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Restricted Euler dynamics along trajectories of small inertial particles in turbulence

  • Perry L. Johnson (a1) and Charles Meneveau (a1)
Abstract

The fate of small particles in turbulent flows depends strongly on the velocity gradient properties of the surrounding fluid, such as rotation and strain rates. For non-inertial (fluid) particles, the restricted Euler model provides a simple low-dimensional dynamical system representation of Lagrangian evolution of velocity gradients in fluid turbulence, at least for short times. Here, we derive a new restricted Euler dynamical system for the velocity gradient evolution of inertial particles, such as solid particles in a gas, or droplets and bubbles in turbulent liquid flows. The model is derived in the limit of small (sub-Kolmogorov-scale) particles and low Stokes number. The system exhibits interesting fixed points, stability and invariant properties. Comparisons with data from direct numerical simulations show that the model predicts realistic trends such as the tendency of increased straining over rotation along heavy particle trajectories and, for light particles such as bubbles, the tendency of reduced self-stretching of the strain rate.

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Corresponding author
Email address for correspondence: pjohns86@jhu.edu
References
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Ashurst Wm. T., Kerstein A. R., Kerr R. M. & Gibson C. H. 1987 Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30 (8), 23432353.
Balkovsky E., Falkovich G. & Fouxon A. 2001 Intermittent distribution of inertial particles in turbulent flows. Phys. Rev. Lett. 86 (13), 27902793.
Batchelor G. K. 1980 Mass transfer from small particles suspended in turbulent fluid. J. Fluid Mech. 98, 609623.
Bec J. 2003 Fractal clustering of inertial particles in random flows. Phys. Fluids 15, 8184.
Bec J., Biferale L., Boffetta G., Cencini M., Musacchio S. & Toschi F. 2006 Lyapunov exponents of heavy particles in turbulence. Phys. Fluids 18, 091702.
Bec J., Biferale L., Lanotte A. S., Scagliarini A. & Toschi F. 2010 Turbulent pair dispersion of inertial particles. J. Fluid Mech. 645, 497528.
Benzi R., Biferale L., Calzavarini E., Lohse D. & Toschi F. 2009 Velocity-gradient statistics along particle trajectories in turbulent flows: the refined similarity hypothesis in the Lagrangian frame. Phys. Rev. E 80, 16.
Betchov R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1 (05), 497504.
Bewley G. P., Saw E. W. & Bodenschatz E. 2013 Observation of the sling effect. New J. Phys. 15, 083051.
Biferale L., Chevillard L., Meneveau C. & Toschi F. 2007 Multiscale model of gradient evolution in turbulent flows. Phys. Rev. Lett. 98, 2528.
Biferale L., Meneveau C. & Verzicco R. 2014 Deformation statistics of sub-Kolmogorov-scale ellipsoidal neutrally buoyant drops in isotropic turbulence. J. Fluid Mech. 754, 184207.
Biferale L., Scagliarini A. & Toschi F. 2010 On the measurement of vortex filament lifetime statistics in turbulence. Phys. Fluids 22 (6), 065101.
Blackburn H. M., Mansour N. N. & Cantwell B. J. 1996 Topology of fine-scale motions in turbulent channel flow. J. Fluid Mech. 310, 269.
Calzavarini E., van den Berg T. H., Toschi F. & Lohse D. 2008 Quantifying microbubble clustering in turbulent flow from single-point measurements. Phys. Fluids 20, 040702.
Cantwell B. J. 1992 Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids 4 (4), 782793.
Chertkov M., Pumir A. & Shraiman B. I. 1999 Lagrangian tetrad dynamics and the phenomenology of turbulence. Phys. Fluids 11 (8), 23942410.
Chevillard L. & Meneveau C. 2006 Lagrangian dynamics and statistical geometric structure of turbulence. Phys. Rev. Lett. 97 (17), 174501.
Chevillard L. & Meneveau C. 2013 Orientation dynamics of small, triaxial-ellipsoidal particles in isotropic turbulence. J. Fluid Mech. 737, 571596.
Chevillard L., Meneveau C., Biferale L. & Toschi F. 2008 Modeling the pressure Hessian and viscous Laplacian in turbulence: comparisons with direct numerical simulation and implications on velocity gradient dynamics. Phys. Fluids 20 (10), 101504.
Chong M. S., Soria J., Perry A. E., Chacin J., Cantwell B. J. & Na Y. 1998 Turbulence structures of wall-bounded shear flows found using DNS data. J. Fluid Mech. 357, 225247.
Eaton J. K. & Fessler J. R. 1994 Preferential concentration of particles by turbulence. Intl J. Multiphase Flow 20, 169209.
Esmaily-Moghadam M. & Mani A. 2016 Analysis of the clustering of inertial particles in turbulent flows. Phys. Rev. Fluids 1 (8), 084202.
Falkovich G., Fouxon A. & Stepanov M. G. 2002 Acceleration of rain initiation by cloud turbulence. Nature 419, 151154.
Girimaji S. S. & Pope S. B. 1990 A diffusion model for velocity gradients in turbulence. Phys. Fluids 2 (2), 242.
Gopalan B., Malkiel E. & Katz J. 2008 Experimental investigation of turbulent diffusion of slightly buoyant droplets in locally isotropic turbulence. Phys. Fluids 20 (9), 095102.
Ireland P. J., Bragg A. D. & Collins L. R. 2016 The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 1. Simulations without gravitational effects. J. Fluid Mech. 796, 659711.
Ishihara T., Gotoh T. & Kaneda Y. 2009 Study of high Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165180.
Jeong E. & Girimaji S. S. 2003 Velocity-gradient dynamics in turbulence: effect of viscosity and forcing. Theor. Comput. Fluid Dyn. 16 (6), 421432.
Johnson P. L. & Meneveau C. 2016 A closure for Lagrangian velocity gradient evolution in turbulence using recent deformation mapping of initially Gaussian fields. J. Fluid Mech. 804, 387419.
Karp-Boss L., Boss E. & Jumars P. A. 1996 Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion. Oceanography Mar. Biol.: An Annual Rev. 34, 71107.
Kerr R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 3158.
Maffettone P. L. & Minale M. 1998 Equation of change for ellipsoidal drops in viscous flow. J. Non-Newtonian Fluid Mech. 78 (2–3), 227241.
Martins Afonso M. & Meneveau C. 2010 Recent fluid deformation closure for velocity gradient tensor dynamics in turbulence: timescale effects and expansions. Physica D 239 (14), 12411250.
Maxey M. R. 1987 The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441.
Maxey M. R. & Riley J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883889.
Meneveau C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219245.
Monchaux R., Bourgoin M. & Cartellier A. 2012 Analyzing preferential concentration and clustering of inertial particles in turbulence. Intl J. Multiphase Flow 40, 118.
Ooi A., Martin J., Soria J. & Chong M. S. 1999 A study of the evolution and characteristics of the invariants of the velocity-gradient tensor in isotropic turbulence. J. Fluid Mech. 381, 141174.
Parsa S., Calzavarini E., Toschi F. & Voth G. A. 2012 Rotation rate of rods in turbulent fluid flow. Phys. Rev. Lett. 109, 14.
Pumir A., Bodenschatz E. & Xu H. 2013 Tetrahedron deformation and alignment of perceived vorticity and strain in a turbulent flow. Phys. Fluids 25, 035101.
Pumir A. & Wilkinson M. 2011 Orientation statistics of small particles in turbulence. New J. Phys. 13, 093030.
Reade W. C. & Collins L. R. 2000 Effect of preferential concentration on turbulent collision rates. Phys. Fluids 12, 25302540.
Soria J., Sondergaard R., Cantwell B. J., Chong M. S. & Perry A. E. 1994 A study of the fine-scale motions of incompressible time-developing mixing layers. Phys. Fluids 6, 871.
Sundaram S. & Collins L. R. 1997 Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations. J. Fluid Mech. 335, 75109.
Toschi F. & Bodenschatz E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech. 41 (1), 375404.
Tsinober A. 2001 An Informal Introduction to Turbulence. Kluwer Academic.
Vieillefosse P. 1982 Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. 43, 837842.
Vieillefosse P. 1984 Internal motion of a small element of fluid in an inviscid flow. Physica A 125, 150162.
Wang L.-P. & Maxey M. R. 1993a Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256, 2768.
Wang L.-P. & Maxey M. R. 1993b The motion of microbubbles in a forced isotropic and homogeneous turbulence. Appl. Sci. Res. 51, 291296.
Wang L.-P., Wexler A. S. & Zhou Y. 2000 Statistical mechanical description and modelling of turbulent collision of inertial particles. J. Fluid Mech. 415, 117153.
Wilczek M. & Meneveau C. 2014 Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. J. Fluid Mech. 756, 191225.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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