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Scalar gradients in stirred mixtures and the deconstruction of random fields

Published online by Cambridge University Press:  05 January 2017

T. Le Borgne ,
P. D. Huck ,
M. Dentz  and
E. Villermaux
T. Le Borgne*
Affiliation:
Université de Rennes 1, CNRS, Geosciences Rennes UMR6118, 35042 Rennes, France
P. D. Huck
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384 Marseille, France
M. Dentz
Affiliation:
IDAEA-CSIC, Barcelona, Spain
E. Villermaux
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384 Marseille, France Institut Universitaire de France, Paris, France
*
†Email address for correspondence: tanguy.le-borgne@univ-rennes1.fr
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Abstract

A general theory for predicting the distribution of scalar gradients (or concentration differences) in heterogeneous flows is proposed. The evolution of scalar fields is quantified from the analysis of the evolution of elementary lamellar structures, which naturally form under the stretching action of the flows. Spatial correlations in scalar fields, and concentration gradients, hence develop through diffusive aggregation of stretched lamellae. Concentration levels at neighbouring spatial locations result from a history of lamella aggregation, which is partly common to the two locations. Concentration differences eliminate this common part, and thus depend only on lamellae that have aggregated independently. Using this principle, we propose a theory which envisions concentration increments as the result of a deconstruction of the basic lamella assemblage. This framework provides analytical expressions for concentration increment probability density functions (PDFs) over any spatial increments for a range of flow systems, including turbulent flows and low-Reynolds-number porous media flows, for confined and dispersing mixtures. Through this deconstruction principle, scalar increment distributions reveal the elementary stretching and aggregation mechanisms building scalar fields.

JFM classification

Mixing: Mixing Low-Reynolds-number flows: Porous media Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 812 , 10 February 2017 , pp. 578 - 610
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© 2017 Cambridge University Press 

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Dover.Google Scholar
de Anna, P., Dentz, M., Tartakovsky, A. & Le Borgne, T. 2014a The filamentary structure of mixing fronts and its control on reaction kinetics in porous media flows. Geophys. Res. Lett. 41, 4586–4593.CrossRefGoogle Scholar
de Anna, P., Jimenez-Martinez, J., Tabuteau, H., Turuban, R., Le Borgne, T., Derrien, M. & Meheust, Y. 2014b Mixing and reaction kinetics in porous media: an experimental pore scale quantification. Environ. Sci. Technol. 48, 508–516.Google Scholar
Antonia, R. A., Hopfinger, E. J., Gagne, Y. & Anselmet, F. 1984 Temperature structure functions in turbulent shear flows. Phys. Rev. A 30 (5), 2704–2707.CrossRefGoogle Scholar
Balkovsky, E. & Fouxon, A. 1999 Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem. Phys. Rev. E 60 (4), 4164–4174.Google Scholar
Batchelor, G. K. 1959 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.CrossRefGoogle Scholar
Battiato, I., Tartakovsky, D. M., Tartakovsky, A. M. & Scheibe, T. 2009 On breakdown of macroscopic models of mixing-controlled heterogeneous reactions in porous media. Adv. Water Resour. 32, 1664–1673.CrossRefGoogle Scholar
Bijeljic, B., Mostaghimi, P. & Blunt, M. 2011 Signature of non-Fickian solute transport in complex heterogeneous porous media. Phys. Rev. Lett. 107, 204502.Google Scholar
Bolster, D. 2014 The fluid mechanics of dissolution trapping in geologic storage of CO2 . J. Fluid Mech. 740, 1–4.Google Scholar
Chiogna, G., Hochstetler, D. L., Bellin, A., Kitanidis, P. K. & Rolle, M. 2012 Mixing, entropy and reactive solute transport. Geophys. Res. Lett. 39, L20405.CrossRefGoogle Scholar
Dentz, M., LeBorgne, T., Englert, A. & Bijeljic, B. 2011 Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120–121, 1–17.Google Scholar
Duplat, J., Innocenti, C. & Villermaux, E. 2010 A nonsequential turbulent mixing process. Phys. Fluids 22, 035104.Google Scholar
Duplat, J. & Villermaux, E. 2008 Mixing by random stirring in confined mixtures. J. Fluid Mech. 617, 51–86.Google Scholar
Engdahl, N., Bolster, D. & Benson, D. A. 2014 Predicting the enhancement of mixing-driven reactions in nonuniform flows using measures of flow topology. Phys. Rev. E 90, 051001.Google Scholar
Falkovich, G., Gawedzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73 (4), 913–975.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.Google Scholar
Fu, X., Cueto-Felgueroso, L., Bolster, D. & Juanes, R. 2015 Rock dissolution patterns and geochemical shutdown of CO2 -brine-carbonate reactions during convective mixing in porous media. J. Fluid Mech. 726, 296–315.Google Scholar
Goodman, J. W. 2007 Speckle Phenomena in Optics. Roberts and Company Publishers.Google Scholar
Greffier, O., Amarouchene, Y. & Kellay, H. 2002 Thickness fluctuations in turbulent soap films. Phys. Rev. Lett. 88 (19), 194101.CrossRefGoogle ScholarPubMed
Haudin, F., Cartwright, J. H. E., Brau, F. & De Wit, A. 2014 Spiral precipitation patterns in confined chemical gardens. Proc. Natl Acad. Sci. USA 111, 17363–17367.Google Scholar
Hidalgo, J. J., Dentz, M., Cabeza, Y. & Carrera, J. 2015 Dissolution patterns and mixing dynamics in unstable reactive flow. Geophys. Res. Lett. 42, 6357–6364.Google Scholar
Jha, B., Cueto-Felgueroso, L. & Juanes, R. 2011 Fluid mixing from viscous fingering. Phys. Rev. Lett. 106, 194–502.CrossRefGoogle ScholarPubMed
Jimenez-Martinez, J., Porter, M. L., Hyman, J. D., Carey, J. W. & Viswanathan, H. S. 2016 Mixing in a three-phase system: enhanced production of oil-wet reservoirs by CO2 injection. Geophys. Res. Lett. 43, 196–205.CrossRefGoogle Scholar
Kalda, J. 2000 Simple model of intermittent passive scalar turbulence. Phys. Rev. Lett. 84 (3), 471–474.CrossRefGoogle ScholarPubMed
Kalda, J. & Morozenko, A. 2008 Turbulent mixing: the roots of intermittency. New J. Phys. 10, 093003.CrossRefGoogle Scholar
Kraichnan, R. H. 1974 Convection of a passive scalar by a quasi-uniform random straining field. J. Fluid Mech. 64, 737–762.CrossRefGoogle Scholar
Kraichnan, R. H. 1994 Anomalous scaling of a randomly advected passive scalar. Phys. Rev. Lett. 72, 1016–1019.Google Scholar
Le Borgne, T., Dentz, M. & Villermaux, E. 2013 Stretching, coalescence and mixing in porous media. Phys. Rev. Lett. 110, 204501.Google Scholar
Le Borgne, T., Dentz, M. & Villermaux, E. 2015 The lamellar description of mixing in porous media. J. Fluid Mech. 770, 458–498.Google Scholar
Le Borgne, T., Ginn, T. & Dentz, M. 2014 Impact of fluid deformation on mixing-induced chemical reactions in heterogeneous flows. Geophys. Res. Lett. 41, 7898–7906.Google Scholar
Meunier, P. & Villermaux, E. 2010 The diffusive strip method for scalar mixing in two dimensions. J. Fluid Mech. 662, 134–172.CrossRefGoogle Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2. MIT Press.Google Scholar
Neufeld, Z. & Hernandez-Garcia, E. 2009 Chemical and Biological Processes in Fluid Flows: A Dynamical Systems Approach. Imperial College Press.Google Scholar
Oboukhov, A. M. 1962 Some specific features of atmospheric tubulence. J. Fluid Mech. 13, 77–81.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Paster, A., Aquino, T. & Bolster, D. 2015 Incomplete mixing and reactions in a laminar shear flow. Phys. Rev. E 92, 012922.Google Scholar
Pumir, A., Shraiman, B. I. & Siggia, E. D. 1991 Exponential tails and random advection. Phys. Rev. Lett. 66 (23), 2984–2987.Google Scholar
Ranz, W. E. 1979 Application of a stretch model to mixing, diffusion and reaction in laminar and turbulent flows. AIChE J. 25 (1), 41–47.Google Scholar
Lord Rayleigh 1880 On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. Phil. Mag. X, 73–78.Google Scholar
Shraiman, B. I. & Siggia, E. D. 2000 Scalar turbulence. Nature 405, 639–646.Google Scholar
Stocker, R. 2012 Marine microbes see a sea of gradients. Science 6107, 628–633.Google Scholar
Tartakovsky, A. M., Tartakovsky, D. M. & Meakin, P. 2008 Stochastic Langevin model for flow and transport in porous media. Phys. Rev. Lett. 101 (4), 044502.CrossRefGoogle ScholarPubMed
Taylor, J. R. & Stocker, R. 2012 Trade-offs of chemotactic foraging in turbulent water. Science 338, 675.Google Scholar
Tel, T., de Mourab, A., Grebogib, C. & Károlyid, G. 2005 Chemical and biological activity in open flows: a dynamical system approach. Phys. Rep. 413, 91–196.CrossRefGoogle Scholar
Vaienti, S., Ould-Rouis, M., Anselmet, F. & Le Gal, P. 1994 Statistics of temperature increments in fully developed turbulence. Part I. Theory. Physica D 73, 99–112.Google Scholar
Vaienti, S., Ould-Rouis, M., Anselmet, F. & Le Gal, P. 1995 Statistics of temperature increments in fully developed turbulence. Part II. Experiments. Physica D 85, 405–424.Google Scholar
Vergassola, M., Villermaux, E. & Shraiman, B. I. 2007 Infotaxis as a strategy for searching without gradients. Nature 445, 406–409.Google Scholar
Vernède, S., Ponson, L. & Bouchaud, J. P. 2015 Turbulent fracture surfaces: a footprint of damage percolation? Phys. Rev. Lett. 114, 215501.Google Scholar
Villermaux, E. 2012a Mixing by porous media. C. R. Méc. 340, 933–943.Google Scholar
Villermaux, E. 2012b On dissipation in stirred mixtures. Adv. Appl. Mech. 45, 91–107.Google Scholar
Villermaux, E. & Duplat, J. 2003 Mixing as an aggregation process. Phys. Rev. Lett. 91 (18), 184501, 1–4.Google Scholar
Villermaux, E. & Duplat, J. 2006 Coarse grained scale of turbulent mixtures. Phys. Rev. Lett. 97, 144506.Google Scholar
Villermaux, E., Stroock, A. D. & Stone, H. A. 2008 Bridging kinematics and concentration content in a chaotic micromixer. Phys. Rev. E 77, 015301 (R).Google Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32 (1), 203–240.CrossRefGoogle Scholar
Yang, X. I. A., Marusic, I. & Meneveau, C. 2016 Hierarchical random additive process and logarithmic scaling of generalized high order, two-point correlations in turbulent boundary layer flow. Phys. Rev. Fluids 1, 024402.Google Scholar
Ye, Y., Chiogna, G., Cirpka, O. A., Grathwohl, P. & Roll, M. 2015 Experimental evidence of helical flow in porous media. Phys. Rev. Lett. 115, 194502.Google Scholar