Page 80 - Mechanical Engineers' Handbook (Volume 4)

9 Viscous Flow and Incompressible Boundary Layers 69

Figure 22 Boundary layer development along a flat plate.

One parameter of interest is the boundary layer thickness , the distance from the bound-
ary in which the flow is retarded, or the distance to the point where the velocity is 99% of
the free stream velocity (Fig. 23). The displacement thickness is the distance the boundary
is displaced such that the boundary layer flow is the same as one-dimensional flow past the
displaced boundary. It is given by (see Fig. 23)
(u u) dy

u
1
1 s 1 dy
u s 0 0 u s
A momentum thickness is the distance from the boundary such that the momentum flux of
the free stream within this distance is the deficit of momentum of the boundary layer flow.
It is given by (see Fig. 23)
u

u
2
0 1 u s u s dy
2
Also of interest is the viscous shear drag D C ( u /2)A, where C is the average skin
ƒ
s
ƒ
friction drag coefficient and A is the area sheared.
These parameters are listed in Table 7 as functions of the Reynolds number Re u x/
s
x
, where x is based on the distance from the leading edge. For Reynolds numbers between
8
7
7
5
1.8 10 and 4.5 10 , C 0.045/Re 1/6 , and for Re between 2.9 10 and 5 10 ,
x
x
ƒ
C 0.0305/Re 1/7 . These results for turbulent boundary layers are obtained from pipe flow
x
ƒ
friction measurements for smooth pipes, by assuming the pipe radius equivalent to the bound-

Figure 23 Definition of boundary layer thickness: (a) displacement thickness; (b) momentum thickness.