**4. Investigation without external excitation**

In the previous section the resonance characteristics of the system was measured experimentally and predicted numerically. In order to reduce the investigation on a few discrete excitation frequencies the eigenfrequency of the system must be approximated firstly. If the geometry is simple Eq. (2) can be used. In order to determine the amplification and the phase shift of the system well defined excitation had to be prescribed. Therefore a partially pulsated mass flow rate with a prescribed frequency near to the eigenfrequency was used in the numerical simulation at the inlet. In this section the simulations were carried out with a constant mass flow rate at the inlet.

The experiences of the investigations showed that the transient waves generated at the start of the computation should be decayed before the excitation with given amplitude and frequency was started. This was necessary to get the real system response at the outlet. A calculation was initialized with homogeneous distribution of each variable. This produces quite strong transient waves. The mass flow rate signal at the outlet of the exhaust gas pipe was used to monitoring the decaying of these waves. As soon as an almost constant mass flow rate was reached the computation could be continued on the second coarsest grid level. The extrapolation of the solution from the coarser on the next finer grid level produces also transient waves because of the sudden change of the shear stress at the walls. These waves are much smaller than the waves generated at the initialization but they are still considerable on the second coarsest grid level. The mass flow rate signal at the outlet of the exhaust gas pipe showed the decaying of these waves but later a certain amount of pulsation was observed and it decayed not at all. The amplitude of this pulsation was not negligible. As the mass flow rate was computed through integration over the whole cross section of the exhaust gas pipe the turbulent fluctuations were mostly filtered out.

At the description of the computational domain it was mentioned that the outlet boundary in the case of the coupled resonators had to be inclined to eliminate standing waves. These standing waves produced a dominant pulsation with large amplitude in the mass flow rate signal therefore the identification of the frequency was relative simple. In the case of the single resonator the computation was shorter so the standing waves were not amplified to a noticeable value. In the mass flow rate signal the frequency of the dominating wave was approximately at the eigenfrequency of the combustion chamber. In order to analyze this signal better the computation without excitation at the inlet was continued to get enough sample for a Fourier transformation.

The frequency spectrum plotted in Fig. 12 was computed from the mass flow rate signal on the second finest and finest mesh. For this computation the time step was increased to Δ*t*=10-4 *s*, to a relative high value to get a better resolution of the spectra in the low frequency range. The samples were taken in each time step, thus the sampling frequency was 10 *kHz* furthermore the sampling length was 32768. The peak at 39 *Hz* agrees very good 198 Advanced Fluid Dynamics

is a significant shearing (see Fig. 9). Unfortunately, in the experiments the plates were perforated. This was necessary to achieve the best velocity distribution at the outlet for the measurement with hot wire. In the simulations the wall condition was used for the plates. The resolution of the holes would yield a tremendous number of grid points. A boundary condition which can model this effect was not available. By the time the geometry data of the configuration were received, it was not possible to replace the plates any more. Probably

In the previous section the resonance characteristics of the system was measured experimentally and predicted numerically. In order to reduce the investigation on a few discrete excitation frequencies the eigenfrequency of the system must be approximated firstly. If the geometry is simple Eq. (2) can be used. In order to determine the amplification and the phase shift of the system well defined excitation had to be prescribed. Therefore a partially pulsated mass flow rate with a prescribed frequency near to the eigenfrequency was used in the numerical simulation at the inlet. In this section the simulations were

The experiences of the investigations showed that the transient waves generated at the start of the computation should be decayed before the excitation with given amplitude and frequency was started. This was necessary to get the real system response at the outlet. A calculation was initialized with homogeneous distribution of each variable. This produces quite strong transient waves. The mass flow rate signal at the outlet of the exhaust gas pipe was used to monitoring the decaying of these waves. As soon as an almost constant mass flow rate was reached the computation could be continued on the second coarsest grid level. The extrapolation of the solution from the coarser on the next finer grid level produces also transient waves because of the sudden change of the shear stress at the walls. These waves are much smaller than the waves generated at the initialization but they are still considerable on the second coarsest grid level. The mass flow rate signal at the outlet of the exhaust gas pipe showed the decaying of these waves but later a certain amount of pulsation was observed and it decayed not at all. The amplitude of this pulsation was not negligible. As the mass flow rate was computed through integration over the whole cross

section of the exhaust gas pipe the turbulent fluctuations were mostly filtered out.

At the description of the computational domain it was mentioned that the outlet boundary in the case of the coupled resonators had to be inclined to eliminate standing waves. These standing waves produced a dominant pulsation with large amplitude in the mass flow rate signal therefore the identification of the frequency was relative simple. In the case of the single resonator the computation was shorter so the standing waves were not amplified to a noticeable value. In the mass flow rate signal the frequency of the dominating wave was approximately at the eigenfrequency of the combustion chamber. In order to analyze this signal better the computation without excitation at the inlet was continued to get enough

The frequency spectrum plotted in Fig. 12 was computed from the mass flow rate signal on the second finest and finest mesh. For this computation the time step was increased to Δ*t*=10-4 *s*, to a relative high value to get a better resolution of the spectra in the low frequency range. The samples were taken in each time step, thus the sampling frequency was 10 *kHz* furthermore the sampling length was 32768. The peak at 39 *Hz* agrees very good

this difference plays the major role in the underprediction of the amplitude ratio.

**4. Investigation without external excitation** 

carried out with a constant mass flow rate at the inlet.

sample for a Fourier transformation.

with the response function of the combustion chamber in Fig. 10. Recent investigations showed that the resonance frequency can be captured already on the coarsest grid level and the signal of the solution on the second coarsest grid can already predict the resonance frequency quite accurately.

Fig. 12. Frequency spectrum of the outlet mass flow rate of the single resonator

It was shown in (Büchner, 2001) that the mass flow rate signal at the outlet and the pressure signal in the combustion chamber can be used as output signal equivalently i.e. the pulsation of the mass flow rate indicates a pulsation of the pressure in the chamber. The Fourier transform of the pressure signal measured at the middle of the side wall of the chamber gives the same result.

The mass flow rate at the inlet for this calculation was kept on a constant value. There was no external excitation in this computation and no turbulence at the inlet was described. The only possible forcing of the pulsation could arise from the turbulent motions inside the combustion chamber. The inflow into the chamber is a jet with strong shear layer which generates a broad band spectrum of turbulent fluctuations (Fig. 9). The combustion chamber then amplifies the pressure fluctuations generated by the turbulence at its eigenfrequency.

In order to investigate the effect of periodic flow instabilities further calculations with different mass flow rate at the inlet were carried out. It was changed to 200% and to 80% of the original value, respectively. The spectra of the mass flow rate of these calculations gave the same distribution in the low frequency range except the amplitude of the pulsation was changing proportional to the mean mass flow rate.

Based on these results the mass flow rate signal in the case of the coupled resonators was also investigated. In Fig. 13 the frequency spectrum of the mass flow rate signal on the second finest mesh is exhibited. The peaks at 27 *Hz* and 54 *Hz* correspond with the eigenfrequencies of the coupled system, which can be read e.g. from Fig. 11 at the phase shift angle 90° and 270°, respectively.

There are some possible mechanisms listed in the literature, which could trigger self-excited instabilities in combustion systems, but they are not sufficiently understood (Büchner, 2001; Joos, 2006; Poinsot & Veynant, 2005; Reynst, 1961). An important achievement of these simulations is that the pressure in the combustion chamber can pulsate already without any

Stability Investigation of Combustion Chambers with LES 201

In this chapter the numerical investigation of the non-reacting flow in a Helmholtz resonator-type model combustion chamber and in a coupled system of burner and combustion chamber is presented briefly. The work was a part of series of investigations to determine the stability limits of combustion systems. The resonance characteristics of the combustion systems were calculated using Large Eddy Simulation. The results are in good agreement with the experimental data and a reduced physical model, which was developed to describe the resonant behaviour of a damped Helmholtz resonator-type combustion

The solution of the fully compressible Navier-Stokes equations was essential to capture the physical response of the pulsation amplification, which is mainly the compressibility of the gas volume in the chamber. Viscous effects play a crucial role in the oscillating boundary layer in the neck of the Helmholtz resonator and, hereby, in the damping of the pulsation. The pulsation and the high shear in the resonator neck produce highly anisotropic turbulent flow. Therefore it is improbable that an URANS simulation can render such flow reliably. The investigation of the case without external excitation showed that the frequency spectrum of the mass flow rate signal at the outlet of the exhaust gas pipe provides a peak at the eigenfrequency of the combustion chamber. The only possible forcing of this pulsation was the turbulent fluctuations generated by the jet in the combustion chamber. The broadband excitation of the turbulent flow can be amplified by the flame and can produce a broadband background heat release rate oscillation as detected also in (Yi & Gutmark, 2008). If the eigenfrequency of the combustion chamber or other vibratory component is in the range of the frequencies of the energetic turbulent eddies a dominant pulsation can occur. The amplitude of this pulsation will be amplified to the limit cycle if the time lag of the flame changed so that the pressure fluctuation and the heat release fluctuation meet the Rayleigh criterion. If the turbulence is modelled statistically (URANS), it cannot excite the flow in the combustion chamber. The use of LES for the investigation of combustion

The present work was a part of the subproject A7 of the Collaborative Research Centre (CRC) 606 – "Unsteady Combustion: Transportphenomena, Chemical Reactions, Technical Systems" at the Karlsruhe Institute of Technology. The project was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft - DFG). We also acknowledge support by DFG and Open Access Publishing Fund of Karlsruhe Institute of Technology. The calculations have been performed using the HP XC4000 of the Steinbuch

Annaswamy, A.M., Ghoniem, A.F. (2002). Active Control of Combustion Instability: Theory

Baade, P. (1974). Selbsterregte Schwingungen in Gasbrennern, *Klima- + Kälteingenieur*, Nr.

Resonator-Type combustion chamber, *Proceedings of the European Combustion* 

and Practice, *IEEE Control Systems Magazine*, Vol.22, Nr.6, pp. 37-54 Arnold, G., Büchner, H. (2003). Modelling of the Transfer Function of a Helmholtz-

chamber (Büchner, 2001).

instabilities is essential.

**6. Acknowledgment** 

**7. References** 

Centre for Computing (SCC) in Karlsruhe.

4/74, Teil 6, pp. 167-176

*Meeting 2003 (ECM2003)*, Orléans, France

external excitation e.g. compressor or other incoming disturbances from ambient or even periodic flow instabilities depending on the design of the burner. Thus the flame is also pulsating. The amplitude of this pulsation will be amplified to the limit cycle if the time lag of the flame changed so that the pressure fluctuation and the heat release fluctuation meet the Rayleigh criterion.

Fig. 13. Frequency spectrum of the outlet mass flow rate of the coupled resonators

The results of the earlier investigations show that the pulsation and the high shear in the resonator neck produce highly anisotropic swirled flow. Therefore it is unlikely that a URANS simulation can render such flow reliably. Furthermore if the turbulence is modelled statistically, it cannot excite the flow in the combustion chamber. The use of LES for the investigation of combustion instabilities is essential.

For the analytical model the eigenfrequency of the system is an important input parameter. If the geometry is rather simple the undamped Helmholtz resonator model can be used. Further important achievement of the present computations is that the eigenfrequency of the system with geometry of high complexity can be predicted without an additional modal analysis. The calculation with constant mass flow rate is a preparation for the investigation with excitation at the inlet. As the amplitude of the excitation is not well defined in the former case only the latter calculation can provide the damping ratio for the analytical model.
