Page 177 - Bird R.B. Transport phenomena

§5.3 The Time-Smoothed Velocity Profile near a Wall 161

This is called the von Kdrmdn-Prandtl universal logarithmic velocity distribution; it is intended
3
to apply only in the inertial sublayer. Later we shall see (in Fig. 5.5-3) that this function
describes moderately well the experimental data somewhat beyond the inertial sublayer.
If Eq. 5.3-1 were correct, then the constants к and Л would indeed be "universal con-
stants/' applicable at any Reynolds number. However, values of к in the range 0.40 to
0.44 and values of Л in the range 5.0 to 6.3 can be found in the literature, depending on
the range of Reynolds numbers. This suggests that the right side of Eq. 5.3-1 should be
multiplied by some function of Reynolds number and that у could be raised to some
4
power involving the Reynolds number. Theoretical arguments have been advanced that
Eq. 5.3-1 should be replaced by

in which B = ^V3, B^ = ™, and /^ = §. When Eq. 5.3-5 is integrated with respect to y, the
o
Barenblatt-Chorin universal velocity distribution is obtained:

г) (5.3-6)
4
Equation 5.3-6 describes regions © and (4) of Fig. 5.3-1 better than does Eq. 5.3-4. Region
® is better described by Eq. 5.3-13.

Taylor-Series Development in the Viscous Sublayer

We start by writing a Taylor series for v x as a function of y, thus

ЭД = o,(0) + ^ 1 ^ d 3! f y=0 y 3 + • • • (5.3-7)
2! f J
d
To evaluate the terms in this series, we need the expression for the time-smoothed shear
stress in the vicinity of a wall. For the special case of the steadily driven flow in a slit of
thickness IB, the shear stress will be of the form r = r^ + 7$ = -т [1 - (у/В)]. Then
yx о
from Eqs. 5.2-8 and 9, we have
+/^-pi^ = T (l-|j (5.3-8)
0
Now we examine one by one the terms that appear in Eq. 5.3-7: 5
(i) The first term is zero by the no-slip condition.
(ii) The coefficient of the second term can be obtained from Eq. 5.3-8, recognizing
that both v' and v' are zero at the wall so that
x
y
= -^ (5.3-9)
/=0 ^
(iii) The coefficient of the third term involves the second derivative, which may be
obtained by differentiating Eq. 5.3-8 with respect to у and then setting у = 0, as follows,

- Л = % (5.3-10)
~
since both v' and v' are zero at the wall.
x
y

5
A. A. Townsend, The Structure of Turbulent Shear Flow, Cambridge University Press, 2nd edition
(1976), p. 163.