Geoscience Reference
In-Depth Information
where
s
is the second layer thickness, and
p
sbk
is the fraction of the
k
th size class of
bed material contained in the second layer.
Eq. (2.162) assumes no exchange between the second and third layers. This is
physically right. In addition, changing the layer thickness or moving the layer divi-
sions up or down during the simulation may induce numerical mixing of sediment
between layers and thus should be avoided, except that the division between the
mixing and second layers may change due to variations in flow, sediment, and bed
conditions.
Rahuel
et al.
(1989) treated the bed-load layer and the mixing layer together as an
active layer in a 1-D bed-load model, and Spasojevic and Holly (1990; 1993) extended
this concept to 2-D and 3-D total-load models. The sediment balance in the active layer
is described as
δ
p
bk
∂δ
p
m
)
∂(δ
m
p
bk
)
∂
−
∂
z
b
∂
m
p
m
)
(
1
−
+∇·
q
bk
+
E
bk
−
D
bk
−
(
1
−
=
0
t
∂
t
t
(2.163)
where
D
bk
and
E
bk
are the deposition and entrainment fluxes of the kth size class of
sediment at the lower bound of the suspended-load zone.
It can be seen that Eq. (2.163) is the sum of Eqs. (2.158), (2.159), and (2.161), with
only the storage term in Eq. (2.158) omitted.
2.7.3 Mixing layer thickness
The mixing layer thickness is related to the time scale under consideration (Bennett and
Nording, 1977; Rahuel
et al
., 1989; Wu, 1991). If a very short, nearly instantaneous
time scale is considered, the mixing layer should be a thin bed surface layer containing
particles susceptible to entrainment due to a momentary increase in the local bed shear
stress. This is called the instantaneous mixing layer. If the time scale is longer, e.g.,
in the order of magnitude of the time it takes for a bed form (ripple or dune) to
traverse its own wavelength, the mixing layer can be the order of magnitude of the
bed form height. If the time scale is much longer, e.g., in the order of magnitude of
the computational time step, the mixing layer includes the layer of material eroded
or deposited and the instantaneous mixing layer.
Since the sand dune height is generally relative to the flow depth, Karim and
Kennedy (1982) evaluated the mixing layer thickness as 0.1-0.2 times the flow
depth. Borah
et al
. (1982) determined the mixing layer thickness under armoring
conditions by
d
L
δ
=
(2.164)
m
p
m
)
(
1
−
p
bm
where
d
L
is the smallest size of the sediment particles that are immobile, and
p
bm
is
the fraction of all the immobile particles in the mixing layer.
