Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-07-04T16:43:45.026Z Has data issue: false hasContentIssue false
  • English
  • Français

The diffusive sheet method for scalar mixing

Published online by Cambridge University Press:  20 December 2017

D. Martínez-Ruiz ,
P. Meunier Open the ORCID record for P. Meunier [Opens in a new window] ,
B. Favier ,
L. Duchemin  and
E. Villermaux Open the ORCID record for E. Villermaux [Opens in a new window]
D. Martínez-Ruiz
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, Marseille, France
P. Meunier
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, Marseille, France
B. Favier
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, Marseille, France
L. Duchemin
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, Marseille, France
E. Villermaux*
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, Marseille, France Institut Universitaire de France, Paris, France
*
†Email address for correspondence: villermaux@irphe.univ-mrs.fr
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Mixing: Mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 230 - 257
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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), 2343–2353.Google Scholar
Batchelor, G. 1952 The effect of homogeneous turbulence on material lines and surfaces. Proc. R. Soc. Lond. A 213, 349–366.Google Scholar
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, 113–144.Google Scholar
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1, 497–504.Google Scholar
Branicki, M. & Wiggins, S. 2009 An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems. Physica D 238 (16), 1625–1657.Google Scholar
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, 21–71.Google Scholar
Cartwright, J. H. E., Feingold, M. & Piro, O. 1996 Chaotic advection in three-dimensional unsteady incompressible laminar flow. J. Fluid Mech. 316, 259–284.Google Scholar
Cocke, W. J. 1969 Turbulent hydrodynamic line stretching: consequences of isotropy. Phys. Fluids 12 (12), 2488–2492.Google Scholar
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, 353–391.Google Scholar
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), 353–361.10.1007/PL00011075Google Scholar
Duplat, J. & Villermaux, E. 2008 Mixing by random stirring in confined mixtures. J. Fluid Mech. 617, 51–86.Google Scholar
Fountain, G. O., Khakhar, D. V. & Ottino, J. M. 1998 Visualization of three-dimensional chaos. Science 281, 683–686.Google Scholar
Girimaji, S. & Pope, S. B. 1990 Material-element deformation in isotropic turbulence. J. Fluid Mech. 220, 427–458.Google Scholar
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), 763–791.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., Huck, P. D., Dentz, M. & Villermaux, E. 2017 Scalar gradients in stirred mixtures and the deconstruction of random fields. J. Fluid Mech. 812, 578–610.10.1017/jfm.2016.799Google Scholar
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, 1471–1480.Google Scholar
Meunier, P. & Leweke, T. 2003 Analysis and treatment of errors due to high velocity gradients in particle image velocimetry. Exp. Fluids 35 (5), 408–421.Google Scholar
Meunier, P. & Villermaux, E. 2003 How vortices mix. J. Fluid Mech. 476, 213–222.Google Scholar
Meunier, P. & Villermaux, E. 2010 The diffusive strip method for scalar mixing in two dimensions. J. Fluid Mech. 662, 134–172.10.1017/S0022112010003162Google Scholar
Moin, P. & Mahesh, K. 1998 Direct numerical simulation: a tool in turbulence research. Annu. Rev. Fluid Mech. 30 (1), 539–578.Google Scholar
Orszag, S. A. 1970 Comments on ‘Turbulent hydrodynamic line stretching: consequences of isotropy’. Phys. Fluids 13, 2203–2204.Google Scholar
Öttinger, H. C. 1996 Stochastic Processes in Polymeric Fluids. Springer.10.1007/978-3-642-58290-5Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport, vol. 3. Cambridge University Press.Google Scholar
Peters, N. 1984 Laminar diffusion flamelet models in non-premixed turbulent combustion. Prog. Energy Combust. Sci. 10 (3), 319–339.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.10.1017/CBO9780511840531Google Scholar
Popinet, S.2010 The GNU Triangulated Surface Library. http://gts.sf.net.Google Scholar
Ramachandran, P. & Varoquaux, G. 2011 Mayavi: 3D visualization of scientific data. Comput. Sci. Engng 13 (2), 40–51.Google Scholar
Ranz, W. E. 1979 Applications of a stretch model to mixing, diffusion, and reaction in laminar and turbulent flows. AIChE J. 25 (1), 41–47.Google Scholar
Richardson, L. F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.Google Scholar
de Rivas, A. & Villermaux, E. 2016 Dense spray evaporation as a mixing process. Phys. Rev. Fluids 1 (1), 014201.Google Scholar
Schumacher, J., Sreenivasan, K. R. & Yeung, P. K. 2005 Very fine structures in scalar mixing. J. Fluid Mech. 531, 113–122.Google Scholar
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), 1204–1214.10.1016/j.ijheatfluidflow.2007.05.012Google Scholar
Souzy, M., Lhuissier, H., Villermaux, E. & Metzger, B. 2017 Stretching and mixing in sheared particulate suspensions. J. Fluid Mech. 812, 611–635.Google Scholar
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), 781–785.Google Scholar
Thom, R. 1993 Prédire n’est pas expliquer. Flammarion.Google Scholar
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), 1017–1028.10.1063/1.858081Google 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, 91–107.10.1016/B978-0-12-380876-9.00003-3Google Scholar
Villermaux, E. & Duplat, J. 2003 Mixing as an aggregation process. Phys. Rev. Lett. 91 (18), 184501.Google Scholar
Yeung, P. K., Donzis, D. A. & Sreenivasan, K. R. 2005 High-Reynolds-number simulation of turbulent mixing. Phys. Fluids 17, 081703.Google Scholar