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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: [email protected]
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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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References

Adalsteinsson, D., Camassa, R., Harenberg, S., Lin, Z., McLaughlin, R. M., Mertens, K., Reis, J., Schlieper, W. & White, B. 2011 Subsurface trapping of oil plumes in stratification: laboratory investigations. Geophys. Mon. Ser. 195, 257–261.Google Scholar
Briggs, G. A. 1965 A plume rise model compared with observations. J. Air Pollut. Control Assoc. 15, 433–438.Google Scholar
Candelier, F. & Vauquelin, O. 2012 Matched asymptotic solutions for turbulent plumes. J. Fluid Mech. 699, 489–499.CrossRefGoogle Scholar
Carazzo, G., Kaminski, E. & Tait, S. 2006 The route to self-similarity in turbulent jets and plumes. J. Fluid Mech. 547, 137–148.Google Scholar
Caulfield, C. P. & Woods, A. W. 1998 Turbulent gravitational convection from a point source in a non-uniformly stratified environment. J. Fluid Mech. 360, 229–248.Google Scholar
Ching, C. Y., Fernando, H. J. S. & Noh, Y. 1993 Interaction of a negatively buoyant line plume with a density interface. Dyn. Atmos. Oceans 19, 367–388.Google Scholar
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic.Google Scholar
Hanna, S. R., Briggs, G. A. & Hosker, R. P. Jr. 1983 Handbook on Atmospheric Diffusion, vol. 17. US Department of Energy.Google Scholar
Hunt, G. R. & Kaye, N. B. 2005 Lazy plumes. J. Fluid Mech. 533, 329–338.Google Scholar
Hunt, G. R. & van den Bremer, T. S. 2010 Classical plume theory: 1937–2010 and beyond. IMA J. Appl. Maths 76, 424–448.Google Scholar
Joint Analysis Group2011 Deepwater Horizon Oil Spill: Review of R/V Brooks McCall data to examine subsurface oil. NOAA Tech. Rep. NOS OR&R 24.Google Scholar
Kaminski, E., Tait, S. & Carazzo, G. 2005 Turbulent entrainment in jets with arbitrary buoyancy. J. Fluid Mech. 526, 361–376.Google Scholar
Kamke, E. 1944 Differentialgleischungen: Lösungensmethoden und Lösungen. Chelsea.Google Scholar
Kaye, N. B. & Scase, M. M. 2011 Straight-sided solutions to classical and modified plume flux equations. J. Fluid Mech. 680, 564–573.Google Scholar
MacIntyre, S., Alldredge, A. L. & Gotschalk, C. C. 1995 Accumulation of marine snow at density discontinuities in the water column. Limnol. Oceanogr. 40, 449–468.Google Scholar
Mariano, A. J., Kourafalou, V. H., Srinivasan, A., Kang, H., Halliwell, G. R., Ryan, E. H. & Roffer, M. 2011 On the modeling of the 2010 Gulf of Mexico oil spill. Dyn. Atmos. Oceans 52, 322–340.Google Scholar
Mehaddi, R., Candelier, F. & Vauquelin, O. 2013 Naturally bounded plumes. J. Fluid Mech. 717, 472–483.Google Scholar
Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Noh, Y., Fernando, H. S. & Ching, C. Y. 1992 Flows induced by the impingement of a two-dimensional thermal on a density interface. J. Phys. Oceanogr. 22, 1207–1220.Google Scholar
Priestley, C. H. B. & Ball, F. K. 1955 Continuous convection from an isolated source of heat. Q. J. R. Meteorol. Soc. 81, 144–157.Google Scholar
Scase, M. M., Caulfield, C. P. & Dalziel, S. B. 2006 Boussinesq plumes and jets with decreasing source strengths in stratified environments. J. Fluid Mech. 563, 463–472.Google Scholar
Wallace, R. & Sheff, B. 1987 Two dimensional buoyant jets in two layer ambient fluid. J. Hydraul. Engng ASCE 113, 992–1005.Google Scholar
Wang, H. & Law, A. W.-K. 2002 Second-order integral model for a round turbulent buoyant jet. J. Fluid Mech. 459, 397–428.Google Scholar
Woods, A. W. 2010 Turbulent plumes in nature. Annu. Rev. Fluid Mech. 42, 391–412.CrossRefGoogle Scholar