Environmental Engineering Reference
In-Depth Information
2.5
2
1.5
1
0.5
0
0
100
200
300
400
500
600
−
0.5
−
1
Figure 6.2
The instantaneous velocity, shown decomposed into its mean and fluctuating components (dotted line).
So, the Navier-Stokes equations become:
∂
assumptions to give the following expression for turbulent
viscosity:
u
j
∂
x
j
=
Cl
2
m
0,
(6.5)
∂
u
∂
y
µ
t
=
,
(6.8)
and
which is based on an assumed length scale,
l
m
, and velocity
scale for turbulence. Here
C
is a constant and
l
m
is derived
from experiment for each flow situation. However, a
constant length scale for turbulence in an environmental
flow is not suitable and more complex turbulence models
were developed, the most common being the
k
-
ε
model,
which is commonly the default in CFD software. Unlike
the mixing length model which assumes that the length
scale and velocity scale is constant everywhere,
k
and
ε
(
u
i
u
j
)
∂
∂ρ
u
i
u
j
∂
+
∂ρ
=−
∂
p
∂
∂
µ
∂
u
i
x
i
+
.
(6.6)
x
j
∂
x
j
∂
x
j
x
j
The last term is an extra one due to the averaging process
and it is known as the Reynolds stress. There are now
more unknowns than there are equations - the set of
equations is not 'closed' and in fluid dynamics this is
known as the 'closure problem'. It is necessary to derive
models for these Reynolds stresses.
A first attempt
a
t closing th
e eq
uations is to say that
the terms
are calculated from partial differential equations and
vary across the domain.
k
represents the kinetic energy
in the turbulent fluctuations and
∂
x
j
∂
x
j
,and
∂ρ
(
u
i
u
j
)
ε
µ
∂
u
i
represents the rate of
dissipation of
k
. A transport equation for
k
can be derived
from physical arguments but that for
∂
∂
x
j
, are similar and that
both represent a diffusion of energy - one through vis-
cosity and the other through turbulence. So a 'turbulent
viscosity' is defined as
relies more on
empirically determined arguments. Interested readers are
referred to CFD texts (Versteeg and Malalasekera, 2007)
for further details and details of variants of this standard
model such as RNG
k
-
ε
and nonlinear
k
-
ε
models. Such
texts also give details of Reynolds Stress models that
do not use the Boussinesq hypothesis (the assumption
about the approximation of the turbulent viscosity) and
ε
µ
t
. Equation 6.6 becomes:
(
,
∂ρ
u
i
u
j
∂
x
j
=−
∂
p
∂
∂
x
j
µ
+
µ
t
)
∂
u
i
∂
x
j
∂
x
i
+
(6.7)
and the problem is now one of finding an expression
for the turbulent viscosity,
µ
t
. Prandtl used various
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