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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramowitz, M., Stegun, I. A.. Handbook of mathematical functions. Technical report, National Bureau of Standards, 1966.
Bender, C. M., Orszag, S. A.. Advanced Mathematical Methods for Scientists and Engineers. MacGraw-Hill, 1978.
Bertolotti, F. P.. Linear and Nonlinear Stability of Boundary Layers with Streamwise Varying Properties. PhD thesis, The Ohio State University, 1991. Herbert, Th. advisor.
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.
Bertolotti, F. P.. An introduction to the parabolized stability equations (eds. T. C. Corke, G. Erlebacher, M. Y. Hussaini). Oxford University Press, 1994.
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.
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.
Bertolotti, F. P., Herbert, Th., Spalart, P. R.. Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech., 242: 441–474, 1992.
Bertolotti, F. P., Joslin, R. D.. Effect of far-field boundary conditions on boundary-layer transition. J. Comp. Phys., Accepted, 1994.
Bertolotti, F.P.. Response of the Blasius boundary layer to weak freestream vortices. Phys. Fluids A, 1995. submitted.
Bouthier, M.. Stabilité linéaire des écoulements presque parallèles: Part ii. la couche limite de Blasius. J. Méchanique, 12: 76–95, 1973.
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.
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.
Craik, A. D. D.. Nonlinear resonant instability in boundary layers. J. Fluid Mech., 50, 1971.
Crouch, J. D.. Non-localized receptivity of boundary-layers. J. Fluid Mech., 244: 567, 1992.
Crouch, J. D.. Distributed excitation of tollrnien-Schlichting waves by vortical free-stream disturbances. Phys. Fluids, 6 (1) 217–223, 1994.
Crouch, J. D., Bertolotti, F. P.. Nonlocalized receptivity of boundary layers to three-dimensional disturbances. Paper 92–0740, Aiaa, 1992.
Davis, D. A. R., Smith, F. T.. On the nonlinear tollmienSchlichting/vortex interaction in three-dimensional boundary layers. Technical Memorandum 106184, Nasa, 1993.
Gaster, M.. On the effects of boundary-layer growth on flow stability. J. Fluid Mech., 66: 465–480, 1974.
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.
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.
Haj-Hariri, H., Characteristics analysis of the parabolized stability equations. Studies in Applied Math, 92, 1994.
Hall, P.. Taylor-Görtler vortices in fully developed or boundary layer flows; linear theory. J. Fluid Mech., 124: 475–494, 1982.
Hall, P.. The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech., 130: 41–58, 1983.
Hall, P., Smith, F. T.. on the strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech., 227: 641–666, 1991.
Herbert, Th.. Secondary instability of boundary layers. Ann. Rev. Fluid Mech., 20, 1988.
Herbert, Th.. Boundary-layer transition — analysis and prediction revised. Paper 91–0737, Aiaa, 1991.
Herbert, Th.. Parabolized stability equations. in Progress in Transition Modelling. Agard Report 793, 1994.
Huerre, P., Monkewitz, P. A.. Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech., 22: 473–537, 1990.
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.
Kachanov, Y. S., Levchenko, V. Y.. The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer. J. Fluid Mech., 138, 1984.
Li, F., Malik, M. R.. Mathematical nature of parabolized stability equations. in Laminar-Turbulent Transition. Proc. 4th Iutam Symp., Sendai, Japan, Springer-Verlag, 1994.
Lin, N, Stuckert, G. K., Herbert, Th.. Boundary layer receptivity to free-stream vortical disturbances. Paper 95–0772, Aiaa, 1995.
Monte Ross, Ed.. Laser Applications. Academic Press, 1971.
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.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
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
Download citation
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