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Bed topography in the bend
Based on the works of ALLEN (1970), BRIDGE further gave a relation for the size of moving bed
particles. The formula was established, neglecting the lift force.
3 ⋅ ρ
d
w
⋅S e
⋅h = --------------------------------------------- s
(3.74)
2 ⋅ tanφ
⋅( ρ s
– ρ w
)
Finally, BRIDGE compared his formula to field data, measured in the River Endrick in Scotland
(BLUCK, 1971), in the River Desna, USSR (ROZOVSKII, 1957) and the River South Esk in Glen
Clova, Scotland. Cross-sections were chosen in areas of fully developed spiral flow. All the rivers
had ripples and dunes, indicating that the flow is in the lower flow regime. The grain size of River
South Esk is of 1 mm, the one of River Endrick is coarser, whereas the one of the Desna River is
finer. The longitudinal bed slope of River Esk is of 0.025 to 0.030 %. Bridge used values of tanφ
between 0.4 and 0.5 in order to obtain a good prediction. He further proposed to adjust the values
of tanφ to increase the precision of the prediction of the radial bed topography.
5) Kikkawa, Ikeda & Kitagawa (1976)
KIKKAWA ET AL. (1976) defined the velocity distribution based on the equation of motion in
radial direction, by means of a simplified stream function of the secondary flow. They assumed a
constant eddy viscosity and a logarithmic velocity distribution, which is modified in radial direction
with a special distribution function. Their equation was established for sand bed rivers and
tested against laboratory data on fixed and mobile bed ( Fr = 0.56 ÷ 0.63 ; B ⁄ h = 16 ÷ 20 ). The
velocity distribution in stream direction at the location with flow depth is described by:
v
------ θ
V*
=
------
V
V*
+
where v θ is the velocity in stream direction at level z, V* the shear velocity averaged in radial
direction and V the average velocity in the cross-section ( V = Q⁄
A)
.
--
1
κ
⋅ ⎛ln-----
z
⎝
+ 1⎞
⎠
(3.75)
The velocity profile at the location r in the cross-section becomes after introduction of a distribution
function fr ():
v
------ θ
f()
r
V 1
= ------ + -- ⋅ ⎛ln z
----- + 1⎞
(3.76)
V* V* κ ⎝ ⎠
where h is the local flow depth. The distribution function f()
r corresponds to the velocity distribution
normalized by the average velocity V .
v θ
h m
h m
h m
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 47