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Direct numerical simulation and biglobal stability investigations of the gaseous motion in solid rocket motors

Published online by Cambridge University Press:  13 July 2012

F. Chedevergne ,
G. Casalis  and
J. Majdalani
F. Chedevergne
Affiliation:
Aerodynamics and Energetics Modelling Department, ONERA, Toulouse 31055, France
G. Casalis
Affiliation:
Aerodynamics and Energetics Modelling Department, ONERA, Toulouse 31055, France
J. Majdalani*
Affiliation:
Department of Mechanical, Aerospace and Biomedical Engineering, University of Tennessee Space Institute, Tullahoma, TN 37388, USA
*
†Email address for correspondence: joe.majdalani@gmail.com
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Abstract

In this article, a biglobal stability approach is used in conjunction with direct numerical simulation (DNS) to identify the instability mode coupling that may be responsible for triggering large thrust oscillations in segmented solid rocket motors (SRMs). These motors are idealized as long porous cylinders in which a Taylor–Culick type of motion may be engendered. In addition to the analytically available steady-state solution, a computed mean flow is obtained that is capable of securing all of the boundary conditions in this problem, most notably, the no-slip requirement at the chamber headwall. Two sets of unsteady simulations are performed, static and dynamic, in which the injection velocity at the chamber sidewall is either held fixed or permitted to vary with time. In these runs, both DNS and biglobal stability solutions converge in predicting the same modal dependence on the size of the domain. We find that increasing the chamber length gives rise to less stable eigenmodes. We also realize that introducing an eigenmode whose frequency is sufficiently spaced from the acoustic modes leads to a conventional linear evolution of disturbances that can be accurately predicted by the biglobal stability framework. While undergoing spatial amplification in the streamwise direction, these disturbances will tend to decay as time elapses so long as their temporal growth rate remains negative. By seeding the computations with the real part of a specific eigenfunction, the DNS outcome reproduces not only the imaginary part of the disturbance, but also the circular frequency and temporal growth rate associated with its eigenmode. For radial fluctuations in which the vorticoacoustic wave contribution is negligible in relation to the hydrodynamic stability part, excellent agreement between DNS and biglobal stability predictions is ubiquitously achieved. For axial fluctuations, however, the DNS velocity will match the corresponding stability eigenfunction only when properly augmented by the vorticoacoustic solution for axially travelling waves associated with the Taylor–Culick profile. This analytical approximation of the vorticoacoustic mode is found to be quite accurate, especially when modified using a viscous dissipation function that captures the decaying envelope of the inviscid acoustic wave amplitude. In contrast, pursuant to both static and dynamic test cases, we find that when the frequency of the introduced eigenmode falls close to (or crosses over) an acoustic mode, a nonlinear mechanism is triggered that leads to the emergence of a secondary eigenmode. Unlike the original eigenmode, the latter materializes naturally in the computed flow without being artificially seeded. This natural occurrence may be ascribed to a nonlinear modal interplay in the form of internal, eigenmode-to-eigenmode coupling instead of an external, eigenmode pairing with acoustic modes. As a result of these interactions, large amplitude oscillations are induced.

JFM classification

Instability: Absolute/convective instability Acoustics: Aeroacoustics Vortex Flows: Vortex shedding
Type
Papers
Information
Journal of Fluid Mechanics , Volume 706 , 10 September 2012 , pp. 190 - 218
Copyright
Copyright © Cambridge University Press 2012

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Footnotes

‡

Present address: Fundamental and Applied Energetics Department, ONERA, Châtillon, France.

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Chedevergne et al. supplementary movie

Evolution of the unsteady radial velocity showing the effects of temporal damping for case 1 with an eigenmode of 40.409 - 9.164i.

Download Chedevergne et al. supplementary movie(Video)
Video 3.2 MB

Chedevergne et al. supplementary movie

Evolution of the unsteady radial velocity showing the effects of temporal damping for case 1 with an eigenmode of 40.409 - 9.164i.

Download Chedevergne et al. supplementary movie(Video)
Video 480.4 KB

Chedevergne et al. supplementary movie

Evolution of the unsteady axial velocity showing the effects of temporal damping for case 1 with an eigenmode of 40.409 - 9.164i.

Download Chedevergne et al. supplementary movie(Video)
Video 4.5 MB

Chedevergne et al. supplementary movie

Evolution of the unsteady axial velocity showing the effects of temporal damping for case 1 with an eigenmode of 40.409 - 9.164i.

Download Chedevergne et al. supplementary movie(Video)
Video 920.3 KB