pdf, 12 MiB - Infoscience - EPFL

Chapter 3 - Theoretical considerations
Figure 3.9: Comparison of tanδ = 11 ⋅ h ⁄ r with laboratory and field data
1-5: Rozovskii’s tests no. I, II, VI, VII, IIX; 6: polygonal channel; 7: sand model;
8: Snov River; 9: Desna River, [Rozovskii. 1957, Fig. 81, p.194]
Therefore BRIDGE proposed the following formula:
tanβ
11 h s
= ⋅ ---- ⋅ tanφ
r
which yields after replacement of tanβ ≈ dh s
⁄ dr and after integration:
h s
= a ⋅ r 11 ⋅ tanφ
(3.72)
(3.73)
BRIDGE used the friction angle given by BAGNOLD (1954, 1956, 1966). Based on tests with small
spheres with diameter 1.3 mm, BAGNOLD indicated that the friction angle tanφ is of the same
order as the ‘static’ friction coefficient for granular solids (see also § 3.1.1 g). If the grains are
sheared over the surface, tanφ ranges from 0.32 (for completely inertial conditions) to 0.75 (for
completely viscous conditions). He found tanφ to be depending on a Reynolds number, but
finally assumed that within the limits 0.32 to 0.75, tanφ was a function of the grain size only. In a
later study on saltation, BAGNOLD (1973) gave an approximation for tanφ , for a rolling movement.
For grains of normal shape and roughness in lower flow regime bed load transport, this
constant value is 0.63. He admitted that tanφ can be as small as 0.32 for fully inertial conditions in
natural streams and greater than 0.63 if viscosity effects are more significant.
The given limit of 0.63 is equivalent to a friction angle of 32°, corresponding well with the friction
angle of sand or finer components. Since the present study analyses gravel bed rivers, the friction
angle is somewhat bigger (see 3.1.1 g).
page 46 / November 9, 2002
Wall roughness effects on flow and scouring