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Chapter 3 - Theoretical considerations
DU BOYS gave the thickness of the considered moving sediment layer for a unit surface (1 ):
1000 ⋅h m
⋅ S
e ≤ ( ------------------------------- s – 1) ⋅ tanφ
(3.37)
This sediment layer is subdivided in horizontal slides. Each of these slides moves with a different
velocity decreasing from the bed surface. DU BOYS gives the sediment transport rate with:
q s
= χ ⋅ D⋅
( D–
D cr
)
(3.38)
where D = 1000 ⋅h m
⋅S
is the drag shear force according to DU BOYS in kg ⁄ m 2 , D cr
the critical
drag shear force and χ a transport coefficient in m 6 ⁄ kg 2 ⁄ s . D cr
and χ need to be determined
for each grain size.
The direct correlation between the sediment transport rate Q s
and the factor h ⋅ S can be found
in most actual formulae. Though the validity of this correlation had been contested by MEYER-
PETER ET AL. (1934), they finally introduced it in their formula of 1948.
m 2
2) Shields (1936)
SHIELDS (1936) compared the “force of the flow on the grain” to the “resistance to the movement
of the grain” based on tests with uniform grain size. Based on theoretical considerations and on
laboratory tests he showed that the inception of the movement is a function of the Shear Reynolds
number Re∗ = ( V∗ ⋅ d) ⁄ ν (see Fig. 3.5).
ρ
----------------- w
ρ s
– ρ w
R h
⋅ S
⋅ ------------- =
g ⋅ d
fct Re∗ ( )
(3.39)
0.1
0.01
θ cr
θ cr
0.050
1 10 100 1000
Re*
Figure 3.5: Shields diagram [Günther, 1971, Fig. 6]
In the fully turbulent area, SHIELDS assumed = . MEYER-PETER & MÜLLER reduced
the critical shear stress to θ cr
= 0.047 . Gessler (1965) pointed out that the critical value for the
dimensionless shear stress given by Shields is systematically too high because he based his work on
tests where bedforms occurred. His values include the effect of theses bedforms.
page 36 / November 9, 2002
Wall roughness effects on flow and scouring