Page 178 - Bird R.B. Transport phenomena

162 Chapter 5 Velocity Distributions in Turbulent Flow

(iv) The coefficient of the fourth term involves the third derivative, which may be
obtained from Eq. 5.3-8, and this is

y dv' y
г- -г Z -г—
2
i/=o ^ V x ду 2 ду ду y ду 1 y=o
ох ^11Г)\ = 0 (5.3-11)
+
Here Eq. 5.2-11 has been used.
There appears to be no reason to set the next coefficient equal to zero, so we find that
the Taylor series, in dimensionless quantities, has the form

The coefficient С has been obtained experimentally, 6 and therefore we have the final result:

y D
, 1 / Г v Y3M_l(j^Y ...l < *<5 (5313)
+ 0
The y 3 term in the brackets will turn out to be very important in connection with turbu-
lent heat and mass transfer correlations in Chapters 13,14, 21, and 22.
For the region 5 < yojv < 30 no simple analytical derivations are available, and
empirical curve fits are sometimes used. One of these is shown in Fig. 5.5-3 for circular
tubes.

§5.4 EMPIRICAL EXPRESSIONS FOR THE
TURBULENT MOMENTUM FLUX

We now return to the problem of using the time-smoothed equations of change in Eqs.
5.2-11 and 12 to obtain the time-smoothed velocity and pressure distributions. As
pointed out in the previous section, some information about the velocity distribution can
(0
be obtained without having a specific expression for the turbulent momentum flux т .
However, it has been popular among engineers to use various empiricisms for т ( 0 that
involve velocity gradients. We mention a few of these, and many more can be found in
the turbulence literature.

The Eddy Viscosity of Boussinesq
By analogy with Newton's law of viscosity, Eq. 1.1-1, one may write for a turbulent shear
flow 1

r<?= V ° ^ (5.4-1)

6
С S. Lin, R. W. Moulton, and G. L. Putnam, Ind. Eng. Chem., 45, 636-640 (1953); the numerical
coefficient was determined from mass transfer experiments in circular tubes. The importance of the y 4
term in heat and mass transfer was recognized earlier by E. V. Murphree, Ind. Eng. Chem., 24, 726-736
(1932). Eger Vaughn Murphree (1898-1962) was captain of the University of Kentucky football team in
1920 and became President of the Standard Oil Development Company.
1
J. Boussinesq, Mem. pres. par div. savants a Vacad. sci. de Paris, 23, #1,1-680 (1877), 24, #2,1-64 (1877).
Joseph Valentin Boussinesq (1842-1929), university professor in Lille, wrote a two-volume treatise on
heat, and is famous for the "Boussinesq approximation" and the idea of "eddy viscosity."