Environmental Engineering Reference
In-Depth Information
ˁ
0
, u
0j
s
,
q
2
s
,
q
K0
and T
0
are base-
ow mixture density,
velocity, pressure (hydrodynamic), partial density of species K and temperature,
respectively, and
q
1
, p
1
,
q
K1
and T
1
are the corresponding acoustic quantities. Re,
Pe, Sc, Da and Da
K
, are the Reynolds, Peclet, Schmidt, Damk
In these equations,
fl
hler
for species K, respectively. With these variables and non-dimensional quantities,
Eqs. (
1
)
ö
hler and Damk
ö
flow with
temperature-dependent density governing the combustion zone, and Eqs. (
6
)
(
5
) can be identi
ed as the NS equations for incompressible
fl
-
(
10
)
as the linearized Euler equations governing the acoustic zone. Explicit coupling
terms, viz., the
-
flow divergence over acoustic length scale (RHS of Eq. (
8
)) and the
acoustic Reynolds stress (ARS, RHS of Eq. (
2
)) are brought out naturally without
any ad hoc treatment.
In this formulation, the acoustic damping does not show up. This is because, the
length scale associated with the acoustic boundary layer where most of the damping
occurs is not considered. Besides, bulk of the acoustic losses could occur at the
boundaries, which are not governed by the above equations. Accounting for
acoustic damping is important to predict the long-time behaviour of the system and
eventually predict the limit cycle amplitude of the system. Towards this end, the
above formulation is extended to include acoustic damping by considering a visco-
thermal friction term (Chakravarthy et al.
2007
) in the acoustic momentum balance
equation Eq. (
7
); the resultant equation is given below,
fl
q
0
x
@
u
0
0
;
i
1
c
@
p
1
@n
i
¼
2
aq
o
x
u
0
0
;
i
;
@s
þ
ð
11
Þ
where
cient
(Chakravarthy et al.
2007
), non-dimensionalised by the duct length L. In com-
bustion
α
is the nondimensional Helmholtz
Kirchhoff wall-attenuation coef
-
fields, acoustic attenuation is increased due to the presence of water
vapour (Chakravarthy et al.
2007
). This is approximately taken into account by
increasing
fl
ow
ʱ
up to 15 times the calculated value.
flow scales span from the integral time/
length to the Kolmogorov time/length scales. There are three possibilities that can
arise in the position of the acoustic time scale relative to the turbulent spectrum. The
acoustic time scale could be shorter than the Kolmogorov time scale, or it can be
longer than the integral time scale, or it can lie in between these two scales. The first
two possibilities are relatively straightforward to handle but rare in occurrence; the
most common candidate is the last possibility, wherein the acoustic time scale falls
in between the integral and Kolmogorov time scales.
With LES, the temporal average is taken over the acoustic time scale t, which
implicitly imposes a
When the
fl
flow is turbulent, the range of
fl
filter width D
a
; the spatial grid should be
ne enough to
resolve this. The temporal average of the convective term in the
fl
ow momentum
equation results in the cross-correlation of the temporal
fl
fluctuations, which can
further be decomposed spatially as
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