Geoscience Reference
In-Depth Information
2.3.4 Two-equation turbulence models
Linear k-
ε
turbulence models
Because of difficulties in specifying the length scale of turbulence in complex flows, the
one-equation turbulence model described above has limitations. Thus, a differential
equation for the length scale is often added, which in conjunctionwith the velocity scale
equation constitute a two-equation turbulence model. The widely used two-equation
models include the
k
-
ε
turbulence mo
dels, which replace the length scale with the
u
i
/∂
u
i
/∂
dissipation rate
ε
=
ν∂
x
j
·
∂
x
j
and assume
k
2
ε
ν
=
c
µ
(2.54)
t
where
c
µ
is a coefficient.
The
k
equation (2.52) is rewritten as
ν
t
∂
k
+
∂
∂
)
=
∂
∂
σ
k
∂
k
x
i
(
¯
u
i
k
+
P
k
−
ε
(2.55)
∂
t
x
i
∂
x
i
ε
equation can be derived from the continuity andNavier-Stokes equations,
but it includes several terms that have little known physical meanings and have to be
modeled drastically (Rodi, 1971). The final modeled
An exact
ε
equation is expressed as
ν
∂ε
∂
t
+
∂
u
i
ε)
=
∂
∂
∂ε
∂
+
ε
k
(
t
σ
ε
x
i
(
¯
c
ε
1
P
k
−
c
ε
2
ε)
(2.56)
∂
x
i
x
i
where
σ
ε
,
c
ε
1
, and
c
ε
2
are coefficients.
Launder and Spalding (1974) suggested a set of values for the coefficients:
c
µ
=
0.09,
c
ε
1
=
1.44,
c
ε
2
=
1.92,
σ
k
=
1.0, and
σ
ε
=
1.3, as listed in Table 2.3. The
k
-
ε
model
using this set of coefficients is often called the standard
k
-
ε
turbulence model.
model overpredicts the spread rate of axisymmetric jet by
about 30% (Rodi, 1993) and underpredicts the flow reattachment length downstream
of a backward-facing step by 15-20% (Abe
et al.
, 1994). Many modifications of it
have been suggested in the literature. Several examples are given below, and more can
be found in Rodi (1993) and other references.
In the standard
k
-
However, the standard
k
-
ε
equation is modeled drastically and may have
limitations. Chen and Kim (1987) added a second time scale of the production
range of turbulence kinetic energy spectrum and modified the
ε
model, the
ε
ε
equation to consider
Table 2.3
Coefficients in linear
k
-
ε
turbulence models
k-
ε
Model
c
µ
c
ε
1
c
ε
2
σ
k
σ
ε
Standard
0.09
1.44
1.92
1.0
1.3
Non-equilibrium
0.09
1.15
+
0.25
P
k
/ε
1.90
0.8927
1.15
3
RNG
0.085
1.42
−
η(
1
−
η/η
0
)/(
1
+
βη
)
1.68
0.7179
0.7179
