Page 176 - Bird R.B. Transport phenomena

160 Chapter 5 Velocity Distributions in Turbulent Flow

Fig. 5.3-1. Flow regions for describing
turbulent flow near a wall: (l) viscous
sublayer, © buffer layer, (3) inertial
sublayer, (4) main turbulent stream.

The Logarithmic and Power Law Velocity Profiles
1
in the Inertial Sublayer " 4
Let the time-smoothed shear stress acting on the wall у = 0 be called r 0 (this is the same
as — Ty \ ). Then the shear stress in the inertial sublayer will not be very different from
X y=0
the value r . We now ask: On what quantities will the time-smoothed velocity gradient
0
dvjdy depend? It should not depend on the viscosity, since, out beyond the buffer layer,
momentum transport should depend primarily on the velocity fluctuations (loosely re-
ferred to as "eddy motion"). It may depend on the density p, the wall shear stress r , and
0
the distance у from the wall. The only combination of these three quantities that has the
dimensions of a velocity gradient is \/т /р/у. Hence we write
о
(5.3-1)

in which к is an arbitrary dimensionless constant, which must be determined experi-
mentally. The quantity Vr /'p has the dimensions of velocity; it is called the friction veloc-
0
ity and given the symbol z;*.When Eq. 5.3-1 is integrated we get

v x = ^kvy + y (5.3-2)

A' being an integration constant. To use dimensionless groupings, we rewrite Eq. 5.3-2 as
V l + A (5.3-3)

in which A is a constant simply related to A'; the kinematic viscosity v was included in
order to construct the dimensionless argument of the logarithm. Experimentally it has
been found that reasonable values of the constants are к = 0.4 and A = 5.5, giving
2

V I + 5.5 >30 (5.3-4)

1
L. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 2nd edition (1987), pp. 172-178.
H. Schlichting and K. Gersten, Boundary-Layer Theory, Springer-Verlag, Berlin, 8th edition (2000),
2
§17.2.3.
3
T. von Karman, Nachr. Ges. Wiss. Gottingen, Math-Phys. Klasse (1930), pp. 58-76; L. Prandtl, Ergeb.
Aerodyn. Versuch., Series 4, Gottingen (1932).
4
G. I. Barenblatt and A. J. Chorin, Proc. Nat. Acad. Sci. USA, 93, 6749-6752 (1996) and SIAM Rev., 40,
265-291 (1981); G. I. Barenblatt, A. J. Chorin, and V. M. Prostokishin, Proc. Nat. Acad. Sci. USA, 94,
773-776 (1997). See also G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asxjmptotics, Cambridge
University Press (1992), §10.2.