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Inequalities and variational principles for turbulent transport

Published online by Cambridge University Press:  19 April 2006

Melvin E. Stern
Melvin E. Stern
Affiliation:
Graduate School of Oceanography, University of Rhode Island, U.S.A.
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Abstract

Two inequalities are proposed for the purpose of bounding from below the mean shear U′o(z) in turbulent channel flow. The first of these inequalities pertains to the energetics of the boundary layer, and the second pertains to the logarithmic form of the asymptote to Uo. These inequalities imply a maximum value of von Kármán's constant, the numerical value of which lies between the measurements of Laufer (1951) and the value obtained from bulk discharge measurements in a pipe. The formalism, which contains no adjustable parameters, is then applied to the turbulent thermal convection problem, and a lower bound for the mean temperature $\overline{T}_o(z)$ is obtained. The minimum value of the latter at large distances from the boundary is in fair agreement with Townsend's (1959) measurements. Although the proposed inequalities have not been deduced from the equations of motion, they provide facts which may be useful in the search for new variational formulations of theturbulent transport problem.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 91 , Issue 3 , 11 April 1979 , pp. 513 - 540
Copyright
© 1979 Cambridge University Press

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References

Busse, F. H. 1969 Z. angew. Math. Phys. 29, 1–14.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Deardorff, J. W. & Willis, G. E. 1967 Quart. J. Roy. Met. Soc. 93, 166–175.
Gol'dshtik, M. A. 1969 Sov. Phys. Dokl. 13, 1008–1010.
Hinze, J. O. 1959 Turbulence. McGraw-Hill.
Howard, L. N. 1972 Ann. Rev. Fluid Mech. 44, 473–494.
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 J. Fluid Mech. 30, 741–776.
Laufer, J. 1951 N.A.C.A. Rep. 1053.
Laufer, J. 1954 N.A.C.A. Tech. Rep. 1174.
Lin, C. C. 1955 Theory of Hydrodynamic Stability. Cambridge University Press.
Linden, P. F. & Shirtcliffe, T. G. L. 1978 J. Fluid Mech. 87, 417–432.
Malkus, W. V. R. 1956 J. Fluid Mech. 1, 521–539.
Malkus, W. V. R. 1963 Outline of a theory of turbulent convection. In Theory and Fundamental Research in Heat Transfer. pp. 203–212. Pergamon.
Orr, W. M. 1907 Proc. Roy. Irish Academy A 27, 69–138.
Reynolds, W. C. & Tiederman, W. G. 1967 J. Fluid Mech. 27, 253–272.
Townsend, A. A. 1959 J. Fluid Mech. 5, 209–241.
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.