Hostname: page-component-84b7d79bbc-l82ql Total loading time: 0 Render date: 2024-07-28T06:17:54.553Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum kinetic energy dissipation and the stability of turbulent Poiseuille flow

Published online by Cambridge University Press:  16 February 2015

J. Bertram
J. Bertram*
Affiliation:
Research School of Biology, Australian National University, Canberra, ACT 0200, Australia
*
†Email address for correspondence: jason.bertram@anu.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Following Malkus’s (J. Fluid Mech., vol. 1, 1956, pp. 521–539) proposal that turbulent Poiseuille channel flow maximises total viscous dissipation $D$, a variety of variational procedures have been explored involving the maximisation of different flow quantities under different constraints. However, the physical justification for these variational procedures has remained unclear. Here we address more recent claims that mean flow viscous dissipation $D_{m}$ should be maximised on the basis of a statistical stability argument, and that maximising $D_{m}$ yields realistic mean velocity profiles (Malkus, J. Fluid Mech., vol. 489, 2003, pp. 185–198). We clarify the connection between maximising $D_{m}$ and other flow quantities, verify Malkus & Smith’s, (J. Fluid Mech., vol. 208, 1989, pp. 479–507) claim that maximising the ‘efficiency’ yields realistic profiles and show that, in contrast, maximising $D_{m}$ does not yield realistic mean velocity profiles as recently claimed. This leads us to revisit Malkus’s statistical stability argument for maximising $D_{m}$ and to address some of its limitations. We propose an alternative statistical stability argument leading to a principle of minimum kinetic energy for fixed pressure gradient, which suggests a principle of maximum $D$ for fixed Reynolds number under certain conditions. We discuss possible ways to reconcile these conflicting results, focusing on the choice of constraints.

JFM classification

Instability: Nonlinear instability Turbulent Flows: Turbulence theory Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 767 , 25 March 2015 , pp. 342 - 363
Check for updates
Copyright
© 2015 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. 1970 Bounds for turbulent shear flow. J. Fluid Mech. 41 (1), 219–240.CrossRefGoogle Scholar
Dewar, R. C. & Maritan, A. 2014 A theoretical basis for maximum entropy production. In Beyond the Second Law (ed. Dewar, R. C., Lineweaver, C. H., Niven, R. K. & Regenauer-Lieb, K.), pp. 49–71. Springer.Google Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69, 1648–1651.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Goody, R. 2007 Maximum entropy production in climate theory. J. Atmos. Sci. 64 (7), 2735–2739.CrossRefGoogle Scholar
Howard, L. N. 1972 Bounds on flow quantities. Annu. Rev. Fluid Mech. 4 (1), 473–494.CrossRefGoogle Scholar
Joseph, D. D.1976 Stability of fluid motions. I, II. NASA STI/Recon Tech. Rep. A 77, 12423.Google Scholar
Kerswell, R. R. 2002 Upper bounds on general dissipation functionals in turbulent shear flows: revisiting the functional. J. Fluid Mech. 461, 239–275.Google Scholar
Lee, M. & Moser, R. D. 2014 Direct numerical simulation of turbulent channel flow up to $R_{{\it\tau}}=5200$ . J. Fluid Mech. (submitted).Google Scholar
Lorenz, E. N. 1960 Generation of available potential energy and the intensity of the general circulation. In Dynamics of Climate (ed. Pfeffer, R. C.), pp. 86–92. Pergamon Press.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225 (1161), 196–212.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1 (5), 521–539.CrossRefGoogle Scholar
Malkus, W. V. R. 1996 Statistical stability criteria for turbulent flow. Phys. Fluids 8 (6), 1582–1587.CrossRefGoogle Scholar
Malkus, W. V. R. 2003 Borders of disorder: in turbulent channel flow. J. Fluid Mech. 489, 185–198.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.CrossRefGoogle Scholar
Paltridge, G. W. 1975 Global dynamics and climate – a system of minimum entropy exchange. Q. J. R. Meteorol. Soc. 101 (429), 475–484.Google Scholar
Pascale, S., Gregory, J. M., Ambaum, M. H. P. & Tailleux, R. 2012 A parametric sensitivity study of entropy production and kinetic energy dissipation using the FAMOUS AOGCM. Clim. Dyn. 38 (5–6), 1211–1227.Google Scholar