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Thermoacoustic instability in a solid rocket motor: non-normality and nonlinear instabilities

  • SATHESH MARIAPPAN (a1) and R. I. SUJITH (a1)

Abstract

An analytical framework is developed to understand and predict the thermoacoustic instability in solid rocket motors, taking into account the non-orthogonality of the eigenmodes of the unsteady coupled system. The coupled system comprises the dynamics of the acoustic field and the propellant burn rate. In general, thermoacoustic systems are non-normal leading to non-orthogonality of the eigenmodes. For such systems, the classical linear stability predicted from the eigenvalue analysis is valid in the asymptotic (large time) limit. However, the short-term dynamics can be completely different and a generalized stability theory is needed to predict the linear stability for all times. Non-normal systems show an initial transient growth for suitable initial perturbations even when the system is stable according to the classical linear stability theory. The terms contributing to the non-normality in the acoustic field and unsteady burn rate equations are identified. These terms, which were neglected in the earlier analyses, are incorporated in this analysis. Furthermore, the short-term dynamics are analysed using a system of differential equations that couples the acoustic field and the burn rate, rather than using ad hoc response functions which were used in earlier analyses. In this paper, a solid rocket motor with homogeneous propellant grain has been analysed. Modelling the evolution of the unsteady burn rate using a differential equation increases the degrees of freedom of the thermoacoustic system. Hence, a new generalized disturbance energy is defined which measures the growth and decay of the oscillations. This disturbance energy includes both acoustic energy and unsteady energy in the propellant and is used to quantify the transient growth in the system. Nonlinearities in the system are incorporated by including second-order acoustics and a physics-based nonlinear unsteady burn rate model. Nonlinear instabilities are analysed with special attention given to ‘pulsed instability’. Pulsed instability is shown to occur with pressure coupling for burn rate response. Transient growth is shown to play an important role in pulsed instability.

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Corresponding author

Email address for correspondence: sujith@iitm.ac.in

References

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Anathakrishnan, N., Deo, S. & Culick, F. E. C. 2005 Reduced-order modeling and dynamics of nonlinear acoustic waves in a combustion chamber. Combust. Sci. Tech. 177, 221247.
Anderson, J. D. 1996 Computational Fluid Dynamics. Springer.
Anthoine, J., Buchlin, J. M. & Hirschberg, A. 2002 Effect of nozzle cavity on resonance in large SRM: theoretical modelling. J. Propul. Power 18, 304311.
Baggett, J. S., Driscoll, T. A. & Trefethen, L. N. 1995 A mostly linear model of transition to turbulence. Phys. Fluids 7, 833838.
Balasubramanian, K. & Sujith, R. I. 2008 a Non-normality and nonlinearity in combustion-acoustic interaction in diffusion flames. J. Fluid Mech. 594, 2957.
Balasubramanian, K. & Sujith, R. I. 2008 b Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity. Phys. Fluids 20, 044103.
Barkley, D. & Tuckerman, L. S. 1999 Stability analysis of perturbed plane Coquette flow. Phys. Fluids 11, 11871195.
Baum, J. D. & Levine, J. N. 1986 Modelling of nonlinear longitudinal instability in solid rocket motors. AIAA J. 13, 339348.
Blomshield, F. S., Mathes, H. B., Crump, J. E., Beiter, C. A. & Beckstead, M. W. 1997 a Nonlinear stability testing of full-scale tactical motors. J. Propul. Power 13, 349355.
Blomshield, F. S., Mathes, H. B., Crump, J. E., & Beiter, C. A. 1997 b Nonlinear stability testing of full scale tactical motors. J. Propul. Power 13, 356366.
Brewster, Q. & Son, S. F. 1995 Quasi-steady combustion modeling of homogeneous solid propellants. Combust. Flame 103, 1126.
Chu, B. T. 1965 On the energy transfer to small disturbances in fluid flow. Acta Mech. 1, 215234.
Cohen, N. S. & Strand, L. D. 1985 Combustion response to compositional fluctuations. AIAA J. 23, 760767.
Cozzi, F., Deluca, L. T. & Novozhilov, B. V. 1999 Linear stability and pressure-driven response function of solid propellants with phase transitions. J. Propul. Power 15, 806815.
Criminale, W. O. & Drazin, P. G. 2000 The initial-value problem for a modeled boundary layer. Phys. Fluids 12, 366374.
Crocco, L. 1956 Theory of combustion instability in liquid propellant rocket motors. AGARDograph Rep 0429886.
Culick, F. E. C. 1963 Stability of high-frequency pressure oscillations in rocket combustion chambers. AIAA J. 1, 10971104.
Culick, F. E. C. 1968 A review of calculations for unsteady burning of a solid propellant. AIAA J. 6, 22412255.
Culick, F. E. C. 1976 a Nonlinear behaviour of acoustic waves in combustion chambers. Part I. Acta Astronaut. 3, 715734.
Culick, F. E. C. 1976 b Nonlinear behaviour of acoustic waves in combustion chambers. Part II. Acta Astronaut. 3, 735757.
Culick, F. E. C. 1994 Some recent results for nonlinear acoustics in combustion chambers. AIAA J. 32, 146169.
Culick, F. E. C. 1997 A note on ordering perturbations and the insignificance of linear coupling in combustion instabilities. Combust. Sci. Tech. 126, 359379.
Culick, F. E. C. 2006 Unsteady motions in combustion chambers for propulsion systems. RTO AGARDograph AG-AVT-039.
Culick, F. E. C., Burnley, V. & Swenson, G. 1995 Pulsed instabilities in solid-propellant rockets. J. Propul. Power 11, 657665.
Culick, F. E. C. & Isella, G. 2000 Modelling the combustion response function with surface and gas phase dynamics. In Thirty-Eighth Aerospace Sciences Meeting and Exhibit. AIAA Paper 2000-0310.
Deluca, L., Disilvestro, R. & Cozzi, F. 1995 Intrinsic combustion instability of solid energetic materials. J. Propul. Power 11, 804815.
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part I. Autonomous operators. J. Atmos. Sci. 53, 20252040.
Flandro, G. A. 1995 a Effects of vorticity on rocket combustion stability. J. Propul. Power 11, 607625.
Flandro, G. A. 1995 b On flow turning. In Thirty-First ASME, SAE, and ASEE, Joint Propulsion Conference and Exhibit. AIAA Paper 1995-2730
Flandro, G. A. 1996 Nonlinear combustion instability data reduction. In Thirty-Second AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit. AIAA Paper 1996-3251.
Flandro, G. A., Fischbach, S. R. & Majdalani, J. 2007 Nonlinear rocket motor stability prediction: limit amplitude, triggering and mean pressure shift. Phys. Fluids 19, 094101.
Flandro, G. A. & Majdalani, J. 2003 Aeroacoustic instability in rockets. AIAA J. 41, 485497.
Friedly, J. C. & Petersen, E. E. 1966 Influence of combustion parameters on instability in solid propellant motors. Part I. Development of model and linear analysis. AIAA J. 4, 16041610.
Gebgart, T. & Grossmann, S. 1994 Chaos transition despite linear stability. Phys. Rev. E 50, 37053711.
Golub, G. H. & Van Loan, C. E. 1989 Matrix Computations. The Johns Hopkins University Press.
Gusachenko, L. K. & Zarko, V. E. 2008 Analysis of unsteady solid-propellant combustion models (review). Combust. Explo. Shock Waves 44, 3142.
Harris, P. G & Champlain, A. D. 1998 Experimental database describing pulse-triggered nonlinear instability in solid rocket motors. J. Propul. Power 14, 429439.
Junye, W. 2000 Non-linear analysis of solid propellant burning rate behaviour. Intl J. Numer. Methods Fluids 33, 627640.
Kedia, K. S., Nagaraja, S. B. & Sujith, R. I. 2008 Impact of linear coupling on thermoacoustic instabilities. Combust. Sci. Tech. 180, 15881612.
Krasnov, D. S., Zienicke, E., Zikanov, O., Boeck, T. & Thess, A. 2004 Numerical study of the instability of the Hartmann layer. J. Fluid Mech. 504, 183211.
Krier, H., T'ien, S. J., Sirignana, W. A. & Summerfield, M. 1968 Nonsteady burning phenomena of solid propellants: theory and experiments. AIAA J. 6, 278285.
Kumar, K. R. & Lakshmisha, K. N. 2000 Nonlinear intrinsic instability of solid propellant combustion including gas-phase thermal inertia. Combust. Sci. Tech. 158, 135166.
Kuo, K. K. & Summerfield, M. 1984 Fundamentals of solid-propellant combustion. Prog. Astronaut. Aeronaut. 90, 760767.
Kourta, A. 1997 Shear layer instability and acoustic interaction in solid propellant rocket motors. Intl J. Numer. Methods Fluids 25, 973981.
Lee, J. G. & Santavicca, D. A. 2005 Experimental diagnostics of combustion instabilities. Prog. Astronaut. Aeronaut. 210, 481529.
Levine, J. N. & Baum, J. D. 1983 A numerical study of nonlinear instability phenomena in solid rocket motors. AIAA J. 21, 557564.
Lin, A. C. & Wang, S. Y. 1995 Investigation of aluminized solid propellant combustion instability by means of a T-burner. In Thirty-Third Aerospace Sciences Meeting and Exhibit. AIAA Paper 95-0606.
Margolis, S. B. & Armstrong, R. C. 1986 Two asymptotic models for solid propellant combustion. Combust. Sci. Tech. 47, 138.
Margolis, S. B. & Armstrong, R. C. 1988 Diffusional/thermal coupling and intrinsic instability of solid propellant combustion. Combust. Sci. Tech. 59, 2784.
Matveev, K. I. 2003 Thermo-acoustic instabilities in the Rijke tube: experiments and modeling. PhD thesis, California Institute of Technology.
Mukhopadhyay, B., Afshordi, N. & Narayan, R. 2006 Growth of hydrodynamic perturbations in accretion disks: possible route to non-magnetic turbulence. Adv. Space Res. 38, 28772879.
Nagaraja, S. B., Kedia, K. S. & Sujith, R. I. 2009 Characterizing energy growth during combustion instabilities: singular values or eigenvalues? Proc. Combust. Inst. 39, 29332940.
Nicoud, F., Benoit, L., Sensiau, C. & Poinsot, T. 2007 Acoustic modes in combustors with complex impedances and multidimensional active flames. AIAA J. 45, 426441.
Padmanabhan, M. S. 1975 The effect of nozzle nonlinearities on the nonlinear stability of liquid rocket motors. PhD thesis, Georgia Institute of Technology, Atlanta, USA.
Price, E. W. 1984 Fundamentals of solid-propellant combustion. Prog. Astronaut. Aeronaut. 90, 479513.
Reddy, S. C. & Trefethen, L. N. 1994 Pseudospectra of the convection–diffusion operator. SIAM J. Appl. Maths 54, 16341639.
Rienstra, S. W. & Hirschberg, A. 2008 An introduction to acoustics. IWDE Rep. 92-06.
Riley, K. F., Hobson, M. P. & Bence, S. J. 2006 Mathematical Methods for Physics and Engineering. Cambridge University Press.
Romanov, O. Y. 1999 Unsteady burning of solid propellants. J. Propul. Power 15, 823836.
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.
Shimada, T., Hanzawa, M., Kata, T., Yoshikawa, T. & Wada, Y. 2007 Stability analysis of solid rocket motor combustion by computational fluid dynamics. In Thirteenth AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2007-3427.
Strogatz, S. H. 2001 Nonlinear Dynamics and Chao: With Applications to Physics, Biology, Chemistry and Engineering. Perseus Books.
Sutton, G. P. 2001 Rocket Propulsion. Wiley-IEEE.
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behaviour of Nonnormal Matrices and Operators. Princeton University Press.
Vuillot, J. 1995 Vortex shedding phenomena in solid rocket motors. J. Propul. Power 11, 626639.
Ward, M. J., Son, S. F. & Brewster, M. Q. 1998 Steady deflagration of HMX with simple kinetics: a gas phase chain reaction model. Combust. Flame 114, 556568.
Wicker, J. M., Greene, W. D., Kim, S.-I. & Yang, V. 1996 Triggering of longitudinal combustion instabilities in rocket motors: nonlinear combustion response. J. Propul. Power 12, 11481158.
Williams, F. A. 1962 Response of a burning solid to small-amplitude pressure oscillations. J. Appl. Phys. 33, 31533166.
Williams, F. A. 1985 Combustion Theory. Addison-Wesley.
Yang, V., Kim, S. I. & Culick, F. E. C. 1990 Triggering of longitudinal pressure oscillations in combustion chambers. Part I. Nonlinear gasdynamics. Combust. Sci. Tech. 72, 183214.
Yoon, H. G., Peddieson, J. & Purdy, K. R. 2001 Non-linear response of a generalized Rijke tube. Intl J. Engng Sci. 39, 17071723.
Zinn, B. T. 1972 Longitudinal mode acoustic losses in short nozzles. Res. Rep. AD0744623. Naval Weapons Center China Lake.
Zinn, B. T. & Lores, M. E. 1971 Application of the Galerkin method in the solution of nonlinear axial combustion instability problems in liquid rockets. Combust. Sci. Tech. 4, 269278.
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