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Wavenumber selection and irregularity of spatially developing nonlinear Dean and Görtler vortices

Published online by Cambridge University Press:  26 April 2006

Y. Guo  and
W. H. Finlay
Y. Guo
Affiliation:
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2G8 Present address: DLR, Institute for Theoretical Fluid Mechanics, Bunsentrasse 10, D-37073 Göttingen, Germany.
W. H. Finlay
Affiliation:
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2G8
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Abstract

Spatially developing vortices found in curved channels (Dean vortices) and in a concave boundary layer (Görtler vortices) asre studied numerically using Legendre spectral-element methods. The linear instability of these vortices with respect to spanwise perturbations (Eckhaus instability) is examined using parabolized spatial stability analysis. The nonlinear evolution of this instability is studied by solving the parabolized Navier–Stokes equations. When the energy level of Dean and Görtler vortices in the flow is low, the spatial growth of the vortices is governed by primary instability (Dean or Görtler instability). At this stage, vortices with different wavelengths can develop at the same time and do not interact with each other significantly. When certain vortices reach the nonlinear stage first and become the dominant wavelength, spatial Eckhaus instability sets in. For all cases studied, spatially developing Dean and Görtler vortices are found to be most unstable to spanwise disturbances with wavelength twice or $\frac{3}{2}$ times that of the dominant one. The nonlinear growth of these perturbations generates a small vortex pair in between two pairs of vortices with long wavelength, but forces two pairs of vortices with short wavelength to develop into one pair. For Görtler vortices, this is manifested mostly by irregular and deformed vortex structures. For Dean vortices, this is manifested by vortex splitting and merging, and spatial Eckhaus instability plays an important role in the wavenumber selection process.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 264 , 10 April 1994 , pp. 1 - 40
Copyright
© 1994 Cambridge University Press

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References

Aihara, Y. & Koyama, H. 1980 Secondary instability of Görtler vortices: formation of periodic three-dimensional coherent structure. Trans. Japan Soc. Aero. Astron. 24(64), 78–94.Google Scholar
Aihara, Y., Tomita, Y. & Ito, A. 1984 Generation, development and distortion of longitudinal vortices in boundary layers along concave and flat plates. In Laminar–Turbulent Transition IUTAM Symp., Novosibirsk, 1984 (ed. V. V. Kozlov), pp. 447–454.
Alfredsson, P. A. & Persson, H. 1989 Instabilities in channel flow with system rotation. J. Fluid Mech. 202, 543–557.Google Scholar
Baker, A. J. 1983 Finite Element Computational Fluid Mechanics. McGraw-Hill.
Bara, B., Nandakumar, K. & Masliyah, J. H. 1992 An experimental and numerical study of the Dean problem: flow development towards two-dimensional, multiple solutions. J. Fluid Mech. 244, 339–376.Google Scholar
Bertolotti, F. P., Herbert, Th. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474.Google Scholar
Bippes, H. 1972 Experimentelle Untersuchung des laminar–turbulenten Umschlags an einer parallel angeströmten konkaven wand. Sizungsberichte der Heidelberger Akademie der Wissenschaften Mathmatisch-naturwissenschaftliche Klasse, 3 Abhandlund, Jahrgang 1972, pp. 103–180; also NASA TM. 75243, 1978.
Bottaro, A. 1993 On longitudinal vortices in curved channel flow. J. Fluid Mech. 251, 627–660.Google Scholar
Bottaro, A., Matsson, O. J. E. & Alfredsson, P. H. 1991 Numerical and experimental results for developing curved channel flow. Phys. Fluids A 3, 1473–1476.Google Scholar
Day, H. P., Herbert, Th. & Saric, W. S. 1990 Comparing local and marching analyses of Görtler instability. AIAA J. 28, 1010–1015.Google Scholar
Dean, W. R. 1928 Fluid motion in a curved channel. Proc. R. Soc. Lond. A 121, 402–420.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Eckhaus, W. 1965 Studies in Nonlinear Stability Theory. Springer.
Finlay, W. H., Keller, J. B. & Ferziger, J. H. 1988 Instability and transition in curved channel flow. J. Fluid Mech. 194, 417–456.Google Scholar
Fletcher, C. A. J. 1988 Computational Techniques for Fluid Dynamics, Vol. II. Springer.
Floryan, J. M. & Saric, W. S. 1984 Wavelength selection and growth of Görtler vortices. AIAA J. 22, 1529–1538.Google Scholar
Görtler, H. 1940 Uber eine driedimensionale instabilitat laminarer Grenzshichten an konkaven wanden. Math. Phys. Kl 2, 1; also NASA TM 1375 (1954).
Guo, Y. 1992 Spanwise secondary instability of Dean & Görtler vortices. PhD thesis, University of Alberta.
Guo, Y. & Finlay, W. H. 1991 Splitting, merging and wavelength selection of vortices in curved and/ or rotating channel flow due to Eckhaus instability. J. Fluid Mech. 228, 661–691.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.Google Scholar
Hall, P. 1988 The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 243–266.Google Scholar
Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37, 151–189.Google Scholar
Herbert, Th. 1976 On the stability of the boundary layer along a concave wall. Arch. Mech. Stos. 28, 1039–1055.Google Scholar
Herbert, Th. 1988 Secondary instability of boundary layers. Ann. Rev. Fluid Mech. 20, 487–526.Google Scholar
Kleiser, L. & Schumann, U. 1984 Spectral simulation of the laminar turbulent transition process in plane Poiseuille flow. In Spectral Methods for Partial Differential Equations (ed. R. G. Voigt, D. Gottlieb & M. Y. Hussaini), p. 141. SIAM.
Lee, K. & Liu, J. T. C. 1992 On the growth of mushroom-like structures in nonlinear spatially developing Görtler vortex flow. Phys. Fluids A 4, 95–103.Google Scholar
Ligrani, P. M., Longest, J. E., Fields, W. A., Kendall, M. R., Fuqua, S. J. & Baun, L. R. 1992 Initial development and structure of Dean vortex pairs in a curved rectangular channel including splitting, merging and spanwise wavenumber selection. Phys. Fluids A submitted.Google Scholar
Ligrani, P. & Niver, R. D. 1988 Flow visualization of Dean vorties in a curved channel with 40 to 1 aspect ratio. Phys. Fluids 31, 3605–3618.Google Scholar
Liu, W. & Domaradzki, J. 1990 Direct numerical simulation of transition to turbulence in Görtler flow. AIAA Paper 90-0114.
Maday, Y. & Patera, A. T. 1989 Spectral element methods for the incompressible Navier–Stokes equations. In State-of-the-Art Surveys on Computational Mechanics (ed. A. T. Noor & J. T. Oden), pp. 71–143. ASME.
Matsson, O. J. E. & Alfredsson, P. H. 1990 Curvature- and rotation-induced instabilities in channel flow. J. Fluid Mech. 202, 543–557.Google Scholar
Matsson, O. J. E. & Alfredsson, P. H. 1992 Experiments on instabilities in curved channel flow. Phys. Fluids A 4, 1666–1676.Google Scholar
Patankar, S. V. & Spalding, D. B. 1972 A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Intl J. Heat Mass Transfer 15, 1787–1806.Google Scholar
Patera, A. T. 1984 A spectral element methods for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488.Google Scholar
Riecke, H. & Paap, H. 1986 Stability and wave-vector restriction of axisymmetric Taylor vortex flow. Phys. Rev. A 33, 547–553.Google Scholar
Rønquist, E. M. 1988 Optimal spectral element methods for the unsteady three-dimensional incompressible Navier–Stokes equations. PhD thesis, Massachusetts Institute of Technology.
Rubin, S. G., Khosla, P. K. & Saari, S. 1977 Analysis of global pressure relaxation for flows with strong interaction and separation. Computers Fluids 11, 281–306.Google Scholar
Sabry, A. S. 1988 Numerical computation of the nonlinear evolution of Görtler vortices. PhD thesis, Brown University.
Sabry, A. S. & Liu, J. T. C. 1991 Longitudinal vorticity elements in boundary layers: nonlinear development from initial Görtler vortices as a prototype problem. J. Fluid Mech. 231, 615–663.Google Scholar
Schlichting, H. 1955 Boundary-Layer Theory. McGraw-Hill.
Spalart, P. R. 1989 Direct numerical study of crossflow instability. In Laminar–Turbulent Transition IUTAM Symp. Toulouse/ France 1989 (ed. D. Arnal & R. Michel), Springer.
Swearingen, J. D. & Blackwelder, R. F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290.Google Scholar
Tani, I. 1962 Production of longitudinal vortices in the boundary layer along a curved wall. J. Geophys. Res. 67, 3075–3080.Google Scholar
Winoto, S. H. & Crane, R. I. 1980 Vortex structure in laminar boundary layers on a concave wall. Intl J. Heat Fluid Flow 2, 221–231.Google Scholar
Yang, K. S. & Kim, J. 1991 Numerical investigation of instability and transition in rotating plane Poiseuille flow. Phys. Fluids A 3, 633–641.Google Scholar
Yu, X. Y. & Liu, J. T. C. 1991 The secondary instabilities in Görtler flow. Phys. Fluids A 3, 1845–1847.Google Scholar