Geoscience Reference
In-Depth Information
non-equilibrium between turbulence generation and dissipation. The modified
k
and
ε
equations are still formulated as Eqs. (2.55) and (2.56), with a functional form of
the coefficient
c
ε
1
as
c
ε
1
=
1.15
+
0.25
P
k
/ε
. The other coefficients are recalibrated
as
c
µ
=
0.09,
c
=
1.90,
σ
k
=
0.8927, and
σ
ε
=
1.15. The modified model is
ε
2
called the non-equilibrium
k
-
turbulence model, which has been tested in the case
of recirculating flows with improved performance over the standard version (Shyy
et al.
, 1997).
Yakhot
et al
. (1992) rederived the
ε
equation using the renormalized group (RNG)
theory. One new term was introduced to take into account the highly anisotropic fea-
tures of turbulence, usually associated with regions of large shear, and to modify the
viscosity accordingly. This term was claimed to improve the simulation accuracy of
the RNG
k
-
ε
turbulence model for highly strained flows. It can be included in the coef-
ficient
c
ε
1
by
c
ε
1
ε
3
1
/
2
k
=
1.42
−
η(
1
−
η/η
)/(
1
+
βη
)
. Here,
β
=
0.015,
η
=
(
2
S
ij
S
ij
)
/ε
,
0
S
ij
=
(∂
¯
u
i
/∂
x
j
+
∂
¯
u
j
/∂
x
i
)/
2, and
η
=
4.38. The other coefficients are
c
µ
=
0.085,
0
c
ε
2
=
1.68,
σ
=
0.7179, and
σ
ε
=
0.7179, as listed in Table 2.3.
k
turbulence model is restricted to high-Reynolds-number flows
and is not applicable in the viscous sublayer near a wall. Jones and Launder (1972)
proposed a low-Reynolds-number
k
-
The standard
k
-
ε
turbulence model, and later many investigators,
e.g., Chien (1982) and Abe
et al.
(1994), suggested revisions. Usually, a damping
function is introduced in the eddy viscosity equation (2.54) to mimic the direct effect of
molecular viscosity on the shear stress, while two damping functions are multiplied to
the production and destruction terms in the equation (2.56) to increase the magnitude
of
ε
(for additional dissipation) near the wall and to incorporate the low Reynolds
number effect on the decay of isotropic turbulence, respectively. Expressions of these
damping functions and their performance can be found in Srikanth and Majumdar
(1992) and Abe
et al.
(1994).
All the above
k
-
ε
ε
turbulence models based on the Boussinesq assumption are often
called linear
k
-
turbulence models. In addition to them, another frequently used
two-equation model is the
k
-
ε
turbulence model established by Wilcox (1993) by
replacing the length scale with the specific dissipation rate
ω
ω
. Because
ω
is related to
ε
ω
=
ε/(β
∗
k
β
∗
=
by
)
with
0.09, the
k
-
ω
model is similar to the standard
k
-
ε
model,
with different coefficients. Its details are left to interested readers.
Nonlinear k-
ε
turbulence models
The Boussinesq assumption, which adopts an isotropic eddy viscosity concept for all
Reynolds stresses, fails for flows with sudden changes in mean-strain rate or with
“extra” strain rates, e.g., curved flows, because the Reynolds stresses adjust to such
changes at a rate unrelated to the mean flow processes. Lumley (1970), Rodi (1976),
Saffman (1976), Wilcox and Rubesin (1980), and Speziale (1987) derivedmore general
relations for the Reynolds stresses. For example, the relation of Speziale (1987) reads
S
ik
S
kj
ij
k
3
ε
τ
2
3
k
1
3
S
mk
S
km
ij
ρ
=
4
C
D
c
2
2
ν
t
S
ij
−
δ
+
−
δ
ij
µ
2
S
ij
ij
k
3
ε
1
3
S
kk
δ
4
C
E
c
2
+
−
(2.57)
µ
2
