Geoscience Reference
In-Depth Information
Eq. (2.49) may be derived by substituti
ng the log
distribution of flow velocity along
the flow depth and the relation
l
m
z
into Eq. (2.48). Therefore, model
(2.49) can be seen as a special case of model (2.48).
=
κ
(
1
−
z
/
h
)
2.3.3 One-equation turbulence models
In the often used one-equation turbulence model, the e
d
dy viscosity is determined using
the Kolmogorov-Prandtl expression, which adopts
√
k
as the velocity scale and reads
√
kL
m
c
µ
ν
=
(2.50)
t
where
c
µ
is a coefficient of about 0.084.
Unlike the mixing length model, the one-equation turbulence model uses a trans-
port equation to determine the turbulent energy
k
and, in turn, the fluctuating
velocity scale. The transport equation of
k
can be derived in exact form from the
continuity and Navier-Stokes equations. For high Reynolds numbers, this equation
reads
⎡
⎣
u
i
⎤
⎦
−
u
j
u
j
2
u
i
u
i
∂
k
+
∂
∂
)
=−
∂
∂
p
ρ
u
i
u
j
∂
¯
u
i
x
j
−
ν
∂
x
j
∂
x
i
(
¯
u
i
k
+
(2.51)
∂
t
x
i
∂
∂
∂
x
j
The three terms on the right-hand side of Eq. (2.51) represent the diffusion, produc-
tion, and dissipation of
k
, respectively. To close this equation, the diffusion term is
treated in analogy to Eq. (2.46), and the dissipation term is determined as
c
D
k
3
/
2
/
L
m
,
thus yielding the modeled
k
equation:
ν
c
D
k
3
/
2
L
m
∂
k
t
+
∂
)
=
∂
∂
σ
k
∂
k
t
x
i
(
¯
+
P
k
−
u
i
k
(2.52)
∂
∂
x
i
∂
x
i
u
i
u
j
∂
¯
where
P
k
is the production of turbulence by shear, defined as
P
k
=−
u
i
/∂
x
j
;
σ
k
c
µ
is a coefficient of about 1.0; and
c
D
is a coefficient, usually set as
c
D
.
For the turbulence in a state of local equilibrium, its production is equal to dissipa-
tion, and then Eq. (2.52) can be simplified as
≈
0.08
/
2
c
D
k
3
/
2
0 in the case
of shear flows. By using Eqs. (2.48) and (2.50), the following relation can be derived
(Rodi, 1993):
ν
(∂
¯
u
/∂
z
)
−
/
L
m
=
t
c
1
/
4
D
c
3
/
4
L
m
l
m
=
(2.53)
µ
Therefore,
L
m
can be determined using simple empirical formulas similar to those
for the mixing length
l
m
.
