Geoscience Reference
In-Depth Information
where
λ
w
is a coefficient. In the
k
-
ε
turbulence models,
λ
w
is determined by
λ
=
w
c
1
/
4
k
1
/
2
P
c
1
/
4
k
1
/
P
y
P
/ν
Ey
P
)
with
y
P
ρ
λ
w
is derived using the
log-law of velocity near the wall and the first relation in Eq. (6.15). Eq. (6.12) is
applied in the region of 11.6
κ/
ln
(
=
. This relation of
µ
µ
y
P
<
∼
300. In the zero-equation turbulence models,
because the turbulent energy
k
is not solved,
<
Ey
P
)
λ
w
is determined by
λ
=
ρ
u
∗
κ/
ln
(
w
with
y
P
=
u
∗
y
P
/ν
. Here,
y
P
is the dist
ance
from the wall to point
P
,
u
∗
is the shear
velocity on the wall defined as
u
∗
=
√
τ
, and
E
is a roughness parameter. For a
smooth wall,
E
is about 8.432. For a rough wall,
E
is related to the roughness Reynolds
number
k
s
/ρ
w
=
u
∗
k
s
/ν
by (Cebeci and Bradshaw, 1977)
E
=
exp
[
κ(
B
0
−
B
)
]
(6.13)
where
k
s
is the equivalent roughness height on the wall,
B
0
is an additive constant of
5.2, and
B
is a function of
k
s
:
⎨
k
s
<
0
2.25
1
κ
ln
k
s
)
ln
k
s
k
s
<
(
B
0
−
+
[
(
−
)
]
≤
8.5
sin
0.4258
0.811
2.25
90
B
=
⎩
1
κ
ln
k
s
k
s
B
0
−
8.5
+
≥
90
(6.14)
turbulence models, the turbulent energy and its dissipation rate at point
P
are specified as (Rodi, 1993)
In the
k
-
ε
u
2
∗
c
1
/
2
u
3
∗
k
P
=
ε
P
=
,
(6.15)
κ
y
p
µ
which are derived by assuming that the local equilibrium of turbulence prevails near
the wall.
However, the turbulent energy
k
P
may also be obtained by actually solving the
k
equation in the control volume near the wall, with the turbulence generation and
dissipation rates specified as
c
3
/
4
k
3
/
2
P
w
κµ
P
k
,
P
=
τ
µ
,
ε
=
(6.16)
P
y
P
κ
y
P
The water level near a rigid wall is usually assumed to have a zero gradient in the
direction normal to the boundary.
Inflow and outflow boundary conditions
As described in Section 5.1.1.2, for subcritical flow, a boundary condition is needed
at each inlet and outlet in order to derive a well-imposed solution for Eqs. (6.1)-(6.3),
while for supercritical flow, two boundary conditions should be specified at each inlet.
For the sake of simplicity, only the subcritical flow case is considered below.
