Page 81 - Mechanical Engineers' Handbook (Volume 4)

70 Fluid Mechanics

Table 7 Boundary Layer Parameters
Laminar Turbulent
Boundary Boundary
Parameter Layer Layer
4.91 0.382
x Re x 1/2 Re x 1/5
1.73 0.048
1
x Re x 1/2 Re x 1/5
0.664 0.037
2
x Re x 1/2 Re x 1/5
1.328 0.074
C ƒ
1/2 1/5
Re x Re x
Re x range Generally not Less than 10 7
over 10 6

ary layer thickness, the centerline pipe velocity equivalent to the free stream boundary layer
flow, and appropriate velocity profiles. Results agree with measurements.
When a turbulent boundary layer is preceded by a laminar boundary layer, the drag
coefficient is given by the Prandtl–Schlichting equation:

0.455 A
C
ƒ
(log Re ) 2.58 Re x
x
where A depends on the Reynolds number Re at which transition occurs. Values of A for
c
various values of Re u x /v are
s c
c
Re c 3 10 5 5 10 5 9 10 5 1.5 10 6
A 1035 1700 3000 4880
Some results are shown in Fig. 24 for transition at these Reynolds numbers for completely
laminar boundary layers, for completely turbulent boundary layers, and for a typical ship
hull. (The other curves are applicable for smooth model ship hulls.) Drag coefficients for
flat plates may be used for other shapes that approximate flat plates.
The thickness of the viscous sublayer in terms of the boundary layer thickness is
b
approximately
b 80
(Re ) 7/10
x
6
At Re 10 , / 0.0050 and when Re 10 , / 0.001, and thus the viscous
7
x
x
b
b
sublayer is very thin.
Experiments show that the boundary layer thickness and local drag coefficient for a
turbulent boundary layer preceded by a laminar boundary layer at a given location are the
same as though the boundary layer were turbulent from the beginning of the plate or surface
along which the boundary layer grows.
10 GAS DYNAMICS
In gas flows where density variations are appreciable, large variations in velocity and tem-
perature may also occur and then thermodynamic effects are important.