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likelihood of droplet recombination downstream of the initial pinching event and thus control the size of droplets generated by liquid jet breakup.

Linear stability theories have also been applied to analyses of liquid sheet breakup processes. The capillary instability of thin liquid sheets was first studied by Squire[258] who showed that instability and breakup of a liquid sheet are caused by the growth of sinuous waves, i.e., sideways deflections of the sheet centerline. For a low viscosity liquid sheet, Fraser et al.[73] derived an expression for the wavelength of the dominant unstable wave. A similar formulation was derived by Li[539] who considered both sinuous and varicose instabilities. Clark and Dombrowski[540] and Reitz and Diwakar[316] formulated equations for liquid sheet breakup length.

Current breakup models need to be extended to encompass the effects of liquid distortion, ligament and membrane formation, and stretching on the atomization process. The effects of nozzle internal flows and shear stresses due to gas viscosity on liquid breakup processes need to be ascertained. Experimental measurements and theoretical analyses are required to explore the mechanisms of breakup of liquid jets and sheets in dense (thick) spray regime.

5.2.2Modeling of Droplet Formation, Breakup, Collision and Coalescence in Sprays

Substantial modeling efforts have been made by numerous investigators to study droplet formation, secondary breakup and coalescence in various processes. Meakin[35] made a comprehensive review of droplet formation processes via deposition growth and coalescence on a surface. Practical examples of condensation stages and droplet pattern formed during the condensation of water vapor on a cold surface, deposition of tin vapor on a hot surface, and deposition of GaAs on Ga-stabilized GaAs surfaces were illustrated. Although the condensation of water vapor may include breath figures or dropwise condensation, the main focus of this review was on the advances in the

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studies of breath figures and related phenomena. Computer models and simulation results were described in detail for all four stages in the breath figure process: (a) droplet growth without coalescence, (b) droplet growth and coalescence without renucleation, (c) droplet growth and coalescence with renucleation, and (d) droplet growth, coalescence and renucleation with large droplet removal. It was demonstrated that the use of scaling ideas was the most successful theoretical approach that had been applied to almost all the theoretical studies of droplet coalescence phenomena.

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O’Rourke and Amsden,[310] two primary approaches have been developed and applied to modeling of physical phenomena in sprays: (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis.[541] Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets.

Two numerical methods have been used for the solution of the spray equation. In the first method, i.e., the full spray equation method,[543][544] the full distribution function f is found approximately by subdividing the domain of coordinates accessible to the droplets, including their physical positions, velocities, sizes, and temperatures, into computational cells and keeping a value of f in each cell. The computational cells are fixed in time as in an Eulerian fluid dynamics calculation, and derivatives of f are approximated by taking finite differences of the cell values. This approach suffersfrom two principal drawbacks: (a) large numerical diffusion and dispersion

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errors[545] associated with convection through the fixed Eulerian mesh, and (b) enormous computer storage requirements.

In the second method, i.e., the particle method,[546][547] a spray is discretized into computational particles that follow droplet characteristic paths. Each particle represents a number of droplets of identical size, velocity, and temperature. Trajectories of individual droplets are calculated assuming that the droplets have no influence on surrounding gas. A later method,[548] that is restricted to steady-state sprays, includes complete coupling between droplets and gas. This method also discretizes the assumed droplet probability distribution function at the upstream boundary, which is determined by the atomization process, by subdividing the domain of coordinates into computational cells. Then, one parcel is injected for each cell.

An important advance in numerical methods for sprays was made by Dukowicz[549] who suggested that the ideas of the Monte Carlo method could be combined with the particle methods for spray calculations. In the method of Dukowicz, i.e., the stochastic particle method, the distribution of droplets at the upstream boundary is sampled stochastically by a relatively small number of computational particles. The droplet distribution function is obtained by averaging over a long time in steady-state calculations, or over many calculations in unsteady problems. The stochastic particle method can calculate unsteady sprays and account for the full coupling between droplets and gas due to mass, momentum, and energy exchanges. The method is robust and economical, and provides a framework for including some important new physical effects in spray calculations. In particular, this method has led to significant progress in discovering the mechanisms governing droplet sizes.

The first major extension of the stochastic particle method was made by O’Rourke[550] who developed a new method for calculating droplet collisions and coalescences. Consistent with the stochastic particle method, collisions are calculated by a statistical, rather than a deterministic, approach. The probability distributions governing the number and nature of the collisions between two droplets are sampled stochastically. This method was initially applied to diesel sprays[317]

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where it was found that coalescences caused a seven-fold increase in the mean droplet size. The importance of droplet collisions in dieseltype sprays has been corroborated by many subsequent studies.[228][551] The second major extension of the stochastic particle method was supplied by Reitz and Diwakar[316] who developed a method for

calculating droplet breakup. Droplet breakup is an important phenomenon in both hollow-cone and full-cone sprays typically formed in direct-injected stratified charge engines. Droplet sizes downstream of the injector are found to be determined primarily by the competition between coalescences and breakups, and influenced by vaporization and condensation. Reitz and Diwakar[316] also proposed a numerical method for calculating atomization using a droplet breakup model. In this method, droplets are injected with a diameter that equals the nozzle exit diameter. The breakup of these large droplets is then accomplished by the breakup model. This method for calculating atomization is based on an assumption that the dynamics and breakup of a liquid jet are indistinguishable from those of a train of droplets with equal diameter. Although this assumption requires further experimental validation, the method promises to remove one of the major weaknesses associated with spray modeling, i.e., the uncertainty in the specification of upstream boundary conditions.

To corroborate the findings of Reitz and Diwakar[316] concerning the importance of droplet breakup, O’Rourke and Amsden[310] developed an alternative model for calculating droplet aerodynamic breakup in spray calculations using the stochastic particle method. This model, called TAB (Taylor Analogy Breakup) model, is based on the analogy, suggested by Taylor,[245] between a spring-mass system and an oscillating-distorting droplet in a gas. The restoring force of the spring is analogous to the surface tension forces of the droplet. The external force on the mass is analogous to the aerodynamic forces of the gas on the droplet. The damping forces due toliquid viscosity were added to the analogy by O’Rourke and Amsden. This TAB model provides a means of predicting distortion, stretching and aerodynamic breakup of liquids such as droplets or ligaments in sprays.

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The TAB model has several advantages. First, as pointed out by Taylor,[245] the model predicts that there is no unique critical Weber number for breakup, and whether or not a droplet breaks up depends on the history of its velocity relative to the gas. The Weber number is a dimensionless measure of the relative importance of the gas aerodynamic forces that distort a droplet and the surface tension forces that restore sphericity. Second, the effects of liquid viscosity are included. Although these effects are negligible for large droplets, liquid viscosity can significantly affect the oscillations of small droplets. Third, the model predicts the state of oscillations and distortions of droplets. Thus, once information is available on how distortions and oscillations affect the exchange rates of mass, momentum, and energy between the droplets and gas, this information can be incorporated in the model. Fourth, the model predicts droplet sizes that are more consistent with experimentally determined mechanisms of liquid jet breakup.[552][553] If this model is further used for calculating liquid jet breakup, an additional advantage is that the model predicts the velocities of the product droplets normal to the

path of the original parent droplet. These normal velocities

—

determine an initial spray angle, tan(θ /2) = Ö3/3(ρ G /ρ L)1/2, that is in good agreement with some measured spray angles.[552] By giving an initial oscillation to the large, injected droplets, the initial spray angle can be varied and the effects of nozzle geometry changes can be included in numerical calculations. This eliminates the need for inputting the spray angle, i.e., the spray angle is automatically calculated in the TAB model.

The major limitation of the TAB model is that it can only keep track of one oscillation mode, while in reality there are many oscillation modes. Thus, more accurately, the Taylor analogy should be between an oscillating droplet and a sequence of spring-mass systems, one for each mode of oscillations. The TAB model keeps track only of the fundamental mode corresponding to the lowest order spherical zonal harmonic[554] whose axis is aligned with the relative velocity vector between the droplet and gas. This is the longest-lived and therefore the most important mode of oscillations. Nevertheless, for large Weber numbers, other modes are certainly excited and contribute to droplet breakup. Despite this

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limitation, good agreement between the model-predicted and experimentally observed breakup times was achieved, as reported in Ref. 310.

The equations in the TAB model contain four dimensionless constants that are determined by some theoretical and experimental results. The model predicts and continuously connects breakup times experimentally observed for the bag and stripping breakup regimes. The bag mode occurs when the Weber number is slightly large than a critical value, while the stripping mode takes place for the Weber numbers much larger than this critical value. The Taylor analogy equations do not predict product droplet sizes; thus an additional equation for the product droplet sizes is included in the TAB model and motivated by an energy conservation argument. A droplet breaks up into smaller droplets with a specified distribution of sizes when droplet distortion reaches a critical value. The product droplet sizes are determined by a Weber number criterion. For example, the Weber number based on the Sauter mean diameter of the product droplets is equal to 12 in the bag breakup regime.

A computer code, called KIVA, has been developed by Amsden et al.[555] at Los Alamos National Laboratory for modeling multidimensional fluid flows with chemical reactions and fuel sprays. The TAB model has been implemented numerically in the KIVA code,[555] along with the wave breakup model of Reitz[232] to describe the breakup process. A comparison of the computational results showed[310] that the two models[310][316] predicted different droplet sizes near the injector because they used different breakup times. Downstream, the two models generated similar results at lowest backpressure; at higher backpressures, however, the TAB model predicted larger droplet sizes than the calculations of Reitz and Diwakar[316] as well as the experiments of Hiroyasu and Kadota.[317] The differences in the implementation of the wave and TAB models have a significant impact on fuel vapor distribution in high-pressure combustion sprays since small product droplets vaporize very rapidly,[232] particularly when breakup time is much longer than numerical time step.

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A modified version of the TAB model, called dynamic drop breakup (DDB) model, has been used by Ibrahim et al.[556] to study droplet distortion and breakup. The DDB model is based on the dynamics of the motion of the center of a half-drop mass. In the DDB model, a liquid droplet is assumed to be deformed by extensional flow from an initial spherical shape to an oblate spheroid of an ellipsoidal cross section. Mass conservation constraints are enforced as the droplet distorts. The model predictions agree well with the experimental results of Krzeczkowski.[311]

The substantial effect of secondary breakup of droplets on the final droplet size distributions in sprays has been reported by many researchers, particularly for overheated hydrocarbon fuel sprays.[557] A quantitative analysis of the secondary breakup process must deal with the aerodynamic effects caused by the flow around each individual, moving droplet, introducing additional difficulty in theoretical treatment. Aslanov and Shamshev[557] presented an elementary mathematical model of this highly transient phenomenon, formulated on the basis of the theory of hydrodynamic instability on the droplet-gas interface. The model and approach may be used to make estimations of the range of droplet sizes and to calculate droplet breakup in high-speed flows behind shock waves, characteristic of detonation spray processes.

Droplet collision and coalescence phenomena become increasingly important in dense sprays.[316] For solid-cone sprays, droplet coalescence is particularly important at high gas densities. For hollow-cone sprays, droplet breakup dominates because droplet coalescence is minimized by the expanding spray geometry. Binary collisions between droplets may lead to coalescence or separation, depending on operation conditions.[558] For a fixed orientation of colliding droplets, coalescence occurs at low collision Weber numbers. Separation of colliding droplets occurs in reflexive mode as the Weber number is increased beyond a critical value. O’Rourke[550] and O’Rourke and Bracco[559] developed a method for calculating droplet collisions in sprays on the basis of the kinetic theory of gases and Langmuir’s work.[560] O’Rourke[550] assumed the probable number of