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On the nonlinear interaction of Görtler vortices and Tollmien-Schlichting waves in curved channel flows at finite Reynolds numbers

Published online by Cambridge University Press:  21 April 2006

Q. Isa Daudpota ,
Philip Hall  and
Thomas A. Zang
Q. Isa Daudpota
Affiliation:
Computational Methods, Branch, NASA Langley Research Center, Hampton, VA 23665, USA
Philip Hall
Affiliation:
Department of Mathematics, Exeter University, Exeter, UK and Institute for Computer Applications in Science and Engineering
Thomas A. Zang
Affiliation:
Computational Methods, Branch, NASA Langley Research Center, Hampton, VA 23665, USA
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Abstract

The flow in a two-dimensional curved channel driven by an azimuthal pressure gradient can become linearly unstable owing to axisymmetric perturbations and/or non-axisymmetric perturbations depending on the curvature of the channel and the Reynolds number. For a particular small value of curvature, the critical Reynolds number for both these perturbations becomes identical. In the neighbourhood of this curvature value and critical Reynolds number, non-linear interactions occur between these perturbations. The Stuart-Watson approach is used to derive two coupled Landau equations for the amplitudes of these perturbations. The stability of the various possible states of these perturbations is shown through bifurcation diagrams. Emphasis is given to those cases that have relevance to external flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 193 , August 1988 , pp. 569 - 595
Copyright
© 1988 Cambridge University Press

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References

Bippes, H. & Görtler, H. 1972 Dreidimensionales Störungen in der Grenzschict an einer konkaven Wand. Acta. Mech. 14, 251–267.Google Scholar
Boyce, W. E. & DiPrima, R. C. 1977 Elementary Differential Equations and Boundary Value Problems. Wiley.
Dean, W. R. 1928 Fluid motion in a curved channel.. Proc. R. Soc. Lond., A 121, 402–440.Google Scholar
Eagles, P. M. 1971 On stability of Taylor vortices by fifth-order amplitude expansions. J. Fluid Mech. 49, 529–550.Google Scholar
Gibson, R. E. & Cook, A. E. 1974 The stability of curved channel flow. Q. J. Mech. Appl. Maths 27, 149–160.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Hall, P. & Mackerrell, S. 1988 On the onset of thre dimensionality and time-dependence in the Görtler vortex problem. J. Fluid Mech. (submitted).Google Scholar
Hall, P. & Malik M. 1986 On the instbaility of a three-dimensional attachment line boundary layer: weakly nonlinear theory and a numerical approach. J. Fluid Mech. 163, 257–282.Google Scholar
Hall, P. & Smith, F. T. 1984 On the effects of nonparallelism, three-dimensionality and mode interaction in nonlinear boundary layer stability. Stud. Appl. Maths 70, 91–120.Google Scholar
Harvey, W. D. & Pride, J. D. 1982 The NASA Langley laminar flow control airfoil experiment. AIAA Paper 82-0567.Google Scholar
Keener, J. P. 1976 Secondary bifurcation in nonlinear diffusion reaction equations. Stud. Appl. Maths 55, 187–211.Google Scholar
Malik, M., Chuang, S. & Hussaini, M. Y. 1982 Accurate numerical solution of compressible linear stability equations. Z. Angew. Math. Phys. 33, 189–201.Google Scholar
Matkowsky, B. J. 1970 Nonlinear dynamic stability: a formal theory. SIAM J. Appl. Maths 18, 872–883.Google Scholar
Meksyn, D. & Stuart, J. T. 1951 Stability of viscous motion between parallel plates for finite disturbances.. Proc. R. Soc. Lond. A 208, 517–526.Google Scholar
Orszag, S. 1971 Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Peerhossaini, H. & Wesfried, J. E. 1987 On the inner structure of streamwise Görtler rolls. Paper presented at Taylor Vortex Working Party Meeting, Tempe, Arizona, March 1987.
Smith, F. T. 1979 Nonlinear stability of boundary layers for disturbances of various sizes.. Proc. R. Soc. Lond. A 368, 573–589.Google Scholar
Stuart, J. T. 1958 On the nonlinear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 1–21.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders.. Phil. Trans. R. Soc. A 223, 289–343.Google Scholar
Walowit, J., Tsao, S. & DiPrima, R. C. 1964 Stability of flow between arbitrary spaced concentric cylindrical surfaces including the effect of a radial temperature gradient. Trans. ASME E: J. Appl. Mech., 31, 585–593.Google Scholar