Geoscience Reference
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where
u
si
u
i
is the “diffusion” velocity of sediment in the mixture. It can be related to
the interphase velocity difference
u
fi
−
−
, based
on Eq. (2.24). Wu and Wang (2000) derived a differential equation for the interphase
velocity difference
u
fi
−
u
si
by
u
si
−
u
i
=−
ρ
f
(
1
−
c
)(
u
fi
−
u
si
)/ρ
u
si
from the momentum equations of the two-fluid model,
but the derived equation is complex and inconvenient to use. For fine sediments, the
particle inertia or the lag between (local) flow and sediment movement is very small,
and nearly no relative motion exists except for the settling due to gravity; thus, the
following relation is assumed:
u
si
−
u
i
=−
ω
δ
(2.32)
sm
3
i
where
sm
the settling velocity of sediment particles in the water-sediment mix-
ture, and the subscript “3” in
ω
δ
3
i
denotes the vertical direction defined by
gravity.
Substituting Eq. (2.32) into the sediment transport equation (2.31) yields the closed
sediment transport equation:
∂
+
∂(
)
=
∂
∂
c
u
i
c
x
i
(ω
sm
c
δ
)
(2.33)
3
i
∂
t
∂
x
i
2.2.3 Simplification in the case of low sediment
concentration
If the sediment concentration is low,
ρ
≈
ρ
≈
constant, 1
−
c
≈
1,
µ
≈
µ
,
m
f
and
s
, the settling velocity of a single particle in clear water. Then
the continuity equation (2.25) and momentum equation (2.26) of the mixture can be
simplified to
ω
sm
is close to
ω
∂
u
i
x
i
=
0
(2.34)
∂
∂
u
i
∂
+
∂(
u
i
u
j
)
∂
1
ρ
1
ρ
∂
p
1
ρ
∂τ
ij
∂
=
F
i
−
x
i
+
(2.35)
t
x
j
∂
x
j
and the constitutive relation (2.27) of Newtonian fluid can be written as
∂
u
i
x
j
+
∂
u
j
τ
=
µ
(2.36)
ij
∂
∂
x
i
Substituting Eq. (2.36) into Eq. (2.35) leads to the following Navier-Stokes equation
widely used in the single-phase fluid mechanics for laminar flows or instantaneous
motions of turbulent flows:
+
∂(
u
i
u
j
)
∂
2
u
i
∂
u
i
∂
1
ρ
1
ρ
∂
p
x
i
+
µ
∂
=
F
i
−
(2.37)
∂
ρ
∂
x
j
∂
t
x
j
x
j
