Application of Method of Lines on the Bifurcation Analysis of a Horizontal Rijke Tube

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1 th ASPACC July 9, 5 Beijing, China Application o Method o Lines on the Biurcation Analysis o a Horizontal Rijke Tube Xiaochuan Yang, Ali Turan, Adel Nasser, Shenghui Lei School o Mechanical, Aerospace and Civil Engineering, The University o Manchester, P.O. Box 88, Manchester M6 QD, UK Thermal Management Research Group, Eicient Energy Transer (ηet), Bell Labs, Alcatel-Lucent, Dublin 5, Ireland Abstract In this study, Method o Lines is implemented or the biurcation analysis o a horizontal Rijke tube. Dierent rom the conventional approach by the Galerkin method, a dynamic system is naturally built up by discretizing the acoustic momentum and energy equations incorporating appropriate boundary conditions using a inite dierence method. Additionally, a modiied orm o King s law is used to model the interaction o Rijke tube velocity with oscillatory heat. A comparison o the numerical results with experimental data and the reported calculations reveals that the current approach can yield very good predictions. Furthermore, subcritical Hop biurcations and old biurcations are captured with the evolution o dierent system parameters to gain an understanding o the intrinsic nonlinear behaviors. Introduction Thermoacoustic instability (or called combustion instability) widely exists in various combustion systems, such as rocket motors, industrial burners, gas turbine engines etc. [-3]. It arises rom the interaction between the heat release and acoustic pressure or velocity oscillations within the combustion system. Rijke tube is an excellent example or the study o thermoacoustic instability [4-6], which contains rich nonlinear behaviors, such as biurcation, limit cycle, quasiperiodicity and chaos. In the past decades, Rijke tube has been extensively studied to understand the intrinsic nonlinear behavior o the thermoacoustic instability. The theoretical work done by Cuclik [7-8] ocused on the nonlinear behavior o acoustic waves within a combustion chamber and was widely utilized by a number o subsequent work on the study o Rijke tube. Heckl [9] developed empirical models or the nonlinear behavior o both heat release and the relection coeicients based on experimental results. Hantschk and Vortmeyer [] investigated sel-excited thermoacoustic instabilities in the Rijke tube using a commercial CFD code. Matveev [-] combined linear theory and thermal analysis to predict the linear stability boundaries in a horizontal Rijke tube. Ananthkrishnan et al. [3] obtained the reduced-order models to capture the global behavior o chamber dynamics via truncating the modal expansions and determined the number o modes required or accurate results. Heckl and Howe [4] conducted stability analysis o the Rijke tube by making use o a Green s unction. Balasubramanian and Sujith [5] studied the role o non-normality and nonlinearity in thermoacoustic system in a Rijke tube. Subramanian et al. [6] conducted biurcation analysis o the dynamic behaviors o a horizontal Rijke tube and obtained biurcation plots as a unction o dierent system parameters. Noble et al. [7] described a data-driven nonlinear and chaos theory based analysis o thermoacoustic instabilities in a simple Rijke tube. Juniper [8] employed adjoint looping o the nonlinear governing equations to ind the lowest initial energy o a Rijke tube. Among the above research, Galerkin method [9] has been the most commonly employed technique to convert the governing PDEs into a series o ODEs or nonlinear analysis. However, in present study, the Method o Lines (MOL) [] approach was employed. It has seldom been used beore in this ield owing to the complexity o discretization and expensive computational cost. In this study, a horizontal Correspondence to: Xiaochuan Yang xiaochuan.yang@postgrad.manchester.ac.uk

2 Yang, X Rijke tube with heat source is employed or stability study, which would provide guidance in the design and operation o combustion system. Physical Model and Governing Equations Figure shows a schematic o a horizontal Rijke tube, which is an open-end cylinder with L in length and R in radium. A base low driven by an external an passes through the tube and is heated up by a hot wire gauze being placed at the position o x rom the inlet. Naturally, the tube displays an ininite number o acoustic modes. It was discovered by Rayleigh [3] that the thermal energy could be transerred to acoustic energy as long as they are in phase and the acoustic oscillations can be urther strengthened. Figure. Schematic o a classical horizontal Rijke tube In this study, it is assumed that the luid is a perect, inviscid and non-heat-conducting gas and thereore the inluence o the mean low and mean temperature gradients can be ruled out and the acoustic damping primarily results rom the acoustic boundary layer and acoustic radiation. Thus, the non-dimensional governing equations or the acoustic momentum and energy are u p () t x p u p D( x x ) u ( t ) () t x 3 3 where is the time delay, ( x x ) is the Dirac Delta unction to narrow the heat-release region D speciically at the wire position x, β is the heat release coeicient deined in a modiied orm o King s law: L T T dw w w cv p u S 3 Clearly, this coeicient as a combination covers all the details o the luid, the hot-wire gauze and the tube. The generic damping coeicient ξ [4] is deined as S L v (4) L R cl where ν is the kinematic viscosity, χ is the thermal diusivity, γ, c, R, L are the heat capacity ratio, speed o sound, radius and length o the tube respectively. In the study, the Rijke tube is open at both ends and thereore the boundary conditions at inlet (denoted as x = ) and outlet (denoted as x = ) become u/ x = and p = at x =,. It should be pointed that these boundary conditions enable the problem o interest to be a well-posed initial value problem within time domain. The acoustic energy per unit volume, E= (u + p )/, as used in Re [8] is selected as the indicator o oscillations. (3) th ASPACC July 9-, 5 - Beijing

3 Amplitude Yang, X 3 Numerical Approaches In order to explore the dynamic behavior o the time-delayed system o Rijke tubes, a dynamic system consisting o a set o ordinary dierential equations (ODEs) should be constructed rom the partial dierential equations Eqns. () and (). Previously, the Galerkin method was broadly employed, or which two sets o presumed basis unctions or u and p were utilized to convert Eqns. () and () to ODEs or each acoustic mode. However, in this study, the approach originally employed is the Method o Lines (MOL) to build up a dynamic system or the Rijke tube. Due to the complexity o discretization and expensive computation cost, it has seldom been used beore in this area. The main idea o the MOL is to directly discretize the spatial variable x in PDEs, but keep the temporal variable t be continuous. Thus, one could obtain a series o ODEs naturally without introducing any approximations. To obtain the unsteady solution structure, the governing equations Eqns. () and () were discretized by a inite dierence method. The x domain along the length o the tube is divided into N points indexed as x i (i =,,, N), which leads to N- parts with an identical interval o /(N- ). Second-order central dierence scheme was used to discretize the spatial derivatives u/ x and p/ x. Particularly, a second-order upwind dierence scheme was applied to deal with the Neumann boundary conditions or u at x and x N, whereas speciic values were given or the Dirichlet boundary conditions o p. Hence, we can obtain the dynamic system written symbolically as dy ( y,, ) (5) dt where y is the dependant variable vector o size N, is the vector unction o size N and θ is the biurcation parameter vector (= β, ξ, x ). A numerical continuation method was then employed to perorm the biurcation analysis o the timedelayed thermoacoustic system. First, the system was linearized around the pre-resolved steady state (equilibrium) solution and the corresponding eigenvalues o the linearized system, namely the roots o the characteristic equation were calculated. These roots were irstly approximated by a linear multistep method (LMS-method) and then corrected using a Newton iteration method []. Mathematically, the rightmost characteristic root, or the characteristic root with the maximal real part, conclusively indicates whether or not the system is stable. I the rightmost characteristic root crosses zero as a biurcation parameter varies, biurcation may arise. Moreover, a prediction-correction approach [5-6] was adopted to capture all the branches starting rom the biurcation point. In addition, the Floquet-multiplier scheme was utilized to determine the stability o each branch, or which the solution is stable only i the moduli o all the multipliers are less than unity. Beore urther biurcation analysis, the grid independence check is conducted to assure the accuracy o prediction. As shown in Figure, there is little change or the biurcation diagram when the number o discretization points increase to N = and beyond. To compromise between the accuracy and speed, the x domain is thus discretized into points or the ollowing biurcation analysis N=8 N= N= N= Figure. Biurcation diagrams as a unction o β using dierent discretization points (ζ=.43, x =.4, τ=.) th ASPACC July 9-, 5 - Beijing 3

4 Emin Emin Yang, X 4 Results and Discussion In this section, the nonlinear behaviors o the Rijke tube thermoacoustic system are analyzed in detail. Dierent biurcation diagrams are obtained to explore the eect o the system parameters including the heat release coeicient β, the generic damping coeicient ζ and the time delay τ. The eect o varying the heat release coeicient β on the dynamic behavior o the system is presented by the biurcation diagram shown in Figure 3. It presents a subcritical Hop biurcation and is similar to the biurcation diagrams reported in [6, 7-8]. The stability o each periodic solution was determined by Floquet multipliers, as discussed in Section 3. The steady state solution is stable or β <.874. At the Hop biurcation point β H (β H =.874), the system loses stability and small amplitude periodic solutions (limit cycle) emerge rom it. These limit cycles are unstable and will be stabilized through a turning point or reerred to as a old biurcation point (β =.85). When β < β, the steady state solution is stable or perturbations o any magnitude. In the region o β < β< β H, linearly stable steady state solution, small-amplitude unstable periodic solutions and large-amplitude stable limit cycles coexist to orm a bistable region, which is susceptible to triggering. This kind o system is called linearly stable but nonlinearly unstable by Zinn and Lieuwen [] Unstable Stable.5 H Figure 3. Biurcation diagram o the minimum acoustic energy on periodic solutions as a unction o β (ζ =.43, x =.3, τ =.), where dashed line indicates the unstable periodic solution, solid line indicates the stable periodic solution H Unstable Stable Figure 4. Biurcation diagrams as a unction o τ (ζ =.43, x =.3, β =.874) th ASPACC July 9-, 5 - Beijing 4

5 Emin Yang, X The biurcation diagram or a varying time delay is shown in Figure 4. Similar to Figure 3, the biurcation is also a subcritical Hop biurcation. A small time delay below τ is always stable, while a large time delay over τ H is always unstable. Between τ and τ H is the bistable region. Since a time delay is the time taken or the change o acoustic velocity to get relected in heat release perturbation[9], a decreased time delay would lead to a aster heat transer between the heat source and air low. Consequently, the air would quickly attain the same temperature as the heat source, which eventually reduces and eliminates the heat transer and diminishes the acoustic oscillation. It can be concluded that decreased time delay has a stabilizing eect on the system. To investigate the inluence o damping on the dynamic behavior, the biurcation diagram is displayed in Figure5. Since the damping eect could attenuate the oscillation, it is expected that the increased damping coeicient would stabilize the system, as provided in Figure5. In the Rijke tube, the damping coeicient could be varied by changing the end conditions o the tube Unstable Stable. H Figure 5. Biurcation diagrams as a unction o ζ (β =.874, x =.3, τ =.) The method is then validated by comparison between the experimental and numerical results as shown in Figure 6. The current predicted stability boundary displays good agreement with both experimental results[,3] and is more accurate than the results provided in [6] using the Galerkin method, which shows the capability o this method in capturing the nonlinear behaviors. The observed minor discrepancy regarding the results may be due to the relatively simple physical model adopted in current study. The predictions obtained are encouraging and could reveal the rich nonlinear behaviors inherent in the Rijke tube thermoacoustic system. In a uture study, it is hoped that the predictions will be more precise using more sophisticated models..5.5 Current Study Exp. Data [] Exp. Data [3] Num. Data [6] Figure 6. Comparison between numerical and experimental results or stability boundary (ζ =.8, x =.5) th ASPACC July 9-, 5 - Beijing 5

6 Yang, X 5. Conclusions In this study, the dynamic behaviors o a horizontal Rijke tube thermoacoustic system are studied in detail. Method o Lines (MOL) technique is employed to discretize the governing equations and numerical continuation method is used to perorm the biurcation analysis. The eects o heat release, damping and time delay are investigated by biurcation diagrams or nonlinear analysis. Subcritical Hop biurcation and old biurcation are captured to reveal the nonlinear behavior o the system. The biurcation diagrams display same trends as the previous results using Galerkin methods. The results agree well with the experimental data, validating the potential o applying the methodology in the analysis o nonlinear stability o a generic time-delayed thermoacoustic combustion system. Reerence [] S. Lei, A. Turan, Int. J. Biurcat. Chaos (4) () [] T. C. Lieuwen; V. Yang, Combustion instabilities in gas turbine engines: operational experience, undamental mechanisms and modeling, American Institute o Aeronautics and Astronautics, 5. [3] A. Morgans, S. Stow, Combust. Flame 5 (4) (7) [4] A. P. Dowling, A. S. Morgans, Annu. Rev. Fluid Mech. 37 (5) 5-8. [5] R. Raun, M. Beckstead, J. Finlinson, K. Brooks, Prog. Energy. Combust. Sci.9 (4) (993) [6] D. Zhao, Combust. Flame 59 (6) () [7] F. Culick, Acta. Astronautica 3 (9) (976) [8] F. Culick, Acta. Astronautica 3 (9) (976) [9] M. A. Heckl, Acta. Acust. United. Acust 7 () (99) [] C. C. Hantschk, D. Vortmeyer, J. Sound. Vib.7 (3) (999) 5-5. [] K. Matveev. Thermoacoustic instabilities in the Rijke tube: experiments and modeling. PhD thesis, Caliornia Institute o Technology, 3. [] K. Matveev, F. Culick, J. Sound. Vib.64 (3) (3) [3] N. Ananthkrishnan, S. Deo, F. E. Culick, Combust. Sci. Technol 77 () (5) -48. [4] M. A. Heckl, M. Howe, J. Sound. Vib.35 (4) (7) [5] K. Balasubramanian, R. Sujith, Phys. Fluids. (994-present) (4) (8) 443. [6] P. Subramanian, S. Mariappan, R. Sujith, P. Wahi, Int. J. Spray Combust. Dyn (4) () [7] A. C. Noble, G. B. King, N. M. Laurendeau; J. R. Gord; S. Roy, Combust. Sci. Technol 84 (3) () [8] M. P. Juniper, J. Fluid. Mech. 667 () [9] S. G. Mikhlin, Variational methods in mathematical physics, Pergamon Press, 964. [] W. E. Schiesser, The Numerical Method o Lines: Integration o Partial Dierential Equations, Academic Press, 99. [] K. Engelborghs, T. Luzyanina, D. Roose, ACM. Trans. Math. Sotw 8 () () -. [] K. Engelborghs, T. Luzyanina; G. Samaey, Department o Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium (). [3] L. Rayleigh, Nature 8 (455) (878) [4] M. S. Howe, Acoustics o luid-structure interactions, Cambridge University Press, 998. [5] E. Doedel, H. B. Keller, J. P. Kernevez, Int. J. Biurcat. Chaos (3) (99) [6] R. Seydel, Practical biurcation and stability analysis, Springer,. [7] I. Waugh. Methods or analysis o nonlinear thermoacoustic systems. PhD thesis, University o Cambridge, 3. [8] N. Noiray, D. Durox, T. Schuller, S. Candel, J. Fluid Mech 65 (8) [9] S. M. Sarpotdar, N. Ananthkrishnan, S. Sharma, Resonance 8 () (3) [3] W.S. Song, S. Lee, D.S. Shin, Y. Na, J Mech. Sci. Technol (6) (6) th ASPACC July 9-, 5 - Beijing 6

Nonlinear Themoacoustic In/Stability of a Rijke Tube with a Distributed Heat Source

Nonlinear Themoacoustic In/Stability of a Rijke Tube with a Distributed Heat Source Xiaochuan Yang,, Ali Turan, Shenghui Lei School of Mechanical, Aerospace and Civil Engineering, The University of Manchester,