Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-04T18:42:13.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Upper bounds for turbulent Couette flow incorporating the poloidal power constraint

Published online by Cambridge University Press:  26 April 2006

R. R. Kerswell  and
A. M. Soward
R. R. Kerswell
Affiliation:
Department of Mathematics and Statistics, University of Newcastle upon Tyne, NE1 7RU, UK Present address: Department of Mathematics, University of Bristol, BS8 1TW, UK.
A. M. Soward
Affiliation:
Department of Mathematics and Statistics, University of Newcastle upon Tyne, NE1 7RU, UK Present address: Department of Mathematics, University of Exeter, EX4 4QE, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

The upper bound on momentum transport in the turbulent regime of plane Couette flow is considered. Busse (1970) obtained a bound from a variational formulation based on total energy conservation and the mean momentum equation. Two-dimensional asymptotic solutions of the resulting Euler-Lagrange equations for the system were obtained in the large-Reynolds-number limit. Here we make a toroidal poloidal decomposition of the flow and impose an additional power integral constraint, which cannot be satisfied by two-dimensional flows. Nevertheless, we show that the additional constraint can be met by only small modifications to Busse's solution, which leaves his momentum transport bound unaltered at lowest order. On the one hand, the result suggests that the addition of further integral constraints will not significantly improve bound estimates. On the other, our optimal solution, which possesses a weak spanwise roll in the outermost of Busse's nested boundary layers, appears to explain the three-dimensional structures observed in experiments. Only in the outermost boundary layer and in the main stream is the solution three-dimensional. Motion in the thinner layers remains two-dimensional characterized by streamwise rolls.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 328 , 10 December 1996 , pp. 161 - 176
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Busse, F. H. 1969a On Howard's upper bound for heat transport by turbulent convection. J. Fluid Mech. 37, 457–477.Google Scholar
Busse, F. H. 1969b Bounds on the transport of mass and momentum by turbulent flow between parallel plates. Z. Angew. Math. Phys. 20, 1–14 (referred to herein as B69).Google Scholar
Busse, F. H. 1970 Bounds for turbulent shear flow. J. Fluid Mech. 41, 219–240 (referred to herein as B70).Google Scholar
Busse, F. H. 1978 The optimum theory of turbulence. Adv. Appl. Mech. 18, 77–121.Google Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.Google Scholar
Howard, L. N. 1972 Bounds on flow quantities. Ann. Rev. Fluid Mech. 4, 473–494.Google Scholar
Ierley, G. R. & Malkus, W. V. R. 1988 Stability bounds on turbulent Poiseuille flow. J. Fluid Mech. 187, 435–449.Google Scholar
Kim, H. T., Kline, S. J. & Reynolds, W. C. 1971 The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50, 133–160.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 185–195.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, 479–507.Google Scholar
Smith, L. M. 1991 Turbulent Couette flow profiles that maximise the efficiency function. J. Fluid Mech. 227, 509–525.Google Scholar
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow.. Cambridge University Press.