CHEMICAL ENGINEERING

308 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

PROBLEM 12.16

Explain the importance of the universal velocity profile and derive the relation between
the dimensionless derivative of velocityuC, and the dimensionless derivative of distance
from the surfaceyC, using the concept of Prandtl’s mixing lengthE.
It may be assumed that the fully turbulent portion of the boundary layer starts at
yCD30, that the ratio of the mixing lengthEto distanceyfrom the surface,E/yD 0 .4,
and that for a smooth surfaceuCD14 atyCD30.
If the laminar sub-layer extends fromyCD0toyCD5, obtain the equation for the
relation betweenuCandyCin the buffer zone, and show that the ratio of the eddy
viscosity to the molecular viscosity increases linearly from 0 to 5 through this buffer
zone.

Solution

The importance of the universal velocity profile is discussed in Section 12.4. From
equation 12.18, for isotropic turbulence, the eddy kinematic viscosity,E ̨EuEwhere
Eis the mixing length anduEis some measure of the linear velocity of the fluid in
the eddies. The momentum transfer rate per unit area in a direction perpendicular to the
surface at positionyis then:

RyDEdux/dy

and for constant density, RyDEdux/dy (equation 12.20)

or: Ry/DEdux/dy

where

√

Ry/, the friction velocity, may be denoted byuŁand thenuŁ^2 DEdux/dy
AssumingEDEuE, that is a proportionality constant of unity, anduEDEjdux/dyj,
then:
uŁ^2 D^2 Edux/dyjdux/dyj

and hence near the surface wheredux/dyis positive:

uŁDEdux/dy

AssumingEDKy:
uŁdy/yDKdux (equation 12.28)

Integrating: ux/uŁD 1 /KlnyCBwhereBis a constant

or: ux/uŁD 1 /KlnyuŁ/CB^0 (equation 12.29)

SinceuŁ/is constant,B^0 is also constant and, writing the dimensionless velocity
term,ux/uŁasuCand the dimensionless derivative ofyyuŁ/asyC, then:

uCD 1 /KlnyCCB^0 (equation 12.30)

Given thatKD 0 .4, then: uCD 2 .5lnyCCB^0