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Bed topography in the bend
The bed profile at equilibrium state and the maximum scour depth are obtained based on equation
3.87:
dh s
k 2
µ ⋅ C
------- – ----------- ⋅ ------------------------ D
(3.90)
dr 2 ⋅ k1
1 µ C ----------------------------------
V
--
1
( fr ()) L ( s – 1) ⋅ g ⋅ d κ
2 ---- h s
–
r
4.167 2.640 1 κ -- V∗
=
⋅ ⋅ ⋅ ⋅ ⋅⎛
+ ⋅ ⋅ ------⎞
⎝
V ⎠
+ ⋅ ------
C D
This equation can be integrated if the function f is known. Assuming that f is given by the forced
vortex distribution and determining the integration constants with the boundary condition
R = R c
; h s
= h m
they found:
k 2
h
----- s
h m
=
exp A
--- ⎛----- r 2
– 1⎞
⎝
⎛ ⋅
2 ⎝ ⎠⎠
⎞
(3.91)
µ ⋅ C
with A – ----------- ⋅ ------------------------ D
(3.92)
2 ⋅ k1
1 µ C ----------------------------------
V
----
V
L ( s – 1) ⋅ g ⋅ d κ V∗
------ – 4.167 2.640 1 κ -- V∗
=
⋅ ⋅ ⋅ ⋅⎛
+ ⋅ ⋅ ------ ⎞
⎝
V ⎠
+ ⋅ ------
C D
R c
2
λ 0
After introduction of different simplifications (eq. 3.88), the constant A
A 1.8955 3.0023 V ∗
= ⎛ – ⋅ ------⎞ ⋅
⎝
V ⎠
Fr d
becomes:
(3.93)
6) Zimmermann (1983)
ZIMMERMANN (see ZIMMERMANN & KENNEDY, 1978; ZIMMERMANN & NAUDASCHER, 1979
and ZIMMERMANN, 1974, 1983) developed a relationship for the boundary shear stress in radial
direction τ r
averaged over the wetted perimeter. He compared the transverse drag on a sediment
particle (computed with τ r ) with the corresponding opposing component of the weight of the
submerged particle (centrifugal force).
His formula is based on the vertical distribution of the streamwise velocity v given by the power
law:
vzr ( , ) = v surf
() r
z
⋅ --
⎝ ⎛ h⎠
⎞ 1 ⁄ n
where v surf
() r is the velocity at the free surface and 1 ⁄ n a dimensionless exponent. The mean
velocity is obtained by integration:
h
z
Vr () ∫v surf
() r ⋅ --
⎝ ⎛ h⎠
⎞ 1 ⁄ n n
= = ----------- ⋅ v
n + 1 surf
() r
0
(3.94)
(3.95)
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 51