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Linear stability analysis in compressible, flat-plate boundary-layers

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Abstract

The stability problem of two-dimensional compressible flat-plate boundary layers is handled using the linear stability theory. The stability equations obtained from three-dimensional compressible Navier–Stokes equations are solved simultaneously with two-dimensional mean flow equations, using an efficient shoot-search technique for adiabatic wall condition. In the analysis, a wide range of Mach numbers extending well into the hypersonic range are considered for the mean flow, whereas both two- and three-dimensional disturbances are taken into account for the perturbation flow. All fluid properties, including the Prandtl number, are taken as temperature-dependent. The results of the analysis ascertain the presence of the second mode of instability (Mack mode), in addition to the first mode related to the Tollmien–Schlichting mode present in incompressible flows. The effect of reference temperature on stability characteristics is also studied. The results of the analysis reveal that the stability characteristics remain almost unchanged for the most unstable wave direction for Mach numbers above 4.0. The obtained results are compared with existing numerical and experimental data in the literature, yielding encouraging agreement both qualitatively and quantitatively.

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References

  1. Lees, L., Lin, C.C.: Investigations of the stability of the laminar boundary layer in a compressible fluid, NACA TN 1115 (1946)

  2. Mack L.M. (1975). Linear stability theory and the problem of supersonic boundary-layer transition. AIAA J. 13(3): 278–289

    ADS  Google Scholar 

  3. Mack, L.M.: Boundary-layer linear stability theory. AGARD R 709 (1984)

  4. Malik M.R. (1989). Prediction and control of transition in supersonic and hypersonic boundary layers. AIAA J. 27(11): 1487–1493

    ADS  Google Scholar 

  5. Arnal, D.: Boundary-layer transition: predictions based on linear theory. AGARD R 793 (1994)

  6. Masad J.A. and Abid R. (1995). On transition in supersonic and hypersonic boundary-layers. Int. J. Eng. Sci. 33(13): 1893–1919

    Article  MATH  MathSciNet  Google Scholar 

  7. Malik M.R. and Anderson E.C. (1991). Real gas effects on hypersonic boundary-layer stability. Phys. Fluids A 3(5): 803–821

    Article  ADS  Google Scholar 

  8. Malik M.R. (2003). Hypersonic flight transition data analysis using parabolized stability equations with chemistry effects. J. Spacecr. Rockets 40(3): 332–344

    Article  ADS  Google Scholar 

  9. Laufer J. and Vrebalovich T. (1960). Stability and transition of a supersonic laminar boundary-layer on a flat-plate. J. Fluid Mech. 9: 257–299

    Article  MATH  ADS  Google Scholar 

  10. Kendall, J.M.: Supersonic boundary-layer experiments. In: McCauley, W.D. (ed.) Proceedings of Boundary Layer Transition Study Group Meeting II, Aerospace Corp. (1967)

  11. Kendall J.M. (1975). Wind tunnel experiments relating to supersonic and hypersonic boundary-layer transition. AIAA J. 13: 290–299

    ADS  Google Scholar 

  12. Coles D. (1954). Measurements of turbulent friction on a smooth flat-plate in supersonic flow. J. Aeronaut. Sci. 21(7): 433–448

    Google Scholar 

  13. Deem, R.E., Murphy, J.S.: Flat-plate boundary-layer transition at hypersonic speeds. AIAA Paper 65–128 (1965)

  14. Laufer, J., Marte, J.E.: Results and a critical discussion of transition-Reynolds number measurements on insulated cones and flat-plates in supersonic wind tunnels. Jet Propulsion Lab, Pasedena, CA, Rept. 20-96 (1955)

  15. Chen F.-J., Malik M.R. and Beckwith I.E. (1989). Boundary-layer transition on a cone and flat-plate at Mach 3.5. AIAA J. 27(6): 687–693

    ADS  Google Scholar 

  16. Schlichting H. (1979). Boundary-layer theory, vol. 7. McGraw-Hill, Inc., New York

    Google Scholar 

  17. The Mathworks, Inc., MATLAB: the language of technical computing (2004)

  18. Mathews J.H. (1987). Numerical methods. Prentice-Hall, London

    Google Scholar 

  19. Mack L.M. (1976). A numerical study of the temporal eigenvalue spectrum of the Blasius boundary-layer. J. Fluid Mech. 73(3): 497–520

    Article  MATH  ADS  Google Scholar 

  20. Press W.H., Teukolsky S.A., Vetterling W.T. and Flannery B.P. (1992). Numerical recipes in Fortran 77. Cambridge University Press, Cambridge

    Google Scholar 

  21. Pruett C.D. and Streett C.L. (1991). A spectral collocation method for compressible, non-similar boundary-layers. Int. J. Numer. Methods Fluids 13: 713–737

    Article  MATH  ADS  Google Scholar 

  22. Obremski, H.J., Morkovin, M.V., Landahl, M.: A portfolio of stability characreristics of incompressible boundary-layers. AGARDograph 136 (1969)

  23. Özgen S. (2004). Effect of Heat Transfer on stability and transition characteristics of boundary-layers. Int. J. Heat Mass Transf. 47: 4697–4712

    Article  MATH  Google Scholar 

  24. Hayes W.D. and Probstein R.F. (1959). Hypersonic flow theory. Academic, New York

    MATH  Google Scholar 

  25. Anderson J.D. (1990). Modern compressible flow. McGraw-Hill, New York

    Google Scholar 

  26. Laufer J. (1964). Some statistical properties of the pressure field radiated by a turbulent boundary layer. Phys. Fluids 7(8): 1191–1197

    Article  MATH  ADS  Google Scholar 

  27. Schubauer, G.B., Skramstad, H.K.: Laminar boundary-layer oscillations and transition on a flat-plate, NACA Report 909 (1946)

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Authors and Affiliations

  1. Department of Aerospace Engineering, Middle East Technical University, 06531, Ankara, Turkey

    Serkan Özgen

  2. Flight Sciences Department, Turkish Aerospace Industries, Inc., 06531, Ankara, Turkey

    Senem Atalayer Kırcalı

Authors
  1. Serkan Özgen
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  2. Senem Atalayer Kırcalı
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Correspondence to Serkan Özgen.

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Communicated by M.Y. Hussaini.

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Özgen, S., Kırcalı, S.A. Linear stability analysis in compressible, flat-plate boundary-layers. Theor. Comput. Fluid Dyn. 22, 1–20 (2008). https://doi.org/10.1007/s00162-007-0071-0

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  • DOI: https://doi.org/10.1007/s00162-007-0071-0

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