Experiments in Fluids

, 46:371 | Cite as

Secondary atomization

  • D. R. Guildenbecher
  • C. López-Rivera
  • P. E. Sojka
Review Article

Abstract

When a drop is subjected to a surrounding dispersed phase that is moving at an initial relative velocity, aerodynamic forces will cause it to deform and fragment. This is referred to as secondary atomization. In this paper, the abundant literature on secondary atomization experimental methods, breakup morphology, breakup times, fragment size and velocity distributions, and modeling efforts is reviewed and discussed. Focus is placed on experimental and numerical results which clarify the physical processes that lead to breakup. From this, a consistent theory is presented which explains the observed behavior. It is concluded that viscous shear plays little role in the breakup of liquid drops in a gaseous environment. Correlations are given which will be useful to the designer, and a number of areas are highlighted where more work is needed.

List of symbols

Dimensional

a

drop acceleration (m/s2)

c

velocity of sound (m/s)

D10

drop or fragment arithmetic mean diameter (m)

D30

drop or fragment volume mean diameter (m)

D32

drop or fragment Sauter mean diameter (m)

D43

drop or fragment de Brouckere mean diameter (m)

d0

drop initial spherical diameter (m)

dcore

diameter of drop core at end of sheet-thinning breakup (m)

dcro

drop cross-stream diameter (m)

dstr

drop stream-wise diameter (m)

FD

aerodynamic drag force (kg m/s2)

Fsurf

net surface force (kg m/s2)

Fμ

shear force (kg m/s2)

f0(D)

fragment number PDF (1/m)

f3(D)

fragment volume PDF (1/m)

K

power-law fluid consistency index (kg/m s(2−n))

k

wave number; 2π/λ (1/m)

MMD

drop or fragment mass median diameter (m)

q

net electrostatic charge (C)

qRa

Rayleigh charge limit (C)

t

time (s)

U0

initial relative velocity between drop and ambient fluid in main flow direction (m/s)

Ucore

velocity of drop core relative to ambient fluid (m/s)

\( \bar{U}_{\text{f}} \)

mean relative velocity of fragments in main flow direction (m/s)

V0

initial relative velocity between drop and ambient fluid perpendicular to main flow direction (m/s)

\( \bar{V}_{\text{f}} \)

mean relative velocity of fragments in cross-stream direction (m/s)

δ

boundary layer thickness (m)

εa

electrical permittivity of ambient (C2/N m2)

λ

wavelength (m)

λ(1)

elastic fluid relaxation time (s)

μa

ambient viscosity (kg/m s)

μd

drop viscosity (kg/m s)

μeff

power-law effective viscosity (kg/m s)

μsol

solvent shear viscosity (kg/m s)

ρa

ambient density (kg/m3)

ρd

drop density (kg/m3)

σ

surface tension (kg/s2)

Non-dimensional

CD

instantaneous coefficient of drag based on drop cross-stream diameter

\( \bar{C}_{\text{D}} \)

average coefficient of drag based on initial spherical diameter

CD-sphere

coefficient of drag of a solid sphere at a given Reynolds number

Eocr

Eötvös number at end of sheet-thinning breakup; \( {{a\left| {\rho_{\rm d} - \rho_{\rm a} } \right|d_{\text{core}}^{2} } \mathord{\left/ {\vphantom {{a\left| {\rho_{\rm d} - \rho_{\rm a} } \right|d_{\text{core}}^{2} } \sigma }} \right. \kern-\nulldelimiterspace} \sigma } \)

La

Laplace number; La = Oh−2

Ma

Mach number

N

viscosity ratio; μd/μa

n

power-law fluid flow behavior index

Oh

Ohnesorge number; \( {{\mu_{\rm d} } \mathord{\left/ {\vphantom {{\mu_{\rm d} } {\sqrt {\rho_{\rm d} d_{0} \sigma } }}} \right. \kern-\nulldelimiterspace} {\sqrt {\rho_{\rm d} d_{0} \sigma } }} \)

Re

gas-phase Reynolds number; \( {{\rho_{\rm a} U_{0} d_{0} } \mathord{\left/ {\vphantom {{\rho_{\rm a} U_{0} d_{0} } {\mu_{\rm a} }}} \right. \kern-\nulldelimiterspace} {\mu_{\rm a} }} \)

ReNN

Reynolds number for a power-law fluid; ρU02−nd0n/K

T

dimensionless time; \( tU_{0} \varepsilon^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}} d_{0}^{ - 1} \)

Tini

breakup initiation time

Ttot

total breakup time

We

Weber number; \( {{\rho_{\rm a} U_{0}^{2} d_{0} } \mathord{\left/ {\vphantom {{\rho_{\rm a} U_{0}^{2} d_{0} } \sigma }} \right. \kern-\nulldelimiterspace} \sigma } \)

Wec

critical Weber number

WecOh→0

critical Weber number at low Ohnesorge number

Wecore

Weber number of drop core at end of sheet-thinning breakup

Wee−

electrostatic Weber number; \( {{\rho_{\rm a} U_{0} d_{0}^{2} } \mathord{\left/ {\vphantom {{\rho_{\rm a} U_{0} d_{0}^{2} } {\left( {\sigma - {{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {8\pi^{2} \varepsilon_{\rm a} d_{0}^{3} }}} \right. \kern-\nulldelimiterspace} {8\pi^{2} \varepsilon_{\rm a} d_{0}^{3} }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sigma - {{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {8\pi^{2} \varepsilon_{\rm a} d_{0}^{3} }}} \right. \kern-\nulldelimiterspace} {8\pi^{2} \varepsilon_{\rm a} d_{0}^{3} }}} \right)}} \)

Wi

Weissenberg number; \( {{\lambda^{(1)} U_{0} } \mathord{\left/ {\vphantom {{\lambda^{(1)} U_{0} } {d_{0} }}} \right. \kern-\nulldelimiterspace} {d_{0} }} \)

y

non-dimensional displacement of drop equator; 1 − (d0/dcro)2

ε

density ratio; ρd/ρa

ω

exponential growth factor

1 Introduction

Spray formation, or the production of drops, is a common phenomenon in a variety of scientific and engineering applications. When an initially spherical drop encounters an ambient flow field moving at a velocity relative to it, aerodynamic forces cause the drop to deform and (perhaps) break apart into fragments. This process is referred to as secondary atomization.

Secondary atomization is in contrast to primary atomization where bulk fluid, typically in the form of a sheet or jet, breaks up for the first time and forms drops. In spray formation, primary atomization occurs at or near the nozzle exit. This may be followed by secondary atomization, which typically occurs further downstream.

Secondary breakup occurs in a wide variety of systems, and is desired in applications as diverse as mass spectrometry (for homeland security portals), internal combustion engines (for land-based power production), injection of gelled hypergolic fuels (for aero-propulsion), coatings (manufacture of pharmaceutical tablets or painting of automobiles), materials processing (thermal barrier coatings), and many more.

Since the goal of many atomization processes is to control the final droplet sizes, one of the most important reasons to study secondary atomization is to determine the conditions that lead to appropriate fragment sizes. In combustion applications, for example, it is desirable to produce small drops in order to increase evaporation and mixing rates. Interestingly, as noted by Tryggvason (1997), the highest ambient velocity does not always lead to the smallest drop diameters. Therefore, by clearly understanding secondary breakup it may be possible to find flow conditions that will produce the desired size drops.

There exists abundant literature on secondary atomization. The first comprehensive review was provided by Pilch and Erdman (1987) over 20 years ago. The most recent comprehensive summary was by Faeth et al. (1995). There have been numerous studies published in the intervening 13 years so a new review is warranted. For the sake of brevity we focus on efforts completed subsequent to the article by Pilch and Erdman (1987).

Another motivation for this review is the number of scientific issues that have arisen in the last dozen or so years. These include the alternative explanations for the physical mechanism that leads to sheet-thinning breakup, and the possibility of placing multi-mode breakup at Weber numbers between those for bag-and-stamen and sheet thinning modes. Competing hypotheses for both are presented here and consistent explanations provided to help resolve disagreements.

This review covers mostly articles published in archival journals due to their wider availability. Conference proceedings and other sources are included when they contain important conclusions.

The work presented here is restricted to secondary atomization when the continuous phase is a gas. For those interested, Pilch and Erdman (1987) and Gelfand (1996) have reviewed liquid-liquid secondary atomization. They found that many aspects are similar to gas-liquid secondary breakup systems at low density ratios.

The article begins by introducing and comparing the four experimental systems commonly used to investigate drop breakup. A criterion for determining when data obtained using the two most widely used techniques are compatible is then developed.

The experimentally observed characteristics of secondary atomization (i.e., the breakup modes) are summarized, and the initiation and breakup times defined. Non-dimensional groups that have been used to quantify those modes are introduced; the critical Weber number is defined and its dependence on Ohnesorge number and other dimensionless groups discussed.

This is followed by a synthesis of previous experimental work on Newtonian drop breakup. The five most commonly agreed upon breakup modes are then listed. In each case a qualitative description precedes an outline of the underlying physical mechanism and breakup behavior. Breakup times for all modes are defined and discussed.

The few non-Newtonian studies are then evaluated using the same framework as for the Newtonian ones (mode identification, qualitative description, physical mechanism, and behavior; breakup times).

Results from studies on fragment size and velocity distributions are introduced. Focus then shifts to results from computational studies. Analytical models are discussed first, and then numerical simulations.

This review closes with suggestions for future work.

2 Characteristics of Newtonian drop secondary atomization

The earliest studies on secondary breakup were experimental. A number of methods emerged. The three most popular are (1) shock tubes, (2) continuous jets, and (3) drop towers. Each method differs by the type of aerodynamic loading on the drop: shock tubes provide a nearly spatially uniform step change in relative velocity, drop towers a gradual change, and continuous jets a shearing effect. These differences are important because they can lead to variations in observed behavior.

2.1 Shock tube

A shock tube is divided into two sections by a diaphragm, as shown in Fig. 1. Pressurized gas is released from the driver section into the driven section, causing a shock wave to develop and travel down the tube. Droplets are inserted into the driven section and their breakup observed.
Fig. 1

A shock tube experimental apparatus

The shock rapidly passes over the drop and causes minimal deformation. Rather, it is the convective flow after the shock which leads to breakup. In most cases, this convective flow is sufficiently slow such that compressibility effects can be neglected.

The major advantage of shock tube experiments is the ability to subject a drop to a nearly step change in ambient flow that is essentially uniform over its surface. This results in a repeatable experiment that is amenable to theoretical and computational study.

One disadvantage of this approach is that drops in many practical systems rarely experience this type of perturbation. Two additional disadvantages of this method are the low data rate, due to the need to reset the experiment after every run, and a limited range of operating conditions. In addition, the need for a compressible ambient fluid means a gas must be used so experimentation at extreme pressures or temperatures is difficult. For this reason, most experiments are conducted using air at or around ambient pressure and temperature.

Studies reviewed here that include original shock tube results are those of Hinze (1955), Ranger and Nicholls (1969), Gelfand et al. (1975), Wierzba and Takayama (1988), Hsiang and Faeth (1992), Hsiang and Faeth (1993), Hsiang and Faeth (1995), Chou et al. (1997), Chou and Faeth (1998), Joseph et al. (1999), Igra and Takayama (2001), Joseph et al. (2002), Igra et al. (2002), Dai and Faeth (2001), and Theofanous et al. (2004).

2.2 Continuous jet

The continuous jet (see Fig. 2) emerged as an alternative to the shock tube due to its simplicity and ability to operate continuously. It also allows fragment size measurement using optical drop sizing techniques such as phase Doppler anemometry (PDA).
Fig. 2

A continuous jet experimental apparatus

In an attempt to make the results equivalent to shock tube experiments, authors shape the nozzle in such a way as to minimize boundary layers in the free jet so that the drops experience a more spatially uniform, step change in velocity. Obviously, if the drop enters the jet too slowly a portion of it may break up before the remainder enters the flow field. Such a situation is seen in the work of Arcoumanis et al. (1996). This can lead to results at odds with those from shock tube studies making direct comparison quite difficult.

To operate a continuous jet in a manner that produces results which closely match those from shock tube experiments, it is necessary that drop distortion and breakup occur almost entirely when the drop is in the jet’s uniform velocity region. Assuming drops are injected at velocity V0 perpendicular to a jet having centerline axial velocity U0 and boundary layer thickness δ, two criteria must be satisfied. First, the initial drop velocity must be low enough to ensure breakup does not occur outside of the jet. This can be expressed as:
$$ \frac{{\rho_{\rm a} V_{0}^{2} d{}_{0}}}{\sigma }\tilde{ < }We_{c} $$
(1)
where Wec is the critical Weber number and is defined in a subsequent section. Second, the time required for the drop to pass through the boundary layer must be less than the time required to initiate breakup. This can be expressed as:
$$ \frac{{\left( {d_{0} + \delta } \right)}}{{V_{0} }}\frac{{U_{0} }}{{\varepsilon^{0.5} d_{0} }} < T_{\rm ini} $$
(2)
where Tini is the initiation time as defined by Eq. 30, and ε is the drop to ambient density ratio. When combined:
$$ \frac{{\left( {1 + {\delta \mathord{\left/ {\vphantom {\delta {d_{0} }}} \right. \kern-\nulldelimiterspace} {d_{0} }}} \right)}}{{T_{\rm ini} \varepsilon^{0.5} }} < \frac{{V_{0} }}{{U_{0} }} < \sqrt {\frac{{We_{\rm c} }}{We}} $$
(3)
where We is the drop Weber number, also defined in a subsequent section.

Using the criteria in Eq. 3 it is possible to evaluate experimental setups. For example, Fig. 4 from Arcoumanis et al. (1996) shows a droplet of diesel oil (d0 = 2.6 mm) breaking up in an air jet (U0 = 86 m/s, ε ≈ 700) such that We = 400. Assuming the nozzle is well designed, δ ≈ 0, using Eq. 30 from below with Tini = 0.43, and Table 2 with Wec = 11 yields 7.6 < V0 < 14 m/s. The actual injection velocity is unknown. However, Arcoumanis et al. (1996) indicated drops were produced using a syringe tip and then allowed to fall due to gravity from 20 mm to 1 m. Assuming zero initial velocity, no drag, and a 1 m fall yields V0~ 4.4 m/s, which by Eq. 3 is insufficient to replicate shock tube behavior.

Papers reviewed here that include original continuous jet results are those of Liu et al. (1993), Liu and Reitz (1993, 1997), Arcoumanis et al. (1996), Hwang et al. (1996), Prevish and Santavicca (1998), Lee and Reitz (1999, 2000, 2001), Gökalp et al. (2000), Park et al. (2006), Cao et al. (2007), Guildenbecher and Sojka (2007), and López-Rivera and Sojka (2008).

2.3 Drop tower

Shock tubes and continuous jets attempt to subject drops to a step change in velocity. Breakup can also occur if drops are accelerated more slowly, as by a constant body force, such as rain drops falling due to gravity. Motivated mostly by applications to atmospheric sciences, many authors have studied secondary atomization using a drop tower. In it drops are allowed to fall under gravity into a quiescent environment and the subsequent breakup is observed.

2.4 Hybrid methods

Many practical applications involve situations which are best characterized as a combination of nearly step acceleration, as seen in shock tubes, and continuous acceleration, as seen in a drop tower. Because of the complexity of such processes little experimental data is available. Two notable exceptions are the works of Shraiber et al. (1996) and Schmelz and Walzel (2003). Shraiber et al. (1996) allowed drops to fall under the action of gravity through various air nozzles specifically designed to produce non-uniform velocity profiles. Schmelz and Walzel (2003) studied the breakup of drops as they fell through a shaped contraction such that the ambient air significantly accelerated during breakup. The results were found to be different than either shock tube or drop tower experiments, and some interesting trends are noted. However, most of the results are presented as a function of the experimental geometry and the applicability to other flow configurations is unclear.

In summary, the drop tower experiment is probably closest to “natural” secondary atomization processes for obvious reasons. However, most man made spray applications are likely best simulated using either the continuous jet (e.g., gas turbine injection in which the drops enter a stream of moving gas) or shock tube (e.g., diesel injection where rapid gas movement can lead to approximate step changes in velocity). Finally, the shock tube is superior from a scientific perspective because it provides repeatable and well-characterized initial and boundary conditions for each drop fragmentation process. For that reason this review concentrates on secondary breakup due to impulsive acceleration as seen in shock tube and some continuous jet experiments. In addition, results will focus on gas-liquid systems where the ambient phase is gas.

2.5 Description

Despite the choice of at least three different types of apparatus, experimental results have revealed common characteristics in all cases of secondary atomization.

The process starts when the drop enters the disruptive flow field. This marks the beginning of the deformation phase. An unequal pressure distribution, due to acceleration of the ambient fluid around the drop, leads to deformation from the initial spherical shape. This deformation is resisted by the interfacial tension and viscous forces. However, if the aerodynamic forces are large enough the drop will enter the fragmentation phase.

Since fragmentation results from ambient/drop interactions, it is a function of the flow conditions. Differing flow conditions lead to differing breakup modes, which are often illustrated by renditions such as those shown in Fig. 3 for Newtonian drops in shock tube experiments. From top to bottom the modes are termed vibrational, bag, multimode (often called bag-and-stamen), sheet-thinning, and catastrophic.
Fig. 3

Newtonian drop breakup morphology

Vibrational breakup is not always observed. It consists of oscillations at the natural frequency of the drop and produces only a few fragments whose sizes are comparable to those of the parent drop.

The bag breakup geometry is composed of a thin hollow bag attached to a thicker toroidal rim. The bag disintegrates first, followed by the toroidal rim. The former results in a larger number of small fragments; the latter a smaller number of large fragments.

Multi-mode (also called bag-and-stamen) breakup is similar to bag breakup, but with the addition of a stamen oriented anti-parallel to the direction of the drop motion. Like bag breakup, the bag is the first to disintegrate, followed by the rim and the stamen. Fragments of multiple sizes are produced.

In sheet stripping (or sheet-thinning), a film is continuously eroded from the drop surface. It disintegrates rapidly after being removed. This results in a plethora of small droplets and, in some cases, a core whose size is comparable to that of the parent drop.

Finally, during catastrophic breakup the drop surface is corrugated by waves of large amplitude and long wavelengths. They form a small number of large fragments that in turn break up into even smaller units. Some authors sub-divide this region into wave-crest stripping and catastrophic. They attribute mass removal from the drop surface via large amplitude-small wavelength waves.

Note that both Pilch and Erdman (1987) and Hsiang and Faeth (1992) provide thorough reviews of early investigations into secondary atomization. Here we adopt the breakup morphology given by Hsiang and Faeth (1992); the morphology of Pilch and Erdman (1987) is nearly identical, differing primarily by the names assigned to each mode.

While the chosen morphology is well established in the literature, a few exceptions exist. Most notably, Theofanous et al. (2004) used a shock tube to study breakup in highly rarefied, supersonic ambient flows and found the breakup morphology to differ significantly from that shown in Fig. 3 (which is derived mostly from experiments at subsonic ambient velocities). Additional testing is needed to confirm their results. Nevertheless, this is an indication that extrapolation of the experimental results should be done with caution.

Renditions such as Fig. 3 give the impression that secondary breakup is an instantaneous process. In reality, mass is first removed from the drop at some time after it is first exposed to the moving ambient fluid. Fragmentation continues until aerodynamic drag has reduced the relative velocity between the drop/fragments and the surrounding flow to a level where disruptive forces are no longer large enough to overcome the restorative forces. The time when all fragmentation has ceased is referred to as the total breakup time, Ttot (Pilch and Erdman 1987).

Figure 4 illustrates some of the breakup modes that are typically observed in experiments and highlights the fact that secondary breakup is a rate process that occurs over a finite time.
Fig. 4

Shadowgraphs of Newtonian drop secondary breakup. Time increases from left to right, disruptive forces increase from top to bottom

2.6 Non-dimensional groups

In general, multiple physical processes and fluid properties are important in secondary breakup phenomena. This is demonstrated in Figs. 3 and 4 by the variety of breakup modes that have been reported.

In addition, the breakup geometries can be highly complicated with multiple entities, re-entrant and other topologically complex surfaces, plus multiple length and time scales. This makes mathematical and numerical analysis very challenging, especially in the early days of this field.

Past researchers therefore described their findings in terms of a number of non-dimensional groups. They are still in use today, and most authors make use of one or more of those listed in Table 1. The logic behind their choice is as follows.
Table 1

Dimensionless groups important in secondary breakup

Weber number

We

\( \frac{{\rho_{\rm a} U_{0}^{2} d_{0} }}{\sigma } \)

Ohnesorge number

Oh

\( \frac{{\mu_{\rm d} }}{{\sqrt {\rho_{\rm d} d_{0} \sigma } }} \)

Reynolds number

Re

\( \frac{{\rho_{\rm a} U_{0} d_{0} }}{{\mu_{\rm a} }} \)

Density ratio

ε

\( \frac{{\rho_{\rm d} }}{{\rho_{\rm a} }} \)

Viscosity ratio

Ν

\( \frac{{\mu_{\rm d} }}{{\mu_{\rm a} }} \)

Mach number

Ma

\( \frac{{U_{0} }}{c} \)

In secondary atomization the aerodynamic forces deform a drop causing it to fragment. This deformation is resisted by the surface tension, which tends to restore the drop to a spherical shape. As a result the Weber number, We, defined as the ratio of the disrupting aerodynamic forces to the restorative surface tension forces, is the most important parameter when describing secondary atomization. A larger We indicates a higher tendency toward fragmentation.

Drop viscosity hinders deformation and also dissipates energy supplied by aerodynamic forces. Both factors reduce the likelihood of fragmentation. To account for this, many authors make use of the Ohnesorge number, Oh, which represents the ratio of drop viscous forces to surface tension forces. A higher Oh indicates a lower tendency toward fragmentation. The Laplace number, La, is used in some works (La Oh−2).

Other important dimensionless groups are the Reyonlds number, Re, which is the ratio of aerodynamic forces to ambient viscous forces, the drop phase-to-ambient phase density ratio, ε, and drop phase-to-ambient phase viscosity ratio, Ν. Note that Re is equal to We0.5Oh−1ε−0.5 N. Finally, the Mach number, Ma, is important when considering compressibility effects.

As pointed out by Shraiber et al. (1996) this list does not encompass all physical processes that may play a role in secondary atomization. Turbulence within the two fluids may create additional forces that destabilize drops. In addition, unsteady ambient flow could be considered by accounting for the time that the disruptive forces act on the drop and/or accounting for the rate of change of these forces. Gelfand (1996) cited experimental evidence that the duration of the disruptive flow must be sufficient to lead to breakup. Furthermore, as noted by Clift et al. (1978), one could consider impurities and particulates that may serve as initiation points for breakup.

When studying temporal phenomena, experimentally observed times are typically made non-dimensional using the characteristic transport time given by Ranger and Nicholls (1969), which is derived from analysis of the drop displacement assuming constant acceleration due to drag:
$$ T = t\frac{{U_{0} }}{{\varepsilon^{0.5} d_{0} }} $$
(4)
Here T is the dimensionless time, t is the dimensional time, U0 is the initial relative velocity between drop and ambient, ε is the drop-to-ambient density ratio, and d0 is the initial spherical diameter. As first noted by Ranger and Nicholls (1969) this choice of characteristic time is not appropriate to describe all temporal phenomena in secondary atomization and authors have proposed alternatives. As examples, Shraiber et al. (1996) suggested non-dimensionalizing by the drop oscillation period while Faeth et al. (1995) suggested using a viscous timescale for drops at high Oh. Scaling by liquid relaxation or retardation time might be more appropriate for non-Newtonian fluids exhibiting elasticity.

2.7 Transition We (low Oh)

For a given drop size and fluid properties, experiments conducted at increased relative velocities result in a continuous transition between breakup modes. However, for simplification most authors have assumed transition occurs abruptly.

Most investigators have found the transition between two modes to be a function of We and Oh and independent of other parameters such as the density ratio (ε) or Re. However, it is important to note that this may be due to the limited ranges of such parameters that can easily be achieved in experiments.

Regardless, numerous experiments have shown that the transition We between pairs of breakup modes are essentially constant for Oh < 0.1 and can be approximated by the values provided in Table 2.
Table 2

Transition We for Newtonian drops with Oh < 0.1

Vibrational

0 < We < ~11

Bag

~11 < We < ~35

Multimode

~35 < We < ~80

Sheet-thinning

~80 < We < ~350

Catastrophic

We > ~350

As noted in the discussion of Fig. 4, regime transition is actually a continuous process, so the values of the transition We are subjective and different authors have reported different magnitudes. For example Pilch and Erdman (1987) reported transition between multimode and sheet-thinning at We = 100, while Hsiang and Faeth (1992) choose We = 80, and Gelfand (1996) found We = 40.

In Table 2 the values of Hsiang and Faeth (1992) are reported with the exception of the transition between sheet-thinning and catastrophic, which was taken from the work of Pilch and Erdman (1987), and the transition between vibrational and bag, which is an average of numerous authors. Reasons for these choices will be discussed in a subsequent section.

2.8 Dependency on Oh

While it is true that Weber numbers demarking breakup mode boundaries are independent of Oh for Oh < 0.1, that is not true for higher Oh conditions. As noted by Faeth et al. (1995), in many high-pressure spray applications the drop phase approaches the thermodynamic critical point where Oh increases rapidly as the surface tension approaches zero and the density ratio decreases. At these elevated Oh, the observed breakup modes remain the same, but experiments have shown an increase in the transitional We values listed in Table 2. According to Hsiang and Faeth (1992) this is because the increased drop viscosity dissipates energy, which slows drop distortion and allows more time for aerodynamic drag to reduce the relative velocity.

No Oh has been observed for which breakup is impossible. Take, for example, the extreme case of Joseph et al. (1999) who performed shock tube experiments at some of the highest recorded values of We and Oh. Bag breakup was observed at We = 160,000 and Oh = 26.6. In contrast, for Oh < 0.1 bag breakup is expected to end at We = 35.

The relation between transitional We and Oh is often plotted as shown in Fig. 5 (Hsiang and Faeth 1995). A number of experimental correlations have been proposed to describe this behavior. Most have focused on the critical Weber number, Wec, defined as the We at the start of bag breakup. However, as can be seen in Fig. 5 the behavior for other transitional We is similar.
Fig. 5

We at transition. Reprinted from Hsiang LP, Faeth GM (1995) Drop deformation and breakup due to shock wave and steady disturbances. International Journal of Muliphase Flow, vol 21(4):545–560, with permission from Elsevier

To describe Wec, Brodkey (1967) proposed the following correlation, which Pilch and Erdman (1987) confirmed for Oh < 10:
$$ We_{\rm c} = We_{{\rm c}Oh \to 0} \left( {1 + 1.077Oh^{1.6} } \right)\quad Oh < 10 $$
(5)
Here WecOh→0 is the critical We at low Oh, as given in Table 2. Similarly, Gelfand (1996) reviewed mostly Russian works and proposed:
$$ We_{\rm c} = We_{{\rm c}Oh \to 0} \left( {1 + 1.5Oh^{0.74} } \right)\quad Oh < 4.0 $$
(6)
These correlations are compared in Fig. 6. Clearly at Oh > 3 they do not agree with one another.
Fig. 6

Wec, from Eqs. 5 and 6

The inaccuracies in experimentally determined correlations have led many to seek relations based at least partially on the assumed underlying physical mechanisms. For example, Cohen (1994) assumed that in the absence of drop viscosity the kinetic energy imparted by the ambient flow to the drop is equal to the surface energy. An extra energy term was added to account for the drop viscosity, therefore increasing the kinetic energy needed to cause breakup. The result was:
$$ We = We_{{\rm c}Oh \to 0} \left( {1 + C \cdot Oh} \right) $$
(7)
where C has a value between 1.0 and 1.8 that is theorized to be dependent on the breakup regime.

Similar to Cohen (1994), Hsiang and Faeth (1995) performed a phenomenological analysis in which they assumed the instantaneous We must reach a certain critical value for regime transition to occur. They also determined that the transition We are approximately linear functions of Oh for high Oh values.

Going further, Aalburg et al. (2003) noted that at very high Oh the effect of surface tension becomes negligible, and at the critical condition drop viscous forces balance aerodynamic forces. They suggested a new regime map complimentary to Fig. 5 where the ratio We1/2/Oh (equivalent to Re based on drop phase viscosity) becomes constant for Oh ≫ 1.

Despite these works, no published correlation is known to be accurate at Oh > 1, so more work is needed.

2.9 Dependency on other non-dimensional parameters

Some authors have observed a dependence of Wec on other quantities. For example, an increase in Wec as the drop density approaches the ambient density was observed in the direct numerical simulations of Han and Tryggvason (1999, 2001) and Aalburg et al. (2003), who also noted that many properties of secondary atomization became essentially independent of density ratio for ε > 32. Gelfand (1996) reviewed liquid–liquid systems where the density ratio is nearly unity and reported a Wec of 17, somewhat higher than the value of 11 for gas–liquid systems given in Table 2. This indicates that gas–liquid systems may behave similarly to liquid–liquid systems at extremely low density ratios. Such density ratios may be found in direct diesel injection. However, no experimental gas-liquid results are known to exist for ε < 32.

In addition, Aalburg et al. (2003) used numerical simulation to study drop deformation at low Re. They found a significant change in Wec in the Stokes flow regime (Re < 100) and almost no dependence on Re for Re > 100.

Hsiang and Faeth (1995) state that some dependence of Wec on Re has been seen for liquid-liquid drop tower experiments. However, no shock tube data is known to exist for such low Re. Consequently, more work is needed to simulate low Re drop motion in gas-liquid systems.

3 Drop deformation and vibrational breakup

3.1 Qualitative description

The earliest stage of secondary atomization is drop deformation into a shape that can be approximated as an oblate ellipsoid. This is illustrated in Fig. 7. Here dstr is the deformed drop diameter in the stream-wise direction and dcro is the deformed drop diameter in the cross-stream direction.
Fig. 7

Rendition of a deformed/vibrating drop

If the aerodynamic forces are large enough the drop will continuously deform until it begins to fragment via one of the modes illustrated in Fig. 3 (Hsiang and Faeth 1992). However, if the aerodynamic forces are insufficient then surface tension may lead to oscillation at the drop natural frequency, which depending on flow conditions may be either stable or unstable (Hsiang and Faeth 1992). When it is unstable the drop eventually breaks apart into a few large fragments. This is referred to as vibrational breakup. As noted by Pilch and Erdman (1987), this breakup mode proceeds much more slowly than other modes and does not lead to small final fragment sizes. As a result, most authors ignore vibrational breakup and consider bag breakup to be the first secondary atomization mode. For that reason few authors have studied virbrational breakup in detail.

Nevertheless, the study of deformation is important because it is the first stage of all aerodynamically induced fragmentation. Also, drop deformation has been shown to significantly affect the drag and hence the trajectory. Therefore, a thorough understanding of the process is necessary to create accurate secondary atomization models.

3.2 Physical mechanism

Deformation is caused by an unequal static pressure distribution over the drop surface. Assuming an inviscid fluid, the forward and rear stagnation points on the drop are at a higher static pressure compared to flow around the drop periphery. This causes the drop to expand laterally and compress in the gas flow direction. The presence of a wake will alter the static pressure distribution; nevertheless, qualitatively similar deformation has been observed.

3.3 Behavior

Early attempts to model the deformation process approximated the changes in drop shape and drag using an average drag coefficient based on the initial spherical diameter, \( \bar{C}_{\rm D} , \) such that the average drag force was given by:
$$ F_{\rm D} = \frac{1}{2}\rho_{\rm a} U_{0}^{2} \bar{C}_{\rm D} \frac{{\pi d_{0}^{2} }}{4} $$
(8)
The reviews of Pilch and Erdman (1987) and Gelfand (1996) report various values of \( \bar{C}_{\rm D} \) that are applicable throughout given ranges of We, Re, and Ma. Ortiz et al. (2004) used such data to create the following correlation:
$$ \bar{C}_{\rm D} = 1.6 + 0.4Oh^{0.08} We^{0.01} $$
(9)
$$ 1000 < We < 162000,\quad Oh < 0.44,\quad 0.95 < Ma < 1.63 $$

Correlations for \( \bar{C}_{\rm D} \) such as Eq. 9 may be useful to predict the drop velocity and position at the end of the deformation stage. However, as first noted by Pilch and Erdman (1987) they do a poor job of predicting the instantaneous acceleration, knowledge of which is needed for some of the instability models used to predict secondary atomization.

3.4 Deformation

To improve accuracy many authors have defined the drag coefficient as a function of deformation, such that the instantaneous drag is given by:
$$ F_{\rm D} = \frac{1}{2}\rho_{\rm a} U_{0}^{2} C_{\rm D} \frac{{\pi d_{\rm cro}^{2} }}{4} $$
(10)
where CD is the instantaneous drag coefficient. However, doing so also requires knowledge of the deformation versus time.
Hsiang and Faeth (1992) measured dcro and found it to increase approximately linearly as a function of time until fragmentation begins at Tini. A phenomenological analysis which considered the interaction between surface tension and pressure forces resulted in the following:
$$ \left( {{{d_{\rm cro} } \mathord{\left/ {\vphantom {{d_{\rm cro} } {d_{0} }}} \right. \kern-\nulldelimiterspace} {d_{0} }}} \right)_{\max } = 1 + 0.19We^{1/2}\quad We < 10^{2},\;Oh < 0.1, $$
(11)
Here (dcro/d0)max is the maximum deformation, which occurs at Tini.

At higher Oh the authors found that the maximum deformation at a given We decreased. It was postulated this was due to the slowing of the rate of deformation which reduces the relative velocity and hence the maximum deformation.

Helenbrook and Edwards (2002) used computational fluid dynamics to simulate over 3000 drops at their terminal velocity and reported the following relation between deformation and flow conditions:
$$ \left( {{{d_{\rm str} } \mathord{\left/ {\vphantom {{d_{str} } {d_{\rm cro} }}} \right. \kern-\nulldelimiterspace} {d_{\rm cro} }}} \right)_{\max } = 1 - 0.11We^{0.82} + 0.013\varepsilon^{0.5} N^{ - 1} Oh^{0.55} We^{1.1} $$
(12)
$$ We < 10,\quad Oh < 10, \quad 5 < \varepsilon < 500, \quad 5 < N < 15,\quad Re < 200, $$

Unfortunately their work is more applicable to drop tower than shock tube experiments. A similar analysis for impulsively accelerated drops is therefore warranted.

Finally, compared to the actual fragmentation event, the physics involved in drop deformation are relatively simple. This has led many to develop analytic predictive models for drop deformation that are widely used in spray simulations. Examples include the Taylor analogy breakup (TAB) model of O’Rourke and Amsden (1987) and the droplet deformation and breakup (DDB) model of Ibrahim et al. (1993). These and other such models are discussed in a subsequent section.

3.5 Drag

No matter how droplet deformation is found, a relation between deformation and drag is needed to determine the acceleration. The literature has identified three main factors that affect overall drag: (1) geometry, (2) internal circulation, and (3) unsteady effects. Here we will assume ε ≫ 1 such that the unsteady effects of virtual mass and Basset history forces can be neglected.

Hsiang and Faeth (1992, 1995) performed shock tube experiments and found that the instantaneous coefficient of drag, CD, can be approximated by a linear interpolation between the steady state value for a solid sphere and for a solid disk with both evaluated at equal Re. For the range of properties considered in their experiment, this indicates that internal circulation effects are minimal.

Liu et al. (1993) proposed:
$$ {{C_{\rm D} } \mathord{\left/ {\vphantom {{C_{\rm D} } {C_{\rm D-sphere} }}} \right. \kern-\nulldelimiterspace} {C_{\rm D-sphere} }} = \left( {1 + 2.632y} \right) $$
(13)
which gives the coefficient of drag as a linear function of deformation. Here y is the non-dimensional displacement of the drop equator, which can be written as y = 1 − (d0/dcro)2,  and CD-sphere is the coefficient of drag for a sphere at the same Reynolds number. Note that for no deformation (y = 0) the coefficient of drag of a sphere is recovered, and for a fully deformed drop (y = 1) the coefficient of drag of a disk is recovered.

Equation 13 can be improved by incorporating results from actual ellipsoidal shapes. A recent example is that of O’Donnell and Helenbrook (2005) who performed numerical calculations for drag over a solid ellipsoid. They proposed a correlation based on interpolation between drag on a sphere, ellipsoid with dcro 2dstr, and a disk, which was shown to be accurate within 1.5% for Re < 200. The interested reader is referred to O’Donnell and Helenbrook (2005) for the complete correlation. Further experimental or numerical results are needed to extend this correlation to Reynolds numbers on the order of 102 to 104, which are present in many spray applications.

Some attempts have been made to quantify the second order effects of internal circulation and unsteady flow on drag. Helenbrook and Edwards (2002) used numerical analysis to divide the deviation from solid sphere drag into the effects of deformation and those of internal circulation. In the limit of low We, the effects of deformation were negligible resulting in the following correlation for the change in coefficient of drag due to internal circulation:
$$ {{C_{\rm D} } \mathord{\left/ {\vphantom {{C_{\rm D} } {C_{\rm D-sphere} }}} \right. \kern-\nulldelimiterspace} {C_{\rm D-sphere} }} = \left( {\frac{2 + 3N}{3 + 3N}} \right)\left( {1 - 0.03N^{ - 1} Re^{0.65} } \right) $$
(14)
$$ 5 < \varepsilon < 500, \quad 5 < N < 15, \quad Re < 200 $$

Because this equation was derived for a spherical drop, its applicability to a highly deformed drop is unclear.

Finally, it is important to emphasize that the above correlations are for steady flow while in reality a drop is continuously accelerating and deforming. Clift et al. (1978) discuss virtual mass and Basset history effects for ellipsoidal shapes, but do not include the effects of transient deformation. Recently these effects were included in numerical simulations such as those by Quan and Schmidt (2006), which have shown that the unsteady drag is higher than the steady drag. To date, no correlations exist to predict this phenomenon and more work is needed.

4 Bag breakup (and critical We)

4.1 Qualitative description

Chou and Faeth (1998) divided bag breakup into four stages: (1) deformation, during which the drop evolves from its initial spherical shape into an oblate spheroid, (2) bag growth, during which the center of the drop gets blown downstream and forms a hollow bag attached to a toroidal ring, (3) bag breakup, where the bag bursts forming a large number of small fragments, and finally (4) ring breakup, where the toroidal ring forms a small number of large fragments. The first row in Fig. 4 illustrates typical bag breakup.

4.2 Physical mechanism

As in all cases of secondary atomization, bag breakup involves times on the order of ms, spatial dimensions on the order of μm, and unsteady, accelerating flows. Therefore no experimental investigations have been capable of measuring the local drop and ambient flow fields that lead to the formation and disintegration of the bag structure. However, recent developments in direct numerical simulation (DNS) of multiphase flows have provided some insight.

Han and Tryggvason (1999, 2001) used DNS to observe the formation of the bag structure due to separation in the ambient flow which leads to a pressure differential between the front stagnation point and the wake (see Fig. 8). This indicates that bag development has some dependency on Re and may not be observed under very low Re conditions.
Fig. 8

Bag breakup mechanism, based on the findings of Han and Tryggvason (1999, 2001)

In another study, Chou and Faeth (1998) discovered that after the bag formed the toroidal ring continued to grow at an expanding rate. This is a result of the outward force exerted on the bag and ring from the high pressure stagnation flow within the bag. After the bag ruptures, the ring continues to grow, but at a constant rate until it is no longer stable and breaks apart.

The mechanism leading to fragmentation of the bag is not well understood. Hwang et al. (1996) noted that the bag first breaks into ligaments that are aligned with the flow direction. For this reason the breakup of the bag cannot be explained as a capillary instability, which would predict ligaments perpendicular to the flow direction. Liu and Reitz (1997) postulated that small holes form in the bag due either to local disturbances in the ambient flow field or particulate impurities in the drop fluid. These holes would serve as inception sites for breakup.

4.3 Critical Weber number

Experimental evidence has consistently shown that once a given drop size and fluid properties are specified bag breakup occurs at a lower ambient velocity than any other breakup mode. As a result the study of bag breakup is especially important because it establishes the criteria for the onset of secondary atomization. For this reason the We value marking the beginning of bag breakup is typically referred to as the critical We, Wec.

As noted in the discussion of breakup morphology, the determination of regime transitions is somewhat arbitrary and different authors have reported different values. However, an important exception is the critical Weber number. For shock tube experiments in the limit of Oh < 0.1 all authors have reported a value of Wec = 11 ± 2. This, along with the fact that Wec marks the start of secondary breakup, is an important means of checking atomization models and direct numerical simulations. Any model or simulation that is unable to reproduce Wec is unlikely to correctly represent the physical mechanisms involved in drop breakup.

Attempts have been made to calculate Wec. Tarnogrodzki (1993) assumed the drop deforms into a flat disk with rounded ends. A duct flow solution was used to estimate the pressure within the flat part of the drop, and the surface tension force was used to find the pressure within the rounded part. Drag coefficients for a disk and sphere were used to approximate the dynamic pressure acting on the drop, which resulted in a solution of deformation versus time. Finally, breakup was assumed to occur when the radial motion of the drop ceased. While the model was able to calculate Wec to within the correct order of magnitude, it predicted that Wec continuously decreased with Oh, even for Oh < 0.1. This is in stark contrast to the experimental observation of constant Wec for Oh < 0.1 and an increase in Wec with an increase in Oh for higher Oh values.

Analyses, such as this, that involve a number of untested assumptions, in general do a poor job of predicting the complicated physics of secondary atomization. At this time, direct numerical simulation of the Navier-Stokes equations is the best known method of theoretically studying the process.

4.4 Behavior

Chou and Faeth (1998) studied the behavior of bag breakup in detail and found the four periods of bag breakup occur within the approximate non-dimensional times given in Table 3.
Table 3

Temporal evolution of bag breakup for Oh < 0.1 (Chou and Faeth 1998)

Deformation

0 < < 2

Bag growth

2 < T < 3

Bag breakup

3 < T < 4

Ring breakup

4 < T < 5

In addition they measured the cross stream diameter, dcro, of the expanding toroidal ring resulting in the following correlations.
$$ \begin{aligned}& {{{d_{\rm cro} }/ {d_{0} }} = 1.0 + 0.5T}\qquad\qquad\qquad {0 < T < 2} \qquad 13 < We < 20, \\& {{{d_{{\rm cro}} } / {d_{0} }} = 0.25T^{2} - 0.18T + 1.43} \quad {2 < T < 4} \qquad 0.0043 < Oh < 0.0427, \\& {{{d_{\rm cro} } / {d_{0} }} = 1.79T - 2.51} \qquad\qquad \quad {4 < T < 5} \quad \quad 633 < \varepsilon < 893,\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\qquad\qquad 1550 < Re < 2150 \end{aligned} $$
(15)

Study of the properties of the toroidal ring is crucial because it contains approximately 60% of the original volume, as reported by Chou and Faeth (1998). Also, the fragments formed from the toroidal ring are much larger than those formed from breakup of the bag; Chou and Faeth (1998) reported the diameter of the fragments formed from ring breakup are on average 30% of the original drop diameter while the mean diameter of the fragments formed from breakup of the bag is approximately 4% of the original drop diameter. The larger fragments dominate subsequent evaporation rates which are crucial to the performance of many spray-related systems.

5 Sheet-thinning breakup

5.1 Qualitative description

Sheet-thinning breakup occurs at higher initial relative velocities than bag breakup, and proceeds in a markedly different fashion. Following initial deformation, ligaments are stripped from the periphery of the drop where they break up into a multitude of small fragments. The process continues until the drop is completely fragmented, or until it has accelerated to the point at which aerodynamic forces are negligible. In the latter case, Hsiang and Faeth (1992) note that a drop core may remain at the completion of secondary atomization. The second row in Fig. 4 illustrates typical sheet-thinning breakup.

5.2 Physical mechanism (shear vs. sheet-thinning)

The physical process responsible for this mode is a point of controversy. Two distinct mechanisms have been put forth: (1) the “shear stripping” mechanism of Ranger and Nicholls (1969) and (2) the “sheet-thinning” mechanism of Liu and Reitz (1997).

Ranger and Nicholls (1969) postulated that shear from the ambient flow over the deformed drop results in the formation of a boundary layer inside its surface. This boundary layer becomes unstable at the drop periphery resulting in stripping of mass. The mechanism is typically referred to as “boundary-layer stripping” or “shear stripping” and is illustrated in Fig. 9.
Fig. 9

Shear stripping breakup mechanism

Chou et al. (1997) performed shock tube measurements of breakup and noted that drop viscosity significantly increased fragment sizes (ligaments and micro-drops), even in the range of Oh < 0.1. They considered this to be evidence in support of the shear stripping mechanism. In addition, they noted a transient period for drops of low viscosity in which the diameter of the fragments increased with time due to temporal growth of the boundary-layer. A phenomenological analysis was conducted which was able to predict the observed dependence on viscosity.

Igra and Takayama (2001) and Igra et al. (2002) studied breakup of a cylindrical water column in a shock tube. Their experimental setup allowed for visualization of density changes inside the liquid column using interferometeric fringes. After the initial deformation stage, no fringes were seen in the liquid column, which the authors interpreted as a uniform pressure field within the drop. They also observed breakup to be qualitatively similar to shear breakup of a spherical drop. These findings lead them to conclude that the drop must be fragmenting due to some effect other than pressure variations, thereby supporting the boundary layer stripping hypothesis. It is important to note that the fringe resolution of their systems was limited; therefore, the validity of these conclusions is questionable.

Using a model of boundary layer development attributed to Taylor (1963), Ranger and Nicholls (1969) proposed a model for the rate of liquid removal from a drop. Using a similar analysis, it is possible to compare the magnitude of the viscous force (shear) to the aerodynamic drag force (sheet-thinning). Here it is assumed that (1) the drop remains spherical, (2) there is zero internal circulation within the drop, (3) the size of the boundary layer is much less than the drop diameter so the drop curvature can be neglected in the momentum integral equations, (4) the pressure gradient term is neglected in the momentum integral equations, (5) the ambient velocity perpendicular to the surface is given by potential flow over a sphere, and (6) shear stripping occurs at the drop periphery. Consequently, only a half drop will be considered.

Figure 10 illustrates the problem. Boundary layer development is assumed to be symmetric such that r = d0 sin (2x/d0)/2. ud(y) and ua(y) are the drop and ambient phase boundary layer velocities, which Taylor (1963) approximated as:
Fig. 10

Boundary layer breakup model

$$ \frac{{u_{\rm d} }}{U} = Ae^{{{{ - y} \mathord{\left/ {\vphantom {{ - y} {\alpha_{\rm d} \sqrt x }}} \right. \kern-\nulldelimiterspace} {\alpha_{\rm d} \sqrt x }}}} $$
(16)
$$ \frac{{u_{\rm a} }}{U} = 1 - (1 - A)e^{{{{ - y} \mathord{\left/ {\vphantom {{ - y} {\alpha_{\rm a} \sqrt x }}} \right. \kern-\nulldelimiterspace} {\alpha_{\rm a} \sqrt x }}}} $$
(17)
U is the ambient velocity perpendicular to the drop surface and is given by U = 3U0 sin (2x/d0)/2.
In this case the momentum integral equations reduce to:
$$ \rho_{\rm a} \frac{\partial }{\partial x}\int\limits_{0}^{\infty } {u_{\rm a} (U - u_{\rm a} )\hbox{\rm d}y} = \mu_{\rm a} \left( {\frac{{\partial u_{\rm a} }}{\partial y}} \right)_{y = 0} $$
(18)
$$ \rho_{\rm d} \frac{\partial }{\partial x}\int\limits_{0}^{\infty } {u_{\rm d}^{2} \hbox{\rm d}y} = - \mu_{\rm d} \left( {\frac{{\partial u_{\rm d} }}{\partial y}} \right)_{y = 0} $$
(19)
Equating the shear stress at the boundary and assuming ≪ 1, Taylor (1963) showed A3 = (Nε)−1 and \( \alpha_{\rm d} = \sqrt {{{\mu_{\rm d} } \mathord{\left/ {\vphantom {{\mu_{\rm d} } {A\rho_{\rm d} U}}} \right. \kern-\nulldelimiterspace} {A\rho_{\rm d} U}}} . \) Finally the shear force on the drop surface is given by:
$$ F_{\mu } = \int\limits_{A} {\tau_{w} \hbox{\rm d}A} $$
(20)
where A is the half drop surface area and τw = −μd( ∂ud/∂y)y=0. Performing the integration and dividing into the drag force given by Equ. 8:
$$ \frac{{F_{\rm D} }}{{F_{\mu } }} = 0.1375\bar{C}_{\rm d} \sqrt {Re } $$
(21)

Refinement of the model to include the effects of the pressure gradient, drop deformation, and internal circulation are expected to affect the leading coefficient in Eq. 21 without altering the functional dependence on Re.

As in the analysis of Ranger and Nicholls (1969), the boundary layer stripping model is found to be a function of Re and can be expected to dominate the aerodynamic forces as Re decreases. However, this contradicts experimental data which indicate this breakup mode is essentially independent of Re.

Most experimental data are reported in terms of We and Oh. For that reason it can be shown that:
$$ \frac{{F_{\rm D} }}{{F_{\mu } }} = 0.1375\bar{C}_{\rm d} We^{1/4} Oh^{-1/2} \varepsilon^{-1/4} N^{1/2} $$
(22)

Assuming constant material properties, the role of the viscous force will increase with decreasing We and increasing Oh. This is opposite to the experimentally observed trends, which indicate this breakup mode occurs at higher values of We and lower values of Oh.

Liu and Reitz (1997) noted the discrepancy between experimentally observed trends and those given in Eqs. 21 and 22. As an alternative, they proposed the “sheet-thinning” mechanism in which the ambient phase inertia causes the periphery of the deformed drop to be deflected in the direction of the flow thereby forming a sheet. Following this, the sheet breaks into ligaments and then individual fragments. As in bag type breakup ambient phase viscosity must be present to cause flow separation and the formation of a wake. However, because the drop is deformed into a disk-like shape flow separation is expected to occur at all practical values of Re. Consequently, the sheet-thinning mechanism is considered an inviscid phenomenon with no dependence on Re. This mechanism is consistent with their experimental observations and is further discussed in the works of Lee and Reitz (1999) and Lee and Reitz (2001). It is illustrated in Fig. 11.
Fig. 11

Sheet-thinning breakup mechanism

In addition to being supported by the experimental observation that this breakup mode does not depend on Re (Liu and Reitz 1997, Lee and Reitz, 2000), the sheet-thinning mechanism is supported by a number of recent numerical simulations.

Han and Tryggvason (1999, 2001) observed flow structures similar to sheet-thinning type breakup, even in the limit of zero drop viscosity. Furthermore, they proposed that strong vorticity and backflow in the wake prevents the formation of the bag structure and the drop edge is eventually pulled back by the flow.

Khosla et al. (2006) performed a volume of fluid (VOF) simulation in which the spatial resolution was fine enough to resolve the drop phase boundary layer so the actual breakup event was simulated. The results were very similar to the sheet-thinning mechanism.

Wadhwa et al. (2007) performed numerical simulation of drop deformation at We = 100. Although the actual breakup event was not simulated, figures provided in the paper seem to show the thin deformed edge of the drop being pinched off as it becomes entrained in the recirculation behind the drop. This interpretation of numerical results supports the sheet-thinning mechanism.

Given the recent insight provided by accurate DNS, it can be concluded that the shear stripping model is incorrect, and this breakup regime is actually the result of sheet-thinning. The dependence of fragment sizes on drop phase viscosity as observed by Chou et al. (1997), which was originally explained using the shear stripping model, can also be attributed to instabilities which lead to the breakup of the sheet into ligaments and fragments and therefore can also be explained using the sheet-thinning mechanism. In addition, the sheet-thinning mechanism may better explain breakup in the transitional multimode regime, a detailed discussion of which is given in the section on multimode breakup.

5.3 Behavior

Hsiang and Faeth (1993) performed a phenomenological analysis to relate the relative velocity of the core drop after breakup, Ucore, to experimental conditions:
$$ \frac{{U_{0} - U_{\rm core} }}{{U_{0} }}\varepsilon^{1/2} \left( {1 + 3C} \right) = \frac{3}{4}\bar{C}_{\rm D} T_{\rm tot} $$
(23)
where \( C = {{3\bar{C}_{\rm D} T_{\rm tot} } \mathord{\left/ {\vphantom {{3\bar{C}_{\rm D} T_{\rm tot} } {4\varepsilon^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} }}} \right. \kern-\nulldelimiterspace} {4\varepsilon^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} }} \) and \( \bar{C}_{\rm D} \) is an average coefficient of drag, which Hsiang and Faeth (1993) suggested be approximately 5. In addition, Chou et al. (1997) gave a correlation for the mean relative velocity of the fragments that is based on the velocity of the core drop:
$$ \bar{U}_{\rm f} = U_{\rm core} + 9.5\varepsilon^{{{\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/ {\vphantom {{ - 1} 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} \left( {U_{0} - U_{\rm core} } \right) \quad \bar{V}_{\rm f} = 0 $$
(24)
$$ 125 < We < 375,\;0.003 < Oh < 0.04,\;3000 < Re < 12000,\;670 < \varepsilon < 990 $$
Here \( \bar{U}_{\rm f} \) is the mean relative velocity of the fragments in the stream-wise direction and \( \bar{V}_{\rm f} \) is the mean relative velocity of the fragments in the cross-stream direction.
Finally, Hsiang and Faeth (1993) noted that the drop core has a final We greater than Wec for the onset of secondary atomization in shock tube experiments. Assuming the criteria for the end of sheet-thinning is more closely related to that for gradually accelerating drops, such as those in a drop tower, Hsiang and Faeth (1993) derived an expression for the drop core We at the end of sheet-thinning breakup:
$$ We_{\rm core} = \frac{{\left( {4Eo_{\rm cr} We/3\bar{C}_{\rm D} } \right)^{1/2} }}{1 + C} $$
(25)
where Eocr is the Eötvös number \( (Eo_{cr} = {{a\left| {\rho_{\rm d} - \rho_{\rm a} } \right|d_{core}^{2} } \mathord{\left/ {\vphantom {{a\left| {\rho_{\rm d} - \rho_{\rm a} } \right|d_{{\rm core}}^{2} } \sigma }} \right. \kern-\nulldelimiterspace} \sigma }) \) at the end of sheet-thinning breakup, which Hsiang and Faeth (1993) suggested could be taken to be 16. Combining Eqs. 23 to 25 with knowledge of the drag coefficient and total breakup time, it is possible to calculate the velocity of the fragments and drop core along with the core diameter after sheet-thinning type breakup. Discussion of the size distribution of fragments is deferred to a later section.

6 Multimode

6.1 Qualitative description

Figure 12 illustrates the transition between bag breakup and sheet-thinning breakup. Here We increases from left to right.
Fig. 12

Transition from bag to sheet-thinning breakup (We increases from left to right)

A number of authors have proposed different modes to describe this transition. Pilch and Erdman (1987) term it “bag-and-stamen” mode. Cao et al. (2007) defined a “dual-bag breakup regime”, which they identified as unique, but can easily be considered one stage of the transition between bag and sheet-thinning breakup. Here the term “multimode” breakup, due to Hsiang and Faeth (1992), will be used.

Dai and Faeth (2001) divided the multimode breakup regime into “bag/plume” breakup and “plume/shear” breakup, which will be referred to here as “plume/sheet-thinning” breakup. During bag/plume breakup a bag forms as in bag breakup. However, the center core is blown downstream more slowly resulting in the formation of a so called plume. This is similar to the bag-and-stamen breakup regime originally described by Pilch and Erdman (1987). Figure 13 shows typical bag/plume breakup.
Fig. 13

Shadowgraph of multimode breakup (bag/plume)

Plume/sheet-thinning breakup differs from bag/plume breakup in that no bag is formed. Rather drops are stripped continuously from the plume in a manner similar to sheet-thinning breakup. Figure 14 shows typical plume/sheet-thinning breakup.
Fig. 14

Shadowgraph of multimode breakup (plume/sheet-thinning)

Dai and Faeth (2001) suggest that bag/plume breakup occurs for ~18 < We < ~40, and plume/sheet-thinning occurs for ~40 < We < ~80, both with Oh < 0.1. These choices provide significant overlap with transition We for bag breakup, as given in Table 2. This again highlights the fact that the transition between breakup modes is actually a continuous process and a single transition We value is an over-simplification.

6.2 Physical mechanism

Given the historically accepted descriptions of shear breakup due to boundary layer growth and bag breakup due to aerodynamic drag, little consideration has been given to the mechanism leading to multimode breakup. Rather it has been assumed to occur when both aerodynamic effects and shear effects are significant.

In contrast, if one adopts the sheet-thinning description, both sheet-thinning and bag breakup result from aerodynamic forces. In this case, a new explanation for the transition regime is needed. In addition, one must address why bag type breakup is seen at low levels of aerodynamic forces and sheet-thinning is seen at higher levels. Currently two theories exist: “the combined Rayleigh–Taylor/aerodynamic drag” mechanism and the “internal flow” mechanism.

Theofanous et al. (2004) studied Rayleigh–Taylor (R–T) instabilities which form on the leading surface of the deformed drop where a heavy fluid is accelerated into a lighter fluid. Assuming that both fluids are inviscid and the density ratio is large, the classical R–T analysis results in the following:
$$ \lambda_{\max } = 2\pi \sqrt {\frac{3\sigma }{{a\rho {}_{\rm d}}}} $$
(26)
Here λmax is the wavelength of the most destructive wave. More details on the R–T analysis are given in the section on catastrophic breakup. Theofanous et al. (2004) further assumed that at the initiation of breakup the drop is deformed into a flat disk of diameter, dcro. Therefore the simplified R–T analysis predicts dcro/λmax wavelengths will form on the leading surface of a deformed drop. Combining Eqs. 26 and 10:
$$ \frac{{d_{\rm cro} }}{{\lambda_{\max } }} = \frac{1}{4\pi }\left( {\frac{{d_{\rm cro} }}{{d_{0} }}} \right)^{2} \sqrt {C_{\rm D} \cdot We} $$
(27)

In this theory, growth of R–T instabilities creates an initial surface disturbance which is intensified by aerodynamic effects; this results in the breakup modes. We call this “the combined R-T/aerodynamic drag” mechanism. Theofanous et al. (2004) used this theory to show good agreement with breakup of drops in rarefied, supersonic flow. Joseph et al. (1999) used a similar analysis to show good agreement with breakup of highly viscous drops.

Figure 15 illustrates breakup when dcro/λmax = 3. This resembles the bag/plume mode observed by Dai and Faeth (2001) for 18 < We < 40. Similarly, bag breakup is expected when dcro/λmax = 1 and more complicated modes are expected for large dcro/λmax.
Fig. 15

Breakup due to combined Rayleigh–Taylor (R–T)/aerodynamic drag mechanism

In Fig. 16, Eq. 27 is used to predict the number of wavelengths on a deformed drop as a function of We. Equation 11 is used to predict the cross-stream diameter at breakup, and Eq. 13 is used to predict the instantaneous drag coefficient, where CD-sphere is taken to be 0.4. Reasonable agreement between the predictions and the experimentally observed ranges of Dai and Faeth (2001) is seen, especially in the bag regime where the predicted number of wavelengths is approximately 1 and in the bag/plume range where the predicted number of wavelengths is between 2 and 3.
Fig. 16

Number of wavelengths on deformed drop as predicted by combined Rayleigh–Taylor (R–T)/aerodynamic drag mechanism

Despite the agreement seen in Fig. 16 a few flaws exist in this theory. First, unstable surface waves are not commonly observed at low We. Also, in the most detailed numerical study known to exist, Khosla et al. (2006) successfully simulated bag, multimode, and sheet-thinning breakup. In all cases, surface waves did form. However, the wavelengths did not appear to control the breakup mode as proposed in the combined R-T/aerodynamic drag mechanism; rather the results of Khosla et al. (2006) are better supported by the “internal flow” mechanism, which is proposed here.

In this mechanism drop deformation leads to internal flow from the poles to the equator. In bag breakup the surface tension is sufficient to resist this flow resulting in the formation of the toroidal ring. However, as the rate of deformation increases with We a critical point is reached where surface tension is insufficient and the drop continuously elongates. Eventually the edges of the drop become so thin that they are carried away by the ambient flow.

Based on this hypothesis, the multimode regime occurs when the effect of the pressure difference across the drop (which tends to result in the formation of the bag structure) and the effect of the rapid deformation (which tends to result in the formation of the sheet-thinning structure) are comparable.

Based on the available experimental and numerical evidence, this mechanism appears to provide the best description of the physics that determine whether a drop deforms into a bag like or sheet-thinning like structure. There is no reliance on phenomena (such as surface wave growth) that has not been observed in experiments. Further numerical or experimental work that focuses on the deformation rate and internal flow may confirm this explanation.

6.3 Behavior

As discussed above, most authors have assumed multimode breakup to have properties which are some combination of those seen in bag and shear type breakup. For this reason, few experiments have been performed that target this regime.

Dai and Faeth (2001) measured the volume fraction of the bag, ring, plume and drop core and found that the volume fraction of the bag and ring decrease with increasing We. The volume fraction of the plume reaches a local maximum at approximately the transition between bag/plume and plume/shear breakup. Finally as We approaches that of the sheet-thinning regime the volume fraction of the drop core dominates.

7 Catastrophic breakup

7.1 Qualitative description

As noted by Faeth et al. (1995) the velocities and drop sizes involved in typical dense sprays are such that catastrophic breakup is not seen. Therefore, this is one of the least studied breakup regimes. Nevertheless, the study of catastrophic breakup is important because this limiting breakup regime occurs at the highest relative velocities. For this reason, analysis of this regime can shed light on the breakup mechanisms.

Unlike other breakup modes, the growth of unstable surface waves on the leading surface of the drop dominates breakup. The disruptive waves grow rapidly with time and eventually penetrate the drop, leading to fragmentation.

To study this phenomenon, Wierzba and Takayama (1988) used holographic interferometry to eliminate the cloud of small fragments that are typically seen in shadowgraphy. They could then observe stripping of drops from a large portion of the surface early on in the breakup process, rather than just the drop periphery as is seen in the sheet-thinning regime. At later times they observed the drop core to break up into large fragments, which in turn underwent stripping breakup.

7.2 Physical mechanism

Liu and Reitz (1993) noted that the wave growth may be described as either a Rayleigh–Taylor (R–T) or Kelvin–Helmholtz (K–H) instability. R–T instabilities occur when a density discontinuity is accelerated toward the lower density, while K–H instabilities occur when high relative velocities exist at an interface.

In secondary atomization R–T instabilities are typically assumed to occur at the front or rear stagnation points while K–H instabilities occur at the drop periphery where the relative velocity between the drop and ambient is the largest. However, because of the extremely large accelerations experienced by small drops, most authors have assumed R-T instabilities dominate. The relevant geometry is provided as Fig. 17.
Fig. 17

Rayleigh–Taylor instability breakup mechanism

Taylor (1950) showed that the acceleration of a heavy fluid into a light fluid will result in the growth of catastrophic surface waves at the interface. Chandrasekhar (1961) expanded Taylor’s analysis to include restorative surface tension effects. If an infinite planar surface is assumed, as shown in Fig. 17, along with a surface disturbance of the form:
$$ f(x,y,t) = Ae^{{i\left( {k_{x} x + k_{y} y} \right) + \omega t}} $$
(28)
where A is an unknown constant, the wavenumber k is \( k = \sqrt {k_{x}^{2} + k_{y}^{2} } \) and is related to the wavelength, λ, by k = 2π/λ, and ω is the exponential growth constant, Chandrasekhar (1961) showed that the ω and k are related:
$$ \begin{aligned} &- \left\{ {\frac{ak}{{\omega^{2} }}\left[ {\left( {\alpha_{1} - \alpha_{2} } \right) + \frac{{k^{2} \sigma }}{{a\left( {\rho_{1} + \rho_{2} } \right)}}} \right] + 1} \right\}\left( {\alpha_{2} q_{1} + \alpha_{1} q_{2} - k} \right) - 4k\alpha_{1} \alpha_{2} \hfill \\ &\quad + \frac{{4k^{2} }}{\omega }\left( {\alpha_{1} \frac{{\mu_{1} }}{{\rho_{1} }} - \alpha_{2} \frac{{\mu_{2} }}{{\rho_{2} }}} \right)\left\{ {\left( {\alpha_{2} q_{1} - \alpha_{1} q_{2} } \right) + k\left( {\alpha_{1} - \alpha_{2} } \right)} \right\} \hfill \\ &\quad + \frac{{4k^{3} }}{{\omega^{2} }}\left( {\alpha_{1} \frac{{\mu_{1} }}{{\rho_{1} }} + \alpha_{2} \frac{{\mu_{2} }}{{\rho_{2} }}} \right)^{2} \left( {q_{1} - k} \right)\left( {q_{2} - k} \right) = 0 \hfill \\ \end{aligned} $$
(29)

Here a is the initial acceleration, α1 = ρ1/(ρ1 + ρ2), α2 = ρ2/(ρ1 + ρ2), \( q_{1} = \sqrt {k^{2} + {{\omega \rho_{1} } \mathord{\left/ {\vphantom {{\omega \rho_{1} } {\mu_{1} }}} \right. \kern-\nulldelimiterspace} {\mu_{1} }}} , \) and \( q_{2} = \sqrt {k^{2} + {{\omega \rho_{2} } \mathord{\left/ {\vphantom {{\omega \rho_{2} } {\mu_{2} }}} \right. \kern-\nulldelimiterspace} {\mu_{2} }}} . \) Of foremost importance is the wave number kmax for which the growth rate is maximum, ωmax. Because it grows the quickest, this wave number (and its corresponding wavelength, λmax) is expected to dominant the instability.

In secondary atomization a drop of higher density is accelerated into the lower density ambient by aerodynamic drag. As a result, the interface is susceptible to R-T instabilities. The above analysis assumes that both fluids are initially at rest. However, in secondary atomization either the drop, the ambient, or both are initially in motion. As pointed out by Hwang et al. (1996) a stationary reference frame would indicate that the flow situation shown in Fig. 17 is expected to occur on the trailing surface of the drop. However, a reference frame fixed on the drop would indicate R–T instabilities should occur on the leading surface of the drop. Most experimental evidence points to the formation of wave like structures on the leading surface; therefore, most authors have assumed R–T waves form there.

R–T instability theory is typically used to characterize breakup in the catastrophic regime. Joseph et al. (1999) considered drops of low viscosity, as well as highly viscous drops. Nitrogen and helium were used as the ambient fluid so Eq. 29 was simplified by neglecting the ambient phase density and viscosity. The predicted wavelengths were compared to highly magnified views of the leading surface of the drop. Surface corrugations were observed that appeared to match the predictions.

Experimental evidence such as this, and similar results by Hwang et al. (1996), indicate that the catastrophic breakup mode is controlled by R–T instabilities.

8 Newtonian drop breakup times

In a review of spray structures, Faeth et al. (1995) noted that during the time they break apart drops may travel as much as 30 to 40 times their initial diameter, and due to the effects of aerodynamic drag, the largest and smallest fragments may be separated by more than 100 initial drop diameters. For this reason, knowledge of the characteristic times of secondary atomization is needed when attempting to create any model of that process. Numerous authors have proposed various characteristic times; not all agree. They are summarized below.

8.1 Initiation time (Tini)

Pilch and Erdman (1987) defined the initiation time as the time required for a drop to deform beyond the oblate ellipsoid shape. For example, the initiation time for bag breakup would be marked by the first sign of the formation of the bag. This time is important because it marks the time when the models of a deforming ellipsoid discussed in the section on deformation and vibration are no longer valid. The correlation proposed by Pilch and Erdman (1987) is given in Eq. 30.
$$ T_{\rm ini} = 1.9\left( {We - We_{\rm c} } \right)^{ - 0.25} \left( {1 + 2.2Oh^{1.6} } \right)\quad We < 10^{4} , \quad Oh < 1.5 $$
(30)
Hsiang and Faeth (1992) have also proposed a correlation for Tini:
$$ T_{\rm ini} = {{1.6} \mathord{\left/ {\vphantom {{1.6} {\left( {1 - Oh/7} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - Oh/7} \right)}}\quad We < 10^{3},\quad Oh < 3.5 $$
(31)
as has Gelfand et al. (1975):
$$ T_{\rm ini} = 1.4(1 + 1.5Oh^{0.74} ) \quad We \approx We_{\rm c} ,\quad Oh < 4.0 $$
(32)
All three expressions are plotted in Fig. 18. Reasonable agreement is seen at low Oh where Tini ≈ 1.5. Clearly at Oh > 2 the correlations do not mach one another. It is unknown which correlation is most accurate and more work is obviously needed.
Fig. 18

Initiation time (Tini) from Eqs. 30 to 32

8.2 Total breakup time (Ttot)

Pilch and Erdman (1987) defined the total breakup time, Ttot, as the time when all fragmentation has ceased. In the limit of low viscosity (Oh < 0.1) they proposed the following correlation to the experimental data that is contained in Fig. 19. [Note that the third equation has been corrected for a typographical error in the Pilch and Erdman (1987) publication.]
Fig. 19

Total breakup time (Ttot) from Eq. 33 (Oh < 0.1)

$$ \begin{array}{*{20}l} {T_{\rm tot} = 6\left( {We - 12} \right)^{ - 0.25} } & {12 < We < 18} \\ {T_{\rm tot} = 2.45\left( {We - 12} \right)^{0.25} } & {18 < We < 45} \\ {T_{\rm tot} = 14.1\left( {We - 12} \right)^{ - 0.25} } & {45 < We < 351} \\ {T_{\rm tot} = 0.766\left( {We - 12} \right)^{0.25} } & {351 < We < 2670} \\ {T_{\rm tot} = 5.5} & {2670 < We\tilde{ < }10^{5} } \\ \end{array} $$
(33)

Note how the transitional Weber numbers for Eq. 33 roughly correspond to the transitional Weber numbers of the breakup morphology given in Table 2. This suggests that the physics governing breakup times is different for each breakup mode, and is further support for dividing secondary atomization into numerous breakup morphologies.

Dai and Faeth (2001) studied the total breakup time in the multimode regime and, like Pilch and Erdman (1987), noticed a local maximum near We = 40 similar to that given in Eq. 33. This local maximum occurs at the transition of bag/plume and plume/sheet-thinning breakup, as defined in the discussion of the multimode breakup regime.

For the case of viscous drops Pilch and Erdman (1987) cited the correlation given by Gelfand et al. (1975). However, they noted that Eq. 33 is more accurate than Eq. 34 for drops of low viscosity (Oh < 0.1). Note that Pilch and Erdman (1987) have a typographical error in their republication of Eq. 34; it has been corrected here.
$$ T_{\rm tot} = 4.5\left( {1 + 1.2Oh^{0.74} } \right)\quad We \approx We_{c} ,\quad Oh < 0.3 $$
(34)
Similarly, Hsiang and Faeth (1992) proposed the following relation:
$$ T_{\rm tot} = {5 \mathord{\left/ {\vphantom {5 {\left( {1 - Oh/7} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - Oh/7} \right)}}\quad We < 10^{3},\quad Oh < 3.5 $$
(35)
Both equations are presented in Fig. 20. Reasonable agreement is seen at low Oh where Ttot ≈ 5.0. At Oh > 0.5 the correlations do not match one another. It is unknown which correlation is most accurate and more work is needed.
Fig. 20

Total breakup time (Ttot) from Eq. 34 and Eq. 35

9 Non-Newtonian drop studies

As mentioned in the previous sections, a vast number of efforts have been made to investigate secondary breakup of Newtonian drops. Here we focus on the far fewer studies where non-Newtonian drop breakup was considered. Our discussion follows the same format as for Newtonian drops.

9.1 Description

A non-Newtonian liquid does not exhibit a linear relationship between shear stress and rate of strain. This feature has made their use popular in a variety of applications where the liquid should have a low effective viscosity during spray formation (high rate of strain) and a higher effective viscosity when on a target (low rate of strain). Examples of non-Newtonian liquids frequently encountered in daily tasks are paints and hair care products. Additional examples include thermal barrier coatings and, most recently, gelled fuels.

It is important to note that this oft-times desirable rheological characteristic causes the secondary breakup behavior of non-Newtonian drops to differ from that of Newtonian liquids. This has been shown by Wilcox et al. (1961), Matta and Tytus (1982), Matta et al. (1983), Arcoumanis et al. (1994, 1996), Joseph et al. (1999, 2002), and most recently by López-Rivera and Sojka (2008).

In contrast to the Newtonian case, there is not enough data to provide a clear consensus as to either common characteristics or processes for non-Newtonian drop secondary breakup. However, all authors observe bag breakup. Bag-and-stamen breakup has been reported by Joseph et al. (2002). In marked contrast to the Newtonian case is the rupture of the bag into a net of filaments that may or may not break up subsequently. In addition, all groups report some form of stripping/shearing. However, in yet another clear departure from Newtonian drop breakup a filament net forms from mass that is stripped/sheared off the drop perimeter. This feature has been observed by all groups, and demonstrates why non-Newtonian secondary atomization is considerably more complex—several stages of breakup are observed instead of ligaments being continuously eroded from the drop surface and then rapidly disintegrating into small droplets.

9.2 Non-dimensional groups

More physical processes and fluid properties are important in the secondary breakup of non-Newtonian drops than for the Newtonian case because of the increased rheological complexity. Non-Newtonian drop breakup geometries are also complicated, as evidenced by the net of filaments that forms when a bag breaks up and the multi-stage ligament breakup during stripping/shearing. These features make mathematical and numerical analysis even more challenging than for the already difficult Newtonian case.

Surprisingly, while previous researchers studying Newtonian drops typically describe their findings in terms of non-dimensional groups (We, Oh, Re, Ma, N and ε), that approach has been rejected by many of those investigating non-Newtonian secondary breakup. As examples, Arcoumanis et al. (1994, 1996) declined to correlate their secondary breakup data using We because they argue that Wec cannot be easily defined due to the shear-dependent viscosity. This is certainly true for the liquids they used since the diffusion coefficient for their polymer (K125) in their solvent (triethyl phosphate) is such that the polymer surface concentration would not be uniform over the drop at any instant in time. However, polymers with molecular weights much lower than that of K125 (~4 × 106) may diffuse rapidly enough to remove surface tension variations from consideration and allow separate investigation of elastic effects. This topic should therefore be investigated.

Perhaps because of concerns about surface tension variations, the possibility of a We versus elastic-Oh regime plot, analogous to the one by Hsiang and Faeth (1995) that is shown here as Fig. 5, has not been mentioned. In fact, only recently has non-Newtonian breakup behavior been related to We (Joseph et al. 1999; López-Rivera and Sojka 2008). Results are preliminary and much work remains to be done.

A list of dimensionless groups that might be expected to play roles in non-Newtonian drop secondary breakup is provided in Table 4. Logic for their choices follows that presented in the discussion after Table 2.
Table 4

Dimensionless groups that might be important in non-Newtonian drop secondary breakup

  

Inelastic liquid

Elastic liquid

Weber number

We

\( \frac{{\rho_{\rm a} U_{0}^{2} d_{0} }}{\sigma } \)

\( \frac{{\rho_{\rm a} U_{0}^{2} d_{0} }}{\sigma } \)

Ohnesorge number

Oh

\( {{We} / {Re_{NN}^{1/2}} } \)

\( {{We} / {Re_{NN}^{1/2}} } \)

Liquid Reynolds or Weissenberg number

 

\( Re_{NN} = \frac{{\rho_{\rm a} U_{0}^{2 - n} d_{0} }}{K} \)

\( Wi = \frac{{\lambda^{\left( 1 \right)} U_{0} }}{{d_{0} }} \)

Ambient phase Reynolds number

Re

\( \frac{{\rho_{\rm a} U_{0} d_{0} }}{{\mu_{\rm a} }} \)

\( \frac{{\rho_{\rm a} U_{0} d_{0} }}{{\mu_{\rm a} }} \)

Density ratio

ε

\( \frac{{\rho_{\rm d} }}{{\rho_{\rm a} }} \)

\( \frac{{\rho_{\rm d} }}{{\rho_{\rm a} }} \)

Viscosity ratio

Ν

\( \frac{{\mu_{\rm eff} }}{{\mu_{\rm a} }} \)

\( \frac{{\mu_{\rm elastic} }}{{\mu_{\rm a} }} \)

Mach number

Ma

\( \frac{{U_{0} }}{c} \)

\( \frac{{U_{0} }}{c} \)

The same aerodynamic forces deform a non-Newtonian drop and can cause it to fragment. They are resisted by the same surface tension force so We is again an important parameter (under restrictions mentioned above).

As for Newtonian drops, “viscosity” hinders deformation and also dissipates (or stores) energy supplied by aerodynamic forces so Oh should be included. In the case of non-Newtonian liquids, there are at least two possible expressions for Oh—one for purely viscous, such as power law liquids and one for visco-elastic, such as Oldroyd B, liquids.

One must again consider the gas Re because it is still the ratio of aerodynamic forces to ambient viscous forces, the drop phase-to-ambient phase density ratio, ε, because inertial effects are present, and the drop phase-to-ambient phase “viscosity” ratio, Ν. There is a different form of N for viscous and visco-elastic non-Newtonian liquids. Finally, the Mach number, Ma, is important when considering compressibility effects.

As was done for Newtonian liquids, experimentally observed times might also be made dimensionless through the scaling given by Eq. 4. It might also be more appropriate to use relaxation or retardation times for elastic liquids.

9.3 Transition We (low Oh), dependency on Oh, and dependency on other non-dimensional parameters

Transition boundaries between breakup modes have yet to be quantified as functions of We, Oh or other parameters such as the density ratio (ε), Wi, ReNN, or gas-phase Re. In fact, at this time it is unclear as to which dimensionless groups play dominant roles in the purely viscous and visco-elastic cases. As such, dependence of transition We on other non-dimensional parameters has yet to be reported. Work on this topic is clearly required.

9.3.1 Influence of polymer concentration

Wilcox et al. (1961) found that increasing polymer concentration from values as low as 0.1% retards breakup in high-velocity airstreams at relative velocities up to Ma = 1. Inhibition was not always observed at Ma = 1, but the authors predict it would be seen if concentration was increased to more than 2%. As expected, higher retardation was obtained for lower velocities.

9.4 Drop deformation and vibrational breakup

9.4.1 Qualitative description and physical mechanism

Oscillation has not been observed for non-Newtonian drops, probably because experiments have yet to be performed at We low enough to observe such behavior. Oscillatory breakup has not been observed either, for the same reason.

While low We drop deformation has not been observed for non-Newtonian liquids, Joseph et al. (1999) did observe that behavior at the start of their high We tests. Their data lead them to propose the same physical mechanism that is accepted for Newtonian drop deformation—an unequal static pressure distribution across the drop.

9.4.2 Behavior

Data from Joseph et al. (1999) demonstrate that early drop core motion obeys a constant acceleration model. As such Eq. 8 could be applied if \( \bar{C}_{\rm D} \) was known. Joseph et al. (1999) do not provide \( \bar{C}_{\rm D} \) values. The authors state that the drop acceleration magnitude falls off as complete breakup approaches.

9.4.3 Deformation and drag

Experimental deformation data are currently unavailable for non-Newtonian drops. Analytical models, such as the TAB-model or its derivatives, could be modified to include purely viscous or visco-elastic non-Newtonian effects. This has yet to be done.

Non-Newtonian drop drag coefficient magnitudes have not been reported. Values might be extracted from the Joseph et al. (1999) acceleration data (their Table 2) and initial condition data (their Table 1), although those authors did not do so.

9.5 Bag breakup

9.5.1 Qualitative description

Like Newtonian liquids, non-Newtonian drops also exhibit bag breakup. The drop deforms with the same thin hollow bag attached to a thicker toroidal rim. The bag is blown downstream and disintegrates first, forming a net of filaments. The filaments undergo breakup, as does the toroidal rim. This behavior has been reported by Arcoumanis et al. (1996). However, Joseph et al. (1999) found this mechanism at very high We for very viscous Newtonian fluids, but not for non-Newtonian liquids. Finally, Joseph et al. (2002) observed bag breakup of visco-elastic drops.

9.5.2 Physical mechanism

Joseph et al. (1999) attribute bag formation to a R–T instability whose wavelength is comparable to or larger than the diameter of the drop. This explanation is similar to that proposed by Theofanous et al. (2004), who also used a shock tube when performing their experiments. Arcoumanis et al. (1996) agree with this interpretation and note the presence of small amplitude short wavelength disturbances on their drops.

All groups report that bag breakup for non-Newtonian drops results in a net of ligaments that form as the bag disintegrates. The ligaments may then undergo breakup.

9.5.3 Wec and behavior

Unlike for Newtonian drops, there is scant information available for non-Newtonian drop Wec. Deformation and breakup time data, plus deformation magnitude results are also largely missing. The exception is the visco-elastic liquid data from Joseph et al. (1999), although these results must be viewed with caution for two reasons. First, combining the characteristic time expression from Bird et al. (1987) with the polyethylene oxide (PEO)-water intrinsic viscosity relationship from Kalashnikov and Askarov (1989), and inserting the very high PEO molecular weights (4 × 106) and concentrations (2%) Joseph et al. (1999) used demonstrates that liquid characteristic times (~1 s) will be several orders of magnitude greater than the experimentally measured breakup times (<1 ms). As such these liquids may exhibit little, if any, elastic behavior. Second, the Oh values provided for these liquids are so high (>80) that viscous effects will almost certainly dominate. Consequently, the Joseph et al. (1999) results are probably more indicative of purely viscous liquid behavior than that for visco-elastic ones.

9.6 Sheet-thinning breakup (critical speed)

9.6.1 Qualitative description

The thinning/stripping mechanism observed for non-Newtonian drops resembles that observed for Newtonian ones in some respects. Ligaments are continuously eroded from the drop surface, which disintegrate rapidly thereafter resulting in numerous small fragments.

9.6.2 Physical mechanism

Arcoumanis et al. (1994) studied the initial stages of weakly visco-elastic drop breakup. Their results showed that drops entering the air flow had a wave appear on their surface, as reported previously by Liu and Reitz (1993), which then peeled away and broke up into ligaments that were joined by a thin sheet. The sheet expanded and the ligaments stretched in the downstream direction where they fragmented.

For more strongly visco-elastic liquids, Arcoumanis et al. (1994) did not observe breakup, although a wave did begin to form on the drop surface and was peeled back. The drop stretched downstream, ligaments were formed and were again joined by a sheet, with the sheet being thicker than for the less visco-elastic drops. Breakup could be prevented by giving the liquids sufficient elastic character.

This behavior is in marked contrast to that reported for Newtonian systems. It is likely due to the elastic nature of the liquids since raising the concentration of polymer increased the number of ligaments formed and the thickness of the sheet. Wilcox et al. (1961) also observed that non-Newtonian liquids form ligaments that break up into larger particles than those produced by Newtonian ones that seem to experience a stripping process leading to very small particles.

There may be additional differences between visco-elastic and Newtonian liquid secondary breakup. Arcoumanis et al. (1994) state that only the initial stages of breakup are shown in their photographs and it is possible that (1) drops are forming in later stages or (2) the resolution of their film is not sufficient to record small fragments.

In a further study, Arcoumanis et al. (1996) extended their previous work (Arcoumanis et al. 1994) to remove uncertainty about the existence of drops as a result of breakup. Their results showed fragments being formed at a distance 20 times the diameter of the original drop. The breakup process that they observed was very similar to that found in their previous work. However, the ligaments were observed to form droplets. The distance from the main droplet over which ligaments are linked was also found to increase with increases in polymer concentration, and to decrease with air velocity.

Finally, Joseph et al. (1999) studied the breakup of viscous and visco-elastic drops (1 mm) in the high speed airstream produced by a shock tube at very high We (11,700–169,000) and Oh (0.002–82.3). These authors also observed that threads and ligaments of liquid arise immediately after breakup, in contrast to Newtonian liquids, for which droplets were seen at Ma as high as 3. Joseph et al. (1999) observed no breakup of some of these threads even at high Ma. In addition, no drops were seen as a result of their breakup.

9.6.3 Critical speed

Arcoumanis et al. (1994) declined to correlate their data in terms of We because they claim that Wec cannot be easily defined due to the shear-dependent viscosity of their fluids. Instead, the authors considered a critical velocity above which there is no further breakup. Since Arcoumanis et al. (1994) observed no drops as a result of breakup, they defined the critical speed as the air jet speed at which many marks were observed on an impaction card (and immediately below which only a few were seen).

These authors also showed that increases in polymer concentration lead to increases in the critical speed of breakup. This supports the retardation in the breakup process reported by Wilcox et al. (1961).

9.7 Multimode breakup

9.7.1 Qualitative description

For the case of non-Newtonian drops, only the first phase of this breakup mechanism, the bag/plume, has been observed. It is qualitatively similar to that seen for Newtonian drops.

One of the first groups of researchers to report it was Joseph et al. (1999), who observed this mode at a very high We (~42,000) for very viscous liquids, but not for viscoelastic drops. This regime was also observed by Joseph et al. (2002), who did observe it for viscoelastic drops.

9.7.2 Physical mechanism

The physical mechanism leading to the multimode breakup is believed by Joseph et al. (1999, 2002) to be the development of R–T instabilities on the surface of the drops.

9.8 Catastrophic breakup

9.8.1 Qualitative description

In contrast to Newtonian liquids, catastrophic breakup is particularly important for non-Newtonian drops because extremely high relative velocities are often required for their fragmentation to occur. This mode has been investigated by Joseph et al. (1999). It may also have been investigated by Matta and Tytus (1982) and Matta et al. (1983).

9.8.2 Physical mechanism

Joseph et al. (1999) argue that R–T instabilities are the cause of non-Newtonian drop catastrophic breakup. Support for their conclusions comes from comparison of experimental results with predictions from a (purely viscous) R–T analysis for both the critical wavelength and its growth rate (Joseph et al. 2002). Agreement is within a few percent.

The Joseph et al. (1999, 2002) agreement between theory and experiment is perhaps surprising since their analysis is purely viscous (in the limit of very short retardation times). This also suggests that their non-Newtonian liquids had characteristic times much greater than those of the breakup events and so should not be categorized as elastic for the purposes of secondary breakup.

9.9 Non-Newtonian drop breakup times

9.9.1 Initiation time (Tini) and total breakup time (Ttot)

Arcoumanis et al. (1994) provide total breakup time data and report Ttot rises with an increase in polymer concentration. They do not provide a relationship between Ttot and We or any more suitable dimensionless group.

Joseph et al. (1999) report Tini for visco-elastic drops. It is defined as the time at which disintegration starts. They observe no, or minimal, variation in Tini throughout the range of rheologies considered.

Joseph et al. (2002) performed an R-T analysis for an Oldroyd-B fluid and used the results to provide a correlation for the Joseph et al. (1999) data. In a manner similar to that of Weber (1931), they defined breakup to occur when a disturbance reached a multiple (10) of its initial amplitude. This lead to Tini = ln (10)/ω where ω is the R–T disturbance growth rate.

9.10 Non-Newtonian fragment size and velocity distributions

One of the few studies supplying non-Newtonian liquid fragment sizes is that performed by Wilcox et al. (1961). These authors observed that fragments of solutions with polymers added were 1.5 orders of magnitude larger than fragments produced by Newtonian liquids.

Matta and Tytus (1982) also studied the breakup of viscoelastic fluids (<0.5 cm) injected into the high velocity airstream (200 m/s) of a wind tunnel. Their experimental results showed that the measured fragment MMD was an order of magnitude larger than values predicted for a Newtonian fluid of similar viscosity magnitude. Their results were found to correlate with the relaxation time obtained from a die swell experiment. From this, it was thought that breakup does not follow a shear mechanism, but an elongational one instead. The first normal stress difference was also found to correlate the breakup results. However, since the breakup deformation rate was unclear, the relaxation time was preferable for predicting particle size.

In a subsequent investigation, Matta et al. (1983) extended their previous work (Matta and Tytus 1982), to identify the proper variable for drop size predictions. For this purpose, heated fluids were considered, since the first normal stress difference is known to decrease more rapidly than the relaxation time with increases in temperature. Instead of a wind tunnel, a helium activated firing device was used. The test conditions were comparable (Ma = 1), although larger diameter (7.6 cm) viscoelastic slugs were employed. It was found that increasing the polymer concentration increased the average drop size, supporting previous findings. Furthermore, results from the tests performed at ambient temperature were observed to correlate with both the relative relaxation time and the first normal stress difference. However, the results of the tests with heated fluids were only correlated using the relative relaxation time, making this parameter the more convenient for drop size predictions.

This concludes the discussion of non-Newtonian drop breakup.

10 Fragment size and velocity distributions

Fragment size distributions are one of the most important but difficult to measure properties of secondary atomization. Historically, techniques to measure fragment sizes have been limited in their accuracy. Among the viable methods were rapid solidification of the fragments and holography. Both methods are time consuming, difficult to set up, and results are hard to analyze.

Recently, the commercial availability of PDA and other optical drop sizing methods have resulted in more rapid and accurate measurements. However, these devices require a continuous process and cannot be easily used in shock tube experiments because their measurement volumes are typically small compared to the region through which fragments pass. As a result, only limited experimental data exist and more research is warranted.

Drop size distributions are often described by two or more characteristic diameters. Here the nomenclature of Mugele and Evans (1951) will be used.

A representative diameter is given by:
$$ D_{\rm pq} = \left[ {\frac{{\int\nolimits_{0}^{\infty } {D^{\rm p} f_{0} (D)\hbox{d}D} }}{{\int\limits_{0}^{\infty } {D^{q} f_{0} (D)\hbox{d}D} }}} \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {p - q}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${p - q}$}}}} $$
(36)
where p and q are positive integers and f0(D) is the number PDF. Common examples include the arithmetic mean diameter, D10, the volume mean diameter, D30, and the Sauter mean diameter, D32.

Simmons (1977a, b) studied the drop size distribution for sprays formed using a large number of aircraft and industrial gas turbine nozzles where secondary atomization was thought to play a crucial role in determining the final size distribution. The fragment mass median diameter (equal to MMD for constant density) and D32 were found to be related by MMD/D32 ∼ 1.2. In addition, given either MMD or D32 the fragment volume PDF, f3(D), could be approximated as root/normal. Finally Simmons (1977a, b) found the maximum fragment size to be approximately three times MMD.

Following the work of Simmons (1977a, b), Hsiang and Faeth (1992, 1993) used holography to measure drop size distributions for Oh < 0.1. In the bag and multimode regimes, the root normal distribution with MMD/D32 ∼ 1.2 proposed by Simmons (1977a, b) was found to fit the data reasonably well. Furthermore, after removal of the drop core, this same distribution was found to be applicable in the sheet-thinning regime. The complete fragment size distribution can be found by using Eqs. 23 to 25 to find the drop core size and velocity.

Having confirmed that the approach of Simmons (1977a, b) is applicable to secondary atomization, the last piece of knowledge needed to determine drop size distributions a priori is either D32 or MMD. To this end Hsiang and Faeth (1992) conducted a phenomenological analysis by considering the size of the drop phase boundary layer, which is thought to determine the size of the fragments in shear breakup. This yielded:
$$ We_{{D_{32} }} = C\varepsilon^{1/4} Oh^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} We^{{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4}}} $$
(37)
$$ We < 1000,\quad Oh < 0.1,\quad 580 < \varepsilon < 1000 $$
where \( We_{{D_{32} }} = {{\rho_{\rm a} D_{32} U_{0}^{2} } \mathord{\left/ {\vphantom {{\rho_{\rm a} D_{32} U_{0}^{2} } \sigma }} \right. \kern-\nulldelimiterspace} \sigma } \) and C is a constant of proportionality. For the range of parameters considered, Hsiang and Faeth (1992) used = 6.2 and Eq. 37 was found to reasonably predict fragment D32. However, they noted that the range of the density ratio was relatively narrow and further testing was needed.
Equation 37 was derived from the assumed physics of shear type breakup; therefore, its applicability to bag and multimode regimes is limited. For this reason, Wert (1995) proposed a new correlation for D32 based on the physics of bag breakup. Because a large portion of the original drop mass is contained in the toroidal rim, D32 was assumed to be governed by the growth of capillary instability waves on this rim. This resulted in the following:
$$ We_{{D_{32} }} = C\left[ {We\left( {T_{\rm tot} - T_{{\rm ini}} } \right)} \right]^{2/3}\quad 12 < We < 80,\quad Oh < 0.1 $$
(38)
where C is a constant of proportionality. Tini and Ttot can be found using Eqs. 30 and 33, respectively.

Based on available data in the bag and multimode regime, Wert (1995) suggested = 0.32 and stated that Eq. 38 outperforms Eq. 37 in these regimes. The authors noted that Eq. 38 may be applicable for Oh > 0.1. However, this has yet to be tested.

The above mentioned distributions were determined experimentally. However, the goal of many researchers has been the determination of drop size (and velocity) distributions from theory. One possibility is the maximum entropy formalism (MEF). Here constraints are placed on the fragment size and velocity distributions. Examples include all drops being spherical, mass being conserved, and estimates for momentum and energy transferred to the drops from the surrounding gas. From this a least biased PDF is computed. Babinsky and Sojka (2002) provide a thorough discussion of the development and application of the MEF to sprays applications. Significant findings will be discussed here, along with works completed subsequent to that review.

MEF has the capability of predicting both fragment size and velocity distributions. However, knowledge of at least two characteristic diameters is required a priori. The requirement of at least two characteristic diameters proves problematical.

Cousin et al. (1996) proposed the use of linear stability theory to predict one characteristic fragment diameter. However, no theoretical method exists to predict the second characteristic diameter so either experimental results or an ad hoc assumption are required.

Dumouchel and Boyaval (1999) expanded on the work of Cousin et al. (1996) by noting that the choice of representative diameter is paramount to the accuracy of the final distribution. For example, D43 is the best choice for determining the volume based distribution because it is very close to the mean of the distribution. Having made these observations, Dumouchel and Boyaval (1999) provide a recommended method to determine the best choice of model constraints based on the distribution being sought.

Li et al. (2005) noted that the MEF is applicable to isolated systems in thermodynamic equilibrium. However, many sprays do not meet these requirements. Therefore, Li et al. (2005) proposed a new model with additional constraints to track the degree of deviation from the equilibrium assumption. The result was a better fit to experimental data. However, this introduced the need for more characteristic diameters which are not easy to predict a priori. This study helps to understand some of the reasons for inaccuracies in the MEF; however, the practical application of this method is limited.

Dumouchel (2006) included an ad hoc physical minimum and maximum drop diameter in their MEF analysis. They were based on the observation that infinitesimally small drops are impossible due to the presence of surface tension, as are infinitely large drops due to instabilities. Their results show that a minimum of three parameters must now be known a priori. This only exacerbates the problem.

In summary, MEF can be used to correlate the fragment size and velocity distributions. However, MEF cannot be considered predictive in practice because constraints are needed a priori, at least some of which must be determined using experimental measurements or come from ad hoc assumptions.

A few other methods have been proposed to predict fragment size distributions. Zhou et al. (2000) studied the fractal characteristics of sprays both theoretically and experimentally. Their model showed some predictive capability. However, some measurements were needed a priori, and more work is needed.

Babinsky and Sojka (2002) discussed the application of the discrete probability function (DPF) approach which uses stability analyses to model the (primary) breakup process coupled with an assumed probability distribution of the input parameters. The DPF method is unlikely to work for secondary atomization because stability analyses, or other closed form predictions of fragment size, are unavailable.

10.1 Non-Newtonian fragment size

The only studies reporting fragment size distributions for non-Newtonian drops are those performed by Matta and coworkers. Their results are contradictory.

In their original study, Matta and Tytus (1982) stated that non-Newtonian fragment diameters followed a normal distribution, although there was some evidence of bi-modality. This was in contrast to their Newtonian liquid data which were log-normally distributed.

In their second investigation, Matta et al. (1983) used the same liquids and claimed that fragment sizes obeyed a log-normal distribution. They did not comment on the contradiction.

A possible explanation for the inconsistency is that fragments formed in their first study were the result of both primary and secondary breakup, since they were injecting a coherent liquid jet into their airstream. This would also explain the evidence of a bi-modal distribution.

It is obvious that much work remains to be done in this area.

11 Modeling efforts

To date no single model has been created that describes all aspects of secondary atomization accurately. Gelfand (1996) considered droplet deformation and breakup with regard to aerodynamic loading, liquid stripping, and stability analyses. None of those models were found to completely explain breakup, and the author surmised that all must be considered in parallel.

Berthoumieu et al. (1999) created a secondary atomization model based entirely on experimentally determined correlations such as those given in the above sections. It did a poor job of predicting the actual distribution of fragments.

Chryssakis and Assanis (2005) had more success by combining experimental correlations for deformation and drag with some theoretical wave growth and boundary layer stripping models. Nevertheless, the model is still only applicable within the range of parameters covered by experiments. To overcome these difficulties much focus has been placed on models based on the assumed underlying physics.

11.1 Analytical

Compared to the fragmentation process seen in other modes of breakup, drop distortion and oscillation is governed by relatively simple physics and therefore lends itself to analytical modeling. One of the first such models was the Taylor analogy breakup (TAB) model proposed by O’Rourke and Amsden (1987). Their model is based on an analogy by Taylor (1963) between an oscillating and distorting droplet and a spring-mass system in which the spring force, external force and dampening are respectively analogous to surface tension, aerodynamic forces, and drop viscosity. Breakup is assumed to occur when dstr → 0. Finally, energy conservation is used to determine the fragment sizes after breakup with the distribution assumed to be χ-squared.

The literature contains many studies in which the TAB model is used to simulate secondary atomization and sprays, including O’Rourke and Amsden (1987), Liu and Reitz (1993), Hwang et al. (1996), Tanner (1997), Lee and Reitz (1999), Park et al. (2002), Apte et al. (2003), Park and Lee (2004), Lee et al. (2004), Trinh and Chen (2006), and Trinh et al. (2007) among others. These studies have pointed to a number of shortcomings in the TAB model.

Hwang et al. (1996) showed that the predicted breakup time most closely matches the initiation time, at which point breakup is assumed to occur instantaneously. However, experimental evidence has shown that breakup actually occurs over a finite time.

Park and Lee (2004) have shown that the accuracy of final fragment size predictions may be a function of operating conditions. In applications to diesel sprays the fragment sizes are typically over predicted for low pressure simulations and under predicted at high pressures.

Hwang et al. (1996) pointed out that the TAB model does not accurately predict the frontal area of the distorted drop. For this reason the calculation of drag may be incorrect leading to poor prediction of drop trajectory.

Finally, the breakup criterion is somewhat arbitrary and experimental data, such as Eq. 11, indicate that the critical deformation is actually a function of We.

An alternative to the TAB model is the droplet deformation and breakup (DDB) model proposed by Ibrahim et al. (1993). In this model drop deformation is calculated by equating the rate of change in kinetic and potential energies to the work done on the drop due to pressure and viscous forces. Breakup is assumed to occur when both kinetic and viscous forces are negligible, resulting in a relation between critical deformation and We. Mass is conserved, therefore an accurate calculation of the drop frontal area is possible.

Again, the literature contains many studies in which the DDB model was used to simulate secondary atomization. These include Ibrahim et al. (1993), Hwang et al. (1996), Liu and Reitz (1997), Park et al. (2002), Pham and Heister (2002), Park and Lee (2004), and Lee et al. (2004) among others.

The DDB model does have shortcomings. Park et al. (2002) noted that the DDB model breakup criterion predicts instantaneous breakup without deformation when We is less than 19. This is clearly unrealistic, so the DDB model as originally proposed by Ibrahim et al. (1993) cannot be applied to low We drops.

Hwang et al. (1996) calculated drop trajectories using both the TAB and DDB models. For the DDB model the drag was calculated using Eq. 13. In this case the DDB model was shown to be superior. In other instances, such as the work of Park et al. (2002), the TAB model has been shown to outperform the DDB model. Currently, both models are used in industrial spray simulations, each with their own advantages and limitations.

A number of authors have proposed improvements to the TAB and DDB models. Tanner (1997) proposed the enhanced TAB (ETAB) model to address two common problems in the original version, namely the instantaneous breakup of the drop at the initiation time and the under prediction of fragment sizes. The breakup criterion remained the same as the original TAB model, but formation of fragments was assumed to occur at a rate proportional to the number of fragments where the constant of proportionality is a function of the breakup regime. For a diesel spray, Tanner (1997) showed that ETAB model predictions better represented the experimental data than those from the TAB model.

Park et al. (2002) proposed an improved TAB model in which the effect of droplet deformation on drag is simulated in a manner similar to the DDB model. Also, a new breakup criterion was proposed that was based on a consideration of both the ambient and drop phase pressure distributions and the surface tension. The result was a more reasonable relation between We and the amount of deformation leading to breakup. Comparison to experimental results showed that the improved TAB model of Park et al. (2002) outperformed both the TAB and DDB model. Nevertheless, all models were shown to be inaccurate near Wec and more work is needed.

Others have used wave instability models as the breakup criteria, rather than a critical deformation level. One example is the breakup model of Lee and Reitz (1999) in which a K–H instability model is used (with limited success). Still others have used hybrid models to simulate the simultaneous effect of breakup due to unstable wave growth and breakup due to aerodynamic deformation, such as is expected in the catastrophic breakup regime. An example is the K–H/DDB competition model of Park and Lee (2004). Finally, Gorokhovski (2001), Apte et al. (2003), and Gorokhovski and Saveliev (2003) used stochastic principles to predict drop breakup and found reasonable agreement with measured spray properties.

These secondary breakup models are combined with primary breakup models and simulation of the ambient phase flow to create complete spray models. Examples include Tanner (1997), Gorokhovski (2001), Pham and Heister (2002), Apte et al. (2003), Gorokhovski and Saveliev (2003), Lee et al. (2004), Trinh and Chen (2006), and Trinh et al. (2007) among others.

11.2 Direct numerical simulation

Direct numerical simulation (DNS) promises the ability to resolve both the drop and ambient phase flow fields and may help answer a number of outstanding questions about secondary breakup. Such a simulation requires solutions to multiphase, unsteady, 3D flow with pinch off of small fragments. This necessitates resolution over a large range of length scales. To date, few simulations have been performed which meet all of these requirements.

Zaleski et al. (1995) performed a 2D simulation of the Navier-Stokes equation with constant density and viscosity in each phase. Their configuration corresponds to the breakup of an infinite cylinder, rather than of a spherical drop. However, Igra and Takayama (2001) and Igra et al. (2002) showed experimentally that the breakup is qualitatively similar. A volume of fluid (VOF) method was used to track the interface, and surface tension effects were included. Fragmentation was simulated at We = 10, 20, and 100. All simulations were performed for ε = 10 and Oh < 0.1. The We = 100 results were qualitatively similar to sheet-thinning regime behavior while the We = 10 and 20 results showed the formation of bag structure. However, the bag formed in the upstream direction rather than downstream as seen in experiments. The authors comment that the discrepancy in the bag regime may be due to the initial conditions they used.

In a series of two papers, Han and Tryggvason (1999, 2001) addressed many of the shortcomings of the Zaleski et al. (1995) study. A front tracking/finite difference method was used to solve the axi-symmetric Navier–Stokes equations. The axi-symmetric assumption allowed for the simulation of a spherical drop rather than a 2D cylinder. Simulations were performed for steady loading, such as seen in a drop tower, as well as impulsively accelerated loading, such as seen in shock tube experiments. Both bag and sheet-thinning type structures were observed, but transitional We values did not match those seen in experiments. This may be due to the fact that calculations were performed for ε < 10. Such a low density ratio was necessary to reduce computational cost, but most experiments are performed for liquid drops in ambient gas environments where the density ratio is much higher. Further experimental data are needed to address the accuracy of these simulations.

Aalburg et al. (2003) expanded the work of Han and Tryggvason (1999, 2001) to simulate drop deformation at much higher density ratios. The level set method of Sussman et al. (1994) was used to track the interface. Although Aalburg et al. (2003) did not have sufficient grid resolution to simulate the breakup event, Sussman et al. (1994) has previously shown that the level set method is capable of resolving such events.. For ε > 128, the predicted trend of Wec versus Oh matched the experimental results of Hsiang and Faeth (1995).

Quan and Schmidt (2006) developed a 3D, finite volume scheme that uses a moving mesh. Again, due to computational cost, simulations were performed at relatively low values of Re and ε, and the actual breakup event was not simulated. Nevertheless, images of the deforming droplet appear to qualitatively match those seen in experiments.

Wadhwa et al. (2005, 2007) developed a code capable of 3D simulations including compressibility effects in the ambient. Due to computational cost, the final simulation of drop deformation was done assuming axi-symmetry. Nevertheless, their results for drop deformation at We up to 100 and Oh up to 0.1 show good agreement with experimental results.

In what is thought to be the most accurate study performed to date, Khosla et al. (2006) used the VOF method to simulate breakup of an ethanol drop in air. Special care was taken to ensure that the grid was fine enough to resolve the internal flow including the drop phase boundary layer. In addition, 3D, reduced 3D, and 2D axi-symmetric cases were analyzed. The 2D axi-symmetric case was shown to be sufficient to resolve the breakup mechanism, although a full 3D case may be needed to accurately determine the final fragment sizes. Finally, unlike moving mesh schemes, VOF was capable of simulating the pinch off and formation of fragments. The results of Khosla et al. (2006) showed excellent agreement with experimental results.

Finally, Chang and Liou (2007) developed a stratified flow model which is capable of incorporating compressible liquids and gases. Therefore, the code can simulate the interaction of a shock wave and liquid drop. Initial results indicate very good agreement with the experimental results of Theofanous et al. (2004) at high Ma.

The above mentioned results assume that the fluids are continuous. Some success has been had using particle methods where fluid packets are tracked via a Lagrangian scheme. One example is the moving-particle semi-implicit (MPS) method, originally proposed by Koshizuka and Oka (1996) and improved by Nomura et al. (2001) with the addition of surface tension. Nomura et al. (2001) and Duan et al. (2003a, b) used the MPS method to simulate secondary atomization. Although their simulations were 2D, they showed good qualitative agreement with experimental results and solutions for large density ratios were possible. Nomura et al. (2001) indicate that they have used the MPS method to perform a 3D simulation of drop breakup, although they give no further details. Finally, Shibata et al. (2004) successfully used the MPS method to simulate primary atomization.

An additional particle method is the Lattice-Boltzmann approach as presented by Sehgal et al. (1999). In this case methodologies from molecular gas dynamics are used to simulate fluid flow. According to Sehgal et al. (1999), the Lattice-Boltzmann method can be shown to be equivalent to solving the incompressible Navier–Stokes equations. The results presented by Sehgal et al. (1999) are qualitatively similar to experimentally observed breakup, but the transition We do not match experimental data so more work is needed.

At this time VOF and the level set method are among the most widely accepted. More work is needed to simulate the entire breakup process including accurately predicting the final fragment sizes.

12 Areas for future research

12.1 Non-Newtonian liquids

This topic has received only cursory attention, despite the fact that non-Newtonian sprays play a key role in so many practical processes, that there are so many interesting physical phenomena to explore, and that the breakup behavior is clearly different than for Newtonian liquids. The limitations of available information have been documented above:
  • Consensus on something as basic as the breakup modes has not been achieved. This is due to a scarcity of studies.

  • Even information as fundamental as which dimensionless groups should be used to describe non-Newtonian drop secondary breakup results is lacking. Again, this is due to a scarcity of studies.

  • The critical Weber number for breakup has not been identified, nor have values that separate regime boundaries. Ohnesorge (and Wiesenberg) number dependencies have yet to be investigated. Valuable figures such as that from Hsiang and Faeth (1985) cannot be constructed until this information has been published.

  • Bag and stripping breakup exhibit a net-like structure that has yet to be studied in detail. It may lead to bi-modal fragment size distributions.

  • Drag expressions are absent.

  • There are no estimates for initiation and total breakup times.

  • There is contradictory information about fragment size distributions.

12.2 Experiments near the thermodynamic critical point

In modern diesel and gas turbine engines the compression ratio is such that the injected fuel may approach the thermodynamic critical point. In such cases, one can expect very low density ratios (on the order of unity) and We and Oh to approach infinity as the surface tension goes to zero.

Currently no experimental secondary breakup data exists at or near the critical point. The only known results come from DNS studies, such as the work of Han and Tryggvason (2001) and Aalburg et al. (2003). These simulations indicate markedly different breakup characteristics at very high We and Oh numbers and very low ε. In addition, current experimentally determined correlations involving high Oh are limited and poor agreement is seen between researchers. Additional experimental and numerical work is needed to fully characterize breakup near the thermodynamic critical point.

12.3 Turbulence

Drops in a turbulent flow field which are larger than the Kolmogorov length scale will be subjected to irregular flow patterns. Hinze (1955) was the first to study such a situation and observed what he described as “bulgy” deformation. Rather than deforming into an oblate ellipsoid as in the case of laminar flow, drop deformation and fragmentation was irregular. Later Prevish and Santavicca (1998) found that Wecr decreased as the turbulence intensity of the ambient flow increased. Also, it appears that turbulence adds randomness to the breakup process. Some drops experience low local velocities and therefore break up slowly, or do not break up at all. Other drops experience local velocities higher than the average and therefore break up faster and at lower We (based on the mean velocity).

Drop phase turbulence may also exist and result from either turbulence generated prior to primary atomization that has not had sufficient time to dissipate or as the result of rapid internal circulation caused by deformation and/or shear from the ambient phase. Trinh and Chen (2006) and Trinh et al. (2007) considered the effects of such a flow situation using modified analytic models and found that drop phase turbulence results in smaller fragment sizes and reduced breakup times. However, due to the difficulties of measuring drop phase flow, no known experimental works exist.

Despite these and a few other works discussed by Lasheras et al. (1998), very little is known about the influence of turbulence on secondary atomization.

12.4 Charged drops

In electrostatic sprays a charge is applied to the liquid to be atomized in order to promote fragmentation and assist in directing the atomized liquid toward a target. Industrial applications include painting, agricultural sprays, internal combustion engines, and others.

In a conductive fluid, electrostatic charge will migrate to the drop surface resulting in a repulsive force that counteracts surface tension. As shown by Shrimpton and Laoonual (2006) the net surface force, Fsurf, thus becomes:
$$ F_{\rm surf} = 4\pi \sigma d_{0} - \frac{{q^{2} }}{{2\pi \varepsilon_{\rm a} d_{0}^{2} }} $$
(39)
where q is the net charge and εa is the permittivity of the surrounding fluid. The net surface force goes to zero when q = qRa, where qRa is the Rayleigh charge limit:
$$ q_{Ra} = \sqrt {8\pi^{2} \sigma \varepsilon_{\rm a} d_{0}^{3} } $$
(40)

Rayleigh (1882) showed that an isolated, stationary drop is unstable at this limit and spontaneously breaks apart.

In processes involving secondary atomization the presence of electrostatic charge will reduce the effective surface tension making the drops more likely to fragment. As first suggested by Shrimpton and Laoonual (2006), an electrostatic Weber number, Wee−, can be defined to account for the effective reduction in surface tension:
$$ We_{e - } = \frac{{\rho_{\rm a} U_{0} d_{0}^{2} }}{{\sigma - {{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {8\pi^{2} \varepsilon_{0} d_{0}^{3} }}} \right. \kern-\nulldelimiterspace} {8\pi^{2} \varepsilon_{0} d_{0}^{3} }}}} $$
(41)

Early experimental results from Guildenbecher and Sojka (2007) indicate that Wee− should be used to define the properties of secondary atomization. Nevertheless, more work is needed to confirm these results and better characterize the properties of secondary breakup of charged drops. Especially important is the determination of final fragment sizes and fragment charge distribution. Also, more work is needed to determine the effect of charge movement on the breakup process. For example, during the atomization of highly conductive drops electrostatic repulsion may lead to locations of high charge concentration. The spatially varying distribution of charge will give rise to a non-uniform surface stress. This may lead to breakup behavior that has not been observed for un-charged drops. Such behavior warrants investigation.

12.5 Others

As discussed in previous sections there are a number of other aspects of secondary atomization where additional research is warranted. Notable examples include: (1) the many outstanding issues related to the breakup of non-Newtonian drops, (2) improving DNS to help answer many of the questions concerning the physical mechanisms leading to breakup, (3) advancing drop deformation and breakup analyses with the goal of creating a model that is accurate for a wide range of applications, (4) improving drop size measurement techniques to better characterize final fragment size, and (5) addressing breakup due to a combination of impulsive and continuous acceleration.

13 Summary and conclusions

The available literature on secondary atomization due to impulsive acceleration has been reviewed with emphasis placed on work completed subsequent to reviews by Pilch and Erdman (1987) and Faeth et al. (1995). For Newtonian liquids, breakup is characterized by a morphology consisting of the following modes: vibrational, bag, multimode, sheet-thinning, and catastrophic. The process is a strong function of Weber number and relatively independent of other parameters such as the Ohnesorge number, Reynolds number, and the density and viscosity ratios. Each mode has been discussed in detail and experimental correlations are reported which are useful for designers of spray systems.

Unfortunately, most of these correlations are purely empirical. As a result, extrapolation outside of the experimental ranges is not advised. To overcome this, models are needed which are based on the underlying physics.

A thorough review of the many mechanisms that have been proposed reveals the following: Bag breakup occurs at the lowest values of Weber number and is the result of the positive pressure difference between the front stagnation point and the wake. This tends to draw the center of the deformed drop downstream faster than the periphery. Contrary to this, sheet-thinning breakup occurs at high values of Weber number when the drop rapidly deforms into a disk with thin edges. These thin edges are blown downstream and the drop fragments before the pressure difference can form the bag structure. At intermediate values of Weber number, a multimode regime is observed in which bag and sheet-thinning structures are seen simultaneously. Finally, at very high values of Weber number unstable surface waves grow rapidly and dominate the breakup mechanism. This is referred to as catastrophic breakup.

A consistent description of the physical mechanisms has been proposed which is supported by experimental observation. No dependence on non-dimensional groups is predicted that has not been observed in experiment. The previous explanation of shear stripping has been disproven and was replaced by the more physically correct description of sheet-thinning. More experimental and numerical results are needed to confirm these mechanisms and create new correlations based on the underlying physics which will be useful to future designers of spray systems.

In addition to the above, this paper highlights a number of other areas in which more experimental and/or numerical results are needed. Surprisingly, despite the fundamental nature of the problem and its many important applications, a thorough understanding of secondary atomization and its outcomes is elusive.

Notes

Acknowledgments

The authors would like to thank Prof. Stephen Heister of Purdue University, Dr. Sachin Khosla of the CFD Research Corporation, Prof. Rolf Reitz of the University of Wisconsin, and Prof. David Schmidt of the University of Massachusetts-Amherst for their stimulating discussions and guidance during the preparation of this review.

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • D. R. Guildenbecher
    • 1
  • C. López-Rivera
    • 1
  • P. E. Sojka
    • 1
  1. 1.Maurice J. Zucrow Laboratories, School of Mechanical EngineeringPurdue UniversityWest LafayetteUSA

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