The calculation and modelling of combustion instabilities has a history reaching back to the mid of the last century, when this problem became important for the development of rocket engines. A large variety of methods has been developed since, mainly focusing on the representation of esentially one-dimensional thermoacoustic problems. So far only little progress has been made in modelling annular combustors typical of modern gas turbines.
One approach is to apply low-order modelling techniques, which basically take advantage of a modal represenation of the thermoacoustic problem in the frequency domain. For further information have a look at our low-order modelling activities. The backdraw of such methods is that they depend on assumptions of the coupling of multiple flames with multi-dimensional acoustics, and thus are restricted to more or less simple geometries.
It is therefore appealing to simulate combustion instabilities in an annular combustor in the time domain. If a description of the unsteady heat release in terms of the acoustic quantitities is available, no further assumptions about the modal coupling have to be made.
Time domain simulation
The idea of our approach is to describe the combustor acoustics as complex as necessary to be able to model realistic geometries, yet at the same time simple enough to keep the computational requirements acceptably low. This can be achieved employing the linearised conservation equations for the fluctuations of mass, momentum, and energy. The unsteady heat release in each individual flame then appears as a source term, related to the acoustic field by a flame model. This flame model can be obtained from experiments, theory, or CFD. The resulting system of equations is solved in the time domain, starting from a random initial perturbation. The evolution of these perturbations yields the desired information about the susceptibility of the combustor to self-excited oscillations.
To demonstrate the feasibility of our approach we use a simple model of an annular combustor as shown in the following figure:
An externally fully premixed gas-air-mixture enters through the inlet (1), which is followed by a supply manifold (2), the burners (3) and the combustion chamber (4), before the exhaust gas exits through the outlet (5). In this simple model, both the supply section and the combustor are represented by annuli, while the burners are straight ducts connecting these two parts. The mean temperature (and thus density) is assumed to be piecewise constant in these sections.
Neglecting entropy waves and assuming uniform mean flow, the acoustics can be described by a linear wave equation. For the flame model we employ a typical time lag approach, which by additional means also accounts for damping of higher frequencies and especially saturation effects. The latter introduces a non-linearity. Together with loss mechanisms in the system like sound transmission at the boundaries, this lets unstable oscillations grow into a limit cycle of finite amplitude.
The aforementioned wave equation is solved numerically with a finite element code. The FE mesh corresponding to our model is shown in the following figure:
Here you can also see an example of the random distribution of the pressure fluctuations used as the inital condition (the pressure fluctuations are normalized with the characteristic impedance and the velocity of sound in the supply). If the parameters of the model (especially the time lag) are chosen suitably, a self-excited oscillation is induced, as can be observed in the following movie (the pressure fluctuations here are normalized with their maximum value):
The initially random perturbations turn into a periodic oscillation of increasing amplitude, until firstly the heat release rate and subsequently the velocity and pressure fluctuations grow into a limit cycle. This can also be seen if you look at the time evolution of the burner exit velocity fluctuations at a specific burner (normalized with the mean burner exit velocity, time normalized with the characterstic time derived from the velocity of sound in the supply and the mean combustor diameter):
If the model parameters are chosen differently, also the stable case can be observed:
Here the initial perturbations decay exponentially and ultimately vanish. Nevertheless, the initially random fluctuations undergo a transition to an ordered oscillation, so that with increasing time a least stable mode becomes prominent.
Also, for different model parameters (time lags) different mode types can be observed. For our model combustor, in addition to the axial mode seen in the movie above, a combined circumferential/axial mode can become unstable. The following figure shows this mode (normalized magnitude of pressure fluctuations) at different phase angles during one period of the limit cycle:
You can also have a look at the movie showing this mode (play the movie as a loop if your player has this option):
The simulations of the model combustor have shown that our approach for a time domain simulation of combustion instabilities is feasible to assess the thermoacoustic behaviour of annular combustors. The method is especially attractive for its ability to model arbitrary geometries in principle. We believe that the technique can bridge the gap between direct numerical simulations (CFD), which are still computationally out of reach for annular combustors, and low-order models, which rely on a correct description of the modal coupling. The task of the ongoing work is to develop the time domain simulation further to a real design tool.
Our technique has to be understood as a hybrid approach, combining the numerical simulation of the acoustics with information about the flame behavior given in terms of a heat release model. Correct modelling therefore closely relys on a sound knowledge about coupling between the flames and the acoustic field. This aspect is part of several other projects at our institute:
For further informations have a look at our paper
C. Pankiewitz and T. Sattelmayer, Time domain simulation of combustion instabilities in annular combustors, ASME Journal of Engineering for Gas Turbines and Power, 2003, in press. Originally presented as ASME Paper No. GT-2002-3063 at ASME TURBO EXPO 2002, June 3-6, 2002, Amsterdam, The Netherlands