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Chapter 3 - Theoretical considerations
2) Van Bendegom (1947)
VAN BENDEGOM (1947) (given in ODGAARD (1981) based his equation on equilibrium considerations
on a grain. A grain on an inclined plane (angle β between horizontal surface and inclined
plane, see Fig. 3.8) is submitted to its weight, the buoyancy and the stream force. Weight and
buoyancy can be summarized by the vertical force:
G′ = ( ρ s
– ρ w
) ⋅ g⋅ k 1
⋅d 3
(3.58)
ρ s
and ρ w
are the densities of the sediment and the fluid, g is the gravity constant and d the characteristic
grain size diameter. The proportionality factor k 1
corresponds to the ratio between the
volume of the grain and .
d 3
The dynamic stream force can be split in two components: the drag force D acting in the plane of
the bed surface and the lift force L acting normally to it. The normal projection D of the dynamic
stream force on the bed surface corresponds to the direction of the flow velocity on the bed surface.
The drag force can be split in a component D θ
in stream direction and a component D r
in
radial direction. δ is the opening angle between D and D θ
(see Fig. 3.8).
L
D
D r
δ
D θ
G’
β
Figure 3.8: Forces acting on a grain on an inclined plane
VAN BENDEGOM assumes that at equilibrium state, the grain moves only in stream direction, that
the radial components of the forces compensate each others and that the dynamic lift force L , can
be neglected. The equilibrium of the forces in radial direction consequently writes:
D r
= G′ sinβ
(3.59)
VAN BENDEGOM put for D θ
the relation:
ρ
D w
⋅ g⋅ V 2 ⋅ A ⊥
θ
= ------------------------------------
(3.60)
In this equation V designates the depth averaged flow velocity, A ⊥
the projection of the surface
of the grain which is exposed to the flow in perpendicular direction to D θ and C is the CHEZY
coefficient which is assumed to take a value of 50 m 1/2 /s.
C 2
page 42 / November 9, 2002
Wall roughness effects on flow and scouring