**3.1 Experimental setup**

188 Advanced Fluid Dynamics

In order to describe the resonance behaviour of combustion systems, they can be treated as single or coupled Helmholtz resonators. In combustion systems the combustion chamber, the burner plenum or other components with larger volume act as resonators. The exhaust gas pipe and the components coupling the resonator volumes together are resonator necks. In industrial combustors the identification of the components of the Helmholtz resonators is easier, in gas turbines more difficult. It is very important which components are assumed to be coupled and which are decoupled. Wrong assumptions can lead to predicting modes

The suppression of the combustion oscillations is not a universal solution. The main goal is

For the prediction of the stability of combustion systems regarding the development and maintaining of self-sustained combustion instabilities the knowledge of the periodic-nonstationary mixing and reacting behaviour of the applied flame type and a quantitative description of the resonance characteristics of the gas volumes in the combustion chamber is conclusively needed. In order to describe the periodic combustion instabilities many attempt have been made to assign the dominant frequency of oscillation to the geometry of the combustion chamber. For the description of the geometry-dependent resonance frequency of the system the equations were derived under the assumption of undamped oscillation (e.g. ¼ wave resonator, Helmholtz resonator). These models predict the resonance frequency quite accurate since the shift due to the moderate damping in the system is negligible. Such a simplified model, however, is not applicable for a quantitative prediction of the stability limit of a real combustion system. On one hand it predicts infinite amplification at the resonance frequency. On the other hand the frequency-dependent phase shift between input and output is described by a step function, hence it cannot be used for the application of a phase criterion (Rayleigh or Nyquist criterion), which is used to predict the occurrence of

A reduced physical model was developed in (Büchner, 2001), which is able to describe the resonance characteristics of combustion chambers, if their geometry satisfies the geometrical conditions of a Helmholtz resonator (Arnold & Büchner, 2003; Büchner, 2001; Lohrmann et al., 2001; Petsch et al., 2005; Russ & Büchner, 2007). The reduced physical model was derived similar to the resonance behaviour of a mass-spring-damper system, which provides a continuous transfer function of the amplification and the phase shift. First the model was developed to describe a single resonator, later it was extended to a coupled system of two resonators. For this reduced physical model scaling laws were developed based on experimental data. The influence of the amplitude of pulsation, the mean mass flow rate, the

In this model the damping in the system is expressed by an integral value. The damping factor cannot be determined by analytical solution. The accurate determination of the damping based on the 2nd Rayleigh-Stokes problem is not possible because of the complexity and non-linearity of the flow motion in the chamber and in the exhaust gas pipe. It was, however, possible to derive a scaling law for the damping in function of the gas temperature. A scaling law for the dependency of the damping on the length of the exhaust

There is a possibility to determine the integral value of the damping ratio by one measurement e.g. at the resonance frequency predicted by the undamped Helmholtz

gas pipe could be also derived but its prediction is less accurate (Büchner, 2001).

to design the combustion system not to be prone to combustion instabilities.

incorrectly or even it is impossible to predict certain modes.

pressure and heat release oscillations in real combustion systems.

temperature of the gas and the geometry were investigated.

**2.4 The reduced physical model** 

Former experimental investigations showed that the combustion chamber has specific impact on the stability of the overall system. As first approximation, if the components upstream to the combustion chamber are decoupled by the pressure loss of the coupling element (e.g. burner), the only vibratory component is the combustion chamber, and the system can be treated as a single resonator.

In Fig. 3 the sketch of the experimental setup is shown. In the experiments the transfer function of the combustion chamber was calculated from the input signal measured with the hot-wire probe 1 at the inlet of the chamber and from the output signal measured with the hot-wire probe 2 at the exit cross section of the exhaust gas pipe (Arnold & Büchner, 2003). An alternative output signal was the pressure measured with a microphone probe at the middle of the side wall in the combustion chamber (Büchner, 2001).

The model of the single Helmholtz resonator describes combustion systems sufficiently precise only in a first approximation, since real combustion systems in general have more vibratory gas volumes in addition to the combustion chamber (mixing device, air/fuel supply, burner plenum and exhaust gas system). The linking of these vibratory subsystems results in a significantly more complex vibration behaviour of the overall system compared to the single combustion chamber. To get closer to real combustion systems the model of the single Helmholtz resonator must be extended to describe more resonators coupled to each other.

For modelling a coupled system the burner plenum was added upstream to the combustion chamber. The reduced physical model was extended for the coupled system of burner and combustion chamber (Russ & Büchner, 2007). In order to prove the prediction of the model for the coupled system different geometric parameters (burner volume, resonator geometry) and operating parameters (mean mass flow rate) were varied in the experimental part. In each case the flow was non-reacting. The transfer function was calculated from the input signal (inlet of the burner plenum) and output signal (exit cross section of the exhaust gas pipe) similar to the case of the single resonator. The sketch of the experimental setup and the analogy of a mass-spring-damper system are shown in Fig. 4.

In order to excite the system at different discrete frequencies a pulsator unit was used. This unit could produce a sinusoidal component of the mass flow rate with prescribed amplitude and frequency (Büchner, 2001). For example in the case of the coupled Helmholtz-resonators in Fig. 4 the mean volume flow rate is partially pulsated by the pulsator unit. The pulsating flow passes through the burner plenum (bp), reaches the combustion chamber (cc) through the resonator neck and leaves the system at the end of the exhaust gas pipe (egp).

Stability Investigation of Combustion Chambers with LES 191

In order to compute the resonance characteristics of the system a series of LESs at discrete

In the case of the single resonator these were taken for a basic configuration corresponding to the experiments, for variation of the geometry of the resonator neck and for variation of the fluid temperature. The compressible flow in the chamber of the basic geometry was simulated for five different frequencies in the vicinity of the resonance frequency. A detailed description of the investigated cases is omitted here as it is not in the focus of this chapter

In the case of the coupled resonators one configuration was investigated. Ten LESs were calculated at different excitation frequencies, because the domain of interest is a broader frequency range than in the case of the single resonator. A detailed description of this

The main goal of the numerical investigation was to predict the damping coefficient of the system which is an important input for the reduced physical model. In order to provide an insight into the flow mechanics inside the system LES were carried out. LES is an approach to simulate turbulent flows based on resolving the unsteady large-scale motion of the fluid while the impact of the small-scale turbulence on the large scales is accounted for by a subgrid scale model. By the prediction of flows in complex geometries, where large, anisotropic vortex structures dominate, the statistical turbulence models often fail. The LES approach is for such flows more reliable and more attractive as it allows more insight into the vortex dynamics. In recent years the rapid increase of computer power has made LES accessible to a broader scientific community. This is reflected in an abundance of papers on the method

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 swirled

The LESs of this system were carried out with the in-house developed parallel flow solver called SPARC (Structured PArallel Research Code) (Magagnato, 1998). The code is based on three-dimensional block structured finite volume method and parallelized with the message

In the case of the combustors the fully compressible Navier-Stokes equations are solved. The spatial discretization is a second-order accurate central difference formulation. The temporal integration is carried out with a second-order accurate implicit dual-time stepping scheme (Zou & Xu, 2000). For the inner iterations the 5-stage Runge-Kutta scheme was used. The time step was Δ*t*=2·10-5 *s* and Δ*t*=2·10-6 *s* for the single resonator and for the coupled resonators, respectively. This was a compromise in order to resolve the turbulent scales and compute the pulsation cycles within the permitted time. The Smagorinsky-Lilly model was chosen as subgrid-scale model (Lilly, 1967). Later investigations with MILES approach and dynamic Smagorinsky model show no significant difference in the results. This proofs that the mesh was sufficiently fine in the regions which are responsible for the damping of the

flow. Therefore it is improbable that a URANS can render such flow reliably.

pulsation, thus the modelling of the SGS structures has a minor influence there.

**3.2 Numerical setup** 

**3.2.1 Numerical method** 

and its applications.

passing interface (MPI).

forcing frequencies had to be completed.

and it can be found in (Magagnato et al., 2005).

investigation can be found in (Pritz et al., 2009).

Fig. 3. The sketch of the test rig and the analogy of the mass-spring-damper system and the combustion chamber as Helmholtz resonator

Fig. 4. Coupled Helmholtz-resonators and oscillating masses connected with springs and damping elements
