Geoscience Reference
In-Depth Information
Figure 2.5
Reynolds' time-averaging procedure.
u
i
u
j
∂
∂
∂
¯
2
∂
¯
u
i
∂
+
∂(
¯
u
i
u
j
¯
)
1
ρ
1
ρ
x
i
+
µ
p
∂
u
i
¯
F
i
=
−
x
j
−
(2.43)
t
∂
x
j
∂
ρ
∂
x
j
∂
x
j
u
i
c
∂
)
−
∂
∂
¯
c
+
∂(
¯
u
i
¯
c
)
=
∂
∂
x
i
(ω
c
¯
δ
(2.44)
s
3
i
∂
t
∂
x
i
x
i
where
u
i
u
j
is the
co
rrelation between the fluctuating velocities in the
x
i
- and
x
j
-directions, and
u
i
c
is the correlation between the
fluct
uating sediment concentra-
tion and velocity in the
x
i
-direction. Physically,
u
i
u
j
−
ρ
represents the momentum
tr
ansp
ort due to turbulent motions;
it is called the turbulent or Reynolds stress.
u
i
c
is the turbulent sediment flux, representing the sediment transport due to
turbulence.
The set of equations (2.42)-(2.44) is not closed, due to the appearance of high-order
correlation terms. In the next subsections, methods are introduced briefly to close
this equation set on the levels of zero-, one-, and two-equation turbulence models.
A detailed review can be found in Rodi (1993).
−
2.3.2 Zero-equation turbulence models
Boussinesq's eddy viscosity concept is widely used to model the turbulent or Reynolds
stresses in Eq. (2.43). This concept assumes that, in analogy to the viscous stresses in
Eq. (2.36), the turbulent stresses are proportional to the mean velocity gradients:
t
∂
¯
x
j
+
∂
¯
u
i
u
j
2
3
k
u
i
u
j
=
ν
−
−
δ
(2.45)
ij
∂
∂
x
i
where
ν
t
is t
he turbulent or eddy viscosity; and
k
is the turbulent kinetic energy, defined
u
i
u
i
/
=
ν
ν
t
is not a
as
k
2. In contrast to the molecular viscosity
, the eddy viscosity
