Geoscience Reference
In-Depth Information
The sediment transport equation (2.33) can be further simplified to
∂
c
+
∂(
u
i
c
)
=
∂
∂
x
i
(ω
s
c
δ
)
(2.38)
3
i
∂
t
∂
x
i
In principle, Eq. (2.38) is applicable only for fine sediments and low concentrations
(see Greimann and Holly, 2001). It is commonly accepted that if the sediment size is
finer than 1mm and the sediment concentration is lower than 0.1 by volume, Eq. (2.38)
can be approximately used.
In summary, the water and sediment two-phase flow model in the case of low sedi-
ment concentration can be simplified to the model of clear water flow with sediment
transport. Because the sediment concentration usually is not high in most natural
rivers, the simplified diffusion model has been widely adopted in river dynamics.
2.3 TIME-AVERAGED MODELS OF TURBULENT FLOW
AND SEDIMENT TRANSPORT
2.3.1 Mean movement equations
Eqs. (2.34), (2.37), and (2.38), which are the exact equations for instantaneous
motions of flow and sediment, cannot be solved directly in most cases, because of
limited computer capacity. Since engineers usually are not interested in the details of the
turbulent fluctuatingmotions, how to describe and solve the meanmotions of turbulent
flow is important in practice. As suggested by Osborne Reynolds, the instantaneous
quantity of a variable
φ
can be divided into mean and fluctuating quantities as
φ
=
φ
+
φ
(2.39)
” denotes the mean quantity, and “
” denotes the fluctuating quantity. The
mean quantity is defined as
where “
−
t
+
T
t
φ
1
T
φ
=
d
τ
(2.40)
where
T
is the time period considered, which should be much longer than the
fluctuation period of turbulence, as shown in Fig. 2.5.
The fluctuating quantity satisfies
t
+
T
t
φ
d
1
T
(φ
)
=
τ
=
0
(2.41)
Reynolds-averaging Eqs. (2.34), (2.37), and (2.38) yields the mean continuity and
momentum equations of flow and the mean transport equation of sediment:
∂
¯
u
i
x
i
=
0
(2.42)
∂
