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Bed topography in the bend
v
v s
v r
vsr
δ
v sθ
v θ
β
Figure 3.11: Velocity components of the flow and of the grain
According to figure 3.11, the flow velocity on the bed is noted v , the velocities in stream and
radial direction are v θ
and v r
. The velocity of the grain is v s
with the corresponding components
v sθ and v sr ; δ is the deviation angle of the grain from the stream direction.
The components of the stream force acting on the grain are:
ρ
D w
v
θ
----- C
2 D k 2 d 2 ( v – v s ) 2 θ
– v
= ⋅ ⋅ ⋅ ⋅ ⋅ ------------------ sθ
v –
v s
D r
=
ρ w
v
----- C
2 D
k 2
d 2 ( v–
v s
) 2 r – v sr
⋅ ⋅ ⋅ ⋅ ⋅ ----------------
v –
v s
(3.80)
L
=
ρ
----- w
⋅ C
2 L
⋅ k 2
⋅ d 2 ⋅ ( v–
v s
) 2
Assuming that the grain slips on the surface 1 and neglecting the longitudinal bed slope the components
of the equilibrium of the grain write:
v sθ
v s
D θ – µ ⋅ ( G′ ⋅ cosβ – L)
⋅ ------- = 0
v sr
v s
G
D r – G′ ⋅ sinβ – µ ⋅( G′ ⋅ cosβ – L)
⋅ ------- – --- ⋅ ---- ⋅ ------- = 0
g r
v s
2
v sθ
v s
(3.81)
(3.82)
where G = ρ s
⋅ g⋅ k 1
⋅ d 3 is the weight of the grain, G′ = ( ρ s
– ρ w
) ⋅ g⋅ k 1
⋅d 3 the weight of
the grain submitted to buoyancy and µ = tanφ
the dynamic friction coefficient.
Since the intensity of the secondary flow on the bed is small compared to the main flow and if the
lateral bed slope is small, the following simplifications can be made:
v sθ
v s
------- ≈ 1 ;
v
----- θ
≈ 1 ;
v
v θ
–
------------------ ≈ 1 ; cosβ ≈ 1 ; sinβ ≈ tanβ
=
v–
v sθ
v s
∂h
-----
∂r
(3.83)
1.This assumption is not confirmed by the tests performed in the present study. The observed
movement of the transported grains has to be qualified as saltation.
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 49