios, showed thatRT, i.e., the baroclinic effect, was not responsible.Instead, vortex tilting and stretching created thestreamwise vorticity and formed the lobes on the KHcone crests similar to side jets of like-density jets.In addition, the computational studies illustratedhole formation on the liquid lobes. These holes in-crease in size with time and reach the tip of thelobe, then tear producing two ligaments from the rimbridge between the crest and the hole. Ligamentscan also develop from elongation of the lobes them-

selves as well as from tearing of the lobes. Lobe tear-ing became more prominent for high gas-to-liquiddensity ratios while lobe elongation was the domi-nant mechanism for lower gas-to liquid density ra-tios and lower Reynolds number (Re). Similar highRe behavior had previously been observed in simula-tions of a round liquid-jet injected into air by Shinjo& Umemura [3] and a planar jet into air by Des-jardins & Pitsch [4] at relatively high gas-to-liquiddensity ratios. However, the mechanisms responsi-ble for lobe perforation and effects of surface tensionand differences in density, velocity, and viscosity be-tween the two phases on lobe tearing were not wellexplained in those prior studies.

Although JS showed the varying qualitative be-havior over a range of Re values, certain importantdetails of these phenomena remain enigmatic. Sev-eral questions must be answered. The sequence ofinstability mechanisms and the time scales of hole,ligament, bridge, and droplet formations must be re-solved so that droplet sizes and volumetric growthrate for the two-phase mixture can be determined.Do ligaments first form from the indentations ap-pearing on the lobes and then holes form on thecone crests at the roots of the ligaments? Or do lig-aments form after lobe perforation occurs? What arethe different behaviors in different parameter ranges,e.g, density and viscosity ratio, surface tension? Thetime scales of the hole, bridge, and ligament forma-tions and the roles of surface tension, densities, andviscosities in determining these time scales are un-known. The cause of hole formation and the locationof the holes on the lobes are not immediately obvi-ous. Do holes form due to collision of the dropletswith the jet core as suggested by Shinjo & Umemura[3] or do they form even before the droplets detachfrom the liquid core? What are the thickness scalesfor ligaments in the different parameter domains?Do droplet sizes relate to the ligament thicknesses?How rapid is the increase of the radial size of thetwo-phase mixture? Answering the aforementionedquestions is formidable experimentally due to limi-tations of optical access to the jet, especially at thehigh gas pressures and densities of interest.

For the computations, very high resolutions arerequired to obtain the details of ligament formation.A less computationally intensive case with a seg-ment of the round liquid jet undergoing temporalinstability allows the required higher resolution. Thethree-dimensional instability, i.e., lobe formation fora wide range of gas-to-liquid density ratios were dis-cussed thoroughly and vorticity dynamics associatedwith interface instability were described previously(JS). Now, compared to that prior research, higher

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resolution simulations have been performed for latertimes after the initiation of the instability. The em-phasis is on the formation of small-scale liquid pro-trusions at the liquid/gas interface, hole creation onthe lobes, and ligament break-up for wider ranges ofliquid viscosity and surface tension coefficient. Viapost-processing, the vorticity dynamics and momen-tum balances responsible for formation of the holesand ligaments have been obtained for various flowconditions to identify the dominating mechanisms.JS showed that there were similarities between thevorticity dynamics of the planar mixing layers of thesame density with two-phase round liquid jet injec-tion due to formation of the hairpin vortices. Thebehavior of a planar mixing layer excited by a time-harmonic wave with spanwise phase variation wasvisualized by means of Schlieren system by Nygaard& Glezer [5]. The primary vortex rollers deformedin the spanwise direction and formed hairpin vor-tices in the braid region. The vortices on the crests,that were induced by the primary deformed vorticesin the braid region, were 180 degrees out of phasewith respect to those in the braid. They finally de-veloped into hairpin vortices extending with an ori-entation opposite to those in the braid. These twoseries of hairpin vortices overlapped and produceda diamond-shaped region arranged in a “chain-linkfence” pattern [6] with smaller scales compared tothe original coherent vortex structures. As noted byComte et al. [7], there is some controversy aboutthe meaning of helical pairing versus true mergerof vortex pairs. A true merger would involve a vis-cous connection of vortices which would require sometime for development. The overlapping and inter-twining was considered as localized pairing althoughthere were no mergers involved between these vor-tices. This localized “helical pairing” occurs whenthe spanwise wavelengths are extremely large, i.e.,a factor of eight larger compared to the streamwisewavelength. For spanwise-to-streamwise wavelengthratios smaller than four, pairing does not happenand the crests of KH waves slightly tilt with respectto the streamwise direction [8]. Therefore, Comte etal. showed overlap from the top view as Figure 1illustrates.

In general, the “helical pairing” instability as in-troduced by Pierrehumbert & Widnall [9] causes os-cillations of opposite orientation on the neighboringvortex roll; however, for small spanwise to stream-wise wavelengths merging between the hairpin vor-tices does not occur. Numerical simulations of Colliset al. [6] with the same conditions of the Nygaard& Glezer [5] experiment showed that no merging oc-curred in the diamond-shaped region between the

Figure 1. Schematic of helical pairing (a) earliertime, (b) later time for moderate spanwise wave-lengths where “helical” pairings do not occur [8].

two hairpin vortices. Computations of Comte etal. [7] demonstrated the “chain-link fence” arrange-ment, i.e. “vortex lattice” (see Figure 2) for planarmixing layers forced by small Gaussian random per-turbations. The core vortex lines wrapped aroundeach other and formed hexagonal regions as shownin Figure 2, in agreement with results of Collis et al.[6]. The vortex lines did not form a double helix;so, “helical pairing” has not been clearly used in theliterature for this phenomenon. Comte et al. [7] re-sults suggested that the rollers did not coalesce into asingle roller in the local pairing region. That is, self-induction acted in opposition to viscous effects thattended to form a single paired vortex in the over-lap region. Forcing the mixing layer to create longerspanwise wavelengths might lead to actual mergingof the vortices into a single pair [6]. The results tobe discussed later will have relatively short spanwisewavelengths so that coalescence is not expected. Thehairpin vortex development and overlaps in Figures1 and 2 will be similar to those for liquid round jets.It will be seen that overlapping without merging canexplain hole formation. A main goal of this researchis to identify mechanisms for ligament and bridgeformations and lobe perforation processes. Specifi-cally, we analyze the lobes, holes, bridges, ligaments,and droplets which appear in a consistent patternfor a given parameter domain. We analyze, in par-ticular, the helical formations of hairpin vortices inthe braid and ring region and compare to results forplanar mixing layers. A study of the vortical struc-tures connects the lobe perforation mechanism in jetflows with vortex lattice formations. However, un-like the single-phase planar mixing layers, densityratio, viscosity ratio, and surface tension play sig-

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Figure 2. Schematic of vortex lattice in planar mix-ing layer [7].

nificant roles in jet flow. The literature reveals noattempt to relate the liquid perforation with vor-tex pairing; however, we will show that “the helicalpairing” mechanism is correlated with the formationof the holes on the liquid sheets. Furthermore, byinspecting the vortical structures of the jet flow, wecan anticipate the locations of the holes on the lobes,bridges, and ligaments. Different primary atomiza-tion regimes for a wide range of flow parameters willbe illustrated. Finally, the break-ups for bridges toligaments and droplets and for ligaments to dropletsare analyzed. Governing equations and computa-tional method can be found in more details in ourearlier paper [2].

Results

Several types of temporal development of the in-stability at the liquid-gas interface and formation ofthe ligaments for low and high density ratios and Reare shown schematically in Figure 3. The instabil-ity starts from the initially axisymmetric KH waves.Time is shown increasing from right to left because,in later figures, the KH cone crests move to the leftin the direction of the co-axial gas flow; thus, we seethe temporal development on a crest as it moves tothe left. For lower gas-to-liquid density ratios andlow Re, a three-dimensional instability deforms theinterface by creating small-scale corrugations at therim of the KH cone crests. Two mechanisms for liga-ment formation are shown in this cartoon: in the toprow, by elongation of the corrugations on the rims;

in the two lower rows, by formation of the holes onthe lobe followed by hole extension and tearing of therim. For low gas-to-liquid density ratios and low Re(top row), a three-dimensional instability deformsthe interface by creating small-scale corrugations atthe rim of the KH cone crests. For this first scenario,the holes form after the first ligaments are createdvia extension from the lobe without relation to holeformation. Then, new ligaments form after theseholes tear the rim. For higher Re (second row), fourlobes are formed without forcing (while more wouldform with the proper initial flow perturbations).

Two holes form off center on both edges of eachlobe. For very high gas-to-liquid density ratios (bot-tom row), the lobe formation is delayed and the lobeedge is less curved and less pronounced. This allowsholes from neighboring lobes to merge and createlarger holes. The liquid entrapped between the rimand the large hole creates a narrow pipe-shaped liq-uid structure named here a “rim bridge”. Capil-lary instability of the rim bridge develops later andeach bridge on the lobe sides breaks into a ligament.When two large holes form on the lobe at higherdensity, a middle bridge as well as the rim bridge isseen. The middle bridge also breaks into two liga-ments and a droplet with the same mechanism as therim bridge. This bridge break-up mechanism is notsignificantly affected by Re for higher gas-to-liquiddensity ratios. The details of the sequence of bridgeformation and break-up followed ligament formationand break-up to droplets are essential to the spraycharacter.

Detailed information concerning length scalesassociated with the structures sketched in Figure3 are relevant to the determination of droplet size.The time scales for the formations of the structuresaffect the growth rate of the two-phase mixture vol-ume which relates to spray cone angle in the prac-tical spatial development of a spray. The thick-nesses of bridges and ligaments relate to dropletsizes. In the following subsections, temporal insta-bility is demonstrated for wide ranges of density ra-tios, Re, andWe. In several figures, vortex lines pro-jected onto the interface, together with streamwise-vorticity contours and streamline projections, showthe relationships between the surface deformationand the vorticity dynamics.

Temporal Development

The initial liquid/gas interface consists of an ax-isymmetric KH wave with a 100 µm wavelength.This initial wavelength is selected based on pre-vious axisymmetric calculations of a liquid jet in-jected at 100 m/s through a 200 µm-diameter ori-

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Figure 3. Schematic of different ligament formationmechanisms and the effects of density ratio ρg/ρl andRe.

fice into an air chamber at 30 atm pressure. So,for the three-dimensional liquid-segment model, a1000 µm-segment of the jet was considered with ini-tial sinusoidal KH surface waves. Thus, a seriesof axisymmetric vortex rings were applied initially.The temporal instability of the liquid segment wasstudied by applying periodic boundary conditionsin the streamwise direction. The three-dimensionalinstability, i.e., deviation from axisymmetric behav-ior, started with formation of hairpin vortices, i.e.,counter-rotating streamwise vortices in the braid re-gion between two adjacent KH cone crests. Soonafter, the KH cone crests also deform and, for theRe in the base calculations, develop naturally intofour lobes. The number of lobes can be affected byRe and/ or forced perturbation of air velocity. (Noforcing is applied in this research.) The lobe loca-tions on the cone crests correlated with locationsof the hairpin-vortices in the ring (crest) region. Aphase difference in the azimuthal variable exists rel-ative to the hairpin vortices that formed earlier inthe braid region. Therefore, two pairs of counter-rotating streamwise vortices form on each lobe.

Figure 4 defines the xyz-coordinate system anddemonstrates at 30 µs the contours of streamwisevorticity magnitude in xz and yz planes for gas-to-liquid density ratio equal to 0.1 for Re=1600 and320, showing clearly evidence of hairpin vortex struc-tures. The pattern is similar to the findings for theplanar mixing layer displayed in Figure 1. How-ever, the spanwise measure is replaced by the az-imuthal angle for liquid round jets. The contours

(a)

(b)

(c)

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Figure 4. Streamwise counter-rotating vortex pairsin xz plane at t = 30 µs, gas-to-liquid density ratiois 0.1, and We = 230, 000: (top) y = 0 meridianplane and (bottom) y = R plane (a) Re = 1600 inxz-plane, (b) Re = 1600 in yz-plane and interface inxz-plane , (c) Re = 320 in xz-plane, (d) Re = 320 inyz-plane and interface in xz-plane. (Gas flows fromright to left.)

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(a)

(b)

(c)

Figure 5. Schematic of overlapping of the two hair-pin vortices from the crest and the braid; (a) sideview (b) top view; (c) thinning of the liquid on thecrest due to the fluid motion with view towards thenegative x-direction. (Gas flows from right to left in(a) and (b).)

are shown in Figure 4 (a) and (c) at two planes, viz.,y = 0 meridian plane and y = R for both Re values.The alternating sign of streamwise vorticity is shownfrom several views in this figure, clearly identifyingthe streamwise overlapping of the hairpin vorticeswith a 180o azimuthal phase difference. Ring hair-pins oriented downstream come above braid hairpinsextended upstream, causing the phase difference ofthe type sketched in Figures 5 and 1(b).

Streamline projections confirm that vorticalstructures of the type identified by Liepmann &Gharib [10] for uniform-density flows as “sidejets”with counter-rotating pairs do appear; see Figure8. The hairpin vortex formations are more orderlyfor higher Re. The views from the downstream sidelooking upstream (i.e., in the negative x-direction)are presented in Figure 4 (b) and (d).

Figure 6 (a) shows one complete lobe and par-tially shows sides of two other lobes for gas-to-liquid

density ratio equal to 0.1 and Re and We equal to1600 and 230,000, respectively. The projected vortexlines on the interface, together with the streamwisevorticity contours of Figure 4, illustrate the hairpinvortices coming onto the lobe from the braid, ori-enting in the opposite direction to the flow, i.e. tothe right. Also, seen are the hairpin vortices com-ing from the upstream lobes, i.e., pointing to theleft. These interpretations follow from comparisonsof Figures 6 with 4.

Later, examination of the streamline projectionson the yz-plane will confirm the presence of counter-rotating axial vortex structures. In Figure 6 (b), theoverlapping-vortex region between two neighboringlobes has been magnified; later, holes will form inthat region. These vortices, including the hairpinvortices, approach each other and form the diamond-shaped region. This overlapping region is very simi-lar to the vortex lattice observed experimentally forsingle-phase planar mixing layers as shown there.Although it might seem from Figure 6 that these vor-tices are merging, no merger is occurring. The hair-pin vortices in the cone crests are above the hairpinvortices in the braid region. This can be explainedby comparing the wavelengths in the azimuthal di-rection with the streamwise wavelength, i.e., that ofthe KH waves. For small spanwise-to-streamwise ra-tio wavelengths, i.e., with four lobes or more, merg-ing is not expected. However, the liquid located be-tween these two vortices in the overlapping regionwill be displaced by the opposing rotational direc-tions from two neighboring vortices. The strandsof the hairpin vortices of opposite orientation standslightly apart at a distance similar to the schematicof Figure 5.

Figure 7 (left) demonstrates the later develop-ment of the lobes of Figure 6 (a). The overlappingvortices have advected downstream with the flowand have moved above the lobe. The lobe has beenextended and its surface area has increased com-pared to Figure 6 (a). Two microseconds later asshown in Figure 7 (b), two holes and four dimplesform on the lobe. Holes form first near the regions oflattice formation identified in Figure 6 and therebyaway from the central azimuthal position where thelobe extension in the downstream direction is great-est. The dimples then become punctures creatingnew holes as shown in Figure 9, where the view hasbeen slightly rotated in the azimuthal direction toillustrate better the new holes formed from the dim-ples in Figure 7 (b) and additional small holes.

The overlapping vortical structures near the tipof the cone produces counter-rotating vortex pairsnear the rim that cause the formation of a small

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(a)

(b)

Figure 6. (a) Iso-surface of liquid-gas interface withprojected vortex lines indicating the hairpin vorticesand two regions of overlapping vortices between thelobes, (b) a magnified picture of the overlapping vor-tex region at t = 38 µs, gas-to-liquid density ratiois 0.1, Re = 1600, We = 230, 000. (Gas flows fromright to left.)

bulge, i.e. pinch. Therefore, the counter-rotatingvortex pairs that form in the overlapping region cancreate dimples if the liquid between the opposite-signvortical structures is pushed downward and pincheswhen it is pushed outward. These surface deforma-tions have been observed without mechanistic expla-nation in computational simulations of Desjardins &Pitsch [4] for a planar jet.

Projected streamlines on the yz-plane crossingthe lobe and the two holes, shown by a dashedline in Figure 7 (right), are demonstrated in Fig-ure 8. The dark color indicates the lobe and the jetcore. The graph is at a fixed x-position; however,the cone crest moves at a sufficiently slow speedto give a meaningful portrayal of transverse veloc-ity relative to the crest. Thereby, useful descrip-tions of counter-rotating structures and radially out-ward flow are given. Streamlines show two pairs of

Figure 7. (left) Iso-surface of liquid-gas interfacewith projected vortex lines indicates the hairpin vor-tices have approached the rims of the lobe at t = 44µs; (right) two holes and four dimples at t = 46µs: gas-to-liquid density ratio is 0.1, Re = 1600,We = 230, 000. (Gas flows from right to left.)

counter-rotating streamwise vortex pairs, i.e., hair-pin vortices adjacent to the lobe near the holes. Thegas flows into the hole from below the lobe in ac-cordance with the direction of the flow circulationat the edge of the lobe. This figure also illustratesthat hairpin vortices overlap without merging. Theradial outward flow by the lobes is consistent withthe results of JS. They also showed a simultaneousradially inward flow in the braid position.

Figure 9 (b) illustrates the extension and merg-ing of the holes observed in 9 (a) and creation ofnew holes at a later time. Figure 10 (a) comparesthe changes in two cone crests at a two-microsecondsinterval. The cone crest on the right side of the pic-ture has developed more than the left crest. Theholes have extended towards the rim forming twoliquid bridges which experience capillary tears whereits cross-sectional area is a minimum. Figure 10 (b)demonstrates two newly created ligaments from thetwo torn bridges on each lobe. Also, the three otherlobes each create two ligaments after tearing. Cap-illary waves develop in the created ligaments as canbe seen in the figure.

Figures 9 and 10 show that the extension of thelobe center in the downstream direction with its re-sulting azimuthal curvature of the crest separatesthe holes on neighboring lobes. Consequently, thoseholes from neighboring lobes do not merge here;however, mergers occur at higher densities.

The surface area of the remaining part of the

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Figure 8. Streamlines projected on the cross-sectional yz plane crossing the lobe and the holesat t = 46 µs: gas-to-liquid density ratio is 0.1,Re = 1600, We = 230, 000.

Figure 9. (Left) Iso-surface of liquid-gas inter-face indicating the formation of additional holes att = 48 µs; (right) extension of the holes at t=50µs: gas-to-liquid density ratio is 0.1, Re = 1600,We = 230, 000. (Gas flows from right to left.)

Figure 10. (Left) Iso-surface of liquid-gas interfaceindicating small bridges at the rim of the lobe att = 52 µs; (right) ligaments form after bridge de-tachment at t = 54 µs: gas-to-liquid density ratiois 0.1, Re = 1600, We = 230, 000. (Gas flows fromright to left.)

lobe tip increases after tearing and formation of theligaments; a crater forms on the center of its sur-face as Figure 11 indicates. The central portion ofthe lobe takes a “spoon” shape after the two off-center holes have opened. The crater is puncturedwhere the sheet thickness at the crater decreases andthe rims become thicker, as shown by 11 (b). Thelocation of the hole is predictable using the vortexlines projected at the liquid interface at the lobe tipas shown in Figure 12. In the fashion of the Fig-ure 1 sketch, hairpin vortices from the braid overlapwith the hairpin vortices at the crest, each with thesame direction for azimuthal vorticity componentsbut opposing streamwise components. As discussedearlier, where two hairpin vortices with same-signcirculation overlap, the liquid sheet becomes thinnerbecause mutual rotation of the vortex lines movesthose vortex (material) lines closer to the same ra-dial position and a hole can form. In summary, thevortex lattice structure (of the type proposed for pla-nar mixing layers in Figure 2) portrayed through theprojected vorticity structures in Figures 6, 7, and 12identifies the hole locations on the cone crests of thejet.

Figure 13 schematically shows the vorticitystructure including the overlapping regions of hair-pin vortices between the lobes. The liquid sheetbecomes thinner at the overlapping hairpin region,as explained in previous discussion related to Fig-ure 5, and the holes form there as indicated with

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(a)

(b)

Figure 11. (a) Top view of the liquid-gas inter-face; showing three of the total four lobes aroundthe circular jet and the extension of the lobes inthe flow direction, curving of the lobe, thinning inthe middle and thickening of the rims and tearing ofthe lobe from the middle. (b) side view. t = 60µs, gas-to-liquid density ratio is 0.1, Re = 1600,We = 230, 000. (Gas flows from right to left.)

Figure 12. Magnified lobe with projected vortexlines, showing two families of hairpin vortices fromthe braid and crest and their pairing at the center ofthe lobe that leads to the lobe puncture and tearing:gas-to-liquid density ratio is 0.1 at t = 60 µs, Re =1600, We = 230, 000. (Gas flows from right to left.)

(a)

(b)

Figure 13. Schematic view of the vortex latticeshowing pairing of the vortices and possible locationsof the holes on lobes. (Gas flows from right to left.)(a) Lower gas density for the same flow condition offigure 12; (b) higher gas density.

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hatched areas. Comparisons of Figures 6 through12 demonstrate a correlation between vortex loca-tion and hole formation. Although the hole locationcan be predicted by hairpin overlap, other flow pa-rameters, e.g. density ratio, viscosity ratio, and sur-face tension, play a role in changing the flow regimeand the hole formation process as will be discussedin the following sub-sections. For example, when thesurface area of the lobe increases and its edge curva-ture decreases, the locations of the holes on neigh-boring lobes will be closer to each other as shown inFigure 13 (b). This happens when the gas densityratio increases and the liquid deformation is delayed.Therefore, instead of three distinct holes on a lobe,i.e., two close to the sides and one near the tip of thelobe, holes merge and create one or two large holeson the lobe.

The hole formation and ligament creation fromthe extension of the holes and tearing of the rimwere observed in the computations of Shinjo &Umemura [3] for Re and We numbers of the sameorder of magnitudes as our results. However, theystated that the lobe punctured due to the im-pact of the droplets that formed earlier from thebreak-up of the mushroom-shaped cap with the jetcore. Our results show that the lobes developedholes before the formation of the droplets sincethe mushroom-shaped cap is not considered in theliquid-segment model. Although the disintegrationof the mushroom-shaped cap might have an effecton the early jet core instability, the hole formationprocess occurs irrespective of the droplet collisionwith the jet; the cap is only a transitional structureappearing during jet start.

Effects of Liquid Viscosity and Re

Figure 16 shows the development of the lobesand holes for gas-to-liquid density ratio equal to 0.1for Re = 5000 and We = 230, 000. Decrease of theliquid viscosity with density and velocity kept thesame as for Figures 4, 6, and 7, causes the numberof lobes to increase from four to five consistent withthe experiments of water injected into water by Liep-mann & Gharib [10] as discussed previously (JS).In general, by lowering the liquid viscosity, smallerscales are allowed and the mode number of azimuthalWidnall instability, which develops into hairpin vor-tices later, increases. Now, with the higher Re cal-culations, qualitative differences do not appear. Thesame mechanisms of lobe formation, lobe surface ex-tension, and hole formation on the lobes are repeatedfor this higher Re case. With the same gas-to-liquiddensity, the only major difference is the formationof the five lobes instead of four.

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(b)

Figure 14. Effects of liquid viscosity on lobe andligament formation at t=50 µs; gas-to-liquid densityratio is 0.1, We = 230, 000 (a) Re = 320 (b) Re =1600. (Gas flows from right to left.)

Figure 15. Surface deformation showing cone andligament formation: gas-to-liquid density ratio is0.05, Re = 320, We = 230, 000. (Gas flows fromright to left.)

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Figure 16. Iso-surface of liquid-gas interface indi-cating the formation of five lobes and later develop-ment of the holes for gas-to-liquid density ratio is0.1; Re = 5000 and t = 36− 42 µs. (Gas flows fromright to left.)

With an increase of the liquid viscosity, the rimsof the primary KH cone crests become thicker andsmall-scale azimuthal corrugations form on the crestas Figure 15 (top) illustrates for gas-to-liquid den-sity ratio equal to 0.05. Due to thicker rims, i.e,55 µm compared to 30 µm thickness of the rim forRe = 1600 with the same gas-to-liquid density ratio,hole formation is delayed and ligaments form fromthe extension of these corrugations on the rims andelongate later in time as shown in Figure 15 (middleand bottom). This agrees with the low-pressure ex-periment [1]. Since ligaments formed by this mech-anism are thicker, they are expected to yield largerdroplets upon their break-up. For the same caseat later times, Figure 17 demonstrates the elonga-tion of the ligaments, development of capillary waveson them, and formation of the holes at the base ofthe ligaments, i.e., where the ligament is attachedto the wave crest. By decreasing Re, ligaments be-come longer before they break into droplets. Thisis consistent with the experimental investigations ofMarmottant & Villermaux [1] for lower gas-to-liquiddensity ratios.

Comparing the streamwise pressure gradientmagnitude induced in the ligament by the externalgas flow, i.e., O(107) kg/m2.s against the local trans-verse gradient magnitude for viscous shear stressbased on the gas viscosity, i.e., O(105−106) kg/m2.s,reveals that the pressure gradient dominates in de-termining the acceleration associated with the liga-ment stretch. Of course, we expect surface tensionto resist the elongation.

Figure 17 also shows the extension of the holewith time and tearing of the rim and formation ofsmall-diameter ligaments after the break up of therim. Therefore, the mechanism that leads to forma-tion of the hole and then its tearing to form newligaments in this lower Re case is qualitatively simi-lar to the findings for higher Re. However, the majordifference is that the more viscous liquid sheet willthin more slowly and require more time to reach thesmall thickness and to puncture. So, this happensafter the formation of the extended ligaments forhigh liquid viscosity.

Figure 18 (a) and (b) show the projected vor-tex lines at the interface for Figure 15 (top) and(bottom), respectively. Unlike the higher Re caseswhere four hairpin vortices formed and four lobeswere created in accordance with them, for low Reas demonstrated in Figure 18 (a), a hairpin vortexformed in less orderly fashion. Simultaneously, a lig-ament grows from the same part of the cone crest.The overlapping vortex region is spread in the braidand cone crests, unlike the high Re case where a

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Figure 17. Formation of a hole at the base of theligaments: gas-to-liquid density ratio is 0.05, Re =320, We = 230, 000. (Gas flows from right to left.)

neat vortex lattice structure formed and hole loca-tion was predicted based on the overlapping vortexregions. The magnitudes of the vorticity variationsfor the lines in this figure indicate that the numeri-cal noise threshold is exceeded and thereby a morechaotic behavior appears at lower Re values. There-fore, although a vortex lattice similar to the struc-ture in the single-phase 3D planar mixing layer isformed in multiphase jet flow for lower liquid vis-cosities, it does not occur in such an orderly mannerat lower Re.

Figures 14 and 19 show the effects of densityratio and liquid viscosity on formation of the liga-ments and holes for gas-to-liquid density ratio equalto 0.1 and 0.5, respectively. By increasing the gasdensity compared to that of Figure 15 with the sameRe = 320 in Figure 14 (a), ligaments form from bothelongation of the corrugations on the cone crests andfrom the tearing of the rims after the hole forma-tion. In general, the liquid protrusions for Re = 320have larger scales compared to those of Figure 14 (b)for Re = 1600 and break into larger droplets. Liga-ments observed in 14 (b) have smaller diameters andthey break at shorter lengths. By increasing the gasdensity to half of the liquid density as shown in Fig-ure 19, the hole and ligament formation is not sig-nificantly affected by the liquid viscosity comparedto flows at lower gas-to-liquid density ratio, i.e, Fig-ure 15. For both low and high Re at higher densityratio, holes form and create a bridge after merging;

(a)

(b)

Figure 18. Projected vortex lines at the inter-face showing the overlapping region and formation ofhairpin vortices after ligaments form: gas-to-liquiddensity ratio is 0.05, Re = 320, We = 230, 000 (a)t=90 µs (b) t=94 µs. (Gas flows from right to left.)

then, the bridge breaks to produce the ligamentsand a droplet. A difference is seen between Fig-ure 19 (a) and (b) for the rim of the lobe whichis less curved and thicker for low Re, as expected,and larger droplets will break from the bridge. TheReynolds number based on gas density and viscos-ity might be more important than our liquid-basedRe in determining the bifurcation in ligament forma-tion. While viscosity was not important in determin-ing the momentum balance for the ligament, it doesaffect the velocity field near the interface and the lig-ament. In conclusion, the mechanism that was ob-served by Marmottant & Villermaux [1] dominatesat low gas-to-liquid density ratio (i.e, lower pressuregas) and lower Re and matches qualitatively withthe conditions of Figure 15.

Spray Character: Droplet Size and Cone Angle

In general, ligaments and bridges are thicker forlarger density ratio and lower Re or lower We. Thesizes for the early formed droplets scale with the

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(a)

(b)

Figure 19. Effects of liquid viscosity on lobe andligament formation; gas-to-liquid density ratio is 0.5at t=50 µs, We = 230, 000 (a) Re = 320 (b) Re =1600. (Gas flows from right to left.)

ligament and bridge sizes. The calculations havenot been carried sufficiently far in time to yield alarge number of droplets or break-up of droplets(secondary atomization). Nevertheless, the compu-tational results are consistent with experimentallymeasured global features of a spray.

Figure 20 shows the temporal radial develop-ment of the mixture of liquid and entrained gas forgas-to-liquid density ratio equal to 0.05, 0.1, and 0.5for Re=320, 1600, and 8000. The outermost radialposition of the continuous liquid defines the radialscale of the two-phase mixture. The furthest loca-tion of the droplets after they detach from the liga-ments and move away from the liquid core for eachcase has also been shown for gas-to-liquid density ra-tio equal to 0.05 and 0.1. In the time duration of ourcalculations, we obtain relatively few droplets, espe-cially for Re=320. These droplets have been formedfrom the middle bridge and rim bridge for 0.5 gas-to-liquid density ratio. Therefore, for the high gasdensity case, the farthest location of the droplets isroughly equal to the outer bound of the liquid andentrained-gas mixture.

The two-phase mixture scale grows faster andextends more radially outward for higher gas den-sity for both Re values. The previously describedradially outward velocity component increases withgas density, thus explaining the faster increasing vol-umetric growth rate with increasing gas density.

For Re = 320 and the lowest gas density, the jet

instability develops more slowly and the cone crestsdo not develop noticeably before the ligaments formfrom the elongation of the corrugations on the cones,becoming longer and extending further radially out-ward. As figure 20 (a) shows, no droplet forms before90 µs. The ligaments must first elongate to producethe droplets. Afterwards, the first droplets formquickly compared to the overall development timeof the cone crests. After the ligaments form, theyspread radially outward. The droplets form at ear-lier times for gas-to-liquid density ratio equal to 0.1compared to 0.05 since the droplets are created fromthe disintegration of the ligaments that, in turn,formed from the tearing of the lobes as discussedearlier. Figure 20 (b) indicates that, by increasingthe Re, the low gas density case behaves similar tohigher gas densities; however, the liquid jet spreadsmore for the highest density ratio, i.e., 0.5, duringthe same time period. The larger entrained mixturevolume at any given time for larger gas density ob-served in our temporal analysis is consistent withthe larger spray cone angle observed with a largergas density in spatially developing spray jets. Thisbehavior is supported by a variety of empirical evi-dence [11, 12, 13].

Figure 21 demonstrates the size of the dropletsmeasured directly within the calculations up to 90µs for gas-to-liquid density ratios of 0.05, 0.1, and0.5 and for Re=320, 1600 and 8000. Larger dropletsare formed for higher gas densities. With increasingliquid viscosity, i.e., decreasing Re, the droplet sizeincreases for all gas-to-liquid density ratios. This isin agreement with the experimental studies of Hiroy-asu and Kadota [14] for a single-hole diesel spray in-jected into gas with pressure varying between 0.1-5.0MPa. They showed that the droplet size increasedwith an increase in gas pressure, i.e., an increasein gas density. Their results demonstrated that anincrease in the liquid viscosity or a decrease in theinjection velocity, i.e., a decrease in Re keeping theliquid density constant, increases the mean dropletsize [14, 15]. An increase in surface tension, i.e.,decrease in We, increases early droplet size, as indi-cated in Figure 21 where a simultaneous decrease inliquid viscosity and increase in surface tension makelittle change. The difference between the droplet sizefor gas-to-liquid density ratios equal to 0.05 and 0.1for Re=1600 is not significant as shown in Figure21. By increasing Re from 1600 to 8000 and reduc-ing the We from 230,000 to 57,500 for density ratioequal to 0.5, no significant difference in droplet sizewas observed.

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(a)

(b)

Figure 20. Radial temporal development of themixture of liquid and entrained gas for gas-to liquiddensity ratio equal to 0.05, 0.1, and 0.5, (a) Re =320, We = 230, 000; (b) Re = 1600, We = 230, 000and Re = 8000, We = 57, 500.

Figure 21. Droplet size for 0.05, 0.1, and 0.5 gas-to-liquid density ratios for Re = 320 and 1600,We = 230, 000; density ratio equal to 0.5, Re =8000,We = 57, 500.

Conclusion and Summary

The development of smaller structures on theoriginally axisymmetric conical wave crests has beendescribed. The formations of lobes, holes, bridges,ligaments, and early droplets have been explainedand quantified. Dependencies on Re,We, liquid vis-cosity, and gas density have been analyzed, coveringthree distinct physical domains. Coherent behaviorwith variations through the parameter domains hasbeen described. Vorticity dynamics has provided auseful explanation for the formations. At higher Re(1600 -16,000) and We (57,500 - 230,000), lobe num-ber is well ordered, increasing with Re, and lobes arelonger in the stream direction; the lobes have morecurvature at the crest edge and a more regular for-mation of holes in the sheet formed by the conicalcrest. The holes enlarge, causing a rim bridge ofliquid at the wave crests on the lobes.

At density ratio of 0.5 throughout the Re range,lobes occur and are regular but develop more slowly,thereby having lower curvature in the azimuthal di-rection at the crest and extending a shorter distancein the gas-stream direction. This allows holes onneighboring lobes to merge before the rims break toform ligaments. Consequently, liquid bridge forma-tions are longer in this domain. At these higher gasdensities throughout the Re range, a middle bridgebetween two holes also forms.

The rim bridges at the crest eventually tear dueto capillary action, transforming the crest rims toligaments and opening the holes. The ligaments re-main connected to the continuous liquid at one endand swing outward radially, thereby affecting spraycone angle. In cases where they exist, middle bridgesalso break by capillary action, forming two ligamentsand a centered droplet. They are not as thick as rimbridges and break earlier. The primary hairpin vor-tices on the lobes deform to create secondary hairpinvortices in a small region between two neighboringlobes that extend on the edge of the cone. Pairs ofcounter-rotating vortices form in the rim bridge af-ter the formation of the hole on the lobe. The samevortex structure is observed in the middle bridge.The bridges break where the two secondary hairpinvortices turn in the opposite direction at the centerof the rim.

In cases where both gas density and Re arelower, the well-ordered lobes are replaced by amore irregular corrugation with a smaller wave-length along the edge of the conical wave crest. Lig-aments form on the lobes by stretching before holesform. The crests are thicker here explaining a de-lay in hole formation. Still, the ligament extensionis driven by pressure gradient rather than shear.

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Detailed experimental results describing the smallerstructures are available only in this lower-valued do-main and agree qualitatively with the computationalresults here.

For Re = 320 and the low gas density, a dif-ferent mechanism affects the early instability; thecone crests do not develop noticeably before the lig-aments form from the elongation of the corrugationson the cones, becoming longer and extending furtherradially outward. By increasing the Re, the low gasdensity case behaves similar to higher gas densities.One must suspect therefore that the Reynolds num-ber based on gas density and liquid viscosity is moreimportant than our liquid-based Re in determiningthe bifurcation in ligament formation. While viscos-ity was not important in determining the momentumbalance for the ligament, it does affect the velocityfield near near the interface and the ligament. In allcases, hole formations align with hairpin and helicalvortices; accordingly, the perforations correlate withresulting fluid motion from these vortices.

Capillary action causes break-up to dropletsfrom the ligaments and bridges. Thicknesses of liq-uid bridges and ligaments and early droplet sizesincrease with increasing gas density and increasingliquid viscosity; i.e., thicker ligaments and largerdroplets form in the low Re, low gas-density range.The droplets form at earlier times for gas-to-liquiddensity ratio equal to 0.1 compared to 0.05 since thedroplets are created from the disintegration of theligaments that, in turn, formed from the tearing ofthe lobes.

The two-phase mixture volume develops fasterand extends more radially outward for higher gasdensity for all Re values tested. This correlates withthe radial velocity increase at the ring with increas-ing gas density. This effect can be explained by theincreasing reduction in circulation magnitude aft ofthe surface wave crest with increasing density dif-ference across the surface. Predicted early dropletsizes and radial dimensions of the emerging sprayshow good qualitative agreement with experimentalevidence. Sizes increase with increasing liquid vis-cosity or surface tension.

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