fluid_mechanics

480 Chapter 9 ■ Flow over Immersed Bodies
E XAMPLE 9.4
Momentum Integral Boundary Layer Equation
GIVEN Consider the laminar flow of an incompressible fluid
past a flat plate at y 0. The boundary layer velocity profile is
approximated as u Uyd for 0 y d and u U for y 7 d,
as is shown in Fig. E9.4.
FIND Determine the shear stress by using the momentum integral
equation. Compare these results with the Blasius results
given by Eq. 9.18.
y
δ
u = U
SOLUTION
From Eq. 9.26 the shear stress is given by
0
u = Uy/δ
U
F I G U R E E9.4
u
while for laminar flow we know that t w m10u0y2 y0 . For the
assumed profile we have
and from Eq. 9.4
or
q
0
d
a y d b a1 y d b dy
0
t w rU 2 d
dx
U
t w m
d
u
U a1 u U b dy d u
U a1 u U b dy
d 6
Note that as yet we do not know the value of d 1but suspect that it
should be a function of x2.
By combining Eqs. 1, 2, and 3 we obtain the following differential
equation for d:
0
mU
d rU 2 dd
6 dx
(1)
(2)
(3)
or
This can be integrated from the leading edge of the plate, x 0
1where d 02 to an arbitrary location x where the boundary layer
thickness is d. The result is
or
d dd 6m
rU dx
d 2
2 6m
rU x
d 3.46 B
nx
U
Note that this approximate result 1i.e., the velocity profile is not actually
the simple straight line we assumed2 compares favorably with
the 1much more laborious to obtain2 Blasius result given by Eq. 9.15.
The wall shear stress can also be obtained by combining Eqs.
1, 3, and 4 to give
rm
t w 0.289U 3 2
B x
(4)
(Ans)
Again this approximate result is close 1within 13%2 to the
Blasius value of given by Eq. 9.18.
t w
Approximate velocity
profiles are used
in the momentum
integral equation.
As is illustrated in Example 9.4, the momentum integral equation, Eq. 9.26, can be used
along with an assumed velocity profile to obtain reasonable, approximate boundary layer results.
The accuracy of these results depends on how closely the shape of the assumed velocity profile
approximates the actual profile.
Thus, we consider a general velocity profile
and
u
g1Y 2 for 0 Y 1
U
u
U 1 for Y 7 1
where the dimensionless coordinate Y yd varies from 0 to 1 across the boundary layer. The
dimensionless function g1Y 2 can be any shape we choose, although it should be a reasonable