Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-14T23:15:38.181Z Has data issue: false hasContentIssue false
  • English
  • Français

Solution of the quasi-one-dimensional linearized Euler equations using flow invariants and the Magnus expansion

Published online by Cambridge University Press:  16 April 2013

Ignacio Duran  and
Stephane Moreau
Ignacio Duran*
Affiliation:
CERFACS, 42 Avenue Gaspard Coriolis, 31057 Toulouse, France
Stephane Moreau
Affiliation:
Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, Canada
*
†Email address for correspondence: ignacio.duran@cerfacs.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The acoustic and entropy transfer functions of quasi-one-dimensional nozzles are studied analytically for both subsonic and choked flows with and without shock waves. The present analytical study extends both the compact nozzle solution obtained by Marble & Candel (J. Sound Vib., vol. 55, 1977, pp. 225–243) and the effective nozzle length proposed by Stow, Dowling & Hynes (J. Fluid Mech., vol. 467, 2002, pp. 215–239) and by Goh & Morgans (J. Sound Vib., vol. 330, 2011, pp. 5184–5198) to non-zero frequencies for both modulus and phase through an asymptotic expansion of the linearized Euler equations. It also extends the piecewise-linear approximation of the velocity profile in the nozzle proposed by Moase, Brear & Manzie (J. Fluid Mech., vol. 585, 2007, pp. 281–304) to any arbitrary profile or equivalently any nozzle geometry. The equations are written as a function of three variables, namely the dimensionless mass, total temperature and entropy fluctuations, yielding a first-order linear system of differential equations with varying coefficients, which is solved using the Magnus expansion. The solution shows that both the modulus and the phase of the transfer functions of the nozzle have a strong dependence on the frequency. This holds for both choked flows and subsonic converging–diverging nozzles. The method is used to compare two different nozzle geometries with the same inlet and outlet Mach numbers, showing that, even if the compact solution predicts no differences between the transfer functions of the two nozzles, significant differences are found at non-zero frequencies. A parametric study is finally performed to calculate the indirect to direct noise ratio for a model combustor, showing that this ratio decreases at higher frequencies.

JFM classification

Acoustics: Acoustics Acoustics: Aeroacoustics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 723 , 25 May 2013 , pp. 190 - 231
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bake, F., Richter, C., Muhlbauer, B., Kings, N., Rohle, I., Thiele, F. & Noll, B. 2009 The Entropy Wave Generator (EWG): a reference case on entropy noise. J. Sound Vib. 574–598.Google Scholar
Bell, W., Daniel, B. & Zinn, B. 1973 Experimental and theoretical determination of the admittances of a family of nozzles subjected to axial instabilities. J. Sound Vib. 30 (2), 179–190.CrossRefGoogle Scholar
Blanes, S., Casas, F., Otero, J. A. & Ros, J. 2009 The Magnus expansion and some of its applications. Phys. Rep. 470 (5–6), 151–238.CrossRefGoogle Scholar
Candel, S. 1972 Analytical studies of some acoustic problems of jet engines. PhD thesis, California Institute of Technology.Google Scholar
Candel, S. & Poinsot, T. 1988 Interactions between acoustics and combustion. In Proceedings of the Institute of Acoustics, vol. 10, pp. 103–153.Google Scholar
Casas, F. 2007 Sufficient conditions for the convergence of the Magnus expansion. J. Phys. A: Math. Theor. 40 (50), 15001–15017.Google Scholar
Crocco, L. & Sirignano, W. A. 1967 Behavior of supercritical nozzles under three-dimensional oscillatory conditions. In AGARD, vol. 117. Butterworth Publications Ltd.Google Scholar
Culick, F. E. C. 1988 Combustion instabilities in liquid-fueled propulsion systems – an overview. In AGARD 72B PEP Meeting, Conference Proceedings 450.Google Scholar
Cumpsty, N. A. & Marble, F. E. 1977 The interaction of entropy fluctuations with turbine blade rows; a mechanism of turbojet engine noise. Proc. R. Soc. Lond. A 357, 323–344.Google Scholar
Dowling, A. P. 1995 The calculation of thermoacoustic oscillations. J. Sound Vib. 180 (4), 557–581.CrossRefGoogle Scholar
Dowling, A. P. 1997 Nonlinear self-excited oscillations of a ducted flame. J. Fluid Mech. 346, 271–290.Google Scholar
Dowling, A. P. & Stow, S. R. 2003 Acoustic analysis of gas turbine combustors. J. Propul. Power 19 (5), 751–764.Google Scholar
Duran, I., Moreau, S. & Poinsot, T. 2013 Analytical and numerical study of combustion noise through a subsonic nozzle. AIAA J. 51 (1), 42–52.Google Scholar
Giauque, A., Huet, M. & Clero, F. 2012 Analytical analysis of indirect combustion noise in subcritical nozzles. Trans. ASME: J. Engng Gas Turbines Power 134 (11), 111202.Google Scholar
Gicquel, L. Y. M., Staffelbach, G. & Poinsot, T. 2012 Large eddy simulations of gaseous flames in gas turbine combustion chambers. Prog. Energy Combust. Sci. 38 (6), 782–817.Google Scholar
Goh, C. S. & Morgans, A. S. 2011a Phase prediction of the response of choked nozzles to entropy and acoustic disturbances. J. Sound Vib. 330 (21), 5184–5198.Google Scholar
Goh, C. S. & Morgans, A. S 2011b The effect of entropy wave dissipation and dispersion on thermoacoustic instability in a model combustor. In Proceedings of the 17th AIAA/CEAS Aeroacoustics Conference, Portland, AIAA-2011-2914.Google Scholar
Gullaud, E., Mendez, S., Sensiau, C., Nicoud, F. & Poinsot, T. 2009 Effect of multiperforated plates on the acoustic modes in combustors. C. R. Acad. Sci. Méc. 337 (6–7), 406–414.Google Scholar
Howe, M. S. 2010 Indirect combustion noise. J. Fluid Mech. 659, 267–288.Google Scholar
Kuo, C.-Y. & Dowling, A. P. 1996 Oscillations of a moderately underexpanded choked jet impinging upon a flat plate. J. Fluid Mech. 315, 267–291.CrossRefGoogle Scholar
Leyko, M., Moreau, S., Nicoud, F. & Poinsot, T. 2011 Numerical and analytical modelling of entropy noise in a supersonic nozzle with a shock. J. Sound Vib. 330 (16), 3944–3958.Google Scholar
Leyko, M., Nicoud, F., Moreau, S. & Poinsot, T. 2008 Numerical and analytical investigation of the indirect noise in a nozzle. In Center for Turbulence Research, NASA AMES, Proceedings of the Summer Program, pp. 343–354. Stanford University, USA.Google Scholar
Leyko, M., Nicoud, F. & Poinsot, T. 2009 Comparison of direct and indirect combustion noise mechanisms in a model combustor. AIAA J. 47 (11), 2709–2716.CrossRefGoogle Scholar
Madhu, P. & Kurur, N. 2006 Fer expansion for effective propagators and Hamiltonians in NMR. Chem. Phys. Lett. 418 (1–3), 235–238.Google Scholar
Magnus, W. 1954 On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7 (4), 649–673.Google Scholar
Marble, F. E. & Candel, S. 1977 Acoustic disturbances from gas non-uniformities convected through a nozzle. J. Sound Vib. 55, 225–243.Google Scholar
Moan, P. C. & Niesen, J. 2006 Convergence of the Magnus series. Tech. Rep. 0316, La Trobe University.Google Scholar
Moase, W. H., Brear, M. J. & Manzie, C. 2007 The forced response of choked nozzles and supersonic diffusers. J. Fluid Mech. 585, 281–304.Google Scholar
Moler, C. & Van Loan, C. 2003 Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45 (1), 3–49.Google Scholar
Motheau, E., Selle, L., Mery, Y., Poinsot, T. & Nicoud, F. 2012 A mixed acoustic–entropy combustion instability in a realistic gas turbine. In Center for Turbulence Research, NASA AMES, Proceedings of the Summer Program, pp. 449–458 Stanford University, USA.Google Scholar
Muthukrishnan, M., Strahle, W. C. & Neale, D. H. 1978 Separation of hydrodynamic, entropy, and combustion noise in a gas turbine combustor. AIAA J. 16 (4), 320–327.Google Scholar
Pickett, G. F. 1975 Core engine noise due to temperature fluctuations convecting through turbine blade rows. In Proceedings of the 2nd AIAA Aeroacoustics Conference, AIAA-1975-528, Hampton, VA.Google Scholar
Poinsot, T. & Veynante, D. 2011 Theoretical and Numerical Combustion, 3rd edn. (www.cerfacs.fr/elearning).Google Scholar
Rayleigh, L. 1878 The explanation of certain acoustic phenomena. Nature 18 July, 319–321.Google Scholar
Ribner, H. S. 1953 Convection of a pattern of vorticity through a shock wave. NACA TN 2864.Google Scholar
Ribner, H. S. 1954 Shock–turbulence interaction and the generation of noise. NACA TN 3255.Google Scholar
Ribner, H. S. 1987 Spectra of noise and amplified turbulence emanating from shock turbulence interaction. AIAA J. 25 (3), 436–442.Google Scholar
Sakurai, J. J. & Tuan, S. F. 1985 Modern Quantum Mechanics, vol. 1. Addison-Wesley.Google Scholar
Silva, C. 2010 Numerical study of combustion noise in gas turbines. PhD thesis, INPT, Toulouse.Google Scholar
Stow, S. R., Dowling, A. P. & Hynes, T. P. 2002 Reflection of circumferential modes in a choked nozzle. J. Fluid Mech. 467, 215–239.Google Scholar
Tsien, H. S. 1952 The transfer functions of rocket nozzles. J. Am. Rocket Soc. 22 (3), 139–143.Google Scholar
Wilcox, R. M. 1967 Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8 (4), 962–982.CrossRefGoogle Scholar
Williams, F. A. 1985 Combustion Theory. Benjamin Cummings.Google Scholar
Zinn, B. T., Bell, W. A., Daniel, B. R. & Smith, A. J. 1973 Experimental determination of three-dimensional liquid rocket nozzle admittances. AIAA J. 11 (3), 267–272.Google Scholar