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Bed topography in the bend
ZIMMERMANN assumed a fully developed secondary flow, which is not always found in natural
river bends. Furthermore, his formula gives a straight line for the transverse bed slope which does
not really correspond to laboratory and field observations.
7) Falcon & Kennedy (1983)
FALCON & KENNEDY (1983) (see also FALCON, 1979) used DU BOYS approach of a sediment
transport in layers. They considered a control volume in the armoring layer at equilibrium state.
Based on the work of KARIM (1981), the authors gave the thickness of the control volume (thickness
of the armoring layer) with:
V∗
z b = d 50 ⋅ ----------
(3.104)
Based on a vertical distribution of radial shear stresses, the primary flow velocity given by the
power law vrz ( , ) ⁄ V = ( n + 1) ⁄ n⋅
z 1 ⁄ n and the definition of the exponent 1 ⁄ n = f ⁄ 8 ⁄ κ,
their equation for the lateral bed profile writes:
dh s 8 ⋅ θ 1 + f h
sinβ ≈ ------- --------------- --------------------- s
= ⋅ ⋅Fr (3.105)
dr ( 1 – p)
d ⋅ ----
1 + 2⋅
f r
with p, the porosity of the armoring layer, f = 8 ⋅ g⋅ h m
⋅ S e
⁄ V2
m
, where V m
is the local depth
averaged velocity. They assumed that κ = 0.354 and therefore n = f (see eq. 3.102).
V∗ cr
The approach of FALCON & KENNEDY is similar to the one of ZIMMERMANN. The difference
resides in the fact that FALCON & KENNEDY considered a vertical control volume, while ZIM-
MERMANN considered the whole cross-section. The formula of Zimmermann also accounts for
the vertical shear stresses.
8) Odgaard (1986)
ODGAARD (1981, 1982, 1984 and 1986) considered the forces acting on a control volume in a stable
armoring layer, without any sediment transport (nor in radial, nor in stream direction). Taking
into account the equation of FALCON (1979) and FALCON & KENNEDY (1983) for the radial drag
force, the following formula is obtained:
β
dh s
≈ -------
3
sin ----------
⋅ α
------
θ n + 1 h
----------- s
= ⋅ ⋅ ⋅ Fr (3.106)
dr 2 κ n + 2 d ⋅ ----
r
where α = 1 ⁄ k 2
is the ratio of the projected area of a sediment grain normalized by d 2 ,
κ = 0.4 the von Karman constant, C the Chezy coefficient and n is given with:
κ κ ⋅ C
n = ------------ = ----------
(3.107)
8 ⁄ f g
Putting α = 4 ⁄ π , κ = 0.4 and ( n + 1) ⁄ ( n + 2)
≈ 1 , an earlier version of the same formula
given in ODGAARD (1984) can be obtained:
β
dh s
h
sin ≈ ------- = 4.8 ⋅ θ ⋅ Fr s
dr
d ⋅ ----
r
(3.108)
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 53