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Passive scalar mixing and decay at finite correlation times in the Batchelor regime

Published online by Cambridge University Press:  11 July 2017

Aditya K. Aiyer Open the ORCID record for Aditya K. Aiyer [Opens in a new window] ,
Kandaswamy Subramanian  and
Pallavi Bhat
Aditya K. Aiyer*
Affiliation:
BITS-Pilani, K. K. Birla Goa campus, Zuarinagar, Goa-403726, India TIFR Centre for Interdisciplinary Sciences Narsingi, Hyderabad 500075, India Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
Kandaswamy Subramanian
Affiliation:
IUCAA, Post Bag 4, Ganeshkhind, Pune 411007, India
Pallavi Bhat
Affiliation:
IUCAA, Post Bag 4, Ganeshkhind, Pune 411007, India Department of Astrophysical Sciences and Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA Plasma Science and Fusion Center, Massachussetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: aditya.kandaswamy@gmail.com
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Abstract

An elegant model for passive scalar mixing and decay was given by Kraichnan (Phys. Fluids, vol. 11, 1968, pp. 945–953) assuming the velocity to be delta correlated in time. For realistic random flows this assumption becomes invalid. We generalize the Kraichnan model to include the effects of a finite correlation time, $\unicode[STIX]{x1D70F}$, using renewing flows. The generalized evolution equation for the three-dimensional (3-D) passive scalar spectrum $\hat{M}(k,t)$ or its correlation function $M(r,t)$, gives the Kraichnan equation when $\unicode[STIX]{x1D70F}\rightarrow 0$, and extends it to the next order in $\unicode[STIX]{x1D70F}$. It involves third- and fourth-order derivatives of $M$ or $\hat{M}$ (in the high $k$ limit). For small-$\unicode[STIX]{x1D70F}$ (or small Kubo number), it can be recast using the Landau–Lifshitz approach to one with at most second derivatives of $\hat{M}$. We present both a scaling solution to this equation neglecting diffusion and a more exact solution including diffusive effects. To leading order in $\unicode[STIX]{x1D70F}$, we first show that the steady state 1-D passive scalar spectrum, preserves the Batchelor (J. Fluid Mech., vol. 5, 1959, pp. 113–133) form, $E_{\unicode[STIX]{x1D703}}(k)\propto k^{-1}$, in the viscous–convective limit, independent of $\unicode[STIX]{x1D70F}$. This result can also be obtained in a general manner using Lagrangian methods. Interestingly, in the absence of sources, when passive scalar fluctuations decay, we show that the spectrum in the Batchelor regime at late times is of the form $E_{\unicode[STIX]{x1D703}}(k)\propto k^{1/2}$ and also independent of $\unicode[STIX]{x1D70F}$. More generally, finite $\unicode[STIX]{x1D70F}$ does not qualitatively change the shape of the spectrum during decay. The decay rate is however reduced for finite $\unicode[STIX]{x1D70F}$. We also present results from high resolution ($1024^{3}$) direct numerical simulations of passive scalar mixing and decay. We find reasonable agreement with predictions of the Batchelor spectrum during steady state. The scalar spectrum during decay is however dependent on initial conditions. It agrees qualitatively with analytic predictions when power is dominantly in wavenumbers corresponding to the Batchelor regime, but is shallower when box-scale fluctuations dominate during decay.

JFM classification

Mixing: Mixing Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 824 , 10 August 2017 , pp. 785 - 817
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© 2017 Cambridge University Press 

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