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Optimal mixing of buoyant jets and plumes in stratified fluids: theory and experiments

Published online by Cambridge University Press:  01 February 2016

R. Camassa ,
Z. Lin ,
R. M. McLaughlin ,
K. Mertens ,
C. Tzou ,
J. Walsh  and
B. White
R. Camassa
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
Z. Lin
Affiliation:
Department of Mathematics, Zhejiang University, Xihu, Hangzhou, Zhejiang 321000, PR China
R. M. McLaughlin
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
K. Mertens
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
C. Tzou*
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
J. Walsh
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
B. White
Affiliation:
Department of Marine Sciences, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
*
†Email address for correspondence: pgtz31@live.unc.edu
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Abstract

The influence of ambient fluid stratification on buoyant miscible jets and plumes is studied theoretically and experimentally. Given a fixed set of jet/plume parameters, and an ambient fluid stratification sandwiched between top and bottom homogeneous densities, a theoretical criterion is identified to show how step-like density profiles constitute the most effective mixers within a broad class of stable density transitions. This is assessed both analytically and experimentally, respectively by establishing rigorous a priori estimates on generalized Morton–Taylor–Turner (MTT) models (Morton et al., Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23; Fischer et al., Mixing in Inland and Coastal Waters. Academic, 1979), and by studying a critical phenomenon determined by the distance between the jet/plume release height with respect to the depth of the ambient density transition. For fluid released sufficiently close to the background density transition, the buoyant jet fluid escapes and rises indefinitely. For fluid released at locations lower than a critical depth, the buoyant fluid stops rising and is trapped indefinitely. A mathematical formulation providing rigorous estimates on MTT models is developed along with nonlinear jump conditions and an exact critical-depth formula that is in good quantitative agreement with the experiments. Our mathematical analysis provides rigorous justification for the critical trapping/escaping criteria, first presented in Caulfield & Woods (J. Fluid Mech., vol. 360, 1998, pp. 229–248), within a class of algebraic density decay rates. Further, the step-like background stratification is shown to be the most efficient mixing profile amongst a broad family of stably stratified profiles sharing the same density transition within a fixed distance. Finally, the analysis uncovers surprising differences between the Gaussian and top-hat profile closures concerning initial mixing of the jet and ambient fluid.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Stratified flows Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 790 , 10 March 2016 , pp. 71 - 103
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Copyright
© 2016 Cambridge University Press 

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