**3.2.2 Computational domain and boundary conditions**

If the flow in the combustion chamber and the resonator neck has to be simulated (grey area in Fig. 5) attention should be paid to some difficulties by the definition of the boundary conditions.

Fig. 5. Sketch of the computational domain and boundary conditions of the single resonator

Even though the geometry of the chamber is axisymmetric no symmetry or periodic condition could be used because the vortices in the flow are three-dimensional and they are mostly on the symmetry axis of the chamber. In the present simulations an O-type grid is used to avoid singularity at the symmetry axis.

At the inflow boundary the fluctuation components should be prescribed for a LES. Furthermore, the boundary must not produce unphysical reflections, if the pressure fluctuations, which move in the chamber back and forth, go through the inlet. A conventional boundary condition can reflect up to 60% of the incident waves back into the flow area. One can avoid these reflections only by the use of a non-reflecting boundary condition. If the inlet would be set at the boundary of the grey area, this problem can be solved hardly. In the experimental investigation a nozzle was used at the inflow into the chamber. The pressure drop of the nozzle ensures that the gas volume in the test rig components upstream of the combustion chamber does not affect the pulsation response of the resonator. It was decided to use this nozzle in the computation also. Although the additional volume of the nozzle increases the number of computing cells, a non-reflecting boundary condition is no more necessary. In addition, the fluctuation components at the inlet can be neglected, since the nozzle decreases strongly the turbulence level downstream.

192 Advanced Fluid Dynamics

The Full Multigrid (FMG) method is used with four grid levels to achieve faster the statistically stationary state. The FMG method implies grid sequencing and a convergence acceleration technique. The number of cells on a grid level is eight time less then on the next

If the flow in the combustion chamber and the resonator neck has to be simulated (grey area in Fig. 5) attention should be paid to some difficulties by the definition of the boundary

Wall

*d*cc

Fig. 5. Sketch of the computational domain and boundary conditions of the single resonator Even though the geometry of the chamber is axisymmetric no symmetry or periodic condition could be used because the vortices in the flow are three-dimensional and they are mostly on the symmetry axis of the chamber. In the present simulations an O-type grid is

*d*egp

20·*d*egp

*l*egp

Ambient pressure

28·*d*egp

At the inflow boundary the fluctuation components should be prescribed for a LES. Furthermore, the boundary must not produce unphysical reflections, if the pressure fluctuations, which move in the chamber back and forth, go through the inlet. A conventional boundary condition can reflect up to 60% of the incident waves back into the flow area. One can avoid these reflections only by the use of a non-reflecting boundary condition. If the inlet would be set at the boundary of the grey area, this problem can be solved hardly. In the experimental investigation a nozzle was used at the inflow into the chamber. The pressure drop of the nozzle ensures that the gas volume in the test rig components upstream of the combustion chamber does not affect the pulsation response of the resonator. It was decided to use this nozzle in the computation also. Although the additional volume of the nozzle increases the number of computing cells, a non-reflecting boundary condition is no more necessary. In addition, the fluctuation components at the inlet can be neglected, since the nozzle decreases strongly the turbulence level downstream.

**3.2.2 Computational domain and boundary conditions** 

*l*cc

used to avoid singularity at the symmetry axis.

finer grid level.

conditions.

Pulsating mass flow rate inlet

At the inlet a partially pulsated mass flow rate was prescribed. The rate of pulsation was set to 25%.

The definition of the outflow conditions at the end of the exhaust pipe is particularly difficult. The resolved eddies can produce a local backflow in this cross section occasionally. In particular, by excitation frequencies in the proximity of the resonant frequency there is a temporal backflow through the whole cross section, which has been observed by the experimental investigations as well.

Fig. 6. Third finest mesh extracted to the symmetry plane (distortions were caused by the extraction in Tecplot)

The change of the direction of the flow changes the mathematical character of the set of equations. For compressible subsonic flow four boundary values must be given at the inlet and one must be extrapolated from the flow area. At the outlet one must give one boundary value and extrapolate four others. Since these values are a function of the space and time, their determination from the measurement is impossible. Further the reflection of the waves must be avoided also at the outlet. For these reasons the outflow boundary was set not at the end of the exhaust gas pipe, but in the far field. In order to damp the waves in direction to the outlet boundary mesh stretching was used.

At the solid surfaces the no-slip boundary condition and an adiabatic wall were imposed. For the first grid point *y*+<1 was obtained, the turbulence effect of the wall was modelled with the van Driest type damping function. The geometry of the computational domain and the boundary conditions are shown in Fig. 5. The entire computational domain contains about 4.3·106 grid points in 111 blocks. A coarsened mesh is shown in Fig. 6.

The definition of the computational domain and the boundary conditions in the case of the coupled resonators were very similar. The geometry of the configuration chosen for the numerical investigation of the coupled resonators is illustrated in Fig. 7. The observation windows (for operations with flame) and the inserted baffle plates increased the complexity of the geometry and hence the generation of the mesh significantly. There were baffle plates placed in the burner plenum and in the combustion chamber to avoid the jet of the nozzle and of the resonator neck to flow directly through the system, furthermore to achieve a

Stability Investigation of Combustion Chambers with LES 195

Fig. 8. The computational domain with block structure in the symmetry plane of the coupled

The aim of the investigations of the single resonator was to identify the main damping mechanisms and estimate their effect on the stability of the system. In order to get an impression about the flow in the resonators iso-surfaces of the *Q*-criterion are plotted in Fig.

In this section the resonance characteristics of the combustion chamber obtained from experiments and computations are compared by means of the amplitude and phase transfer

> ˆ ˆ *out in*

The amplitude ratios and phase shift were identified in the numerical simulations if the

In Fig. 10 experimental data sets with the analytical model and the results of the computation are exhibited. In one case of the experiments the exhaust pipe was manufactured from a turned steel tube, in the other case the tube was polished. The LES data compare more favourable with the experimental data of polished tube, because the wall in the simulation was aerodynamically smooth, just like the polished resonator neck. The computation predicts the damping factor quite well; the deviation is about 7%. If the results of the measurement of the turned steel tube are compared with the simulation, the

(4)

*<sup>m</sup> <sup>A</sup> m*

9. A detailed investigation of the pulsating flow is shown in (Pritz, 2010).

functions. The amplitude ratio of the mass flow rates is defined as:

resonators

**3.3 Comparison of the results** 

cycle limit was reached.

deviation is about 40%.

homogeneous distribution of the velocity in the cross-section of the measuring point at the end of the exhaust gas pipe.

Fig. 7. Geometry of the test rig (left) and the 3D block-structure of the mesh (right)

The outlet boundary had to be modified somewhat compared to the case of the single resonator. The size of this outflow region is 50·*degp* in axial direction and 40*degp* in radial direction. At the outlet surface at *x*=5 *m* the static pressure outlet condition is used and the surface is inclined based on the observation explained next (Fig. 8). In order to obtain a statistically steady solution before applying the excitation at the inlet a long time calculation on the multigrid level 4 (coarsest mesh) and 3 was carried out. The entropy waves generated by the transient of the initialization must be advected through the burner plenum and the combustion chamber and finally out of the system. This needed a relative long time as the convection velocity behind the baffle plates is quite small. After the acoustic waves generated also by the transient of the initialization were decayed, it was detected, that acoustic waves of a discrete frequency were amplified to extreme high amplitudes. The wave length coincided width the length of the computational domain. After the outlet surface was slanted these standing waves decayed.

For the distribution of the control volumes a very important aspect was to apply the findings of the investigations of the single resonator. Thus much more computational cells were arranged in the regions of the resonator neck and of the exhaust gas pipe, respectively, and in this case around the baffle plates. The final version of the mesh consists of approx. 27·106 control volumes distributed among 612 blocks.

194 Advanced Fluid Dynamics

homogeneous distribution of the velocity in the cross-section of the measuring point at the

Fig. 7. Geometry of the test rig (left) and the 3D block-structure of the mesh (right)

surface was slanted these standing waves decayed.

27·106 control volumes distributed among 612 blocks.

The outlet boundary had to be modified somewhat compared to the case of the single resonator. The size of this outflow region is 50·*degp* in axial direction and 40*degp* in radial direction. At the outlet surface at *x*=5 *m* the static pressure outlet condition is used and the surface is inclined based on the observation explained next (Fig. 8). In order to obtain a statistically steady solution before applying the excitation at the inlet a long time calculation on the multigrid level 4 (coarsest mesh) and 3 was carried out. The entropy waves generated by the transient of the initialization must be advected through the burner plenum and the combustion chamber and finally out of the system. This needed a relative long time as the convection velocity behind the baffle plates is quite small. After the acoustic waves generated also by the transient of the initialization were decayed, it was detected, that acoustic waves of a discrete frequency were amplified to extreme high amplitudes. The wave length coincided width the length of the computational domain. After the outlet

For the distribution of the control volumes a very important aspect was to apply the findings of the investigations of the single resonator. Thus much more computational cells were arranged in the regions of the resonator neck and of the exhaust gas pipe, respectively, and in this case around the baffle plates. The final version of the mesh consists of approx.

end of the exhaust gas pipe.

Fig. 8. The computational domain with block structure in the symmetry plane of the coupled resonators

#### **3.3 Comparison of the results**

The aim of the investigations of the single resonator was to identify the main damping mechanisms and estimate their effect on the stability of the system. In order to get an impression about the flow in the resonators iso-surfaces of the *Q*-criterion are plotted in Fig. 9. A detailed investigation of the pulsating flow is shown in (Pritz, 2010).

In this section the resonance characteristics of the combustion chamber obtained from experiments and computations are compared by means of the amplitude and phase transfer functions. The amplitude ratio of the mass flow rates is defined as:

$$A = \frac{\hat{\dot{m}}\_{out}}{\hat{\dot{m}}\_{in}} \tag{4}$$

The amplitude ratios and phase shift were identified in the numerical simulations if the cycle limit was reached.

In Fig. 10 experimental data sets with the analytical model and the results of the computation are exhibited. In one case of the experiments the exhaust pipe was manufactured from a turned steel tube, in the other case the tube was polished. The LES data compare more favourable with the experimental data of polished tube, because the wall in the simulation was aerodynamically smooth, just like the polished resonator neck. The computation predicts the damping factor quite well; the deviation is about 7%. If the results of the measurement of the turned steel tube are compared with the simulation, the deviation is about 40%.

Stability Investigation of Combustion Chambers with LES 197

The results of the coupled resonators on different grid levels are plotted in Fig. 11. The difference in the resonance characteristics on the finest and second finest grid is negligible. It was tested only at the lower resonant frequency, at the highest amplitude ratio, because the calculation on the finest mesh was very time consuming. The higher is the amplitude ratio the higher are the demands on the mesh. This result shows that the flow phenomena, which influence the damping, are adequate resolved on the second finest mesh. It is important to take into consideration that the mesh was optimized on the results of the investigation of the

Fig. 11. Amplitude response (top) and phase shift function (bottom) of the coupled

The plotted results in Fig. 11 show generally a very good prediction of the resonance frequencies and of the phase shift, respectively. In the gain, however, there is a discrepancy of approx. 20% in the prediction of the amplitude ratio at the highest peak, at *f*ex=28 *Hz*. It was mentioned at the experimental setup that baffle plates were implemented in the burner plenum and in the combustion chamber. On these plates the flow is strongly deflected, there

single resonator.

resonators

Fig. 9. Flow pattern in the resonators: iso-surfaces of the *Q*-criterion at 5·104 *s*-2 in the single resonator (top) and at 104 *s*-2 in the coupled resonators (bottom)

Fig. 10. Amplitude response (left) and phase transfer function (right) of the single resonator

196 Advanced Fluid Dynamics

Fig. 9. Flow pattern in the resonators: iso-surfaces of the *Q*-criterion at 5·104 *s*-2 in the single




**Phase shift angle [ ° ]**

Fig. 10. Amplitude response (left) and phase transfer function (right) of the single resonator


0

**Phase shift**

20 25 30 35 40 45 50 55 60 **Frequency [Hz]**

resonator (top) and at 104 *s*-2 in the coupled resonators (bottom)

Exp. polished wall Model, D=0.055

Exp. not polished wall Model, D=0.0816

LES

**Ratio of amplitude**

20 25 30 35 40 45 50 55 60 **Frequency [Hz]**

**A [-]**

The results of the coupled resonators on different grid levels are plotted in Fig. 11. The difference in the resonance characteristics on the finest and second finest grid is negligible. It was tested only at the lower resonant frequency, at the highest amplitude ratio, because the calculation on the finest mesh was very time consuming. The higher is the amplitude ratio the higher are the demands on the mesh. This result shows that the flow phenomena, which influence the damping, are adequate resolved on the second finest mesh. It is important to take into consideration that the mesh was optimized on the results of the investigation of the single resonator.

Fig. 11. Amplitude response (top) and phase shift function (bottom) of the coupled resonators

The plotted results in Fig. 11 show generally a very good prediction of the resonance frequencies and of the phase shift, respectively. In the gain, however, there is a discrepancy of approx. 20% in the prediction of the amplitude ratio at the highest peak, at *f*ex=28 *Hz*. It was mentioned at the experimental setup that baffle plates were implemented in the burner plenum and in the combustion chamber. On these plates the flow is strongly deflected, there

Stability Investigation of Combustion Chambers with LES 199

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

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

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

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

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

frequency quite accurately.

chamber gives the same result.

changing proportional to the mean mass flow rate.

shift angle 90° and 270°, respectively.

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