Environmental Engineering Reference
In-Depth Information
2
3
q
k
r
v
2
3
l
t
ðr
v
Þ
2
2
G
k
¼
l
t
f
c
S
ð
21
Þ
2
3
ðr
v
Þ
2
3
xr
v
2
2
G
x
¼
qc
½
ð
S
Þ
ð
22
Þ
D
ˉ
in Eq. (
20
) is a cross-derivative term de
ned in Eq. (
23
)
F
l
Þqr
x
2
1
D
x
¼
2
ð
1
x
r
k
rx
ð
23
Þ
4.4 Reynolds Stress Model
The Reynolds stresses model (RSM) was proposed by Launder et al. (
2006
) and is a
seven-equation model solving six-component symmetric Reynolds stress tensor and
turbulent dissipation equation. In two-dimensional computation, only four of
Reynolds stress components need to be evaluated; therefore, the total number of
equation is reduced to
five. RSM can be expressed as in Eq. (
24
).
@
t
ðq
u
i
u
j
Þþ
@
@
x
k
ðq
u
k
u
i
u
j
Þ
¼
@
@
x
k
½
q
u
i
u
j
u
0
k
þ
p
0
ðd
kj
u
i
þ d
ik
u
j
Þ
ð
ð
I
Þ
ð
II
Þ
Þ
þ
@
x
k
½
l
@
x
k
ð
u
i
u
j
Þ qð
u
i
u
0
k
@
u
j
III
@
x
k
þ
u
j
u
0
k
@
u
i
@
x
k
Þ
ð
Þ
qbð
g
i
u
j
h þ
g
j
u
i
hÞþ
p
0
ð
@
u
i
IV)
ð
V
@
u
j
@
u
j
@
x
k
2
l
@
u
i
@
x
k
@
x
j
þ
@
x
i
Þ
ð
VI
Þ
ð
VII
Þ
ð
VIII
Þ
ð
24
Þ
ˁ
In Eq. (
24
), the term (I) is the local time derivative term with
as the density and
u
0
as the turbulent
fluctuating velocity component in RSM; the term (II) is the
convection term; term (III) is the turbulent diffusion term; term (IV) is the molecular
diffusion term with
fl
ʲ
as the viscosity of
fl
fluid; term (V) is the stress production term;
term (VI) is the buoyancy term with
ʲ
as the thermal expansion coef
cient, g as the
acceleration direction and
ʸ
as the temperature; term (VII) is the pressure strain term
of turbulence pressure
fl
fluctuation; and term (VIII) is the dissipation term.
5 Simulation Setup
The simulation modeling setting is shown in Table
1
. The initial and boundary
condition are shown in Table
2
.
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