**1. Introduction**

24 Fluid Dynamics

182 Advanced Fluid Dynamics

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[27] Williamson, C. H. K. (1996). Vortex dynamics in the cylinder wake, *Ann. Rev. Fluid Mech.*,

[25] Strang, G. & Nguyen, T. (1997). *Wavelets and Filter Banks,* Wellesley-Cambridge Press. [26] Pastoor, M., Henning, L., Noack, B.R., King, R., & Tadmor, G., (2008). Feedback shear

layer control for bluff body drag reduction, *J. Fluid Mech.*, Vol. 608:161-196.

New York.

Vol. 28:477-526.

Our primary energy consumption is supported in 81% by the combustion of fossil energy commodities (IEA, 2010). The demand on energy will grow by about 60% in the near future (Shell, 2008). The efficiency of the combustion processes is crucial for the environment and for the use of the remaining resources. At the Karlsruhe Institute of Technology the longterm project Collaborative Research Centre (CRC) 606: "Non-stationary Combustion: Transport Phenomena, Chemical Reactions, Technical Systems" was founded to investigate the basics of combustion and for the implementations relevant processes coupled to combustion (Bockhorn et al., 2003; SFB 606, 2002).

Modern combustion concepts comprise lean premixed (LP) combustion, which allows for the reduction of the pollutant emissions, in particular oxides of nitrogen (NOx) (Lefebvre, 1995). Lean premixed combustors are, however, prone to combustion instabilities with both low and high frequencies. These instabilities result in higher emission, acoustical load of the environment and even in structural damage of the system.

A subproject in CRC 606 was dedicated to investigate low frequency instabilities in combustion systems. The main goal of this subproject was to validate an analytical model, which was developed to describe the resonant characteristics of combustion systems consisting of Helmholtz resonator type components (burner plenum, combustion chamber) (Büchner, 2001). The subproject included experimental and numerical investigations as well. The goal of the numerical part was to find a reliable tool in order to predict the damping ratio of the system. The damping ratio is a very important input of the analytical model. The combination of the numerical prediction of the damping ratio and the analytical model enables the stability investigation of a system during the design phase.

In the numerical part Large Eddy Simulation (LES) was used to predict the damping ratio as previous investigations with unsteady Reynolds-averaged Navier-Stokes simulation (URANS) failed to predict the damping ratio satisfactorily (Rommel, 1995). The results of LES showed a very good agreement with the experimentally measured damping ratio. The focus of this chapter is to show results of further numerical investigations, which sheds light on a very important source of self-excited combustion instabilities, and to show how can provide LES the eigenfrequencies of a system.

In this chapter firstly a short description to combustion instabilities is given. After it the experimental and the numerical investigations of the resonant characteristics of the combustion systems will be shown briefly. In these investigations the system was excited

Stability Investigation of Combustion Chambers with LES 185

low- and high-cycle fatigue of system components (Joos, 2006; Lieuwen & Yang, 2005). The oscillation of flow parameters can increase the amplitude of flame movements. This can cause blowoff of the flame or, in worst case, a flashback of the flame into the burner plenum. There are several mechanisms suspected of leading to combustion instabilities, such as periodic inhomogeneities in the mixture fraction, pressure sensitivity of the flame speed and

The coupling of flame and acoustics can produce self-excited thermo-acoustic pulsation. The pulsation will be amplified then to the "limit cycle". Thermo-acoustic or thermal acoustic oscillations (TAO) were observed at first by Higgins in 1777 during his investigation of a "singing flame" (Higgins, 1802). The computation of self-excited thermo-acoustic oscillations began with the investigation of the Rijke-tube in (Lehmann, 1937). A short overview about the history of simulations of TAO is given in (Hantschk, 2000). It shows that most of the investigators wanted to compute oscillations excited by the flame or the system with flames excited by an external force at least. Because of the complexity of the problem many

Lord Rayleigh proposed for the first time a criterion, which, regardless of the source of the instabilities, describes the necessary condition for instabilities to occur (Rayleigh, 1878). The criterion expresses that a pressure oscillation is amplified if heat is added at a point of maximum amplitude or extracted at a point of minimum amplitude. If the opposite occurs, a pressure oscillation is damped. The mathematical representation of this criterion was first

() () 0

respectively, *t* is the time and *T* is the period of the pulsation. The condition will be satisfied for a given frequency if the phase difference between the heat release oscillation and the pressure oscillation is less than ±90°. Additionally, the amplitude of the pressure oscillation will be amplified if the losses through the damping effects are less than the energy fed into the oscillation. More appropriate forms of the Rayleigh criterion and similar criterions can

In combustion systems of highly complex shape there can be more various modes: low frequency bulk mode, transversal, tangential, radial and longitudinal modes. In such a combustion system it is almost impossible today to predict all the unstable operating points. There are more strategies in practice to suppress the combustion oscillations in the unstable

Passive or static control methods tune the resonance characteristics of the combustion system with additional devices as quarter-wave tube, Helmholtz resonators, soundabsorbing batting, orifice, ports and baffles (Putnam, 1971). Resonators can be placed in the fuel system (Richards & Robey, 2008), in the combustor (Gysling et al., 2000) or in other components. Perforates can be used at the premixer inlet (Tran et al., 2009), which is also an additional resonator to tune the resonant characteristics of the system. Instabilities can also be suppressed by means of injection of aluminium (Heidmann & Povinelli, 1967). Passive or

operating points. These can be grouped into passive and active control methods.

and *p* are the fluctuating parts of the heat release rate and the pressure,

*q t p t dt* (1)

0

*T*

the formation of large-scale turbulent structures.

computations could not predict the limit cycle.

proposed in (Putnam, 1971) as:

be found in (Poinsot & Veynant, 2005).

**2.1 Suppression of combustion-driven oscillations** 

where *q*

with a sinusoidal mass flow rate at the inlet and the system response was captured at the outlet. Contrarily in the ensuing numerical investigations there is no excitation at the inlet and the system is still pulsating. The source of this pulsation and the consequences will be discussed.

It is important to notice that in these investigations the flow is non-reacting. There is no combustion, thus no flame in the combustion chamber. Hence there is no self-excited thermo-acoustic oscillation. In the subproject of CRC 606 the investigations of the lowfrequency oscillations in the range of a few *Hz* up to several 100 *Hz* were focused on the passive parts of the system: the combustion chamber and the burner plenum. The determination of the flame resonant characteristics is the object of other works (Büchner, 2001; Giauque et al., 2005; Lohrmann et al., 2004; Lohrmann & Büchner, 2004, 2005), and also of an other subproject within the CRC 606.

It is also important to clarify here that in these investigations the ignition stability of the flame will not be concerned. The combustion instabilities mentioned here are driven by thermo-acoustic self-excited oscillations. If there is no pulsation in the combustion chamber the flame is stable. Furthermore pulse combustors designed for oscillations are also not dealt within this chapter (Reynst, 1961; Zinn, 1996).

On the other hand, if the flow in a combustion system without flame is investigated the mostly used terms to express this are "cold flow", "non-reacting flow" or "isothermal condition". The last one neglects any changes in the temperature of the gas beyond the one occurred by the heat release of the flame. This is however misleading for peoples who do not investigate flames and physically incorrect. The LES results showed temperature changes due to the pulsation nearly 100 *K* in the exhaust gas pipe, which is then in the range of 10% of the temperature changes produced by the flame.
