Length-scale cascade and spread rate of atomizing planar liquid jets

Arash Zandiana,∗, William A. Sirignanoa, Fazle Hussainb

aDepartment of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697, USAbDepartment of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA

Abstract

The primary breakup of a planar liquid jet is explored via direct numerical simulation (DNS) of the incom-pressible Navier-Stokes equation with level-set and volume-of-fluid interface capturing methods. PDFs of thelocal radius of curvature and the local cross-flow displacement of the liquid-gas interface are evaluated overwide ranges of the Reynolds number (Re), Weber number (We), density ratio and viscosity ratio. The tempo-ral cascade of liquid-structure length scales and the spread rate of the liquid jet during primary atomization areanalyzed. The formation rate of different surface structures, e.g. lobes, ligaments and droplets, are comparedfor different flow conditions and are explained in terms of the vortex dynamics in each atomization domainthat we identified recently. With increasing We, the average radius of curvature of the surface decreases, thenumber of small droplets increases, and the cascade and the surface area growth occur at faster rates. Thespray angle is mainly affected by Re and density ratio, and is larger at higher We, at higher density ratios,and also at lower Re. The change in the spray spread rate versus Re is attributed to the angle of ligamentsstretching from the jet core, which increases as Re decreases. Gas viscosity has negligible effect on boththe droplet-size distribution and the spray angle. Increasing the wavelength-to-sheet-thickness ratio, however,increases the spray angle and the structure cascade rate, while decreasing the droplet size. The smallest lengthscale is determined more by surface tension and liquid inertia than by the liquid viscosity, while gas inertiaand liquid surface tension are the key parameters in determining the spray angle.

Keywords: Gas/liquid flow, spray angle, primary atomization, length-scale distribution

1. Introduction

Atomization, the process by which a liquid streamdisintegrates into droplets, has numerous industrial,automotive, environmental, as well as aerospace ap-plications. In the field of engineering, atomizationof liquid fuels in energy conversion devices is im-portant because it governs the spread rate of the liq-uid jet (i.e. spray angle), size of fuel droplets, anddroplet evaporation rate. The spray angle also mea-sures the liquid sheet instability growth, which con-trols the rate and intensity of the atomization.

The common purpose of breaking a liquid streaminto spray is to increase the liquid surface area so that

∗Corresponding authorEmail address: azandian@uci.edu (Arash Zandian)

subsequent heat and mass transfer can be increasedor a coating can be obtained. The spatial distribution,or dispersion, of the droplets is important in combus-tion systems because it affects the mixing of the fuelwith the oxidant, hence the flame length and thick-ness. The size, velocity, volume flux, and numberdensity of droplets in sprays critically affect the heat,mass, and momentum transport processes, which, inturn, affect the flame stability and ignition character-istics. Smaller drop size leads to higher volumetricheat release rates, wider burning ranges, higher com-bustion efficiency, lower fuel consumption and lowerpollutant emission. In some other applications how-ever, small droplets must be avoided because theirsettling velocity is low and under certain meteoro-logical conditions, they can drift too far downwind(Negeed et al., 2011).

Preprint submitted to International Journal of Multiohase Flow August 29, 2018

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Most previous research has attempted to assessthe manner by which the final droplet-size distri-bution – after the atomization is fully developed– is affected by the gas and liquid properties andby the nozzle geometry (Dombrowski and Hooper,1962; Senecal et al., 1999; Mansour and Chigier,1990; Stapper et al., 1992; Lozano et al., 2001; Car-valho et al., 2002; Varga et al., 2003; Marmottantand Villermaux, 2004; Negeed et al., 2011). Gen-eral conclusions are that the Sauter Mean Diameter(SMD = ΣNid3i /ΣNid

2i , where Ni is the number of

droplets per unit volume in size class i, and di is thedroplet diameter) decreases with increasing relativegas-liquid velocity, increasing liquid density, and de-creasing surface tension, while viscosity is found tohave little effect (Stapper et al., 1992). The mainfocus has been on the final droplet-size distributionand the spatial growth of the spray (spray angle),but little emphasis has been placed on the tempo-ral cascade of the surface length scales – growth ofKelvin-Helmholtz (KH) waves and their cascade intosmaller structures leading to final breakup into liga-ments and then droplets – and the spread rate of thespray (spray angle) in primary atomization. In orderto better control and optimize the atomization effi-ciency, the transient process from the point of injec-tion to the fully-developed state needs to be betterunderstood. Hence the spray angle and the rate atwhich the length scales cascade at different flow con-ditions must be analyzed. This is the main focus ofthe current study. The data presented herein will becrucial in the analysis and design of atomizers.

In recent years, more computational studies haveaddressed the liquid-jet breakup length scales andspray width. Most of these, however, have qualita-tively investigated the effects of fluid properties andflow parameters on the final droplet size distributionor the spray angle. Only a few studies quantified thespatial variation of the droplet/ligament size along oracross the spray axis. However, there are no detailedstudy of the temporal variation of the liquid-structuresize distribution and the spray width (spray angle)during primary atomization. Most recent computa-tional studies analyze the spatially developing insta-bility leading to breakup of liquid streams; however,all of them are at relatively low values of the We-ber number (We < 2000). That is, although some of

those works are described as “atomization” studies,they all fit better under the classical “wind-inducedcapillary instabilities” defined by Ohnesorge (1936)and Reitz and Bracco (1986). Most of these stud-ies are linear (Otto et al., 2013), two-dimensional(2D) inviscid (Matas et al., 2011), two-dimensional(Fuster et al., 2013; Agbaglah et al., 2017), or three-dimensional (3D) large-eddy simulations (Agbaglahet al., 2017). Of course, these did not resolve thesmaller structures that form during the cascade pro-cess of the breakup. An analysis with spatial devel-opment offers some advantage with practical realismover temporal analysis. At the same time, the ad-ditional constraints imposed by the boundary condi-tions remove generality in the delineation of the im-portant relevant physics. For these reasons, we fol-low the path of temporal-instability analysis in theclassical atomization (high We range) provided byJarrahbashi and Sirignano (2014). The goal is to re-veal and interpret the physics in the cascade processknown as atomization. Note that some spatial de-velopment is provided when the temporal analysiscovers a domain that is several wavelengths in size.Relations between spatially developing and temporalresults are discussed for single-phase and two-phaseflows by Gaster (1962) and Fuster et al. (2013), re-spectively.

Jarrahbashi et al. (2016) studied wide rangesof density ratio (0.05–0.5), Reynolds number(320 < Re < 8000), and We of O(104–105) in roundliquid jet breakup. They defined the radial scale ofthe two-phase mixture as the outermost radial loca-tion of the continuous liquid and showed that the ra-dial spray growth increases with increasing gas-to-liquid density ratio. They showed that droplets arelarger for higher gas densities and lower We values,and form at earlier times for higher density ratios.However, the effects of Re and We were not fullystudied there.

In a similar study, Zandian et al. (2016) exploredthe effects of We on the droplet size as well as theliquid sheet expansion rate. They also showed thevariation in the size and number density of dropletsfor We in the range 3000–72, 000; however, boththese quantities were obtained from visual post-processing, which incur non-negligible errors. Theirqualitative comparison showed that droplet and lig-

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Figure 1: Schematic showing the spray width basis as defined by Zandian et al. (2016).

ament sizes decrease with increasing We, while thenumber of droplets increases. Their results appliedfor a limited time after the injection and did not showhow fast the length scales cascade. Zandian et al.(2016) similar to Jarrahbashi et al. (2016) defined thedistance of the farthest point on the continuous liquidsheet surface from the jet centerplane as the cross-flow width of the spray, as schematically depicted inFig. 1. Even though this definition provides a sim-ple description of the spray growth, it lacks statisti-cal information about the number density of liquidstructures at different cross-flow distances from thejet centerplane. Thus, this definition is not optimalfor the spray width analysis and failed to show theeffects of We on the spray expansion rate.

More recently, Zandian et al. (2017) identifiedthree atomization mechanisms and defined their do-mains of dominance on a gas Weber number (Weg)versus liquid Reynolds number (Rel) diagram, shownin Fig. 2. The liquid structures seen in each of theseatomization domains are sketched in Fig. 3, wherethe evolution of a liquid lobe is shown from a topview. These domain classifications were shown toapply to both planar and round jets. At high Rel, theliquid sheet breakup characteristics change based ona modified Ohnesorge number, Ohm ≡

√Weg/Rel, as

follows: (i) at high Ohm and high Weg, lobes be-come thin and puncture, creating holes and bridges.Bridges break as perforations expand and create lig-aments, which then stretch and break into dropletsby capillary action. This domain is indicated as At-omization Domain II in Fig. 2, and its process iscalled LoHBrLiD based on the cascade of structuresin this domain (Lo ≡ Lobe, H ≡ Hole, Br ≡ Bridge,Li ≡ Ligament, and D ≡ Droplet); (ii) at low Ohm

Figure 2: The breakup characteristics based on Weg and Rel,showing the LoLiD mechanism (Atomization Domain I) de-noted by diamonds, the LoHBrLiD mechanism (AtomizationDomain II) denoted by circles, the LoCLiD mechanism (At-omization Domain III) denoted by squares, and the transitionalregion denoted by triangles. The cases with density ratio of 0.1(ρ̂ = 0.1) are shaded. The ρ̂ = 0.1 and ρ̂ = 0.5 cases that overlapat the same point on this diagram are noted. – · – · –, transitionalboundary at low Rel; and – – –, transitional boundary at highRel (Zandian et al., 2017). The red symbols denote the casesfor lower ρ̂ = 0.05, added to the original diagram by Zandianet al. (2018).

and high Rel, holes are not seen at early times; in-stead, many corrugations form on the lobe front edgeand stretch into ligaments. This process is called Lo-CLiD (C ≡ Corrugation) and occurs in AtomizationDomain III (see Fig. 2), and results in ligaments anddroplets without having the hole and bridge forma-tion steps. The third is a LoLiD process that occursat low Rel and low Weg (Atomization Domain I in

3

Figure 3: Sketch showing the cascade of structures on a liquid lobe from a top view, for the LoLiD (top), LoCLiD (center), andLoHBrLiD (bottom) processes. The gas flows on top of these structures from left to right, and time increases to the right (Zandianet al., 2017).

Fig. 2), but with some difference in details from theLoCLiD process.

The main difference between the two processes(Domains I and III) at low and high Rel is that,at higher Rel the lobes become corrugated beforestretching into ligaments. Hence, each lobe may pro-duce multiple ligaments, which are typically thinnerand shorter than those at lower Rel. At low Rel, onthe other hand, because of the higher viscosity, theentire lobe stretches into one thick, usually long lig-ament. There is also a transitional region betweenthe atomization domains, shown by the dashed-linesin Fig. 2. As seen in Fig. 3, the primary atomizationfollows a cascade process, where larger liquid struc-tures, e.g. waves and lobes, cascade into smaller andsmaller structures; e.g. bridges, corrugations, liga-ments, and droplets. The time scales of the struc-ture formations were also determined and were cor-related to Rel and Weg. However, the length scale ofthe structures observed in each domain and their cas-cade rates were not discussed. The main focus of thisstudy is on quantifying the cascade rate of differentsurface structures at the three atomization domainsthat were identified above. Moreover, the growth

rate of the spray width in primary atomization is alsoquantified for the first time and is compared for dif-ferent atomization domains. In the remainder of thissection, some of the older methods used for quantifi-cation of the mean droplet size and spray angle areintroduced and their pertinent results are presented.

Desjardins and Pitsch (2010) defined the half-width of the planar jets as the distance from the jetcenterline to the point at which the mean stream-wise velocity excess is half of the centerline velocity.They showed that high We jets grow faster, whichsuggests that surface tension stabilizes the jets. Theyalso demonstrated that, regardless of We, dropletsare generated through the creation and stretching ofliquid ligaments (pertinent to Domain I in Fig. 2).Ligaments are longer, thinner, and more numerousas We is increased. The corrugation length scalesappear larger for lower Rel – attributed to lesser en-ergy contained in small eddies at lower Rel. Conse-quently, earlier deformation on the smaller scales ofthe interface is more likely to take place on relativelylarger length scales as Rel is reduced. Their workwas limited to We of O(102–103), which were lowcompared to our range of interest for common liq-

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uid fuels and high-pressure operations. The effectsof density ratio were also not studied in their work.

Dombrowski and Hooper (1962) developed the-oretical expressions for the size of drops producedfrom fan spray sheets, and showed that under certainoperating conditions in super atmospheric pressures,the drop size increases with ambient density. Veri-fied experimentally, the drop size initially decreases,passing through a minimum with further increase ofambient density. They also showed that, for rela-tively thin sheets (h/λ < 1.25; h is sheet thicknessand λ is the unstable wavelength), the drop size in-creases with the surface tension and the liquid den-sity, but depends inversely on the liquid injectionpressure and the gas density. For relatively thicksheets (h/λ > 1.5), however, the drop size is inde-pendent of surface tension or injection pressure andincreases with increasing gas density. In another lin-ear stability analysis, Senecal et al. (1999) derivedexpressions for the ligament diameter for short andlong waves. They showed that the ligament diame-ter is directly proportional to the sheet thickness, butinversely to the square root of Weg. They also analyt-ically related the final droplet size to the ligament di-ameter and the Ohnesorge number (Oh =

√We/Re),

and showed that the SMD decreases with time. Thegas viscosity and the nonlinear physics of the prob-lem, however, were neglected.

Effects of mean drop size and velocity on liquidsheets were measured using a phase Doppler parti-cle analyzer (PDPA) by Mansour and Chigier (1990).The spray angle decreased notably with increasingliquid 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. Increasing the air pressure for afixed liquid flow rate resulted in an increase in thespray angle. Mansour and Chigier (1990) also mea-sured the SMD along and across the spray axis, andshowed a significant reduction in droplet size by in-creasing the air-to-liquid mass-flux ratio.

Carvalho et al. (2002) performed detailed mea-surements of the spray angle versus gas and liq-uid velocities in a planar liquid film surroundedby two air streams in a range of Re from 500 to5000. For low gas-to-liquid momentum ratio, thedilatational wave dominates the liquid film disinte-

gration mode, and the atomization quality is ratherpoor, with a narrow spray angle (pertinent to DomainIII). For higher gas-to-liquid momentum ratios, sinu-soidal waves dominate and the atomization quality isconsiderably improved, as the spray angle increasessignificantly (Domain I). Regardless of the gas ve-locity, a region of maximum spray angle occurs, fol-lowed by a sharp decrease for higher values of theliquid velocity, and the maximum value of the sprayangle decreases with the gas flow rate (correspondingto a transition from Domain I to III).

At present, it is very challenging to predict thedroplet-diameter distribution as a function of injec-tion conditions. For combustion applications, manyempirical correlations relate the droplet size to theinjection parameters (Lefebvre, 1989); however, de-tailed studies of fundamental breakup mechanisms– especially during the initial period – are neededto construct predictive models. A summary of sev-eral of these expressions for airblast atomization wascompiled by Lefebvre (1989). The dependence ofthe primary droplet size (d) on the atomizing gas ve-locity (Ug) is most often expressed as a power law,d ∝ U−ng , where 0.7 ≤ n ≤ 1.5. Physical expla-nations for particular values of the exponent n aregenerally lacking. Varga et al. (2003) found that themean droplet size is not very sensitive to the liq-uid jet diameter – actually opposite to intuition; thedroplet size is observed to increase slightly with de-creasing nozzle diameter. This effect was attributedto the slightly longer gas boundary-layer attachmentlength. Varga et al. (2003) also observed a clear re-duction in droplet size with lowering surface tension(transition from Domain I to II). They suggested thescaling d ∝ We−1/2g , where Weg is the Weber numberbased on gas properties and jet diameter. Their studywas limited to very low Oh of O(10−3).

Marmottant and Villermaux (2004) presentedprobability density functions (PDF) of the ligamentsize and the droplet size in their experimental studyof the round liquid spray formation. The ligamentsize d0 was found to be distributed around the meanin a nearly Gaussian distribution, but the droplet di-ameter d was more broadly distributed and skewed.Rescaled by d0, which depends on the gas veloc-ity Ug, the size distribution keeps roughly the sameshape for various gas flows. The average droplet size

5

after ligament breakup was found to be d ' 0.4d0,with its distribution P(d) having an exponential fall-off at large diameters parameterized by the averageligament size 〈d0〉, namely P(d) ∼ exp(−nd/ 〈d0〉).The parameter n ≈ 3.5 slowly increased with Ug.The mean droplet size in the spray decreases likeU−1g . Most of their experiments were at low Re andlow We ranges (Domain I).

More recently, Negeed et al. (2011) analyticallyand experimentally studied the effects of nozzleshape and spray pressure on the liquid sheet char-acteristics. They showed that the droplet mass meandiameter decreases by increasing the Rel or by in-creasing the water sheet We (transition from DomainI to Domain III), since the inertia force increases byincreasing both parameters. The spray pressure dif-ference was also shown to have a similar effect on themean droplet size as Rel. Their study was limited tovery high Rel values (10, 000–36, 000) and very lowWeg values of O(101); i.e. Domain III.

The scarcity of studies in Domain II is especiallynotable since the trend towards much higher oper-ating pressures has started in engine designs. Thestudies introduced above indicate that there is noproper method in the literature for quantification ofthe cascade rate and spray spread rate at early liquid-jet breakup. Therefore, implementation of a newmethodology – as will be introduced here – is es-sential for evaluation of these quantities. Quantifica-tion of these parameters is very important for under-standing and identifying the most significant causes,which are helpful in controlling the atomization pro-cess.

1.1. Objectives

Our objectives are to (i) study the effects of the keynon-dimensional parameters, i.e. Rel, Weg, gas-to-liquid density ratio and viscosity ratio, and also thewavelength-to-sheet-thickness ratio, on the temporalvariation of the spray width and the liquid-structureslength scale; (ii) establish a new model and defini-tion for measurement of the liquid surface length-scale distribution and spray width for the early periodof spray formation; and (iii) explain the roles dif-ferent breakup regimes play in the cascade of lengthscales and spray development during the early atom-ization period. The results are separated by atom-

ization domains to clarify the effects of each atom-ization mechanism on the cascade process and sprayexpansion. Moreover, the time portions of the be-haviors are related to the various structures formedduring the primary atomization period.

To address these objectives, two PDFs are ob-tained from the numerical results, for a wide rangeof liquid-structure size and transverse location ofthe liquid-gas interface at different times to give abroader understanding of the temporal variation ofthe length-scale distribution and the spray width. Thetemporal evolution of the distribution functions isgiven rather than just showing the fully-developedasymptotic length scales, as is well explored and ex-amined in the literature. The first PDF indicates thesize of the small liquid structures through the localradius of curvature of the gas-liquid interface. Thenovelty of this model is that we do not address onlythe droplets that are formed, nor present only thesize of the droplets as the length scale of the atom-ization problem – as what the SMD presents. In-stead, the length-scale distribution here comprisesthe size of all the liquid structures – from the ini-tial surface waves, to the lobes, bridges, ligaments,and finally droplets. Thus, this analysis reveals thechange in the overall length scale of the jet surfaceeven before the droplets are formed – not examinedor quantified before. The second PDF indicates thelocation of the liquid-gas interface, giving more thanjust the outermost displacement of the spray, as wastypically measured in most past computational stud-ies. Rather, we present the “density” of the liquidsurface at any transverse location, which providesa more meaningful and realistic presentation of thespray width. “Density” here means the relative liquidsurface area at a given control volume at any distancefrom the jet midplane.

In Section 2, the numerical methods that havebeen used and the most important flow parametersinvolved in this study are presented along with thepost-processing methods used to obtain the PDFs.Section 3 is devoted to the analysis of the effectsof We (Section 3.2), Re (Section 3.3), density ratio(Section 3.4), viscosity ratio (Section 3.5), and sheetthickness (Section 3.6) on the spray width and theliquid-structure size. Conclusions are given in Sec-tion 4, accompanied by a short summary of our most

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important findings.

2. Numerical Modeling

The 3D Navier-Stokes equations with level-set andvolume-of-fluid interface capturing methods yieldcomputational results for the liquid segment whichcaptures the liquid-gas interface deformations withtime.

The incompressible continuity and Navier-Stokesequations, including the viscous diffusion and sur-face tension forces and neglecting the gravitationalforce are as follows:

∇ · u = 0, (1)

∂(ρu)∂t

+ ∇ · (ρuu) =

− ∇p + ∇ · (2µD) − σκδ(d)n,(2)

where D is the rate of deformation tensor,

D =12

[(∇u) + (∇u)T

]. (3)

u is the velocity vector; p, ρ, and µ are the pres-sure, density and dynamic viscosity of the fluid, re-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) isthe Dirac delta function, d is the distance from theinterface, and n is the unit vector normal to the liq-uid/gas interface.

Direct numerical simulation is done with an un-steady 3D finite-volume solver for the incompress-ible Navier-Stokes equations describing the planarliquid sheet segment (initially stagnant), which issubject to instabilities due to a gas stream that flowspast it on both sides. A uniform staggered gridis used with the mesh size of ∆ = 2.5 µm and atime step of 5 ns – finer grid resolution of 1.25 µmis used for the cases with higher Weg (≥ 36, 000)and higher Rel (= 5000). Spatial discretization andtime marching are given by the third-order accurateQUICK scheme and the Crank-Nicolson scheme, re-spectively. The continuity and momentum equationsare coupled through the SIMPLE algorithm.

The level-set method developed by Osher and hiscoworkers (Zhao et al., 1996; Sussman et al., 1998;Osher and Fedkiw, 2001) captures the liquid-gas in-terface. The level set φ is a distance function withzero value at the liquid-gas interface, positive val-ues in the gas phase and negative values in the liquidphase. All the fluid properties for both phases in theNavier-Stokes equations are defined based on the φvalue and the equations are solved for both phases si-multaneously. Properties such as density and viscos-ity vary continuously but with a very large gradientnear the liquid-gas interface. The level-set functionφ is also advected by the velocity field;

∂φ

∂t+ u · ∇φ = 0. (4)

The interface curvature needed to calculate thesurface tension force can also be obtained from thelevel-set function (κ = ∇. ∇φ|∇φ| ), which for a 3D do-main results in the following equation.

κ =φxx(φ2y + φ

2z ) + φyy(φ

2x + φ

2z ) + φzz(φ

2x + φ

2y)

(φ2x + φ2y + φ2z )3/2

−2φxφyφxy + 2φyφzφyz + 2φzφxφxz

(φ2x + φ2y + φ2z )3/2 . (5)

For detailed descriptions of this interface captur-ing see Sussman et al. (1998).

At low density ratios, a transport equation simi-lar to Eq. (4) is used for the volume fraction f , alsocalled the Volume-of-Fluid (VoF) variable, in orderto describe the temporal and spatial evolution of thetwo-phase flow (Hirt and Nichols, 1981).

∂ f∂t

+ u · ∇ f = 0. (6)

where the VoF-variable f represents the volume of(liquid phase) fluid fraction at each cell.

The fully conservative momentum convection andvolume fraction transport, the momentum diffusion,and the surface tension are treated explicitly. To en-sure a sharp interface of all flow discontinuities andto suppress numerical dissipation of the liquid phase,the interface is reconstructed at each time step by thePLIC (piecewise linear interface calculation) methodof Rider and Kothe (1998). The normal direction of

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the interface is found by considering the volume frac-tions in a neighborhood of the cell considered (simi-lar to Eq. 5), where f changes most rapidly. A least-square method introduced by Puckett et al. (1997)is used for normal reconstruction, where the inter-face is approximated by a straight line in the cellblock. Once interface reconstruction has been per-formed, direction-split geometrical fluxes are com-puted for VoF advection (Popinet, 2009). This advec-tion scheme preserves sharp interfaces and is close tosecond-order accurate for practical atomization ap-plications. We use the paraboloid fitting techniqueof Popinet (2009) to compute an accurate estimate ofthe curvature from the discrete volume fraction val-ues. The capillary effects in the momentum equa-tions are represented by a capillary tensor introducedby Scardovelli and Zaleski (1999).

2.1. Flow Configuration

The computational domain, shown in Fig. 4(a),consists of a cube, which is discretized into uniform-sized cells. The liquid segment, which is a sheetof thickness h0 (h0 = 50 µm for the thin sheet and200 µm for the thick sheet), is located at the cen-ter of the box and is stationary in the beginning.The domain size in terms of the sheet thickness is16h0 × 10h0 × 10h0, in the x, y and z directions, re-spectively, for the thin sheet, and 4h0 × 4h0 × 8h0 forthe thick sheet. The liquid segment is surrounded bythe gas zones on top and bottom. The gas moves inthe positive x- (streamwise) direction with a constantvelocity (U = 100 m/s) at top and bottom bound-aries, and its velocity diminishes to the interface ve-locity with a boundary layer thickness obtained from2D full-jet simulations. In the liquid, the velocitydecays to zero at the center of the sheet with a hy-perbolic tangent profile. For more detailed descrip-tion of the initial velocity profile and boundary layerthickness, see figure 12 of Zandian et al. (2016). Thenormal gradient (∂/∂z) is set to zero for the normaland spanwise velocity components (w and v) at thetop and bottom boundaries, so that the gas would befree to flow into the domain following the entrain-ment by the jet. Periodic boundary conditions for allcomponents of velocity as well as the level-set/VoFvariable are imposed on the four sides of the compu-tational domain; i.e. the x- and y-planes.

The computational box resembles a segment of theliquid sheet far upstream of the jet cap, as shown inFig. 4(b). This sub-figure shows the starting behaviorof a spatially developing full liquid jet injected withconstant velocity Ul into quiescent gas. A Galileantransformation of the velocity field shows that the gasstream flows upstream with respect to the liquid jetwith a relative velocity Urel = Ul (denoted by thered arrows) in the box shown in Fig. 4(b). Our tem-poral study focuses on this region of the jet streamaway from the jet cap, where previous studies indi-cate that surface deformations are periodic (Shinjoand Umemura, 2010).

This study involves a temporal computationalanalysis with a relative velocity between the twophases. Due to friction, the relative velocity de-creases with time. Furthermore, the domain is sev-eral wavelengths long in the streamwise directionso that some spatial development occurs. In orderto reduce the dependence of the results on detailsof the boundary conditions, specific configurations(e.g., air-assist or air-blast atomization) are avoided.However, calculations are made with the critical non-dimensional parameters in the ranges of practical ap-plications.

The liquid-gas interface is initially perturbed sym-metrically on both sides with a sinusoidal profile andpredefined wavelength and amplitude obtained fromthe 2D full-jet simulations (see Zandian et al., 2016).Two analyses without forced or initial surface per-turbations – the full-jet 2D simulations (figure 11 ofZandian et al., 2016) and the initially non-perturbed3D simulations (figure 17 of Zandian et al., 2017) –show KH wavelengths in the moderate range of 80–125 µm over a wide range of Rel, Weg and ρ̂ stud-ied here. In order to expedite the appearance andgrowth of the KH waves, initial perturbations withwavelength of 100 µm are imposed on the interfacewith a small amplitude of 4 µm for the thick sheetand 2 µm for the thin sheet. This amplitude is smallenough that subharmonics would have a chance toform and grow. Similar to Jarrahbashi and Sirignano(2014), our results show that at higher Rel and lowerρ̂, smaller waves appear superimposed on the initialperturbations. The waves also merge to create largerwaves at lower Weg. Both streamwise (x-direction)and spanwise (y-direction) perturbations are consid-

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Figure 4: The computational domain with the initial liquid and gas zones (a), and a spatially developing full liquid jet (b).

ered in this study.The most important dimensionless groups in this

study are the Reynolds number (Re), the Weber num-ber (We), as well as the gas-to-liquid density ratio(ρ̂) and viscosity ratio (µ̂), as defined below. The ini-tial wavelength-to-sheet-thickness ratio (Λ) is also animportant parameter.

Re =ρUh0µ

, We =ρU2h0σ

, (7a)

ρ̂ =ρg

ρl, µ̂ =

µg

µl, Λ =

λ0h0. (7b)

The initial sheet thickness h0 is considered as thecharacteristic length, and the relative gas–liquid ve-locity U as the characteristic velocity. The subscriptsl and g refer to the liquid and gas, respectively. The-oretically, if the flow field is infinite in the stream-wise direction (as in our study), a Galiliean trans-formation shows that only the relative velocity be-tween the two streams is consequential. Spatiallydeveloping flow fields, however, are at most semi-infinite so that both velocities at the flow-domain en-try and their ratio (or their momentum-flux ratio) areimportant. A wide range of Re and We at high andlow density and viscosity ratios is covered in this re-search. Kerosene with density of 800 kg m−3 is used

as the liquid. The liquid viscosity and surface ten-sion coefficient are obtained from the given Reynoldsand Weber numbers, respectively, and the gas proper-ties are calculated from the desired density and vis-cosity ratio. For a typical case at Rel = 2500 andWeg = 5000, µl ≈ 0.005 Pa s and σ ≈ 0.03 kg s−2are obtained. The range of dimensionless parame-ters analyzed in this study are given in Table 1. Therange of parameters considered here covers the morepractical ranges usually seen in most atomization ap-plications; however, higher ρ̂ (high pressures) andhigher Weg values are also studied to fully exploreand portray their effects. Even though current prac-tical applications usually perform at density ratioslower than 0.05, high pressure (high density) applica-tions are currently being discussed for future gas tur-bines/engines. The analysis performed in this studyis in line with this trend of future injection systemstowards higher density ratios.

The grid independency tests were performed indetail by Zandian et al. (2018). They showed thatthe errors in the size and location of ligaments anddroplets, and the magnitude of the velocity andboundary layer thickness computed using differentmesh resolutions were within an acceptable range.The effects of mesh resolution on the most impor-tant flow parameters, e.g. surface structures, veloc-

9

Table 1: Range of dimensionless parameters.

Parameter Rel Weg ρ̂ µ̂ Λ

Range 1000–5000 1500–36,000 0.05–0.9 0.0005–0.05 0.5–2.0

ity and vorticity profiles were studied in detail byZandian et al. (2018) and the mass conservation ofthe LS method was confirmed. The domain-size in-dependency were also checked in both streamwiseand spanwise directions to ensure that the resolvedwavelengths were not affected by the domain lengthand width. The transverse (cross-flow) dimensionof the domain was chosen such that the top andbottom boundaries remain far from the interface atall times, so that the surface deformation is not di-rectly affected by the boundary conditions. Zan-dian et al. (2018) showed that, as the KH waves am-plify, the convective velocity of the interface at thebase of the waves was in very good agreement withthe Dimotakis speed (Dimotakis, 1986) defined asUD = (Ul+

√ρ̂Ug)/(1+

√ρ̂). The effects of the fuzzy

zone thickness – where properties have large gradi-ents to approximate the discontinuities between thetwo phases – have been previously addressed by Jar-rahbashi and Sirignano (2014). The effect of meshresolution on the PDFs and length scale measure-ments is detailed in Section 3.1.

2.2. Data analysis

Twice the inverse of the liquid surface curvature(κ = 1/R1 + 1/R2, where R1 and R2 are the two radiiof curvature of the surface in a 3D domain) has beendefined as the local length scale in this study. Thislength scale represents the local radius of curvatureof the interface, and is a proper quantity enabling usto monitor the overall size of the surface structures.It would eventually asymptote to the droplet radiusafter the entire jet is broken into approximate spher-ical droplets. Based on this definition, a length scaleis obtained at each computational cell in the fuzzyzone at the interface. The curvature of each cell con-taining the interface is computed at every time step;

then, the structure length scale (L) is defined as

Li jk =2|κi jk|

, (8)

where κ is the curvature, and the i jk indices denotethe coordinates of the cell in a 3D mesh. The lengthscale Li jk of each cell is used to create a PDF of thelength scales. The κi jk value is measured per com-putational cell and weighted by the interface area inthat cell to obtain the PDFs. Only cells containingthe interface are included in this analysis. The binsize used for the length-scale analysis is dL = ∆ =2.5 µm. The probability of the length scale in the in-terval (L, L + dL) is obtained by multiplying the PDFvalue at that length scale, f (L), by the bin size:

prob(L ≤ L′ ≤ L + dL) ≡ P(L) = f (L)dL. (9)

This is an operational definition of the PDF. Sincethe probability is unitless, f (L) has units of the in-verse of the length scale; i.e. m−1. However, in ourstudy, the length scale is nondimensionalized by theinitial wavelength. Thus, f (L/λ0) becomes unitless.The relation between the length-scale PDF and itsprobability could be derived from Eq. (9);

f (L/λ0) =P(L/λ0)dL/λ0

= 40P(L/λ0), (10)

where dL is the bin size and λ0 = 100 µm is the initialKH wavelength. The probability of having the lengthscale in the finite interval [a, b] can be determined byintegrating the PDF;

prob(a ≤ Lλ0≤ b) ≡ P(a ≤ L

λ0≤ b)

=

∫ ba

f (L′

λ0)dL′

λ0. (11)

In order to obtain the average length scale at eachtime step, the length scales are integrated along the

10

liquid-gas interface and divided by the total interfacearea. The average length scale δ is nondimension-alized using the initial perturbation wavelength, λ0.The average length scale is defined as

δ =1λ0

∫Lds

S≈ 1λ0

∑Lisi∑si, (12)

where S is the total surface area of the interface, si isthe interface area in cell i and Li is the length scaleof that particular cell.

Since L has a wide range from a few microns toinfinity (if the curvature is zero at a cell), we neglectthe length scales that are very large, i.e. L > 4λ0, sothat δ would not be biased towards large scales dueto those off values. δ can show the overall change inthe size of the structures on the liquid surface; there-fore, one can track the stretching of the surface – if δgrows – or its cascade into smaller structures and ap-pearance of subharmonic instabilities – as the meandecays.

Similar to the length scale, a PDF is obtainedfor the transverse distance of the interface from thecenterplane. This is related to the interface densitymodel introduced by Chesnel et al. (2011), where theinterface density was defined as the ratio of the inter-face area within the considered control volume. Inour study, however, the interface density is measuredat different transverse locations to form the PDFs.This PDF also has units of m−1, as discussed before;however, it is normalized by the initial sheet thick-ness. This gives a better statistical data about thedistribution of the transverse location of the spray in-terface rather than just presenting the outermost lo-cation of the liquid surface. Thereby, the concen-tration, or density, of the liquid surface at any trans-verse plane is measured. This quantity also shows thebreakup of surface structures and demonstrates howuniformly the spray spreads; i.e. the quality of spraydevelopment. In fact, this PDF is a more generalizedversion of the average liquid volume fraction distri-bution. To account for both sides of the liquid sheetin this analysis, the cross-flow distance h of each lo-cal point at the interface is defined as the absolutevalue of its z-coordinate (z = 0 at the centerplane);

hi jk = |zi jk|. (13)

The probability of the spray width is obtained froman equation similar to Eq. (9), where L is replacedby h. The relation between the spray-width PDF,f (h/h0), and its probability, P(h/h0), is

f (h/h0) =P(h/h0)dh/h0

, (14)

where dh is the bin size for the spray-width PDF,taken to be equal to the mesh size, i.e. dh = ∆ =2.5 µm, and h0 is the initial sheet thickness; h0 =50 µm for the thin sheet, and 200 µm for the thicksheet.

The mean spray width (sheet thickness) ζ is alsoobtained by integrating h along the interface, anddividing it by the total interface area. The meanspray width is nondimensionalized by the initialsheet thickness h0;

ζ =1h0

∫hds

S≈ 1

h0

∑hisi∑si, (15)

where hi is the cross-flow distance of the interfacein cell i from the midplane. This definition repre-sents how dense the surface is at any distance fromthe jet centerplane. Therefore, it gives a more real-istic representation of the jet growth in a way thatis more useful for many applications such as com-bustion and coating. A simple height function forthe spray would be a single-valued function of down-stream distance and time; our PDF accounts for themulti-valued nature of the real situation.

3. Results and discussion

3.1. Data analysis verification

The choice of bin size and the sensitivity ofthe measurements against the mesh resolution andcomputational-domain size are tested and verified inthis section. The results of these tests are given inFigs. 5–7. Fig. 5(a) compares the temporal variationof δ with two different bin sizes used for measure-ment of this parameter. The solid line is the resultobtained by the actual bin size used in our analysis(dL = ∆ = 2.5 µm), and the dashed line denotes thevariation of δ with time using a bin size twice as big;i.e. dL = 2∆ = 5 µm. Both cases converge at about40 µs and are in good agreement thereafter. Both test

11

cases result in the same asymptotic length scale at theend of the computation; however, in the early stages,the larger bin size produces slightly larger scales, es-pecially around the maximum point (t ≈ 10 µs). Themaximum error in the larger bin size is around 13%.The error gradually decreases after 10 µs and be-comes less than 1% at 40 µs. The temporal trendof the length scales, i.e. when the scales are grow-ing or declining, predicted by both test cases, matchvery well. The only difference is that the larger binoverpredicts the length scales at early stages. Since

Figure 5: Effect of bin size (a) and mesh resolution (b) on thetemporal evolution of δ; Rel = 2500, Weg = 7250, ρ̂ = 0.5,µ̂ = 0.0066, and Λ = 2.0.

the magnitude in the early stage of the length scalegrowth (around the time when the maximum errorsoccur) is not the main goal of our research, it can beconcluded that choosing dL = 2.5 µm as the bin sizeis acceptable for the purposes of this study. Smallerbin size is not recommended in this analysis sincethe smallest length scale that the simulations are ableto capture is as small as the mesh size; thus, choos-ing a bin smaller than the mesh resolution would bemeaningless.

Fig. 5(b) shows the temporal evolution of δ forthree different grid resolutions. The result for thegrid size used in this study (∆ = 2.5 µm) is denotedby dashed line in this plot. Two other grid resolu-tions – one twice as big (∆ = 5 µm) and the otherone half of the current grid size (∆ = 1.25 µm) –are also plotted in dash-dotted and solid lines, re-spectively. The larger grid is unable to predict theasymptotic length scale and has about 5% error nearthe asymptote. The maximum error of this large gridis about 10% and occurs near the maximum scale.The result of the finest grid however, matches per-fectly with the current grid after 15 µs. The maxi-mum difference between the results of the two finergrids is less than 1%. To further delineate the effectsof mesh resolution on the length scales population,

Figure 6: Effect of mesh resolution on the length scale PDF,f (L/λ0), at t = 70 µs; Rel = 2500, Weg = 7250, ρ̂ = 0.5,µ̂ = 0.0066, and Λ = 2.0.

12

Figure 7: Effect of computational domain size on the temporalevolution of ζ; Rel = 2500, Weg = 7250, ρ̂ = 0.5, µ̂ = 0.0066,and Λ = 2.0.

the PDF of length scales, f (L/λ0), obtained by thesethree meshes are compared in Fig. 6 at 70 µs. Foreach of these cases, the bin size is equal to their meshresolution. The PDF results of the coarsest grid arequite overpredicted, with more than 20% error nearthe smallest scale. The difference between the twofiner grids however, is much smaller. The largest er-ror at the smallest L is about 5%. The error in massdistribution would be even much smaller since thesmallest scales have much smaller volumes too. Thedifference in the PDFs also does not impact the re-ported results for δ. Therefore, this comparison veri-fies that the 2.5 µm grid resolution is justified for ourstudy.

The size of the computational domain (especiallyin the transverse direction) has a major influence onthe evaluation of the mean spray width. Fig. 7 com-pares the temporal growth of the ζ for three domainsizes – the original domain used in this study (1X),a domain half the original size (0.5X), and another1.5 times larger (1.5X). The largest domain predictsa very similar result compared to the original case,with very slight difference in ζ after 70 µs. How-ever, the difference in the results of the two largerdomains never exceeds 1.5%, which verifies the ap-propriateness of our chosen domain size. However,a larger domain is definitely required if one wants

to study the process further in time. The predictedspray width of the 0.5X Domain is acceptable until55 µs, but it departs from the correct trend henceforthand becomes noticeably underpredicted. The error ofthe smallest domain reaches about 14% by the end ofthe simulation. This clearly shows that the boundaryconditions have a major impact on the predicted re-sults for the small domain.

3.2. Weber number effects

The temporal variation of δ for low, medium, andhigh Weg and moderate Rel are illustrated in Fig. 8.The density ratio is kept the same (0.5) among allthese cases; hence, Weg is only changed through thesurface tension. The effects of ρ̂ are analyzed in Sec-tion 3.4, and the combined effects of ρ̂ and We are re-vealed there. The symbols on the plot denote the firstinstant at which different liquid structures form dur-ing the atomization process. The definition of eachsymbol is introduced above the plot in Fig. 8 and willbe used hereafter in the proceeding plots. The sym-bols help us compare the rate of formation of eachstructure, say ligament or droplet, at different flowconditions. The relation between the formation ofeach structure and the behavior of the length scale orspray width can also be better understood using thesesymbols. The atomization domain for each processis also denoted on the plot. Note that the zigzag sym-bol denoting the “corrugation” formation does notappear in Fig. 8 because this structure forms only inDomain III (a Domain III result will be discussed inthe next sub-section). All computations are stoppedat 100 µs, which is sufficient for the quantities of in-terest to reach a steady state. It will be shown laterthat further continuation of the computations is notjustified because the interface gets too close to thetop and bottom computational boundaries, so thatthe length scale and sheet width get affected by theboundary conditions.

The lowest Weg falls in the ligament stretching(LoLiD) category in Atomization Domain I, whilethe two higher Weg cases follow the hole-formationmechanism (LoHBrLiD) in Domain II; see Fig. 2.δ decreases with time for all cases, to be expectedfor the cascade of structures presented in Fig. 3 –lobes to holes and bridges, to ligaments and then todroplets. δ becomes smaller as Weg increases, and

13

Figure 8: Effect of Weg on the temporal variation of δ; Rel =2500, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0. The symbols indicatethe first time when different liquid structures form.

its cascade is hastened by increasing Weg. At 40 µs,the average length scale for Weg = 36, 000 is al-most 0.22λ0 = 22 µm, while it increases to 0.3λ0for Weg = 7250, and to 0.5λ0 at the lowest Weg.This shows the clear influence of surface tension onthe length-scale cascade and on the size of the liga-ments and droplets. An increase in surface tensionsuppresses the instabilities and increases the struc-ture size. This trend is consistent with both analyti-cal (Negeed et al., 2011) and numerical (Desjardinsand Pitsch, 2010) results. The droplets and ligamentsformed in Domain II are generally smaller than inDomain I.

The length scale starts from 0.9λ0 in all three casesdue to the initial perturbations, and δ increases forthe first 10 µs, until it reaches a maximum. Weg,and hence surface tension, does not notably affectthe initial length scale growth. This growth involvesthe initial stretching of the waves, which createsflat regions near the braids – they have low curva-tures, hence large length scales. As lobes and lig-aments form later, δ decreases because (i) the ra-dius of curvature of these structures is much smallerthan the initial waves, and (ii) the total interface areaincreases by the formation of lobes, bridges, liga-

ments, and droplets, smearing out the influence ofthe large length scales. Fig. 8 also shows that all thestructures – especially ligaments and droplets – formsooner with increasing Weg. Moreover, ligamentsand droplets form slower in Domain I than in Do-main II; that is, the LoHBrLiD mechanism is moreefficient than the LoLiD process in terms of cascaderate at the same Rel range.

Zandian et al. (2017) showed that two distinctcharacteristic times exist for the formation of holesand the stretching of lobes and ligaments. At agiven Rel, as surface tension increases (i.e. decreas-ing Weg), the characteristic time for hole formationincreases, thereby delaying the hole formation. Thus,for lower Weg, most of the earlier ligaments areformed by direct stretching of the lobes and/or cor-rugations, while the hole formation is inhibited. Onthe other hand, at relatively large Rel (> 3000), as liq-uid viscosity is increased (i.e. decreasing Rel), at thesame Weg, the ligament-stretching time gets larger.In this case, hole formation prevails over the liga-ment stretching mechanism, resulting in more holeson the lobes. As Weg increases, the time at whichthe first hole forms decreases. This indicates that thehole formation time should be inversely proportionalto Weg.

At low Rel (< 3000), the liquid viscosity has anopposite effect on the hole formation and ligamentstretching (Zandian et al., 2017). As shown in Fig. 2,near the left transitional boundary, the time scale ofthe stretching becomes relatively smaller than thehole-formation time scale as Rel is reduced at a con-stant Weg. Therefore, there is a reversal to liga-ment stretching as Rel is decreased at a fixed Weg.Keeping all these effects in mind, the following twonondimensional characteristic times were proposedby Zandian et al. (2017);

Uτhh0∝ 1

Weg

(1 +

kRel

)(16a)

Uτsh0∝ 1

Rel, (16b)

where τh and τs are the dimensional characteristictimes for hole formation and ligament stretching, re-spectively, and k is a dimensionless constant. The

14

Figure 9: PDF of the normalized length scales at different timesfor Weg = 1500 (a), Weg = 7250 (b), and Weg = 36, 000 (c);Rel = 2500, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0. The brokengreen lines indicate the cell size.

results in Fig. 8 are consistent with Eq. (16), whichsuggests the hole formation time scale to be inverselyproportional to Weg. Therefore, the hole formationtime for Weg = 36, 000 should be nearly 5 timessmaller than for Weg = 7250 since Rel is the samefor both cases. From Fig. 8, the first instant whena hole is formed is almost 5 µs for Weg = 36, 000(solid line), and 22 µs for Weg = 7250 (dashed line).Thus, the ratio of τh for these two cases is about4.4, in good agreement with the result obtained fromEq. (16a).

Even though δ gives a good insight into the tem-poral variation of the overall scale of the liquid struc-tures, it does not show the distribution of the scales,for which the length-scale PDFs are needed. Fig. 9compares the PDFs of the length scales of the threecases of Fig. 8 at different times on a log-scale.

All cases start from the same initial distributionindicated by the red line. The length scales in therange 0.5λ0–0.6λ0 have the maximum probability ofapproximately 10%. Later, the most probable lengthscale becomes smaller, while the probability of thedominant scale increases. The transition towardssmaller scales is faster as Weg increases since lower-ing surface tension reduces the resistance of the liq-uid surface against deformations. At 50 µs, the mostprobable length scale becomes 0.1λ0 for Weg = 1500(dashed-line in Fig. 9a). Higher Weg cases reachthe same most probable length scale at 40 µs (dash-dotted line in Fig. 9b) for Weg = 7250, and at lessthan 10 µs (not shown) for Weg = 36, 000.

The PDFs also show an increase in the probabil-ity of smallest scales at higher Weg. At t = 70 µs,for example, the lowest Weg has a probability ofabout 27% for the smallest computed length scale of2.5 µm; see where the blue curve intersects the ver-tical axis in Fig. 9(a). That probability at the sametime increases to 59% as Weg increases to 7250. Ateven higher Weg = 36, 000, the probability of 2.5 µmor lower is slightly more than 64% at 70 µs. Since thesmaller length scale also implies smaller volume, theconclusion is that the number density of the smalldroplets also increases with Weg. The green bro-ken lines in Fig. 9 indicate the normalized cell size.In all cases and at all times, more than 98% of thecomputed length scales lie to the right of this lineand are larger than the mesh size, which justifies the

15

Figure 10: Liquid-jet surface at 70 µs for Weg = 7250; Rel = 2500, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

Figure 11: Schematic of the radii of curvatures on a typicalligament (a), and its resulting droplet (b).

sufficient resolution of the current grid. The effectsof grid resolution on the liquid structures scale wasdemonstrated in detail by Zandian et al. (2018) and itwas shown that the grid resolution used for the cur-rent analysis is fine enough to capture the smallestradii of curvature.

Fig. 10 shows the liquid surface for the moderateWeg = 7250 case at 70µs. As shown in the magnifiedimage, at this time the liquid surface is mainly com-prised of ligaments (either broken or still attached)and droplets, while little or almost no lobes with verylarge scales are present. Therefore, the evolution ofthe surface is summarized mainly in stretching andbreakup of the ligaments henceforth.

As shown in Fig. 11, there are two radii of curva-ture in a typical perturbed ligament. R1 is the smaller

azimuthal radius which is initially equal to the radiusof the cylindrical ligament. R2, the radius of curva-ture of the streamwise arc of the ligament after it un-dergoes Rayleigh-Plateau (RP) instability, is muchlarger than R1. Theoretical analyses of Rayleigh(1879) show that for a cylindrical liquid segment ofradius R1, unstable components are only those wherethe product of the wave number with the initial radiusis less than unity; i.e. kR1 < 1. Thus, the minimumunstable RP wavelength for a ligament of radius R1is λRP = 2πR1. The volume of a cylinder of radiusR1 and length λRP is V = 2π2R31. If this segment ofthe cylinder (ligament) breaks into a droplet, a sim-ple mass balance shows that the resulting droplet ra-dius (Rd in Fig. 11b) would be Rd ≈ 1.67R1. R2 isnever smaller than R1; R2 ≈ R1 based on an approx-imate sinusoidal surface shape at the instant of lig-ament breakup, and R2 = ∞ in case of unperturbedligament; i.e. R1 ≤ R2 < ∞. Considering these limitsand using the relation between R1 and Rd, the extentsof ligament length scale Ll = 2/(1/R1 + 1/R2) as afunction of Rd follow

if R2 = ∞ → Ll =2

1/R1= 2R1 ≈ 1.2Rd, (17a)

if R2 = R1 → Ll =2

1/R1 + 1/R1= R1 ≈ 0.6Rd (17b)

thus, 0.6Rd < Ll < 1.2Rd. The droplet length scale16

Figure 12: Temporal variation of the standard deviation of thedimensionless length-scale PDFs of Fig. 9 for Weg = 1500,7250, and 36, 000; Rel = 2500, ρ̂ = 0.5, µ̂ = 0.0066, andΛ = 2.0.

is Ld = Rd. Therefore, the time-averaged ligamentlength scale is approximately equal to the dropletlength scale during the short period of RP instabil-ity growth and ligament breakup. This simple anal-ysis explains why the average length scale becomesalmost constant after 70µs (see Fig. 8) while the lig-ament breakup is still occurring and the number ofdroplets is increasing. Since the ligament formationis delayed (about 30 µs) at lower Weg, the asymp-totic length scale is also expected to occur later forWeg = 1500. This is consistent with Fig. 8, wherethe length scale asymptotes about 27 µs later forWeg = 1500 compared to Weg = 7250.

The standard deviation of the PDFs of Fig. 9, il-lustrated in Fig. 12, provides a more accurate quan-titative measure of the length-scale distribution. Thelength scales range from a few microns, i.e. a smallfraction of the initial wavelength, to several hun-dred microns, e.g. four times the initial wavelength.Therefore, the standard deviation of δ (σδ) is verylarge, even in the beginning. δ is about 0.9λ0 =90 µm at the start of the computations (see Fig. 8),but σδ is about 0.85λ0 = 85 µm at this time.

At the early stage, σδ increases as larger scalesbecome more probable following the flattening and

stretching of the waves. Later, the flow field getsfilled with more small ligaments and droplets andmore curved surfaces, which reduce both the meanand the standard deviation. However, even at the endof the process, σδ is still around 0.2–0.4λ0 = 20–40 µm; so, a wide range of length scales is stillpresent in the flow. σδ decreases with increasingWeg, as the smaller capillary force allows the largerscales to deform easily and cascade more quicklyinto smaller scales with higher curvatures; this re-duces the deviation of the scales. σδ becomes almostconstant when δ asymptotes to its ultimate value.

The length-scale PDFs show that: (i) the asymp-totic length scale (ligament and droplet size) de-creases with increasing Weg; (ii) the number of smalldroplets increases with increasing Weg; and (iii) thecascade of length scales occurs faster at higher Weg.The last item is also implied by the temporal evolu-tion of δ. The first two items are consistent with theliterature, but the third item is a new finding.

Temporal growth of the non-dimensional liquidsurface area (S ∗) for the three Weg cases is plotted inFig. 13. The surface area (S ) is non-dimensionalizedby the initial liquid surface area; i.e. S ∗ = S/S 0.Therefore, S ∗ grows monotonically in time from 1at t = 0; S 0 ≈ 0.8 mm2.

As expected, the surface area growth rate is higherfor higher Weg, where the surface deformations oc-cur and grow faster and ligaments and droplets formearlier. S ∗ grows very gradually in the first 30 µs, butexperiences a sudden increase in the growth rate af-ter the formation of first ligaments and droplets. Thisabrupt growth in surface area occurs sooner and at ahigher rate for higher Weg following the earlier for-mation of ligaments in those cases. The S ∗ growthrate decreases slightly towards the end of the compu-tations, while the area still keeps growing. The causeof this gradual decrease in growth rate is speculatedto be mainly due to the breakup of ligaments intodroplets. Surface tension minimizes the surface areaafter ligament breakup. The surface deformation atlater times mainly consists of formation of ligamentsand their breakup into droplets. Following the sim-plified ligament and droplet radii model shown inFig. 11, a simple calculation reveals that the surfacearea of the resulting droplet is 10% smaller than thesurface area of its mother ligament; i.e. S d = 0.9S l.

17

Therefore, the ligament breakup decreases the S ∗

growth rate associated with mere stretching of theligaments. Even though the mean length scale hasreached an asymptote at the final stage (t > 70 µs),the increase in S ∗ indicates that the surface dynamicsare still in progress and ligament and droplet forma-tion still continues. The rate of growth of S ∗ becomesalmost constant in the asymptotic phase. At 100 µs,the surface area has grown more than 11 times forWeg = 36, 000, while for Weg = 1500, the surfacehas less than 6 times its initial area.

As mentioned earlier, defining the jet width (thick-ness) as the distance to the farthest liquid point fromthe centerplane might render results which are proneto misinterpretation. This definition does not takeinto account the interface location distribution andonly considers the farthest liquid location. For thispurpose, the PDFs of the transverse interface loca-tion are used to describe the expansion of the liquidsheet in a more meaningful form.

The effect of Weg on the temporal evolution of theaverage spray width (ζ) is shown in Fig. 14 alongwith the liquid-jet interface picture at 70 µs of eachprocess. The expansion rate of the liquid jet can bedistinguished better with the new definition of the jet

Figure 13: Temporal growth of the non-dimensional liquid sur-face area (S ∗) for Weg = 1500, 7250, and 36, 000; Rel = 2500,ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

width (compare this plot with figure 40 of Zandianet al., 2016). The jet images at 70 µs show that thetwo higher Weg jets would have a very similar widthif the widths were presented as the farthest liquidpoints from the centerplane. However, our methodclearly shows the difference between these two cases,and indicates that a larger portion of the liquid sur-face is in fact located at a farther transverse distancefrom the jet center for the higher Weg.

The spray expands faster at higher Weg and resultsin a wider spray at the end – a result which is inagreement with numerical results of Desjardins andPitsch (2010). ζ remains close to 1.0 for the first 50microseconds for Weg = 1500 since the high surfacetension suppresses instability waves, lobe stretch-ing, and ligament breakup. The jet expands muchsooner at higher Weg; for example, at about 20 µs forWeg = 7250, and at 7 µs for Weg = 36, 000. Thestructures stretch much more quickly at higher Wegand are less suppressed by the surface tension forces;thus, they can expand more freely and are carriedaround more easily by the gas flow, after breakup.The expansion of the jet at the lowest Weg (dash-dotted line in Fig. 14) coincides with the formation ofthe first ligament (at 50 µs). Therefore, the lobes aremuch less amplified at such a low Weg, and the lig-ament formation and stretching are primarily in thenormal direction in Domain I. Generally, the sprayangle is larger in Domain II than in Domain I at acomparable Rel range.

The spikes and oscillations in ζ are caused by thedetachment of a liquid structure, e.g. bridges, liga-ments or droplets, from the jet core. The first decayin ζ coincides with the formation of the first droplets;i.e. the black circles in Fig. 14. The spray-widthPDFs, given in Fig. 15, support this view.

All three cases in Fig. 15 start from a bell-shapeddistribution around h = h0, denoted by the red solidlines. The two peaks on the two sides of h/h0 = 1are due to the initial perturbations amplitude of 5 µmimposed on the surface of the sheet. Since there aremore computational cells near the peak and troughof the perturbations compared to the neutral plane,i.e. h/h0 = 1, the probability of those sizes areslightly higher. The probability of the initial thick-ness value increases in all cases during the first 20 µssince the wave amplitude decays as the waves get

18

Figure 14: Effect of Weg on the temporal variation of ζ; Rel = 2500, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0. The liquid jet surface at70 µs is shown for each process.

stretched in the flow direction, in the initial stage.

Later, when the waves start to grow due to the KHinstability and roll-up of the lobes over the primaryKH vortices, the jet width increases and the distri-bution becomes wider and skews towards larger val-ues on the right. Meanwhile, smaller lengths are alsoobserved. With time, the peak of the initial distri-bution curve decreases while the distribution broad-ens. This decline occurs faster at higher Weg, sincethe spray grows faster for lower surface tension. Att = 50 µs, only 8% of the surface, i.e. computa-tional cells at the interface, lie near the initial thick-ness for Weg = 36, 000 (dashed-line in Fig. 15c),and the spray has grown upto three times the initialsheet thickness; i.e. h/h0 ≈ 3. At the same time,the farthest transverse distance reached by the liquidis about 2.8h0 from the centerplane for Weg = 7250;see where the dashed-line in Fig. 15(b) meets the hor-izontal axis. At still lower Weg, the maximum spraywidth is just slightly more than 2h0 at 50 µs, and still

more than 16% of the surface lies around the initialsheet thickness; see Fig. 15(a).

Fig. 15(b) shows a missing section (having zeroprobability) around 1.4 < h/h0 < 1.8 at t = 30 µs.The same missing section moves to the right (to1.5 < h/h0 < 2.0) at t = 40 µs, and finally vanishes at50 µs. These sections – marked as the “first breakup”– coincide with the places where the sudden declinein ζ was seen in Fig. 14 for Weg = 7250, explain-ing the oscillations in the average spray width. Themissing section appears since some part of the liquidjet (ligament, bridge or droplet) detaches from the jetcore into the gas flow, leaving behind a vertical gapempty of any liquid surface at the breakup location,as shown in the sequential liquid surface images inFig. 16.

The lobes form and stretch until about 25 µs, re-sulting in an increase in ζ. At 30 µs (Fig. 16b), thebridge breaks from the lobe and creates a gap nearthe detachment location, where there are no liquid el-

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Figure 15: Nondimensional spray-width PDF at different timesfor Weg = 1500 (a), Weg = 7250 (b), and Weg = 36, 000 (c);Rel = 2500, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

ements. Therefore, ζ suddenly drops at that instant,even though the distance of the farthest liquid ele-ment from the centerplane is still growing; i.e. theintersection of the PDF curves with the horizontalaxis in Fig. 15(b) moves to the right. While the de-tached liquid blob moves away from the interface,the jet surface stretches outward again due to KHinstability; the missing zone moves outward follow-ing this motion (Fig. 16c). ζ grows again when thenew lobes and ligaments stretch enough to compen-sate for the broken (missing) section. At this time,the lobes and ligaments fill in the missing gap whilethe broken liquid structures advect farther from theinterface, as shown in Fig. 16(d) at 50 µs. Fig. 16also shows that the instabilities start from a symmet-ric distribution but gradually move towards an anti-symmetric mode (Fig. 16d). The transition towardsantisymmetry is seen in all cases studied here and isexplained in detail via vortex dynamics analysis byZandian et al. (2018). It is shown that transition to-wards antisymmetry is faster for thin liquid sheetsdue to the higher induction of the KH vortices on theopposite sides of the liquid surface. In practical at-omization conditions, the antisymmetric mode has ahigher growth rate and thus eventually dominates thesymmetric mode.

The PDFs and the average spray-width plots in-dicate the first instance of ligament/bridge breakup,when ζ starts to decline. Since the lower Weg has lessstretching and fewer ligament detachments at earlytime – due to the high surface tension – its spikeis less intense and also appears much later (about55 µs). The higher Weg, however, breaks muchsooner, at about 27 µs.

There are also some later oscillations near themaximum spray width of the PDF plots (see Fig. 15),for all three cases. The reason for these spikes isexplained using the liquid isosurface at 70 µs forWeg = 1500, illustrated in Fig. 17. The planes2.5h0 and 3.0h0 away from the centerline are markedwith the red lines. Undulations in the PDF curveoccur in this range; see the blue line in Fig. 15(a)marked as “containing droplets”. This range ismostly empty of liquids, i.e. filled with gas, exceptfor rare cells which contain the occasional droplets ordetached ligaments. In the spray-width PDF plot, theempty spaces have zero or almost negligible proba-

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Figure 16: Side view of the liquid sheet surface at t = 20 µs (a), 30 µs (b), 40 µs (c), and 50 µs (d); Rel = 2500, Weg = 7250,ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0. Gas flows from left to right.

Figure 17: Side-view of the liquid surface at t = 70 µs; Weg =1500, Rel = 2500, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0. Gas flowsfrom left to right.

bility, while other transverse heights containing liq-uid droplets and broken ligaments have greater prob-ability. Thus, waviness is seen in the spray-widthPDF at later times and at greater distances from thecenterplane. Those spikes represent the droplets thatare thrown outward from the jet core, as the sprayspreads.

The reason for having non-zero probability forzero width in the PDF plots of Fig. 15 can also beassessed in Fig. 17. Since the antisymmetric modeis dominant in the considered range of Rel and Weg,the trough of the surface wave reaches the center-plane (indicated by the broken black line in Fig. 17)and even crosses it. Thus, after some time, non-zeroprobabilities occur for the surfaces that intersect thecenterplane.

Fig. 18 shows the standard deviation of the spray-width (σζ) PDFs of Fig. 15. Initially, σζ is slightlybelow 0.2h0 = 10 µm, which is exactly equal to thepeak-to-peak amplitude of the initial perturbations.At early times, as the waves stretch in the streamwisedirection, their amplitude decreases; hence, a largerportion of the interface gets closer to the initial sheetsurface. This reduces σζ . This reduction is greaterfor lower Weg because of the stabilizing role of sur-face tension. Later, as the spray expands, the distance

Figure 18: Temporal variation of the standard deviation of thedimensionless spray-width PDFs of Fig. 15 for Weg = 1500,7250, and 36, 000; Rel = 2500, ρ̂ = 0.5, µ̂ = 0.0066, andΛ = 2.0.

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between the outermost and the innermost liquid sur-face grows, and the spray-width PDF gets wider; seeFig. 15. σζ increases with increasing Weg. The rateof increase of σζ decreases at about 70 µs for thehighest Weg and at 90 µs for the lowest one. The rea-son for this change in pace is that, beyond this point,some kind of saturation occurs in the computationalbox by the broken liquid blobs that get too close tothe top and bottom boundaries; notice that the liq-uid particles cannot leave the box from the normalboundaries. This could clearly influence the dynam-ics of atomization. Thus, the computations are notcontinued beyond 100 µs. A larger computationalbox is required if one would want to continue theanalysis in time, but this is not plausible for this studydue to its computational cost.

3.3. Reynolds number effects

Practically, Rel should have a major effect on boththe final droplet size and the spray angle as well asthe cascade rate, since both the inertia and viscouseffects are involved in these quantities. These effectsare studied quantitatively here. Three different Relvalues are compared in this study; Rel = 1000, 2500,and 5000 – each representing one of the domains inthe Weg vs. Rel plot with a particular breakup char-acteristic, as defined in Figs. 2 and 3. Rel = 5000(in Domain III) and Rel = 1000 (in Domain I)have the stretching characteristics during the primarybreakup, with and without the corrugation forma-tion, respectively. The Rel = 2500 case follows thehole/bridge formation mechanism in Domain II.

Figs. 19(a) and 19(b) respectively demonstrate theeffects of Rel on the average length scale (δ) and theaverage cascade rate (dδ/dt) over time. The aver-age cascade rate is obtained by calculating the aver-age slope of the curvatures in Fig. 19(a) in 5 µs and10 µs time spans. As expected, the liquid surface isstretched more in the streamwise direction with in-creasing Rel, creating flatter surfaces early on. Thus,as Rel increases, relatively larger length scales oc-cur at early times. For higher Rel, the lobes stretchfor a longer time before breakup, yielding maximumlength scale at a later time. The breakup process andcascade of length scales occur later for higher Rel(see Fig. 19b).

Figure 19: Effect of Rel on the temporal variation of δ (a), andthe average cascade rate (b); Weg = 7250, ρ̂ = 0.5, µ̂ = 0.0066,and Λ = 2.0.

After the early injection period, the rate of cascadeof the large liquid structures, e.g. lobes and bridges,into smaller structures, e.g. corrugations, ligamentsand droplets, is greater at higher Rel, as observedfrom the mean slopes in Fig. 19(a) in the downfallportion of the plot. As shown in Fig. 19(b), only onemaximum cascade rate is observed in Domain I atabout 18 µs, with an average rate of −4 µm/µs, whichcorresponds to the stretching of lobes into ligaments.

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In Domain II (dashed line), two major rates are ob-served – a larger rate at ≈ 20 µs, which correspondsto the stretching of lobes, and another lower rate at≈ 35 µs, corresponding to the formation of holes andligaments. In Domain III (solid line), there are twolocal minimums. There is a slower rate (−1.3 µm/µs)which spans from 10 µs to about 25 µs, and a largerrate of ≈ −3.8 µm/µs, which occurs between 25 µsand 50 µs. The first rate corresponds to the stretch-ing of the lobe, and the second (larger) rate corre-sponds to the formation of corrugations on the lobeedges. Since corrugations result in much smallerscales compared to the lobes, this transition resultsin a sudden growth in the cascade rate. The aver-age cascade rate goes to zero with the formation ofdroplets, as was discussed before.

By decreasing the liquid viscosity, i.e. increas-ing Rel, with the other properties held constant, thebreakup occurs faster but later. This can be attributedto the stabilizing effects of viscosity, which dampsthe small scale instabilities at low Rel. Fig. 19(a)also shows that Rel affects the ultimate δ value. Eventhough the effect of Rel on the final length scale isnot as significant as Weg, the magnified subplot ofFig. 19(a) shows that the asymptotic scale decreasesfrom 0.11λ0 to 0.09λ0 as Rel increases from 1000to 5000. Even though the rate of cascade of lengthscales is larger at higher Rel, the asymptotic lengthscale is achieved later for higher Rel since the largestlength scales are also larger for higher Rel. δ forthe highest Rel case just becomes smaller than thetwo lower Rel cases at about 75 µs. Negeed et al.(2011) and Desjardins and Pitsch (2010) also quali-tatively showed that the final droplet size decreaseswith increasing Rel; however, they did not quantifythe droplet size nor its cascade rate.

Even though the formation of lobes is not affectedmuch by Rel, the ligaments and droplets form no-tably later as Rel increases. This counter-intuitivefact is related to the process of ligament and dropletformation. At a constant Weg, the ligament forma-tion in the LoCLiD process (Domain III) is slowerthan in the LoHBrLiD process (Domain II), andboth are slower than that in the LoLiD process (Do-main I). All these cascade processes are explainedvia vortex dynamics by Zandian et al. (2018). Theyshow that the formation of corrugations at higher Rel

Figure 20: PDF of the normalized length scales at t = 5 µs(solid lines) and 10 µs (dashed-lines) for Rel = 1000 (red line),Rel = 2500 (black line), and Rel = 5000 (blue line); Weg =7250, ρ̂ = 0.5, µ̂ = 0.0066, and Λ = 2.0.

takes longer and requires downstream advection ofthe split KH vortex by a distance of one wavelength(≈ 100 µm), until the hairpin vortices get undulatedand induce the corrugations. The LoHBrLiD pro-cess, on the other hand, requires only stretching andoverlapping of the hairpins over the KH vortices,which occur faster. The LoLiD mechanism involvescross-flow advection of the KH vortices, which oc-curs slightly more quickly. However, the dropletsformed in the LoCLiD process are smaller than in theother two processes, and the LoLiD process results inthe thickest ligaments and the largest droplets.

To better understand the difference in δ for thethree Rel cases at early injection stage, the lengthscale PDFs for these cases are plotted at 5 µs and10 µs in Fig. 20. Both times are within the initialinjection period when the maximum δ occurs; seeFig. 19(a). The Rel = 1000, 2500, and 5000 casesare shown by red, black, and blue lines in Fig. 20,respectively. The PDFs are denoted by solid lines at5 µs and by dashed lines at 10 µs.

The highest Rel (blue line) has the highest prob-ability of larger length scales (L/λ0 > 2.0) at both5 µs and 10 µs, confirming the earlier claim thathigher Rel causes more stretched surfaces and largerscales (less curved surfaces) early on. Even though

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Figure 21: Liquid surface at t = 5 µs from a side view (a), andtop view (b); Rel = 1000, Weg = 7250, ρ̂ = 0.5, µ̂ = 0.0066,and Λ = 2.0.

the maximum distribution of the length scales movesto smaller scales from 5 µs to 10 µs for Rel = 5000,δ still grows, as shown in Fig. 19(a). In particu-lar, a large population of the cells contains surfaceswith very large length scales. As Rel decreases, tran-sition towards smaller scales occurs faster at theseearly times. Therefore, δ decreases sooner for lowerRel. Clearly, there are two factors in determiningthe mean length scale: the sizes of the smallest andlargest length scales, and the population of thosescales. At an early stage, i.e. t < 10 µs, the small-est as well as the largest length scales are the samefor all cases; however, it is the population of thosesmall scales compared to the large ones that con-trols δ. Since there are more large scales at higherRel, more time is needed for those structures to cas-cade to smaller scales; thereby, δ keeps growing fora longer period at higher Rel (Fig. 19b). After thisinitial period, however, the cascade is faster for thehigher Rel because of the lower viscous resistanceagainst surface deformation; therefore, the smallestbin gets populated at a higher rate.

There are two distinct spikes in the length-scalePDFs of Fig. 20 at 5 µs – one at a low scale of approx-imately L = 0.08λ0 = 8 µm, and another with lowerprobability but larger scale of L = 0.8λ0 = 80 µm forRel = 1000. Similar two spikes are seen for other Relcases at an early time. The cause of the two spikesis shown in Fig. 21. The smaller scale represents theradius of curvature of the streamwise KH wave crest,which has the largest probability because this lengthscale exists for many cells along the spanwise direc-tion on the front edge of the waves. This is shown inthe side-view of Fig. 21(a). The other spike with the

larger length scale but smaller probability applies tothe radius of curvature of the spanwise waves. Thesepoints are indicated on the top-view of Fig. 21(b).This length scale occurs for all the cells near the tipof the protruded liquid lobe.

The conclusion from the above results is that thereare two stages in the liquid-sheet distortion: (i) theinitial stage of distortion when the lengths grow, and(ii) the final asymptote in time. Viscosity but not sur-face tension is dominant in the first stage, which isinertially driven, while surface tension more than vis-cosity affects the mean scale at the latter stage.

Figs. 22(a) and 22(b) show the effects of liquid vis-cosity on ζ and its growth rate, respectively. Sinceliquid inertia dominates the viscous effects at higherRel, the spray is oriented more in the streamwise di-rection, and ζ and accordingly the spray angle aresmaller at higher Rel. This is consistent with bothnumerical simulations (Jarrahbashi et al., 2016) andexperimental results (Carvalho et al., 2002; Mansourand Chigier, 1990). Mansour and Chigier (1990) andCarvalho et al. (2002) showed that the spray angle isreduced by increasing the liquid velocity (or massflowrate); i.e. increasing Rel. However, they onlyreported the final spray angle, but did not addressits temporal growth. We show here that the growthrate of the spray width (dζ/dt) is lower at higher Rel(Fig. 22b). The expansion of the spray starts muchsooner in Domain I than in Domains II and III, andthe asymptotic spray growth rate is lower for higherRel.

Even though both the highest and the lowest Relcases produce similar lobe stretching mechanisms(with and without corrugation formation, respec-tively), there is a significant difference in their sprayexpansion. At high Rel, the corrugations on the lobesstretch into streamwise ligaments, resulting in thin-ner and hence shorter ligaments. The time lapsebetween ligament formation and ligament breakupis much shorter in Domain III since the ligamentsare generally shorter and need less time to thin andbreak. At low Rel, on the other hand, the lobes di-rectly stretch into ligaments, and the stretching isoriented in the transverse (normal) direction as theviscous forces resist streamwise stretching caused bythe inertia. It takes more time for the ligaments tobreak up into droplets. The ligaments are generally

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Figure 22: Effect of Rel on the temporal variation of ζ (a), andthe average spread rate (b); Weg = 7250, ρ̂ = 0.5, µ̂ = 0.0066,and Λ = 2.0.

thicker and longer, as also shown for low Rel by Mar-mottant and Villermaux (2004). This difference orig-inates from the difference in the vortex structures ofthese two regimes, and shows that these two mech-anisms have different causes from vortex dynam-ics perspective. Zandian et al. (2018) explain thatstreamwise vortex stretching is stronger at higherRel, resulting in streamwise oriented ligaments.

The difference in the jet spread rate at low andhigh Rel manifests in the angle of ligaments that

grow out of the liquid surface. This is clearly illus-trated in Fig. 23, where the liquid surface at a lowRel (Fig. 23a) and high Rel (Fig. 23b) are shown at70 µs. The overall shape of ligaments is denotedby the red lines in this figure, and the average an-gle of the ligament tips (measured from the stream-wise direction) are also indicated. At high Rel, thelower viscous forces on the ligaments are not ableto overcome the relatively high gas momentum, andthe ligament angle decreases as it penetrates furtherinto the gas. Thus, the transverse velocity is muchsmaller than the streamwise velocity and the angleis small (about 20◦). On the other hand, the higherliquid viscosity at lower Rel balances the gas inertiaand opposes the streamwise stretching of the liga-ments. Consequently, the ligament angle increasesas it penetrates further into the gas, resulting in a50◦ angle at the ligament tip. The ligament shapes

Figure 23: Liquid surface at t = 70 µs from a side view forRel = 1000 (a), and Rel = 5000 (b); Weg = 7250, ρ̂ = 0.5,µ̂ = 0.0066, and Λ = 2.0. The red lines indicate the qualitativeform and angle of the ligaments.

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can be attributed to the velocity profile – the bound-ary layer becomes thinner and the velocity gradientin the z-direction becomes steeper as Rel increases.These results are consistent with the temporal vari-ation of ζ shown in Fig. 22. Our results correctlypredict that the asymptotic jet spread rate is higher atlower Rel. The non-dimensional spread rate is 2.1,1.6, and 1.26 1/s for Rel = 1000, 2500 and 5000,respectively (Fig. 22b). This was not the case forjet spread rate versus Weg. This confirms that thereason for faster growth of jet width at higher Wegwas mainly due to the fact that the droplets break upfaster and can be carried away by the vortices nearthe interface, which is completely different from thecause-and-effect of Rel, explained here.

3.4. Density-ratio effects

The effect of density ratio (ρ̂) on δ and its cascaderate are shown in Fig. 24. Three density ratios arestudied in a range of 0.05 (low gas pressure) to 0.9(high pressure gas). The liquid Weber number, andthereby the surface tension coefficient, is kept thesame for all three cases; thus, Weg is also differentfor these three cases through gas density. The low-est ρ̂ falls in Domain I, where lobes stretch directlyinto ligaments, and the other two higher density ra-tios belong to Domain II, and involve hole and bridgeformation in their breakup.ρ̂ has only a slight effect on the final length scale.

All cases asymptotically reach a nearly similar meanlength scale of about 0.1λ0, with the final length scalebeing slightly smaller (≈ 0.09λ0) for the lowest ρ̂(dash-dotted line in Fig. 24a). Besides, ρ̂ clearlyaffects the rate at which the ultimate δ is achieved.The rate of length-scale cascade is higher at lowergas densities, with the highest rate of approximately−5.5 µm/µs occurring between 10 µs and 20 µs forρ̂ = 0.05. This rate slowly decreases as the lobestretches into ligaments. Between 30 µs and 45 µsa slower cascade rate (≈ −1 µm/µs) is dominant,which corresponds to the formation of first ligamentsby the end of this period. As ρ̂ increases, the cascadeof length scales becomes slower for the early stages(t < 30 µs). With transition into Domain II, a fewdistinct cascade rates are seen in the process, eachcorresponding to the formation of a new liquid struc-ture: e.g. holes, bridges, and ligaments. The cascade

Figure 24: Effect of density ratio on the temporal variation ofδ (a), and the average cascade rate (b); Rel = 2500, Wel =14, 500, µ̂ = 0.0066, and Λ = 2.0.

rate corresponding to droplet formation for t > 35 µsis larger in Domain II. The reason for this differenceis in the mechanisms of droplet formation in thesetwo domains. In Domain I, each lobe results in a sin-gle qualitatively large droplet, whereas in Domain II,several smaller droplets are formed from the breakupof ligaments and bridges. Therefore, the average cas-cade rate is larger for Domain II in those later periods(t = 35–45 µs).

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The asymptotic length scale is achieved at 64 µsfor ρ̂ = 0.05, but at about 68 µs for ρ̂ = 0.5, andat 70 µs for ρ̂ = 0.9. As denoted by the symbols inFig. 24(a), the rate of ligament and droplet formationis notably affected by ρ̂. As ρ̂ grows, so that the Do-main changes from I to II, the ligament and dropletformation rates undergo a large jump. As ρ̂ keepsincreasing in the same Domain (II), the ligamentsand droplets form sooner, but the difference is notas notable as the jump during the domain transition.Jarrahbashi et al. (2016) also showed that the dropsform earlier at higher ρ̂; however, they did not ad-dress the relation between this trend and the changein the breakup process. As gas density increases,the higher gas inertia intensifies the hole formation,therefore expediting the formation of ligaments anddroplets.ρ̂ does not alter the initial length-scale growth rate

significantly – the maximum scale occurs at nearlythe same time and the maximum growth rates arealso very close (see Fig. 24b) – which proves it tobe correlated with the liquid inertia and not the gasinertia. The asymptotic stage is also driven by theliquid inertia and is almost independent of the gasdensity in the ρ̂ range considered here. As shown