Geoscience Reference
In-Depth Information
0.8
0.8
Gravity line
Gravity line
0.4
0.4
Immobile
Entrainment
0
0
0
0.5
1
0
0.5
1
2
2
-
u
w
/
u
*
-
u
w
/
u
*
u
0
w
0
/
u
*
2
versus
z
/
h
for immobile and entrainment threshold beds
Fig. 6 Plots of
have a departure from the gravity line. On the other hand, in the upper flow zone,
they are reasonably consistent with the gravity line, although they have
a s
light
tendency to overestimate the gravity line. For
z
/
h
2
for entrainment threshold beds diminishes more tha
n tha
t for immobile beds,
although there is a general nea
r-be
d damping in the
<
0.1 (near the bed),
u
0
w
0
=
u
u
0
w
0
due to roughness. The
reduction in magnitude of
u
0
w
0
for entrainment threshold beds is attributed to
the fact that a portion of the fluid turbulent stress is transferred to the bed particles to
overcome the frictional resistance at the contacts of the entrained sediment parti-
cles. This is analogous to the concept of Grass (
1970
). The damping of the Reynolds
shear stress can also be explained that the bed particles are associated with the
provided momentum for the flow to maintain their motion.
Turbulent energy
budg
et in two-dimensional flows is constituted by the turbul-
ent production
t
P
½¼
u
0
w
0
ð@
u
=@
z
Þ
that is balanced by the summation of the turbulent
dissipation
e
, tu
rbulent energy diffusion
t
D
(
¼
∂
f
kw
/
z
), pressure energy diffusion
∂
2
k
/
z
2
)]; where
f
kw
¼
p
D
½¼ @ð
p
0
w
0
=rÞ=@
z
, and viscous diffusion
v
D
[
¼u
(
∂
∂
:
ð
w
0
w
0
w
0
þ
w
0
u
0
u
0
;
p
0
¼
pressure fluctuations; and
k
¼
0
turbulent kinetic energy.
In turbulent flows
, the visco
us diffusion
v
D
is insignificant. To evaluate
75
e
, the relation-
2
was used. The pressure energy diffusion
p
D
was estimated
(15
u
/
u
2
)
ship
e ¼
ð@
u
0
=@
t
Þ
as
p
D
¼
t
D
. Figure
7a, b
illustrates the energy budget in flows over immobile
and entrainment threshold beds having uniform sediment size of 4.1 mm. The nondi-
mensional form of these parameters are
T
P
,
E
D
,
T
D
,
P
D
¼
t
P
e
3
).
(
t
P
,
e
,
t
D
,
p
D
)
(
h
/
u
In general,
T
P
increases near the bed with an increase in
z
/
h
up to
z
/
h
>
0.05 and then
decreases rapidly becoming nearly constant for
z
/
h
0.3. The distributions of
E
D
have
a distinct lag from those of
T
P
. The influence of a sediment entrainment is apparent in
the near-bed distributions of
T
P
and
E
D
, where the lag is reversed, which means
E
D
>
>
T
P
. Essentially, the difference of
T
P
and
E
D
at any depth is balanced by the
combination of
T
D
and
P
D
.InFig.
7a, b
,
T
D
decreases with an increase in
z
/
h
within the
wall-shear layer and then it becomes almost invariant of
z
/
h
. On the other hand,
P
D
attends a positive peak at
z
/
h
0.05 and then gradually decreases with an increase in
