Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T07:22:05.802Z Has data issue: false hasContentIssue false
  • English
  • Français

Fundamental and subharmonic secondary instabilities of Görtler vortices

Published online by Cambridge University Press:  26 April 2006

Fei Li  and
Mujeeb R. Malik
Fei Li
Affiliation:
High Technology Corporation, PO Box 7262, Hampton, VA 23666, USA
Mujeeb R. Malik
Affiliation:
High Technology Corporation, PO Box 7262, Hampton, VA 23666, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The nonlinear development of stationary Görtler vortices leads to a highly distorted mean flow field where the streamwise velocity depends strongly not only on the wall-normal but also on the spanwise coordinates. In this paper, the inviscid instability of this flow field is analysed by solving the two-dimensional eigenvalue problem associated with the governing partial differential equation. It is found that the flow field is subject to the fundamental odd and even (with respect to the Görtler vortex) unstable modes. The odd mode, which was also found by Hall & Horseman (1991), is initially more unstable. However, there exists an even mode which has higher growth rate further downstream. It is shown that the relative significance of these two modes depends upon the Görtler vortex wavelength such that the even mode is stronger for large wavelengths while the odd mode is stronger for short wavelengths. Our analysis also shows the existence of new subharmonic (both odd and even) modes of secondary instability. The nonlinear development of the fundamental secondary instability modes is studied by solving the (viscous) partial differential equations under a parabolizing approximation. The odd mode leads to the well-known sinuous mode of break down while the even mode leads to the horseshoe-type vortex structure. This helps explain experimental observations that Görtler vortices break down sometimes by sinuous motion and sometimes by developing a horseshoe vortex structure. The details of these break down mechanisms are presented.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 297 , 25 August 1995 , pp. 77 - 100
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aihara, Y. & Koyama, H. 1981 Secondary instability of Görtler vortices: formation of periodic three-dimensional coherent structures. Trans. Japan Soc. Aero. Space Sci. 24, 78–94.Google Scholar
Beckwith, I. E., Malik, M. R., Chen, F.-J. & Bushnell, D. M. 1984 Effects of nozzle design parameters on the extent of quiet test flow at Mach 3.5. IUTAM Symp. on Laminar-Turbulent Transition, Novosibirsk, USSR (ed. V. V. Kozlov), pp. 589–600.
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. 1978 Experimental study of the laminar-turbulent transition of a concave wall in a parallel flow. NASA TM-75243.Google Scholar
Chang, C.-L., Malik, M. R., Erlebacher, G. & Hussaini, M. Y. 1991 Compressible stability of growing boundary layers using parabolized stability equations. AIAA Paper 91-1636.Google Scholar
Floryan, J. M. 1991 On the Görtler vortex instability of boundary layers. Prog. Aerospace Sci. 28, 235–271.Google Scholar
Gaster, M. 1962 A note on the relation between temporarlly increasing and spatially increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222–224.Google Scholar
Guo, Y. & Finlay, W. H. 1994 Wavenumber selection and irregularity of spatially developing nonlinear Dean and Görtler vortices. J. Fluid Mech. 264, 1–40.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 30, 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 nonlinear breakdown stage. Mathematika 37, 151–189.CrossRefGoogle Scholar
Hall, P. & Horseman, N. J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375.Google Scholar
Hall, P. & Lakin, W. D. 1988 The fully nonlinear development of Görtler vortices in growing boundary layers. Proc. R. Soc. Lond. A 415, 421–444.Google Scholar
Herbert, Th. 1991 Boundary-layer transition - analysis and prediction revisited. AIAA Paper 91- 0737.Google Scholar
Herbert, Th., Bertolotti, F. P. & Santos, G. R. 1985 Floquet analysis of secondary instability in shear flows. In Stability of Time Dependent and Spatially Varying Flows (ed. D. L. Dwoyer & M. Y. Hussaini), pp. 48–57. Springer.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473–537.Google Scholar
Ito, A. 1985 Breakdown structure of longitudinal vortices along a concave wall. J. Japan Soc. Aero. Space Sci. 33, 116–173.Google Scholar
Lee, K. & Liu, J. T. C. 1992 On the growth of mushroomlike structure in nonlinear spatially developing Görtler vortex flow. Phys. Fluids A 4, 95–103.Google Scholar
Liu, W. & Domaradzki, J. A. 1993 Direct numerical simulation of transition to turbulence in Görtler flow. J. Fluid Mech. 246, 267–299.Google Scholar
Malik, M. R., Chuang, S. & Hussaini, M. Y. 1982 Accurate numerical solution of compressible stability equations. Z. Angew. Math. Phys. 33, 189–201.Google Scholar
Malik, M. R. & Li, F. 1992 Three-dimensional boundary layer stability and transition. SAE Paper No. 921991. Presented at SAE Aerotech '92, Anaheim, CA.
Malik, M. R. & Li, F. 1993 Secondary instability of Görtler and crossflow vortices. In Proc. Int. Symp. on Aerospace and Fluid Science, pp. 460–477. Institute of Fluid Science, Tohoku University, Sendai, Japan.
Malik, M. R., Li, F. & Chang, C.-L. 1994 Crossflow disturbances in three-dimensional boundary layers: nonlinear development, wave interaction and secondary instability. J. Fluid Mech. 268, 1–36.CrossRefGoogle Scholar
Masuda, S., Hori, D. & Matubara, M. 1994 Secondary instability associated with streamwise vortices in a rotating boundary layer. IUTAM Symp. on Laminar-Turbulent Transition, Sendai, Japan (to appear).Google Scholar
Nayfeh, A. H. & Padhye, A. 1979 The relation between temporal and spatial stability in three-dimensional flows. AIAA J. 17, 1084–1090.Google Scholar
Peerhossaini, H. & Wesfreid, J. E. 1988 On the inner structure of streamwise Görtler rolls. Intl J. Heat Fluid Flow 9, 12–18.Google Scholar
Sabry, A. S. & Liu, J. T. C. 1991 Longitudinal vorticity elments in boundary layers: nonlinear development from initial Görtler vortices as a prototype problem. J. Fluid Mech. 231, 615–663.Google Scholar
Saric, W. S. 1994 Görtler vortices. Ann. Rev. Fluid Mech. 26, 379–409.Google Scholar
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 (referred to herein as SB).Google Scholar
Yu, X. & Liu, J. T. C. 1991 The secondary instability in Görtler flow. Phys. Fluids A 4, 1825–1827.Google Scholar
Yu, X. & Liu, J. T. C. 1994 On the mechanism of sinuous and varicose modes in three-dimensional viscous secondary instability of nonlinear Görtler rolls. Phys. Fluids 6, 736–750.Google Scholar