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Chapter 3 - Theoretical considerations
3.5 Bed topography in the bend
3.5.1 Introduction
In general scour formulae are established based on equilibrium considerations of the forces acting
on a grain. Taking the water depth h (free water surface to bed surface) and the radius r as
variables, the local lateral bed slope is usually given by:
sinβ
K h s
= ⋅ ----
r
(3.54)
h s
h m
β
r
R c
B
B
θ
r
R c
Figure 3.6: Definition sketch; situation and cross-section
In order to obtain the cross-section at the location of maximum scour,
sinβ≈
tanβ
= dh ⁄ dr and yields (after integration):
h s
r
----- = ⎛-----
⎝ ⎠
h m
R c
⎞ K
sinβ
is approached with
(3.55)
3.5.2 Scour formulae
1) Fargue (1868) and Williams (1899)
LOUIS JÉRÔME FARGUE (1872-1910) (cited in FARGUE, 1868 and in WILLIAMS, 1899) was probably
the first to establish a scour formula for river bends. He established six laws based on observations
of a 22 km long reach of the Garonne River in France, derived with momentum
considerations. Two of them are given hereafter:
1. Law of displacement: The deepest and the highest point of the bed topography in a bend are
located towards half the opening angle between beginning and end of the bend or in downstream
direction of this point.
2. Law of depth: The greater the curvature, the greater the depth.
page 40 / November 9, 2002
Wall roughness effects on flow and scouring