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Bed topography in the bend
VAN BENDEGOM further determined the deviation angle δ between the direction of the shear
stress D (respectively the velocity) on the bed surface and the longitudinal axis of the channel D θ
as tanδ
= 10 ⋅ h ⁄ r. Taking into account that D r
= D θ
⋅ tanδ
, he obtained:
10
D r
------------
⋅ g
ρ w
g V 2 h
= ⋅ ⋅ ⋅ ⋅A ⊥ ⋅ --
(3.61)
r
C 2
Replacing A ⊥
with A ⊥
= k 2
⋅ d 2 , where k 2
is a factor similar to k 1
for the volume of the grain
in equation 3.57, he obtained after introducing equation 3.61 in equation 3.59:
sinβ
10 ⋅ g
------------ ------------------------------ ---- ----
10 ⋅ g 2 k
⋅ ⋅ ⋅ ------------ 2
2 h
⋅Fr ( s – 1) ⋅ g⋅
d r
d ⋅ ---- ⋅ ----
s
= = = 0.0589 ⋅ Fr
r
d ⋅ ----
r
C 2
V 2
k 2
k 1
h s
(3.62)
where s = ρ s
⁄ ρ w
. For spheres k 1
= π ⁄ 6 and if the grain is completely exposed to the stream
force (sphere on a smooth surface) k 2
= π ⁄ 4 .
This formula is based on spherical grains which are completely exposed to the flow. The equilibrium
state is reached if no transport in radial direction occurs, i.e. if the grains only move in stream
direction.
The assumption of a complete exposure of the grains is conservative and leads to underestimated
values for β - therefore the lateral bed slope will be too small. The omission of the buoyancy
causes an overestimation of the radial force in the downwards direction. This leads to an additional
underestimation of the lateral bed slope β .
ODGAARD (1981) compared the predicted lateral bed slope of different formulae and compared
them with laboratory and field data. For small slopes (only), a good correlation with the formula of
VAN BENDEGOM can be obtained if the constant in equation 3.62 is increased from 0.059 to about
0.20 for laboratory data and even to 0.8 for field data.
C 2
k 1
h s
3) Engelund (1974)
ENGELUND (1974) also established his formula considering the forces acting on a sediment grain.
He assumed that the dynamic lift force L acts in vertical direction and that the grain slips on the
bed. He expressed the equilibrium state, at which the grain just moves, by means of the slide friction
law as:
( G′ – L) ⋅ cosβ
⋅ tanφ
= D θ
(3.63)
( G′ – L) ⋅ cosβ
corresponds to the force acting on the grain perpendicularly to the bed, tanφ
is
the coefficient of friction and φ is the dynamic shear angle. The equilibrium condition in radial
direction can be expressed as:
( G′ – L) ⋅ sinβ
=
(3.64)
D r
Dividing equation 3.64 by equation 3.63, yields:
----------
tanβ
tanφ
D r
= ------ = tanδ
D θ
(3.65)
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 43