Page 150 - Bird R.B. Transport phenomena

134 Chapter 4 Velocity Distributions with More Than One Independent Variable

boundary layer. The success of the method depends on the thinness of the boundary
layer, a condition that is met at high Reynolds number.
We consider the steady, two-dimensional flow of a fluid with constant p and /x
around a submerged object, such as that shown in Fig. 4.4-1. We assert that the main
changes in the velocity take place in a very thin region, the boundary layer, in which the
curvature effects are not important. We can then set up a Cartesian coordinate system
with x pointing downstream, and у perpendicular to the solid surface. The continuity
equation and the Navier-Stokes equations then become:

dv
x
(4.4-1)
dv x to x \ _ _1№, (d V x d v x
2
2
P^X \ S x 2 dy 2 (4.4-2)
dV d<3>
(4.4-3)

Some of the terms in these equations can be discarded by order-of-magnitude argu-
ments. We use three quantities as "yardsticks": the approach velocity v , some linear di-
a
mension / 0 of the submerged body, and an average thickness S o of the boundary layer.
The presumption that S o « l allows us to make a number of rough calculations of or-
0
ders of magnitude.
Since v x varies from zero at the solid surface to v a at the outer edge of the boundary
layer, we can say that

where О means "order of magnitude of." Similarly, the maximum variation in v x over
the length of the surface will be у , so that
/
0
ж
- = o and ^r^ = О \ — (4.4-5)
dx
Here we have made use of the equation of continuity to get one more derivative (we are
concerned here only with orders of magnitude and not the signs of the quantities). Inte-
gration of the second relation suggests that v y = СЖ^/УгО < < v . The various terms in
x
Eq. 4.4-2 may now be estimated as
£4
dV x = Jv
\',v. 2 2 •<*% (4.4-6)
dx \ ll dy

Approximate outer limit
of boundary layer where
v r -> vlx)
Fig. 4.4-1. Coordinate system
for the two-dimensional flow
around a submerged object.
The boundary-layer thickness
is greatly exaggerated for pur-
poses of illustration. Because
the boundary layer is in fact
quite thin, it is permissible to
use rectangular coordinates lo-
cally along the curved surface.