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Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

Published online by Cambridge University Press:  16 October 2008

UWE EHRENSTEIN and
FRANÇOIS GALLAIRE
UWE EHRENSTEIN
Affiliation:
IRPHÉ UMR 6594, Aix-Marseille Université, CNRS, 49 Rue Joliot-Curie, F-13384 Marseille Cedex 13, France ehrenstein@irphe.univ.mrs.fr
FRANÇOIS GALLAIRE
Affiliation:
Laboratoire J. A. Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 02, France francois.gallaire@unice.fr
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Abstract

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A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 614 , 10 November 2008, pp. 315 - 327
Copyright
Copyright © Cambridge University Press 2008

References

REFERENCES

Åkervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D. S. 2008 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. B/Fluids 27, 501–513.CrossRefGoogle Scholar
Åkervik, E., Hœpffner, J., Ehrenstein, U. & Henningson, D. S. 2007 Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314.CrossRefGoogle Scholar
Alizard, F. & Robinet, J.-C. 2007 Spatially convective global modes in a boundary layer. Phys. Fluids 19, 114105.CrossRefGoogle Scholar
Barkley, D., Gomes, M. & Henderson, D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–189.CrossRefGoogle Scholar
Beaudoin, J.-F., Cadot, O., Aider, J.-L. & Wesfreid, J. 2004 Three-dimensional stationary flow over a backward-facing step. Eur. J. Mech. B/Fluids 23, 147–155.CrossRefGoogle Scholar
Bernard, A., Foucaut, J. M., Dupont, P. & Stanislas, M. 2003 Decelerating boundary layer: a new scaling and mixing length model. AIAA J. 41, 248–255.CrossRefGoogle Scholar
Cherry, N. J., Hiller, R. & Latour, M. P. 1984 Unsteady measurements in a separating and reattaching flow. J. Fluid Mech. 144, 13–46.CrossRefGoogle Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Dallmann, U., Herberg, Th., Gebing, H., Su, W.-H. & Zhang, H.-Q. 1995 Flow field diagnostics: topological flow changes and spatio-temporal flow structure. AIAA Paper 95-791.Google Scholar
Dovgal, A. V., Kozlov, V. V. & Michalke, A. 1994 Laminar boundary layer separation: instability and associated phenomena. Prog. Aerospace Sci. 30, 61–94.CrossRefGoogle 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
Gallaire, F., Marquillie, M. & Ehrenstein, U. 2007 Three-dimensional transverse instabilities in detached boundary-layers. J. Fluid Mech. 571, 221–233.CrossRefGoogle Scholar
Häggmark, C. P., Bakchinov, A. A. & Alfredsson, P. H. 2000 Experiments on a two-dimensional laminar separation bubble. Phi. Trans. R. Soc. Lond. A. 358, 3193–3205.CrossRefGoogle Scholar
Hammond, D. A. & Redekopp, L. G. 1998 Local and global instability properties of separation bubbles. Eur. J. Mech. B/Fluids 17, 145–164.Google Scholar
Joslin, R. D., Streett, C. L. & Chang, C.-L. 1993 Spatial direct numerical simulation of boundary-layer transition mechanisms: validation of pse theory. Theoret. Comput. Fluid Dyn. 4, 271–288.Google Scholar
Kaiktsis, L., Karniadakis, G. E. & Orszag, S. A. 1991 Onset of three-dimensionality, equilibria and early transition in flow over a backward-facing step. J. Fluid Mech. 231, 501–528.CrossRefGoogle Scholar
Kaiktsis, L., Karniadakis, G. E. & Orszag, S. A. 1996 Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step. J. Fluid Mech. 321, 157–187.CrossRefGoogle Scholar
Marquet, O., Sipp, D., Chomaz, J.-M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429–443.CrossRefGoogle Scholar
Marquillie, M. & Ehrenstein, U. 2003 On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid Mech. 490, 169–188.CrossRefGoogle Scholar
Pauley, L. L., Moin, P. & Reynolds, W. C. 1990 The structure of two-dimensional separation. J. Fluid Mech. 220, 397–411.CrossRefGoogle Scholar
Peyret, R. 2002 Spectral Methods for Incompressible Flows. Springer.Google Scholar
Rist, U. & Maucher, U. 2002 Investigations of time-growing instabilities in laminar separation bubbles. Eur. J. Mech. B/Fluids 21, 495–509.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2002 On the stability of a falling liquid curtain. J. Fluid Mech. 463, 163–171.CrossRefGoogle Scholar
Stoer, J. & Bulirsch, R. 1992 Introduction to Numerical Analysis. Springer.Google Scholar
Theofilis, V., Hein, S. & Dallmann, U. 2000 On the origins of unsteadiness and three-dimensionality in a laminar separation bubble. Phil. Trans. R. Soc. Lond. A 358, 3229–3246.Google Scholar
Trefethen, N. & Embree, M. 2005 Spectra and Pseudospectra; The Behaviour of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Williams, P. T. & Baker, A. J. 1997 Numerical simulations of laminar flow over a 3d backward-facing step. Intl J. Numer. Meth. Fluids 24, 1159–1183.Google Scholar