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Atomization and Sprays

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ANALYSIS OF THE EFFECT OF THE 2D PROJECTION ON DROPLET SHAPE PARAMETERS

Volume 32, Issue 8, 2022, pp. 59-98
DOI: 10.1615/AtomizSpr.2022040525
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ABSTRACT

The characterization of the shape of liquid elements in a spray is a good way to analyze atomization processes. Experimental approaches based on common imaging techniques suffer from partial information given by 2D images to characterize intrinsically 3D objects such as liquid droplets. In this paper, we address the question, To what extent can the shape parameters measured on 2D images reveal the 3D content of the droplet shape? An analytical approach is first adopted to determine relationships between 2D and 3D shape parameters for two families of objects, the oblate and prolate spheroids. Two numerical databases obtained from direct numerical simulation of two-phase flows are then explored for their ability to give complete 3D shape information and extrapolate 2D parameter values from projections on planes selected according to flow characteristics. Focus is put on one shape parameter particularly sensitive to 3D to 2D projection effects, namely the uniformity parameter introduced by Blaisot and Yon (2005). Statistics obtained from the numerical databases are used to guide the analysis of results extracted from experimental images. It is shown that statistics on a 3D shape parameter could be induced from the ones for the 2D parameters. Two conditions are necessary: (1) the 2D projection is performed perpendicularly to the main flow direction, which is always the case in experiments; and (2) a particular care must be put on the determination of the statistics of orientations of the main axis of the liquid elements. This last point should be tackled in future experimental analyzes to estimate 3D shape parameter statistics from 2D images.

Figures

  • Shadowgraphy image sequence (25,000 frames/s) of a liquid jet (Dumouchel and Blaisot, 2014).
The field of view is 3.5 mm × 6.4 mm. Refer to text for roman letters.
  • Definition of distances and shape parameters construction for the projection of a peanut-like droplet
(in red) and its equivalent circle filled in black. The union of these two shapes, AD∪C contains these two
areas and the intersection of these two shapes, AD∩C, are white.
  • Projection of two specific 3D shapes: a prolate spheroid and a tube, respectively left and right.
Morphological values for the 3D and each projection are given, respectively the aspect ratio; α, the uniformity; η, the irregularity; ι, and the SDS parameter; ψ, with initial length a3 = 4 b3.
  • Prolate and oblate spheroids (left and right). θ is the polar or projection angle and ϕ the azimuthal
angle. The lengths λk and λ⊥ are the length along the axisymmetry axis and its perpendicular.
  • Evolution of a2 , and b2 • scaled by λk, against θ for a spheroid with an initial uniformity
parameter of η3 = 0.25; left, prolate spheroid; right, oblate spheroid. 3D information is shown as a3
and b3 . (a) Prolate spheroid and (b) oblate spheroid.
  • Evolution of RQ
2 , R2Q,q=Q (reconstruction of the radius based on the family of the
spheroid) and R2Q,q6=Q (reconstruction of the radius based on the wrong family of the spheroid), scaled
by R3 , against θ for a spheroid with an initial uniformity parameter of η3 = 0.25; left, prolate
spheroid; right, oblate spheroid. (a) Prolate spheroid and (b) oblate spheroid.
  • Evolution of the 2D uniformity, ηQ 2 , ηQ 2,q=Q and ηQ 2,q6=Q , for a range of direction
of projection θ = [0, π]. The 3D uniformity value is fixed, η3 = 0.25 ; left, prolate spheroid; right,
oblate spheroid. (a) Prolate spheroid and (b) oblate spheroid.
  • Plots of η3 versus η2. Markers represent random spheroids sets based on the initial spheroid family
• η
Q2
(left prolate, right oblate). The top row presents results of reconstruction with the same family
η
Q2
,q=Q and bottom row with the wrong family ◭ ηQ 2,q6=Q. Average η¯Q 2 , and maximum values
η
Q2
,max, are shown for the random population (ηQ 2 ). The first bisector, corresponding to η2 = η3, is also
shown as . (a) Prolate spheroid (• ηP 2 and ηP 2,P ); (b) oblate spheroid (• ηO 2 and ηO 2,O), (c) prolate
spheroid (• ηP 2 and ◭ ηP 2,O), and (d) oblate spheroid (• ηO 2 and ◭ ηO 2,P )
  • Left: instantaneous X–Z projection of the planar prefilming airblast atomizer liquid volume fraction; the plate is shown in gray and the liquid fuel in black. Right: droplets database from the airblast
atomizer based on their aspect ratio, α3, and numerical resolution, R3/(∆x).
  • 3D morphological parameters as a function of the aspect ratio, respectively, from left to right: (a)
the uniformity η3; (b) irregularity ι3; (c) and SDS parameter ψ3 for isotropic and anisotropic droplets (black
triangles ◭ and blue circles •, respectively). Analytical prolate spheroids relations (ηP 3 , ιP 3 , ψP 3 ),
and oblate spheroids (ηP 3 , ιP 3 , ψP 3 ). The four droplets extracted from Appendix B are represented
by orange square, circle, diamond, and down triangle symbols, from A to D, respectively
  • 2D morphological parameters as a function of the aspect ratio, respectively: (a) the uniformity
η2 [Fig. 10(a)]; (b) irregularity ι3 [Fig. 10(b)]; (c) and SDS parameter ψ3 [Fig. 10(c)] for isotropic and
anisotropic droplets (black triangles ◭, X–Y –Z anisotropic are •, ◭, ⋆, respectively). Analytical relation
shown for the ellipse (ηE 2 , ιE 2 , ψE 2 ). The four droplets extracted from Appendix B are represented
by orange square, circle, diamond, and down triangle symbols, from A to D, respectively
  • Probability density functions for R2 in black, R2,P in red (prolate spheroid assumption), and
R2,O in green (oblate spheroid assumption)
  • Distribution of b3 against b2 scaled by the equivalent 3D radius R3 (left). Distribution of a3
against a2 scaled by the equivalent 3D radius R3 (right). The colormap is associated with R2/R3 values,
see Fig. 12.
  • Distribution of the 3D uniformity, η3, against η2. The colormap is associated to R2/R3 values;
see Fig. 12. Lines: 2D averaged uniformity for each kind of spheroid family, η¯Q 2 , maximum values,
η¯Q 2,max, and family correction, η¯Q 2,Q and its maximum η¯Q 2,max,Q (in green, oblatebased; in red, prolate-based)
  • From left to right: histogram of the deviation of the projected equivalent radius: R2 in black,
and their correction based on the prolate spheroid assumption: R2,P , in red, and for the oblate spheroid
assumption: R2,O, in green, for direction of projections: (a) X; (b) Y ; (c) Z.
  • Distribution of b3 against b2 scaled by the equivalent 3D radius R3 (top). Distribution of a3 against
a2 scaled by the equivalent 3D radius R3 (bottom). The colormap is based on the scaling of the equivalent
2D and 3D radius; see Fig. 15. An individual scattering of each direction of projection is given for both
quantities; from left to right: X–Y –Z.
  • Distribution of the 3D uniformity, η3, against η2, for the three direction of projections, left to
right: X − Y − Z. Lines: 2D averaged uniformity for each kind of spheroid family, η¯Q 2 , maximum
values η¯Q 2,max , family correction, η¯Q 2,Q , and its maximum η¯Q 2,max,Q (it is recalled that
Q represents both prolate and oblate spheroid shapes, green is used for oblate and red for prolate relations).
The colormap scales equivalent 2D and 3D radius; see Fig. 15.
  • Images of the spray: air-assist jet on the left and an image of the experimental database on the
right. The red rectangle shows the region covered by images of the database.
  • Droplet size distribution of the experimental database. Data removed by applying contrast; outof-focus and morphological filters correspond to the hashed zone.
  • Morphological parameters as a function of the 2D aspect ratio: (a) the uniformity η2, (b) irregularity ι2, and (c) SDS parameter ψ2. Analytical relations are shown by Eqs. (1)–(3) . The colormap
is based on the equivalent 2D radius expressed in logarithmic scale.
  • (a) η2,Q versus η2 for prolate •, and oblate , reconstruction assumptions with the isotropic
(green), anisotropic X–Y (blue) and experimental (red) sets. Average uniformity evolution for prolate η¯P 2
, and oblate η¯O 2 , is shown (details in Fig. 8), as for the first bisector . (b) Probability
density function of uniformity (direct and with prolate or oblate assumptions) for each database.
  • Probability density function of the uniformity of the numerical anisotropic experiment. Reconstruction of 2D is given from 3D
REFERENCES
  1. Adrian, R.J., Particle-Imaging Techniques for Experimental Fluid Mechanics, Ann. Rev. Fluid. Mech., vol. 23, no. 1, pp. 261-304, 1991.

  2. Adrian, R.J., Twenty Years of Particle Image Velocimetry, Exper. Fluids, vol. 39, no. 2, pp. 159-169, 2005.

  3. Bachalo,W. and Houser,M., Phase/Doppler Spray Analyzer for Simultaneous Measurements of Drop Size and Velocity Distributions, Optical Eng., vol. 23, no. 5, p. 235583, 1984.

  4. Baert, L., Fanara, D., Remon, J.P., and Massart, D., Correlation of Extrusion Forces, Raw Materials and Sphere Characteristics, J. Pharm. Pharm., vol. 44, no. 8, pp. 676-678, 1992.

  5. Blaisot, J. and Yon, J., Droplet Size and Morphology Characterization for Dense Sprays by Image Processing: Application to the Diesel Spray, Exper. Fluids, vol. 39, no. 6, pp. 977-994, 2005.

  6. Blott, S.J. and Pye, K., Particle Shape: A Review and New Methods of Characterization and Classification, Sedimentol., vol. 55, no. 1, pp. 31-63, 2008.

  7. Bothell, J.K., Machicoane, N., Li, D., Morgan, T.B., Aliseda, A., Kastengren, A.L., and Heindel, T.J., Comparison of X-Ray and Optical Measurements in the Near-Field of an Optically Dense Coaxial Air-Assisted Atomizer, Int. J. Multif. Flow, vol. 125, p. 103219, 2020.

  8. Chan, W.H.R., Dodd, M.S., Johnson, P.L., and Moin, P., Identifying and Tracking Bubbles and Drops in Simulations: A Toolbox for Obtaining Sizes, Lineages, and Breakup and Coalescence Statistics, J. Comput. Phys., vol. 432, p. 110156, 2021.

  9. Charpentier, J.B., Brandle de Motta, J., and Menard, T., Capillary Phenomena in Assemblies of Parallel Cylindrical Fibers: From Statics to Dynamics, Int. J. Multif. Flow, vol. 129, p. 103304, 2020.

  10. Chen, T., Cheron, V., Guo, Z., Brandle De Motta, J.C., Menard, T., and Wang, L.P., Simulation of Immiscible Two-Phase Flows Based on a Kinetic Diffuse Interface Approach, 10th Int. Conf. Multiphase Flow, Rio de Janeiro, Brazil, May 19-24, 2019.

  11. Cheron, V., Brandle de Motta, J.C., Vaudor, G., Menard, T., and Berlemont, A., From Droplets to Particles: Transformation Criteria, 29th European Conf. on Liquid Atomization and Spray Systems, Paris, France, September 2-4, 2019.

  12. Clift, R., Grace, J.R., and Weber, M., Bubbles, Drops and Particles, New York, NY: Academic Press, 1978.

  13. Dodd,M.S. and Ferrante, A., On the Interaction of Taylor Length Scale Size Droplets and Isotropic Turbulence, J. Fluid. Mech., vol. 806, pp. 356-412, 2016.

  14. Dumouchel, C. and Blaisot, J.B., Multi-Scale Analysis of Liquid Atomization Processes and Sprays, 25th European Conf. on Liquid Atomization and Spray Systems, Chania, Greece, September 1-4, 2013.

  15. Dumouchel, C. and Blaisot, J.B., Laser Diffraction Measurement of Nonspherical Drop Sprays, Atomization Sprays, vol. 24, no. 3, pp. 223-249, 2014.

  16. Dumouchel, C., Blaisot, J.B., Bouche, E., Menard, T., and Vu, T.T., Multi-Scale Analysis of Atomizing Liquid Ligaments, Int. J. Multif. Flow, vol. 73, pp. 251-263, 2015.

  17. Duret, B., Luret, G., Reveillon, J., Menard, T., Berlemont, A., and Demoulin, F.X., DNS Analysis of Turbulent Mixing in Two-Phase Flows, Int. J. Multif. Flow, vol. 40, pp. 93-105, 2012.

  18. Eriksson, M., Alderborn, G., Nystrom, C., Podczeck, F., and Newton, J., Comparison between and Evaluation of Some Methods for the Assessment of the Sphericity of Pellets, Int. J. Pharmaceutics, vol. 148, no. 2, pp. 149-154, 1997.

  19. Fdida, N. and Blaisot, J., Morphological Characterization of Droplets. Application to Atomization of Sprays, Proc. 13th Int. Symp. on Flow Visualization, Nice, France, July 1-4, 2008.

  20. Fdida, N., Blaisot, J.B., Floch, A., and Dechaume, D., Drop-Size Measurement Techniques Applied to Gasoline Sprays, Atomization Sprays, vol. 20, no. 2, 2010.

  21. Fedkiw, R.P., Aslam, T., Merriman, B., and Osher, S., A Non-Oscillatory Eulerian Approach to Interfaces inMultimaterial Flows (The Ghost Fluid Method), J. Comput. Phys., vol. 152, no. 2, pp. 457-492, 1999.

  22. Ficuciello, A., Blaisot, J., Richard, C., and Baillot, F., Investigation of Air-Assisted Sprays Submitted to High Frequency Transverse Acoustic Fields: Droplet Clustering, Phys. Fluids, vol. 29, no. 6, p. 067103, 2017.

  23. Ghaemi, S., Rahimi, P., and Nobes, D., Measurement of Droplet Centricity and Velocity in the Spray Field of an Effervescent Atomizer, 14th Int. Symp. on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 7-10, 2008.

  24. Ghaemi, S., Rahimi, P., and Nobes, D.S., Assessment of Parameters for Distinguishing Droplet Shape in a Spray Field Using Image-Based Techniques, Atomization Sprays, vol. 19, no. 9, 2009.

  25. Gonzalez, H. and Garcia, F., The Measurement of Growth Rates in Capillary Jets, J. Fluid.Mech., vol. 619, p. 179, 2009.

  26. Green, A., An Approximation for the Shapes of Large Raindrops, J. Appl. Meteorol., vol. 14, no. 8, pp. 1578-1583, 1975.

  27. Han, L., Luo, H., and Liu, Y., A Theoretical Model for Droplet Breakup in Turbulent Dispersions, Chem. Eng. Sci., vol. 66, no. 4, pp. 766-776, 2011.

  28. Herrmann, M., Detailed Numerical Simulations of the Primary Atomization of a Turbulent Liquid Jet in Crossflow, J. Eng. Gas Turbin. Power, vol. 132, no. 6, 2010a.

  29. Herrmann,M., A Parallel Eulerian Interface Tracking/Lagrangian Point ParticleMulti-Scale Coupling Procedure, J. Comput. Phys., vol. 229, no. 3, pp. 745-759, 2010b.

  30. Lamb, H., On the Vibrations of an Elastic Sphere, Proc. London Math. Soc., vol. s1-13, no. 1, pp. 189-212, 1881.

  31. Lebas, R., Menard, T., Beau, P.A., Berlemont, A., and Demoulin, F.X., Numerical Simulation of Primary Break-Up and Atomization: DNS and Modelling Study, Int. J. Multif. Flow, vol. 35, no. 3, pp. 247-260, 2009.

  32. Lohse, D., Fundamental Fluid Dynamics Challenges in Inkjet Printing, Ann. Rev. Fluid. Mech., vol. 54, pp. 349-382, 2022.

  33. Lopez, J. and Hernandez, J., Analytical and Geometrical Tools for 3D Volume of Fluid Methods in General Grids, J. Comput. Phys., vol. 227, no. 12, pp. 5939-5948, 2008.

  34. Malot, H. and Blaisot, J.B., Droplet Size Distribution and Sphericity Measurements of Low-Density Sprays through Image Analysis, Part. Part. Sys. Charact., vol. 17, no. 4, pp. 146-158, 2000.

  35. Masuk, A.U.M., Salibindla, A., and Ni, R., A Robust Virtual-Camera 3D Shape Reconstruction of Deforming Bubbles/Droplets with Additional Physical Constraints, Int. J. Multif. Flow, vol. 120, p. 103088, 2019.

  36. Mayor, L., Silva, M., and Sereno, A., Microstructural Changes during Drying of Apple Slices, Drying Technol., vol. 23, nos. 9-11, pp. 2261-2276, 2005.

  37. Menard, T., Tanguy, S., and Berlemont, A., Coupling Level Set/VOF/Ghost Fluid Methods: Validation and Application to 3D Simulation of the Primary Break-Up of a Liquid Jet, Int. J. Multif. Flow, vol. 33, no. 5, pp. 510-524, 2007.

  38. Moallemi, N., Li, R., and Mehravaran, K., Breakup of Capillary Jets with Different Disturbances, Phys. Fluids, vol. 28, no. 1, p. 012101, 2016.

  39. Mukherjee, S., Safdari, A., Shardt, O., Kenjeres, S., and Van den Akker, H.E.A., Droplet-Turbulence Interactions and Quasi-Equilibrium Dynamics in Turbulent Emulsions, J. Fluid. Mech., vol. 878, 2019.

  40. Mukundan, A.A., Menard, T., Berlemont, A., de Motta, J.C.B., and Eggels, R., Validation and Analysis of 3D DNS of Planar Pre-Filming Airblast Atomization Simulations, in Proc. of ILASS Americas, 30th Annual Conf. on Liquid Atomization and Spray Systems, Tempe, AZ, USA, May 12-15, 2019.

  41. Mukundan, A.A., Tretola, G., Menard, T., Herrmann, M., Navarro-Martinez, S., Vogiatzaki, K., de Motta, J.C.B., and Berlemont, A., DNS and LES of Primary Atomization of Turbulent Liquid Jet Injection into a Gaseous Crossflow Environment, Proc. Combust. Inst., vol. 38, no. 2, pp. 3233-3241, 2021.

  42. Peano, G., Applicazioni Geometriche Del Calcolo Infinitesimale, Turin, Italy: Fratelli Bocca Editori, 1887.

  43. Perlekar, P., Biferale, L., Sbragaglia,M., Srivastava, S., and Toschi, F., Droplet Size Distribution in Homogeneous Isotropic Turbulence, Phys. Fluids, vol. 24, no. 6, p. 065101, 2012.

  44. Perrard, S., Riviere, A., Mostert, W., and Deike, L., Bubble Deformation by a Turbulent Flow, J. Fluid. Mech., vol. 920, 2021.

  45. Podczeck, F., Rahman, S., and Newton, J., Evaluation of a Standardised Procedure to Assess the Shape of Pellets Using Image Analysis, Int. J. Pharm., vol. 192, no. 2, pp. 123-138, 1999.

  46. Pope, S.B., Turbulent Flows, Cambridge, UK: Cambridge University Press, 2000.

  47. Ravelet, F., Colin, C., and Risso, F., On the Dynamics and Breakup of a Bubble Rising in a Turbulent Flow, Phys. Fluids, vol. 23, no. 10, p. 103301, 2011.

  48. Rorato, R., Arroyo, M., Ando, E., and Gens, A., Sphericity Measures of Sand Grains, Eng. Geology, vol. 254, pp. 43-53, 2019.

  49. Rosales, C. and Meneveau, C., Linear Forcing in Numerical Simulations of Isotropic Turbulence: Physical Space Implementations and Convergence Properties, Phys. Fluids, vol. 17, no. 9, p. 095106, 2005.

  50. Schober, P.,Meier, R., Schafer, O., and Wittig, S., Visualization and Phase Doppler Particle Analysis Measurements of Oscillating Spray Propagation of an Airblast Atomizer under Typical Engine Conditions, Ann. New York Acad. Sci., vol. 972, no. 1, pp. 277-284, 2002.

  51. Sussman, M., Fatemi, E., Smereka, P., and Osher, S., An Improved Level Set Method for Incompressible Two-Phase Flows, Comput. Fluids, vol. 27, nos. 5-6, pp. 663-680, 1998.

  52. Tanguy, S. and Berlemont, A., Application of a Level Set Method for Simulation of Droplet Collisions, Int. J. Multif. Flow, vol. 31, no. 9, pp. 1015-1035, 2005.

  53. Trontin, P., Vincent, S., Estivalezes, J.L., and Caltagirone, J.P., Direct Numerical Simulation of a Freely Decaying Turbulent Interfacial Flow, Int. J. Multif. Flow, vol. 36, nos. 11-12, pp. 891-907, 2010.

  54. van Beeck, J. and Riethmuller, M., Rainbow Phenomena Applied to the Measurement of Droplet Size and Velocity and to the Detection of Nonsphericity, Appl. Optics, vol. 35, no. 13, pp. 2259-2266, 1996.

  55. Vaudor, G., Menard, T., Aniszewski, W., Doring, M., and Berlemont, A., A Consistent Mass and Momentum Flux Computation Method for Two Phase Flows. Application to Atomization Process, Comput. Fluids, vol. 152, pp. 204-216, 2017.

  56. Warncke, K., Gepperth, S., Sauer, B., Sadiki, A., Janicka, J., Koch, R., and Bauer, H.J., Experimental and Numerical Investigation of the Primary Breakup of an Airblasted Liquid Sheet, Int. J. Multif. Flow, vol. 91, pp. 208-224, 2017.

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