fluid_mechanics

9.2 Boundary Layer Characteristics 481
1
= g(Y)
U u
Y
0
0 1
U u
approximation to the boundary layer profile, as shown by the figure in the margin. In particular,
it should certainly satisfy the boundary conditions u 0 at y 0 and u U at y d.
That is,
The linear function g1Y 2 Y used in Example 9.4 is one such possible profile. Other conditions,
such as dgdY 0 at Y 1 1i.e., 0u0y 0 at y d2, could also be incorporated into the function
g1Y 2 to more closely approximate the actual profile.
For a given g1Y 2, the drag can be determined from Eq. 9.22 as
or
d rbU 2 dC 1
where the dimensionless constant C 1 has the value
Also, the wall shear stress can be written as
where the dimensionless constant
d
1
d rb u1U u2 dy rbU 2 d g1Y 231 g1Y 24 dY
1
C 1 g1Y 231 g1Y 24 dY
t w m 0u
0y `
y0
mU d
has the value
C 2 dg
dY `
Y0
By combining Eqs. 9.25, 9.27, and 9.28 we obtain
0
C 2
g102 0 and g112 1
0
0
dg
dY ` mU
Y0 d C 2
(9.27)
(9.28)
d dd mC 2
rUC 1
dx
which can be integrated from d 0 at x 0 to give
or
d B
2nC 2 x
UC 1
d
x 12C 2C 1
1Re x
By substituting this expression back into Eqs. 9.28 we obtain
(9.29)
Approximate
boundary layer results
are obtained
from the momentum
integral equation.
t w B
C 1 C 2
2
rm
U 3 2
A x
(9.30)
To use Eqs. 9.29 and 9.30 we must determine the values of C 1 and C 2 . Several assumed velocity
profiles and the resulting values of d are given in Fig. 9.12 and Table 9.2. The more closely
the assumed shape approximates the actual 1i.e., Blasius2 profile, the more accurate the final results.
For any assumed profile shape, the functional dependence of d and t w on the physical parameters
r, m, U, and x is the same. Only the constants are different. That is, d 1mxrU2 1 2
or
dRe 1 2
x x constant, and t w 1rmU 3 x2 12 , where Re x rUxm.
It is often convenient to use the dimensionless local friction coefficient, c f , defined as
c f
t w
1
(9.31)
2 rU 2