Page 153 - Bird R.B. Transport phenomena

§4.4 Flow near Solid Surfaces by Boundary-Layer Theory 137

SOLUTION We know intuitively what the velocity profile v {y) looks like. Hence we can guess a form for
x
v (y) and substitute it directly into the von Karman momentum balance. One reasonable choice
x
is to let v (y) by a function of у 18, where 8{x) is the "thickness" of the boundary layer. The
x
function is so chosen that v x = 0 at у = 0 and v x = v at у = 8. This is tantamount to assuming
c
geometrical similarity of the velocity profiles for various values of x. When this assumed pro-
file is substituted into the von Karman momentum balance, an ordinary differential equation
for the boundary-layer thickness 8(x) is obtained. When this equation has been solved, the 8(x)
so obtained can then be used to get the velocity profile and other quantities of interest.
For the present problem a plausible guess for the velocity distribution, with a reasonable
shape, is
ЗУ
25 for 0 < у < 8(х) (boundary-layer region) (4.4-14)
for у >8(х) (potential flow region) (4.4-15)

This is "reasonable" because this velocity profile satisfies the no-slip condition at у = 0, and
Sv /dy = 0 at the outer edge of the boundary layer. Substitution of this profile into the von
x
Karman integral balance in Eq. 4.4-13 gives
a ( i 39 2 *
8 = V Tx[2S6 pVx8 (4.4-16)

This first-order, separable differential equation can now be integrated to give for the bound-
ary-layer thickness

(4.4-17)

Therefore, the boundary-layer thickness increases as the square root of the distance from the up-
stream end of the plate. The resulting approximate solution for the velocity distribution is then

(4.4-18)

From this result we can estimate the drag force on a plate of finite size wetted on both sides.
For a plate of width W and length L, integration of the momentum flux over the two solid sur-
faces gives:

dxdz = 1.293 (4.4-19)
./ = 0
The exact solution, given in the next example, gives the same result, but with a numerical co-
efficient of 1.328. Both solutions predict the drag force within the scatter of the experimental
data. However, the exact solution gives somewhat better agreement with the measured veloc-
ity profiles. 3 This additional accuracy is essential for stability calculations.

EXAMPLE 4.4-2 Obtain the exact solution for the problem given in the previous example.

Laminar Flow along SOLUTION
a Flat Plate (Exact
Solution)' This problem may be solved by using the definition of the stream function in Table 4.2-1. In-
serting the expressions for the velocity components in the first row of entries, we get
ъ
2
2
дф д ф дф д ф _ д ф
(4.4-20)
ду дхду дХ Jy 2 А У Ъ
7
This problem was treated originally by H. Blasius, Zeits. Math. Phys., 56,1-37 (1908).