Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T02:17:55.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Deformation statistics of sub-Kolmogorov-scale ellipsoidal neutrally buoyant drops in isotropic turbulence

Published online by Cambridge University Press:  30 July 2014

L. Biferale ,
C. Meneveau  and
R. Verzicco
L. Biferale*
Affiliation:
Department of Physics and INFN, Università di Roma ‘Tor Vergata’, Roma, Italy
C. Meneveau
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
R. Verzicco
Affiliation:
Department of Industrial Engineering, Università di Roma ‘Tor Vergata’, Roma, Italy PoF and MESA+, University of Twente, Enschede, The Netherlands
*
†Email address for correspondence: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Small droplets in turbulent flows can undergo highly variable deformations and orientational dynamics. For neutrally buoyant droplets smaller than the Kolmogorov scale, the dominant effects from the surrounding turbulent flow arise through Lagrangian time histories of the velocity gradient tensor. Here we study the evolution of representative droplets using a model that includes rotation and stretching effects from the surrounding fluid, and restoration effects from surface tension including a constant droplet volume constraint, while assuming that the droplets maintain an ellipsoidal shape. The model is combined with Lagrangian time histories of the velocity gradient tensor extracted from direct numerical simulations (DNS) of turbulence to obtain simulated droplet evolutions. These are used to characterize the size, shape and orientation statistics of small droplets in turbulence. A critical capillary number is identified associated with unbounded growth of one or two of the droplet’s semi-axes. Exploiting analogies with dynamics of polymers in turbulence, the critical capillary number can be predicted based on the large deviation theory for the largest finite-time Lyapunov exponent quantifying the chaotic separation of particle trajectories. Also, for subcritical capillary numbers near the critical value, the theory enables predictions of the slope of the power-law tails of droplet size distributions in turbulence. For cases when the viscosities of droplet and outer fluid differ in a way that enables vorticity to decorrelate the shape from the straining directions, the large deviation formalism based on the stretching properties of the velocity gradient tensor loses validity and its predictions fail. Even considering the limitations of the assumed ellipsoidal droplet shape, the results highlight the complex coupling between droplet deformation, orientation and the local fluid velocity gradient tensor to be expected when small viscous drops interact with turbulent flows. The results also underscore the usefulness of large deviation theory to model these highly complex couplings and fluctuations in turbulence that result from time integrated effects of fluid deformations.

JFM classification

Drops and Bubbles: Breakup/coalescence Drops and Bubbles: Drops and Bubbles Turbulent Flows: Homogeneous turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 754 , 10 September 2014 , pp. 184 - 207
Copyright
© 2014 Cambridge University Press 

References

Arora, D., Behr, M. & Pasqualis, M. 2006 Hemolysis estimation in a centrifugal blood pump using a tensor-based measure. Artif. Organs 30, 539–547.Google Scholar
Babler, M. U., Biferale, L. & Lanotte, A. S. 2012 Breakup of small aggregates driven by turbulent hydrodynamical stress. Phys. Rev. E 85, 025301R.CrossRefGoogle ScholarPubMed
Balkovsky, E., Fouxon, A. & Lebedev, V. 2000 Turbulent dynamics of polymes solutions. Phys. Rev. Lett. 84, 4765–4768.Google Scholar
Bec, J., Biferale, L., Boffetta, G., Cencini, M., Musacchio, S. & Toschi, F. 2006 Lyapunov exponents of heavy particles in turbulence. Phys. Fluids 18, 091702.Google Scholar
Bec, J., Biferale, L., Lanotte, A. S., Toschi, F. & Scagliarini, A. 2010 Turbulent pair dispersion of inertial particles. J. Fluid Mech. 645, 497–528.Google Scholar
Benzi, R., Biferale, L., Fisher, R., Lamb, D. Q. & Toschi, F. 2010 Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence. J. Fluid Mech. 653, 221–244.CrossRefGoogle Scholar
Benzi, R., Biferale, L., Paladin, G., Vulpiani, A. & Vergassola, M. 1991 Multifractality in the statistics of the velocity gradients in turbulence. Phys. Rev. Lett. 67, 2299–2302.Google Scholar
Boffetta, G., Celani, A. & Musacchio, S. 2003 Two-dimensional turbulence of dilute polymer solutions. Phys. Rev. Lett. 91, 034501.Google Scholar
Can, E. & Prosperetti, A. 2012 A level set method for vapour bubble dynamics. J. Comput. Phys. 231, 1533–1552.Google Scholar
Cencini, M., Bec, J., Biferale, L., Boffetta, G., Celani, A., Lanotte, A. S., Musacchio, S. & Toschi, F. 2006 Dynamics and statistics of heavy particles in turbulence. J. Turbul. 7, 1–16.Google Scholar
Chen, S., Doolen, G. D., Kraichnan, R. H. & She, Z.-S. 1993 On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence. Phys. Fluids A 5, 458–463.CrossRefGoogle Scholar
Chertkov, M. 2000 Polymer stretching by turbulence. Phys. Rev. Lett. 84, 4761–4764.Google Scholar
Chevillard, L. & Meneveau, C. 2013 Orientation dynamics of small, triaxial-ellipsoidal particles in isotropic turbulence. J. Fluid Mech. 337, 187–193.Google Scholar
Cristini, V., Awzdziewicz, J. B., Loewenberg, M. & Collins, L. R. 2003 Breakup in stochastic Stokes flows: sub-Kolmogorov drops in isotropic turbulence. J. Fluid Mech. 492, 231–250.CrossRefGoogle Scholar
Davies, J. T. 1985 Drop sizes of emulsions related to turbulent energy dissipation rates. AIChE J. 40, 839–842.Google Scholar
Eckmann, J.-P. & Procaccia, I. 1986 Fluctuations of dynamical scaling indices in nonlinear systems. Phys. Rev. A 34, 659–661.Google Scholar
Frisch, U. 1995 Turbulence, The Legacy of A.N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Guala, M., Liberzon, A., Tsinober, A. & Kinzelbach, W. 2007 An experimental investigation on Lagrangian correlations of small-scale turbulence at low Reynolds number. J. Fluid Mech. 574, 405–427.Google Scholar
Guido, S., Minale, M. & Maffettone, P. L. 2000a Drop shape dynamics under shear flow reversal. J. Rheol. 44, 1385–1399.CrossRefGoogle Scholar
Guido, S., Minale, M. & Maffettone, P. L. 2000b Drop shape dynamics under shear-flow reversal. J. Rheol. 44, 1385–1399.Google Scholar
Guido, S. & Villone, M. 1998 Three dimensional shape of a drop under simple shear flow. J. Rheol. 42, 395–415.Google Scholar
Hinze, O. 1955 Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AIChE J. 1, 289–295.Google Scholar
Kolmogorov, A. N. 1949 On the disintegration of drops in turbulent flow. Dokl. Akad. Nauk 66, 825–828.Google Scholar
Lasheras, J. C., Eastwood, C., Martinez-Bazan, C. & Montanes, J. L. 2002 A review of statistical models for the break-up of an immiscible fluid immersed into a fully developed turbulent flow. Intl J. Multiphase Flow 28, 247–278.Google Scholar
Lefebvre, A. H. 1989 Atomization and Sprays, Combustion: An International Series, vol. 1040, No. 2756. CRC Press.Google Scholar
Li, M. & Garrett, C. 1998 The relationship between oil droplet size and upper ocean turbulence. Marine Poll. Bull. 36, 961–970.Google Scholar
Lund, T. S. & Rogers, M. M. 1994 An improved measure of strain state probability in turbulent flows. Phys. Fluids 6, 1838–1847.Google Scholar
Maffettone, P. L. & Minale, M. 1998 Equation of change for ellipsoidal drops in viscous flow. J. Non-Newtonian Fluid Mech. 78, 227–241.Google Scholar
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219–245.Google Scholar
Minale, M. 2004 Deformation of a non-Newtonian ellipsoidal drop in a non-Newtonian matrix: extension of Maffettone–Minale model. J. Non-Newtonian Fluid Mech. 123, 151–160.Google Scholar
Minale, M. 2008 A phenomenological model for wall effects on the deformation of an ellipsoidal drop in viscous flow. Rheol. Acta 47, 667–675.Google Scholar
Minale, M. 2010 A phenomenological model for wall effects on the deformation of an ellipsoidal drop in viscous flow. Rheol. Acta 49, 789–806.CrossRefGoogle Scholar
Paladin, G. & Vulpiani, A. 1987 Anomalous scaling laws in multifractal objects. Phys. Rep. 4, 147–225.Google Scholar
Parsa, S., Calzavarini, E., Toschi, F. & Voght, G. A. 2012 Rotation rate of rods in turbulent fluid flow. Phys. Rev. Lett. 109, 134501.Google Scholar
Perlekar, P., Biferale, L., Sbragaglia, M., Shrivastava, S. & Toschi, F. 2012 Droplet size distribution in homogeneos and isotropic turbulence. Phys. Fluids 24, 065101.Google Scholar
Pozrikidis, C. 2003 Numerical simulation of the flow-induced deformation of red blood cells. Ann. Biomed. Engng 31, 1194–1205.CrossRefGoogle ScholarPubMed
Shin, M. & Koch, D. L. 2005 Rotational and translational dispersion of fibres in isotropic turbulent flows. J. Fluid Mech. 540, 143–173.Google Scholar
Sundaresan, S. 2000 Modeling the hydrodynamics of multiphase flow reactors: current status and challenges. AIChE J. 46, 1102–1105.CrossRefGoogle Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. A 138, 41–48.Google Scholar
Terashima, H. & Tryggvason, G. 2009 A front–tracking/ghost–fluid method for fluid interfaces in compressible flows. J. Comput. Phys. 228, 4012–4037.Google Scholar
de Tullio, M. D., Nam, J., Pascazio, G., Balaras, E. & Verzicco, R. 2012 Computational prediction of mechanical hemolysis in aortic valved prostheses. Eur. J. Mech. (B/Fluids) 35, 47–53.Google Scholar
Yang, D., Chamecki, M. & Meneveau, C. 2014 Inhibition of oil plume dilution in Langmuir ocean circulation. Geophys. Res. Lett. 41, 1632–1638.Google Scholar
Yeung, P. K., Pope, S. B., Kurth, E. A. & Lamorgese, A. G. 2007 Lagrangian conditional statistics, acceleration and local relative motion in numerically simulated isotropic turbulence. J. Fluid Mech. 582, 399–422.Google Scholar
Yu, H. & Meneveau, C. 2010a Lagrangian refined Kolmogorov similarity hypothesis for gradient time evolution and correlation in turbulent flows. Phys. Rev. Lett. 104, 084502.Google Scholar
Yu, H. & Meneveau, C. 2010b Scaling of conditional Lagrangian time correlation functions of velocity and pressure gradient magnitudes in isotropic turbulence. Flow Turbul. Combust. 85, 457–472.Google Scholar