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Structural changes of laminar separation bubbles induced by global linear instability

Published online by Cambridge University Press:  12 May 2010

D. RODRÍGUEZ  and
V. THEOFILIS
D. RODRÍGUEZ
Affiliation:
School of Aeronautics, Universidad Politécnica de Madrid, Pza. Cardenal Cisneros 3, E-28040 Madrid, Spain
V. THEOFILIS*
Affiliation:
School of Aeronautics, Universidad Politécnica de Madrid, Pza. Cardenal Cisneros 3, E-28040 Madrid, Spain
*
†Email address for correspondence: vassilis@torroja.dmt.upm.es
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Abstract

The topology of the composite flow fields reconstructed by linear superposition of a two-dimensional boundary layer flow with an embedded laminar separation bubble and its leading three-dimensional global eigenmodes has been studied. According to critical point theory, the basic flow is structurally unstable; it is shown that in the presence of three-dimensional disturbances the degenerate basic flow topology is replaced by a fully three-dimensional pattern, regardless of the amplitude of the superposed linear perturbations. Attention has been focused on the leading stationary eigenmode of the laminar separation bubble discovered by Theofilis et al. (Phil. Trans. R. Soc. Lond. A, vol. 358, 2000, pp. 3229–3324); the composite flow fields have been fully characterized with respect to the generation and evolution of their critical points. The stationary global mode is shown to give rise to a three-dimensional flow field which is equivalent to the classical U-shaped separation, defined by Hornung & Perry (Z. Flugwiss. Weltraumforsch., vol. 8, 1984, pp. 77–87), and induces topologies on the surface streamlines that are resemblant to the characteristic stall cells observed experimentally.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 655 , 25 July 2010 , pp. 280 - 305
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Abdessemed, N., Sherwin, S. J. & Theofilis, V. 2004 On unstable 2d basic states in low pressure turbine flows at moderate Reynolds numbers. In Thirty-fourth Fluid Dynamics Conference and Exhibit. Portland, OR, Paper 2004-2541. AIAA.CrossRefGoogle Scholar
Abdessemed, N., Sherwin, S. J. & Theofilis, V. 2009 Linear instability analysis of low pressure turbine flows. J. Fluid Mech. 628, 57–83.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 modes. J. Fluid Mech. 579, 305–314.CrossRefGoogle Scholar
Allen, T. & Riley, N. 1995 Absolute and convective instabilities in separation bubbles. Aerosp. J. 99, 439–448.Google Scholar
Barkley, D., Gomes, M. G. M. & Henderson, R. D. 2002 Three-dimensional instability in a flow over a backward-facing step. J. Fluid Mech. 473, 167–190.CrossRefGoogle Scholar
Bippes, H. & Turk, M. 1982 Half model testing applied to wing above and below stall. In Recent Contributions to Fluid Mechanics (ed. Haase, W.), pp. 22–30. Springer.CrossRefGoogle Scholar
Blumen, W. 1970 Shear layer instability of an inviscid compressible fluid. J. Fluid Mech. 40, 769–781.CrossRefGoogle Scholar
Boin, J.-P., Robinet, J.-C., Corre, C. & Deniau, H. 2006 3d steady and unsteady bifurcations in a shock-wave/laminar boundary layer interaction: a numerical study. Theor. Comput. Fluid Dyn. 20, 163–180.CrossRefGoogle Scholar
Briley, W. R. 1971 A numerical study of laminar separation bubbles using the Navier–Stokes equations. J. Fluid Mech. 47, 713–736.CrossRefGoogle Scholar
Carter, J. E. 1975 Inverse solutions for laminar boundary-layer flows with separation and reattachment. Tech. Rep. TR R-447. NASA.CrossRefGoogle Scholar
Cebeci, T. & Cousteix, J. 2001 Modeling and Computation of Boundary-Layer Flows: Solutions Manual and Computer Programs. Springer.Google Scholar
Cebeci, T., Keller, H. B. & Williams, P. G. 1979 Separating boundary-layer calculations. J. Comput. Phys. 31, 363–378.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
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids 2 (5), 765–777.CrossRefGoogle Scholar
Crouch, J. D., Garbaruk, A. & Magidov, D. 2007 Predicting the onset of flow unsteadiness based on global instability. J. Comput. Phys. 224, 924–940.CrossRefGoogle Scholar
Dallmann, U. 1982 Topological structures of three-dimensional flow separations. Tech. Rep. DFVLR-IB 221-82 A07.Google Scholar
Dallmann, U. C., Vollmers, H. & Su, W.-H. 1997 Flow topology and tomography for vortex identification in unsteady and in three-dimensional flows. In IUTAM Symposium on Simulation and Identification of Organized Structures in Flows, pp. 223–238, Lyngby, Denmark.Google Scholar
Dobrinsky, A. & Collis, S. S. 2000 Adjoint parabolized stability equations for receptivity prediction. Paper 2000-2651. AIAA.CrossRefGoogle Scholar
Fasel, H. & Postl, D. 2004 Interaction of separation and transtion in boundary layers: direct numerical simulations. In Proceedings of the IUTAM Laminar-Turbulent Symposium V (ed. Govindarajan, R.), pp. 71–88. Springer.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
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.CrossRefGoogle Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.CrossRefGoogle Scholar
Gurbacki, H. M. 2003 Ice-induced unsteady flowfield effects on airfoil performance. PhD thesis, University of Illinois, Urbana-Champaign, IL.Google Scholar
Hammond, D. A. & Redekopp, L. G. 1998 Local and global instability properties of separation bubbles. Eur. J. Mech. B/Fluids 17, 145–164.CrossRefGoogle Scholar
Hill, D. C. 1995 Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292, 183–204.CrossRefGoogle Scholar
Hornung, H. G. & Perry, A. E. 1984 Some aspects of three-dimensional separation. Part I. Streamsurface bifurcations. Z. Flugwiss. Weltraumforsch. 8, 77–87.Google Scholar
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Jacobs, J. & Bragg, M. B. 2006 Particle image velocimetry measurements of the separation bubble on an iced airfoil. Paper 2006-3646. AIAA.CrossRefGoogle Scholar
Kitsios, V., Rodríguez, D., Theofilis, V., Ooi, A. & Soria, J. 2009 Biglobal stability analysis in curvilinear coordinates of massively separated lifting bodies. J. Comput. Phys. 228, 7181–7196.CrossRefGoogle Scholar
Luchini, P. 2003 Adjoint methods in transient growth, global instabilities and turbulence control. In Proceedings of the 2nd Symposium on Global Flow Instability and Control, Crete, Greece, June 11–13, 2003 (ed. Theofilis, V., Colonius, T. & Chomaz, J.-M.). Fundacion General U.P.M. ISBN-13: 978-84-692-6245-0.Google Scholar
Lundbladh, A., Schmid, P., Berlin, S. & Henningson, D. 1994 Simulation of by-pass transition in spatially evolving flow. In CP-551, pp. 18.1–18.13. AGARD.Google Scholar
Marquet, O., Lombardi, M., Chomaz, J.-M., Sipp, D. & Jacquin, L. 2009 Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 1–21.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
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent velocity profile. J. Fluid Mech. 19, 543–556.CrossRefGoogle Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, Parts I, II. McGraw-Hill.Google Scholar
Perry, A. & Fairlie, B. 1975 A study of turbulent boundary-layer separation and reattachment. J. Fluid Mech. 69, 657–672.CrossRefGoogle Scholar
Perry, A. E. & Chong, M. S. 1987 A description of eddying motions and flow patterns using critical point concepts. Annu. Rev. Fluid Mech. 19, 125–156.CrossRefGoogle Scholar
Perry, A. E. & Hornung, H. G. 1984 Some aspects of three-dimensional separation. Part II. Vortex skeletons. Z. Flugwiss. Weltraumforsch. 8, 155–160.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
Robinet, J.-C. 2007 Bifurcations in shock-wave/laminar-boundary-layer interaction: global instability approach. J. Fluid Mech. 579, 85–112.CrossRefGoogle Scholar
Rodríguez, D. & Theofilis, V. 2008 On instability and structural sensitivity of incompressible laminar separation bubbles in a flat-plate boundary layer. In Thirty-eighth Fluid Dynamics Conference, Seattle, WA. Paper 2008-4148. AIAA.CrossRefGoogle Scholar
Rodríguez, D. & Theofilis, V. 2009 Massively parallel numerical solution of the biglobal linear instability eigenvalue problem using dense linear algebra. AIAA J. 47 (10), 2449–2459.CrossRefGoogle Scholar
Rodríguez, D. & Theofilis, V. 2010 On the birth of stall cells on airfoils. Theor. Comput. Fluid Dyn. Special Issue on Global Flow Instability and Control (in press) (doi:10.1007/s00162-010-0193-7).CrossRefGoogle Scholar
Salwen, H. & Grosch, C. 1981 The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfunction expansions. J. Fluid Mech. 104, 445–465.CrossRefGoogle Scholar
Schewe, G. 2001 Reynolds-number effects in flow around more-or-less bluff bodies. J. Wind Engng Ind. Aerodyn. 89, 1267–1289.CrossRefGoogle Scholar
Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Spalart, P. R. 1988 Direct numerical study of leading-edge contamination. In CP-438 Fluid Dynamics of Three-Dimensional Turbulent Shear Flows and Transition, pp. 5.1–5.13. Cesme, Turkey. AGARD.Google Scholar
Theofilis, V. 2000 Global linear instability in laminar separated boundary layer flow. In Proceedings of the IUTAM Laminar-Turbulent Symposium V (ed. Fasel, H. & Saric, W.), pp. 663–668. Springer. Sedona, AZ.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
Theofilis, V. 2009 The role of instability theory in flow control. In Fundamentals and Applications of Modern Flow Control. (ed. Joslin, R. D. & Miller, D.), AIAA Progress in Aeronautics and Astronautics, vol. 231, pp. 73–116. AIAA.CrossRefGoogle Scholar
Theofilis, V., Barkley, D. & Sherwin, S. J. 2002 Spectral/hp element technology for flow instability and control. Aeronaut. J. 106, 619–625.CrossRefGoogle Scholar
Theofilis, V., Fedorov, A., Obrist, D. & Dallmann, U. C. 2003 The extended Görtler–Hämmerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow. J. Fluid Mech. 487, 271–313.CrossRefGoogle 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, 3229–3324.CrossRefGoogle Scholar
Tumin, A. 2003 Multimode decomposition of spatially growing perturbations in a two-dimensional boundary layer. Phys. Fluids 15 (9), 2525–2540.CrossRefGoogle Scholar
Vorobieff, P. & Rockwell, D. 1996 Wavelet filtering for topological decomposition of flow fields. Intl J. Imaging Syst. Technol. 7, 211–214.3.0.CO;2-B>CrossRefGoogle Scholar
Winkelmann, A. & Barlow, B. 1980 Flowfield model for a rectangular planform wing beyond stall. AIAA J. 8, 1006–1008.CrossRefGoogle Scholar