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Chapter 3 - Theoretical considerations
ODGAARD indicated that his formula may not apply in channels with heavy sediment load. Both
equations 3.106 and 3.108 are valid for fully developed secondary flows in bends.
ODGAARD (1984) also developed an equation for the lateral bed slope in the development region,
that is the zone where the bed slope in radial direction passes from a horizontal bed to the maximum
scour bed profile. He used the exponential function given by ROZOVSKII (1957) for the
growth of the secondary flow. ROZOVSKII assumed that the bed shear stress depends linearly on
the radial velocity component at the free surface and on the lateral bed slope.
The lateral bed slope in the zone between the beginning of the bend and the first scour (called
development region) is consequently given by:
β
dh s
2 ⋅ κ
sin ≈ ------- 4.8 θ Fr (3.109)
dr
d
1
2 R
------------ c
– exp – ⋅ ----- ⋅φ
⎝
⎛ n ⎠
⎞ h s
= ⋅ ⋅ ⋅ ⋅ ----
r
h c
9) Bazilevich (1982)
BAZILEVICH (1982) simplified a model established by the Institute of Fluid Mechanics, Academy
of Science of the Ukrainian SSR 1 . He defined the equilibrium condition of scour as the absence of
noticeable deepening of the bottom. At this state, the friction velocity V∗ at the bottom has to be
equal to the friction velocity V∗ dest corresponding to the destruction of the erosion pavement.
V∗ dest is given for a gravel-bed channel (with d ≥ 1.5 mm) by:
0.189 ⋅ ( s – 1) ⋅g ⋅d V∗ m
dest = -------------------------------------------------------
(3.110)
( d 25
⁄ d 75
) 1 ⁄ 4
The friction velocity at the bottom at the given point is determined by:
V∗ =
V
---------------
⋅ g
= ---------------------
V⋅
g ⋅f
=
C
⁄
h 1 6
---------------------
V⋅
g
K S
⋅ h 1 ⁄ 6
BAZILEVICH recommended to use the CHEZY-MANNING relation n = ( h2 ⁄ 3
m
⋅ S1 ⁄ 2 ) ⁄ V 0
determine the depth averaged velocity at the maximum scour location:
1 – A
V hmax
= ------------ ⋅ V
1 + A
2 + 2 ⋅ g ⋅ ∆z
(3.111)
to
(3.112)
where A
g ⋅ ln 2 β
1
= ----------------- , and h (3.113)
⁄
m
= -- ⋅ ( h
2 in
+ h)
h m
4 3
The length of the calculated reach is given as a function of the local water depth h
the inlet h in :
L = 10 ⋅ ( h – h in
)
and the one at
(3.114)
1. He does not give any reference to this complete model. Unfortunately the translated paper does
not allow a detailed analysis.
page 54 / November 9, 2002
Wall roughness effects on flow and scouring