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Stability and sensitivity analysis in a simplified solid rocket motor flow

Published online by Cambridge University Press:  28 March 2013

G. Boyer ,
G. Casalis  and
J. L. Estivalèzes
G. Boyer
Affiliation:
CNES - Direction des Lanceurs, 52 rue Jacques Hillairet, 75612 Paris CEDEX, France
G. Casalis
Affiliation:
ONERA, The French Aerospace Lab, 2 avenue douard Belin, B.P. 74025, 31055 Toulouse CEDEX 4, France
J. L. Estivalèzes
Affiliation:
ONERA, The French Aerospace Lab, 2 avenue douard Belin, B.P. 74025, 31055 Toulouse CEDEX 4, France
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Abstract

The present article aims at enhancing the computation of the global stability modes of the internal flow of solid rocket motors (SRMs) approximated by the Taylor–Culick solution. This modal approach suffers from the consequences of the non-normality of the global linearized incompressible Navier–Stokes operator, namely the lack of robustness of the eigenvalues that can lead to the computation of pseudo-modes rather than actual eigenmodes. In this respect, the effects of non-normality associated with strongly amplified eigenfunctions are highlighted on a simplified convective–diffusive stability problem with uniformly accelerated base state, the latter property being a typical characteristic of the Taylor–Culick flow. Non-convergence zones for the eigenvalues are exhibited for large Reynolds numbers and are related to the critical sensitivity to disturbances applied to one of the boundary conditions. For this reason, and according to experimental and numerical data related to the stability of simplified SRMs, a global stability analysis is performed assuming that the hydrodynamic fluctuations emerge from a geometrical defect applied at the sidewall. This comes to fix the upstream boundary condition at the abscissa of the sidewall disturbance. The resulting eigenmodes are shown to be discrete, numerically converged, well identified by a finite number of points of undefined phase of the velocity fluctuations. They marginally depend on Reynolds number variations, but are modified by changes on the boundaries location. As in the simplified problem, the inflow boundary condition is the most critical in terms of sensitivity to numerical errors, although not dramatic. Finally, the sensitivity analysis to infinitesimal base flow changes indicates that the variations applied close to the inflow boundary condition induce the largest move of the eigenvalues. In spite of the large non-normal effects induced by the large polynomial growth of the eigenfunctions, this paper shows that discrete instabilities may emerge from a wall defect, in contrast to configurations without such a geometrical perturbation whose dynamics may be rather driven by pseudo-modes.

JFM classification

Instability: Absolute/convective instability Vortex Flows: Vortex shedding
Type
Papers
Information
Journal of Fluid Mechanics , Volume 722 , 10 May 2013 , pp. 618 - 644
Copyright
©2013 Cambridge University Press

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References

Airiau, C. 2000 Non-parallel acoustic receptivity of a Blasius boundary layer using an adjoint approach. Flow Turbul. Combust. 65 (3/4), 347–367.CrossRefGoogle Scholar
A˚kervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D. S. 2008 Global two-dimensional stability measures of flate plate boundary layer flow. Eur. J. Mech. (B/Fluids) 27, 501–513.CrossRefGoogle Scholar
Alizard, F. & Robinet, J.-C. 2007 Spatially convective global modes in a boundary layer. Phys. Fluids 19.Google Scholar
Arnoldi, W. E. 1951 The principle of minimized iteration in the solution of the matrix eigenvalue problem. Q. Appl. Maths 9, 17–29.Google Scholar
Avalon, G. & Josset, T. 2006 Cold gas experiments applied to the understanding of aeroacoustic phenomena inside solid propellant rockets. In 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and exhibit, Sacramento, California.Google Scholar
Blackford, L. S., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D. & Whaleyy, R. C. 1997 Scalapack: a linear algebra library for message-passing computers.Google Scholar
Bottaro, A., Corbett, P. & Lucchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293–302.Google Scholar
Boyer, G., Casalis, G. & Estivalèzes, J.-L. 2012 Theoretical investigation of the parietal vortex shedding in solid rocket motors. In 48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Atlanta, Georgia.Google Scholar
Brandt, L., Sipp, D., Pralits, J. O. & Marquet, O. 2011 Effects of base-flow variation in noise amplifiers: the flat-plate boundary layer. J. Fluid Mech. 687, 503–528.CrossRefGoogle Scholar
Brown, R. S., Dunlap, R., Young, S. W. & Waugh, R. C. 1981 Vortex shedding as a source of acoustic energy in segmeted solid motor rockets. J. Spacecraft 18, 312–319.CrossRefGoogle Scholar
Canuto, C., Hussaini, Y. M., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Casalis, G., Avalon, G. & Pineau, J. Ph. 1998 Spatial instability of planar channel flow with fluid injection through porous walls. Phys. Fluids 10, 2258–2568.Google Scholar
Chedevergne, F. 2007 Instabilités intrinsèques des moteurs à propulsion solide. PhD thesis, ENSAE.Google Scholar
Chedevergne, F., Casalis, G. & Feraille, T. 2006 Biglobal linear stability analysis of the flow induced by wall injection. Phys. Fluids 18.Google Scholar
Chedevergne, F., Casalis, G. & Majdalani, J. 2012 Direct numerical simulation and biglobal stability investigations of the gaseous motion in solid rocket motors. J. Fluid Mech. 706, 190–218.Google Scholar
Culick, F. 1966 Rotational axisymmetric mean flow and damping of acoustic waves in a solid propellant rocket. AIAA J. 4, 1462–1464.Google Scholar
Don, W. S. & Solomonoff, A. 1997 Accuracy enhancement for higher derivatives using Chebyshev collocation and a mapping technique. SIAM J. Sci. Comput. 18 (4), 1040–1055.CrossRefGoogle Scholar
Dunlap, R., Blackner, A. M., Waugh, R. C., Brown, R. S. & Willoughby, P. G. 1990 Internal flow field studies in a simulated cylindrical port rocket chamber. J. Propulsion 6 (6), 290–704.Google Scholar
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218.Google Scholar
Flandro, G. A. 1995 Effects of vorticity on rocket combustion stability. J. Propul. Power 11 (4), 607–625.Google Scholar
Flandro, G. A. & Majdalani, J. 2003 Aeroacoustic instability in rockets. AIAA J. 41, 485–497.Google Scholar
Griffond, J. & Casalis, G. 2001 On the nonparallel stability of the injection induced two-dimensional Taylor flow. Phys. Fluids 13 (1635).CrossRefGoogle Scholar
Hill, D. C. 1995 Adjoint systems and their role in the receptivity problems for boundary layers. J. Fluid Mech. 292, 183–204.Google Scholar
Majdalani, J. 2009 Multiple asymptotic solutions for axially travelling waves in porous channels. J. Fluid Mech. 636 (1), 59–89.Google Scholar
Majdalani, J., Fischbach, S. R. & Flandro, G. A. 2006 Improved energy normalization function in rocket motor stability calculations. Aerosp. Sci. Technol. 10, 495–500.CrossRefGoogle Scholar
Marquet, O., Lombardi, M., Chomaz, J.-M., Sipp, D. & Jacquin, L. 2009 Direct and adjont global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622.Google Scholar
Marquet, O., Sipp, D. & Jacqin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.Google Scholar
Monokrousous, A., A˚kervik, E., Brandt, L. & Henningson, D. S. 2010 Global three-dimensional optimal disturbances in the Blasius boundary layer flow using time-steepers. J. Fluid Mech. 650.Google Scholar
Prévost, M. & Godon, J.-C. 2005 Thrust oscillations in reduced scale solid rocket motors. Part 1. Experimental investigations. In 41nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and exhibit, Tucson, Arizona.Google Scholar
Rodríguez, D. & Theophilis, V. 2009 Massively parallel solution of the BiGlobal eigenvalue problem using dense linear algebra. AIAA J. 47, 2449–2459.Google Scholar
Sipp, D., Marquet, O. & Meglia, Ph. 2010 Dynamics and control of global instabilities in open flows; a linearized approach. Appl. Mech. Rev. 63.Google Scholar
Taylor, G. I. 1956 Fluid flows in regions bounded by porous surfaces. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 234 (1199), 456–475.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249–315.Google Scholar
Trefethen, L.-N. & Embree, M. 2005 Spectra and Peudospectra. Princeton University Press.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261.Google Scholar
Tumin, A. 2003 Multimode decomposition of spatially growing perturbations in a two-dimensional boundary layer. Phys. Fluids 15 (9), 2525–2540.Google Scholar
Varapaev, V. N. & Yagodkin, V. I. 1969 Flow instability in a channel with porous wall. Izv. Acad. Nauk SSSR Mekh. Zhidk. Gaza 4 (5).Google Scholar
Vuillot, F. 1995 Vortex shedding phenomena in solid propellant motors. J. Propul. Power 11 (4), 626–639.CrossRefGoogle Scholar
Yagodkin, V. I. 1967 Use of channels with porous walls for studying flows which occur during combustion of solid propellants. In 18th International Astronautical Congress AGARD.Google Scholar