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CONTROL OF COMBUSTION-INSTABILITIES THROUGH 
VARIOUS PASSIVE DEVICES 



Abdelkader Frendi 
University of Alabama in Huntsville 
Huntsville, AL 35899 

Tom Nesman and Francisco Canabal 

NASA Marshall Space Flight Center 

Huntsville, AL 35812 



Abstract 

Results of a computational study on the effectiveness of various passive devices 
for the control of combustion instabilities are presented. An axi-symmetric combustion 
chamber is considered. The passive control devices investigated are, baffles, tfelmholtz 
resonators and quarter-waves. The results show that a Helmholtz resonator with a smooth 
orifice achieves the best control results, while a baffle is the least effective for the 
frequency tested. At high sound pressure levels, the Helmholtz resonator is less effective. 
It is also found that for a quarter wave, the smoothness of the orifice has the opposite 
effect than the Helmholtz resonator, i.e. results in less control. 



I. Introduction 

It is well known that under some operating conditions, rocket engines (using solid or 
liquid fuels) exhibit unstable modes of operation that can lead to engine malfunction and 
shutdown. The sources of these instabilities are diverse and are dependent on fuel, 
chamber geometry and various upstream sources such as pumps, valves and injection 
mechanism. It is believed that combustion-acoustic instabilities occur when the acoustic 
energy increase due to the unsteady heat release of the flame is greater than the losses of 
acoustic energy from the system [1, 2]. 



Giammar and Putnam [3] performed a comprehensive study of noise generated by 
gas- fired industrial burners and made several key observations; flow noise was 
sometimes more intense than combustion roar, which tended to ha\e a characteristic 
frequency spectrum. Turbulence was amplified by the flame. The noise power varied 
directly with combustion intensity and also with the product of pressure drop and heat 
release rate. Karchmer [4] correlated the noise emitted from a turbofan jet engine with 
that in the combustion chamber. This is important, since it quantified how much of the 
noise from an engine originates in the combustor. 

A physical interpretation of the interchange of energy between sound waves and 
unsteady heat release rates was given by Rayleigh [5] for inviscid, linear perturbations. 
Bloxidge et al [6] extended Rayleigh's criterion to describe the interaction of unsteady 
combustion with one-dimensional acoustic waves in a duct. Solutions to the mass, 
momentum and energy conservation equations in the pre- and post-flame zones were 
matched by making several assumptions about the combustion process. They concluded 
that changes in boundary conditions affect the energy balance of acoustic waves in the 
combustor. Abouseif et al [7] also solved the one -dimensional flow equations, but they 
used a one-step reaction to evaluate the unsteady heat release rate by relating it to 
temperature and velocity perturbations. Their analysis showed that oscillations arise from 
coupling between entropy waves produced at the flame and pressure waves originating 
from the nozzle. Yang and Culick [8] assumed a thin flame sheet, which is distorted by 
velocity and pressure oscillations. Conservation equations were expressed in integral 
form and solutions for the acoustic wave equations and complex frequencies were 
obtained. The imaginary part of the frequency indicated stability regions of the flame. 

Activation energy asymptotics together with a one -step reaction were used by 
Mcintosh [9] to study the effects of acoustic forcing and feedback on unsteady, one- 
dimensional flames. He found that the flame stability was altered by the upstream 
acoustic feedback. Shyy et al [10] used a high-accuracy TVD scheme to simulate 
unsteady, one-dimensional longitudinal, combustion instabilities. However, numerical 
diffusion was not completely eliminated. Recently, Prasad [11] investigated numerically 
the interactions of pressure perturbations with premixed flames. He used complex 
chemistry to study responses of pressure perturbations in one-dimensional combustors. 



His results indicated that reflected and transmitted waves differed significantly from 
incident waves. 

In the present paper, an extension of the work performed by Frendi [12] is carried- 
out. In particular, an attempt is made at understanding the onset of combustion 
instabilities as well as the use of passive devices to control them. Following the 



i-i : 



l r-i syt 



i i • _i ii 



systematic CApcnmciiiai iiivcsugauuu ui i^auuicn ei ui. iuj, wiiu siuuicu uic ucsign ui 
acoustic cavities, a numerical investigation is performed. The remainder of the paper is 
organized as follows; a brief description of the mathematical model is given in the next 
section followed by the numerical techniques used to solve it. Details of the results are 
then discussed followed by the concluding remarks. 



II. Mathematical Model 



The model used to describe this problem is based on the nonlinear Euler 
equations, which can be written as 

3U dF, 



dt dx. 



B 



(1) 



where 



U = 



P 

P v ; 
P E . 



andF. = 



pv, 

pv i v j +pd ij 

p Vi (E + p/p) 



(2) 



and E = c Y T + / 2 v t v • . The closure is obtained through the equation of state for an ideal 
gas, p - pRT . For a two-dimensional axi-symmetric model the definition of the vector B 

results from the transformation to cylindrical coordinates. If x\ is the axis of symmetry, 
the vector B can be written as 



B= 



pv 2 

P V 2 V 1 

Pv 2 v 2 
pv 2 (E + p/p) 



(3) 



III. Method of Solution 

The numerical methodology employed to solve Equation (1) can be described as a 
combination of a temporal discretization scheme, a spatial discretization scheme and, a 
discontinuity-capturing scheme. In the time discretization scheme the value of U is 
sought given its value at a previous time by means of a Taylor series expansion. The 
spatial discretization is done using the Galerkin finite element method in which the 
integrated weighted-residual is minimized. To resolve discontinuities in the flow field, a 
convective flux correction term is devised. A second order accuracy is achieved in time 
while a second order or higher accuracy is achieved in space. Details of the numerical 
scheme can be found in [14, 15]. The boundary conditions used are rigid slip wall along 
the chamber walls and characteristic boundaries at the inlet (upstream boundary) and exit 
(downstream boundary). When acoustic disturbances are introduced in the chamber, the 
inlet pressure is specified as 

p = p (l + erni((dt)) or p = p (l+eR(t)) (4) 

with e being the excitation amplitude, to = 2nf the frequency and R(t) a random rumber. 
In Equation (4), p is the chamber pressure. The other inlet quantities are specified using 
the method of characteristics. 



IV. Results and Discussions 

The numerical investigation carried-out in this paper follows the detailed 
experimental investigation performed by Laudien et al. [13]. In particular, acoustic tests 
are performed in an axi- symmetric combustion chamber having the shape shown on 
Figure 1. A loudspeaker is placed at the center of the inlet boundary and emits plane 
harmonic or random acoustic waves. As described in section III, a finite element 
technique is used. Figure 1 also shows a typical grid used. A denser grid is used near the 
acoustic source. The conditions in the chamber are those of air at T = 300 K and P = 1 
atm. 




Figure 1: Typical unstructured grid used. 



At first, the natural frequencies of the chamber are determined. To this end a 
random acoustic disturbance is introduced by the loudspeaker and the pressure 
fluctuations are monitored at several points in the chamber. Figure 2 shows the sound 
pressure level (SPL) as a function of the frequency at a point along the top chamber wall. 
Several peaks corresponding to the various chamber modes are obtained. In order to 
determine the mode shapes, one can excite the individual modes by introducing a plane, 
harmonic acoustic wave at the given frequency. Following the determination of the 
natural frequencies of the chamber, a frequency of 1880 Hz, corresponding to a chamber 
mode, is selected for extensive tests. The first such test is to obtain the chamber response 
to a high (i.e. nonlinear) and low (i.e. linear) excitation amplitude. Figures 3(a)-(b) show 
the time histories of the pressure at an observation point along the top wall. 



140 r 



120 - 



CO 

_r 

a 

CO 




100 - 



2500 
Frequency, Hz 



5000 



Figure 2: Frequency response of the combustion chamber to a random excitation. 



Figure 3(a) shows rapid growth of the pressure oscillations and a slowly varying mean 
pressure, whereas Figure 3(b) shows a slow growth of the oscillations around a zero mean 
pressure. The corresponding power spectra are shown on Figs. 4(a)-(b). For the high 
excitation amplitude case, e = 0.5 , strong harmonics are shown to exist along with the 
fundamental frequency. It is believed that the nonlinear interaction between the fundamental 
and its harmonics is responsible in part for the rapid growth of the fluctuations. This point is 
confirmed by Fig. 4(b) which shows weak harmonics and therefore a quasi- linear behavior 
and weaker growth as shown by Fig. 3(a). 



a. 

< 



* o 



-0.002 




0.002 r 



Figure 3fe Time history of the pressure at a point along the top wall (a) e = 0.5 , (b) 
e =0.001 



150 



(a) 




5000 

Frequency, Hz 



10000 



150 r 



(b) 




5000 

Frequency, Hz 



10000 



Figure 4: Sound pressure level at a point along the top wall (a)e = 0.5 , (b) e = 0.001 . 

In order to assess the] level of damping of a given mode by the chamber, the 
loudspeaker is turned on and rthen off and the pressure oscillations in the chamber are 
monitored at a given point.(^igure6^hows the damping level for two different modes 1880 
Hz (Fig. 5(a)) and 512 Hz (Fig. 5(b)). In addition to the pressure oscillations, a decay curve 
described by Xe^°" is also shown, where T| is the decay factor. The Figure shows that the 
mode corresponding to 1880 Hz decays faster than the mode corresponding to 512 Hz. 



o.ooi 5 r 



(a) 




-0.001 5 



0.02 



Q. 



-0.02 




Time 



Figure 5: Decay rate of the pressure oscillations for two different chamber modes 
corresponding to (a) 1880 Hz (b) 512 Hz. 



IV.l Baffles 

Following the preliminary studies presented above, the effect of adding a baffle in the 
radial direction of the inlet boundary, Figure 6, on the pressure oscillations in the chamber, is 
investigated. Figure 7 shows the pressure oscillations at the observation point along the top 
wall obtained with and without the baffle. The Figure shows a significant reduction of the 
oscillation amplitude of the pressure. The excitation amplitude used is£ = 0.1 . 



(a) 



(b) 




Figure 6: Sketch of the inlet boundary with a circular baffle; (a) front view, (b) side view 



0.15 r 



Q. 
< 



No Control 
. Baffle 




Figure 7: Effect of a radial baffle on the pressure oscillations in the chamber. 



10 



Figure 8 shows the corresponding power spectra. This figure answers a critical question that 
can be asked based on Figure 7, which is: where did the excitation energy go? The answer 
based on Figure 8 is: the energy is redistributed on a broader frequency spectrum. The reason 
for this redistribution is that the excitation frequency, 1880 Hz, is no- longer a fundamental 
mode of the chamber, therefore there is no resonance. 



140 r 



No Control 
Baffle 




5000 
Frequency, Hz 



10000 



Figure 8: Power spectra of the pressure oscillations in the chamber with and without a baffle. 

The effect of the baffle on the mode corresponding to the 512 Hz frequency is shown on 
Figs. 9(a)-(b). Figure 9(a) shows that after the excitation source is turned-off, Time > 30, a 
weak decay of the pressure oscillations is obtained without using the baffle. When a baffle is 
used, Fig. 9(b), the decay rate is larger as indicated by the steeper exponential curve. 
Therefore, one can conclude that even though the mode corresponding to the 512 Hz 
frequency is not suppressed by the baffle, its damping is increased. This results is in 
agreement with the experimental measurements of Laudien et al. [13]. The overall sound 
pressure level in the chamber based on the root- mean- square (rms) pressure has also dropped 
from 170 dB to 156.5 dB. 



11 



IV.2 Helmholtz Resonators 

Helmholtz resonators, known also as acoustic cavities, are very popular passive control 
devices. Their use is wide spread; from controlling instability waves to noise reduction 
devices such as iiners. A Helmholtz resonator is composed of an inlet of area S and length £ 



0X)2|- 




-0132 



Time 



0.1 i- 




(b) 



Figure 9: Time history of the chamber pressure along the top wall with the excitation source 
turned-on and then off (Time > 30), (a) without a baffle, (b) with a baffle. 



12 



followed by a cavity of volume V, as shown in Fig. 10. For such a resonator, the resonar.. 
frequency is given by 



/.-TT- 



2niv(e + A£) 



(5) 




Figure 10: Typical geometry of a Helmholtz resonator. 



where c is the speed of sound in the chamber and A£ = 0.85 J with d being the inlet 
diameter. The Helmholtz resonator used is tuned to a resonant frequency of 1880 Hz, which 
corresponds to a chamber mode. When all the corners are sharp, Figure 11 shows that the 
presence of the resonator has reduced the pressure fluctuations in the chamber significantly. 
Similar to the baffle case, the power spectrapshow the presence of more frequencies when the 
resonator is used, supporting the argument that energy has been diverted from the 
fundamental to other frequencies. The overall sound pressure level in the chamber based on 
the rms-pressure has also dropped from 170 dB to 154 dB (2.5 dBs better than the baffle). 



13 



0.15 i- 



u. 

< 



No Control 

Helmholtz Resonator 



..jHJJ^IJS 6!UkamS5ww»J fin , 




Figure 11: Effect of a Helmholtz resonator on the pressure oscillations in the combustion 

chamber. 



140 



m 

■o 

_r 

o. 

(0 



100 



No Control 
Helmholtz Resonators 




5000 
Frequency, Hz 



10000 



Figure 12: Power spectra of the pressure oscillations in the chamber with and without a 

resonator. 



14 



The effect of smoothing the various comers of the resonator on the overall sound pressure 
level in the chamber is shown on Figure 13. The figure shows that the best result is obtained 
when both inlet corners (referred to as throat corners on the Figure) are smoothed. Smoothing 
the other interior resonators corners did not achieve any additional improvement, as shown 
by the figure. Notice that "inlet corner" on the figure refers to the corner facing the 
combustion chamber. 



180 i- 




N.C. : No Control 
Sh.C. : Sharp Corners 
Sm.l.C. : Smooth Inlet Corner 
Sm.T.C. : Smooth Throat Corners 
A.C.Sm. : All Comers Smooth 



_LL 



Sh.C. Sm.l.C. Sm.T.C A.C.Sm. 



Figure 13: Effect of smoothing the resonator corners on the overall sound pressure level. 



According to the experimental results of Laudien et al. [13], the resonator becomes less 
effective at higher noise levels in the chamber. This result is confirmed by our computations, 
which show less-drop in dB- level, Figure 14, even with the best configuration of Fig. 13. 



15 



180 



m 

■o 

_T 
a. 

(0 



160 



140 



N.C. 



N.C. : No Control 
Sh.C: Sharp Comers 
Sm.C: Smooth Comers 



Sh.C. 



Sm.C. 



Figure 14: Effect of a Helmholtz resonator on the overall sound pressure level of a 

combustion chamber at high noise levels. 



The problem in designing a Helmholtz resonator is the required knowledge of the 
tuning or resonant frequency. Therefore, one question we needed to answer is how effective a 
resonator is when the design frequency is slightly off? This is of practical importance since in 
an actual combustion experiment the frequency to be controlled is not known accurately. To 
answer this question, a resonator tuned to a frequency of 1800 Hz was used to control a mode 
of frequency 1880 Hz. Figure 15 shows the overall sound pressure level in the chamber with 
an "On- Design" and an "Off- Design" resonator. The "Off- Design" resonator achieves 13.5 
dB less reduction than the "On- Design" one. This is significant since the "Off- Design" 
frequency is only 80 Hz off the design value. 



16 



180 



GO 
T3 

_r 

a. 
a) 



160 



140 



N.C. 



1 



D. 



N.C. : No Control 
D.: Design Resonator 
Off D. : Off Design 



OffD. 



Figure 15: Effect of resonator design frequency on the overall sound pressure level in the 

combustion chamber. 



I V.3 Quarter- Waves 

Another passive control device commonly used is a quarter-wave. It is somewhat 
similar to a Helmholtz resonator except for the absence of the cavity part, Figure 16. The 
resonant frequency of a quarter- wave is given by 



/o = 



4(L + A£) 



(6) 



with c being the speed of sound and A£ ~ 0.85c? with d the quarter-wave diameter. 



17 




Figure 16: Typical geometry of a quarter- wave. 

In practice, several shapes of quarter-waves are used. In this study two of these shapes will 
be investigated; an L-shaped quarter- wave and a straight quarter- wave. In addition, the effect 
of smoothing the corners will be analyzed. Figure 17 shows the time history of the chamber 
pressure with and without a tuned quarter-wave. A significant reduction in pressure 
oscillations is achieved when using a quarter-wave. Figure 18 shows the corresponding 
power spectra. Similar to the baffle and resonator, the quarter- wave effect is to detune the 



0.15 r 



t 



No Control 

Q uarter Wave 



.iii''i(!8 






M 



» 



.V' 

i'tf 



(i'iii ' " " ft"" l'» 

6 &awS2p*Wi«i 



ml 






"' iSWftw 






mm 



n:zmnn 



to 



'.'iM 
I * J" ' 



1 I 1 f 1 t 



10 



' i ■ ■ ■ i < i i i i i i i i i 



20 30 

Time 



40 



50 



Figure 17: Effect of a quarter-wave on the pressure oscillations in the combustion chamber. 



18 



140 



m 

■o 



Q. 
C/> 



100 



. No Control 
Quarter Wave 



1 1 ..jJLi. ,! 



m 



60 




5000 

Frequency, Hz 



10000 



Figure B: Power spectra of the pressure oscillations in the chamber with and without a 

quarter-wave. 



I80 r 



N.C. 




140 



Sm.C. 



Sh.C. 



N.C. : No Control 
Sh.C. : Sharp Comers 
Sm.C. : Smooth Comers 
N.I. : Narrow Inlet 




L-ShapedQ.W. 



Straight Q.W. 



Figure 19: Effect of Lshaped and Straight quarter-waves with different corner geometries 

on the overall sound pressure level in the chamber. 



19 



chamber and spread the energy to other frequencies. A comparison of an L-shaped quarter- 
wave to a straight one with various corner geometries is shown on Figure 19. For all corner 
geometries, the straight quarter-wave achieves more reduction in SPL than the L-shaped one. 
The figure also shows that smoothing the inlet corners has an adverse effect on the overall 
SPL in the chamber for both quarter-wave geometries. For the straight quarter-v»a\ 
narrow inlet resulted in a higher SPL, as expected. 



ravu. a 



Figure 20 shows a comparison of overall SPL in the combustion chamber for all the 
passive devices tested in this study. The figure shows that the Helmholtz resonator with a 
smooth inlet gives the best results. 



180 



m 

_j 160 

Q. 

CO 



140 



N.C. 



B 



N.C. : No Control 

B : Baffle 

Q.W. : Quarter Wave 

H.R. : Helmholtz Resonator 



U 



Q.W. 



H.R. 



Figure 20: Comparison of overall sound pressure level in the chamber obtained with the 

various passive devices. 

Figures 21-24 show the instantaneous pressure contours in the chamber obtained with the 
various passive devices. A different pattern can be observed for each device. 



20 




Figure 21: Instantaneous pressure contours in the combustion chamber obtained without a 

passive device. 




Figure 22: Instantaneous pressure contours in the combustion chamber obtained with a 

baffle. 



21 




Figure 23: Instantaneous pressure contours in the combustion chamber obtained with a 

Helmholtz resonator. 




Figure 24: Instantaneous pressure contours in the combustion chamber obtained with a 

quarter- wave. 



22 



Concluding Remarks 

Numerical experiments have been carried-out to study the effectiveness of various 
passive devices to control combustion instabilities. The results show that, for the given 
mode studied, a Helmholtz resonator with a smooth inlet achieves the best control results. 
However, at high sound pressure levels in the chamber, the resonator becomes less 
effective. For the case of a quarter-wave, it is found that a straight shape performs better 
than an L-shaped quarter-wave. In addition, smoothing the corners of a quarter- wave inlet 
may have an adverse effect on the overall sound pressure level in the chamber. The 
computational results presented here are in good qualitative agreement with the 
experimental measurements of Laudien etal. [13]. 



Acknowledgements 

This work was supported by a summer faculty fellowship from NASA Marshall 
Space Flight Center with Tom Nesman as the technical monitor. 



23 



References 

[1] Yang, V., and Anderson, W.E., "Liquid Rocket Engine Combustion Instability", 
Progress in Aeronautics and Astronautics, Vol. 169, Editor-in-Chief Paul Zarchan, 
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[2] "Liquid Propellant Rocket Combustion Instability", NASA SP 194, Edited by David T. 
Harrje, 1972. 

[3] Giammar, R.D., and Putnam, A.A., "Combustion Roar of Premix Burners, Singly and 
in Pairs," Combustion and Flame, Vol, 18, 1972, pp. 435-438. ' 

[4] Karchmer, A.M., "Identification and Measurement of Combustion Noise from a 
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[5] Rayleigh, J.W.S., "The Explanation of Certain Acoustical Phenomena," Nature, Vol. 
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[6] Bloxidge, G.J., Dowling, A.P., Hooper, N., and Langhorne, P.J., "Active Control of 
Reheat Buzz," A/AA Journal, Vol. 26, 1988, pp. 783-790. 

[7] Abouseif, G.E., Keklak, J.A., and Toong, T.Y., "Ramjet Rumble: The Low Frequency 
Instability Mechanism in Coaxial Dump Combustors," Combustion Science and 
Technology, Vol. 36, 1984, pp. 83-108. 

[8] Yang, V. and Culick, F.E.C., "Linear Theory of Pressure Oscillations in Liquid- 
Fueled Ramjet Engines," A/AA Paper 83-0574, 1983. 

[9] Mcintosh, A.C., "The Effect of Upstream Acoustic Forcing and Feedback on the 
Stability and Resonance of Anchored Flames," Combustion Science and 
Technology, Vol. 49, 1986, pp. 143-167. 



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[10] Shyy, W., Thakur, S., Udaykumar, H.S., "A High Accuracy Sequential Solver for 
Simulation and Active Control of Longitudinal Combustion Instability," 
Computing Systems in Engineering, Vol. 4, No. 1, 1993, pp. 27-41. 

[11] Prasad, K., "Interaction of Presure Perturbations with Premixed Flames", 
Combustion and Flames, Vol. 97, pp. 713-200, 1994. 

[12] Frendi, A., "On the Role of Acoustic Coupling on Combustion Instabilities", AIAA- 
2002-3181, Hilton Head, South Carolina, 2003. 

[13] Laudien, E., Pongratz, R., Pierro, R., and Preclik, D., "Experimental Procedure Aiding 
the Design of Acoustic Cavities", Liquid Rocket Engine Combustion Instability, edited 
by Vigor Yang and William Anderson, Progress in Astronautics and Aeronautics, Vol. 
169, pp.377-399, 1995. 

[14] Canabal, F., and Frendi, A, "A Finite Element Based Flux Correction Scheme for the 
Euler Equations", AIAA-2003-3121. 

[15] Canabal, F., "Prediction and Suppression of Ignition Overpressure", PhD Thesis, 
University of Alabama in Huntsville, 2004. 



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