ILASS-Americas 23rd Annual Conference on Liquid Atomization and Spray Systems, Ventura, CA, May 2011

Simulation and Experimental Studies of Shear-Thinning Droplet Breakup

Kyle G. Mooney† , Sharon Snyder‡, P. E. Sojka‡, and David P. Schmidt†,∗

†Department of Mechanical and Industrial Engineering,

University of Massachusetts Amherst‡Maurice J. Zucrow Laboratories,

Purdue University

Abstract

CFD simulations of non-Newtonian droplet breakup are presented alongside experimental observations.Liquid phase rheology is modeled with the Carreau-Yasuda shear thinning viscosity model and the gas-liquid interface is handled using the volume of fluids (VOF) interface capturing method. Because the totaldroplet translation due to drag forces is significant, a centroid tracking moving mesh is used to greatlyreduce the required domain size and computational expense. Numerical validation is performed againstexperimental results for both Newtonian and non-Newtonian breakup data. We note the trends for breakuptime and morphology as a function of Ohnesorge number. Trailing side wake dynamics as well as effects ofthe non-Newtonian viscosity on breakup time and deformation morphology are discussed.

∗Corresponding Author: schmidt@ecs.umass.edu

1 Introduction

Secondary atomization can be described as thedeformation and fragmentation of a droplet dueto a surrounding dispersed phase moving at somevelocity relative to the drop, as stated by Guilden-becher et al [1]. Dimensionless groups that governthis phenomenon are shared with droplet collisions(We,Oh,Re) with the addition of the Eotvos num-ber [2] which represents the interplay of buoyancyforces and surface tension forces as shown in Eq.1.Here ∆ρ = (ρliq − ρgas). The Weber numberforumlation is shown in Eq.2 where ρgas is thedensity of the ambient surrounding fluid, σ is thesurface tension, U0 is the relative velocity of dropand ambient fluid, and d0 is the initial drop diameterassuming a perfect sphere.

Eo =∆ρgd204σ

(1)

We =ρgasU

20d0

σ(2)

One of the earliest works on secondary at-omization was by Pilch and Erdman [3] in 1987.Through experimental observation they were able tocharacterize several distinct breakup modes as wellas non-dimensionalize the flow conditions byWe andOh (defined in Eq.3). Pilch also established multiplenon-dimensional time scales (defined in Eq.4) thatwill be of use in numerical validation, discussed laterin the work. These quantities are Tini, the time fromthe last perfect sphere to an oblate spheroid, Ttot,the time from the last perfect sphere to completedrop breakup, Tbaggrowth, the time from the lastperfect sphere to the largest unbroken bag, andTbagbreakup, the time from the last perfect sphereto the point where the bag is completely broken upwith or without the rim destabilizing.

Oh =µliq√ρliqd0σ

(3)

T = tU0

d0

√ρgas

ρliq(4)

There have been many studies of numericalmodeling of this process using a variety of in-terface modeling methods. The work by Helen-brook and Edwards [4] used arbitrary-Lagrangian-Eulerian (ALE) interfacial modeling in conjunc-tion with the finite element method to performan extensive parametric study on the effects ofdeformation on effective drag coefficients. A front-tracking study by Han and Tryggvason [5] exploredthe effect of Eo on drop deformation transience by

parametrically varying an applied body force to thedroplet. While deformation and bag inflation hasbeen successfully simulated in multiple publishedworks, a predictive method of final fragmentationsizes as a function of flow conditions and rheologyhas yet to be accomplished [1].

As reviewed by Lopez-Rivera and Sojka [6],there has been limited study concerning non-Newtonain breakup [7, 8, 9, 10, 11, 12, 13].Lopez-Rivera proceeded to study the breakupmorphology of drops composed of a shear-thinningxanathan gum solution concluding that shear-thinning rheology does not strongly effect regimeboundaries but does tend to increase total bagand ligament growth magnitude when comparedto Newtonian drop deformation. The experimentaland modeling work presented here is an extensionof that research.

2 Methodology

Elastic non-Newtonian drop breakup morphol-ogy and breakup times were obtained for dropscomposed of xanthan gum (XG) water solutionsthat were exposed to high speed air flows. The mainexperimental components are the air flow system,the liquid flow system plus droplet generator, andthe high-speed imaging system. Fig.1 is a schematicof this experimental apparatus.

Figure 1: System Schematic: [1] air supply, [2]shutoff valve, [3] regulator, [4] air flow meter, [5]drop generator, [6] air nozzle, [7] stainless steel tank,[8] high speed camera

Videos used to determine drop morphology andbreakup times were recorded using a Vision ResearchPhantom 7.1 high speed digital camera outfittedwith a 105 mm Nikon lens. 800 x 600 pixel imageswere recorded at 4700 fps. A 1000W Xe arc lampwas used to produce a collimated beam that served

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as the illumination source. The collimated beampassed through a plano-convex lens that focused thebeam onto an opal glass diffuser. This backlit thedrop breakup process. The optical setup is shownin Fig.2. To determine drop sizes directly from theimages, a calibration image of a 5 mm square gridwas taken to relate camera pixels to length.

Figure 2: Optical imaging system: [1] arc lamp,[2] dichromatic mirror, [3] plano-convex lens, [4]diffusing plate, [5] 105 mm focal length lens, [6] highspeed camera

The open-source CFD package, OpenFOAM[14], served as the basis for the two-phase simulationcapability. This framework has been used for similardroplet dynamics applications in previous works[15, 16]. Using finite volume discretization, theunsteady incompressible Navier-Stokes equation issolved using the pressure implicit split operator(PISO) method developed by Issa [17].

The volume of fluids (VOF) method [18] is anEulerian approach to interface modeling which usesa bounded scalar function γ = f(x, y, z, t) that isadvected with the flow. From Rusche [16], γ isdefined as

γ(x) =

1; for a point inside fluid a0 < γ < 1; for a point in the

transitional region0; for a point inside fluid b

The unsteady evolution of the γ field follows thematerial conservation equation shown in Eq.5. Dueto the continuity of the gamma field across the fluidinterface (as opposed to the discontinuous interfaceof an explicit/mesh motion method) density andviscosity are calculated via the cell centered colorfunction value using a harmonic mean weightingscheme as shown in Eq.6 and Eq.7 respectively [19,20]. The multidimensional universal limited ex-plicit solver (MULES) is used to calculate the γ

advection solution and maintain the 0 ≤ γ ≤ 1restriction. Interface sharpness is maintained withthe counter-gradient compression method describedin Berberovic et al [15]. The case is run decomposedin a 4 processor parallel fashion using the opensource message passing interface (OpenMPI) libraryto handle inter-processor communication protocols.

∂γ

∂t+ u · ∇γ = 0 (5)

ρ =ρ1ρ2

γρ2 + (1− γ)ρ1(6)

µ =µ1µ2

γµ2 + (1− γ)µ1(7)

Due to the continuum nature of the inter-face treatment, the continuum surface force (CSF)method developed by Brackbill [21] is used to ac-count for interface surface forces and include themas a volumetric force. Formally, the curvature of asurface, κ, is defined as the divergence of the unitsurface normal vector (κ = −(∇ · n)). The unitnormal n can be approximated using the normalizedgradient of γ as shown in Eq.8. The capilarypressure, σκn, is then added to the cell centeredpressure field.

n =∇γ

| ∇γ |(8)

3 Simulation Setup and Validation

The secondary atomization code is benchmarkedagainst Newtonian experimental results before beingextended to the more complex non-Newtonian vis-cosity models. The mesh used for all simulationsis comprised of 561250 hexahedra and 35766 prismsas shown in Fig.3. The original 2D mesh is rotated5 °about the axis of symmetry to represent an ax-isymmetric flow field. Having a mixed element meshhas two advantages in this setting. 1. A triangularmesh can grow more rapidly and anisotopically thana hexahedral mesh, lowering computation cost. 2.

A uniform hexahedral mesh provides more accurateinterpolation as well as avoids the need for a non-orthogonal correction sub-cycle in the pressure PISOloop. Neither the liquid phase of the simulation northe trailing wake leaves the uniform hexahedral zone.

The inlet boundary condition is fixed valueuniform velocity and zero gradient pressure. Theoutlet boundary condition is zero gradient velocityand fixed value zero pressure as illustrated in Fig.4.The maximum blockage ratio of the drop throughoutthe simulation is less than 2% which justifies thefull slip, no-penetration condition at the far wall. A

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significant blockage ratio (< 5% as a rule of thumb)could invalidate the free-stream assumptions at thatboundary. The droplet is initialized as a perfectsphere.

Figure 3: Mixed hexahedral-prism mesh used fordrop-in-cross flow simulations

Total drop displacement throughout thebreakup process is on order of the mesh domainwidth. To avoid extending the domain in thestream-wise direction, the mesh is translated withthe drop via centroid tracking, solid body meshdisplacement, and absolute/relative flux correction.The centroid Ci

drop of the liquid phase at the ithtimestep is calculated with a volume weightedγ-average as shown in Eq.9 where j is the cell index,n is the number of cells in the domain, Vj is thevolume of the cell, xj is the cell center positionvector, and γj is the value of the color function insaid cell.

Cidrop =

n∑

j=0

(γjxjVj)

n∑

j=0

(γjVj)

(9)

The uniform mesh displacement vector xmesh

is calculated by xmesh = Cidrop − Ci−1

drop. Torestrict displacement along the axis of symmetry,the final mesh displacement vector is obtained byxmesh−final = (xmesh · xProj) · xProj where xProj

is unit vector parallel to the axis of symmetry.xmesh−final is then added to all mesh point positionvectors, displacing the mesh.

Figure 4: Illustration of the boundaries of thecomputational domain

Parameter Value Units

ρgas 1.13 kg/m3

ρliq 997 kg/m3

νgas 1.48e-5 m2/s

νliq 1e-6 m2/s

σ 0.0729 kg/s2

VcrossF low 20.08 m/s

DropDiameter 2.5 mm

Table 1: Case configuration details for theNewtonian droplet (H2O) in cross flow simulation

The code is bench-marked for a Newtonianwater drop with experimentally observed quantita-tive breakup details from the Zucrow PropulsionLaboratory at Purdue University. Case details areoutlined in Table.1. Experimental results fromZucrow labs show a range of 8000 µs to 25000 µs baginflation time while VOF simulation shows a 6850 µs

inflation time for water droplets in a crossflow.Simulated droplet deformation is shown in Fig.5and pressure and velocity flow fields are illustratedin Fig.6. These profiles agree well with numericalstudies presented by Han [5] and Theofanous [22].

The Carreau Yasuda viscosity model, as shownin Eq.10 [23], was used to capture non-Newtonianshear thinning viscosity effects. Here, η0 is the zeroshear-rate viscosity and ηinf is the infinite shear rateviscosity. The variables λ, a, and n, are constantsused to fit rheological behavior. A plot of the

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Figure 5: Simulated water droplet deformation dueto cross flow. Note the fluid bag rupture in the finalframe.

Figure 6: Pressure and velocity fields from t =0.0033s. Droplet liquid phase is colored black.

measured effective viscosity of a 0.05% XG/watersolution is shown in Fig.7 along with its Carreau-Yasuda model fit.

η = ηinf + (η0 − ηinf)[(1 + λγshear)a](n−1)/a (10)

Figure 7: Profile of the Carreau-Yasuda fit for a0.05% Xanathan Gum / water solution (red line)and raw rheometry data (2).

4 Results

Non-Newtonian drops exhibit similar breakupregimes as Newtonian drops yet tend to have dif-ferent morphological behavior such as the formationof persistent ligaments, thicker rims, and a largerbag structure with a longer lifetime. Time lapseimagery of the breakup of a drop composed of 0.05%XG solution is shown in Fig.9. A compilation ofexperimentally observed breakup initiation (Tini)times are illustrated in Fig.10.

Simulation results of the breakup of a 0.05%XG/water droplet is shown in Fig.8. While thebreakup time is somewhat under predicted, the nu-merical solutions can elucidate some of the externalflow structures taking part in the disintegrationprocess. The most prominent flow feature is thestable toroidal wake behind the trailing edge of thedrop. This phenomenon, also present in the waterbased simulations (Fig.6), has also been observedin simulations presented in Theofanos [22]. Simu-lations show that this toroidal wake destabilizes inconcert with disintegration of the liquid bag.

5 Conclusions

Under-prediction of the breakup time may beattributed to inaccuracies in rheological modelingor under-resolution of the thin bag structure. Inaddition, the droplet has a finite residence time in

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Figure 9: Break up of a droplet of 0.05% XG /water mixture. Oh = 0.00170,We = 9.89.

Figure 10: Observed dimensionless breakup initiation time Tini vs. Oh for various Xanathan Gum/waterconcentrations.

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Figure 8: Simulated breakup of a Carreau-Yasudadroplet colored by effective viscosity (Oh =0.00164,We = 11.03). The time duration betweenimages is 1.5 [ms]. Image spacing does not reflectdroplet displacement

the mixing layer of the crossflow jet as it decends.Due to this the droplet experiences a decreasedinitial crossflow which increases to the steady valueas it enters the core of the jet. Further simulationswill impose a ramped inlet velocity to represent thistransition through the mixing layer.

References

[1] D. R. Guildenbecher, C. Lopez-Rivera, andP. E. Sojka. Experiments in Fluids, 2009.

[2] R. Clift, J. R. Grace, and M. E. Webber. Bub-bles, Drops, and Particles. Dover Publications,2005.

[3] M. Pilch and C. A. Erdman. International

Journal of Multiphase Flow, 1987.

[4] B. T. Helenbrook and C. F. Edwards. Interna-tional Journal of Multiphase Flow, 2001.

[5] Jaehoon Han and Gretar Tryggvason. Physics

of Fluids, 1999.

[6] C. Lopez-Rivera and P. E. Sojka. Eleventh

Triennial International Conference on Liquid

Atomization and Spray Systems, Vail, Col-orado, USA, July 2009.

[7] C. Arcoumanis, L. Khezzar, D. S. Whitelaw,and B. C. H Warren. Experiments in Fluids,17, 1994.

[8] C. Arcoumanis, D. S. Whitelaw, and J. H.Whitelaw. Atomization and Sprays, 6, 1994.

[9] D. D. Joseph, G. S. Beavers, and T. Funada.Journal of Fluid Mechanics, 453, 2002.

[10] D. D. Joseph, J. Belanger, and G. S. Beavers.International Journal of Multipase Flow, 25,1999.

[11] J. E. Matta and R. P. Tytus. Journal of AppliedPolymer Science, 27, 1982.

[12] J. E. Matta, R. P. Tytus, and J. L. Harris.Chemical Engineering Communications, 19,1983.

[13] J. D. Wilcox, R. K. June, H. A. Brown, andR. C. Kelly. Journal of Applied Polymer

Science, 5, 1961.

[14] H. Jasak H. G. Weller, G. Tabor and C. Fureby.Computers in Physics, 12(6):620–631, 1998.

[15] E. Berberovic, N. P. van Hinsberg, S. Jakirlica-cute, I. V. Roisman, and C. Tropea. Physical

Review E, 79(3), 2009.

[16] Henrik Rusche. PhD thesis, Imperial CollegeLondon, 2002.

[17] R. I. Issa. J. Comput. Phys., 62(1):40–65, 1986.

[18] C. W. Hirt and B. D. Nichols. Journal of

Computational Physics, 1981.

[19] S. V. Patankar. Numerical Heat Transfer and

Fluid Flow. McGraw-Hill Book Company, 1980.

[20] A. V. Coward, Y. Y. Renardy, and M. Renardy.Journal of Computational Physics, 1997.

[21] J. U. Brackbill, D. B. Kothe, and C. Zemach.Journal of Computational Physics, 1992.

[22] T. G. Theofanous. Annual Review of Fluid

Mechanics, 43, 2011.

[23] K. Yasuda, R. C. Armstrong, and R. E. Cohen.Phys Rev. E, 2009.

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