Environmental Engineering Reference
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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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