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Global stability of swept flow around a parabolic body: features of the global spectrum

Published online by Cambridge University Press:  14 January 2011

CHRISTOPH J. MACK  and
PETER J. SCHMID
CHRISTOPH J. MACK*
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS-École Polytechnique, F-91128 Palaiseau, France Department of Numerical Mathematics, Universität der Bundeswehr (UniBw), D-85577 Munich, Germany
PETER J. SCHMID
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS-École Polytechnique, F-91128 Palaiseau, France
*
†Email address for correspondence: christoph.mack@mytum.de
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Abstract

The global temporal stability of three-dimensional compressible flow about a yawed parabolic body of infinite span is investigated using an iterative eigenvalue technique in combination with direct numerical simulations. The computed global spectrum provides a comprehensive picture of the temporal perturbation dynamics of the flow, and a wide and rich variety of modes has been uncovered for the investigated parameter choices: stable and unstable boundary-layer modes, different types of stable and unstable acoustic modes, and stable wavepacket modes have been found. A parameter study varying the spanwise perturbation wavenumber and the sweep Reynolds number reproduced a preferred spanwise length scale and a critical Reynolds number for a boundary-layer or acoustic instability. Convex leading-edge curvature has been found to have a strongly stabilizing effect on boundary-layer modes but only a weakly stabilizing effect on acoustic modes. Furthermore, for certain parameter choices, the acoustic modes have been found to dominate the boundary-layer modes.

JFM classification

Compressible Flows: Compressible boundary layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 669 , 25 February 2011 , pp. 375 - 396
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Åkervik, E., Brandt, L., Henningson, D. S., Hœpfner, J., Marxen, O. & Schlatter, P. 2006 Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18, 068102.CrossRefGoogle Scholar
Bagheri, S., Schlatter, P., Schmid, P. J. & Henningson, D. S. 2009 Global stability of a jet in crossflow. J. Fluid Mech. 624, 33–44.CrossRefGoogle Scholar
Balakumar, P. & Malik, M. R. 1992 Discrete modes and continuous spectra in supersonic boundary layers. J. Fluid Mech. 239, 631–656.CrossRefGoogle Scholar
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Bertolotti, F. P. 1999 On the connection between cross-flow vortices and attachment-line instabilities. In IUTAM Symposium on Laminar–Turbulent Transition, Sedona, AZ, pp. 625–630.Google Scholar
Bippes, H. 1999 Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instability. Prog. Aerosp. Sci. 35, 363–412.CrossRefGoogle Scholar
Bonfigli, G. & Kloker, M. 2007 Secondary instability of crossflow vortices: validation of the stability theory by direct numerical simulation. J. Fluid Mech. 583, 229–272.CrossRefGoogle Scholar
Gaster, M. 1965 A simple device for preventing turbulent contamination on swept leading edges. J. R. Aero. Soc. 69, 788.CrossRefGoogle Scholar
Gray, W. E. 1952 The effect of wing sweep on laminar flow. Tech. Rep. RAE TM Aero 255. British Royal Aircraft Establishment.Google Scholar
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disc. Phil. Trans. R. Soc. Lond. A 248, 155–199.Google Scholar
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part I. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 33–54.CrossRefGoogle Scholar
Hall, P., Malik, M. & Poll, D. I. A. 1984 On the stability of an infinite swept attachment-line boundary layer. Proc. R. Soc. Lond. A 395, 229–245.Google Scholar
Joslin, R. D. 1995 Direct simulation of evolution and control of three-dimensional instabilities in attachment-line boundary layers. J. Fluid Mech. 291, 369–392.CrossRefGoogle Scholar
Joslin, R. D. 1996 Simulation of nonlinear instabilities in an attachment-line boundary layer. Fluid Dyn. Res. 18, 81–97.CrossRefGoogle Scholar
Kazakov, A. V. 1990 Effect of surface temperature on the stability of swept attachment line boundary layer. Fluid Dyn. 25 (6), 875–878.CrossRefGoogle Scholar
Knoll, D. A. & Keyes, D. E. 2004 Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193 (2), 357–397.CrossRefGoogle Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users' Guide. Software, Environments and Tools: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM.Google Scholar
Lin, R. S. & Malik, M. R. 1995 Stability and transition in compressible attachment-line boundary-layer flow. Tech. Rep. 952041. SAE.Google Scholar
Lin, R. S. & Malik, M. R. 1996 On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz flow. J. Fluid Mech. 311, 239–255.CrossRefGoogle Scholar
Lin, R. S. & Malik, M. R. 1997 On the stability of attachment-line boundary layers. Part 2. The effect of leading edge curvature. J. Fluid Mech. 333, 125–137.CrossRefGoogle Scholar
Liu, Y., Zaki, T. A. & Durbin, P. A. 2008 Boundary-layer transition by interaction of discrete and continuous modes. J. Fluid Mech. 604, 199–233.CrossRefGoogle Scholar
Mack, C. J. & Schmid, P. J. 2010 a Direct numerical study of hypersonic flow about a swept parabolic body. Comput. Fluids 39, 1932–1943.CrossRefGoogle Scholar
Mack, C. J. & Schmid, P. J. 2010 b A preconditioned Krylov technique for global hydrodynamic stability analysis of large-scale compressible flows. J. Comput. Phys. 229 (3), 541–560.CrossRefGoogle Scholar
Mack, C. J., Schmid, P. J. & Sesterhenn, J. L. 2008 Global stability of swept flow around a parabolic body: connecting attachment-line and crossflow modes. J. Fluid Mech. 611, 205–214.CrossRefGoogle Scholar
Mack, L. M. 1984 Boundary layer linear stability theory. Tech. Rep. 709. AGARD.Google Scholar
Moretti, G. 1987 Computations of flows with shocks. Annu. Rev. Fluid Mech. 19, 313–337.CrossRefGoogle Scholar
Obrist, D. & Schmid, P. J. 2010 Algebraically decaying modes and wave packet pseudomodes in swept Hiemenz flow. J. Fluid Mech. 643, 309–332.CrossRefGoogle Scholar
Pfenninger, W. 1965 Some results form the X-21 programm. Part I. Flow phenomena at the leading edge of swept wings. Tech. Rep. 97. AGARDograph.Google Scholar
Poll, D. I. A. 1979 Transition in the infinite swept attachment-line boundary layer. Aeronaut. Q. 30, 607–628.CrossRefGoogle Scholar
Saad, Y., ed. 2003 Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Saric, W. S., Reed, H. L. & White, E. B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413–440.CrossRefGoogle Scholar
Semisynov, A. I., Fedorov, A. V., Novikov, V. E., Semionov, N. V. & Kosinov, A. D. 2003 Stability and transition on a swept cylinder in a supersonic flow. J. Appl. Mech. Tech. Phys. 44, 212–220.CrossRefGoogle Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.CrossRefGoogle Scholar
Sorensen, D. C. 1992 Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13 (1), 357–385.CrossRefGoogle Scholar
Spalart, P. R. 1988 Direct numerical study of leading-edge contamination. In AGARD-CP-438, pp. 5/1–5/13.Google Scholar
Spalart, P. R. 1989 Direct numerical study of crossflow instability. In IUTAM Laminar-Turbulent Symposium III, pp. 621–630.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249–315.CrossRefGoogle Scholar
Trefethen, L. N. 2005 Wave packet pseudomodes of variable coefficient differential operator. Proc. R. Soc. Lond. A 461 (2062), 3099–3122.Google Scholar
van der Vorst, H. A. 1992 Bi-CGStab: A fast and smoothly converging variation of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644.CrossRefGoogle Scholar
Wassermann, P. & Kloker, M. 2003 Transition mechanisms induced by travelling crossflow vortices in a three-dimensional boundary layer. J. Fluid Mech. 483, 67–89.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. A. 2005 Mode interaction and the bypass route to transition. J. Fluid Mech. 531, 85–111.CrossRefGoogle Scholar