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Chapter 3 - Theoretical considerations
The centrifugal force is not constant over the depth because of the velocity gradient. The induced
force T c (due to the centrifugal force) in the center of the channel on the flow for an angular
increment of dφ is
r o
h
v
dT c
ρ 2 ( zr , )
= ⎛ ----------------- w
r
z –
h
⋅ ⋅⎛ --⎞ ⋅r⎞ ∫∫
dφdzdr
⎝
⎝ 2⎠
⎠
r i
0
Introducing equations 3.94 and 3.95, Zimmermann obtained:
ρ
dT w
c
-----
n + 1
= ⋅
2
----------------------- ⋅ (
n ⋅ ( n + 2)
R – R ) ⋅ h 2 ⋅ V 2 dφ
o i
(3.96)
(3.97)
The centrifugal force is primarily balanced by the momentum induced by . Assuming that
the shear stress τ cr is uniformly distributed around the channel perimeter and that the cross-section
is rectangular, ZIMMERMANN got:
dT r
τ r
R o
( R o
– R c
) d R i
⋅( R c
– R i
) ⋅d
R2 o
R2
d
= – ⋅ ⋅ ⋅ +
+ ( – i
) ⋅ -- dφ (3.98)
4
in which R c
= ( R o
– R i
) ⁄ 2 . Combining 3.97 and 3.98 he finally obtained the boundary shear
stress in radial direction:
ρ
τ r
----- w n + 1 h
= ⋅ ----------------------- ⋅ ----- m
⋅V (3.99)
3 n⋅
( n+
2)
2
Considering now the balance of the forces acting in radial direction on a spherical particle moving
along the inclined bed plane ( β ) upwards toward the inner bank, ZIMMERMANN obtained:
dh n + 1 h
----- α (3.100)
dr z
------------------------------- m
= ⋅ ⋅ ----- ⋅Fr2
2 ⋅ n⋅
( n+
2)
d
α z
is the ratio of the projected surface of sediment particle to the projected area of a sphere. Integrating
v given by the logarithmic velocity defect law in streamwise direction (between 0 and h)
(( v–
v s
) ⁄ V∗ = 1 ⁄ κ ⋅ ln( z ⁄ h)
), and putting this relation equal to equation 3.95, yields:
-----------
n
= 1 – -----------
V∗
(3.101)
n + 1 κ ⋅ v s
Eliminating v s
from the previous relation and introducing the definition of the Darcy-Weissbach
friction factor f lead to:
R c
R c
T r
τ r
f = 8 ⋅
κ -- n
(3.102)
After introduction of κ = 0.4 , Zimmermann found f = 1.13 ⁄ n , which can be compared to
the experimental relation of NUNNER (1956): f = 1 ⁄ n
ZIMMERMANN finally gave the following formula for the lateral bed slope:
f+
f h
sinβ ---------------------------------- Fr2
s
=
⋅ d ⋅ -----
(3.103)
2 ⋅ ( 1+
2⋅
f)
R c
page 52 / November 9, 2002
Wall roughness effects on flow and scouring