Velocity proﬁle relaxation is commonly believed to be acause of jet breakup. This claim is critically reevaluated in this work. Contrary to common belief, laminar liquid jets with parabolic velocity proﬁles are actually more stable than laminar jets with ﬂatter velocity proﬁles. This is shown using prior theory and experiments. For turbulent jets, theinﬂuenceofthevelocityproﬁleismorediﬃculttodetermine. Previousexperimentalistsclaimedtoshowthatthe velocityproﬁlehasaneﬀectbyvaryingthenozzlelength. Theclaimisthattheboundarylayerthicknessgrowswith nozzle length, and that the larger the boundary layer, the less stable the jet. In this work, nozzle length is shown to be a poor proxy for velocity proﬁle eﬀects because the turbulence intensity also increases as the nozzle length increases. Studies with this confounding were ignored in this work. Thinner boundary layers have greater shear, yet experiments have shown that if the boundary layer were made thinner (all else constant), the jet often is more stable. This is termed the“shearparadox”. Apotentialresolutiontotheshearparadoxisdevelopedbyconsideringthattheareawithshear also decreases as the boundary layer thickness is decreased, and by non-dimensionalizing the turbulent production rate by the dissipation. This theory shows an interaction between the integral scale and velocity proﬁle relaxation which has not been previously discussed. The theoretical prediction that a smaller integral scale leads to more stable jets (due to increased turbulent dissipation) is shown to be somewhat consistent with the limited experimental and DNS data available. ∗https://trettelresearch.com/contact.html Introduction Velocity proﬁle relaxation has been proposed as a causeofjetbreakupsincetheearlyworkofSchweitzer[1] in1937,andindependentlyafewyearslaterbyLittaye[2] in1942. Thepresentexplanationisthatnon-uniformve- locityproﬁleshaveexcessenergy,whichcanacceleratethe breakup process [3–7], though the details of the physical mechanism remain vague. This eﬀect couldalternatively be called a boundary layer or shear instability [8, 9]. McCarthy and Molloy [5, p. 7] proposed the velocity proﬁle kinetic energy coeﬃcient as a measure of the tendency for velocity proﬁle relaxation to cause breakup: ¯ A0U3 0dA U3 0A0: (1) Above, the 0subscript refers to a quantity located at thenozzleexitplane, U0isthe(local)meanaxialvelocity at the nozzle exit, A0is the nozzle exit area ( A0=d2 04, where d0is the nozzle diameter), and U0is the mean plane-averaged axial velocity at the nozzle exit. The larger the kinetic energy coeﬃcient is, the strongerbreakupduetovelocityproﬁlerelaxationisac- cordingtoMcCarthyandMolloy. Thekineticenergycoef- ﬁcient issmallest( =1)forauniform/ﬂatvelocitypro- ﬁle, and largest for a parabolic proﬁle ( =2). For a fully developed turbulent pipe velocity proﬁle 1:06[10, p.219] 1,which,ifthecriteriaistrue,suggeststhatfully developed turbulent pipe ﬂows are not likely to see major velocity proﬁle relaxation eﬀects. (Note that this does mean that turbulent ﬂows in general won’t see velocity proﬁle relaxation eﬀects.) McCarthy and Molloy justiﬁed this with ﬁgure 1, which shows jets which are identical asidefromchangingthenozzlelength,whichchangesthe velocityproﬁlefromrelativelyﬂattomoreparabolicasthe ﬂowdevelops. Theﬂowisinitiallylaminarineachcase shown, though it likely would be turbulent for the fully developed case as the Reynolds number is 4750. In this work, the Reynolds number is denoted Re`0U0d0`, where d0isthenozzlediameterand `istheliquidviscos- ity. Similarly, the Weber number is We`0`U2 0d0, where`istheliquiddensityand isthesurfacetension. To avoid velocity proﬁle relaxation and consequently enhancethestability ofturbulentliquidjets, specialnoz- zleshavebeendesignedtoproduceﬂatvelocityproﬁles[8, 11],andtheseeﬀortsappeartoproducemorestablejets. Additionally, the largest breakup lengths I am aware of, hxbi=O¹1000º,wereobtainedwithshortnozzles(nozzle oriﬁce length L0=d0) [12, ﬁg. 4] which produce rela- tivelyﬂatvelocityproﬁles[13,ﬁg.4,p.283]. However, 1McCarthy and Molloy [5, p. 7L] incorrectly suggest that is in the range of 1:1to1:2, but direct computation indicates lower. This suggests that barely changes at all with nozzle length in the turbulent case.there are ﬂaws in this narrative. McCarthyandMolloy’sphotographsarequalitative atbest. Objectivemeasurementsofthebreakuplengthare necessary tovalidatethe theory. Theapparent worst-case- scenario, a parabolic velocity proﬁle, appears to be more stable than previously thought. Linear stability analysis for laminar jets shows that parabolic velocity proﬁles are actually more stable than uniform [14–16]. Indeed, theexperimentaldatashowsthatforlaminarjets,asthe oriﬁce(development) lengthincreases(again, changing the velocity proﬁle from roughly ﬂat to parabolic), the breakuplengthincreasesby30% 2,qualitativelyconsistent with linear stability theory. See ﬁgure 2. For this plot I compiled data from Arai et al. [12] for breakup length as a function of nozzle length, and interpolated the data to get lines of constant Weber number. Each line is also a constant, but diﬀerent, Reynolds number 3. The trend is not asexpected forthe two laminarcases —the breakup lengthisroughlyconstantasthenozzleaspectratio L0d0 increases, and slightly increasing for the lowest Reynolds numbercase. Here, L0isthelengthofthenozzleoriﬁce. A longer nozzle aspect ratio for the laminar case would indicate that the velocity proﬁleis closer to parabolic. If velocityproﬁlerelaxationweresocatastrophicforlaminar jets as McCarthy and Molloy [5] claim, then why is the this not observed in this experiment? Clues come from the nozzle aspect ratios and hy- drodynamic regime in the example used by McCarthy and Molloy [5, ﬁg. 5]. The example of McCarthy and Molloy is for an initially laminar but transitional jet at high Weber number ( We`01:5104), which is close to the We`0=1104case in ﬁgure 2 4. For this case, as the nozzle aspect ratio increased, the breakup length decreasesthrough L0d010,whichisthelongestaspect ratio considered by McCarthy and Molloy. The trend McCarthy and Molloy observed applies only for high Weber number liquid jets with L0d0.10. The breakup length trend at higher nozzle aspect ratios is slightly in- creasing, contrary to what one might expect from reading McCarthy and Molloy. Indeed, Debler and Yu [15] exam- ine only L0d0&10and come to the opposite conclusion as McCarthy and Molloy at lower Reynolds numbers. Analternativemechanismcanexplaintheobserved trends. ThetransitionalandturbulentdataofMcCarthy andMolloy[5]andAraietal.[12]bothsuﬀerfrom con- 2Thediﬀerenceappearstobestatisticallysigniﬁcant. Araietal.[12] providenouncertaintyestimates,butassumingthestatisticaluncertainty is negligible for electrical conductivity measurements (as they are essentially a large number of pulses), the main source of uncertainty is the precision of the length measurement. If the experimental setup of Arai et al. was similar to that of Phinney and Humphries [17, p. 9], the measurement was within 2 mm. Then for L0d0=1,hxbid0= 72:46:7, and for L0d0=50,hxbid0=93:86:7. 3See the caption of ﬁgure 2 for the precise Reynolds numbers. 4AlsocomparetheReynoldsnumbers: Re`0=4750forMcCarthy and Molloy [5, ﬁg. 5], 16 000in ﬁgure 2. 2 founding between the turbulence intensity and velocity proﬁle 5. (A “plane-averaged” turbulence intensity in this workisdeﬁnedas Tu0q 2k0¹3U2 0ºwhere k0isaplane- averaged turbulent kinetic energy.) I discuss confounding indetailinanotherpaperatthisconference[18]. Essen- tially,boththeturbulenceintensityandvelocityproﬁleare changingwhenthenozzlelengthchanges[19]. McCarthy and Molloy [5, p. 10] do not seem to be aware of this giventhattheystatethat“thesejets,onissuingfromthe nozzles, diﬀer from each other onlyin the value of ” (the emphasis is my own). An increase in disturbances in the ﬂow as measured by turbulence intensity would obviously aﬀect the stability of the jet and its transition to turbulence. Others agree on this point — at higher ReynoldsnumbersDeblerandYuattributeMcCarthyand Molloy’s observations to the onset of turbulence. To be clear, both the turbulence intensity and velocity proﬁle are factors, but the relative contributions of each cannot bedeterminedbasedontheexperimentsofMcCarthyand Molloy and Arai et al. The case where the breakup length hxbiin Arai et al. [12]wasO¹1000ºlikelyhadlowturbulenceintensityas the diameter contraction ratio was dind0=10. Large contractionstendto reducetheturbulenceintensity[20]. As the nozzle length increases, the turbulence intensity at the nozzle exit increases due to the eﬀects of the shear at the walls 6. The increase in breakup length as the nozzlelengthincreasesfrom L0d0=10to50couldbe explained by turbulence transition moving to inside the nozzleasnozzlebecomesbasicallyafullydevelopedpipe ﬂow. Turbulence transition inside the nozzle appears to stabilize jets, as will be discussed more below. An unambiguous test would try to maintain the tur- bulence intensity as close as possible between the two velocity proﬁles tested to isolate the eﬀect of the velocity proﬁle. Likely the breakup lengths were much longer for shorter nozzle aspect ratios in the data of Arai et al. due to the low turbulence intensity for their short nozzles. Including other studies, the data for short nozzle aspect ratiosseemstovarygreatly,likelyduetotheturbulence intensity being a strong function of the inﬂow. See, for example,thatthebreakuplengthsreportedbyChenand Davis [22] for their short nozzles are much lower than 5Jets which will remain laminar even in the fully developed state, e.g., the cases for which Re`0<2000in ﬁgure 2 do not suﬀer from this confounding problem because the disturbances introduced from the wall are damped by viscosity. Hence the low Reynolds number (always laminar regardless of L0d0) case mentioned earlier does not suﬀer from this confounding. 6ThenozzledesignapproachofTheobald[11]camefromWhitehead etal.[21],andsuﬀersfromasimilarproblem. Inadditiontohavingaﬂat velocityproﬁle,Whiteheadetal.alsoavoidboundarylayerseparation tokeeptheturbulenceintensitylow. Consequently,theapparentsuccess ofTheobald’snozzledesignisnotnecessarilyduetothevelocityproﬁle relaxation eﬀects discussed in Theobald [11].those of Arai et al. [12]. AqualitativeunambiguoustestwasconductedbyWu etal.[9]. Theturbulenceintensitywaslowinthiscase,but notquantiﬁed. Whilebreakuplengthswerenotmeasured, the photographs show that breakup was suppressed when theboundarylayerwasthinner. TheDNSstudyofSander andWeigand[23,ﬁg.11],anditmakesclearthatbreakup at least qualitatively is inﬂuenced by the velocity proﬁle. The early studies of Eisenklam and Hooper [3] (at Re`03000to12 000) and Rupe [4, ﬁg. 4, p. 12] (at Re`02100)havephotosshowinginitiallylaminarjets withnearcompletebreakupoccurringoverashortdistance. While these researchers suggested that the parabolic ve- locity proﬁle was responsible for this observation, as was previouslydetailed,laminarjetswithparabolicvelocity proﬁles show no particular inclination towards instability at Reynolds numbers below the critical Reynolds number. Instead,Ihypothesizethatturbulencetransitionoutsideof the nozzle causes the strong breakup, and that turbulence transitionissensitivetothevelocityproﬁle. TheReynolds numbers where this form of breakup occurs coincides well with where transition to turbulence occurs in fully developed pipe ﬂows 7. This idea is consistent with the observationsofHoytandTaylor[8,p.96L]whosuggest that when a liquid jet transitions to turbulence after it has exited the nozzle, the transition is more violent thanif the jetwerealreadyturbulent. Further,HoytandTaylorpro- pose that reducing boundary layer thickness at the nozzle exit can reduce these eﬀects, which again, is consistent withtheideathatthevelocityproﬁleinﬂuencesturbulence transition. The earlier mentioned DNS study of Sander andWeigand[23]showedaclearsensitivitytothevelocity proﬁle. Sander and Weigand used low Reynolds numbers from 3000to7000,whichareroughlyinthetransitional rangeforfullydevelopedpipeﬂows,ultimatelyconsistent withtheideathatturbulencetransitioncancausestrong breakupandalsocanbeinﬂuencedbythevelocityproﬁle. The reader is referred to Portillo et al. [25] and Umemura [26] for the latest experimental and theoretical research on transitional liquid jets, including detailed explanationsofthetransitionmechanismwhichmaybe usefulfornozzledesigninthisregime. Moreresearchis needed to explain why transitional liquid jets appear so unstable. The criteria developed later in this paper may explain why transition can produce such violent breakup, buttheﬁtwiththetransitionalbreakupdataavailableat present (which is largely qualitative) is mixed. Note that many of the trends discussed previously seem to applyonlyat low ambient densities, i.e., density 7Somereadersmay believethataReynolds numberof 12 000isfar too highto be transitional, butlaminar fully developedpipe ﬂows have been maintained at Reynolds numbers as high as 100 000depending on the quality of the experimental setup [24, p. 7]. Indeed, Reynolds himself was able to establish laminar pipe ﬂow at a Reynolds number of 13 000. 3 ratios`g>500. The data of Arai et al. [12, ﬁg. 7] seems tosuggest that athigh ambient densitiesand mod- erate jet velocities, nozzle length has little eﬀect, but at higher jet velocities, longer nozzles have much longer breakuplengthsthanshorternozzles. AtpresentIcannot explainthesetrends. Thispaperfocusesonlyonthelow ambient density case. Physicsofturbulentvelocity-proﬁle-inducedbreakup The mechanism by which the velocity proﬁle con- tributes to turbulentbreakup has so far notbeen detailed. For turbulent jets, if turbulent ﬂuctuations are indeed a major cause of breakup, then velocity proﬁle relaxation destabilizes jets by increasing the transverse RMS tur- bulent velocity v0(p hv2i) at the surface 8. The most natural mechanism is turbulent production. Consider a jet wherehUiis the mean convective velocity, hViis the mean radial velocity, and hWi=0(no swirl). The production terms for the uRMS velocity, u0, and vRMS velocity, v0, are [28, appendix 5] 9: Pu2hu2i@hUi @x2huvi@hUi @r 2huwi r@hUi @; (2) Pv2huvi@hVi @x2hv2i@hVi @r 2hvwi r@hVi @hWi ;(3) where the subscript on Prefers to the direction. Velocityproﬁle relaxation isoften claimedto work through the creation of radial velocity components [4, p. 13L, 29, p. 3385R, 6, p. 512L], which is possible, however, the process is not direct. The production of v0term,Pv, is not a direct function of @hUi@rj0s. No other terms allow for energy transfer from the mean axial velocityhUito the turbulent radial velocity v0through variation in r. Consequently, in turbulent velocity proﬁle relaxation, u0is produced ﬁrst and then the energy is redistributed to v0, leading to breakup. Ratherthanmodeltheredistributionprocess,Iwill solelyexaminetheamountofproductionof u0forsimplic- ity. To determine an appropriate dimensionless group for the velocity gradient, I normalized the production rate for u0by the dissipation rate, as production will need to ex- 8One goal of this work is to put all of turbulent breakup theory into a more consistent framework. As turbulent velocity ﬂuctuations are the generally accepted cause of turbulent breakup, I prefer explanations usingturbulentﬂuctuationsratherthanstabilitytheoryorwave-based arguments, even if the alternatives are in some sense equivalent. See Trettel [27] for my earlier theoretical work on turbulent breakup. 9The single-phase production is used here because I am assuming that when velocity proﬁle relaxation is occurring, breakup is not yet signiﬁcant, so the two phases are separate. Favre averaged equations would be more general.ceeddissipationforturbulencetobegeneratedfromshear. If I use the common dissipation model "_v0 030, the expressionfortheproductionof u0(equation2)neglecting all terms other than the mean radial gradient term, and assume thathuvi_v0 02(a common turbulence modeling approximation [30, p. 121]), then I obtain: P "/huvi "@hUi @r_v0 02 v0 030@hUi @r=0 v0 0@hUi @r:(4) Theusubscript onPhas been dropped for brevity. Insteadofthinkingintermsofvelocitygradients,itiscom- moninthejetbreakupliteraturetothinkintermsofbound- ary layer thickness. Let’s use the notation @hUi@rj0s to refer to the liquid velocity gradient located at the nozzle exit plane (0) and free surface (s). If I assume that@hUi@rj0sU00where0is a measure of the boundary layer thickness at the nozzle exit then 0 v0 0@hUi @r 0s0U0 v0 00=0 0Tu1 0= Tu00 01 :(5) Consequently,¹Tu000º1would presumably need to beminimizedtopreventvelocityproﬁlerelaxationeﬀects. However, this criteria leads to a paradox. Shear instability paradox Shearcancauseinstabilityinﬂuidﬂowsandgenerate turbulent kinetic energy. The larger the velocity gradient, thestrongertheproduction. Equation5agreeswiththis view. Or equivalently: the thinner the boundary layer, the stronger the production. This seems to be in direct contradictionwiththecommonsuggestionintheliterature toreducetheboundarylayerthicknesstoimprovestability. The experiments of Wu et al. [9] are clear: a nozzle designed to have thinner boundary layers while changing nothing else does seem to result in more stable liquid jets. So why does the theory suggest the opposite? One possibly resolution of this paradox is that the overalllevel of production is reduced with thinner bound- ary layers because the total area with gradients is smaller. Onemightexpectthetwoeﬀectstoroughlycanceleach other out. That would seem to lead to another problem about why thinner boundary layers would be better. Iattempttoresolvetheparadoxbyestimatingtheratio oftheplane-averagedproductiontotheplane-averageddis- sipation. Thisratioismotivatedbythefactthatproduction needs to exceed dissipation for turbulent kinetic energy to increase. A plane average is used to take into account the factthattheregionofhighproductionshrinksasthebound- ary layer shrinks, ultimately reducing the total amount of production. The plane-averaged production is relevant because the turbulence naturally becomes more homo- geneous downstream, distributing the production over theentirejetcrosssection. Unfortunately,thisapproach 4 appears to be sensitive to the modeling approximations used. Most approximations lead to inﬁnite production as the boundary layer thickness decreases 10. Those ap- proximations were rejected as implausible as empirically, thinner boundary layers do not appear to cause extreme instabilityin these jets. Thesimplest speciﬁcationwhich is stable uses the Reynolds stress model huvi=Cuvk (which Durbin and Pettersson Reif [30, p. 121] notes is acceptableintheloglawregionoftheboundarylayer)and thedissipationmodel "=C"v0 030. Ialsoapproximate the velocity proﬁle as linear in the boundary layer, with velocity Ucin the ﬂat center region: hUi¹rº=8>>>> < >>>>:Uc ifrd0 20 Uc 0d0 2r ifrd0 20:(6) Thisformisparticularlyconvenientastheproductionwill bezerointhecenterregionoftheﬂow,sothatonlyone integral is required in the computationofthe production. The plane-averaged velocity U0can be computed as U0=Uc" 120 d0 +4 30 d02# :(7) Consideringonlythe rgradientofhUiforsimplicity, applying thehuvimodel, and assuming that k0equals its plane-averaged value k0everywhere, the plane-averaged production is P0=¹ Ahuvi0@hUi @rdA d2 04 =8Cuvk0 d2 0¹d02 d020Uc rdr =4Cuvk0Uc d0 10 d0 : (8) Applying the "model, the overall production-dissipation ratio is P0 "0=6Cuv C"Uc v0 00 d0 10 d0 (9) _U0 v0 00 d0 10 d0 120 d0 +4 30 d02; (10) 10In particular, an eddy viscosity model was unstable in all cases I tried, including one where kincreased quadratically from zero at the wall to take into account the no-slip condition. I had thought that would kill any instability, but it did not.so P0 "0/Tu1 00 d0 1+0 d0 for small0 d0:(11) Equation 11 will be termed the “overall production- dissipation ratio”. Contrary to equation 5, this criteria does suggest that thinner boundary layers have less pro- duction and consequently lead to liquid jets which are more stable. However, this analysis would suggest that in the limit as the boundary layer thickness approaches zero, production would not go to zero. This seems unusual and may be the result of unrealistic modeling assumptions. It alsosuggeststhatvelocityproﬁlerelaxationisrelatively weakinturbulentjetsasthe 1termisalwaysessentially much larger than 0d0. This conclusion is consistent withtheexperimentsofDurbinetal.[31],whofoundthat havingathinboundarylayerislessimportantthanhaving lowturbulenceforreducingjetbreakup. Additionally,the overall production-dissipation ratio is controlled by more than the boundary layer thickness (or equivalently, the velocity proﬁle), as will be discussed in the next sections. The eﬀect of the turbulence intensity on velocity pro- ﬁle relaxation The production-dissipation ratio theory would sug- gest that ﬂows with low turbulent RMS velocities v0 0(i.e., laminarornearlylaminarﬂows)wouldtendtohavepro- portionallyworseproductionofturbulencethanﬂowswith higherturbulenceintensities. Thisisconsistentwiththe observationthatvelocityproﬁlerelaxationseemsstrongest fortransitionalﬂows(whichhavelowturbulenceintensity) andisweakenedforcompletelyturbulentﬂows. However, the trend shown by McCarthy and Molloy [5, ﬁg. 5] does notseemtoobviouslyconﬁrmthetheory. Typicallythe turbulenceintensityincreasesasnozzlelengthincreases (unless the inﬂow was particularly turbulent). The theory would predict that a slight increase in turbulence inten- sity should result in less, not dramatically more, breakup. Possibly the turbulence intensity increase is small if even measurableasthenozzlelengthincreasesfrom L0d0=0 to10and the ﬂow remains laminar. In that case, the boundary layer thickness increase may indeed explain the dramatic increase in breakup, ampliﬁed by the fact that the turbulence intensity is low. This hypothesis needs experimental validation. Some researchers have reported that when the tur- bulence intensity is low, increasing turbulence intensity can lead to increased stability of liquid jets, contrary to the suggestion that increased turbulence intensity only de- creases jet stability. This has been shown experimentally for both circular [32] and ﬂat [33–35] nozzle geometries. The researchers have hypothesized that velocity proﬁle relaxation occurs more quickly in these instances because turbulent diﬀusion would smooth out the velocity proﬁle, 5 in turn reducing total production. The overall production- dissipationratiotheorydevelopedherealsocanexplain how increasing turbulence intensity can stabilize liquid jets in special circumstances using a diﬀerent mechanism (turbulentdissipation). Bothmechanismsarepresentin liquid jet breakup. In future work I will examine the evolution of turbu- lence in the jet to develop a criteria taking into account turbulent diﬀusion as well. It seems most reasonable to developanestimateforthemaximumturbulencekinetic energyinthejetandusethatasameasureofthetendency for velocity proﬁle relaxation to cause breakup. Turbu- lence production is countered by not only dissipation but also turbulent diﬀusion. The eﬀect of the integral scale on velocity proﬁle re- laxation Previousresearchershavespeculatedaboutintegral scale eﬀects, most often in the form of using the inte- gral scale 0in place of the nozzle diameter d0in an equation [36, 37]. While likely part of the picture, that assumptionmissestheroleoftheintegralscaleindissi- pation. In the common dissipation model "_k32, it’s clear that smaller integral scales would lead to greater dissipation. The greater dissipation would reduce the overallproductiondissipationratio, P",ascanbeseen as the integral scale is reduced in equation 11. This has not been recognized in the turbulent jet breakupliterature,butitisconsistentwiththeexperiments of Durbin et al. [31]. Durbin et al. compared the breakup ofturbulentliquid(sheet)jetsproducedbynozzleswith andwithoutascreenplacedimmediatelyupstream. The addition of a screen reduced surface ﬂuctuations [31, ﬁg.11,p.317],whichpresumablyareaproxyforbreakup. They concluded was that the screen reduced turbulence intensity, leading to reduced breakup. However, the integral scale also changes appreciably as the screen is removed. To ﬁrst-order, the integral scale will be proportional to the mesh size of the screen. Durbin et al. [31, p. 310] state that the honeycomb in their nozzle immediately upstream of the screen has 0:32 cmdiameter openings. Thescreenshaveopeningsofwidth 0:51 mm, whichwouldsuggestthattheintegralscaleimmediately after the screen is about 6times smaller than without the screen. Unfortunately this is unlikely to represent the change in integral scale at the nozzle as approximately 15 nozzle thicknesses pass before entering the contraction. The integral scale presumably also increases over this distance, so this experiment is not entirely conclusive. Fortunately,theDNSstudyofSanderandWeigand [23, ﬁg. 11], where the integral scale was speciﬁed pre- cisely,qualitativelyshowsthestabilizingeﬀectofreducing the integral scale. However, the eﬀect appears to be weak. The laterDNS study of Salvadoret al. [38] looksfurtherinto the eﬀect of the integral scale. Salvador et al. [38, ﬁg. 7] shows qualitatively that a reduced integral scale leads to a more stable jet. A plot of axial mass concentra- tion Salvador et al. [38, ﬁg. 7], which is analogous to the breakup length, also shows that a smaller integral scale is morestable. Thenumberofdropletsgeneratedfromthejet (ameasure ofhow muchthejet hasbroken up)increases astheintegralscaleincreases[38,ﬁg.11a]. Thedroplet size distribution changes relatively little as the integral scale is increased from 0to0:17d0with other variables heldconstantorapproximatelyso[38,ﬁg.11a],though thismaynotbeclearintheoriginalwork. Asanexample, using the droplet size histograms given by Salvador et al., I computed that for 0d0=0,D32=9:17µm, and for 0d0=0:17d0,D32=9:45µm. This contradicts the hypothesis that characteristic droplet sizes (e.g., D32) are proportional to the integral scale in turbulent breakup, as statedbyHuhetal.[37]. Theturbulenceintensityinthese cases was set to that of a fully developed pipe ﬂow, which isnotlow,sothedestabilizingeﬀectsoflowturbulence intensity are not seen in these cases. Ultimately, these integral scale results are prelimi- nary. Furtherdetailedcomputationsandexperimentsare required to make clear conclusions. Conclusions When strongly stable liquid jets are desired, one frequently reads recommendations to make the velocity proﬁleasﬂataspossible. Thisdesigncriteriashouldbe reevaluated. Forlaminarliquidjets,parabolicvelocityproﬁlesare morestablethanﬂatvelocityproﬁles,contrarytocommon belief. Forliquidjetswhicharelaminaratthenozzleexitbut transitiontoaturbulentstatedownstream,thetransition canbeparticularlyviolent,leadingtostrongbreakup. Hoyt andTaylor[8]recommendthatifstablejetsaredesired,it is better tobring the transition pointinto the nozzle than to allow the jet to transition outside of the nozzle. The transition process appears to be sensitive to the boundary layerthickness. TheturbulenttheoryIdevelopedmayhelp explaincertainaspectsofthetransitionalbreakupprocess includingwhytransitionalbreakupissostrong,butitdoes not explain whether or where a jet will transition. For liquid jets which areturbulent at the nozzle exit, thefollowingcriteriamaycharacterizehowmuchvelocity proﬁle relaxation contributes to breakup (higher means more breakup, all else equal): Tu1 00 d0 1+0 d0 : (11) The overall production-dissipation ratio,equation 11, ap- pears to explain the qualitative trends observed in the velocityproﬁlerelaxationliterature(whythinnerbound- 6 ary layers are more stable, why transitional breakup is so strong), including some lesser known eﬀects (stabi- lization at moderate turbulence intensities, integral scale eﬀects). However, the theory is still only a hypothesis as no quantitative data is available to fully validate the theory at present. New experiments and detailed com- putationsdirectlytestingthishypothesisareencouraged. The sensitivity to the overall production-dissipation ratio is at present unknown, but I believe for most breakup quantities the Weber number and turbulence intensity are more important. The theory in this paper is preliminary. Future work will improve the theory with more accurate turbulence modeling approximations. As much of the present data is qualitativeorvague,newexperimentalandcomputational studiesarealsoneededtobetterexaminetheroleofthe velocity proﬁle and its interactions with the integral scale in turbulent jet breakup. Acknowledgments AERADE is thanked for posting the report by Eisen- klam and Hooper [3] online. The NASA STI Information Desk is thanked for removing the distribution limitations on the report byRupe [4] so that it may beposted online. Andrew Trettel is thanked for reviewing this paper and for a discussion about potential future improvements to the theory which will be the subject of future work due to time constraints. Nomenclature Aarea D32Sauter mean diameter d0nozzle exit diameter L0nozzle oriﬁce length k1 2¹u02+v02+w02º, turb. kinetic energy kplane-averaged turbulent kinetic energy Pturbulent production Re`0U0d0`, liquid Reynolds number rradial coordinate Tuu0U, local turbulence intensity Tu0q 2k0¹3U2 0º, turbulence intensity using plane-averaged k Uvelocity in the xdirection uUhUi,Uvelocity ﬂuctuation u0p hu2i,uRMS velocity Vvelocity in the rdirection vVhVi,Vvelocity ﬂuctuation v0p hv2i,vRMS velocity Wvelocity in the direction wWhWi,Wvelocity ﬂuctuation w0p hw2i,wRMS velocityWe`0`U2 0d0, liquid Weber number xaxial coordinate hxbiaverage breakup length kinetic energy coeﬃcient, equation 1 boundary layer thickness "turbulent dissipation rate integral scale of turbulence `liquid kinematic viscosity azimuth ggas mass density `liquid mass density surface tension Operators x¹¯ AxdAºA, plane average of x hxiensemble average of x Subscripts 0at nozzle exit g using gas properties `using liquid properties s at free surface