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A prediction for the optimal stratification for turbulent mixing

Published online by Cambridge University Press:  26 August 2009

W. TANG ,
C. P. CAULFIELD  and
R. R. KERSWELL
W. TANG
Affiliation:
School of Mathematical & Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA
C. P. CAULFIELD*
Affiliation:
BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
R. R. KERSWELL
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
*
†Email address for correspondence: [email protected]
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Abstract

By identifying the stratification which leads to maximal buoyancy flux in a stably-stratified plane Couette flow, we make a prediction of what bulk stratification (as a function of the shear) is optimal for turbulent mixing. A previous attempt to do this (Caulfield, Tang & Plasting, J. Fluid Mech., vol. 498, 2004, p. 315) failed due to an unexpected degeneracy in the variational problem. Here, we overcome this issue by parameterizing the variational problem implicitly with the overall mixing efficiency which is then optimized across to return a rigorous upper bound on the buoyancy flux. We find that the bulk Richardson number quickly approaches 1/6 in the asymptotic limit of high shear with the associated mixing efficiency tending to 1/3. The predicted mean profiles associated with the bound appear to have a layered structure, with the gradient Richardson number being low both in the interior, and in boundary layers near the walls, with a global maximum, also equal to 1/6, occurring at the edge of the boundary layers.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 634 , 10 September 2009 , pp. 487 - 497
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Armenio, V. & Sarkar, S. 2002 An investigation of stably stratified turbulent channel flow using large-eddy simulation. J. Fluid Mech. 459, 1–42.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.CrossRefGoogle Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.CrossRefGoogle Scholar
Busse, F. H. 1969 On Howard's upper bound for heat transport by turbulent convection. J. Fluid Mech. 37, 457–477.CrossRefGoogle Scholar
Busse, F. H. 1970 Bounds for turbulent shear flow. J. Fluid Mech. 41, 219–240.CrossRefGoogle Scholar
Caulfield, C. P. & Kerswell, R. R. 2001 Maximal mixing rate in turbulent stably stratified Couette flow. Phys. Fluids 13, 894–900.CrossRefGoogle Scholar
Caulfield, C. P., Tang, W. & Plasting, S. C. 2004 Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in a plane stratified Couette flow. J. Fluid Mech. 498, 315–332.CrossRefGoogle Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69, 1648–1651.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49, 4087–4099.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53, 5957–5981.CrossRefGoogle ScholarPubMed
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455–493.CrossRefGoogle Scholar
Hopf, E. 1941 Ein allgemeiner endlichkeitssatz der hydrodynamik. Math. Ann. 117, 764–775.CrossRefGoogle Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.CrossRefGoogle Scholar
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.CrossRefGoogle Scholar
Joseph, D. D. 1976 Stability of Fluid Motions I. Springer.Google Scholar
Kerswell, R. R. 2000 Lowering dissipation bounds for turbulent flows using a smoothness constraint. Phys. Lett. A 272, 230–235.CrossRefGoogle Scholar
Kerswell, R. R. & Soward, A. M. 1996 Upper bounds for turbulent Couette flow incorporating the poloidal power constraint. J. Fluid Mech. 328, 161–176.CrossRefGoogle Scholar
Krommes, J. A. & Smith, R. A. 1987 Rigorous upper bounds for transport due to passive advection by inhomogeneous turbulence. Ann. Phys. 177, 246–329.CrossRefGoogle Scholar
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 2–23.CrossRefGoogle Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. 225, 196–212.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory for turbulent shear flow. J. Fluid Mech. 1, 521–539.CrossRefGoogle Scholar
Osborn, T. R. 1980 Estimates of the local-rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.2.0.CO;2>CrossRefGoogle Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.CrossRefGoogle Scholar
Phillips, O. M. 1972 Turbulence in a strongly stratified fluid is unstable? Deep Sea Res. 19, 79–81.Google Scholar
Plasting, S. C. & Kerswell, R. R. 2003 Improved upper bound on the energy dissipation rate in plane Couette flow: the full solution to Busse's problem and the Constantin–Doering–Hopf problem with one-dimensional background field. J. Fluid Mech. 477, 363–379.CrossRefGoogle Scholar
Posmentier, E. S. 1977 The generation of salinity fine structures by vertical diffusion. J. Phys. Oceanogr. 7, 298–300.2.0.CO;2>CrossRefGoogle Scholar
Prastowo, T., Griffiths, R. W., Hughes, G. O. & Hogg, A. M. 2008 Mixing efficiency in controlled exchange flows. J. Fluid Mech. 600, 235–244.CrossRefGoogle Scholar
Riley, J. J. & de Bruyn Kops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047–2059.CrossRefGoogle Scholar
Saiki, E. M., Moeng, C.-H. & Sullivan, P. P. 2000 Large-eddy simulation of the stably stratified planetary boundary layer. Bound.-layer Meteorol. 95, 1–30.CrossRefGoogle Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar