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Turbulent Couette flow profiles that maximize the efficiency function

Published online by Cambridge University Press:  26 April 2006

L. M. Smith
L. M. Smith
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA 02139, USA
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Abstract

A previous study revealed that only maximization of the efficiency function, from among a large class of mean field moments, results in both a logarithmic law and a velocity defect law in turbulent Poiseuille channel flow. The efficiency function, [Escr ], is the product of a drag coefficient and the ratio of the fluctuation and mean dissipation rate integrals. Here, maximum [Escr ] is explored in Couette flow to test its generality as a statistical stability criterion for turbulent shear flows. The optimal flow exhibits a logarithmic law but does not have a velocity defect law. A decreasing velocity defect is predicted for Reynolds numbers up to 30 000. This prediction is shown to be supported by the existing data, which are limited to Reynolds numbers less than 20 000.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 227 , June 1991 , pp. 509 - 525
Copyright
© 1991 Cambridge University Press

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References

Arnol'D, V. I. 1965 Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSSR 162, 975.Google Scholar
Bayly, B. J., Orszag, S. A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359.Google Scholar
Busse, F. H. 1969a Bounds on the transport of mass and momentum by turbulent flow between parallel plates. Z. Angew. Math. Phys. 20, 1.Google Scholar
Busse, F. H. 1969b On Howard's upper bound for heat transport by turbulent convection. J. Fluid Mech. 37, 457.Google Scholar
Busse, F. H. 1970 Bounds for turbulent shear flows. J. Fluid Mech. 41, 219.Google Scholar
Busse, F. H. 1978 The optimum theory of turbulence. Adv. Appl. Mech. 18, 77.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
El Telbany, M. M. M. & Reynolds, A. J. 1982 The structure of turbulent plane Couette flow. Trans. ASME I: J. Fluids Engng 104, 367.Google Scholar
Howard, L. N. 1972 Bounds on flow quantities. Ann. Rev. Fluid Mech. 4, 473.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions, Vols. 1 and 2. Springer.
Lee, M. J. 1989 CTR Annual Research Briefs. Center for Turbulence Research, Stanford University.
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence.. Proc. Roy. Soc. Lond. A 225, 196.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521.Google Scholar
Malkus, W. V. R. 1968 A proof of the non-existence of steady two dimensional disturbances in unbounded plane parallel flows (unpublished).
Malkus, W. V. R. 1979 Turbulent velocity profiles from stability criteria. J. Fluid Mech. 90, 401.Google Scholar
Malkus, W. V. R. & Smith, L. M. 1989 Upper bounds on functions of the dissipation rate in turbulent shear flow. J. Fluid Mech. 208, 479 (referred to herein as I).Google Scholar
Nickerson, E. C. 1969 Upper bounds on the torque in cylindrical Couette flow. J. Fluid Mech. 38, 807.Google Scholar
Reichardt, H. 1959 Gesetzmässigkeiten der geradlinigen turbulenten Couetteströmung. Mitt. Max Planck Inst. Strömungsforschung, Göttingen 22, 45.Google Scholar
Robertson, J. M. & Johnson, H. F. 1970 Turbulence structure in plane Couette flow. J. Engng Mech. Div. ASCE 96, 1171.Google Scholar
Smith, G. P. & Townsend, A. A. 1982 Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mech. 123, 187.Google Scholar