Page 180 - Bird R.B. Transport phenomena

164 Chapter 5 Velocity Distributions in Turbulent Flow
The Modified van Driest Equation

There have been numerous attempts to devise empirical expressions that can describe
the turbulent shear stress all the way from the wall to the main turbulent stream. Here
7
we give a modification of the equation of van Driest. This is a formula for the mixing
length of Eq. 5.4-4:
— exp(—\
I = 0.4y (5.4-7)
Vl - exp(-026yvJv)
This relation has been found to be useful for predicting heat and mass transfer rates in
flow in tubes.
In the next two sections and in several problems at the end of the chapter, we illus-
trate the use of the above empiricisms. Keep in mind that these expressions for the
Reynolds stresses are little more than crutches that can be used for the representation of
experimental data or for solving problems that fall into rather special classes.

EXAMPLE 5.4-1 Obtain an expression for т { ух = pv v as a function of у in the neighborhood of the wall.
x y
Development of the SOLUTION
Reynolds Stress
Expression in the (a) We start by making a Taylor series development of the three components of v':
Vicinity of the Wall
(5.4-8)
y = 0 2!

J + 2! sf (5.4-9)
y
dv.
v' (y) = v' У + 2?- ir + (5.4-10)
z
y = 0 ^*
The first term in Eqs. 5.4-8 and 10 must be zero because of the no-slip condition; the first term
in Eq. 5.4-9 is zero in the absence of mass transfer. Next we can write Eq. 5.2-11 at у = 0,
dv[ Sv'
z = 0 (5.4-11)
dx y=0 ./ = 0
The first and third terms in this equation are zero because of the no-slip condition. Therefore
we have to conclude that the second term is zero as well. Hence all the dashed-underlined
terms in Eqs. 5.4-8 to 10 are zero, and we may conclude that

rf = v' v' = Ay + By 4 (5.4-12)
3
x P x y
This suggests—but does not prove —that the lead term in the Reynolds stress near a wall
8
9
3
should be proportional to y . Extensive studies of mass transfer rates in closed channels have,
however, established that А Ф 0.
7
E. R. van Driest, /. Aero. Sci., 23,1007-1011 and 1036 (1956). Van Driest's original equation did
not have the square root divisor. This modification was made by O. T. Hanna, О. С Sandall, and
P. R. Mazet, AIChE Journal, 27, 693-697 (1981) so that the turbulent viscosity would be proportional to y 3
as у —> 0, in accordance with Eq. 5.4-2.
8
H. Reichardt, Zeits. f. angew. Math. u. Mech., 31, 208-219 (1951). See also J. O. Hinze, Turbulence,
McGraw-Hill, New York, 2nd edition (1975), pp. 620-621.
9
R. H. Notter and C. A. Sleicher, Chem. Eng. Sci., 26,161-171 (1971); О. С Sandall and О. Т. Hanna,
AIChE Journal, 25,190-192 (1979); D. W. Hubbard and E. N. Lightfoot, Ind. Eng. Chem. Fundamentals, 5,
370-379 (1966).