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Linear and nonlinear stability of the Blasius boundary layer

Published online by Cambridge University Press:  26 April 2006

F. P. Bertolotti ,
Th. Herbert  and
P. R. Spalart
F. P. Bertolotti
Affiliation:
ICASE NASA Langley Research Center, Hampton, VA 23665-5225, USA
Th. Herbert
Affiliation:
Department of Mechanical Engineeering, The Ohio State University, Columbus, OH 43210, USA
P. R. Spalart
Affiliation:
NASA Ames Research Center, Moffet Field, CA 94035, USA Boeing Commercial Airplane Group, Seattle, WA 98124, USA.
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Abstract

Two new techniques for the study of the linear and nonlinear instability in growing boundary layers are presented. The first technique employs partial differential equations of parabolic type exploiting the slow change of the mean flow, disturbance velocity profiles, wavelengths, and growth rates in the streamwise direction. The second technique solves the Navier–Stokes equation for spatially evolving disturbances using buffer zones adjacent to the inflow and outflow boundaries. Results of both techniques are in excellent agreement. The linear and nonlinear development of Tollmien–Schlichting (TS) waves in the Blasius boundary layer is investigated with both techniques and with a local procedure based on a system of ordinary differential equations. The results are compared with previous work and the effects of non-parallelism and nonlinearly are clarified. The effect of nonparallelism is confirmed to be weak and, consequently, not responsible for the discrepancies between measurements and theoretical results for parallel flow. Experimental uncertainties, the adopted definition of the growth rate, and the transient initial evolution of the TS wave in vibrating-ribbon experiments probably cause the discrepancies. The effect of nonlinearity is consistent with previous weakly nonlinear theories. White nonlinear effects are small near branch I of the neutral curve, they are significant near branch II and delay or event prevent the decay of the wave.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 242 , September 1992 , pp. 441 - 474
Copyright
© 1992 Cambridge University Press

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References

Ashpis, D. E. & Reshotko, E. 1990 The vibrating ribbon problem revised. J. Fluid Mech. 213, 534–547.Google Scholar
Bayliss, A., Maestrello, L., Parikh, P. & Turkel, E. 1985 Wave phenomena in a high Reynolds number compressible boundary layer. ICASE Rep. 85–56.Google Scholar
Bertolotti, F. P. 1991 Linear and nonlinear stability of boundary layers with streamwise varying properties. Ph.D. thesis, The Ohio-State University, Columbus, Ohio.
Bertolotti, F. P. & Herbert, T. 1991 Analysis of the linear stability of compressible boundary layers using the PSE. J. Theor. Comput. Fluid Dyn. 3, 117–124.Google Scholar
Bouthier, M. 1972 Stabilité linéaire des écoulements presque parallèles. J. Méc. 11, 599–621.Google Scholar
Bouthier, M. 1973 Stabilité linéaire des écoulements presque parallèles: Part II. La couche limite de Blasius. J. Méc. 12, 76–95.Google Scholar
Bridges, T. J. & Morris, P. J. 1986 Boundary layer stability calculations. Phys. Fluids 30, 3351–3358.Google Scholar
Crouch, J. D. 1990 A nonlinear mode-interaction model for boundary-layer receptivity. Bull. Am. Phys. Soc. 35, 2262.Google Scholar
Crouch, J. D. & Bertolotti, F. P. 1992 Nonlocalized receptivity of boundary layers to three-dimensional disturbances. AIAA Paper 92-0740.Google Scholar
Drazin, A. P. & Reid, A. W. 1981 Hydrodynamic Stability. Cambridge University Press.
Fasel, H. F. 1976 Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equation. J. Fluid Mech. 78, 355–383.Google Scholar
Fasel, H. F. & Konzelmann, U. 1990 Non-parallel stability of a flat-plate boundary layer using the complete Navier—Stokes equations. J. Fluid Mech. 221, 311–347.Google Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222–224.Google Scholar
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22, 433–441.Google Scholar
Gaster, M. 1974 On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66, 465–480.Google Scholar
Gohberg, I., Lancaster, P. & Rodman, L. 1982 Matrix Polynomials. Academic.
Goldstein, M. E. 1983 The evolution of Tollmien—Schlichting waves near a leading edge. J. Fluid Mech. 127, 59–81.Google Scholar
Goldstein, M. E. & Durbin, P. A. 1986 Nonlinear critical layers eliminate the upper branch of spatially growing Tollmien—Schlichting waves. Phys. Fluids 29, 2344–2345.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.Google Scholar
Herbert, Th. 1974 Über endliche Amplituden periodischer Störungen der Grenzschicht an der ebenden Platte. DLR-FB 74-53 (ESA Tech. Transl. TT-169, 1975).
Herbert, Th. 1980 Nonlinear stability of parallel flows by high-order amplitude expansions. AIAA J. 18, 243–248.Google Scholar
Herbert, Th. 1991 Boundary-layer transition — analysis and prediction revised. AIAA Paper 91-0737.Google Scholar
Herbert, Th. & Bertolotti, F. P. 1992 Analysis of boundary-layer transition with parabolized stability equations. J. Fluid Mech. In preparation.Google Scholar
Herbert, Th. & Morkovin, M. V. 1980 Dialogue on bridging some gaps in stability and transition research. In Laminar—Turbulent Transition (ed. R. Eppler & H. Fasel), pp. 47–72. Springer.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473–537.Google Scholar
Itoh, N. 1974 Spatial growth of finite wave disturbances in parallel and nearly parallel flows. Part 2. Trans. Japan Soc. Aeron. Space Sci. 17, 175–186.Google Scholar
Itoh, N. 1986 The origin and subsequent development in space of Tollmien-Schlichting waves in a boundary layer. Fluid Dyn. Res. 1, 119–130.Google Scholar
Kachanov, Yu. S., Kozlov, V. V. & Levchenko, V. Ya. 1977 Nonlinear development of a wave in a boundary layer. Fluid Dyn. 3, 383–390.Google Scholar
Konzelmann, U. & Fasel, H. 1991 Numerical simulation of a three-dimensional wave packet in a growing flat plate boundary layer In Boundary Layer Transition and Control Conference, 8—12 April 1991, Cambridge. R. Aeronaut. Soc.
Lancaster, P. 1969 Theory of Matrices. Academic.
Mcaninch, G. L. 1986 Higher order parabolic approximations for sound propagation in stratified moving media. AIAA J. 24, 253–260.Google Scholar
Morkovin, M. V. 1985 Guide to experiments on instability and laminar—turbulent transition in shear layers. Course notes, AIAA Professional Study Series: Instabilities and Transition to Turbulence. Cincinnati, Ohio.
Rist, U. & Fasel, H. 1991 Spatial three-dimensional numerical simulation of laminar-turbulent transition in a flat-plate boundary layer In Boundary Layer Transition and Control Conf. 8–12 April 1991, Cambridge, R. Aeronaut. Soc.
Ross, J. A., Barnes, F. H., Burns, J. G. & Ross, M. A. S. 1970 The flat plate boundary layer. Part 3. Comparison of theory with experiment. J. Fluid Mech. 43, 819–832.Google Scholar
Saric, W. S. 1990 Low-speed experiments: requirements for stability measurements. In Instability and Transition, Vol. 1 (ed. M. Y. Hussaini & R. G. Voigt). Springer.
Saric, W. S. & Nayfeh, A. II. 1977 Nonparallel stability of boundary layers with pressure gradients and suction In Laminar—Turbulent Transition, AGARD CP-224, pp. 6/1–21.
Schubauer, G. B. & Skramstad, H. K. 1947 Laminar boundary-layer oscillations and transition on a flat plate. J. Res. Natl Bur. Stand. 38, 251–292.Google Scholar
Smith, A. M. O. & Gamberoni, N. 1956 Transition, pressure gradient and stability theory. Rep. ES 26388. Douglas Aircraft Co., El Segundo, CA.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 layers for disturbances of various sizes.. Proc. R. Soc. Lond. A 368, 573–589 (and corrections: Proc. R. Soc. Lond. A 371, 1980, 439).Google Scholar
Smith, F. T., Papageorgiou, D. & Elliot, J. W. 1984 An alternate approach to linear and nonlinear stability calculations at finite Reynolds numbers. J. Fluid Mech. 146, 313–330.Google Scholar
Spalart, P. R. 1984 A spectral method for external viscous flows. In Fluids and Plasmas: Geometry and Dynamics (ed. J. E. Marsden). Contemporary Mathematics Series. Vol. 28. Am. Math. Soc., Providence RI.
Spalart, P. R. 1988 Direct numerical study of leading-edge contamination In Fluid Dyn. of 3D Turbulent Shear Flows and Transition, AGARD-CP-438.
Spalart, P. R., Moser, R. D. & Rogers, M. M. 1991 Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions J. Comput. Phys. (in press).Google Scholar
Spalart, P. R. & Yang, K. S. 1987 Numerical study of ribbon-induced transition in Blasius flow. J. Fluid Mech. 178, 345–365.Google Scholar
Strazisar, A. J., Prahl, J. M. & Reshotko, E. 1975 Experimental study of the stability of heated laminar boundary layers in water FTAS/TR-75-113. Case Western Univ., Dept. Fluid Thermal Aerosp. Sci.
Tollmien, W. 1929 Über die Entstehung der Turbulenz. Nachr. Ges. Wiss. Göttingen, Math-phys. Kl., pp. 21–44.
Van Dyke, A. M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.
Van Ingen, J. L. 1956 A suggested semi-empirical method for the calculation of the boundary layer transition region. Rep. UTHL-74. University of Technology, Dept. Aero. Engng.
Van Stijn, Th. L. & Van de Vorren, A. I. 1983 On the stability of almost parallel boundary layer flows. Computers Fluids 10, 223–241.Google Scholar
Wortmann, F. X. 1955 Untersuchung instabiler Grenzschichtschwingen in einem Wasserkanal mit der Tellurmethode. In: 50 Jahre Grenzschichtforschung (ed. H. Görtler & W. Tollmien). Vieweg.