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Transition Modeling Based on the PSE

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Turbulence and Transition Modelling

Part of the book series: Series ((ERCO,volume 2))

Abstract

The PSE are a set of nonlinear parabolic partial differential equations used to study the transition of a flow from a laminar state to a turbulent state. Following the lines of the classical stability analysis for flows with parallel streamlines, the PSE assume that the transition process starts with the amplification of small disturbances. The PSE equations remain valid when the disturbances reach finiteamplitudes, and can be applied to flows that have slowly changing properties in the streamwise direction, such as, for example, diverging streamlines, temperature, and chemical composition. If a boundary is present, then slow changes in the boundary geometry, such as curvature, and roughness distribution, or in the boundary conditions, such as variable transpiration velocity, are also allowed.

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References

  1. Abramowitz, M., Stegun, I. A.. Handbook of mathematical functions. Technical report, National Bureau of Standards, 1966.

    Google Scholar 

  2. Bender, C. M., Orszag, S. A.. Advanced Mathematical Methods for Scientists and Engineers. MacGraw-Hill, 1978.

    MATH  Google Scholar 

  3. Bertolotti, F. P.. Linear and Nonlinear Stability of Boundary Layers with Streamwise Varying Properties. PhD thesis, The Ohio State University, 1991. Herbert, Th. advisor.

    Google Scholar 

  4. Bertolotti, F. P.. Vortex generation and wave-vortex interaction over a concave plate with roughness and suction. Report 93–101, Icase, Nasa Langley, Hampton Va., 1993. Submitted to Theo. Comp. Fluid Dyn.

    Google Scholar 

  5. Bertolotti, F. P.. An introduction to the parabolized stability equations (eds. T. C. Corke, G. Erlebacher, M. Y. Hussaini). Oxford University Press, 1994.

    Google Scholar 

  6. Bertolotti, F. P. on the birth and evolution of disturbances in threedimensional boundary layers. in Nonlinear stability and transition in threedimensional boundary-layers, Manchester, U.K., July 1995. Iutam, Kluwer Publishers.

    Google Scholar 

  7. Bertolotti, F. P., Crouch, J. D.. Simulation of boundary-layer transition: receptivity to spike stage. in Proc. First Euro. Comp. Fluid Dynam. Conference, pages 183–190, Brussels, Belgium, Sept 1992. Elsevier Science Publishers B.V.

    Google Scholar 

  8. Bertolotti, F. P., Herbert, Th., Spalart, P. R.. Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech., 242: 441–474, 1992.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Bertolotti, F. P., Joslin, R. D.. Effect of far-field boundary conditions on boundary-layer transition. J. Comp. Phys., Accepted, 1994.

    Google Scholar 

  10. Bertolotti, F.P.. Response of the Blasius boundary layer to weak freestream vortices. Phys. Fluids A, 1995. submitted.

    Google Scholar 

  11. Bouthier, M.. Stabilité linéaire des écoulements presque parallèles: Part ii. la couche limite de Blasius. J. Méchanique, 12: 76–95, 1973.

    Google Scholar 

  12. Chang, C. L., Malik, M. R.. Oblique mode breakdown in a supersonic boundary layer using nonlinear Pse. in Proc. First Euro. Comp. Fluid Dynam. Conference, Brussels, Belgium, 1992. Elsevier Science Publishers B.V.

    Google Scholar 

  13. Chang, C. L., Malik, M. R., Erlebacher, G., Hussaini, M. Y.. Compressible stability of growing boundary layers using the parabolized stability equations. Paper 91–1636, Aiaa, 1991.

    Google Scholar 

  14. Craik, A. D. D.. Nonlinear resonant instability in boundary layers. J. Fluid Mech., 50, 1971.

    Google Scholar 

  15. Crouch, J. D.. Non-localized receptivity of boundary-layers. J. Fluid Mech., 244: 567, 1992.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Crouch, J. D.. Distributed excitation of tollrnien-Schlichting waves by vortical free-stream disturbances. Phys. Fluids, 6 (1) 217–223, 1994.

    Article  ADS  MATH  Google Scholar 

  17. Crouch, J. D., Bertolotti, F. P.. Nonlocalized receptivity of boundary layers to three-dimensional disturbances. Paper 92–0740, Aiaa, 1992.

    Google Scholar 

  18. Davis, D. A. R., Smith, F. T.. On the nonlinear tollmienSchlichting/vortex interaction in three-dimensional boundary layers. Technical Memorandum 106184, Nasa, 1993.

    Google Scholar 

  19. Gaster, M.. On the effects of boundary-layer growth on flow stability. J. Fluid Mech., 66: 465–480, 1974.

    Article  ADS  MATH  Google Scholar 

  20. Goldstein, M. E., Leib, S. J.. Three-dimensional boundary-layer instability and separation induced by small-amplitude streamwise vorticity in the upstream flow. J. Fluid Mech., 246: 21–41, 1993.

    Article  ADS  MATH  Google Scholar 

  21. Goldstein, M. E., Leib, S. J., Cowley, S. J.. Distortion of a flat-plate boundary layer by free-stream vorticity normal to the plate. J. Fluid Mech., 237: 231–260, 1992.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Haj-Hariri, H., Characteristics analysis of the parabolized stability equations. Studies in Applied Math, 92, 1994.

    Google Scholar 

  23. Hall, P.. Taylor-Görtler vortices in fully developed or boundary layer flows; linear theory. J. Fluid Mech., 124: 475–494, 1982.

    Article  ADS  MATH  Google Scholar 

  24. Hall, P.. The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech., 130: 41–58, 1983.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Hall, P., Smith, F. T.. on the strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech., 227: 641–666, 1991.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Herbert, Th.. Secondary instability of boundary layers. Ann. Rev. Fluid Mech., 20, 1988.

    Google Scholar 

  27. Herbert, Th.. Boundary-layer transition — analysis and prediction revised. Paper 91–0737, Aiaa, 1991.

    Google Scholar 

  28. Herbert, Th.. Parabolized stability equations. in Progress in Transition Modelling. Agard Report 793, 1994.

    Google Scholar 

  29. Huerre, P., Monkewitz, P. A.. Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech., 22: 473–537, 1990.

    Article  MathSciNet  ADS  Google Scholar 

  30. Itoh, N.. Spatial growth of finite wave disturbances in parallel and nearly parallel flows. part 2. Trans. Japan Soc. Aeron. Space Sci., 17: 175–186, 1974.

    Google Scholar 

  31. Kachanov, Y. S., Levchenko, V. Y.. The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer. J. Fluid Mech., 138, 1984.

    Google Scholar 

  32. Li, F., Malik, M. R.. Mathematical nature of parabolized stability equations. in Laminar-Turbulent Transition. Proc. 4th Iutam Symp., Sendai, Japan, Springer-Verlag, 1994.

    Google Scholar 

  33. Lin, N, Stuckert, G. K., Herbert, Th.. Boundary layer receptivity to free-stream vortical disturbances. Paper 95–0772, Aiaa, 1995.

    Google Scholar 

  34. Monte Ross, Ed.. Laser Applications. Academic Press, 1971.

    Google Scholar 

  35. Morkovin, M. V.. Critical evaluation of transition from laminar to turbulent shear layers with emphasis on hypersonically traveling bodies. Tr 68–13c, Martin Marietta, 1968.

    Google Scholar 

  36. Thumm, A., Wolz, W., Fasel, H. F.. Numerical simulation of spatially growing three-dimensional disturbance waves in compressible boundarylayers. in Proceedings of the third Iutam Symposium on Laminar-Turbulent Transition, toulouse, France, 1989. published 1990.

    Google Scholar 

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© 1996 Springer Science+Business Media Dordrecht

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Bertolotti, F.P. (1996). Transition Modeling Based on the PSE. In: Hallbäck, M., Henningson, D.S., Johansson, A.V., Alfredsson, P.H. (eds) Turbulence and Transition Modelling. ERCOFTAC Series, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8666-5_8

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  • DOI: https://doi.org/10.1007/978-94-015-8666-5_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4707-6

  • Online ISBN: 978-94-015-8666-5

  • eBook Packages: Springer Book Archive

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