Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-27T14:12:17.852Z Has data issue: false hasContentIssue false

Outline of a theory of turbulent shear flow

Published online by Cambridge University Press:  28 March 2006

W. V. R. Malkus
Affiliation:
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

Abstract

In this paper the spatial variations and spectral structure of steady-state turbulent shear flow in channels are investigated without the introduction of empirical parameters. This is made possible by the assumption that the non-linear momentum transport has only stabilizing effects on the mean field of flow. Two constraints on the possible momentum transport are drawn from this assumption: first, that the mean flow will be statistically stable if an Orr-Sommerfeld type equation is satisfied by fluctuations of the mean; second, that the smallest scale of motion that can be present in the spectrum of the momentum transport is the scale of the marginally stable fluctuations of the mean. Within these two constraints, and for a given mass transport, an upper limit is sought for the rate of dissipation of potential energy into heat. Solutions of the stability equation depend upon the shape of the mean velocity profile. In turn, the mean velocity profile depends upon the spatial spectrum of the momentum transport. A variational technique is used to determine that momentum transport spectrum which is both marginally stable and produces a maximum dissipation rate. The resulting spectrum determines the velocity profile and its dependence on the boundary conditions. Past experimental work has disclosed laminar, ‘transitional’, logarithmic and parabolic regions of the velocity profile. Several experimental laws and their accompanying constants relate the extent of these regions to the boundary conditions. The theoretical profile contains each feature and law that is observed. First approximations to the constants are found, and give, in particular, a value for the logarithmic slope (von Kármán's constant) which is within the experimental error. However, the theoretical boundary constant is smaller than the observed value. Turbulent channel flow seems to achieve the extreme state found here, but a more decisive quantitative comparison of theory and experiment requires improvement in the solutions of the classical laminar stability problem.

Type
Research Article
Copyright
© 1956 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

Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Goldstein, S. (ed.) 1938 Modern Developments in Fluid Dynamics, Vol. 2. Oxford University Press.
Heisenberg, W. 1924 Ann. Phys. 74, 577627.
Laufer, J. 1950 Nat. Adv. Comm. Aero., Wash., no. 2123.
Lin, C. C. 1945–46 Quart. Appl. Math. 3, 117142, 218234, & 277301.
Malkus, W. V. R. 1954 Proc. Roy. Soc. A, 225, 185195 & 196212.
Orr, W. M. F. 1906–7 Proc. Roy. Irish Acad. 27, 926, 69138.
Sommerfeld, A. 1908 Proc. 4th Internat. Cong. Math., Rome, 116124.
Synge, J. L. 1938 Semicent. Publ. Amer. Math. Soc. 2, 227269.