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The diffusive sheet method for scalar mixing

  • D. Martínez-Ruiz (a1), P. Meunier (a1), B. Favier (a1), L. Duchemin (a1) and E. Villermaux (a1) (a2)...

Abstract

The diffusive strip method (DSM) is a near-exact numerical method for mixing computations initially developed in two dimensions (Meunier & Villermaux, J. Fluid Mech., vol. 662, 2010, pp. 134–172). The method, which consists of following stretched material lines to compute the resulting scalar field a posteriori, is extended here to three-dimensional flows. We describe the procedure and its three-dimensional peculiarity, which relies on the Lagrangian advection of a triangulated surface from which the stretching rate is extracted to infer the scalar field. The method is first validated at moderate Péclet number against a classical pseudospectral method solving the advection–diffusion equation for a Batchelor vortex, and then applied to a simple Taylor–Couette experimental configuration with non-rotating boundary conditions at the top-end disk, bottom-end disk and outer cylinder. This motion, producing an elaborate although controlled steady three-dimensional flow, relies on Ekman pumping arising from the rotation of the inner cylinder. A recurrent two-cell structure is separated by the horizontal mid-plane and formed by stream tubes shaped as nested tori under laminar flow conditions. A scalar blob in the flow experiences a Lagrangian oscillating dynamics undergoing stretchings and compressions, driving the mixing process. The DSM enables the calculation of the blob elongation and scalar concentration distributions through a single variable computation along the advected blob surface, capturing the rich evolution observed in the experiments. Interestingly, the mixing process in this axisymmetric and steady three-dimensional flow leads to a linear growth of surfaces in time similar to the one obtained in a two-dimensional shear. The potentialities, limits and extension of the method to more general flows are finally discussed.

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Corresponding author

Email address for correspondence: villermaux@irphe.univ-mrs.fr

References

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Ashurst, W. 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.
Batchelor, G. 1952 The effect of homogeneous turbulence on material lines and surfaces. Proc. R. Soc. Lond. A 213, 349366.
Batchelor, G. K. 1959 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, 113144.
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1, 497504.
Branicki, M. & Wiggins, S. 2009 An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems. Physica D 238 (16), 16251657.
Buch, K. A. Jr & Dahm, W. J. A. 1996 Experimental study of the fine-scale structure of conserved scalar mixing in turbulent shear flows. Part 1. Sc ≫ 1. J. Fluid Mech. 317, 2171.
Cartwright, J. H. E., Feingold, M. & Piro, O. 1996 Chaotic advection in three-dimensional unsteady incompressible laminar flow. J. Fluid Mech. 316, 259284.
Cocke, W. J. 1969 Turbulent hydrodynamic line stretching: consequences of isotropy. Phys. Fluids 12 (12), 24882492.
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. M. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353391.
Duplat, J. & Villermaux, E. 2000 Persistency of material element deformation in isotropic flows and growth rate of lines and surfaces. Eur. Phys. J. B 18 (2), 353361.
Duplat, J. & Villermaux, E. 2008 Mixing by random stirring in confined mixtures. J. Fluid Mech. 617, 5186.
Fountain, G. O., Khakhar, D. V. & Ottino, J. M. 1998 Visualization of three-dimensional chaos. Science 281, 683686.
Girimaji, S. & Pope, S. B. 1990 Material-element deformation in isotropic turbulence. J. Fluid Mech. 220, 427458.
Krauskopf, B., Osinga, H. M., Doedel, E. J., Henderson, M. E., Guckenheimer, J., Vladimirsky, A., Dellnitz, M. & Junge, O. 2005 A survey of methods for computing (un)stable manifolds of vector fields. Intl J. Bifurcation Chaos 15 (03), 763791.
Le Borgne, T., Dentz, M. & Villermaux, E. 2015 The lamellar description of mixing in porous media. J. Fluid Mech. 770, 458498.
Le Borgne, T., Huck, P. D., Dentz, M. & Villermaux, E. 2017 Scalar gradients in stirred mixtures and the deconstruction of random fields. J. Fluid Mech. 812, 578610.
Lesur, G. & Longaretti, P.-Y. 2007 Impact of dimensionless numbers on the efficiency of magnetorotational instability induced turbulent transport. Mon. Not. R. Astron. Soc. 378, 14711480.
Meunier, P. & Leweke, T. 2003 Analysis and treatment of errors due to high velocity gradients in particle image velocimetry. Exp. Fluids 35 (5), 408421.
Meunier, P. & Villermaux, E. 2003 How vortices mix. J. Fluid Mech. 476, 213222.
Meunier, P. & Villermaux, E. 2010 The diffusive strip method for scalar mixing in two dimensions. J. Fluid Mech. 662, 134172.
Moin, P. & Mahesh, K. 1998 Direct numerical simulation: a tool in turbulence research. Annu. Rev. Fluid Mech. 30 (1), 539578.
Orszag, S. A. 1970 Comments on ‘Turbulent hydrodynamic line stretching: consequences of isotropy’. Phys. Fluids 13, 22032204.
Öttinger, H. C. 1996 Stochastic Processes in Polymeric Fluids. Springer.
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport, vol. 3. Cambridge University Press.
Peters, N. 1984 Laminar diffusion flamelet models in non-premixed turbulent combustion. Prog. Energy Combust. Sci. 10 (3), 319339.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Popinet, S.2010 The GNU Triangulated Surface Library. http://gts.sf.net.
Ramachandran, P. & Varoquaux, G. 2011 Mayavi: 3D visualization of scientific data. Comput. Sci. Engng 13 (2), 4051.
Ranz, W. E. 1979 Applications of a stretch model to mixing, diffusion, and reaction in laminar and turbulent flows. AIChE J. 25 (1), 4147.
Richardson, L. F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.
de Rivas, A. & Villermaux, E. 2016 Dense spray evaporation as a mixing process. Phys. Rev. Fluids 1 (1), 014201.
Schumacher, J., Sreenivasan, K. R. & Yeung, P. K. 2005 Very fine structures in scalar mixing. J. Fluid Mech. 531, 113122.
Schwertfirm, F. & Manhart, M. 2007 DNS of passive scalar transport in turbulent channel flow at high Schmidt numbers. Intl J. Heat Fluid Flow 28 (6), 12041214.
Souzy, M., Lhuissier, H., Villermaux, E. & Metzger, B. 2017 Stretching and mixing in sheared particulate suspensions. J. Fluid Mech. 812, 611635.
Sutherland, W. 1905 A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin. Lond. Edin. Dublin Phil. Mag. J. Sci. 9 (54), 781785.
Thom, R. 1993 Prédire n’est pas expliquer. Flammarion.
Varosi, F., Antonsen, T. M. J. & Ott, E. 1991 The spectrum of fractal dimensions of passively convected scalar gradients in chaotic fluid flows. Phys. Fluids A 3 (5), 10171028.
Villermaux, E. 2012a Mixing by porous media. C. R. Méc. 340, 933943.
Villermaux, E. 2012b On dissipation in stirred mixtures. Adv. Appl. Mech. 45, 91107.
Villermaux, E. & Duplat, J. 2003 Mixing as an aggregation process. Phys. Rev. Lett. 91 (18), 184501.
Yeung, P. K., Donzis, D. A. & Sreenivasan, K. R. 2005 High-Reynolds-number simulation of turbulent mixing. Phys. Fluids 17, 081703.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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