Evaporation of dilute droplets in a turbulent jet: clustering and entrainment effects
EEvaporation of dilute droplets in a turbulent jet: clustering and entrainmenteffects
Federico Dalla Barba and Francesco Picano University of Padova, Department of Industrial Engineering, Via Venezia 1, 35131 Padova,Italy (Dated: 20 March 2017)
Droplet evaporation in turbulent sprays involves unsteady, multiscale and multiphase processes which makeits comprehension and model capabilities still limited. The present work aims to investigate droplet vapor-ization dynamics within a turbulent spatial developing jet in dilute, non-reacting conditions. We addressthe problem using a Direct Numerical Simulation of jet laden with acetone droplets using an hybrid Eule-rian/Lagrangian approach based on the point droplet approximation. A detailed statistical analysis of bothphases is presented. In particular, we show how crucial is the preferential sampling of the vapour phaseinduced by the inhomogeneous localization of the droplets through the flow. The preferential segregation ofdroplets develops suddenly downstream the inlet both within the turbulent core and in the mixing layer. Twodistinct mechanisms have been found to drive these phenomena, the inertial small-scale clustering in the jetcore and the intermittent dynamics of droplets across the turbulent/non-turbulent interface in the mixinglayer where dry air entrainment occurs. These phenomenologies strongly affect the overall vaporization pro-cess and lead to a spectacular widening of droplets size and vaporization rate distributions in the downstreamevolution of the turbulent spray.
I. INTRODUCTION
Turbulent sprays are complex multiphase flows involv-ing unsteady and multi-scale phenomena such as turbu-lence coupled with phase transition processes. The pres-ence of two distinguished phases which mutually interactexchanging mass, momentum and energy makes the de-scription of the problem extremely challenging. If com-bustion is considered, chemical reactions and heat releaseadd some clear complexities. In this scenario, a satis-factory comprehension of turbulent spray dynamics hasnot yet been achieved and existing models capabilitiesfor applications are still limited . Nevertheless, theprogress of the research in this field is crucial for severalindustrial applications as well by an environmental pointof view. A typical example can be found in the devel-opment of high efficiency and low emission internal com-bustion engines. In these applications the liquid fuel canbe directly injected into the combustion chamber wherethe vaporization of fuel droplets occurs together withchemical reactions within the turbulent gaseous environ-ment. The pollutants formation in turbulent spray com-bustion is related to complex multi-scale phenomenonsthat involve fluctuations of temperature and reactantsconcentrations. In particular, the soot formation occursthrough a pyrolysis process in fuel-rich regions that ex-perience high temperature without enough oxidizer to re-act . This can be observed within droplets clusters,where the concentration of fuel droplets can be even thou-sand times higher then its bulk value leading to a peak inthe fuel vapor concentration. Hence, in order to predictand model soot formation an improved understanding ofthe mechanisms that govern the distribution of dropletsand fuel/oxidizer mixture within a turbulent jet spray ismandatory.A phenomenological description of the overall evolu- tion of the spray dynamics can be found in the reviewof Jenny et al. . The process starts with the pri-mary atomization of a high velocity liquid jet. As theliquid flow is ejected from a duct into a gaseous envi-ronment, interface instabilities such as Kelvin-Helmholtzand Rayleigh-Taylor, fragmentize the jet into large dropsand ligaments . In the downstream evolution the liquidligaments and the drops are subjected to aerodynamicforces induced by the velocity difference at liquid/gas in-terface. The stresses induced by the aerodynamic forcesproduce a further brake-up of the liquid phase ( secondaryatomization ) giving origin to a system of small dropletsdispersed within the gaseous turbulent phase. The at-omization process occurs in a so called dense regime and terminate when surface tension prevails on aerody-namic stresses preventing further fragmentation. At thisstep a dilute regime establishes: droplets mutual interac-tions (e.g. collisions and coalescence) are negligible butthe effect of droplets on the carrier flow is still signif-icant . While in the dense regime the vaporizationrate is negligible, in dilute conditions the vaporizationprocess becomes significant. In this phase of the turbu-lent spray evolution, the most part of the liquid evapo-rates. Moreover, the small droplets evolve preserving aspherical shape due to the dominance of surface tensionon aerodynamic stresses. Even if the presence of dropletsexerts a significant effect on the flow in terms of mass,momentum and energy balance, at this step the dimen-sion of droplets is below or comparable to the smallestscales of the turbulent flow so point-droplet approxima-tion applies . Hence, in dilute conditions the mathe-matical description of droplet-laden flows lends itself par-ticularly well to an hybrid Eulerian/Lagrangian descrip-tion. The Navier-Stokes equations govern the continuousphase dynamics if distributed sink-source terms are con-sidered in order to represent the mass, momentum and a r X i v : . [ phy s i c s . f l u - dyn ] M a r energy exchange between the Eulerian carrier phase andLagrangian point-droplets.One of the most peculiar phenomenology that char-acterize dispersed multiphase turbulent flows in di-lute conditions is the preferential segregation of parti-cles/droplets as a result of the interaction of their inertiawith the carrier phase turbulent dynamics (see e.g. ).The mechanisms at the origin of preferential segregationin free flows have been widely investigated in literatureboth for solid particles and liquid droplets. The inten-sity of small scale segregation of solid particles in homo-geneous turbulence is found to be driven by the Stokesnumber St η = τ p /τ η , with τ p the particle relaxation timeand τ η the Kolmogorov time scale. More specifically, theintensity of small scale clustering is maximum when theparticle relaxation time is of the order of the Kolmogorovtime scale such that the Stokes number, St η (cid:39) .The same behavior is observed for evaporating dropletswhich behave as inertial particles . Concerning turbu-lent jets, a mean accumulation of the dispersed phasehas been observed at specific distances from the inflowboth experimentally and numerically . Even if pref-erential concentration of a dispersed phase has been wellcharacterized in homogeneous turbulence, the effect ofthis phenomenology on the overall vaporization processwithin turbulent jet sprays is still object of research andconstitute one of the main focuses of this paper.By a theoretical and numerical point of view one of thefirst description of the vaporization of spherical dropletsdragged by a gaseous phase flow was addressed in theseminal works of Spalding and Godsave . Fixingthe environmental vapor concentration, they found thatdroplets surface decreases linearly with time ( D law).Maxey & Riley, report an equation for the motionof a finite size spherical particle/droplets at low dropletReynolds number. The equation accounts for the Stokesdrag, added-mass effect and buoyancy force. Dealingwith a dispersed phase with a density much higher thanthat of the fluid the Stokes drag is sufficient to describeits dynamics . Abramzon & Sirignano proposedan improved model for droplets vaporization, consider-ing non-uniform and time-dependent environmental con-ditions, taking into account forced convection, moleculardiffusion and the Stefan flow contribution to the vaportransport from droplet surface to neighboring environ-ment. Even if several models can be found in liter-ature for the simulation of turbulent evaporating spraysin Reynolds Averaged Navier-Stokes (RANS) or Large-Eddy Simulation (LES) frameworks, these models lackin capabilities to accurately reproduce complex phenom-ena such as droplets small scale clustering . Despitethe highly demanding computational resources, the useof Direct Numerical Simulation (DNS) allows to cap-ture the whole physics of the spray vaporization pro-cess in order to understand the complex phenomenonsinvolved. In this context, Mashayek adopted an Eu-lerian/Lagrangian approach in order to perform a DNSof low Mach number, homogeneous shear turbulent flows laden with droplets . Miller & Bellan report a DNSof a confined three-dimensional, temporally developinggas mixing layer laden with evaporating hydrocarbondroplets at subsonic Mach number. Reveillon and co-workers studied the effect of preferential droplet ac-cumulation on the evaporation in isotropic turbulenceshowing that different regime takes place depending ondroplet concentration. Recently, Bukhvostova et al. consider the DNS of a turbulent channel flow of a mix-ture of air and water vapor laden with water droplets.The work focuses on the comparison between the per-formances of an incompressible and a low Mach numberasymptotic formulation in reproducing the flow dynam-ics. Even if the two formulations show good qualitativeagreement, the low Mach formulation is found to be cru-cial in order to obtain a reliable quantitative predictionsof heat and mass transfer.The prototypal flow for an evaporating spray is consti-tuted by a turbulent free jet which is characterized by theeffect of environmental gas entrainment. In more details,a turbulent jet is constituted by a rotational turbulentcore which is continuously entrained by the surroundingirrotational fluid . In sprays, the entrained dry flowdilutes the vapor concentration and controls the vapor-ization process. This phenomenology was found to beof critical importance also in natural phenomena. Oneexample consists in the effect of entrainment in stratocu-mulus clouds where it constitutes a driving parametersin the determination of cloud lifetimes and in turns evenregulating planetary-scale properties . The fast growrate of the droplet size spectrum in warm cloud is a chal-lenging, still not understood, problem in cloud physics , despite its importance in determining raining con-ditions.To the best of authors knowledge, a fundamental studyon the effects of the entrainment in an evaporating tur-bulent jet spray together with an analysis of the prefer-ential segregation effect is still missing. This work aimsto cover this lack considering DNS data of an evaporat-ing turbulent spray using a 2-way coupling approach be-tween the two phases and accounting for the entrainmenteffect. The numerical algorithm adopts a hybrid Eule-rian/Lagrangian approach and point droplets approxi-mation. In addition, the effect of density variation is ac-counted by a low Mach number formulation of the Navier-Stokes equations. A strong preferential segregation ofdroplets is observed over the whole downstream evolu-tion of the spray which induces a preferential samplingof vapor concentrated regions. Two different mechanismsare found to drive this process, the former is due to in-ertial clustering, the second is related to the dynamicsof the jet entrainment. This last mechanism is foundto be crucial in the outer part of the jet core where theevaporation peaks and strongly impacts the vaporizationdynamics which is characterized by a strong widening ofthe droplet size spectrum. II. NUMERICAL METHOD
In this paper we report a direct numerical simula-tion of a turbulent evaporating spray in an hybrid Eu-lerian/Lagrangian framework adopting the point dropletapproximation in the 2-way coupling conditions. Thegoverning equations for the Eulerian gaseous phase con-sist in a low Mach number formulation of the Navier-Stokes equations in an open environment where arbitrarydensity variations can be accounted neglecting acous-tics . Consistently with previous studies , the ef-fect of the dispersed phase on the gaseous phase is ac-counted by sink/source coupling terms appearing in theright hand side of the mass, momentum and energy equa-tions, S ρ , S m and S e (see ), ∂ρ∂t + ∇ · ( ρ u ) = S ρ (1) ∂∂t ( ρY V ) + ∇ · ( ρY V u ) = ∇ · ( ρ D∇ Y V ) + S ρ (2) ∂∂t ( ρ u ) + ∇ · ( ρ u ⊗ u ) = ∇ · τττ − ∇ P + S m (3) ∇ · u = 1 p (cid:20) γ − γ ∇ · ( k ∇ T ) + S e (cid:21) (4) p = ρ R G (1 + M Y V ) T (5)where ρ , u , p , P and T are respectively the density,the velocity, the thermodynamic pressure, the hydrody-namic pressure and the temperature of the carrier va-por/gas mixture that will be denoted in the following asthe carrier phase. The vapor mass fraction, Y V , is definedas the ratio of the vapor partial density and the totaldensity of the carrier phase, Y V = ρ V /ρ . The vapor/gasbinary diffusion coefficient, the thermal conductivity ofthe mixture and the specific constant of gas are denotedrespectively as D , k and R G . The parameter M is definedas M = W G /W L − W G and W L the molar weightof the gas and liquid phases. The ratio of the constantpressure coefficient, C P , and the constant volume coeffi-cient, C V , of the mixture is denoted by γ . The viscousstress tensor is τττ = 2 µ ( ∇ u + ∇ u T ) − µ b / ∇ · uI , with µ and µ b the dynamic and bulk viscosities. It should be re-marked that the thermodynamic pressure p is constantin space, due to the low-Mach number asymptotic expan-sion and in time, due to the open space conditions.Droplets are treated as rigid evaporating spheres andthe liquid phase properties (e.g. temperature) are as-sumed to be uniform inside the droplets. Droplet ro-tation, distortion and mutual interactions (e.g. collision,coalescence) are neglected considering dilute volume frac-tions. d u d dt = u − u d τ d (6) dm d dt = − m d τ d ShSc ln (1 + B m ) (7) dT d dt = 13 τ d (cid:18) N uP r C
P,G C L ( T − T d ) − ShSc L V C L ln (1 + B m ) (cid:19) (8)where u d , m d and T d are droplet velocity, mass andtemperature, C P,G and C L are the constant pressurecoefficient of the gas and the liquid specific heat and τ d = 2 ρ L r d / (9 µ ) is the droplet relaxation time, with ρ L the liquid phase density. The mass diffusivity and thethermal conductivity are accounted through the Schmidtand Prandtl numbers respectively, Sc = µ/ ( ρ D ) and P r = µ/ ( C p k ). The Nusselt number, N u , and the Sher-wood number, Sh , are estimated as a function of thedroplets Reynolds number, Re d = ρ || uuu − uuu d || /µ , accord-ing to the Fr¨ossling correlation: N u = 2 + 0 . Re d P r (9) Sh = 2 + 0 . Re d P r (10)A correction is then applied to N u and Sh in orderto account for the Stefan flow : N u = 2 + (
N u − F T , F T = (1 + B t ) . B t ln (1 + B t )(11) Sh = 2 + ( Sh − F M , F M = (1 + B m ) . B m ln (1 + B m )(12)The parameter B m and B t are the Spalding mass andheat transfer number respectively, the former being thedriven parameter for the vaporization rate, B m = ( Y V,s − Y V )(1 − Y V,s ) (13) B t = C P,V L V ( T − T d ) (14)where C P,V is the constant pressure coefficient of thevapor, Y V is the vapor mass fraction in the carrier phaseevaluated at droplet center and Y V,s is the vapor massfraction evaluated at droplet surface. This latter corre-sponds to mass fraction of vapor in a saturated vapor/gasmixture at droplet temperature. In order to estimate Y V,s we assume the equilibrium hypothesis such that theClausius-Clapeyron relation applies: χ V,s = p ref p exp (cid:20) L V R V (cid:0) T ref − T d (cid:1)(cid:21) (15)with χ V,s the vapor molar fraction at droplet surface,p the thermodynamic pressure, p ref and T ref arbitraryreference pressure and temperature and R V the vaporgas constant. The saturated vapor mass fraction is then: Y V,s = χ V,s χ V,s + (1 − χ V,s ) W g W L (16)We have performed the Direct Numerical Simulationof the evaporation of liquid acetone droplets dispersedwithin a turbulent air/acetone vapor mixture. The nu-merical code is constitute by two different modules. AnEulerian algorithm directly evolves the gaseous phase dy-namics solving the Low-Mach number formulation of theNavier-Stokes equations (1)-(5) (see e.g. and refer-ences therein for validation and tests). A second ordercentral finite differences scheme is adopted on the stag-gered grid for space discretization, while temporal evolu-tion is performed by a low-storage third order Runge-Kutta scheme. A Lagrangian solver evolves dropletsmass, momentum and temperature laws (6)-(8). Thetemporal integration uses the same Runge-Kutta schemeof the Eulerian phase and second-order accurate polyno-mial interpolations are used to calculate Eulerian quan-tities at droplet positions. We have preliminarily testedthe evaporation dynamics in the numerical code in twodifferent test cases. The former case concerns a liquid wa-ter droplet carried by a laminar dry jet. In this extremelydilute conditions the Spalding D law is a valid analyt-ical solution for the droplet radius evolution over time(Fig. 1(a)). In the latter case, a water droplets freelyfalling in wet air is considered and the numerical solu-tion for the droplet temperature evolution is comparedto an experimental dataset (Fig. 1(b)).The computational domain consists of a cylinder ex-tending for 2 π × R × R in the azimuthal, θ , radial, r and axial, z , directions. The domain has been dis-cretized by N θ × N r × N z = 128 × ×
640 points usinga non-equispaced, staggered mesh in the radial and ax-ial direction. The mesh has been stretched in order tobe of the order of the Kolmogorov length in the down-stream evolution. The flow is injected at the center ofone base of the cylindrical domain and streams out at theother end. Time-dependent inflow boundary conditionsare prescribed. A fully turbulent velocity is assigned atthe jet inflow section (Dirichlet condition) by means of across-sectional slice of a fully developed companion pipeflow DNS. The flow is injected through a center orificewhile the remaining part of the domain base is imper-meable and adiabatic. A convective condition is adoptedat the outlet and an adiabatic traction-free condition isprescribed at the side boundary. This side boundary con-dition makes the entrainment of external fluid possiblewhich in the present case is dry air. The gas/vapor mix-ture is injected at a bulk velocity U = 9 m/s througha nozzle of radius R = 5 10 − m . The ambient pres-sure is set to p = 101300 P a while the initial tempera-ture is fixed to T = 275 . K . The injection flow rate r d / r d , t / t SimulationD Law (a) T d / T d , t / τ t SimulationExperiments (b)
FIG. 1. (a) Evolution over time of the radius of a single waterdroplet in a dry air laminar jet. The ambient pressure andtemperature are set to p = 101300 P a and T = 273 . K . Theinlet radius and the bulk velocity of the jet are respectively R = 5 10 − m and U = 1 . m/s . The droplet is injected atthe local carrier phase velocity. The initial droplet radius isset to r d, = 5 µm and the temperature to T d, = 273 . K .In the figure, the droplet radius and the time are scaled re-spectively by the droplet initial radius and the reference timescale t = R/U . The continuous line represents the analyticalsolution computed by the Spalding D law, r d = (cid:113) r d, − kt ,where k = 2 ρ D ln (1 + B m ) /ρ L . Dots represents the numer-ical results. (b) Temperature evolution over time of a waterdroplet freely falling in air at pressure p = 101300 P a , temper-ature T = 301 . K and relative humidity χ = 0 .
22. Droplet isinitially at rest with an initial temperature, T d, , equal to theenvironmental air one. In the figure the temperature is scaledby the initial droplet temperature, T d, , while time is scaled bydroplet thermal relaxation time, τ t . This latter is defined asthe time required by droplet temperature to change by the 63% of its total change between initial temperature and regimetemperature. The continuous line represents the result of thesimulation while dots report an experimental dataset . Theregime temperature of droplet corresponds to the ventilatedwet bulb temperature at prescribed environmental pressureand actual temperature, which is T wb = 288 . K . of the gas is kept constant fixing a bulk Reynolds num-ber Re = 2 U R/ν = 6000, with ν = 1 . − m /s thekinematic viscosity. At the inflow section a near satu-ration condition is prescribed for the air/acetone vapormixture, S = Y V /Y V,s = 0 .
99, with Y V,s the vapor massfraction saturation level at the actual temperature. Theacetone mass flow rate is set by the mass flow rate ratioΦ m = ˙ m act / ˙ m air = 0 .
23, with ˙ m act = ˙ m act,L + ˙ m act,V the sum of liquid and gaseous acetone mass flow rates and˙ m air the gaseous one. Liquid monodisperse droplets withradius r d, = 6 µm are injected within the saturated vaporcarrier phase. All the droplet characteristics have beenchosen to reproduce acetone liquid. The injected dropletsare distributed randomly over the inflow section with ini-tial velocity equals to the local turbulent gas phase ve-locity. Before the injection of droplets, the simulation isstarted considering only the single-phase flow until sta- (a) (b) FIG. 2. (a) A radial-axial slice of the turbulent spray. Theblack points represent a subset of droplets formed by 1/5 ofthe whole population. Only droplets located inside a slice ofwidth w/R = 0 .
01 are visible. Each point size is proportionalto the corresponding droplet radius (scale factor 100). Thecarrier phase is contoured according to the instantaneous va-por mass fraction field, Y V , which is bounded between 0 and0 .
18, the former corresponding to the dry condition and thelatter to the 99% saturation level prescribed at inlet. (b) En-largements of two different jet regions centered at z/R = 10(lower panel) and z/R = 20 (upper panel). tistical steady conditions have been attained. Then thesimulation is run for about 200
R/U time scales in orderto reach a statistical steady condition for the two-phaseevaporating flow before to collect the dataset. The statis-tics considers around one hundred samples separated intime by R/U = 1. III. RESULTS AND DISCUSSIONS
A general overview of the instantaneous vapor massfraction field and droplet population distribution is pro-vided in Fig. 2. Droplets populate only the turbu-lent jet core, while are not present in the outer dry re-gion. The distribution of droplets is strongly inhomo-geneous and clustering is also apparent. In particular,droplets seem to preferentially segregate in regions char-acterized by high vapor concentration while only few iso-lated droplets can be found in poorly saturated areas (seethe enlargements shown in Fig. 2(b)). There are severalmechanisms driving droplets preferential concentrationin a turbulent flow, the most relevant of them being thesmall-scale clustering , droplets accumulation along jet axis and as we will show the droplet dynamicsin the mixing layer. The small scale clustering concernsthe interaction of droplets distribution with the smalleststructures of turbulence that promotes the segregation ifthe dissipative flow time scale is of the same order of theparticle/droplet (inertial) time scale. In turbulent jets orsprays, a mean accumulation of the dispersed phase hasbeen observed at specific distances from the inflow. Thislocation is determined by the matching of particle timescale and the local large scale jet time which quadrati-cally decreases with the downstream axial distance .Independently by the mechanisms driving droplets seg-regation, the vapor mass fraction increases rapidly insideclusters of evaporating droplets due to their high concen-tration. As the vapor concentration increases, the localevaporation rate is reduced. The vaporization processmay be even completely blocked if the vapor concentra-tion reach the saturation level, Y V,s . In this case a nonevaporating, fully saturated core appears inside the clus-ter . It is then clear how the clustering phenomenoncan strongly affect the overall vaporization process (e.g.evaporation length) by reducing locally the rate of va-porization.The Fig. 3(a) provides the average distribution of liq-uid mass fraction over the spray, Ψ = m L /m G , where m L is the mean mass of liquid acetone and m G is themean mass of the gaseous carrier phase inside an arbi-trary small control volume, ∆ V . The overall vaporiza-tion length can be defined as the axial distance from inletwhere the 99% of the injected liquid mass has transit tovapor phase. According to this definition, the vaporiza-tion process is terminated at about z/R (cid:39)
43. It shouldbe noted that the liquid phase mass fraction is signifi-cantly higher in the spray core then in the outer layer,such that the turbulent gaseous phase dynamics is espe-cially affected by the presence of the dispersed phase inthis region. This is consistent with the distribution ofthe average saturation field, S , reported in Fig. 3(b).We note an almost-saturated flow near the inlet causedby the prescribed inflow conditions, then the saturationlevel gently decreases in the downstream evolution whilesharply towards the outer jet region. The turbulent sprayis constituted by a spreading and slowly decaying turbu-lent core which is surrounded by the dry and irrotationalenvironmental air. The turbulent core is continuouslyentrained by the environmental air which mixes up withthe turbulent air/vapor mixture thus reducing the vaporconcentration. Since the inner core fluid cannot reach theouter region, the spray core shows higher saturation levelover the whole downstream evolution of the flow. Theeffect of dry air entrainment is crucial on the overall va-porization process. The dilution of vapor concentrationis indeed fundamental in order to allow the vaporizationprocess to advance.The average droplets radius and evaporation rate dis-tributions are reported respectively in Fig. 4(a) and Fig.4(b). According to the discussed entrainment effect, thevaporization rate is maximum in the mixing layer sep- (a) (b) FIG. 3. (a) Average liquid phase mass fraction, Ψ = m L /m gas where m L is the mean mass of liquid acetone and m G is the mean mass of the gaseous carrier phase.The la-bels show different distances from the jet inlet, z/R , in cor-respondence of which the 50%, 90%, 99% and 99.9% of theinjected liquid mass is evaporated. (b) Average saturationfield, S = Y V /Y V,s , where Y V is the actual vapor mass frac-tion field and Y V,s = Y V,s ( p, T ) is the value of vapor massfraction corresponding to the local saturation condition. arating the jet outer and core regions. The peak valueis reached in the shear layer immediately downstreamthe inflow section, where large droplets enter in directcontact with the dry environmental air. Consequently,at each axial distance form inlet, larger droplets can befound in the spray core where the vaporization processis slowed down by the high vapor concentration, whilesmaller droplets can be found towards the outer regionwhere the vaporization proceeds faster.The transition of liquid phase to vapor phase requiresan amount of energy per unit mass equal to the latentheat of vaporization of acetone so that the power requiredby the vaporization is proportional to the evaporationrate. The overall energy required by the vaporizationprocess is provided by the internal energy of both thegaseous carrier phase and the liquid dispersed phase, thusresulting in an overall cooling of the spray in the down-stream evolution. The average distribution of the carrierphase and the droplets temperature is reported in Fig. 5.In the outer spray region the smaller droplets surroundedby low-saturated gas are colder than core droplets dueto the higher evaporation rate. Nevertheless, the car-rier phase shows an opposite behavior: the spray coreis sensibly colder then the shear layer and the minimumgas temperature can be observed between z/R = 25 and z/R = 45. This phenomenon is due to the distribution ofthe liquid phase mass fraction. In the spray core the liq-uid mass represent a significant part of the overall spraymass. Hence, the cooling effect due to vaporization ismuch more intense in this region where a large amount (a) (b) FIG. 4. (a) Average droplet radius rescaled by the initialdroplets radius r d, . (b) Average droplet vaporization ratedivided by the reference scale defined as ˙ m d, = m d, /τ d, with m d, the initial droplet mass and τ d, the initial dropletrelaxation time. (a) (b) FIG. 5. (a) Average gas phase temperature, T , rescaled by theinjection temperature, T . (b) Average droplet temperature, T d rescaled by droplet initial temperature, T d, . The referencetemperature scales are equal T d = T d, . of droplets slowly evaporate.We have highlighted the existence of a strong preferen-tial segregation of droplets, focusing in particular on theeffect of this inhomogeneous distribution on the overallvaporization process dynamics. The intensity of dropletssegregation can be measured in different ways, e.g. . Wewill measure the intensity of clustering in each point ofthe inhomogeneous turbulent jet spray using the clus-tering index K , which is defined as K = ( δn ) n − n and ( δn ) are the mean and variance of thenumber of droplets in given small volume ∆ V . If dropletsare completely randomly located, their distribution is de-termined by a Poisson process in which mean and vari-ance coincide. Hence if clustering is not present and par-ticles are random distributed K = 0. On the opposite,if K > K computed over thewhole spray domain. The large positive value of K cor-responds to strong preferential segregation of droplets.We note that droplets are initially random distributedand then develop clustering. In particular, near the in-flow K assumes positive value only in the mixing layerwhere the local droplet concentration is intermittent be-cause of the fluctuation of the turbulent/non-turbulentinterface which separates the turbulent core populatedby droplets and the outer region without droplets. Itshould be noted that the air regions entrained from theenvironment in the core are almost droplet-free and en-hance the fluctuation level of the droplet concentrationeven in the jet core, see the snapshots reported in Fig 2.Downstream the clustering appears in the whole turbu-lent jet core. We attribute this phenomenon also to thedeveloping of small-scale turbulent clustering.The main mechanism driving the small scale clusteringrelies in the competition between inertia and Stokes drag.The drag tends to trail droplets according to the highlyconvoluted local turbulent structures while droplets fi-nite inertia prevents them to follow exactly the turbu-lent flow motion. By this mechanism droplets heavierthen the fluid tend to be ejected from vortex cores .The small-scale droplet distribution is governed by theStokes number, St η , which is defined as the ratio ofdroplets response time, τ d = 2 ρ L r d / (9 ρν ), and the char-acteristic time of the dissipative scales, τ η = ( ν/ε ) / .Droplets with St η (cid:29) St η (cid:28) St η ∼
1. The Fig. 6(b) provides theStokes number, St η , of droplets located within a radialdistance r/R = 0 . z/R (cid:39) (a) z/R S t η (b) FIG. 6. (a) Droplet clustering index, K . (b) Evolution ofthe mean droplets Stokes number, St η = τ d /τ η , based on theKolmogorov dissipative scales on the jet axis. rich of droplet and the outer region which is depleted ofdroplets.To quantify the importance of the droplet clustering inthe evaporation process we compare the mean vapor con-centration field felt by the droplets, Y V,dc , and the uncon-ditioned Eulerian one, Y V . Y V,dc is the vapor concentra-tion field obtained by a conditional average on the dropletpresence in a given point. Hence if droplets preferentiallyaccumulate in locations where the vapor concentration isrelatively high it results Y V,dc > Y V . Figure 7 reports theradial profiles of Y V,dc and Y V at different axial distances z/R .The vapor mass fraction felt by droplets is usuallyhigher then the correspondent unconditional value. At z/R = 10 the droplet conditioned and unconditioned va-por concentration are similar with the exception of theouter part. This behaviour is expected since we have ob-served the clustering to be small near the inflow with theexception of the mixing layer. The clustering associatedto the mixing layer, separating the outer and inner jetregions, will be discussed in details in the following. Athigher z/R the preferential sampling of the vapor phaseoperated by segregating droplet is significant with anoversampling of about 10 ÷
40% more the unconditionedvalue even in the inner jet core. To characterize dropletdynamics in the mixing layer, we need to discern betweenthe inner turbulent jet core and the irrotational outer re-gion. The two regions are separated by an almost sharpfluctuating layer, so-called turbulent/non-turbulent in-terface , that is highly convoluted over a wide range ofturbulent scales. The most used observable to character-ize the two regions is the local enstrophy, ζ = ||∇ × (cid:126)u || .The inner turbulent core is characterized by large fluctua-tions of enstrophy while in the outer region the enstrophyis null. Thus, fixing an entrophy threshold, ζ th , it is pos-sible to distinguish if a point is located into the turbulent r/R Y V , Y V , D S , Y V , E C V Y V,DS Y V,EC (a) r/R Y V , Y V , D S , Y V , E C V Y V,DS Y V,EC (b) r/R Y V , Y V , D S , Y V , E C V Y V,DS Y V,EC (c) r/R Y V , Y V , D S , Y V , E C V Y V,DS Y V,EC (d)
FIG. 7. The figure report the radial profiles of the averagevapor mass fraction field at four different axial distances fromthe origin: (a) z/R = 10, (b) z/R = 20, (c) z/R = 30 and(d) z/R = 40. Each plot shows the enstrophy-threshold con-ditional average, Y v,ec , the droplet-presence conditional av-erage, Y v,dc and the unconditional Eulerian one, Y v . Theenstrophy-threshold conditional average is calculated by sam-pling the vapor mass fraction only over turbulent core events( I = 1), that is when local enstrophy exceeds a fixed thresh-old. Y v,dc is the vapor concentration field obtained by a con-ditional average on the droplet presence in a given point. region or not: I ( (cid:126)x, t ) = H [ ζ ( (cid:126)x, t ) − ζ th ] (18)with H the heaviside function. I = 1 denotes a turbu-lent event, while I = 0 an irrotational one. The value of ζ th has been shown to play a weak influence . Using I we can define an enstrophy-threshold conditional averagefor the vapor concentration, Y V,ec , by sampling the vapormass fraction only over turbulent core events. Besidesthe unconditioned and the droplet conditioned statistics,Fig. 7 provides also Y V,ec which can be seen as the meanconcentration field of the turbulent core region. In themixing layer, this turbulent conditional average shows anexcellent agreement with the vapor concentration felt bythe droplets which is significantly higher than the un-conditioned value. Since the unconditioned value Y V isdetermined both by irrotational dry outer and turbulentvapor-concentrated events, the present analysis indicatesthat the droplet dynamics in the mixing layer is mainlydetermined by turbulent events. In other words, in themixing layer, droplets moving towards the outer regionfrom the inner turbulent core are surrounded by highly concentrated vapor gas ejected with the droplets. Onthe contrary, in a point of the mixing layer, a low vaporconcentration event is associated to an engulfment of en-trained dry air which is depleted of droplets. Becauseof this dynamics, in average, droplets evaporating in themixing layer do not feel the unconditioned mean vaporconcentration, but a higher level. At further downstreamdistance the effect of small-scale clustering previously dis-cussed adds its contribution to this dynamics.Hence, since droplet vaporization rate is driven by thevapor concentration sampled by droplets, the oversam-pling of the vapor concentration field slows down theoverall vaporization process, thus increasing the overallvaporization length and time. We find that droplets pref-erential sampling is the results of two different contribu-tions, originated by distinct mechanisms. The first mech-anism is induced by the fluctuation of the turbulent/non-turbulent interface in the mixing layer. In this areadroplets are entrapped in turbulent structures with highvapor concentration that protract into the irrotational,droplet-free ambient gas. The second contribution isgiven by the inertial small scale clustering. Clusters ofevaporating droplets move together with their own highlysaturated atmosphere induce an oversapling of the vaporconcentration with respect to the neighboring environ-ment. The contribution of this latter mechanism is moreevident in the spray core and tend to increase in inten-sity in the downstream evolution of the spray, while theformer is dominant in the mixing layer. A. Probability density function of droplet observables
In order to further characterize droplet vaporizationdynamics, we consider the probability density functionof the vaporization length and time computed over thewhole droplets population. In analogy with the overallvaporization length definition, one single droplet vapor-ization length can be defined as the axial distance frominlet, z e , necessary for the the droplets radius to decreasefrom r d, to a threshold radius r d,th = . r d, . The va-porization time t e is the corresponding amount of time.The PDFs for droplets evaporation length and time arereported in Fig. 8(a) and 8(b) respectively. The mean,standard deviation, skewness and kurtosis are reportedin table III A and show a nearly Gaussian behavior withsignificant standard deviations. The Gaussian behaviorin turbulent flows is usually associated to fluctuations in-duced by the large-scale motions. It is remarkable howdifferent are the histories of the droplets: half of the in-jected droplets is still present at about z/R (cid:39)
32 whereabout 90% of the injected liquid mass fraction is evapo-rated. This aspect is connected to the high polydispersitydeveloped by the droplets.Fig. 9(a) shows the probability density function of thedroplet radius at different axial distances from the ori-gin. Even starting from a monodisperse suspension, wesuddenly observe a radius distribution which spans for µ σ K Sz e /R t e /t µ , standard de-viation, σ , kurtosis, K and skewness, S , of the PDFsof droplets evaporation length and time with µ = E [ X ], σ = (cid:112) E [( X − µ ) ], K = E [( X − µ ) ] /E [( X − µ ) ] and S = E [( X − µ ) ] /E [( X − µ ) ] / . All variables are non-dimensional. P D F z e / R PDFGaussian (a) P D F t e / t PDFGaussian (b)
FIG. 8. (a) Probability density function of non-dimensionaldroplet vaporization length, z e /R , with R the jet inlet ra-dius. (b) Probability density function of non-dimensionaldroplet vaporization time, t e /t , with t the reference timescale, t = R/U b . The PDFs are computed over the entiredroplets population injected into the computational domain. around one decade after 10 jet radii from the inlet. Itshould be remarked that this quantity amounts in dif-ferences of droplet volumes of about 10 . This intensespread of the droplets size spectrum may be attributedto the complex preferential segregation dynamics whichhas been previously discussed. Indeed, the evolution ofdroplets, which is made up of aggregates of different sizeand dynamics, induces extremely different surrounding conditions on droplets themselves as can be observed bythe vapor saturation level felt by the droplets (see Fig.9(b)). Near the inlet at z/R = 0 .
25, droplets sample thealmost saturated vapor phase. From z/R = 10 the PDFshows a wide range of sampled value caused by the com-plex droplet dynamics previously discussed. We also ob-serve that few droplets show condensation as denoted bysaturation values above 1. Further downstream z/R = 20the spreading trend inverts and droplets are subjected toprogressively more uniform saturation levels. The differ-ent evaporation dynamics caused by the saturation levelfelt by the particles induces a similar statistical behav-ior for the temperature PDF, see Fig. 9(c). The highpolydispersity combined to the wide spectrum of satura-tion levels sampled by the droplets induces a non-trivialbehavior of the vaporization rate PDF. Even though wecannot provide arguments, we find that the PDFs of thevaporization rate, shown in Fig. 9(d), appears to follow a power-law with exponent about −
3, independently bythe axial distance from inlet.
IV. FINAL REMARKS
The dynamics of a turbulent evaporating spray is in-vestigated by means of a Direct Numerical Simulation.The simulation reproduces an acetone/air spray evolvingin an open environment considering dilute, non-reactingconditions and accounting for the full coupling betweenthe two phases due to mass, momentum and energy ex-change. The entrainment of external dry air is also ac-counted. Liquid acetone monodisperse droplets are con-tinuously injected within the turbulent gaseous phase ata bulk Reynolds number Re R = U R/ν = 6000. A com-plete description of both instantaneous and average fieldsof Eulerian and Lagrangian observables is provided. Thedistribution of droplets is strongly inhomogeneous withclustering apparent. In particular, droplets seem to pref-erentially persist in high vapor concentration regions thusbeing affected by a reduction of the vaporization rate.The intensity of the preferential segregation is estimatedby the evaluation of the clustering index. Preferentialsegregation develops downstream the jet inlet first in themixing layer and then in the turbulent core. In par-ticular, two different mechanisms driving the inhomoge-neous droplet distribution are identified: inertial smallscale clustering and droplet segregation induced by theturbulent/non-turbulent interface. The former one is theresults of the competition between inertia and Stokesdrag and is found to be responsible for droplets prefer-ential accumulation mainly in the spray core and in thefar field evolution of the flow. The latter one mainly af-fects the mixing layer and consists in the entrapment ofdroplets in turbulent structures with high vapor concen-tration which are originated in the core and protract to-wards the droplet-free dry environment. Simultaneously,droplet-free dry air regions are engulfed in the jet coreenhancing the fluctuation of the droplet concentration.Both these mechanisms affect droplets dynamics and re-sult in an oversampling of the vapor concentration expe-rienced by each droplet, hence affecting the overall vapor-ization length. Probability Density Function of dropletobservable have been reported at different axial distancesfrom the inlet. A spectacular increase of the droplet poly-dispersity is found to arise in the downstream evolution ofthe spray resulting in an extreme widening of the dropletssize spectrum. This intense spread is attributed to theheavy-tail PDF of the droplet vaporization rate which isthe result of the complex dynamics coupling droplet andvapor concentration fields This mechanisms is expectedto be important in all turbulent flows characterized by amixing layer with entrainment of dry air, e.g. clouds.The proper modeling of this phenomenon is critical inorder to improve LES and RANS model capabilities toaccurately reproduce the turbulent vaporization dynam-ics both for reacting and non-reacting sprays.0 -5 -3 -1
0 0.2 0.4 0.6 0.8 1 P D F r d / r d,0 z/R=0.25z/R=10z/R=20z/R=30z/R=40z/R=50 (a) -5 -3 -1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 P D F S d z/R=0.25z/R=10z/R=20z/R=30z/R=40z/R=50 (b) -5 -3 -1 P D F T d / T d,0 z/R=0.25z/R=10z/R=20z/R=30z/R=40z/R=50 (c) -9 -8 -7 -6 P D F -m. d τ d / m d,0 z/R=10z/R=20z/R=30z/R=40 (d) FIG. 9. Probability Density Function (PDF) of Lagrangian variables. (a): PDF of non-dimensional droplets radius, r d /r d, ,where r d, is the initial radius of injected droplets. (b): PDF of the saturation field at droplets surface, S d = Y v,s /Y v,d , where Y v,s = Y v,s ( T d , p ) is the vapor mass fraction at saturation computed as a function of droplet actual temperature and the carrierphase thermodynamics pressure, p. Y v,ds is the actual vapor mass fraction in the carrier gaseous mixture evaluated at dropletcenter position. (c): PDF of non-dimensional droplets temperature, T d /T d, , where T d, is the initial temperature of injecteddroplets. (d): PDF of non-dimensional droplets vaporization rate, − ˙ m d τ d, /m d, , where τ d, and m d, are the initial relaxationtime and mass of injected droplets. The PDF plots (a), (b) and (c) are log-linear, while plot (d) is log-log. ACKNOWLEDGEMENT
The authors acknowledge financial supportthrough the University of Padova Grant PRAT2015(CPDA154914), as well as the computer resourcesprovided by CINECA ISCRA C project: TaStE(HP10CCB69W). B Abramzon and WA Sirignano. Droplet vaporization model forspray combustion calculations.
International journal of heat andmass transfer , 32(9):1605–1618, 1989. Vincenzo Armenio and Virgilio Fiorotto. The importance of theforces acting on particles in turbulent flows.
Physics of Fluids ,13(8):2437–2440, 2001. Antonio Attili, Fabrizio Bisetti, Michael E Mueller, and HeinzPitsch. Formation, growth, and transport of soot in a three-dimensional turbulent non-premixed jet flame.
Combustion andFlame , 161(7):1849–1865, 2014. F Battista, F Picano, G Troiani, and Carlo Massimo Casciola.Intermittent features of inertial particle distributions in turbulentpremixed flames.
Physics of Fluids , 23(12):123304, 2011. Jeremie Bec, Luca Biferale, Massimo Cencini, Alessandra Lan-otte, Stefano Musacchio, and Federico Toschi. Heavy particleconcentration in turbulence at dissipative and inertial scales.
Physical review letters , 98(8):084502, 2007. A Bukhvostova, E Russo, JGM Kuerten, and BJ Geurts. Com-parison of dns of compressible and incompressible turbulentdroplet-laden heated channel flow with phase transition.
Inter-national journal of multiphase flow , 63:68–81, 2014. Enrico Calzavarini, Massimo Cencini, Detlef Lohse, and FedericoToschi. Quantifying turbulence-induced segregation of inertialparticles.
Physical review letters , 101(8):084504, 2008. Carlos B da Silva, Julian CR Hunt, Ian Eames, and Jerry West-erweel. Interfacial layers between regions of different turbulenceintensity.
Annual review of fluid mechanics , 46:567–590, 2014. John K Eaton and JR Fessler. Preferential concentration of par-ticles by turbulence.
International Journal of Multiphase Flow ,20:169–209, 1994. S Elghobashi. On predicting particle-laden turbulent flows.
Ap-plied scientific research , 52(4):309–329, 1994. G Falkovich, A Fouxon, and MG Stepanov. Acceleration of raininitiation by cloud turbulence.
Nature , 419(6903):151–154, 2002. Antonino Ferrante and Said Elghobashi. On the physical mecha-nisms of two-way coupling in particle-laden isotropic turbulence.
Physics of fluids , 15(2):315–329, 2003. GAE Godsave. Studies of the combustion of drops in a fuelspray?the burning of single drops of fuel. In
Symposium (In-ternational) on Combustion , volume 4, pages 818–830. Elsevier,1953. Paolo Gualtieri, F Picano, and Carlo Massimo Casciola.Anisotropic clustering of inertial particles in homogeneous shearflow.
Journal of Fluid Mechanics , 629:25–39, 2009. Paolo Gualtieri, F Picano, Gaetano Sardina, and Carlo MassimoCasciola. Exact regularized point particle method for multiphaseflows in the two-way coupling regime.
Journal of Fluid Mechan-ics , 773:520–561, 2015. Patrick Jenny, Dirk Roekaerts, and Nijso Beishuizen. Modelingof turbulent dilute spray combustion.
Progress in Energy andCombustion Science , 38(6):846–887, 2012. Ian M Kennedy. Models of soot formation and oxidation.
Progress in Energy and Combustion Science , 23(2):95–132, 1997. Gilbert D Kinzer and Ross Gunn. The evaporation, temperatureand thermal relaxation-time of freely falling waterdrops.
Journalof Meteorology , 8(2):71–83, 1951. Jonas Kr¨uger, Nils EL Haugen, Dhrubaditya Mitra, and TereseLøv˚as. The effect of turbulent clustering on particle reactivity.
Proceedings of the Combustion Institute , 2016. Timothy CW Lau and Graham J Nathan. The effect of stokesnumber on particle velocity and concentration distributions in awell-characterised, turbulent, co-flowing two-phase jet.
Journalof Fluid Mechanics , 809:72–110, 2016. ANDFREW MAJDA and James Sethian. The derivation andnumerical solution of the equations for zero mach number com-bustion.
Combustion science and technology , 42(3-4):185–205,1985. C Marchioli, MV Salvetti, and A Soldati. Some issues concerninglarge-eddy simulation of inertial particle dispersion in turbulent bounded flows.
Physics of Fluids , 20(4):040603, 2008. Philippe Marmottant and Emmanuel Villermaux. On spray for-mation.
Journal of fluid mechanics , 498:73–111, 2004. F Mashayek. Direct numerical simulations of evaporating dropletdispersion in forced low mach number turbulence.
Internationaljournal of heat and mass transfer , 41(17):2601–2617, 1998. Martin R Maxey and James J Riley. Equation of motion for asmall rigid sphere in a nonuniform flow.
The Physics of Fluids ,26(4):883–889, 1983. Juan Pedro Mellado. Cloud-top entrainment in stratocumulusclouds.
Annual Review of Fluid Mechanics , 49:145–169, 2017. Richard S Miller and J Bellan. Direct numerical simulation of aconfined three-dimensional gas mixing layer with one evaporatinghydrocarbon-droplet-laden stream.
Journal of Fluid Mechanics ,384:293–338, 1999. Stefano Olivieri, Francesco Picano, Gaetano Sardina, Daniele Iu-dicone, and Luca Brandt. The effect of the basset history forceon particle clustering in homogeneous and isotropic turbulence.
Physics of fluids , 26(4):041704, 2014. F Picano, F Battista, G Troiani, and Carlo Massimo Casciola.Dynamics of piv seeding particles in turbulent premixed flames.
Experiments in Fluids , 50(1):75–88, 2011. F Picano, G Sardina, Paolo Gualtieri, and Carlo Massimo Casci-ola. Anomalous memory effects on transport of inertial particlesin turbulent jets.
Physics of Fluids , 22(5):051705, 2010. Venkat Raman and Rodney O Fox. Modeling of fine-particle for-mation in turbulent flames.
Annual Review of Fluid Mechanics ,48:159–190, 2016. J Reveillon and FX Demoulin. Evaporating droplets in turbu-lent reacting flows.
Proceedings of the Combustion Institute ,31(2):2319–2326, 2007. Julien Reveillon and Fran¸cois-Xavier Demoulin. Effects of thepreferential segregation of droplets on evaporation and turbulentmixing.
Journal of Fluid Mechanics , 583:273–302, 2007. G Rocco, F Battista, F Picano, G Troiani, and Carlo Mas-simo Casciola. Curvature effects in turbulent premixed flamesof h2/air: a dns study with reduced chemistry.
Flow, Turbulenceand Combustion , 94(2):359–379, 2015. Gaetano Sardina, Francesco Picano, Luca Brandt, and RodrigoCaballero. Continuous growth of droplet size variance dueto condensation in turbulent clouds.
Physical review letters ,115(18):184501, 2015. Ewe Wei Saw, Raymond A Shaw, Sathyanarayana Ayyalaso-mayajula, Patrick Y Chuang, and ´Armann Gylfason. Inertialclustering of particles in high-reynolds-number turbulence.
Phys-ical review letters , 100(21):214501, 2008. RA Shaw, AB Kostinski, and ML Larsen. Towards quantify-ing droplet clustering in clouds.
Quarterly journal of the royalmeteorological society , 128(582):1043–1057, 2002. Dudley Brian Spalding. The combustion of liquid fuels. In
Sym-posium (international) on combustion , volume 4, pages 847–864.Elsevier, 1953. Federico Toschi and Eberhard Bodenschatz. Lagrangian proper-ties of particles in turbulence.