Environmental Engineering Reference
In-Depth Information
In eddy viscosity turbulence models, the Reynolds stresses are linked to the velocity
gradients via the turbulent viscosity, and this relation is called the Boussinesq assumption,
where the Reynolds stress tensor in the time averaged Navier-Stokes equation is replaced by
the turbulent viscosity multiplied by the velocity gradients. The two-equation models
assume an eddy-viscosity relationship for the Reynolds stresses in Eq. (1) given by
∂
u
∂
u
2
j
'
'
=
μ
(
i
+
)
−
ρ δ
k
(3)
−
ρ
uu
t
ij
i
j
3
∂
x
∂
x
i
j
where
μ
t
is the eddy viscosity and
δ
ij
is the Kronecker delta.
The
k-ε
two equations model has become the workhorse for practical engineering modeling
of turbulent flows due to its robustness, economy and reasonable accuracy for a wide range
of flows. In the derivation of the
k-ε
model, it is assumed that the flow is fully turbulent and
that the effects of molecular viscosity are negligible. Therefore, the model is only valid for
fully turbulent flows. An improved version of the
k-ε
turbulence model, the renormalization
group (RNG)
k-ε
model, is adopted here, which is derived from the transient Navier-Stokes
equation and employs a new mathematical method called a “renormalization group”. RNG
k-ε
model provides an option to account for the effects of swirl or rotation by modifying the
turbulent viscosity appropriately
The new turbulent equations for “
k
” and “
ε
” are introduced as follows.
k
equation
∂
∂
∂
∂
k
()
ρ
k
+
(
ρ
ku
)
=
(
α μ
)
+
G
−
ρε
(4)
i
k
eff
k
∂
t
∂
x
∂
x
∂
x
i
j
j
ε
equation
2
∂
∂
∂
∂
ε
ε
ε
()
ρε
+
(
ρε
u
)
=
(
α μ
)
+
C
G
−
C
ρ
−
R
(5)
i
ε
eff
1
ε
k
2
ε
ε
∂
t
∂
x
∂
x
∂
x
k
k
i
j
j
In Eqs. (4-5),
α
k
and
α
ε
are the inverse effective Brandt numbers for the
k
and
ε
equations,
respectively;
G
k
is expressed by the mean flow velocity gradient arising from the turbulent
kinetic energy;
C
1ε
and
C
2ε
are constants with values of 1.42 and 1.68, respectively; and
μ
eff
is
the validity viscosity coefficient defined as follows:
2
k
μ==
ρ
C
(6)
t
eff
μ
ε
C
μ
is a constant equal to 0.0845.
The additional term
R
ε
is the major difference between the RNG
k-ε
model and the standard
k-ε
, which significantly improves the accuracy for a high strain rate and large degree of
crook flows. This term is given as
3
C
ρη
(1
−
η
/
η
)
2
ε
μ
0
R
=
(7)
ε
3
1
+
βη
k
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