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Fractal iso-level sets in high-Reynolds-number scalar turbulence

Kartik P. Iyer1, Jörg Schumacher2,1, Katepalli R. Sreenivasan1,3,*, and P. K. Yeung4

  • 1Tandon School of Engineering, New York University, New York, New York 11201, USA
  • 2Institut für Thermo-und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
  • 3Department of Physics and the Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
  • 4Schools of Aerospace and Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

  • *krs3@nyu.edu
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Phys. Rev. Fluids 5, 044501 – Published 27 April, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.044501

Abstract

We study the fractal scaling of iso-level sets of a passive scalar mixed by three-dimensional homogeneous and isotropic turbulence at high Reynolds numbers. The scalar field is maintained by a linear mean scalar gradient, and the Schmidt number is unity. A fractal box-counting dimension DF can be obtained for iso-levels below about three standard deviations of the scalar fluctuation on either side of its mean value. The dimension varies systematically with the iso-level, with a maximum of about 8/3 for the iso-level at the mean scalar value; this maximum dimension also follows as an upper bound from the geometric measure theory. We interpret this result to mean that mixing in turbulence is incomplete. A unique box-counting dimension for all iso-levels results when we consider the spatial support of the steep cliffs of the scalar conditioned on local strain rate; that unique dimension, independent of the iso-level set, is about 4/3.

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References (42)

  1. A. M. Obukhov, Structure of the temperature field in a turbulent flow, Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 13, 58 (1949).
  2. S. Corrsin, On the spectrum of isotropic temperature fluctuations in an isotropic turbulence, J. Appl. Phys. 22, 469 (1951).
  3. G. K. Batchelor, Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity, J. Fluid Mech. 5, 113 (1959).
  4. K. R. Sreenivasan, On local isotropy of passive scalars in shear flows, Proc. R. Soc. London A 434, 165 (1991).
  5. Z. Warhaft, Turbulence in nature and in the laboratory, Proc. Natl. Acad. Sci. USA 99, 2481 (2002).
  6. P. E. Dimotakis, Turbulent mixing, Annu. Rev. Fluid Mech. 37, 329 (2005).
  7. T. Gotoh and P. K. Yeung, Passive scalar transport in turbulence: A computational perspective, in Ten Chapters in Turbulence, edited by P. A. Davidson, Y. Kaneda, and K. R. Sreenivasan (Cambridge University Press, Cambridge, 2013), p. 87.
  8. K. R. Sreenivasan, Turbulent mixing: A perspective, Proc. Natl. Acad. Sci. USA 116, 18175 (2019).
  9. F. Holzer and E. D. Siggia, Turbulent mixing of a passive scalar, Phys. Fluids 6, 1820 (1994).
  10. K. P. Iyer, J. Schumacher, K. R. Sreenivasan, and P. K. Yeung, Steep Cliffs and Saturated Exponents in Three-Dimensional Scalar Turbulence, Phys. Rev. Lett. 121, 264501 (2018).
  11. A. Celani, A. Lanotte, A. Mazzino, and M. Vergassola, Universality and Saturation of Intermittency in Passive Scalar Turbulence, Phys. Rev. Lett. 84, 2385 (2000).
  12. A. Celani, A. Lanotte, A. Mazzino, and M. Vergassola, Fronts in passive scalar turbulence, Phys. Fluids 13, 1768 (2001).
  13. G. Falkovich, K. Gawędzki, and M. Vergassola, Particles and fields in fluid turbulence, Rev. Mod. Phys. 73, 913 (2001).
  14. R. H. Kraichnan, Small-scale structure of a scalar field convected by turbulence, Phys. Fluids 11, 945 (1968).
  15. B. B. Mandelbrot, Fractals: Form, Chance and Dimension (W. H. Freeman & Co., New York, 1977).
  16. K. R. Sreenivasan and C. Meneveau, The fractal facets of turbulence, J. Fluid Mech. 173, 357 (1986).
  17. K. R. Sreenivasan, R. Ramshankar, and C. Meneveau, Mixing, entrainment and fractal dimension of surfaces in turbulent flows, Proc. Roy. Soc. London A 421, 79 (1989).
  18. K. R. Sreenivasan, Fractals and multifractals in turbulence, Annu. Rev. Fluid Mech. 23, 539 (1991).
  19. C. Meneveau and K. R. Sreenivasan, Interface dimension in intermittent turbulence, Phys. Rev. A. 41, 2246(R) (1990).
  20. L. P. Dasi, F. Schuerg, and D. R. Webster, The geometric properties of high-Schmidt-number passive scalar iso-surfaces in turbulent boundary layers, J. Fluid Mech. 588, 253 (2007).
  21. E. Villermaux and C. Innocenti, On the geometry of turbulent mixing, J. Fluid Mech. 393, 123 (1999).
  22. E. Villermaux, On shapes and forms: Population balance dynamics of corrugated stirred fronts, C. R. Phys. 19, 306 (2018).
  23. K. P. Shete and S. M. de Bruyn Kops, Area of scalar isosurfaces in homogeneous isotropic turbulence as a function of Reynolds and Schmidt numbers, J. Fluid Mech. 883, A38 (2020).
  24. D. A. Donzis, K. R. Sreenivasan, and P. K. Yeung, The Batchelor spectrum for the mixing of passive scalars in isotropic turbulence, Flow, Turb. Combust. 85, 549 (2010).
  25. P. K. Yeung, D. A. Donzis, and K. R. Sreenivasan, Dissipation, enstrophy and pressure statistics in turbulence simulations at high Reynolds numbers, J. Fluid Mech. 700, 5 (2012).
  26. A. Brandenburg, I. Procaccia, D. Segel and A. Vincent, Fractal level sets and multifractal fields in direct simulations of turbulence, Phys. Rev. A 46, 4819 (1992).
  27. I. San Gil, Fractal character of iso-scalar surfaces in shear free turbulence and some effects of shear on the turbulence structure, Ph.D. thesis, Yale University, 2001.
  28. J. Schumacher and K. R. Sreenivasan, Statistics and geometry of passive scalars in turbulence, Phys. Fluids 17, 125107 (2005).
  29. C. Allain and M. Cloitre, Characterizing the lacunarity of random and deterministic fractal sets, Phys. Rev. A 44, 3552 (1991).
  30. P. Constantin, I. Procaccia, and K. R. Sreenivasan, Fractal Geometry of Isosurfaces in Turbulence: Theory and Experiment, Phys. Rev. Lett. 67, 1739 (1991).
  31. F. Morgan, Geometric Measure Theory: A Beginner's Guide (Academic Press, San Diego, 2000).
  32. S. Grossmann and D. Lohse, Fractal-dimension crossovers in turbulent passive scalar signals, Europhys. Lett. 27, 347 (1994).
  33. B. Eckhardt and J. Schumacher, Structure function of passive scalars in two-dimensional turbulence, Phys. Rev. E 60, 4185 (1999).
  34. M. R. Overholt and S. B. Pope, Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence, Phys. Fluids 8, 3128 (1996).
  35. U. Frisch, Turbulence—The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
  36. D. Kushnir, J. Schumacher, and A. Brandt, Geometry of Intensive Scalar Dissipation Events in Turbulence, Phys. Rev. Lett. 97, 124502 (2006).
  37. E. Villermaux, Mixing versus stirring, Annu. Rev. Fluid Mech. 51, 245 (2019).
  38. P. Götzfried, M. S. Emran, E. Villermaux, and J. Schumacher, Comparison of Lagrangian and Eulerian frames of passive scalar turbulent mixing, Phys. Rev. Fluids 4, 044607 (2019).
  39. W. T. Ashurst, A. R. Kerstein, R. M. Kerr, and C. H. Gibson, Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence, Phys. Fluids 30, 2343 (1987).
  40. J. Schumacher, K. R. Sreenivasan, and P. K. Yeung, Very fine structures in scalar mixing, J. Fluid Mech. 531, 113 (2005).
  41. J. Schumacher, Scalar gradient fields by geometric measure theory, Phys. Rev. E 69, 047301 (2004).
  42. D. A. Donzis, P. K. Yeung and K. R. Sreenivasan, Scalar dissipation rate and dissipative anomaly in isotropic turbulence, J. Fluid Mech. 532, 199 (2005).

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Phys. Rev. Fluids 5, 044501– Published 27 April, 2020
  • Received 1 October 2019
  • Accepted 14 February 2020
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