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Velocity Distributions in Turbulent Flow 153

cannot be simply related to velocity gradients in the way that the momentum flux is
given by Newton's law of viscosity in Chapter 1. At the present time the turbulent mo-
mentum flux is usually estimated experimentally or else modeled by some type of em-
piricism based on experimental measurements.
Fortunately, for turbulent flow near a solid surface, there are several rather general
results that are very helpful in fluid dynamics and transport phenomena: the Taylor se-
ries development for the velocity near the wall; and the logarithmic and power law ve-
locity profiles for regions further from the wall, the latter being obtained by dimensional
reasoning. These expressions for the time-smoothed velocity distribution are given in
§5.3.
In the following section, §5.4, we present a few of the empiricisms that have been
proposed for the turbulent momentum flux. These empiricisms are of historical interest
and have also been widely used in engineering calculations. When applied with proper
judgment, these empirical expressions can be useful.
The remainder of the chapter is devoted to a discussion of two types of turbulent
flows: flows in closed conduits (§5.5) and flows in jets (§5.6). These flows illustrate the
two classes of flows that are usually discussed under the headings of wall turbulence and
free turbulence.
In this brief introduction to turbulence we deal primarily with the description of the
fully developed turbulent flow of an incompressible fluid. We do not consider the theo-
retical methods for predicting the inception of turbulence nor the experimental tech-
niques devised for probing the structure of turbulent flow. We also give no discussion of
the statistical theories of turbulence and the way in which the turbulent energy is distrib-
uted over the various modes of motion. For these and other interesting topics, the reader
1 6
should consult some of the standard books on turbulence. " There is a growing litera-
ture on experimental and computational evidence for "coherent structures" (vortices) in
turbulent flows. 7
Turbulence is an important subject. In fact, most flows encountered in engineering
are turbulent and not laminar! Although our understanding of turbulence is far from sat-
isfactory, it is a subject that must be studied and appreciated. For the solution to indus-
trial problems we cannot get neat analytical results, and, for the most part, such
problems are attacked by using a combination of dimensional analysis and experimental
data. This method is discussed in Chapter 6.

S. Corrsin, "Turbulence: Experimental Methods," in Handbuch der Physik, Springer, Berlin (1963),
1
Vol. VIII/2. Stanley Corrsin (1920-1986), a professor at The Johns Hopkins University, was an excellent
experimentalist and teacher; he studied the interaction between chemical reactions and turbulence and
the propagation of the double temperature correlations.
A. A. Townsend, The Structure of Turbulent Shear Flow, Cambridge University Press, 2nd edition
2
(1976); see also A. A. Townsend in Handbook of Fluid Dynamics (V. L. Streeter, ed.), McGraw-Hill (1961)
for a readable survey.
3
J. O. Hinze, Turbulence, McGraw-Hill, New York, 2nd edition (1975).
H. Tennekes and J. L. Lumley, A First Course in Turbulence, MIT Press, Cambridge, Mass. (1972);
4
Chapters 1 and 2 of this book provide an introduction to the physical interpretations of turbulent flow
phenomena.
M. Lesieur, La Turbulence, Presses Universitaires de Grenoble (1994); this book contains beautiful
5
color photographs of turbulent flow systems.
Several books that cover material beyond the scope of this text are: W. D. McComb, The Physics of
6
Fluid Turbulence, Oxford University Press (1990); Т. Е. Faber, Fluid Dynamics for Physicists, Cambridge
University Press (1995); U. Frisch, Turbulence, Cambridge University Press (1995).
P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems, and
7
Symmetry, Cambridge University Press (1996); F. Waleffe, Phys. Rev. Lett., 81, 4140^148 (1998).