**2. Combustion instabilities**

It is an indispensable prerequisite for the successful implementation of advanced combustion concepts to avoid periodic combustion instabilities in combustion chambers of turbines and in industrial combustors (Büchner et al., 2000; Külsheimer et al., 1999). For the elimination of the undesirable oscillations it is important to know the mechanisms of feedback of periodic perturbations in the combustion system. If the transfer characteristics of the subsystems (in a simple case burner, flame and chamber) furthermore of the coupled subsystems are known, the oscillation disposition of the combustion system can be evaluated during the design phase for different, realistic operation conditions (desired load range, air ratio, fuel type, fuel quality and temperature).

In order to get a high density of heat release flux i.e. power density and simultaneously low NOx emission highly turbulent lean premixed or partially premixed flames are mostly used (Lefebvre, 1995). Significant property of these flames is that any disturbances in the equivalence ratio through turbulence or in the air/fuel mixture supply produce a very fast change in the heat release. Compared to axial jet flames the premixed swirl flames can significantly amplify the disturbances (Büchner & Külsheimer, 1997). The combustion process is increasingly sensitive to perturbation in the equivalence ratio under lean operating conditions.

Unsteady heat release involves pressure and velocity pulsation in the combustion chamber. These can result in thrust/torque oscillation, enhanced heat transfer and thermal stresses to combustor walls and other system components, oscillatory mechanical loads that results in 184 Advanced Fluid Dynamics

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

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

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

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

It is an indispensable prerequisite for the successful implementation of advanced combustion concepts to avoid periodic combustion instabilities in combustion chambers of turbines and in industrial combustors (Büchner et al., 2000; Külsheimer et al., 1999). For the elimination of the undesirable oscillations it is important to know the mechanisms of feedback of periodic perturbations in the combustion system. If the transfer characteristics of the subsystems (in a simple case burner, flame and chamber) furthermore of the coupled subsystems are known, the oscillation disposition of the combustion system can be evaluated during the design phase for different, realistic operation conditions (desired load

In order to get a high density of heat release flux i.e. power density and simultaneously low NOx emission highly turbulent lean premixed or partially premixed flames are mostly used (Lefebvre, 1995). Significant property of these flames is that any disturbances in the equivalence ratio through turbulence or in the air/fuel mixture supply produce a very fast change in the heat release. Compared to axial jet flames the premixed swirl flames can significantly amplify the disturbances (Büchner & Külsheimer, 1997). The combustion process is increasingly sensitive to perturbation in the equivalence ratio under lean

Unsteady heat release involves pressure and velocity pulsation in the combustion chamber. These can result in thrust/torque oscillation, enhanced heat transfer and thermal stresses to combustor walls and other system components, oscillatory mechanical loads that results in

discussed.

of an other subproject within the CRC 606.

within this chapter (Reynst, 1961; Zinn, 1996).

**2. Combustion instabilities** 

operating conditions.

of 10% of the temperature changes produced by the flame.

range, air ratio, fuel type, fuel quality and temperature).

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 formation of large-scale turbulent structures.

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 computations could not predict the limit cycle.

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 proposed in (Putnam, 1971) as:

$$\int\_{0}^{T} \tilde{\dot{q}}(t) \cdot \tilde{p}(t) dt > 0 \tag{1}$$

where *q* and *p* are the fluctuating parts of the heat release rate and the pressure, 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 be found in (Poinsot & Veynant, 2005).

#### **2.1 Suppression of combustion-driven oscillations**

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

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

Stability Investigation of Combustion Chambers with LES 187

the phase and gain relationship between pressure and heat release oscillation is a key issue

Helmholtz resonators are mostly used as passive devices for attenuations of pulsations in combustion systems. Furthermore the resonance behaviour of the combustion system can be

If a cavity is coupled to the ambient through a port (Fig. 2), the gas in this system can be forced into resonance if excited with a certain frequency. Such a geometrical configuration is named Helmholtz resonator after Hermann von Helmholtz, who investigated such devices

The mechanical counterpart of the Helmholtz resonator is a mass-spring-damper system (Fig. 2). The gas in the neck acts as the mass, the gas in the cavity acts as the spring. The identification of the damping is more difficult. There are linear and non-linear effects in the flow. Damping is provided by the bulk viscosity during the pressure-volume work, the laminar viscosity in the oscillating boundary layer in the resonator neck, the vortex shedding at the ends of the resonator neck at the inflow and outflow and the dissipation of the kinetic energy through turbulence generation. Which source is dominating in the

in the 1850s. The port is the resonator neck, the cavity is the resonator.

pulsating flow in the combustion system is discussed in (Pritz, 2010).

Fig. 2. The Helmholtz resonator and a mass-spring-damper system The eigenfrequency of the Helmholtz resonator can be predicted as:

heat ratio *γ* and the specific gas constant *R* of an ideal gas as:

with different geometries.

<sup>0</sup> 2

*L*

*c A <sup>f</sup>*

4

(2)

*x* 

. (3)

*m*

*F*

*k B* 

*VL d*

*d* 

where *c* is the speed of sound and can be calculated from the temperature *T*, the specific

*c RT* 

Furthermore in Eq. (2) *d* is the diameter, *L* is the length and *A* is the cross section area of the neck, *V* is the volume of the resonator. The second term in the parenthesis in the denominator is a length correction term, which can be different for Helmholtz resonators

to design stable combustion systems.

described if it bears analogy to this resonator.

*V T* 

**2.3 Helmholtz resonator** 

static control strategies methods are more robust and need a minimum of maintenance. Their disadvantage is that while an unstable operating point is removed, another may arise.

An overview about theory and practice of active control methods is given in (Annaswamy & Ghoniem, 2002). Active control methods can be subdivided into open-loop (Richards et al., 2007) and closed-loop design (Kim et al., 2005). Active control is achieved by a sensor in the combustion chamber, which measures frequency and phase of the combustion oscillation. The measured signal is analyzed and a proper periodic response is determined. The response is either an acoustic perturbation (Sato et al., 2007) or a modulation of the fuel injection (Guyot et al., 2008). Active control is able to suppress combustion instabilities substantially and is already in use for numerous practical applications. However, the apparatus is rather expensive and needs continuous maintenance. A failure of the control system can lead to a break down of the combustion system.

Based on the investigations of combustion instabilities (Culick, 1971; Zinn, 1970) there is also an approach to keep off unstable regimes during altering operation conditions. Online prediction of the onset of the combustion instabilities can help the operator to avoid, that the system becoming unstable (Johnson et al., 2000; Lieuwen, 2005; Yi & Gutmark, 2008). This technique is very useful if the ambient conditions vary in wide range e.g. for aircraft gas turbine. For stationary gas turbines with approximately constant ambient conditions, however, this cannot help to design the system for operation conditions, where combustion instabilities are not present.

#### **2.2 System analysis**

In order to analyse the stability of the system control theory can be used. The combustion system can be divided in subsystems as burner plenum, flame and combustion chamber (Baade, 1974; Büchner, 2001; Lenz, 1980; Priesmeier, 1987). The simplified feedback loop of these subsystems is depicted in Fig. 1. A perturbation of the pressure in the combustion chamber influences the mass flow rate at the burner outlet. This changes the heat release rate of the flame, which results in an alteration of the pressure in the combustion chamber. The transfer function of this closed loop and the subsystems can be determined by system identification furthermore the stability can be investigated by e.g. the Nyquist criterion (Deuker, 1994; Sattelmayer & Polifke, 2003a, 2003b).

Fig. 1. Feedback loop of a combustion system with mass flow rate, pressure and heat release rate signals

If the system is built from these elements, a thermoacoustic network can be modelled to predict the unstable modes (Bellucci, 2005). Here, however, some information from measurement is needed.

If the phase shift and gain of the components is known the amplification of the pulsation can be predicted by means of the Rayleigh criterion. This shows that the accurate knowledge of the phase and gain relationship between pressure and heat release oscillation is a key issue to design stable combustion systems.

#### **2.3 Helmholtz resonator**

186 Advanced Fluid Dynamics

static control strategies methods are more robust and need a minimum of maintenance. Their disadvantage is that while an unstable operating point is removed, another may arise. An overview about theory and practice of active control methods is given in (Annaswamy & Ghoniem, 2002). Active control methods can be subdivided into open-loop (Richards et al., 2007) and closed-loop design (Kim et al., 2005). Active control is achieved by a sensor in the combustion chamber, which measures frequency and phase of the combustion oscillation. The measured signal is analyzed and a proper periodic response is determined. The response is either an acoustic perturbation (Sato et al., 2007) or a modulation of the fuel injection (Guyot et al., 2008). Active control is able to suppress combustion instabilities substantially and is already in use for numerous practical applications. However, the apparatus is rather expensive and needs continuous maintenance. A failure of the control

Based on the investigations of combustion instabilities (Culick, 1971; Zinn, 1970) there is also an approach to keep off unstable regimes during altering operation conditions. Online prediction of the onset of the combustion instabilities can help the operator to avoid, that the system becoming unstable (Johnson et al., 2000; Lieuwen, 2005; Yi & Gutmark, 2008). This technique is very useful if the ambient conditions vary in wide range e.g. for aircraft gas turbine. For stationary gas turbines with approximately constant ambient conditions, however, this cannot help to design the system for operation conditions, where combustion

In order to analyse the stability of the system control theory can be used. The combustion system can be divided in subsystems as burner plenum, flame and combustion chamber (Baade, 1974; Büchner, 2001; Lenz, 1980; Priesmeier, 1987). The simplified feedback loop of these subsystems is depicted in Fig. 1. A perturbation of the pressure in the combustion chamber influences the mass flow rate at the burner outlet. This changes the heat release rate of the flame, which results in an alteration of the pressure in the combustion chamber. The transfer function of this closed loop and the subsystems can be determined by system identification furthermore the stability can be investigated by e.g. the Nyquist criterion

Fig. 1. Feedback loop of a combustion system with mass flow rate, pressure and heat release

cc *p* ( )*t*

Flame Combustion

chamber (cc)

cc *p* ( )*t*

If the system is built from these elements, a thermoacoustic network can be modelled to predict the unstable modes (Bellucci, 2005). Here, however, some information from

If the phase shift and gain of the components is known the amplification of the pulsation can be predicted by means of the Rayleigh criterion. This shows that the accurate knowledge of

system can lead to a break down of the combustion system.

(Deuker, 1994; Sattelmayer & Polifke, 2003a, 2003b).

Burner plenum (bp)

*m mt* ( ) bp *q t* ( ) *<sup>p</sup>* ( )*<sup>t</sup>* ss *<sup>p</sup>*

instabilities are not present.

**2.2 System analysis** 

rate signals

*m*

measurement is needed.

Helmholtz resonators are mostly used as passive devices for attenuations of pulsations in combustion systems. Furthermore the resonance behaviour of the combustion system can be described if it bears analogy to this resonator.

If a cavity is coupled to the ambient through a port (Fig. 2), the gas in this system can be forced into resonance if excited with a certain frequency. Such a geometrical configuration is named Helmholtz resonator after Hermann von Helmholtz, who investigated such devices in the 1850s. The port is the resonator neck, the cavity is the resonator.

The mechanical counterpart of the Helmholtz resonator is a mass-spring-damper system (Fig. 2). The gas in the neck acts as the mass, the gas in the cavity acts as the spring. The identification of the damping is more difficult. There are linear and non-linear effects in the flow. Damping is provided by the bulk viscosity during the pressure-volume work, the laminar viscosity in the oscillating boundary layer in the resonator neck, the vortex shedding at the ends of the resonator neck at the inflow and outflow and the dissipation of the kinetic energy through turbulence generation. Which source is dominating in the pulsating flow in the combustion system is discussed in (Pritz, 2010).

Fig. 2. The Helmholtz resonator and a mass-spring-damper system The eigenfrequency of the Helmholtz resonator can be predicted as:

$$f\_0 = \frac{c}{2 \cdot \pi} \cdot \sqrt{\frac{A}{V \cdot \left(L + \frac{\pi}{4} \cdot d\right)}}\tag{2}$$

where *c* is the speed of sound and can be calculated from the temperature *T*, the specific heat ratio *γ* and the specific gas constant *R* of an ideal gas as:

*c RT* . (3)

Furthermore in Eq. (2) *d* is the diameter, *L* is the length and *A* is the cross section area of the neck, *V* is the volume of the resonator. The second term in the parenthesis in the denominator is a length correction term, which can be different for Helmholtz resonators with different geometries.

Stability Investigation of Combustion Chambers with LES 189

resonator model. This is, however, feasible only if the combustion system already exists. In order to determine the value of the damping factor in the design stage numerical simulation

As mentioned in the Introduction the investigations focused on the passive parts of the system: burner plenum and combustion chamber (including exhaust gas pipe). Here two configurations will be discussed. A single combustion chamber as a single resonator, and a

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

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

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

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

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

coupled system of burner plenum and combustion chamber as coupled resonators.

**3. Resonant characteristics of combustion systems** 

middle of the side wall in the combustion chamber (Büchner, 2001).

analogy of a mass-spring-damper system are shown in Fig. 4.

should be carried out.

**3.1 Experimental setup** 

system can be treated as a single resonator.

resonators coupled to each other.

the exhaust gas pipe (egp).

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 incorrectly or even it is impossible to predict certain modes.

#### **2.4 The reduced physical model**

The suppression of the combustion oscillations is not a universal solution. The main goal is to design the combustion system not to be prone to combustion instabilities.

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 pressure and heat release oscillations in real combustion systems.

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 temperature of the gas and the geometry were investigated.

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

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 resonator model. This is, however, feasible only if the combustion system already exists. In order to determine the value of the damping factor in the design stage numerical simulation should be carried out.
