integration of cfd and low-order models for combustion ... - IWR

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momentum flux and energy flux can be written as
d ˆ ©
m
dx
d ˆ f
dx
d
ʤ
dx
iωH ˆ ©
ρ*¦ (31)
¤
iω ˆ ©
m ¤
q+ ˆ
ρ ¯
c v
ˆ
© ¡
p
iωH
© ¢
ρ ˆ
T ˆ
¯ © ¡
dH
dx (32)
¦ ¡
c v T¯
1
2 ¯ 2 u ¯F£
u ˆ ¯F 1 u
2 ¯ 2 ˆF£#§ (33)
¡ u
The unsteady heat release rate is the main driving force of
combustion instability, and when predicting combustion oscillations,
it is important to describe the coupling between this rate of
heat release and the flow. Many models for flame transfer functions
have been described in the literature [4, 5]. In our recent
work, a method for calculating flame transfer functions with CFD
has been developed. Here, CFD was used to calculate the unsteady
flow in the combustor. Through calculations of the forced
unsteady combustion resulting from a specified time-dependent
variation in the air supply, we were able to obtain information
on the transfer function between rate of heat release and the air
flow rate through the atomizer. For one-dimensional analysis, the
heat release rate is obtained by integration across the combustior
section. The ARX model is used in the transfer function calculation
[6]. This means the transfer function at axial position x is
written in the form of an IIR filter, with additional noise, i.e.
q
x¦ nT £¥¤
1
¢ I,
∑ a i x£ m a x¦
0 i-
K
∑
¢
b k
x£
1 k-
q
n i N£ T
n
k£ T
ε ¢ nT £§ (34)
The error ε ¢ t£ is assumed to be uncorrelated to q ¢ x¦ t£ and independent
of frequency. The input signal we used in this work
is the sum of sinusoidal signals, before normalization it can be
written as
¢
m a £¤ nT
K
∑
k- 1
sin ¢ π ¢ n 1£ ω k
t ¡ 2πw ¢ k££¦
and n ¤ 1 . .. N ¦ (35)
where K is the number of sinusoids which are equally spread
over the passband. ω 1
and ω 2
are the lower and upper limits of
¡
the passband, and ω k
¤ ω 1
k ω 2
ω 1
K¦ k ¤ 1 . . . K. w ¢ k£
£¨
describes the phase of the kth sinusoid at ¢ t 0 and is a ¤
random
number chosen from a uniform distribution on the interval
¢ ¢
. After the 0¦ 1 a i
x£ and b k
are obtained, the x£ frequency domain
transfer function of the flame model can be written as
H ¢ ω ¦ x£¤
q ˆ
ω ¦ x£
ˆm a
¢
ω£
1 ¢ I,
∑ a i
x£ z i , i- 0
K ¢
1 ∑ b k
x£ z k , § (36)
1 k-
The integrated mean heat release rate is shown in Fig. 6. The
flame transfer functions at 50 Hz, calculated with I¦ K ¤ 10 and
I¦ K ¤ 30 are shown in Fig. 7. The transfer functions of the shape
factors can be calculated with the same method.
4 RESULTS AND DISCUSSION
The mean values of mass, momentum and energy fluxes can
be calculated through integration of Eqs. (17)-(19) with appropriate
inlet boundary conditions. The results are shown in Figs. 8-
10 are with inlet boundary conditions derived from the CFD results
to aid comparison in this paper. In these figures, three steps
are clearly demonstrated, which correspond to the contributions
from the primary and secondary air injections. In current study,
the source terms S m , S f
and S e are kept as constants in both the
CFD and in the linear analysis. The mean values of flow variables
can be obtained according to Eqs. (20)-(23). The mean values
of temperature and velocity are shown in Figs 11 and 12, respectively.
From Figs. 8-12, we can see that the mean flow results
from one-dimensional analysis are in good agreement with those
from CFD calculations. From Fig. 11, it can be seen that the integrated
temperature is much lower than that within combustion
zone. This is because, at this idle condition, the temperature of
most of the air within combustor is relatively low, although the
local temperature can be as high as 2000 K. For the integrated
velocity, as shown in Fig. 12, three steps due to additional air
injected are clearly demonstrated. The decrease following each
step in mean velocity is the subsequence of the relatively high
density of the cool flow in the recirculation induced by the flow
injection.
The aim of this study is to develop a method whereby a linear
stability analysis can provide a useful tool to predict ‘rumble’.
Our first calculation involved keeping the shape factors
constant (independent of time) and using the flame transfer functions
identified from CFD calculation. In the one-dimensional
analysis, the flame transfer function ˆ describes the relationship
between the unsteady heat release rate with inlet oscilla-
q
tions. We found that this approach led to one-dimensional disturbances
which were significantly different from the area-averaged
CFD results. Therefore, it is necessary to investigate the sensitivities
of the results to the time dependence of the three nondimensional
shape factors. In Eq. (11), the non-dimensional
shape factor J 1 describes the effects of oscillation on the shape