Transactions on Modelling and Simulation vol 20, 1998 WIT Press, ISSN X

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1 Application of boundary element methods in modeling multidimensional flame-acoustic interactions Tim Lieuwen & Ben T. Zinn 5c/zWj ofmecaamw aw^em^ace Engmecrmg Georgia Institute of Technology, Atlanta, GA 30318, USA Abstract Detrimental combustion instabilities are often excited in high intensity combustion systems. These instabilities manifest themselves as large amplitude acoustic oscillations, which are driven by complex interactions between acoustic waves and unsteady combustion processes. To predict the onset of these instabilities, capabilities for modeling the interactions between the combustion process and the multidimensional acoustic field must be developed. Due to the complexity of the problem, previous analyses of such problems have generally assumed that the acoustic field in the vicinity of the flame is one-dimensional or quasi one-dimensional, thus producing results whose validity is uncertain. This paper re-examines this problem by investigating the effects of multi-dimensional acoustic oscillations in the nearfield of a flame upon the flames' response. This is accomplished by developing an integral formulation that is solved via boundary element techniques. Specifically, the solution domain is divided into two regions that are separated by the flame. The Helmholtz Integral is used to determine the acoustic oscillations in each region that are coupled by conservation relations across the flame "interface". The solutions of the problem show that even for compact flames the oscillations in the flame's near field are inherently multidimensional, and that results based on one-dimensional analyses may significantly affect the accuracy of predictions of the flame's response. In summary, this study shows that the multidimensional nature of the unsteady flow in the near field of the flame must be taken into consideration in studies of acoustic-flame interactions.

2 542 Boundary Elements BACKGROUND The occurrence of combustion instabilities has long been a problem in the development of combustors used in air breathing propulsion systems, rocket motors, ramjets and more recently, power generating gas turbines [1]. These instabilities manifest themselves as large amplitude, organized oscillations of the flow field that are driven by complex feedback type interactions between the acoustic field and unsteady combustion processes. Elimination of these instabilities requires an understanding of the feedback mechanism responsible for the initiation and maintenance of the oscillations. This, in turn, requires capabilities for modeling the dominant processes that are involved. However, these instability mechanisms are inherently complex because they often involve interactions between a number of different processes such as unsteady flame propagation, acoustic wave propagation, and natural or forced hydrodynamic instabilities, that lead to the formation of large scale, coherent vortical structures [1]. One of the fundamental problems facing workers in the field is developing models that accurately account for the interactions and response of the unsteady flow and combustion processes. A useful model to describe the influence and interaction of the flame front and the mean flow field was developed by earlier workers[2, 3,4]. This model treats the flame front as a discontinuity that separates the cold reactants from the hot combustion products, and matches the flow fields on either side by conservation relations, see Fig. 1. A discussion of this model can be found in Emmons [2], and some numerical and analytical results can be found in Tsien [3] or Fabri et al [4]. Extension of this model to include effects of flow unsteadiness in realistic combustor geometries was first considered by Marble and Candel [5], and Subbaiah [6], and actually incorporated into a combustion instability model by Yang and Culick [7]. These studies considered the linear response and stability of a ducted flame subjected to longitudinal acoustic disturbances where, because the frequency was below the cutoff frequency of the duct, the acoustic disturbances were assumed to be one-dimensional or quasi one-dimensional. All three of these studies found that unsteadiness in the rate of heat release could be induced by changes in the surface area of the flame, (and, thus, its heat release rate) due to flame movement in response to velocity perturbations. Similar results have been obtained more recently by Fleifel et al. [8] and Dowling [9].

3 Boundary Elements 543 Incident Wave Reflected Wave \ Transmitted Wave Cold Reactants Flame Hot Combustion Products Figure 1: A schematic of the modeled combustor configuration. This study was motivated by the observation that although the acoustic field in a duct rapidly becomes one- dimensional after encountering some scattering object (for frequencies below cutoff), it nonetheless is multidimensional in the near field of the scatterer (e.g., the flame). Since the flame responds to the local perturbations in field variables, it seems essential to resolve the acoustic nearfield in order to determine the flame's response. That is, it is not clear that the flame response can be qualitatively or quantitatively predicted by assuming a one-dimensional acoustic nearfield. The objective of this study is to attempt to develop an understanding of the acoustic nearfield of aflame and to assess the validity of the quasi one-dimensional approximations in the prior studies. The Analysis section of this paper discusses the principal assumptions made in the study and shows how the acoustic field can be described in terms of two coupled integral equations. The solution of these integral equations using the BEM is then discussed in the Numerical Implementation section. Finally, some results from the study are presented in the Results section, and a summary of significant findings and recommendations for future research are included in the Summary and Conclusions section. ANALYSIS The geometrical configuration investigated in this study is shown in Fig. 1 and reflects a simplified geometry of modern gas turbine combustors. It consists of a flame anchored on a dump plane between two constant area ducts, where premixed reactants are supplied from the left. The domain is assumed to be two dimensional and is divided into two regions

4 544 Boundary Elements that are separated by the flame. Each region is assumed to be isothermal, with cold reactants on the left and hot products on the right. Since chemical reactions are confined to the flame, isentropic conditions are assumed to persist in either region. Also, since the Mach number of the mean flow on either side of the flame is often very small, the effect of terms of O(M) or higher on acoustic wave propagation in either region will be neglected. In addition, for this initial study, the effect of flame movement in response to flow perturbations will be neglected, and thus the flame will be considered as a fixed surface in space. The additional effects of flame movement (which may be substantial) will be reported in a future publication. Finally, the effects of viscosity and other molecular transport processes are assumed to be negligible. With these assumptions, the equation describing the propagation of harmonic acoustic disturbances in either region can be shown to be [10]: VY+&V=0 (1) It should be noted that the corresponding unsteady velocity field will be irrotational. Thus, this simplified analysis cannot address such issues as the unsteady vorticity production by the flame. Boundary and matching conditions are necessary to close the problem. For this study, the walls of the combustor are assumed to be rigid. Also, since the flame surface is assumed to be stationary, it can be shown in a similar manner as in [11] that momentum and energy conservation across the flame requires that locally: PC where the subscripts 1 and 2 denote the value of the quantity on the up and downstream sides of the flame surface (see Fig. 1), respectively, and q' denotes the fluctuating combustion process heat release. For this study, the specific case where the effects of q' are and are not included will be referred to as active and passive flames, respectively. For this problem, it is convenient to use the Helmholtz integral solution of the wave equation [10]. For two dimensional domains, it can be shown [13] that the appropriate free space Green's function is given by G(x,Xs)=i7iHo(kr), where HO is the zeroth order cylindrical Hankel function. The solution for the interior wave field in each region is then given by: (2)

5 Boundary Elements 545 s It has been shown by Seybert et al [12] that for an interior problem: * (3) where r = x-x^. For smooth boundary points C(x)=2%, for points inside the domain C(x)=47i, and for points outside the fluid C(x)=0. The solutions of Eq. (3) in each region are coupled by the conservation laws that must be satisfied across the flame discontinuity, see Eq. (2). Their numerical solution is described in the following section. NUMERICAL FORMULATION To numerically solve the coupled integral equations we use the BEM. A full discussion of the method and its implementation for acoustics problems can be found in Brebbia et al [14] and Meyer et al [15]. The implementation and notation used in this section closely follows those of Brebbia et al In each region, the boundary is discretized into N elements, where the constant unknown value is taken to be in the middle of the element. Applying this procedure to Eq. (3) yields: 7=1 j rr (* ~*/)-HV ^/(^)^(^)_J! 1 s, H/ Following Brebbia et al [14], we define the "influence" coefficients: i(,r;-.-) -ds (6) iv,/// / w ~j v These quantities are readily evaluated using Gaussian quadrature. For this study, except for the case i=j, a four point scheme was found to yield

6 546 Boundary Elements satisfactory results. The case i=j requires special care because of the singularity of the Hankel function as r tends to zero. For this study, the GH term was evaluated analytically using the first order expansion of the Hankel function about kr-0; that is, H<>(kr)= l+2i/7i(log (kr/2)+y)+o(kr)^ (where y= is Eulers constant). Using this expansion, it can be shown that: G = A(l + - (log A 4- y -1) + 0(^A)') (7) n 4 where A is width of the elements. The HJJ terms are identically zero because the normal vector and the element coordinate are always perpendicular. By treating the i=j case separately as discussed here, none of the low frequency difficulties in implementing the BEM discussed by Hussain and Peat [16] were encountered. The discretized equations constitute two sets of coupled linear equations. Their solution can be determined by standard techniques, see Brebbia et al [14]. In order to assess the accuracy of the numerical implementation and to determine a reasonable number of nodes necessary to accurately resolve the field, the numerical solutions were compared with various approximate and analytical solutions. These included situations with area discontinuities to evaluate the effects of the sharp corner. It was found that errors in the local value of the acoustic pressure of less than 5 per cent were generally obtained by using kja < For more complex problems where no analytical solutions existed, the "global" accuracy of the solution was checked by comparing the net flux of acoustic energy into, reflected by, and transmitted through the flame. It was found that with kia < 0.15, the "global" errors in the solution were generally less than 0.5 per cent. RESULTS This section presents typical results obtained in this study for the following parameter values (see Fig.1): k]li=0.38 ('Case I'), 1.14 ('Case IF), k,!^ 2k,Lj,, k,lf= 0.87kiL,, T,=300 K, ^2 =2000 K. These values reflect ranges of combustor size to wavelength ratios that might be encountered in observed longitudinal instabilities. These calculations also assumed that the flame had the shape of a triangle. The primary objective of the study was to resolve the acoustic field in the vicinity of the flame since these will be the actual flow

7 Boundary Elements 547 perturbations that disturb the flame and, consequently, the rate of heat release. To obtain these results, a plane disturbance (of unit magnitude and zero phase at the flame base) was imposed on the flame from the right, where it was partially reflected and partially transmitted. Instantaneous acoustic velocity vectors and pressure isobars at four different phases of a cycle for Case I are presented in Fig. 2. The figure clearly shows the multidimensional nature of the acoustic velocity in the vicinity of the flame (while the acoustic pressure also appears to be strongly multidimensional, this is due to the scaling of the isobars). Due to the ultra low frequency oscillations in this case (i.e., LAi=0.05), both axial and transverse velocity components are essentially in phase along the length of the flame, and the fluid basically sloshes back and forth in a bulk motion. More "exotic" results are obtained at higher frequencies, such as in Case II (although the flame length is still much smaller than a wavelength), particularly if the effects of fluctuating heat release are added (see Eq. 2). Figure 3 shows a result for a case with heat addition, where it has been assumed that the heat release responds to velocity perturbations through the relationship: %,, (8) The ratio of the magnitude of the transverse and axial velocity components shown in Fig. 2, which provides a measure of the "departure" from quasi one-dimensionality, is presented in Fig. 4 for cases with and without a temperature jump at the flame surface. Observe that the two velocity components are of the same order of magnitude at the base of the flame, and that this ratio approaches zero at the downstream tip of the flame. It can be seen that both the area and temperature jump contribute to the observed two-dimensional oscillations. Examination of Figs. 2-3 shows that although the velocity field is two dimensional near the base of the flame, the acoustic field rapidly becomes one dimensional as the downstream end of the flame is approached. Evidently, this behavior is due to the large temperature jump, which significantly increases the wavelength of the oscillations, and rapidly damps all multidimensional modes. This is even more evident in the pressure field. Figure 5 shows a plot of the magnitude of the acoustic pressure along the surface of the combustor and the flame for the same conditions as in Fig. 2. It shows that the two are "close", implying a nearly one-dimensional pressure field. The multidimensionality of the pressure field is enhanced somewhat by

8 548 Boundary Elements 45 degrees 135 degrees 225 degrees Figure 2: Instantaneous acoustic velocity vectors and pressure isobars in the nearfield of a passive flame subjected to a plane disturbance for Case I (k,l,=0.38, k,l^2k,l,, k,l=0.87k,l,, T-300 K, T, =2000 K) at four phases during the cycle. 45 degrees 135 degrees 225 degrees p' -V ii. Figure 3: Instantaneous acoustic velocity vectors and pressure isobars in the nearfield of an active flame subjected to a plane disturbance for Case II (k,l,=1.14, k,l^2k,l,, k,l=0.87k,l,,t,=300 K, T; =2000 K) at four phases during the cycle.

9 Boundary Elements 549 including the fluctuating heat release model in Eq. (8) (rather than treating the flame simply as a passive surface), see Fig. 5 for a plot under the same conditions as Fig. 3. Although the heat addition increases the difference between the pressure at the combustor wall and flame, the two remain qualitatively similar. Whether this result will hold when a more accurate flame response model is used is yet to be determined and will be addressed in a future publication. Area Discontinuity and Flame(i.e., temperature jump) X^ Area Discontinuity (No Flame) X Figure 4: Distributions of the ratio of the transverse and axial velocity component over the flame surface, in the presence and absence of a temperature discontinuity for Case I With Oscillating Heat Release E O? o ^ 0.65 Passive Flame X Figure 5: Comparison of the magnitude of the acoustic pressure along the combustor wall and flame surface for Cases I (with passive flame) and II (with active flame).

10 550 Boundary Elements SUMMARY AND CONCLUSIONS This paper has demonstrated the utility of the BEM in solving problems related to acoustic-flame interactions. The results of this paper have shown that even for low frequency problems the acoustic velocity field produced by a plane incident wave is highly multidimensional in the nearfield of the flame. It has also been shown that this multidimensionality is produced by the area discontinuity at the base of the flame, by the temperature jump across the two dimensional flame, and by the presence of unsteady heat addition. In contrast to the acoustic velocity field, the corresponding acoustic pressure field in the flame nearfield appears to remain nearly one-dimensional. These results indicate that the results of past investigations of these interactions that assumed both a quasi one-dimensional pressure and velocity field should be viewed with some caution. Finally, it should be pointed out that the current study assumed that the flame remains stationary, which is contrary to experimental observations and expectations. The effect of flame oscillations is currently under study and the results will be described in a future publication. ACKNOWLEDGEMENTS This research was supported by AGTSR under Contract No SR031; Dr. Daniel B. Fant, Contract Monitor. REFERENCES [1] McManus, K.R., Poinsot, T., Candel, S.M., A Review of Active Control of Combustion Instabilities, Prog. Energy Combust. Sci., Vol. 19,1993. [2]. Emmons, H.W., Flow Discontinuities Associated with Combustion, Fundamentals of Gas Dynamics (High Speed Aerodynamics and Jet Propulsion, Vol. Ill, H.W., Emmons, Ed.), Princeton University Press, Princeton, p.584, [3] Tsien, H.S., Influence of Flame Front on the Flow Field, Journal of ^j9p/fwmgc/z^/cj, Vol. 73, pp , [4] Fabri, J., Siestrunck, R., Foure, L., On the Aerodynamic Field of Stabilized Flames, Fourth International Symposium on Combustion, The Combustion Institute, Pittsburgh, pp , 1953.

11 Boundary Elements 551 [5] Marble, F.E., Candel, S. M, An Analytical Study of the Non-Steady Behavior of Large Combustors, Seventeenth International Symposium on Combustion, The Combustion Institute, Pittsburgh pp [6] Subbaiah, M.V., Nonsteady Flame Spreading in Two Dimensional Ducts, AIAA Journal, Vol. 21, No. 11, pp , [7] Yang, V., Culick, F.E.C, Analysis of Low Frequency Combustion Instabilities in a Laboratory Ramjet Combustor, Comb. Sci and Tech., vol. 45, pp. 1-25, [8] Fleifel, M., Annaswamy, A.M., Ghoniem, Z.A., Ghoniem, A.F., Response of a Laminar Premixed Flame to Flow Oscillations: A Kinematic Model and Thermoacoustic Instability Results, Comb and Flame, 106: , [9] Dowling, A., Nonlinear Self Excited Oscillations of a Ducted Flame, J. Fluid Mech, Vol. 346, pp , [10] Peirce, A., Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, New York, [11] Lieuwen, T., Zinn, B.T., Theoretical Investigation of Combustion Instability Mechanisms in Lean Premixed Gas Turbines, AIAA paper #P&OW7, [12] Seybert, A.F., Soenarko, B., Rizzo, F.J., Shippo, D.J., An Advanced Computational Method for Radiation and Scattering of Acoustic Waves in Three Dimensions, J. Acoust. Soc. Am., 77(2), [13] Morse, P.M., Ingard, K.U., 7%gorff;W /(cowjfl, Princeton University Press, Princeton, [14] Brebbia, C.A., Silva, J.J.R., Partridge, P.W., Computational Formulation, ^ozwary ^menf A/g/Ao66 m /4co%jf/cj, (Ciskowski, R.D., Brebbia, C.A., Eds.), Computational Mechanics Publications- Boston, pp , [15] Meyer, W.L., W.A. Bell, Zinn, B.T., Boundary Integral Solutions of 3-D Acoustic Radiation Problems, J. Sound and Vibr. 59(2) pp , [16] Hussain, K.A., Peat, K.S., Boundary Element Analysis of Low Frequency Cavity Acoustical Problems, J. Sound and Vibration V 169,No. 2, 1994.