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

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.

[1]  H. B. Mathes,et al.  Nonlinear Stability Testing of Full-Scale Tactical Motors , 1997 .

[2]  Ar-Cheng Lin,et al.  Investigation of aluminized solid propellant combustion instability by means of a T-burner , 1995 .

[3]  Gary A. Flandro,et al.  Aeroacoustic Instability in Rockets , 2001 .

[4]  R. Sujith,et al.  Non-normality and nonlinearity in combustion–acoustic interaction in diffusion flames , 2007, Journal of Fluid Mechanics.

[5]  F. Culick,et al.  REDUCED-ORDER MODELING AND DYNAMICS OF NONLINEAR ACOUSTIC WAVES IN A COMBUSTION CHAMBER , 2005 .

[6]  F. Culick,et al.  Modeling the combustion response function with surface and gas phase dynamics , 2000 .

[7]  Robert C. Armstrong,et al.  Two Asymptotic Models for Solid Propellant Combustion , 1986 .

[8]  J. Levine,et al.  A numerical study of nonlinear instability phenomena in solid rocket motors , 1981 .

[9]  P. Schmid.  Nonmodal Stability Theory , 2007 .

[10]  Toru Shimada,et al.  Stability Analysis of Solid Rocket Motor Combustion by Computational Fluid Dynamics , 2008 .

[11]  L. Trefethen,et al.  Spectra and pseudospectra : the behavior of nonnormal matrices and operators , 2005 .

[12]  Ramesh Narayan,et al.  Growth of hydrodynamic perturbations in accretion disks: Possible route to non-magnetic turbulence , 2006 .

[13]  V. Yang,et al.  Triggering of longitudinal combustion instabilities in rocket motors - Nonlinear combustion response , 1996 .

[14]  M. E. Lores,et al.  Application of the Galerkin Method in the Solution of Non-linear Axial Combustion Instability Problems in Liquid Rockets , 1971 .

[15]  Luigi T. DeLuca,et al.  Linear Stability and Pressure-Driven Response Function of Solid Propellants with Phase Transition , 1999 .

[16]  Gebhardt,et al.  Chaos transition despite linear stability. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  Hyun-Gull Yoon,et al.  Non-linear response of a generalized Rijke tube , 2001 .

[19]  Kenneth K. Kuo,et al.  Fundamentals of Solid-Propellant Combustion , 1984 .

[20]  Junye Wang.  Non-linear analysis of solid propellant burning rate behavior , 2000 .

[21]  Gary A. Flandro,et al.  Nonlinear Rocket Motor Stability Prediction: Limit Amplitude, Triggering, and Mean Pressure Shift , 2007 .

[22]  J. N. Levine,et al.  A numerical study of nonlinear instability phenomena in solid rocketmotors , 1983 .

[23]  Lloyd N. Trefethen,et al.  Pseudospectra of the Convection-Diffusion Operator , 1993, SIAM J. Appl. Math..

[24]  T. Driscoll,et al.  A mostly linear model of transition to tur , 1995 .

[25]  L. Tuckerman,et al.  Stability analysis of perturbed plane Couette flow , 1999, physics/0312048.

[26]  L. D. Strand,et al.  Combustion response to compositional fluctuations , 1983 .

[27]  Raman Sujith,et al.  Characterizing energy growth during combustion instabilities: Singularvalues or eigenvalues? , 2009 .

[28]  B. Chu.  On the energy transfer to small disturbances in fluid flow (Part I) , 1965 .

[29]  G. Flandro,et al.  Nonlinear combustion instability data reduction , 1996 .

[30]  Petros,et al.  Generalized Stability Theory . Part I : Autonomous Operators , 2022 .

[31]  Ben T. Zinn.  Longitudinal mode acoustic losses in short nozzles , 1972 .

[32]  Fred E. C. Culick,et al.  A review of calculations for unsteady burning of a solid propellant. , 1968 .

[33]  Raman Sujith,et al.  Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity , 2007 .

[34]  Steven F. Son,et al.  Quasi-steady combustion modeling of homogeneous solid propellants , 1995 .

[35]  F. Cozzi,et al.  Intrinsic combustion instability of solid energetic materials , 1995 .

[36]  G. A. Flandro,et al.  Effects of vorticity on rocket combustion stability , 1995 .

[37]  Brian F. Farrell,et al.  Generalized Stability Theory. Part II: Nonautonomous Operators , 1996 .

[38]  K. N. Lakshmisha,et al.  Nonlinear Intrinsic Instability of Solid Propellant Combustion Including Gas-Phase Thermal Inertia , 2000 .

[39]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering , 1994 .

[40]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[41]  Fred E. C. Culick,et al.  STABILITY OF HIGH-FREQUENCY PRESSURE OSCILLATIONS IN ROCKET COMBUSTION CHAMBERS , 1963 .

[42]  Domenic A. Santavicca,et al.  Experimental Diagnostics of Combustion Instabilities , 2005 .

[43]  F. Nicoud,et al.  Acoustic modes in combustors with complex impedances and multidimensional active flames , 2007 .

[44]  François Vuillot,et al.  Vortex-Shedding Phenomena in Solid Rocket Motors , 1995 .

[45]  Mysore Srikantiah Padmanabhan,et al.  The effect of nozzle nonlinearities on the nonlinear stability of liquid rocket motors , 1975 .

[46]  W. B. LITTLER,et al.  Rocket Propulsion , 1959, Nature.

[47]  J. Anthoine,et al.  Effect of nozzle cavity on resonance in large SRM : theoretical Modeling , 2002 .

[48]  Fred E. C. Culick,et al.  Nonlinear behavior of acoustic waves in combustion chambers. I, II. [stability in solid propellant rocket engine and T burner , 1976 .

[49]  V. Zarko,et al.  Analysis of unsteady solid-propellant combustion models (review) , 2008 .

[50]  K. F. Riley,et al.  Mathematical Methods for Physics and Engineering , 1998 .

[51]  V. Yang,et al.  Triggering of Longitudinal Pressure Oscillations in Combustion Chambers. I: Nonlinear Gasdynamics , 1990 .

[52]  Raman Sujith,et al.  Impact of Linear Coupling on Thermoacoustic Instabilities , 2008 .

[53]  H. B. Mathes,et al.  Stability Testing of Full-Scale Tactical Motors , 1997 .

[54]  F. Culick.  Some recent results for nonlinear acoustics in combustion chambers , 1994 .

[55]  Paul Kuentzmann,et al.  Unsteady Motions in Combustion Chambers for Propulsion Systems , 2006 .

[56]  Gary A. Flandro,et al.  On flow turning , 1995 .

[57]  James S. T'ien,et al.  Nonsteady burning phenomena of solid propellants - theory and experiments , 1968 .

[58]  William O. Criminale,et al.  The initial-value problem for a modeled boundary layer , 2000 .

[59]  F. Spellman.  Combustion Theory , 2020 .

[60]  Fred E. C. Culick.  A Note on Ordering Perturbations and the Insignificance of Linear Coupling in Combustion Instabilities , 1997 .

[61]  Stephen B. Margolis,et al.  Diffusional/Thermal Coupling and Intrinsic Instability of Solid Propellant Combustion , 1988 .

[62]  D. Birchall,et al.  Computational Fluid Dynamics , 2020, Radial Flow Turbocompressors.

[63]  J. C. Friedly,et al.  Influence of combustion parameters on instability in solid propellant motors. I - Development of model and linear analysis. , 1966 .

[64]  F. A. Williams,et al.  Response of a Burning Solid to Small-Amplitude Pressure Oscillations , 1962 .

[65]  L. Crocco,et al.  Theory of Combustion Instability in Liquid Propellant Rocket Motors , 1956 .

[66]  Azeddine Kourta.  Shear layer instability and acoustic interaction in solid propellant rocket motors , 1997 .

[67]  Fred E. C. Culick,et al.  Pulsed Instabilities in Solid-Propellant Rockets , 1995 .

[68]  P. Schmid,et al.  Stability and Transition in Shear Flows. By P. J. SCHMID & D. S. HENNINGSON. Springer, 2001. 556 pp. ISBN 0-387-98985-4. £ 59.50 or $79.95 , 2000, Journal of Fluid Mechanics.

[69]  Steven F. Son,et al.  Steady Deflagration of HMX With Simple Kinetics: A Gas Phase Chain Reaction Model , 1998 .

[70]  Thomas Boeck,et al.  Numerical study of the instability of the Hartmann layer , 2004, Journal of Fluid Mechanics.