Geoscience Reference
In-Depth Information
Eq. (7.67) considers only the effect of bed slope along the flow direction; it was
applied in the simulation of the vertical 2-D headcut migration by Wu and Wang
(2005). For general applications, the effect of bed slope in the direction normal to the
flow can also be added, as shown in Eq. (3.47). However, the side slope affects sedi-
ment transport within the scour hole in two counteractive ways: reducing the critical
shear stress for sediment incipient motion, but tending to move sediment toward the
scour hole center.
The effective tractive force
τ be includes the bed shear stress and the horizontal
component of the dynamic-pressure-difference force:
6 d
a
p d
τ be = τ b
(7.68)
s
τ b
where
s is the streamwise
gradient of dynamic pressure near the bed, and a is a coefficient assumed as 4
is the bed shear stress due to grain roughness,
p d /∂
.
τ b d 2
Eq. (7.68) is derived by assuming that the shear stress on a sediment particle is
a
and determing the dynamic-pressure-difference-force on this particle with Eq. (7.64).
Because the method for computing
/
τ b used by van Rijn (1984a & b) for uniform
flow is not appropriate for rapidly varied flows,
τ b is directly set to the bed shear stress
calculated by the flow model. However, to be consistent with the original van Rijn
formulas, the equivalent bed roughness height k s used in the wall-boundary approach
in the flowmodel is set to the grain roughness 3 d 90 . Because only the grain roughness is
considered, this approach is applicable to situations without bed forms. Usually, most
clear-water scour cases belong to such situations. For more general applications, all
roughness elements may be considered in the wall-boundary approach, and then, the
grain shear stress is separated from the computed total bed shear stress using the
approaches introduced in Section 3.3.2.
In addition, the effect of gravity on sediment transport over a steep slope may be
considered by adding the streamwise component of gravity to the effective tractive
force
be rather than applying the correction factor K g to the critical shear stress, as
shown in Eq. (3.132). Thus, Eqs. (7.63) and (7.68) can be modified as
τ
τ
=
K p K d τ
(7.69)
cr
c
a
6 d
p d
τ be , i = τ b , i + λ 0 τ
c sin
ϕ i /
sin
φ
x i (
i
=
x , y
)
(7.70)
r
In analogy to Eq. (6.98), an equation can be derived from Eq. (7.70) for the bed-load
transport direction cosines
α by , e .
If the slope angle in the scour hole is larger than the repose angle of sediment, a
loose bed will collapse due to gravity. This physical phenomenon can be calculated
by adjusting the steeper bed slope to the repose angle according to mass conservation.
The non-cohesive bank or bed sliding algorithm introduced in Section 6.3.5 can be
adopted here.
The approaches presented in Eqs. (7.63) and (7.68)-(7.70) were tested in the sim-
ulation of local scour process at bridge piers by this author using the 3-D flow
model introduced in Section 7.1.3.2 and the sediment transport model introduced in
α bx , e and
 
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