Geoscience Reference
In-Depth Information
The density of the mixture,
ρ
, is determined by
ρ
=
ρ
(
1
−
c
)
+
ρ
s
c
(2.23)
f
and the specific weight of the mixture is correspondingly given by
γ
=
ρ
g
.
The velocity of the mixture,
u
i
, is defined as
1
ρ
[
ρ
f
(
u
i
=
1
−
c
)
u
fi
+
ρ
s
cu
si
]
(2.24)
where
u
fi
is the
i
-component of water velocity,
u
si
is the
i
-component of sediment
velocity, and
i
denotes three spatial directions (
=
1, 2, 3).
2.2 GOVERNING EQUATIONS OF WATER AND SEDIMENT
TWO-PHASE FLOW
Because the stochastically averaged properties of a group of sediment particles are
mainly concerned in river engineering, sediment is often assumed to be a kind of
continuous medium. Two mathematical models can be used to describe the water
and sediment two-phase flow based on this assumption. One is the two-fluid model
that considers water and sediment as two fluids and establishes the continuity and
momentum equations for each phase. The other is the diffusion model that consid-
ers the movement of sediment particles as a phenomenon of diffusion in the water
flow and hence establishes the continuity and momentum equations for the water-
sediment mixture and the transport (diffusion) equation for sediment particles. The
two-fluid model is more general, from which the diffusion model can be derived, as
described by Wu and Wang (2000). Detailed discussions on the two-fluid model can
also be found in Soo (1967), Ni
et al
. (1991), Liu (1993), and Greimann and Holly
(2001). However, the two-fluid model is not introduced here because it is quite com-
plex. The flow and sediment transport equations used in this topic are based on the
diffusion model.
2.2.1 Hydrodynamic equations
Applying the mass and momentum conservation laws leads to the continuity and
momentum equations for the instantaneous movement of the water-sediment mixture.
These equations are written in Cartesian tensor notations as follows:
∂ρ
∂
t
+
∂(ρ
u
i
)
=
0
(2.25)
∂
x
i
∂(ρ
u
i
)
+
∂(ρ
u
i
u
j
)
−
∂
x
i
+
∂τ
p
ij
=
F
i
(2.26)
∂
t
∂
x
j
∂
∂
x
j
where
t
is the time;
x
i
is the
i
-coordinate in the Cartesian coordinate system;
p
is the
