Advertisement

Instability of the flat-plate boundary layer

  • Andrey V. BoikoEmail author
  • Alexander V. Dovgal
  • Genrih R. Grek
  • Victor V. Kozlov
Chapter
  • 1.5k Downloads
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 98)

Abstract

The linear stability of the Blasius boundary layer which is a quasi-parallel rather than a strictly parallel flow is addressed in this chapter. At high Reynolds numbers, the wall-normal velocity V is small and the growth of the boundary layer is slow compared to distances characteristic for the laminar–turbulent transition induced by small disturbances. Hence, it is not too much surprising that the parallel-flow approximation works quite satisfactorily for two-dimensional waves observed in experiments.

Keywords

Boundary Layer Streamwise Velocity Linear Stability Theory Neutral Stability Curve Streamwise Pressure Gradient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

References

  1. Bertolotti FP, Herbert T, Spallart PR (1992) Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech 242:441–474 MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. Bouthier M (1973) Stabilité linéaire des écoulements presque parallèles. Part 2. La couche limite de Blasius. J Mécanique 12(1):75–95 Google Scholar
  3. Gaster M (1974) On the effects of boundary-layer growth on flow stability. J Fluid Mech 66:465–480 ADSzbMATHCrossRefGoogle Scholar
  4. Jordisson R (1970) The flat plate boundary layer. Part 1. Numerical integration of the Orr–Sommerfeld equation. J Fluid Mech 43:801–811 ADSCrossRefGoogle Scholar
  5. Kachanov YS, Kozlov VV, Levchenko VY (1979) The development of small-amplitude oscillations in a laminar boundary layer. Fluid Mech – Soviet Res 8(2):152–156 Google Scholar
  6. Klingmann BGB, Boiko AV, Westin KJA, Kozlov VV, Alfredsson PH (1993) Experiments on the stability of Tollmien–Schlichting waves. Eur J Mech B/Fluids 12(4):493–514 Google Scholar
  7. Kozlov LF, Babenko VV (1978) Experimental investigations of boundary layer. Naukova dumka, Kiev. In Russian. Google Scholar
  8. Levchenko VY, Volodin AG, Gaponov SA (1975) Stability characteristics of boundary layers. Nauka. Sib. Otd., Novosibirsk. In Russian. Google Scholar
  9. Libby PA, Liu TM (1967) Further solutions of the Falkner-Skan equation. AIAA J 5(5):1040–1042 ADSzbMATHCrossRefGoogle Scholar
  10. Lin CC (1955) The theory of hydrodynamic stability. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  11. Maslowe SA, Spiteri RJ (2001) The continuous spectrum for a boundary layer in a streamwise pressure gradient. Phys Fluids 13(5):1294–1299 ADSCrossRefGoogle Scholar
  12. Ross JA, Barnes FH, Burns JG, Ross MAS (1970) The flat plate boundary layer. Part 3. Comparison of theory with experiment. J Fluid Mech 43:819–832 ADSzbMATHCrossRefGoogle Scholar
  13. Saric WS, Nayfeh AH (1977) Nonparallel stability of boundary layers with pressure gradients and suction. In: AGARD-CP–224 Laminar–Turbulent Transition, Copenhagen, pp 6–1621 Google Scholar
  14. Schlichting H (1933) Zur Entstehung der Turbulenz bei der Plattenströmung. In: Math. Phys. Klasse, Nachr. Ges. Wiss. Göttingen, Göttingen, pp 181–208 Google Scholar
  15. Schlichting H (1935) Amplitudenverteilung und Energiebilanz der kleinen Störungen bei der Plattenströmung. In: Math. Phys. Klasse, Fachgruppe I, vol 1, Nachr. Ges. Wiss. Göttingen, Göttingen, pp 47–78 Google Scholar
  16. Schubauer GB, Skramstad HK (1948) Laminar-boundary layer oscillations and transition on a flat plate. TN 909, NACA Google Scholar
  17. Stewartson K (1954) Further solutions of the Falkner–Skan equation. Proc Camb Phil Soc 50:454–465 MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. Tollmien W (1929) Über die Entstehung der Turbulenz. 1. Mitteilung. In: Math. Phys. Klasse. Nachr. Ges. Wiss. Göttingen, Göttingen, pp 21–44, translated as NACA TM 609, 1931. Google Scholar
  19. Tumin AM, Shepelev VE (1980) Numerical analysis of disturbance development in incompressible flat plate boundary layer. Chisl Met Mekhan Splosh Sredy 1(3):141–152, In Russian. Google Scholar
  20. Westin KJA (1997) Laminar–turbulent boundary layer transition influenced by free stream turbulence. TRITA-MEK TR 1997:10, Royal Institute of Technology, Stockholm, Doctoral Thesis. Google Scholar
  21. Wortmann FX (1953) Eine Methode zur Beobachtung und Messung Wasserströmung mit Tellur. Z Angew Phys 5(6):200–206 Google Scholar
  22. Wortmann FX (1964) Experimental investigation of vortex occurence at transition in unstable boundary-layers. AFOSR 64–1280 Google Scholar

Further Reading

  1. Benney DJ, Rosenblatt S (1964) Stability of spatially varying and time dependent flows. Phys Fluids 7(8):1385 ADSCrossRefGoogle Scholar
  2. Boiko AV, Westin KJA, Klingmann BGB, Kozlov VV, Alfredsson PH (1994) Experiments in a boundary layer subjected to free stream turbulence. Part 2. The role of TS-waves in the transition process. J Fluid Mech 281:219–245 ADSCrossRefGoogle Scholar
  3. Bouthier M (1971) Stabilité linéaire des écoulements presque parallèles des échelles multiples. Compt Rendus Acad Sci, Sèr A 273:1101–1104 MathSciNetzbMATHGoogle Scholar
  4. Bouthier M (1972) Stabilité linéaire des écoulements presque parallèles. Part 1. J Mécanique 11(4):599–621 Google Scholar
  5. Bouthier M (1973) Stabilité linéaire des écoulements presque parallèles. Part 2. La couche limite de Blasius. J Mécanique 12(1):75–95 Google Scholar
  6. van Dyke M (1964) Perturbation methods in Fluid Mechanics. Academic, New York zbMATHGoogle Scholar
  7. 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 ADSzbMATHCrossRefGoogle Scholar
  8. Fasel H, Konzelmann U (1990) Non-parallel stability of a flat-plate boundary layer using the complete Navier–Stokes equations. J Fluid Mech 221:311–347 ADSzbMATHCrossRefGoogle Scholar
  9. Haj-Hariri H (1994) Characteristics analysis of the parabolized stability equations. Stud Appl Math 92:41–53 MathSciNetzbMATHGoogle Scholar
  10. Herbert T (1997) Parabolized stability equations. Ann Rev Fluid Mech 29:245–283 MathSciNetADSCrossRefGoogle Scholar
  11. Herbert T, Bertolotti FP (1987) Stability analysis of nonparallel boundary layers. Bull Amer Phys Soc 32(10):2079 Google Scholar
  12. Itoh N (1986) The origin and subsequent development in space of Tollmien–Schlichting waves in a boundary layer. Fluid Dyn Res 1:119–130 ADSCrossRefGoogle Scholar
  13. Kachanov YS, Kozlov VV, Levchenko VY, Maksimov VP (1979) Transformation of external disturbances into the boundary layer waves. In: Proc. Sixth Intern. Conf. on Numerical Methods in Fluid Dyn., Springer–Verlag, Berlin, Lecture Notes in Physics, 19, pp 299–307 CrossRefGoogle Scholar
  14. Li F, Malik MR (1997) Spectral analysis of parabolized stability equations. Computers & Fluids 23(3):279–297 MathSciNetCrossRefGoogle Scholar
  15. Murdock JW (1977) A numerical study of nonlinear effects on boundary layer stability. AIAA J 15(8):1167–1173 ADSCrossRefGoogle Scholar
  16. Nayfeh AH (1973) Perturbation methods. Wiley-Interscience, New York zbMATHGoogle Scholar
  17. Nayfeh AH, Saric WS, Mook DT (1974) Stability of non-parallel flows. Arch Mech 26(3):401–406 zbMATHGoogle Scholar
  18. Saric WS (1990) Low-speed experiments: Requirements for stability measurements. In: Hussaini MY, Voight RG (eds) Instability and Transition, Springer–Verlag, Berlin, ICASE/NASA LaRC Series, vol 1, pp 162–172 Google Scholar
  19. Saric WS (1994) Low-speed boundary-layer transition experiments. In: Corke TC, Erlebacher G, Hussaini MY (eds) Transition, Experiments, Theory & Computations. Oxford University Press, Oxford, pp 1–113 Google Scholar
  20. Volodin AG (1973) Stability of plane boundary layer with account of nonparallelity. Izv Sib Otd Akad Nauk SSSR, Ser Tekhn Nauk 8(2):14–17 MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Andrey V. Boiko
    • 1
    Email author
  • Alexander V. Dovgal
    • 1
  • Genrih R. Grek
    • 1
  • Victor V. Kozlov
    • 1
  1. 1.Inst. Theoretical & Applied MechanicsRussian Academy of SciencesNovosibirskRussia

Personalised recommendations