fluid_mechanics

9.2 Boundary Layer Characteristics 473
viscous effects on the plate. This idea allows us to simulate the presence that the boundary layer
has on the flow outside of the boundary layer by adding the displacement thickness to the actual
wall and treating the flow over the thickened body as an inviscid flow. The displacement
thickness concept is illustrated in Example 9.3.
E XAMPLE 9.3
Boundary Layer Displacement Thickness
GIVEN Air flowing into a 2-ft-square duct with a uniform velocity
of 10 fts forms a boundary layer on the walls as shown in
Fig. E9.3a. The fluid within the core region 1outside the boundary
layers2 flows as if it were inviscid. From advanced calculations it
is determined that for this flow the boundary layer displacement
thickness is given by
U 1 =
10 ft/s
2-ft square
Viscous effects important
Inviscid core
δ ∗
U(x)
where d* and x are in feet.
d* 0.00701x2 1 2
FIND Determine the velocity U U1x2 of the air within the
duct but outside of the boundary layer.
12
10
8
SOLUTION
6
If we assume incompressible flow 1a reasonable assumption because
of the low velocities involved2, it follows that the volume
4
flowrate across any section of the duct is equal to that at the entrance
1i.e., Q 1 Q 2 2. That
2
is,
(1)
U, ft/s
(1) (2)
x
1a2
0
0 20 40 60 80 100
According to the definition of the displacement thickness, d*, the
flowrate across section 122 is the same as that for a uniform flow
with velocity U through a duct whose walls have been moved inward
by d*. That is,
By combining Eqs. 1 and 2 we obtain
or
U 1 A 1 10 fts 12 ft2 2 40 ft 3 s
122u dA
40 ft 3 s u dA U12 ft 2d*2 2
122
40 ft 3 s 4U11 0.0070x 1 2 2 2
U
10
11 0.0070x 1 2 2 2 ft s
(2)
(Ans)
COMMENTS Note that U increases in the downstream direction.
For example, as shown in Fig. E9.3b, U 11.6 fts at
x 100 ft. The viscous effects that cause the fluid to stick to the
walls of the duct reduce the effective size of the duct, thereby
1from conservation of mass principles2 causing the fluid to accelerate.
The pressure drop necessary to do this can be obtained
by using the Bernoulli equation 1Eq. 3.72 along the inviscid
streamlines from section 112 to 122. 1Recall that this equation is
not valid for viscous flows within the boundary layer. It is, how-
x, ft
1b2
F I G U R E E9.3
ever, valid for the inviscid flow outside the boundary layer.2
Thus,
Hence, with r 2.38 10 3 slugsft 3 and p 1 0 we obtain
or
p 1 2 r 1U 2 1 U 2 2
1 2 12.38 103 slugsft 3 2
c110 fts2 2
p 0.119 c 1
p 1 1 2 rU 2 1 p 1 2 rU 2
10 2
11 0.0079x 1 2 2 4 ft2 s 2 d
1
11 0.0070x 1 2 2 4 d lb ft 2
For example, p 0.0401 lbft 2 at x 100 ft.
If it were desired to maintain a constant velocity along the
centerline of this entrance region of the duct, the walls could be
displaced outward by an amount equal to the boundary layer displacement
thickness, d*.