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Spatially varying mixing of a passive scalar in a buoyancy-driven turbulent flow

Published online by Cambridge University Press:  24 February 2014

Daan D. J. A. van Sommeren ,
C. P. Caulfield  and
Andrew W. Woods
Daan D. J. A. van Sommeren
Affiliation:
BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
C. P. Caulfield*
Affiliation:
BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Andrew W. Woods
Affiliation:
BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

We perform experiments to study the mixing of passive scalar by a buoyancy-induced turbulent flow in a long narrow vertical tank. The turbulent flow is associated with the downward mixing of a small flux of dense aqueous saline solution into a relatively large upward flux of fresh water. In steady state, the mixing region is of finite extent, and the intensity of the buoyancy-driven mixing is described by a spatially varying turbulent diffusion coefficient $\kappa _v(z)$ which decreases linearly with distance $z$ from the top of the tank. We release a pulse of passive scalar into either the fresh water at the base of the tank, or the saline solution at the top of the tank, and we measure the subsequent mixing of the passive scalar by the flow using image analysis. In both cases, the mixing of the passive scalar (the dye) is well-described by an advection–diffusion equation, using the same turbulent diffusion coefficient $\kappa _v(z)$ associated with the buoyancy-driven mixing of the dynamic scalar. Using this advection–diffusion equation with spatially varying turbulent diffusion coefficient $\kappa _v(z)$, we calculate the residence time distribution (RTD) of a unit mass of passive scalar released as a pulse at the bottom of the tank. The variance in this RTD is equivalent to that produced by a uniform eddy diffusion coefficient with value $\kappa _e= 0.88 \langle \kappa _v \rangle $, where $\langle \kappa _v \rangle $ is the vertically averaged eddy diffusivity. The structure of the RTD is also qualitatively different from that produced by a flow with uniform eddy diffusion coefficient. The RTD using $\kappa _v$ has a larger peak value and smaller values at early times, associated with the reduced diffusivity at the bottom of the tank, and manifested mathematically by a skewness $\gamma _1\approx 1.60$ and an excess kurtosis $\gamma _2\approx 4.19 $ compared to the skewness and excess kurtosis of $\gamma _1\approx 1.46$, $\gamma _2 \approx 3.50$ of the RTD produced by a constant eddy diffusion coefficient with the same variance.

JFM classification

Mixing: Turbulent mixing Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 742 , 10 March 2014 , pp. 701 - 719
Copyright
© 2014 Cambridge University Press 

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