Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-27T20:21:53.240Z Has data issue: false hasContentIssue false
  • English
  • Français

On a degeneracy of temporal secondary instability modes in Blasius boundary-layer flow

Published online by Cambridge University Press:  26 April 2006

W. Koch
W. Koch
Affiliation:
DLR Institute for Theoretical Fluid Mechanics, D-3400 Göttingen, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Using the parallel-flow approximation the prechaotic bifurcation behaviour of Blasius boundary-layer flow is studied at finite Reynolds numbers. The objective of this investigation is to search for qualitative solution changes which might be linked to the rapid breakdown at transition. As a first step this requires the computation of the two-dimensional primary equilibrium surface of nonlinear Tollmien-Schlichting waves. This two-dimensional neutral surface exhibits a period-halving bifurcation which is qualitatively different from the situation for plane Poiseuille flow. At the same time the numerically computed equilibrium solution offers the possibility of assessing the range of convergence of weakly nonlinear results.

In a second step the stability of this nonlinear equilibrium solution is investigated with respect to three-dimensional disturbances. Of particular importance is the existence of a modal degeneracy between amplified secondary instability modes, implying locally algebraic growth. On decreasing the Reynolds number, the amplification rate of this direct resonance point switches from being amplified to being damped. Interestingly, the Reynolds number corresponding to this zero-amplification point seems to be in the vicinity of the experimentally observed transition Reynolds number for Blasius flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 243 , October 1992 , pp. 319 - 351
Copyright
© 1992 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barkley, D. 1990 Theory and predictions for finite-amplitude waves in two-dimensional plane Poiseuille flow.. Phys. Fluids A 2, 955–970.Google Scholar
Benney, D. J. & Gustavsson, L. H. 1981 A new mechanism for linear and nonlinear hydrodynamic instability. Stud. Appl. Maths 64, 185–209.Google Scholar
Bertolotti, F. P. 1991a Linear and nonlinear stability of the boundary layers with streamwise varying properties. Ph.D. thesis, The Ohio State University.
Bertolotti, F. P. 1991b Compressible boundary layer stability analyzed with the PSE equations. AIAA Preprint 91–1637.
Boyd, J. P. 1989 Chebyshev & Fourier Spectral Methods. Lecture Notes in Engineering, vol. 49. Springer.
Bridges, T. J. 1991 A dynamical systems approach to boundary layer transition. Warwick Preprint 2/1991.
Chang, C.-L., Malik, M. R., Erlebacher, G. & Hussaini, M. Y. 1991 Compressible stability of growing boundary layers using parabolized stability equations. AIAA Preprint 91–1636.Google Scholar
Cliffe, K. A. 1988 Numerical calculations of the primary-flow exchange process in the Taylor problem. J. Fluid Mech. 197, 57–79.Google Scholar
Conlisk, A. T., Burggraf, O. R. & Smith, F. T. 1987 Nonlinear neutral modes in the Blasius boundary layer In Forum on Unsteady Separation. ASME FED vol. 52, pp. 119–121.
Dhawan, S. 1953 Direct measurements of skin friction. NACA Rep. 1121.
Dhawan, S. & Narasimha, R. 1958 Some properties of boundary layer flow during transition from laminar to turbulent flow. J. Fluid Mech. 3, 418–436.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Eckhaus, W. 1965 Studies in Non-Linear Stability Theory. Springer.
Ehrenstein, U. 1988 Lösung der linearen, inkompressiblen Stördifferentialgleichungen mittels Chebyshev-Kollokation. Internal DFVLR-Rep. IB 221–88 A 09.
Ehrenstein, U. & Koch, W. 1989 Nonlinear bifurcation study of plane Poiseuille flow. DLR Res. Rep. FB 89–42.
Ehrenstein, U. & Koch, W. 1991 Three-dimensional wave-like equilibrium states in plane Poiseuille flow. J. Fluid Mech. 228, 111–148.Google Scholar
Fasel, H. 1976 Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations. J. Fluid Mech. 78, 355–383.Google Scholar
Fischer, T. M. 1990 Ein mathematisch-physikalisches Modell zur Beschreibung transitioneller Grenzschichtströmungen I. Die linearen und nichtlinearen Störungsdifferentialgleichungen. DLR Res. Rep. FB 90–24 (with corrigendum).
Fischer, T. M. 1991 On the nonlinear development of disturbances in boundary layers. Eur. J. Mech. B Fluids 10 (suppl.), 310).Google Scholar
Fischer, T. M. 1992 On the nonlinear development of disturbances in boundary layers. Stud. Appl. Maths. (submitted).Google Scholar
Gajjar, J. & Smith, F. T. 1985 On the global instability of free disturbances with a time-dependent nonlinear viscous critical layer. J. Fluid Mech. 157, 53–77.Google Scholar
Gottlieb, D., Hussaini, M. Y. & Orszag, S. A. 1984 Theory and applications of spectral methods. In Spectral Methods for Partial Differential Equations (ed. R. G. Voigt, D. Gottlieb & M. Y. Hussaini), pp. 1–54. Philadelphia: SIAM.
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer.
Gustavsson, L. H. 1986 Excitation of direct resonances in plane Poiseuille flow. Stud. Appl. Maths 75, 227–248.Google Scholar
Herbert, T. 1975 On finite amplitudes of periodic disturbances of the boundary layer along a flat plate. In Proc. Fourth Intl. Conf. on Num. Methods in Fluid Dynamics (ed. R. D., Richtymer), Lecture Notes in Physics, vol. 35, pp. 212–217, Springer. (See also Res. Rep. DLR-FB 74–53.)
Herbert, T. 1977 Die Neutralfläche der ebenen Poiseuilleströmung. Habilitationsschrift, Universität Stuttgart.
Herbert, T. 1988 Secondary instability of boundary layers. Ann. Rev. Fluid Mech. 20, 487–526.Google Scholar
Iooss, G. & Joseph, D. D. 1980 Elementary Stability and Bifurcation Theory. Springer.
Itoh, N. 1974 Spatial growth of finite wave disturbances in parallel and nearly parallel flows. part 2. The numerical results for the flat plate boundary layer. Trans. Japan Soc. Aerospace Sci. 17, 175–186.Google Scholar
Jang, P. S., Benney, D. J. & Gran, R. L. 1986 On the origin of streamwise vortices in a turbulent boundary layer. J. Fluid Mech. 169, 109–123.Google Scholar
Jones, C. A. 1988 Multiple eigenvalues and mode classification in plane Poiseuille flow. Q. J. Mech. Appl. Maths 41, 363–382.Google Scholar
Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. H., Rabinowitz), pp. 359–384. Academic.
Kleiser, L. & Zang, T. A. 1991 Numerical simulation of transition in wall-bounded shear flows. Ann. Rev. Fluid Mech. 23, 495–537.Google Scholar
Koch, W. 1986 Direct resonances in Orr—Sommerfeld problems. Acta Mech. 59, 11–29.Google Scholar
Laurien, E. & Kleiser, L. 1989 Numerical simulation of boundary-layer transition and transition control. J. Fluid Mech. 199, 403–440.Google Scholar
Lifshits, A. M. & Shtern, V. N. 1986 Gauss method in nonlinear stability problems. In Proc. Sixth GAMM Conf. on Numerical Methods in Fluid Mechanics (ed. D. Rues & W. Kordulla), pp. 241–247. Vieweg & Sohn.
Lifshits, A. M., Rakhmatullaev, R. D. & Shtern, V. N. 1989 Threshold of development of three-dimensional structures in Blasius flow. Fluid Dyn. 24, 515–520.Google Scholar
Likhachev, O. A. & Shtern, V. N. 1975 Self-oscillating flow in the boundary layer. J. Appl. Mech. Tech. Phys. 16, 564–569.Google Scholar
Mahalov, A. & Leibovich, S. 1991 Weakly nonlinear analysis of rotating Hagen—Poiseuille flow. Eur. J. Mech. B: Fluids 10 (suppl), 55–60.Google Scholar
Milinazzo, F. A. & Saffman, P. G. 1985 Finite-amplitude steady waves in plane viscous shear flows. J. Fluid Mech. 160, 281–295.Google Scholar
Morkovin, M. V. 1991 Panoramic view of changes in vorticity distribution in transition instabilities and turbulence. In Boundary Layer Stability and Transition to Turbulence (ed. D. C. Reda, H. L. Reed & R. Kobayashi). ASME FED, vol. 114, pp. 1–12.
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 374–385.Google Scholar
Pugh, J. D. & Saffman, P. G. 1988 Two-dimensional superharmonic stability of finite amplitude waves in plane Poiseuille flow. J. Fluid Mech. 194, 295–307.Google Scholar
Saffman, P. G. 1983 Vortices, stability and turbulence. Ann. N.Y. Acad. Sci. 404, 12–24.Google Scholar
Schlichting, H. 1958 Grenzschicht-Theorie (3rd edn). Braun.
Schubauer, G. B. & Skramstad, H. K. 1948 Laminar boundary-layer oscillations and transition on a flat plate. NACA Rep. 909.
Sen, P. K. & Vashist, T. K. 1989 On the nonlinear stability of boundary-layer flow over a flat plate.. Proc. R. Soc. Lond. A 424, 81–92.Google Scholar
Shanthini, R. 1989 Degeneracies of the temporal Orr—Sommerfeld eigenmodes in plane Poiseuille flow. J. Fluid Mech. 201, 13–34.Google Scholar
Smith, F. T. 1979a On the nonparallel flow stability of the Blasius boundary layer.. Proc. R. Soc. Lond A 366, 91–109.Google Scholar
Smith, F. T. 1979b Nonlinear stability of boundary layersr for disturbances of various sizes.. Proc. R. Soc. Lond. A 368, 573–589 (and corrections Proc. R. Soc. Lond. A 371 (1980), 439–440).Google Scholar
Smith, F. T. 1988 A reversed-flow singularity in interacting boundary layers.. Proc. R. Soc. Lond. A 420, 21–52.Google Scholar
Smith, F. T. 1991 Nonlinear 3D interactions in open-flow transition. Eur. J. Mech. B Fluids 10 (suppl.), 301–306.Google Scholar
Smith, F. T. & Burggraf, O. R. 1985 On the development of large-sized short-scaled disturbances in boundary layers.. Proc. R. Soc. Lond A 399, 25–55.Google Scholar
Smith, F. T., Doorly, D. J. & Rothmayer, A. P. 1990 On displacement-thickness, wall-layer and mid-flow scales in turbulent boundary layers, and slugs of vorticity in channel and pipe flows.. Proc. R. Soc. Lond. A 428, 255–281.Google Scholar
Smith, F. T. & Khorrami, F. 1991 The interactive breakdown in supersonic ramp flow. J. Fluid Mech. 224, 197–215.Google Scholar
Soibelman, I. & Meiron, D. I. 1991 Finite-amplitude bifurcations in plane Poiseuille flow: two-dimensional Hopf bifurcations. J. Fluid Mech. 229, 389–416.Google Scholar
Spalart, P. R. 1984 A spectral method for external viscous flows. Contemp. Maths 28, 315–335.Google Scholar
Stuart, J. T. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353–370.Google Scholar