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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
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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 

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