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On Random Mixing

  • J. Duplat
  • C. Innocenti
  • E. Villermaux
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 510)

Abstract

We study the relaxation of initially segregated scalar mixtures in randomly stirred media, aiming at describing the overall concentration distribution of the mixture, its shape, and rate of deformation as it evolves towards uniformity. Two distinct experiments, one involving an ever dispersing mixture, the other a mixture confined in a channel, both in high Reynolds, three dimensional flows, behave very di erently. We show how these differences are reminiscent of two concomitant aspects of the process of mixing, namely the distribution of individual histories on one hand examined in the present review, and the interaction between the fluid particles on the other, examined separately in [2008]. The particles are stretched sheets whose rates of di usive smoothing and coalescence build up the overall mixture concentration distribution. The randomness of the particle’s net elongation at a given instant of time induces a distribution of the mixing time from which molecular di usion becomes e ective in erasing the concentration di erences. This ingredient is shown to rule the composition of an ever dispersing mixture, providing a detailed analytic description of the overall concentration distribution. It compares favorably with experiments using three di erent passive scalars suggesting that the mixture composition results from a one step lengthening process distributed among the sheets. Consequences of these processes on the spectral, and some geometrical facets of random mixtures are also examined.

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Bibliography

  1. M. Abramowitz and A. Stegun, I. Handbook of Mathematical Functions. Dover Publications, Inc., New York., 1964.MATHGoogle Scholar
  2. C. J. Allègre and D. L. Turcotte. Implications of a two-component marblecake mantle. Nature, 323: 123–127, 1986.CrossRefGoogle Scholar
  3. E. Balkovsky and A. Fouxon. Universal long-time properties of lagrangian statistics in the batchelor regime and their application to the passive scalr problem. Phys. Rev. E., 60((4)):4164–4174, 1999.CrossRefMathSciNetGoogle Scholar
  4. G. K. Batchelor. Small-scale variation of convected quantities like temperature in a turbulent fluid. part 1. general discussion and the case of small conductivity. J. Fluid Mech., 5:113–133, 1959.MATHCrossRefMathSciNetGoogle Scholar
  5. D. Beigie, A. Leonard, and S. Wiggins. A global study of enhanced stretching and diffusion in cahotic tangles. Phys. Fluids A, 3((5)):1039–1050, 1991.CrossRefMathSciNetGoogle Scholar
  6. R. Betchov. An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech., 1:497–504, 1956.MATHCrossRefMathSciNetGoogle Scholar
  7. G. Boffetta and I. M. Sokolov. Statistics of two-particle dispersion in twodimensional turbulence. Phys. Fluids, 14((9)):3224–3232, 2002.CrossRefGoogle Scholar
  8. J.-P. Bouchaud and M. Potters. Theory of Financial Risks and Derivative Pricing. Cambridge University Press., 2003.Google Scholar
  9. B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Z. Wu, S. Zaleski, and G. Zanetti. Scaling of hard thermal turbulence in rayleigh-benard convection. J. Fluid Mech., 204:1–30, 1989.CrossRefGoogle Scholar
  10. H. J. Catrakis and P. E. Dimotakis. Mixing in turbulent jets: scalar measures and isosurface geometry. J. Fluid Mech., 317:369–406, 1996.CrossRefGoogle Scholar
  11. A. Celani, M. Cencini, M. Vergassola, E. Villermaux, and D. Vincenzi. Shear effects on passive scalar spectra. J. Fluid Mech., 523:99–108, 2005.MATHCrossRefGoogle Scholar
  12. M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev. Statistics of a passive scalar advected by a large scale two-dimensional velocity field: Analytic solution. Physical Review E, 51(6):5609, 1995.CrossRefMathSciNetGoogle Scholar
  13. W. J. Cocke. Turbulent hydrodynamic line-stretching: The random walk limit. Phys. Fluids, 14((8)):1624–1628, 1971.CrossRefMathSciNetGoogle Scholar
  14. S. N. Coppersmith, C. h. Liu, S. Majumdar, O. Narayan, and T. A. Witten. Model for force fluctuations in bead packs. Phys. Rev. E, 53((5)):4673–4685, 1996.CrossRefGoogle Scholar
  15. S. Corrsin. On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys., 22:469–473, 1951.MATHCrossRefMathSciNetGoogle Scholar
  16. R. L. Curl. Dispersed phase mixing: I. theory and effect in simple reactors. AIChE J., 9((2)):175–181, 1963.CrossRefGoogle Scholar
  17. P. E. Dimotakis. Turbulent mixing. Annu. Rev. Fluid Mech., 37:329–356, 2005.CrossRefMathSciNetGoogle Scholar
  18. P. E. Dimotakis and H. J. Catrakis. Turbulence, fractals and mixing. In H. Chaté, E. Villermaux, and J. M. Chomaz, editors, Mixing Chaos and Turbulence. Kluwer Academic/Plenum Publishers, New York., 1999.Google Scholar
  19. C. Dopazo. Recent developments in pdf methods. In P. A. Libby and F. A. Williams, editors, Turbulent Reacting Flows chapter 7. Academic Press, 1994.Google Scholar
  20. J. Duplat and E. Villermaux. Persistency of material element deformation in isotropic flows and growth rate of lines and surfaces. Eur. Phys. J. B, 18:353–361, 2000.CrossRefGoogle Scholar
  21. J. Duplat and E. Villermaux. Mixing by random stirring in confined mixtures. J. Fluid Mech., 617:51–86, 2008.MATHCrossRefMathSciNetGoogle Scholar
  22. C. Eckart. An analysis of the stirring and mixing processes in incompressible fluids. J. Mar. Res., 7:265–275, 1948.Google Scholar
  23. G. Falkovich, K. Gawedzki, and M. Vergassola. Particles and fields in fluid turbulence. Rev. Mod. Phys., 73((4)):913–975, 2001.CrossRefMathSciNetGoogle Scholar
  24. A. Fannjiang, S. Nonnenmacher, and L. Wolonski. Dissipation time and decay of correlations Nonlinearity, 17:1481–1508, 2004.MATHCrossRefMathSciNetGoogle Scholar
  25. D. R. Fereday and P. H. Haynes. Scalar decay in two-dimensional chaotic advection and batchelor-regime turbulence. Phys. Fluids, 16(12):4359–4370, 2004.CrossRefMathSciNetGoogle Scholar
  26. J. Fourier. Théorie analytique de la chaleur. Firmin Didot, Paris., 1822.Google Scholar
  27. R. O. Fox. Computational Models for Turbulent Reacting Flows. Cambridge University Press., 2004.Google Scholar
  28. Y. Gagne, E. Villermaux, J. Duplat, and C. Auriault. Etude expérimentale du scalaire passif et du mélange en turbulence. DGA Report, DGA/DSP, 97/1045, 1999.Google Scholar
  29. J. W. Gibbs. Elementary Principles in Statistical Mechanics. Reprint Ox Bow Press, Woodbridge, 1981., 1901.Google Scholar
  30. S. S. Girimaji and S. B. Pope. Material-element deformation in isotropic turbulence. J. Fluid Mech., 220:427–458, 1990.CrossRefGoogle Scholar
  31. A. Groisman and V. Steinberg. Efficient mixing at low reynolds numbers using polymer additives. Nature, 410:905–908, 2001.CrossRefGoogle Scholar
  32. E. J. Hinch. Mixing, turbulence and chaos-an introduction. In H. Chaté, E. Villermaux, and J. M. Chomaz, editors, Mixing Chaos and Turbulence. Kluwer Academic/Plenum Publishers, New York, 1999.Google Scholar
  33. M. Holzer and E. D. Siggia. Turbulent mixing of a passive scalar. Phys. Fluids, 6((5)):1820–1837, 1994.MATHCrossRefMathSciNetGoogle Scholar
  34. Jayesh and Z. Warhaft. Probability distributions of a passive scalar in gridgenerated turbulence. Phys. Rev. Letters, 67((25)):3503–3506, 1991.CrossRefGoogle Scholar
  35. Jayesh and Z. Warhaft. Probability distributions, conditional dissipation, and transport of passive temperature fluctuations in grid-generated turbulence. Phys. Fluids A, 4((10)):2292–2307, 1992.CrossRefGoogle Scholar
  36. M. C. Jullien, P. Castiglione, and P. Tabeling. Experimental observation of batchelor dispersion of passive tracers. Phys. Rev. Letters, 85((17)):3636–3639, 1980.CrossRefGoogle Scholar
  37. J. Kalda. Simple model of intermittent passive scalar turbulence. Phys. Rev. Letters, 84:471–474, 2000.CrossRefGoogle Scholar
  38. S. Kida and S. Goto. Line statistics: Stretching rate of passive lines in turbulence. Phys. Fluids, 14((1)):352–361, 2002.CrossRefMathSciNetGoogle Scholar
  39. R. H. Kraichnan. Convection of a passive scalar by a quasi-uniform random field. J. Fluid Mech., 64(4):737–762, 1974.MATHCrossRefMathSciNetGoogle Scholar
  40. A. Lavertu and L. Mydlarski. Scalar mixing from a concentrated source in a turbulent channel flow. J. Fluid Mech., 528:135–172, 2005.MATHCrossRefGoogle Scholar
  41. M. A. Levèque. Les lois de la transmission de la chaleur par convection. Ann. Mines, 13:201–239, 1928.Google Scholar
  42. F. E. Marble. Mixing, diffusion and chemical reaction of liquids in a vortex field. In M. Moreau and P. Turq, editors, Chemical Reactivity in Liquids: Fundamental Aspects. Plenum Press., 1988.Google Scholar
  43. F. E. Marble, and J. E. Broadwell. The coherent flame model for turbulent chemical reactions. Project SQUID, Tech. Rep. TRW-9-PU, 1977.Google Scholar
  44. P. Meunier and E. Villermaux. How vortices mix. J. Fluid Mech., 476: 213–222, 2003.MATHCrossRefMathSciNetGoogle Scholar
  45. P. L. Miller and P. E. Dimotakis. Measurements of scalar power spectra in high schmidt number turbulent jets. J. Fluid Mech., 308:129–146, 1996.CrossRefGoogle Scholar
  46. W. D. Mohr, R. L. Saxton, and C. H. Jepson. Mixing in laminar-flow systems. Industrial and Engineering technology, 49((11)):1855–1856, 1957.CrossRefGoogle Scholar
  47. L. Mydlarski and Z. Warhaft., Passive scalar statistics in high peclet number grid turbulence. J. Fluid Mech., 358:135–175, 1998.CrossRefGoogle Scholar
  48. S. Nagata, Mixing, Principles and Applications. John Wiley & sons, New York., 1975.Google Scholar
  49. A. M. Obukhov. Structure of the temperature field in a turbulent flow. Izv. Acad. Nauk SSSR, Geogr. i Geofiz, 13:58–69, 1949.Google Scholar
  50. L. Onsager. Statistical hydrodynamics. Nuovo Cimento, VI((IX)):279–287, 1949.MathSciNetGoogle Scholar
  51. J. M. Ottino. The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press., 1989.Google Scholar
  52. J. M. Ottino. Mixing, chaotic advection and turbulence. Annu. Rev. Fluid Mech., 22:207–253, 1990.CrossRefMathSciNetGoogle Scholar
  53. S. B. Pope. Turbulent Flows. Cambridge University Press., 2000.Google Scholar
  54. S. B. Pope. Pdf methods for turbulent reacting flows. Prog. Energy Combust. Sci., 11:119–192, 1985.CrossRefMathSciNetGoogle Scholar
  55. A. Pumir, B. I. Shraiman, and E. D. Siggia. Exponential tails and random advection. Phys. Rev. Letters, 66((23)):2984–2987, 1991.CrossRefGoogle Scholar
  56. W. E. Ranz. Application of a stretch model to mixing, diffusion and reaction in laminar and turbulent flows. AIChE Journal, 25((1)):41–47, 1979.CrossRefGoogle Scholar
  57. L. F. Richardson. Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. A, 110:709–737, 1926.CrossRefGoogle Scholar
  58. M. K. Rivera and R. E. Ecke. Pair dispersion and doubling time statistics in two-dimensional turbulence. Phys. Rev. Letters, 95:194503., 2005.CrossRefGoogle Scholar
  59. H. Schlichting. Boundary Layer Theory. McGraw-Hill, Inc., New York., 1987.Google Scholar
  60. B. I. Shraiman and E. D. Siggia. Scalar turbulence. Nature, 405:639–646, 2000.CrossRefGoogle Scholar
  61. B. I. Shraiman and E. D. Siggia. Lagrangian path integrals and fluctuations in random flows. Phys Rev. E, 49:2912–2927, 1994.CrossRefMathSciNetGoogle Scholar
  62. C. Simonet and A. Groisman. Chaotic mixing in a steady flow in a microchannel. Phys. Rev. Letters, 94:134501., 2005.CrossRefGoogle Scholar
  63. D. T. Son. Turbulence decay of a passive scalar in the batchelor limit: Exact results from a quantum-mechanical approach. Phys. Rev. E., 59 ((4)):R3811–R3814., 1999.CrossRefMathSciNetGoogle Scholar
  64. K. R. Sreenivasan. Fractals and multifractals in fluid turbulence. Annu. Rev. Fluid Mech., 23:539–600, 1991.CrossRefMathSciNetGoogle Scholar
  65. K. R. Sreenivasan. The passive scalar spectrum and the obukhov-corrsin constant. Phys. Fluids, 8((1)):189–196, 1996.MATHCrossRefMathSciNetGoogle Scholar
  66. A. D. Stroock, S. K. W. Dertinger, A. Adjari, I. Mezic, H. A. Stone, and G. M. Whitesides. Chaotic mixer for microchannels. Science, 295:647–651, 2002.CrossRefGoogle Scholar
  67. J. Sukhatme and R. T. Pierrehumbert. Decay of passive scalars under the action of single scale smooth velocity fields in bounded two-dimensional domains: From non-self-similar probability distribution functions to self-similar eigenmodes. Phys. Rev. E, 66:056302, 2002.CrossRefMathSciNetGoogle Scholar
  68. E. S. Szalai and F. J. Muzzio. Predicting microstructure in three-dimensional chaotic systems. Phys. Fluids, 15((11))3274–3279, 2003.CrossRefMathSciNetGoogle Scholar
  69. G. I. Taylor. Diffusion by continuous movements. Proc. Lond. Math. Soc., 20:196–212., 1921.CrossRefGoogle Scholar
  70. G. I. Taylor. Statistical theory of turbulence, part i. Proc. Roy. Soc. A, CLI:421–444, 1935.Google Scholar
  71. S. T. Thoroddsen and C. W. Van Atta. Exponential tails and skewness of density-gradient probability density functions in stably stratified turbulence. J. Fluid Mech., 244:547–566, 1992.CrossRefGoogle Scholar
  72. E. Villermaux and J. Duplat. Mixing as an aggregation process. Phys. Rev. Letters, 91((18)):184501., 2003.CrossRefGoogle Scholar
  73. E. Villermaux and Y. Gagne. Line dispersion in homogeneous turbulence: Stretching, fractal dimensions and micromixing. Phys. Rev. Letters, 73((2)):252–255, 1994.CrossRefGoogle Scholar
  74. E. Villermaux and C. Innocenti. On the geometry of turbulent mixing. J. Fluid Mech., 393:123–145, 1999.MATHCrossRefMathSciNetGoogle Scholar
  75. E. Villermaux and H. Rehab. Mixing in coaxial jets. J. Fluid Mech., 425: 161–185, 2000.MATHCrossRefGoogle Scholar
  76. E. Villermaux, C. Innocenti, and J. Duplat. Histogramme des fluctuations scalaire dans le mélange turbulent trasitoire. C. R. Acad. Sci. Paris, 326(Série IIb):21–26, 1998.MATHGoogle Scholar
  77. E. Villermaux, H. Chaté, and J. M. Chomaz. Why mixing? In H. Chaté, E. Villermaux, and J. M. Chomaz, editors, Mixing Chaos and Turbulence. Kluwer Academic/Plenum Publishers, New York., 1999.Google Scholar
  78. E. Villermaux, C. Innocenti, and J. Duplat. Short circuits in the corrsinoboukhov cascade. Phys. Fluids, 13(1):284–289, 2001.CrossRefGoogle Scholar
  79. E. Villermaux, A. D. Stroock, and H. A. Stone. Bridging kinematics and concentration content in a chaotic micromixer. Phys. Rev. E, 77 (1, Part 2), JAN 2008. ISSN 1539-3755. doi: 10.1103/PhysRevE.77.015301.Google Scholar
  80. M. von Smoluchowski. Versuch einer mathematischen theorie der koagulationskinetik kolloider losungen. Z. Phys. Chem., 92:129–168, 1917.Google Scholar
  81. G. A. Voth, G. Haller, and J. P. Gollub. Experimental measurements of stretching fields in fluids. Phys. Rev. Letters, 88((25)):254501, 2002.CrossRefGoogle Scholar
  82. Z. Warhaft. Passive scalars in turbulent flows. Annu. Rev. Fluid Mech., 32: 203–240, 2000.CrossRefMathSciNetGoogle Scholar
  83. Z. Warhaft. Probability distributions of a passive scalar in grid-generated turbulence. Phys. Rev. Letters, 67:3503–3506, 1991.CrossRefGoogle Scholar
  84. P. Welander. Studies on the general development of motion in a twodimensional, ideal fluid. Tellus, 7((2)):141–156, 1955.MathSciNetCrossRefGoogle Scholar
  85. B. S. Williams, D. Marteau, and J. P. Gollub. Mixing of a passive scalar in magnetically forced two-dimensional turbulence. Phys. Fluids, 9((7)):2061–2080, 1997.MATHCrossRefMathSciNetGoogle Scholar
  86. P. K. Yeung. Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations. J. Fluid Mech., 427:241–274, 2001.MATHCrossRefGoogle Scholar

Copyright information

© CISM, Udine 2009

Authors and Affiliations

  • J. Duplat
    • 1
  • C. Innocenti
    • 2
  • E. Villermaux
    • 3
    • 4
  1. 1.IUSTIAix-Marseille UniversitéMarseille Cedex 13France
  2. 2.Department of PhysicsUniversity of FlorenceFlorenceItaly
  3. 3.IRPHEAix-Marseille UniversitéMarseille Cedex 13France
  4. 4.Institut Universitaire de FranceFrance

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