Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-01T02:48:32.816Z Has data issue: false hasContentIssue false
  • English
  • Français

On the breakdown of Rayleigh’s criterion for curved shear flows: a destabilization mechanism for a class of inviscidly stable flows

Published online by Cambridge University Press:  07 October 2013

Philip Hall
Philip Hall*
Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of a high-Reynolds-number flow over a curved surface is considered. Attention is focused on spanwise-periodic vortices of wavelength comparable with the boundary layer thickness. The wall curvature and the Görtler number for the flow are assumed large, so that stable or unstable vortices of wavelength comparable with the boundary layer thickness are dominated by inviscid effects. The growth or decay rate determined by using the viscous correction to the inviscid prediction is found to give a good approximation to the exact value for a range of wavenumbers. For the stable configuration, it is shown that the flow can be destabilized by arbitrarily small wall waviness, with the critical configuration involving only geometric properties of the flow. This destabilization mechanism has consequences for a number of flows of aerodynamic interest, particularly those where surface waviness under loading occurs. Control strategies based on how the curvature is distributed are discussed, and it is demonstrated how small regions of varying curvature can be used to remove the exponentially growing modes. The development of the parametric instabilities is continued into the strongly nonlinear regime. The nonlinear evolution is found to be governed by a generalized form of the cubic amplitude equation, generally referred to as the Stuart–Landau equation. The novel feature of the new equation is that the nonlinearity is not in the form of a power of the amplitude but is fixed by the solution of an associated nonlinear partial differential equation boundary value problem. Numerical solutions of the nonlinear partial differential system that fix the disturbance amplitude are given. The mechanism of destabilization described is shown to be directly relevant to a number of flows that are inviscidly stable in the presence of body forces. Thus it is shown how Rayleigh–Bénard convection or Taylor vortices can, at high Rayleigh or Taylor numbers, be destabilized by asymptotically small modulation in time or space.

JFM classification

Boundary Layers: Boundary layer stability Instability: Nonlinear instability Instability: Parametric instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 734 , 10 November 2013 , pp. 36 - 82
Copyright
©2013 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

Bassom, A. P. & Hall, P. 1994 The receptivity problem for $O(1)$ wavelength Görtler vortices. Proc. R. Soc. A 446, 499–518.Google Scholar
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. A 225, 505–515.Google Scholar
Blennerhassett, P. J. & Smith, F. T. 1992 Nonlinear interaction of oblique Tollmien–Schlichting waves and longitudinal vortices in channel flows and boundary layer flows. Proc. R. Soc. A 436, 585–602.Google Scholar
Bottaro, A. & Luchini, P. 1999 Görtler vortices: Are they amenable to local eigenvalue analysis? Eur. J. Fluids B 18, 47–65.Google Scholar
Chandrasehkar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Cambridge University Press.Google Scholar
Choudhari, M., Hall, P. & Street, C. 1994 On spatial evolution of long wavelength Görtler vortices governed by a viscous–inviscid interaction. Part 1. The linear case. Q. J. Mech. Appl. Maths 47, 207–230.Google Scholar
Deguchi, K., Hall, P. & Walton, A. G. 2013 The emergence of localized vortex–wave interaction states in plane Couette flow. J. Fluid Mech. 720, 582–617.Google Scholar
Denier, J., Hall, P. & Seddougui, S. O. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. A 334, 51–85.Google Scholar
Floryan, J. M. & Saric, W. S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20 (3), 316–324.Google Scholar
Gaster, M. 1974 On the effects of boundary layer growth on flow stability. J. Fluid Mech. 66, 465–480.Google Scholar
Görtler, H. 1941 Instabilität laminarer Grenzschichten an konkaven Wänden gegenüber gewissen dreidimensionalen Störungen. Z. Angew. Math. Mech. 21, 250–252.Google Scholar
Gresho, P. M. & Sani, R. L. 1970 The effects of gravity modulation on the stability of a heated fluid layer. J. Fluid Mech. 40, 783–806.Google Scholar
Hall, P. 1982 Taylor–Görtler vortices in fully-developed or boundary layer flows. J. Fluid Mech. 124, 475–494.CrossRefGoogle Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.CrossRefGoogle 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
Hall, P. & Horseman, N. 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. A 415, 421–444.Google Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.Google Scholar
Hall, P. & Smith, F. T. 1989 Nonlinear Tollmien–Schlichting/vortex interaction in boundary layers. Eur. J. Mech. B 8, 179–205.Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.Google Scholar
Hall, P. & Wu, X. 2013 The role of the exact inviscid eigenmode in the Görtler vortex receptivity problem (in preparation).Google Scholar
Lee, S. Y. 1980 The inhomogeneous Airy functions $Gi(z), Hi(z)$ . J. Chem. Phys. 72, 332–337.Google Scholar
Li, F. & Malik, M. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77–100.Google Scholar
McLachlan, N. W. 1947 Theory and Application of Mathieu Functions. Oxford University Press.Google Scholar
Niemla, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837–840.CrossRefGoogle Scholar
Park, D. S. & Huerre, P 1995 Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate. J. Fluid, Mech. 283, 249–272.Google Scholar
Rozhko, S. B. & Ruban, A. I. 1987 Longitudinal–transverse interaction in a three-dimensional boundary layer. Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 3, 42–50 in Russian.Google Scholar
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26, 379–409.CrossRefGoogle Scholar
Smith, A. M. O. 1955 On the growth of Taylor–Görtler vortices along highly concave walls. Q. J. Math. 13, 213–262.Google Scholar
Smith, F. T. & Walton, A. G. 1989 Nonlinear interaction of near planar TS waves and longitudinal vortices in boundary layer transition. Mathematika 36, 262–289.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.Google Scholar
Wu, X., Zhao, D. & Luo, J. 2013 Excitation of steady and unsteady Görtler vortices by free stream vortical disturbances. J. Fluid Mech. 682, 66–100.Google Scholar
Yu, X. & Liu, J. T. C. 1991 The secondary instability in Görtler flow. Phys. Fluids 3, 1845–1847.Google Scholar