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

1 INTRODUCTION
The pressure oscillations that can occur within the combustion
chambers of aeroengines are a major technical challenge to
the development of high performance and low emission propulsion
systems. The oscillations are driven by the resonant interaction
between acoustic waves and unsteady combustion. The
acoustic waves vary the air and fuel supplies and make the combustion
unsteady. In return, unsteady combustion generates yet
stronger waves. The pressure waves can become so intense that
they cause unbearable noise, vibration and even structural damage.
It is known that the combustion oscillation is a systemcoupled
phenomenon, which involves the interaction of pressure
waves, unsteady combustion, fuel/air supplies, and geometry of
combustors. In order to fully investigate the phenomenon, a combination
of theoretical analysis, computational fluid dynamics
and experimental work is needed. These techniques combine together
to provide comprehensive information to understand combustion
oscillations and to develop control strategies.
For aeroengines gas turbines, combustors with fuel spray atomizers
are particularly susceptible to a low frequency oscillation
at idle and sub-idle conditions. The frequency of this oscillation
is typically in the range 50-120 Hz and is commonly
called ‘rumble’. In recent years, a systematic program of work
has been undertaken to investigate the ‘rumble’ phenomenon. In
our study, CFD has been employed to calculate unsteady combustion
flows at idle conditions. Through harmonic forcing of
the air or fuel supplies, the main source of the unsteady heat release
has been identified [1]. Excitation by broad-band signals
gives fuller information on the ‘flame transfer function’ between
the rate of unsteady combustion and change in the air mass flow
rate [2]. In the primary zone, an increase in the inlet air mass flow
rate increases the turbulence scalar dissipation rate and hence the
rate of combustion. The increase of air mass flow at the atomizer
leads to a decrease in the inlet fuel droplet size. These smaller
droplets evaporate quickly, increasing the gaseous mixture fraction
in the recirculation zone with a recirculation time delay, and
this also influences the instantaneous rate of combustion and the
location of the burning region. When the boundary conditions
of the combustor are modified to describe the inlet air and fuel
supplies and to include a downstream nozzle, the self-excited oscillations
are established [3]. The causes of the oscillation have
been identified: pressure variations in the combustor chamber alter
the inlet air and fuel spray characteristics, and thereby change
the rate of combustion and generate ‘hot spots’, which convect
downstream. As these ‘hot spots’ accelerate through the downstream
nozzle, an upstream propagating acoustic wave is generated,
which at certain discrete frequencies reinforces the pressure
oscillation.
Although CFD provides a powerful tool to understand combustion
oscillations, there are two drawbacks to relying entirely
on this method. One is that the calculations are very timeconsuming,
and it is not feasible to investigate a wide range of
combustor geometries and flow conditions. The second is that
the calculations are carried out in the time domain. An unstable
operating condition is recognized by tracking the time evolution
until a self-excited oscillation is established. Long simulation
times are necessary to demonstrate stability. These time domain
calculations provide detailed information on the flow motion, but
there is no direct information on the damping factor. The latter
is very useful in determining how close an operating point is to a
instability boundary and in developing control strategies. These
self-excited oscillations involve an almost one-dimensional interaction,
consisting of the convection of ‘hot spots’ downstream
and upstream propagation of a plane acoustic wave. We might
therefore hope to describe it much more quickly through a onedimensional
analysis. Moreover, the stability can be investigate
the linearized equations of motion, where one is interested in
whether disturbances with time dependence e iωt grow or decay
with time.
In this paper, a one-dimensional linear stability analysis and
CFD are integrated to study combustion oscillations. With this
approach, CFD simulation and stability analysis work together to
provide comprehensive information on the mechanism of combustion
oscillation. Results from the CFD calculation provide
the flame transfer function to describe unsteady heat release rate.
Departure from ideal one dimensional flows are described by
shape factors and we describe a procedure through which they
can be determined. Combined with this information, low-order
models can work out the possible oscillation modes and their initial
growth rates. This work is a further development of our systematic
investigation into the ‘rumble’ phenomenon, which occurs
in aeroengine combustors at idle and sub-idle conditions and
is typically in the range 50-120 Hz. However, the approach developed
here can be used in more general situations for the analysis
of any combustion oscillations. In the next section, the areaaveraged
equations for a quasi-one-dimensional stability analysis
are derived through integrating the equations of motion. Three
non-dimensional shape factors are defined to account the effects
of non-uniformity in the radial direction. In section 3, based on
the integrated equations, the equations for mean flow and linear
perturbations are derived from these area-averaged equations.
The determination of the flame transfer function and shape factors
from a single forced CFD calculation are briefly introduced.
In section 4, we integrate the models from this CFD calculation
with the linear perturbation equations. The results are compared
with those from CFD calculation performed with the same inlet
boundary conditions. The sensitivities of the shape factor perturbations
are also investigated. Finally, the conclusions are presented
in section 5.
2 INTEGRATED EQUATIONS
As the ‘rumble’ problem we consider here is a low frequency
oscillation and the wavelengths are long in comparison