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Bed topography in the bend
WILLIAMS (1899) gives a derivation of FARGUE’s laws using his own shock pulse surface theory.
This theory of shock pulse surfaces does not only allow the computation of the maximum scour
depth, but also the location of the scour holes even for meandering rivers (see Fig. 3.7)
Figure 3.7: Location of scour holes by means of the shock pulse surface theory [Williams 1899, Fig. 4 and 5]
Based on impulse consideration on the stream bank WILLIAMS demonstrates the validity of FAR-
GUE’s equation for the maximum scour depth given by the following equation:
,
= c ⋅ 1 + m 2 ⋅ ⎛1
– ⎛----------------------
⎞ 2
⎞
⎝ ⎝R c
+ B ⁄ 2⎠
⎠
= c ⋅ 1 + m 2 ⋅ sin--
φ
2
h smax
(3.56)
2
⎛ V
with c 1 0
⎞
⎜ + --------- ⎟ -------------------
1
µ A 2
shock
⋅ V m
= ⋅ ⋅ ⋅ ---------------------------
(3.57)
2
⎝ V m
⎠ 1 + m 2 2 ⋅ g
The coefficient c depends on the size of the shock surface A shock
, its roughness µ , the average
flow velocity V m
, the flow velocity on the ground V 0
and the bank slope m : 1 ( m = 0 for vertical
side walls). φ is the angle between the tangents at the beginning and the end of the bend
(= the opening angle between beginning and end of the bend). The maximum scour is supposed
to be located at about φ ⁄ 2 .
For a test section at the white Elster in Eastern Germany, WILLIAMS admitted that c = 0.5 is
constant. Equation 3.57 depends on the radius of curvature, the geometry of the cross-section (B,
m) and the velocity distribution ( , )
V 0
V m
R c
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 41