Understandingthephysicsofthebreakupofturbulentliquidjetsisimportantforavarietyofapplicationsincluding engine sprays, ﬁre suppression systems, and water jet cutting. Models of turbulent jet breakup allow predictions of quantities of interest like the droplet size distribution and breakup length of the jet. These models are compared against experimental data in a process called validation. If the model predictions are within the experimental uncertainty, then themodelis“validated”andbelievedtobeaccurate,andpossiblycanexplainthephysics. Uncertaintyquantiﬁcation isnecessaryformodelvalidation. Whileunfortunatelyrelativelyfewexperimentalstudiesquantifyuncertainty,that is not the most pressing validation issue in turbulent jet breakup. I detail 3 additional problems that can make the apparent validation of a model actually an illusion, regardless of how well the model appears to match the data. These problems include: 1. important variables being omitted or guessed in experiments and models, 2. confounding between independent variables, that is, two variables changing simultaneously, making determining cause and eﬀect impossible, and 3. testing only combinations of submodels and not each submodel in isolation. To avoid these problems and others, I developed validation guidelines that are detailed inthis work. Following these guidelines, I compiled a large experimental database. Only 28 out of 47 experimental studies considered met my data quality guidelines. Only 18 studies had quantiﬁed uncertainty, and only 3 studies had substantial variation in the turbulence intensity. ∗https://trettelresearch.com/contact.html 1 Introduction Liquidjetbreakuphasbeenmodeledinawidevariety of diﬀerent ways since the theoretical study of liquid jets began in the 19th century. The precise modeling techniques are not a concern in this paper, rather, how the success of models is evaluated is. The comparison ofmodelpredictionsagainstexperimentaldataiscalled “validation”[1]. Ifthecomparisonisasuccess,thenthe modelisdeemed“validated”andbelievedtoaccurately predict cases not yet measured. Often, models appear to work well in published works, but their accuracy is stillregardedwithsuspicion. Thegoalisthispaperisto highlightsomeofthereasonswhyanapparentlysuccessful validationofaturbulentjetbreakupmodelmayactually be illusory. For example, the KH-RT (Kelvin- Helmholtz/Rayleigh-Taylor) jet and droplet breakup model [2] is popular, but regarded as not fully pre- dictive [3, p. 34L] despite the seemingly favorable ﬁt betweenthemodelandexperimentaldataintheoriginal paper. Thelackofpredictabilityisdemonstratedbyone of the model coeﬃcients, B1, taking calibrated values ranging from 1:73to40[4, p. 14]. Magnotti and Genzale attribute this to the model not taking into account all the physical mechanisms involved. This is just one of several possibilities. Even if the model considers all of the physical mechanisms involved, it’s possible that a particularsubmodelis inaccuratefor itsintendedpurpose. Whiletheprecisecriteriawhichdetermineswhethera modelhasbeenvalidatedisnottrivial[1],forthepurposes of this work I’ll call a model validated if its estimates are withintheerrorbounds(say,95%)roughlyasfrequentlyas theerrorbounditself. Inotherwords,roughly95%ofthe model predictions need to be within the 95% intervals of thedata. Thesemodelpredictionsareconsideredthemost likely cases estimated by the model. Model uncertainty are multi-modal distributions are not considered in this work. This simpliﬁed approach is suﬃcient for this work because, as will be discussed in more detail, relatively few turbulent jet breakup experimental data sources have quantiﬁed uncertainty, many data sources which have quantiﬁeduncertaintyhavelargeuncertaintiesthatarenot diﬃcultformodelstostaywithin,andevenifthesedata sourcesdidhavequantiﬁedandsmalluncertainties,it’sstill possiblefora“bad”modeltomatchtheexperimentaldata wellduetovalidationproblemsunrelatedtouncertainty quantiﬁcation. These problems largely will be resolved through bet- ter and more comprehensive data. Consequently, for validating turbulent jet breakup models, I developed a large data compilation for turbulent jet breakup. This datacompilationwasspeciﬁcallydesignedtobechalleng- ing and diagnostic for turbulent jet breakup models. A summaryof theproblems withexistingdata isin table1.These problemsare elaboratedin this work. Ultimately, only 18ofthe 44(41.9%)experimentalstudieswiththe quantitiesofinterestwereused. 10studieswereneglected despite being acceptable because the data collected ap- pearedtomainlyduplicatealreadytranscribeddata. For reference,allstudiesconsideredarecitedattheendofthis sentence[5–51]. Thestudiesusedarecitedattheendof this sentence [6, 12, 13, 15, 17, 18, 22, 26, 29, 31–33, 39, 41–43,47,48]. Theindividualbreakdownofstudiesused for each quantity of interest is discussed in Trettel [52]. Some regression analysis from this data compilation hasbeenpublishedpreviously[52], butthemotivations behind the data selected and problems found with the previous literature are published here ﬁrst. For additional details,seethedissertationassociatedwiththiswork[53]. Tokeepthisworksimple,Ifocusprimarilyoncircular non-cavitating turbulent Newtonian liquid jets injected into still gases at high density ratios `g&500. In thisregimethemainphysicalmechanismsresponsiblefor breakupareturbulenceandvelocityproﬁlerelaxation,the latter of which can be explained through turbulence [54]. Morebroadly,makingamodelwhichworksinthisregime is necessary for the model to work in more challeng- ingregimesinvolvingcavitationandhigheratmospheric densities. Seeﬁgure1foranillustrationofthebasicjetbreakup quantities of interest (QoIs) considered in this work. The liquid jet travels to the right from the nozzle on the left surrounded by still gas. The nozzle oriﬁce diameter is d0; the0subscriptindicatesavariablelocatedatthenozzle. Similarly, theensemble averagedbreakup onset location hxiiand the spray angle iuse anisubscript to indicate their location as well. The initial breakup location is where jet breakup (i.e., droplet formation from the jet) isﬁrstobservedtooccur. Thislocationisaveragedasit varies over time. Similarly, as jet breakup depletes the liquid core, the liquid core eventually ends at a location calledthe“breakuplength”,whichisalsoantime-varying quantity, so the average breakup length hxbiis measured throughvariousmeanstobediscussedlaterinthiswork. Forbrevitytheaveragebreakuplengthwilloftenbecalled simply the breakup length. Not in the illustration are various measures of the droplet diameter. The Sauter mean diameter is a common measure of the “average” droplet diameter; this is denoted D32. How frequently eachQoIismeasuredinexperimentscanbeseenintable1. The most common independent variables involved in this problem are the nozzle bulk (average) velocity U0, the surface tension , the liquid viscosity `, the liquid density `, and the gas density g. From these the Reynolds number Re`0U0d0`, Weber number We`0`U2 0d0,anddensityratio `gcanbeformed. Asbefore,the 0subscriptindicatesaquantitymeasured atthe nozzleexit, and `indicatesthis isa liquidquantity. 2 description number percentage total considered 47 — acceptable, used 18 41.9% acceptable, not used 10 23.3% D327 16.3% hvdi1 2.3% hxii8 18.6% hxbi14 32.6% i7 16.3% uncertainty quantiﬁcation 18 41.9% rough pipes 3 7.0% far-ﬁeld droplets 5 71.4% (of 7) just-formed droplets 2 28.6% (of 7) no QoI 4 — ambiguous data 2 4.7% curves, not data points 2 4.7% inconsistencies found 3 7.0% foreign language 3 7.0% Table 1: Summary of reasons why studies were neglected from the data compilation. The numbers are the total number of studies ﬁtting a criteria on the left. The percentage does not include studies which did not have any of the quantities of interest (QoIs). The percentage for droplet quantities is taken out of all studies which measured droplet quantities. Disclaimer Following Rider [55], a disclaimer is warranted. I will use certain published papers as examples of poor model validation. The issues identiﬁed in this work are notanindictmentoftheresearchers. Instead,theyshow ﬂaws in the accepted practices of the atomization com- munity.In this work, a accepted practice is deﬁned as a practice which appears in the recent (and up-to-date) liter- aturerepeatedlyorinpaperswhicharewidelyaccepted atpresentinthecommunity. AlloftheresearchIcitewas conducted in good faithto my knowledge. Theproblems Idiscussarenotobvious. Inparticular,confoundingcan be particularly challenging to identify. This study is by no means comprehensive, and reﬂects my own judgment about which validation problems are more pressing based on examination of a large fraction of the published litera- tureonturbulentjetbreakupatlowatmosphericdensities. Other regimes and other atomization problems may not suﬀerfromthesameissuesorsuﬀerfromanymajorissue at all. Uncertainty quantiﬁcation A minority of the data sources (41.9%; see table 1) consideredinthisworkquantifyuncertainty 1[6,13,15, 1Todeterminewhetherastudyhadquantiﬁeduncertainty,Iuseda generous deﬁnition: If it was possible using data in the work and some mild assumptions, I considered the work to have quantiﬁed uncertainty.18, 19, 22, 26, 32, 37, 41–43, 45, 47, 48, 51]. When uncertainties are estimated, they are often large. For example, the percent error of the Sauter mean diameter measurementsofWu[56,p.139]wasestimatedas33%. Large uncertainties make validation easier than it should be. New experiments with small, known uncertainties are needed for rigorous validation of turbulent jet breakup models. Thereareestablishedproceduresforuncertaintyquan- tiﬁcation for droplet size[57–60, 61, p. 128]. Therealso appears to be a wide spread in spray angle data due to thesensitivity ofthesprayangle toitsdeﬁnition [62,63, p.114,64,p.12,65],aproblemwhichcanonlybesolved by standardization of the deﬁnition of the spray angle 2. Aside from a brief discussion by Osta [51, p. 108] little has been written on uncertainty quantiﬁcation for the breakup length, which is used for the examples in this paper. Therearetwomainmethodstomeasurebreakup length: electricalconductivityofthejetandphotographic measurement of where the jet core ends. The electrical conductivitymeasurementsdeﬁnethebreakuplengthas the point where the jet conducts electricity through itself 50% of time. Photographic measurement deﬁnes the breakup length as the average location of the end of the jet’s core. Because the distribution of breakup location is highly symmetric, these two numbers are essentially equal [19, 66], and consequently I use the notation hxbi for the breakup length irrespective of how it was mea- sured. The two components of the uncertainty in this case are the measurement precision and the statistical error from taking a ﬁnite number of data points 3. The electrical conductivity case is essentially taking a very large number of measurements, making the statistical componentnegligible,sothemainsourceofuncertainty is the precision of the measurements. For photographic measurements, the main source of error is typically sta- tistical. For turbulent jets this can be estimated using the fact that the standard deviation of the jet breakup location (b)iswellpredictedbyaconstantcoeﬃcientofvariation Cbbhxbi=0:12910:0019, as is discussed in my dissertation [53]. This along with the t-distribution can be used to estimate the uncertainty in photographic mea- surements ofthe breakup length. The statistical errorfor photographicmeasurementstendstoberatherlargeatthe samplesizesusedinthepreviousliterature,afactwhich hasnotbeenappreciatedtomyknowledge. Thiscanbe seeninﬁgure2;thelargeerrorsinthemeasurementsof 2New experiments may be necessary to determine how to obtain roughlyequivalentsprayanglesusingdiﬀerenttechniques,e.g.,diﬀerent thresholdsforphotographicormassfractionmeasurementswhichresult in approximately the same spray angles. 3A third component, transcription error from the conversion of data to graphical plotsback to data, is morediﬃcult to characterize andhas not been included in this work. However, I intend to examine this in future work. 3 all We`0We`0,Tu0Re`0 Cb3:785 8:051 3:891 9:478 CTu0:272 00:269 0 CWe 0:313 0:318 0:327 0 CRe 0:014 0 0 0 :253 R20:980 0:877 0:979 0:709 Table 2: Table showing the eﬀects of using diﬀerent vari- ables in a regression analysis for the breakup length, hxbi. All the available data ﬁtting the data quality guidelines is used. The regression equation is the same as in table 3. Regression equation: hxbid0=CbTuCTu 0WeCWe `0ReCRe `0. Total number of data points: 145. all We`0We`0,Tu0Re`0 Cb1:061 9:356 1:488 0:396 CTu0:132 00:862 0 CWe 0:114 0:303 0:254 0 CRe 0:364 0 0 0 :607 R21:000 1:000 1:000 1:000 Table 3: Table showing the eﬀects of using diﬀerent variablesinaregressionanalysisforthebreakuplength, hxbi. OnlydatafromAraietal.[33]isused. Totalnumber of data points: 5. Grant[67] areparticularlynoticeable 4. Itislikelythatif the uncertainty were quantiﬁed in some of the earlier pho- tographic breakup length measurements, the researchers would have conducted more trials to reduce the statistical error in their measurements. Electrical conductivity mea- surementsarepreferred,thoughinprinciplephotographic measurements can have small statistical errors at larger sample sizes. Omitted and qualitative independent variables If a model does not include an important variable, it is intuitive that the model may perform poorly. The modelmaymatchitscalibrationdatawell,butbeseverely inaccurate in other situations where the neglected vari- able diﬀers substantially from the values it took in the calibration data. Unfortunately, this is often the case in turbulent jet breakup for an entire class of variables: turbulence quan- tities. It is uncontroversial that some measure of the “strength”of turbulenceisa major factorin turbulentjet 4Neither Kusui [17] or Arai etal. [33] report measurementprecision. These were estimated as 1 cmand0:2 cm, respectively. Also, note that the R2value in the corner of ﬁgure 2 diﬀers from that of table 2 because the table only uses the latter 3 data sources, neglecting the noisierphotographicmeasurementsofChenandDavis[13]andGrant [67]. This was necessary to obtain a clear turbulence intensity exponent in the regression procedure. The large error washed out any turbulence intensity eﬀects.breakup,withthebreakupbeingmoreseverefor“stronger” turbulence[68,p.14,69,p.512,70,p.72L]. Themostnat- uralmeasureofthestrengthofturbulenceistheturbulence intensity, Tu5. Todeﬁne the turbulence intensityﬁrst the the RMS velocity ﬂuctuation u0p hu2imust be deﬁned. Inthedeﬁnitionof u0,uisthevelocityﬂuctuation,deﬁned asuUhUiwherehUiisthetimeorensembleaverage of the velocity. If the reader is familiar with statistics, the turbulentRMSvelocityissimplythestandarddeviation ofaparticularvelocitycomponent. ThelargertheRMS velocity,thelargertheﬂuctuations. Thisquantitycanbe deﬁnedinotherdirections,e.g.,theradialRMSvelocity v0is believed to be particularly important in turbulent jet breakup as the radial ﬂuctuations can directly cause breakup. Often it is convenient to measure the strength ofﬂuctuationsinalldirections. Inthiscaseonecanuse the turbulent kinetic energy, k1 2¹u0+v0+w0º, which considersﬂuctuationsineachdirection. Theturbulence intensityisanon-dimensionalversionoftheRMSvelocity: Tuu0hUi. Inthiswork,I’llusethenotation Tu0torefer toaturbulenceintensitydeﬁnedusingaplane-averaged turbulent kinetic energy: Tu0¹2k03º12¹3U0º. While this choice may seem peculiar, it is chosen because it is believed to capture the ﬁrst-order eﬀects of the strength of turbulence while ignoring the eﬀects of inhomogeneity in the radial direction and anisotropy. A more recent regression analysis by this author [52] suggested moreprecise sensitivities tothe turbulencein- tensity for the breakup length, hxbi, and spray angle, i. The previous study of Kusui [17] also showed a clear turbulence intensity eﬀect on the transition to the atom- ization regime. Most QoIs show a turbulence intensity dependence in turbulent jet breakup to my knowledge. The turbulence intensity is not typically constant. Measurements of the turbulence intensity in large-scale air models of nozzles varied between roughly 4% and 11%fornozzlessimilartodieselnozzles[71,ﬁg.4]and roughly 4%and 13%for suddenand smoothcontraction nozzles with oriﬁce lengths of L0d0=4[72]. These measurements neglected cavitation, which presumably couldincreasetheturbulencelevelfurther. Inapplications whereparticularly stablejetsare desired,lowturbulence intensities on the order of 1% are expected. Unfortunately, the vast majority of previous liquid jet experiments did not characterize turbulence quantities. Thiswasobservedasearlyasthe1967surveyofLapple et al. [73, pp. 9–10] and unfortunately the situation has not changed since then. In 2010, Osta and Sallam [74, p.945]notethatturbulencequantitiesarestillneglected 5Many researchers believe that the Reynolds number ( Re`0 U0d0`)isthemostnaturalmeasureofthestrengthofturbulence. The Reynoldsnumberabsolutelyisafactorinturbulence,however,itwillbe shown later in this paper that in the second wind-induced regime, the breakup length is nearly insensitive to the Reynolds number. 4 inexperiments,despite theirimportance. Theneglectof turbulencequantitiesisunderstandableasturbulencemea- surementsinfreesurfaceﬂowsarediﬃcult[75,p.345]. With that being said, there have been several turbulent jet breakup studies which varied the turbulence intensity, often by avoiding the need for measurement of the tur- bulence level in a free surface ﬂow. The 1948 study of Bogdanovich [76] used large-scale air models of nozzles to get credible estimates of the turbulence intensity at the nozzle outlet. In 1963, Skrebkov [12] used long pipes ofvaryingroughnesstocontroltheturbulenceintensity relatively precisely with a known relationship between theturbulenceintensityinfullydevelopedpipeﬂowand thefrictionfactor. Theﬁrsttheoreticalstudytoconsider the turbulence intensity was made by Natanzon [77] in 1938. Unfortunately these studies are little known, likely because they were originally written in Russian. Theneglectofturbulenceintensitywasexaminedina review of dimensional analysis of turbulent jet breakup in my dissertation [53]. Only 20% ofthe 45 studies consid- eredtheRMSvelocity(andbyextension,theturbulence intensity) in their dimensional analyses. A further 20% considered nozzle geometry as a factor, which could be considered a proxy for turbulence intensity (though not a good one; see the next section on confounding). As adding a variable is easy in dimensional analysis, this indicatesthatturbulenceintensityeﬀectsareunderstudied in turbulent jet breakup in general. Theneglectofturbulencequantitiespresentsmajor issues from a modeling perspective. The data can not be compared fairly against models because the turbulence intensity is now a free parameter. Its precise value is unknown,anditisfrequentlyestimatedatpreciselywhere it needs to be to make the model work regardless of whether that value is credible. One example of this problem is breakup length model of Lafrance [78], which uses an implausibly low value for the turbulence intensity (0.8% for fully developed smooth pipe ﬂow, vs. about 5% in reality, depending on the Reynolds number) because that’s what matches the data best. If a realistic value of the turbulence intensity were used, the model would not produce a realistic breakup length. The model is miscalibrated. Another example is the breakup length modelofErvineetal.[79]whichusesanarbitrarilychosen turbulence intensity value of 3%, which, of course, ﬁts the data very nicely. There is no reason to believe that choice is appropriate and possibly the model suﬀers from the same problem as Lafrance’s model. TheworkofWu[56]alsoreliesonestimatesofthe turbulence intensity implicitly as the turbulence intensity was assumed roughly constant. This hides the problem inempiricallydeterminedcoeﬃcients,whichultimately are functions of the turbulence intensity. By treating these coeﬃcients are constants, the model assumes thattheturbulenceintensityisconstant,limitingitsabilityto generalize. BeingbasedontheworkofWu[56],therecent modelofMagnottietal.[80]itselfneglectstheturbulence intensity, despite Magnotti and Genzale’s criticism of the KH-RT model for not considering turbulence eﬀects [3, p. 34L]. It’s not even necessary to quantify a variable (even implicitlyasinthecoeﬃcientcase)to“prove”thevalidity of a model. Bergwerk [81, p. 655] rejects the idea that turbulence can cause breakup because “turbulent veloc- ity components [...] are hardly likely to be of suﬃcient magnitude”tocausebreakup. However,Bergwerkdidnot quantifythemagnitudeoftheturbulentvelocitycompo- nents or the velocity magnitude needed to cause breakup, making their argument simply an assertion. Admittedly, the inverse problem of determining a model input from the outputs can often be perfectly valid. But it relies entirely on the model being validated with knownvaluesoftheinputs. Ifthemodelwasnotvalidated with known values of the inputs, avoiding the issues mentioned in the previous paragraph, then there is little reason to be conﬁdent in the inversion. I don’t believe that current models have been validated due to the issues mentioned in this paper. And even if a model passed a series of good validation tests for turbulent jet breakup, I am not convinced that any present models (including the modelIdevelopinTrettel[52])aresuﬃcientlytrustworthy to be used for inverse modeling purposes. The parameter space explored by existing data is too small; I’d need to be conﬁdent outside of the ranges of the source data. If an inverse problem can be avoided entirely (through measurement,pre-existingdata,etc.), avoidinginversion isobviouslymuchpreferred. InthisworkIintentionally onlyselectdatawherethisinverseproblemcanbeavoided entirely. Estimating the turbulent kinetic energy with a model seems prudent if measurement is diﬃcult. Unfortunately, the popular nozzle turbulence model developed by Huh etal.[82]aspartofalargerspraymodelisinsevereerror when compared against experiment data, as I detailed in a previous paper [83]. For typical nozzle lengths (L0d04), Huh et al.’s model predicts turbulent kinetic energieswhicharemorethananorderofmagnitudetoo high. Despite this severe error, the combined nozzle- spray model was successfully validated for predicting spray angles. This suggests either that the turbulence levelofthejetisunimportant, whichseemsunlikelyand contradicts the claims of Huh et al., or it suggests that the spray model is mis-calibrated due to the poor estimates of the turbulence level. This type of problem (integration tests being insuﬃcient) will be discussed later in this paper. Qualitative trends are also not suﬃcient. There are many studies which compare jets with presumably “low” 5 turbulenceintensityproducedbysmoothandshortnozzles againstagainstjetsofpresumably“high”turbulenceinten- sity produced by jets from longer nozzles 6. These studies can be used for only qualitative validation of models at best. For example, Reitz and Bracco [84, p. 1741L] reject the idea that turbulence alone could be responsible for jet breakup in a high density environment because the trendofaparticularmodelcoeﬃcientasthenozzlelength increases(presumablyincreasingtheturbulenceintensity) istheoppositeofexpectationsifturbulencecontributedto breakup. AmorerecentexampleisthestudyofOstaetal. [85], which examined injectors of nozzle aspect ratios L0d0=10andL0d0=40as a proxy for turbulence level. Osta et al. conclude that the longer nozzle has a faster rate of breakup, but diﬀerent models can predict thatwithoutgettingtheprecisesensitivitytoturbulence intensitycorrectbecausethenozzlelengthalsochanges thevelocityproﬁle,aswillbediscussedinthenextsection. Becausequalitativetrendsaresoeasytomatch,theyare not suﬃcient for validation. The conclusion from the examples over the last page is that if a variable is not empirically quantiﬁed, it can be used to “validate” essentially any model or “conﬁrm” any hypothesis. From an experimental perspective, neglecting impor- tantvariablesalsomeansthatanexperimentcouldbeless reproducible. Alaterexperimentercouldtrytoreproduce theexperimentbutbeunableto,andhavenowaytoverify that their setup is producing the same jets. Aside from fully developed pipe ﬂows, the turbulence intensity at theoutletofaninternalﬂowcomponentisafunctionof the inlet turbulence intensity. Consequently, even using the same nozzles and same upstream pipework may not be suﬃcient for reproducibility. The inﬂows to the test system must also be standardized. Giventhatfullydevelopedturbulentpipeﬂowshavea universalandwell-understoodstate,theymakeanexcellent basis for experimentation. “Pipe” nozzles are the de facto standardnozzleforbasicturbulentjetbreakupresearch. The friction factor fof a long pipe is strongly correlated withtheaveragedturbulenceintensity Tu0oftheﬂow[52]: TuFD=0:3655 f0:4587: (1) See ﬁgure 5. Consequently, the turbulence intensity of any pipe nozzle can be estimated. The data compilation I made was restricted solely to pipe nozzles for this reason. Even if the researchers did not measure the turbulence intensity,itcanstillbecrediblyestimatediftheyuseda pipe nozzle. For smooth pipes with turbulent ﬂows, the friction factorisarelativelyweakfunctionoftheReynoldsnumber. 6Anadditionalcommonproblemwiththesestudiesisconfounding between the velocity proﬁle and turbulence intensity, which will be discussed in a later section of this paper.The previously mentioned study by Skrebkov [12] used pipesofvaryingroughnesstochangetheturbulenceinten- sity independent of the Reynolds number. Unfortunately, as can be seen in table 1, only 3studies I am aware of usedroughpipes[12,17,31],sothereisverylittledata with appreciable variation in turbulence intensity. With this beingsaid, pipe nozzles are not apanacea; theyareapoorchoiceforstudyinglowturbulenceintensity scenariosasasmoothpipehasaturbulenceintensityof roughly 5%, while some nozzles designed to produce highlystablejetsmayhaveturbulenceintensitiesbelow 1%. Care must also be taken to have a smooth outlet to separate the eﬀects of imperfections in the oriﬁce and turbulence intensity [86, p. 1162, 13, p. 179, 18, p. 6]. When the turbulence intensity can be taken into account, the accuracy of a turbulentjet breakup model is improved. Theresultsofregressionanalysisofbreakup lengthdataundervariousconditionsisshownintable2. The regression equation hxbi d0=CbTuCTu 0WeCWe `0ReCRe `0(2) is ﬁtted to 145 data points from 5 diﬀerent studies, as shown in ﬁgure 2. These studies all used long pipes which produce fully developed turbulent ﬂow as their nozzles. The data has been limited to the second wind- induced regime where a power law for breakup length has been shown to hold. The study of Kusui [17] had varyingroughnesswhichallowstheturbulenceintensity to vary from about 5% for a smooth pipe to about 13% for a very rough pipe as can be seen in ﬁgure 4 7. This variation in turbulence intensity is much wider than is typical and provides a strong challenge to turbulent jet breakup models. Thecolumnsoftable2indicatewhichofthe3vari- ables ( We`0,Re`0, and Tu0) are considered. The left columnliststheexponentsoftheregressionequation. The bottomrowis R2,asimplemeasureofhowwellthemodel matchesthedata 8. Higher R2valuesindicateabetterﬁt, with 1being the maximum. Comparing the We`0and We`0,Tu0casesshowsthatincludingtheturbulenceinten- sity in the model does appreciably improve the accuracy, increasing R2from 0:877to0:979. Adding Re`0only oﬀers a marginal improvement with an R2value of 0:980, indicating that the turbulence intensity is indeed more 7UnfortunatelyKusui[17]hadamoderatelengthsmoothsectionafter their rough pipe, which complicated the estimation of the turbulence intensity. The turbulence intensity was estimated as if the length of the shortsectionwaszero. Thisselectionwasfoundtobemostconsistent with breakup length data from non-pipe nozzles. This issue is discussed inapreviouspaper[52]. Newexperimentswithoutthisissueareneeded. 8In theintroduction, I hadrecommended examininghow frequently the model estimates are within the error bounds of the data instead. R2 implicitlyassumesthatthedatahasnouncertainty,andischosenhere for simplicity. 6 important than the Reynolds number in the second wind- induced regime. As will be discussed in the next section, thisapparent(small)Reynoldsnumbereﬀectmayactually be a turbulence intensity eﬀect due to confounding. Confounding and spurious correlation Confoundingbetweenvariablesoccurswhenanex- perimenter can not diﬀerentiate between the eﬀects of changing one variable and the eﬀects of changing an- other. Iftwoindependentvariablesarechangedatonce, it is impossible to know the relative contributions each independent variable to the change seen in the dependent variable. Any change seen could have been due to one variable, the other, or both. The vast majority of previous turbulent jet breakup experiments considered in this work suﬀeredfromconfoundingbetweenvariables,makingdata analysis ambiguous unless steps are taken to avoid the confounding. Unfortunately,thatwasdoneinfrequently for the confounding between We`0andRe`0, rarely for the confounding between Re`0andTu0, and rarely for the confounding between the velocity proﬁle and Tu0. The diﬃculty of distinguishing between We`0andRe`0 eﬀectsinjetbreakupexperimentswasﬁrstnotedbyAsset andBales[6, p.2]in1951,later independentlynotedby Dodu [87, 88] in 1959, but appears to have received little attention since. Themostobviousexampleofconfoundinginturbu- lentjet breakup isbetweenthe Reynoldsnumber Re`0 U0d0`and theWebernumber We`0`U2 0d0. For most experiments, the researcher runs a series of tests withaparticularnozzleandﬂuid,varyingonlythepres- sure. This, in turn, varies the bulk velocity of the jet, U0. Changing only the bulk velocity of the jet changes both theWeberandReynoldsnumberssimultaneously. Seeﬁg- ure3;theapparentlinescomefromdiﬀerentexperimental trials using the same nozzle and ﬂuid. 9 Again,iftwovariablesarechangedsimultaneously, it’simpossibletoattributetheeﬀectsseentoeithervariable unambiguously. Perhapsthejetisinsensitivetochanges intheReynoldsnumberaslongastheReynoldsnumberis highenoughtoestablishturbulentﬂow 10. Then,allofthe changes seen would be due solely to the Weber number changes. However, that can not be determined from a singlenozzlewithasingleﬂuid. Onemustusediﬀerent nozzle diameters and/or ﬂuids to break the correlation between We`0andRe`0. This is what I did by compiling datafrommanydiﬀerentdiameternozzlesandﬂuids,as canbeseeninﬁgure3. Breakingtheconfoundingrequires varyingthenozzlediameterandtheﬂuid(tochangethe 9The earliest example of a We`0-Re`0plot that I am aware of is due to Dodu [87, p. 500, ﬁg. 4] in 1959. 10For convenience, when I write “independent of Re`0”, I mean “independent of Re`0provided it is high enough that the hydrodynamic ﬂow regime is turbulent”.viscosity and/or surface tension). Less obvious is the confounding between Re`0and Tu0. For a particular nozzle geometry and surface rough- ness,theReynoldsnumberatthenozzleoutletdetermines the turbulence intensity at the nozzle outlet. The relation- shipbetweenthetwoisnotuniversal[71],butisknownfor longpipenozzles,asdiscussedpreviously. Thisconfound- ing can be seen in ﬁgure 4. The confounding between the velocity proﬁle and Tu0was also discussed in Trettel [54]. Confoundingisoftencausedbynondimensionaliza- tion, as it is in the We`0andRe`0case. Dimensionless variables frequently have common dimensional variables. Evenifallofthedimensionalvariableswereuncorrelated, there may now exist a correlation between the dimension- lessvariables. Asubsetofthisissuehasbeendiscussed extensively in the dimensional analysis literature as “spu- rious correlation” [89]. However, spurious correlation is only a consequence of a particular type of confounding. Inspuriouscorrelation,thedependent(output)dimension- lessvariablescontaindimensionalvariablesincommon with the independent (input, not statistically independent) dimensionlessvariables. Incontrast,confoundingismore general, and applies between independent variables. Estimatesofthecorrelationbetweenvariableswith commontermscanbecomputedassumingthatthedimen- sional variables are uncorrelated [89]. The correlation between dimensionless variables often shouldbe consid- ered when quantifying uncertainty, as typical approaches assume that all variables are uncorrelated 11. To be clear, the confounding seen in turbulent jet breakup is not necessarily caused by nondimensional- ization. For example, while the dimensionless velocity proﬁle and turbulence intensity both have the average velocity U0in common, confounding can still occur in caseswhere U0isheld constant,asitroughly isinmany experiments. The confounding is actually an artifact of how many experiments are conducted. Multiple variables changeasthenozzleaspectratio L0d0ischanged,where L0is the nozzle oriﬁce length: 1. the velocity proﬁle changes (and consequently, the boundarylayerthickness increases);2.theﬂowcantransitionfromlaminartotur- bulent; 3. the turbulent kinetic energy typically increases (as would the magnitude of the Reynolds shear stress); 4. swirl decays; and 5. the ﬂow could have separated at the nozzle inlet but reattach further downstream. Con- sequently, if one uses L0d0as a proxy for any of the 5 mentioned eﬀects, one can not distinguish between these eﬀects. Asimilarproblemcausesconfoundingbetween theturbulenceintensity Tu0andReynoldsnumber Re`0, asforsmoothpipenozzlestheturbulenceintensityisonly a function of the Reynolds number. If one variable is a 11Note that even if the dimensional variable in common between two dimensionlesstermswereheldconstantexperimentally,theerrorswould still be correlated. 7 function of the other only, then regardless of the com- position of those variables (i.e., into dimensional terms), theywillbehighlycorrelated. Thetypeofconfounding caused speciﬁcally by nondimensionalization is, however, the cause of confounding between We`0andRe`0. Avoiding confounding Confoundingingeneralisbestavoidedbycovering the relevant parameter spaces relatively completely. This could be accomplished through factorial experimental de- signs. Factorialexperimentsappeartoberareinturbulent jet breakup; I am aware of only the studies of Ruiz and Chigier [90–92]. The experimenter is also required to not missanyimportantvariables. Non-experimentalistsare limited by existing experiments in this regard. For turbu- lent jet breakup, existing data can avoid confounding only forthebreakuplength,becausetheparameterspacesare widely sampledenough inthat case. To detectconfound- ing,checkparameterspaceplots(e.g.,ﬁgures3and4)for correlations between diﬀerent variables. If these are seen, then cover the parameter space more comprehensively by changing the experimental conditions or looking for new data in diﬀerent parts of the parameter space. As previously mentioned for the We`0andRe`0case, this may require changing the nozzle diameter and ﬂuid. Consequences of confounding The potential consequences of confounding can be seen in tables 2 and 3. These tables show the results of regression analyses of breakup length data under various conditions. As previously discussed, table 2 shows the eﬀect of using diﬀerent variables in a regression analysis. Table3showstheeﬀectofusingdiﬀerentvariablesina regression analysis of confounded data . Due to the con- founding between We`0andRe`0, and also Re`0andTu0, is is impossible to say whether the observed trends are duetochanges inanyofthosethreevariables inthecon- foundedcase. Indeed,the R2valuesforallfourconditions considered are essentially 1in this case. Confounding between the Reynolds number and turbulence intensity occurs in most turbulent jet breakup experiments. And confoundingbetweentheWeberandReynoldsnumbers is not uncommon either. As can be seen in ﬁgure 3, only 2experimental studies (Grant and Middleman [15] and Kusui[17])outof 5consideredavoidedconfoundingby using diﬀerent nozzle diameters and/or ﬂuids to cover the parameter space more widely. Compiling data in this case helped avoid confounding, but just adding data is not a solution to confounding. The data must avoid a correlation between the two variables of interest to avoid confounding. Confoundingmayhelpexplainwhysomanyregres- sions in turbulent jet breakup seem to be contradictory 12. 12ThecontradictionslikelycanbepartlyexplainedbydiﬀerencesinFor the breakup length, many diﬀerent functional de- pendencies have been used. A small sample can give the reader an idea of the variety. Miesse [93, p. 1698] proposed hxbi d0=538We0:5 `0Re0:625 `0: (3) Grant and Middleman [15, p. 184] suggested hxbi d0=8:51We0:32 `0: (4) Wu and Faeth [47, p. 2917L] suggested a similar form usingadatacompilationincludingdatafromGrantand Middleman. Shavlovsky [94, p. A6-82] oﬀered hxbi d0=A68106Re`0; (5) where A85to112. Finally,DeJarlaisetal.[35,p.87R] obtained the best ﬁt equation hxbi d0=480We0:5 `0Re0:53 `0: (6) Some researchers try both Weber and Reynolds numbers, whilesomepreferjustoneofeither. Itispossiblethateach oftheseexpressionsdoesinfactﬁtthesourcedatawell, butconfoundingmakestheprecisefunctionaldependency morediﬃculttoidentify. Consideringconfounding,the data most closely matches the general form ﬁrst proposed by Grant and Middleman [15], albeit with a turbulence intensity modiﬁcation. The data presented here does not clearly eliminate aReynoldsnumberdependence,rather,itmerelyshows that any Reynolds number dependence on the breakup length in the second wind-induced regime is weak. If I assume that the experimental data has no uncertainty and neglect the (presumably small) eﬀects of confounding, thenusingthestandarderrorforthecoeﬃcientinthecase whereallthreevariablesareconsidered(table2)Iﬁndthat CRe`0=0:013770:0003(95% interval). Thisdoesnot overlapwithzero,thoughitmightiftheuncertaintyinthe experimentaldataisconsidered. Futureworkwillexamine if there still is some confounding between Re`0andTu0 which might make CRe`0statistically indistinguishable from zero. Integration tests are not suﬃcient Another common problem is the use of easily mea- suredquantitieswhichincludedropletbreakup(secondary breakup), droplet coalescence, and droplet transport to “validate”modelswhichonlypredictprimary 13breakup quantities like the droplet diameter at formation. For simplicityI’llcallmeasurementswhichincludesecondary regimes as well. 13“Primary” breakup and “jet” breakup are identical. 8 breakup,dropletcoalescence,anddroplettransport“far- ﬁeld” measurements. Kastengren et al. [95, p. 132L] rightfullynotethat“simplycomparingmodeledtomea- sured droplet size in the far-ﬁeld is insuﬃcient to validate the physical breakup model; data in the “near-ﬁeld” are needed, since this is the region in which primary breakup actuallyoccurs.” MagnottiandGenzale[3,p.34L]have alsoexpressedskepticismabouttheusefulnessoffar-ﬁeld measurements. Convertingbetweennear-ﬁeldandfar-ﬁeldquantities nowrequiresadditionaldropletbreakup,coalescence,and transport models. As such, this attempt at validation doesnottesttheprimarybreakupmodeldirectly. Nowa validationfailurecouldmeanafailureofeitherthedroplet breakupmodel,thedropletcoalescencemodel,thedroplet transport model, the primary breakup model, or any com- bination of the four. The result is ambiguous, making the models diﬃcult to falsify. And a validation success does notnecessarilymeanthattheprimarybreakupmodelis correct, as the droplet breakup and coalescence models couldhypotheticallycompensateforproblemsinthepri- marybreakupmodelinawaywhichmakeseachmodel wrong whentaken inisolation. This possibilitybecomes muchmorelikelywhenoneconsidersthatthesemodels are usually tuned to the data. Forthisreason,itisstronglypreferredtovalidateeach model individually (like a “unit test”) in addition to the “integration test” for all of the submodels in combination. In software testing, a unit test tests a speciﬁc part of a software. An integration test tests the combination of the parts of the software. The same terminology can be applied to testing models in isolation vs. testing the largercollectionofmodels. Similarterminologyhasbeen adoptedinmodelvalidationpreviously[96,pp.37L-38R]. Inthedropletdiametermeasurementcasethereisone additional subtlety that is often missed in the literature. It isnotstrictlycorrecttomerelycomparepredictionsofpri- mary breakup droplet diameters to droplet measurements inthenear-ﬁeld,thatis,dropletmeasurementsataparticu- lar location. One must measure only droplets which were just formed from the jet, that is, droplets formed through primary breakup without any other inﬂuences. I’ll call these “just-formed” droplets for brevity. The measure- mentsoftheFaethgroup(e.g.Wuetal.[42],WuandFaeth [43], and Wu et al. [97]) are the only which I am aware of forjust-formeddroplets. Thesemeasurementsaremore diﬃcult and limited than either near-ﬁeld or far-ﬁeld mea- surements,astheyrequireanalyzingmanyphotographs of the breakup process to select only just-formed droplets. Photography is unfortunately limited in the near-ﬁeld due tothedensityofthesprayinmanysituations. Fortunately, new DNS data analysis techniques are being developed toobtainsizedistributionsforjust-formeddroplets[98], however, this processed DNS data is not available as ofthis writing. Itispossiblethatthedistributionofnear-ﬁelddroplets is similar to the distribution of just-formed droplets. If true, this would greatly simplify experimentation while maintaining rigor. To my knowledge, this hypothesis has yet to be validated. The easiest way to test the hypothesis would be to measure droplet diameters in the near-ﬁeld in a casedirectly comparableto data collectedby theFaeth group. Several sources of data were neglected in my data compilation because only far-ﬁeld quantities were mea- sured. Of the 7pipe jet studies with droplet diameter measurements, only 2had droplets in the near-ﬁeld. Both weremeasurementsofjust-formeddroplets. Thenon-pipe data of Bogdanovich [76] and Dumouchel et al. [99] in- cludesturbulentkineticenergyatthenozzleexit,however, the droplet diameter measurements are in the far-ﬁeld, making these data sources less attractive for validation. The insuﬃciency of integration tests also was pre- viously mentioned in the case of the combined nozzle turbulenceandjetbreakupmodelsofHuhetal.[82]. The nozzle turbulence model is in severe error, making one of the inputs of the spray model far oﬀ where it should be, likely making the model poorly calibrated for the true turbulence intensity despite the apparent validation of the model. Other common model validation problems Apples-to-oranges comparisons Strictlyspeaking,itisincorrecttonaivelycombine data from nozzles of diﬀerent geometries, but this is still frequentlydoneinliquidjetbreakupresearch. Ifnotall theimportantvariablesarequantiﬁed,toproperlycombine dataa researcherneedstobe conﬁdentthatthevariables whicharenotquantiﬁeddonotvarymuch,andassuchcan not have a major inﬂuence on the results. This is diﬃcult todoinatomizationresearchingeneral. Assumingthat onehastheturbulenceintensity,forexample,thatisnot suﬃcient to fully characterize the turbulence. One would need at the very least some measure of the integral scales. However, integral scale measurements are even more rare than turbulence intensity measurements. Consequently, whencompilingdata,forthemomentitwouldbeuseful to know that the integral scales are roughly ﬁxed and consequently will not be aﬀecting the results appreciably. This may be true for pipe jets and is another reason to preferpipesasnozzles. Inanearlierpaper[52]Icompared aregressionagainstdatafromnon-pipenozzlesandfound deviations. The prime suspects for the error are of course variableswhichIhavenoestimatesfor,suchastheintegral scales. To give some examples of this problem, consider the breakuplengthmodelofGorokhovski[100]in2001. The breakuplengthisplottedasafunctionofthesquarerootof 9 the density ratiop `gin ﬁgure 6. Gorokhovski would have had to selected a Weber and Reynolds number for thisplotbasedontheirmodel(ifnotothervariableslike turbulence intensity not considered in the model). This selection was not discussed in the paper, and it seems unlikely that the Weber and Reynolds numbers match the experimental data cited (Lee and Spencer [101] and Hoyt and Taylor [102]). Consequently, this appears to be an apples-to-oranges comparison. The problem is actually worse than it appears at ﬁrst glance, as neither Lee and Spencer[101]orHoytandTaylor[102]reportwhatismost commonly known as breakup length. Their studies are photographicanddonotreportenoughphotostoallowfor estimationofthebreakuplength. Gorokhovski’smodel computesthebreakuplength,butitisnotbeingcompared againstthebreakuplength. And,ﬁnally,theﬁtbetween themodelanddataisnotparticularlygood. PossiblyHoyt andTaylor’ssingledatapointﬁtspoorlyduetotheapples- to-oranges comparison. Gorokhovski does not discuss thisdiscrepancy,butcharacterizestheﬁtas“satisfactory”, thoughthelackofuncertaintyquantiﬁcationmakeshow adequate the ﬁt is diﬃcult to determine. This problem was not isolated to Gorokhovski’s model. Over a decade later in 2017, Movaghar et al. [103]usedaverysimilarplottovalidatetheirturbulent jet breakup model. See ﬁgure 7. Movaghar et al.’s model was calibrated on pipe nozzles, so it seems inappropriate forthemodeltomatchdatafromdiﬀerentnozzleswithout mention of changing the turbulence intensity, which is a factor in Movaghar et al.’s model. This second ex- ample shows that these poor validation practices have been accepted as suﬃcient for the past 20 years in several well-regardedjournals,AtomizationandSpraysandthe International Journal of Multiphase Flow. Validationagainstonlyasmallamountofdatawhenmore is available TheexamplesfromGorokhovski[100]andMovaghar et al. [103] demonstrate another common issue: Valida- tionagainstasmallamountofdata. Gorokhovski[100] compared against a total of 5data points. For the density ratio eﬀect, Movaghar et al. [103] used only 4of the previousdatapoints. Fortunately,Movagharetal.[103] usedthedatacompilationofWuandFaeth[47]forvali- dation oftheir breakup lengthmodel at low atmospheric densities (high `g). Aside from the Faeth group, data compilation is rare. As previously mentioned, turbulence intensityis fre- quently neglected. Comparison against data where the turbulence intensity varies over the range expected in practice is necessary to validate turbulent jet breakup models. Aside from my CDRSV model [52], Skrebkov’s model [12]’s, and Bogdanovich’s scaling model [76], I am not aware of comparison of a model against data withvarying turbulence intensity 14. Moredata,aslongasitdoesnotduplicateexistingdata orcontributetoconfounding,typicallychallengesamodel. Jet breakup has been studied for over a century. While many of the early experimental studies are of poor quality bymodernstandards,studiesfromthe1950sthroughnow aregenerallyworthconsideration. Validationisbestdone withasmuchdataaspossibletoidentifyasmanyfaults with a model as possible. Not measuring the most typical quantity of interest Considering onlypipe jets, 4studies wereneglected because the QoIs were not measured at all [8, 11, 14, 27]. These studies tended to be old and were often qualitative (e.g. Hoyt and Taylor [27] or focused on diﬀerent QoIs than I do. Consideringnon-pipejets,anotherproblemappeared. Some researchers used an alternative QoI which is analo- goustobutnotthesameasanothercommonQoI.Asan example, the DNS study of Salvador et al. [104, ﬁg. 7] has a plot of axial mass concentration. It is reasonable to believe this is analogous to the breakup length, but the two are not directly comparable. There is no reason why thetime-averagedor50thpercentilebreakuplengthcould not have been measured from the same DNS data that produced the axial mass concentration plot. Theseissueswouldbereducedbyadoptionofstan- dardquantitiesofinterest. InthispaperIfocusonthemost commonlymeasuredquantitiesofinterest(seeﬁgure1), which I believe are physically meaningful and useful. Data presentation issues Ambiguous data 2studies were neglected due to the data being pre- sented ambiguously [7, 20]. For example, to my knowl- edge it is not possible to determine the Reynolds number, Weber number, etc. from the regime data presented by the study of Kusui [17] 15. This is because the data was onlyplottedincoordinatesthatdidnotallowtheReynolds number and Weber number to be determined indepen- dently. This problem is likely to persist due to space consid- erationsinjournals. However,inthefutureitisstrongly recommended to include tabulated raw data (in primi- tive variables, e.g., U0,d0, and`rather than Re`0) in reportsanddissertations. Postingrawdataonlineisalso highly recommended in addition. To increase the prob- abilityofthedatabeingavailableinthefuture,bothare recommended. To identify whether something is ambiguous in the 14TheregressionmodelofDumoucheletal.[99]isworthmentioning at this point, but it is not a phenomenological model like the others mentioned. 15Note that as I was able to obtain some data from Kusui [17], this study was not included among the two neglected. 10 data,itisrecommendedthatsomeoneelsereproducesome basic plots of the data given the tabulated or digitized version. Ona relatednote, 2studieswere neglectedfrom the datacompilationbecausetheypresentedcurvesforexperi- mentaldataratherthandatapoints[10,36]. Thesestudies wereneglectedbecauseitisimpossibletodeterminewhich precise points were tested from a curve. Data neglected due to inconsistency with others A few data sources were neglected due to inconsis- tency with other data sources deemed reliable. Breakup onset location ( hxii) measurements from Eisenklam and Hooper [9] and Reitz [26] were neglected becausetheywereinconsistentwithothermeasurements 16. Possiblythisisduetotheuseofsmallsamplesizes,making the error in the mean very large. Breakuplengthmeasurements ( hxbi) fromMansour and Chigier [105, p. 595, ﬁg. 4] diﬀered from others and were neglected. This may be due to a diﬀerence in how Mansour and Chigier deﬁned the breakup length, as Mansour and Chigier used a deﬁnition based on the wavelength of disturbances [105, p. 594]. Breakup length measurements from Phinney [19, p.695,ﬁg.3]werefoundtobeinconsistentwithothersand were neglected. However, the breakup length probability distribution from Phinney [19, p. 698, ﬁg. 6] is consistent with other data. Only Phinney’s ﬁgure 3 appears to be in error. Private communication [106] with the author and the fact that the author’s other data is consistent with otherresearchersindicatesthislikelywasadatareduction error limited solely to ﬁgure 3 which does not indicate unreliability of any of the author’s other data. On this note,thedeﬁnitionofthestabilityparameterinPhinney [19,p.692,ﬁg.2]hasatypographicalerror;thecorrect deﬁnition is given on page 690 in the text, veriﬁed by computing the stability parameter for data from Chen andDavis[13]andcomparingagainsttheplot. Another potentially related problem is that the surface tension for ﬂuid II appears to be much lower than seems possible forsaltwater. FortunatelyﬂuidIIwasnotusedforﬁg.6 of Phinney [19, p. 698], which may help explain why thatplotappearsconsistentwithotherdatabuttheother breakup length plot does not. Foreign language 3studies wereneglected becausethey were written in a foreign language and did not appear to contain useful data [5, 24, 28]. While foreign language does not disqual- ify a study 17, if a foreign language study does not appear to contain valuable data from a superﬁcial examination, I 16BecausethesprayanglefromReitz[26]wasused,thisstudydoes not count towards the count of inconsistent studies. 17I published English translations of Russian papers by Lebedev [71] and Natanzon [77], so the language barrier is not impenetrable.didnotdeemitimportantenoughtoexaminefurther. Itis possible that these studies do contain useful data that is obscured by the language barrier. Conclusions and recommendations Improving the validation of turbulent jet breakup models requires not only changes to how models are validated but also new experimental data that is more challenging for model validation. Consequently recom- mendationsaremadeforbothmodelersandexperimental- ists/computationalists. Theseguidelinesaredesignedto address issues speciﬁcally in turbulent jet breakup, and they complement existing validation guidelines [96]. Recommendations for model developers 1.Compile as much data as possible. Actively look for data thatﬁllsin gapsin yourparameter spaces(e.g., ﬁgure 3) and avoids confounding. 2.Consider the uncertainty of the source data. If necessary, neglect data which is too uncertain. 3.Consider turbulence intensity in new models, as it is an important variable in turbulent jet breakup that is frequently neglected. 4.Forprimarybreakupdropletdiameterandvelocity models,itisnecessarytocompareagainstprimary breakup data, i.e.,data on the diameterand velocity of droplets just-formed from the jet. Far-ﬁeld mea- surementsarediscouraged. Near-ﬁeldmeasurements may be acceptable, but they need to be compared against just-formed measurements to check that they are similar. 5.Match all variables ( We`0,Re`0,Tu0, etc.) when comparing model estimates to data. And ensure that the model estimate is for the same quantity as the data. Otherwise the comparisons are invalid; they would be apples-to-oranges comparisons. Recommendations for experimentalists and computation- alists 1. Estimate uncertainty for every measurement. 2.Measure or estimate turbulence intensity at the very least,ifnototherturbulencequantities(integralscale, Reynolds stress). 3.If turbulence intensity will not be measured, use a standardized setup where the turbulence intensity can be credibly estimated. “Pipe” nozzles which producefullydevelopedturbulentﬂowareoneway todothis. Ifthepipesareroughenedandthefriction factor of the pipe is measured, then the turbulence intensity can be credibly estimated [52]. 11 4.Coverthe We`0-Re`0andRe`0-Tu0parameterspaces well enough to avoid confounding. Don’t use nozzle oriﬁcelengthasaproxyforturbulencelevel orthe velocity proﬁle. 5.Measurecommonquantitiesofinterest: dropletdi- ameter distribution f¹Dºor Sauter mean diameter D32, average droplet velocity at formation hvdi, av- eragebreakuponsetlocation hxii,averagebreakup lengthhxbi,andsprayangle i. Otherquantitiesof interest may be useful in speciﬁc applications. 6.Forquantitieswithoutclearstandarddeﬁnitions(e.g., the spray angle), deﬁne the quantity in a precise way. Preferably this deﬁnition does not require expensive equipment. Forthesprayanglethestandardcouldbe aspeciﬁcthresholdforphotographicormassfraction measurements. 7. Distinguish between quantities of interest which de- pend on only primary breakup and those which include other eﬀects. For testing primary breakup models, it is better to measure the diameters of droplets “just-formed” from the jet. If only near- ﬁeld measurements are possible, replicate one of the papers with just-formed droplets and compare the near-ﬁeld and just-formed measurements. If the two areclose,thennear-ﬁeldisanacceptableproxyfor just-formed measurements. 8. Release raw data with each publication. Acknowledgments Prof. O. A. Ezekoye of the University of Texas at Austin is thanked for asking several times “How do we know that this is a function of the Weber number or the Reynoldsnumber?” Dr.WilliamRiderofSandiaNational Labs and Nick Brown of the University of Groningen are thanked for advice on how to best present this work.