Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-19T17:06:23.763Z Has data issue: false hasContentIssue false
  • English
  • Français

Model for unstable global modes in the rotating-disk boundary layer

Published online by Cambridge University Press:  28 September 2010

J. J. HEALEY
J. J. HEALEY*
Affiliation:
Department of Mathematics, Keele University, Keele, Staffordshire ST5 5BG, UK
*
†Email address for correspondence: j.j.healey@maths.keele.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent simulations and experiments appear to show that the rotating-disk boundary layer is linearly globally stable despite the existence of local absolute instability. However, we argue that linear global instability can be created by local absolute instability at the edge of the disk. This argument is based on investigations of the linearized complex Ginzburg–Landau equation with weakly spatially varying coefficients to model the propagation of a wavepacket through a weakly inhomogeneous unstable medium. Therefore, this mechanism could operate in a variety of locally absolutely unstable flows. The corresponding nonlinear global mode has a front at the radius of onset of absolute instability when the disk edge is far from the front. This front moves radially outwards when the Reynolds number at the disk edge is reduced. It is shown that the laminar–turbulent transition front also behaves in this manner, possibly explaining the scatter in observed transitional Reynolds numbers.

JFM classification

Instability: Absolute/convective instability Boundary Layers: Boundary layer stability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 663 , 25 November 2010 , pp. 148 - 159
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Abramowitz, M. & Stegun, J. A. 1964 Handbook of Mathematical Functions. National Bureau of Standards.Google Scholar
Briggs, R. J. 1964 Electron–Stream Interaction with Plasmas. MIT Press.Google Scholar
Chin, D. & Litt, M. 1972 An electrochemical study of flow instability on a rotating disk. J. Fluid Mech. 54, 613–625.CrossRefGoogle Scholar
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1987 Models of hydrodynamic resonances in separated shear flows. In Proceedings of the 6th Symposium on Turbulent Shear Flows Toulouse, France, pp. 119–144.Google Scholar
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1991 A frequency selection criterion in spatially developing flows. Stud. Appl. Math. 84, 119–144.Google Scholar
Clarkson, M. H., Chin, S. C. & Shacter, P. 1980 Flow visualizations of inflexional instabilities on a rotating disk. AIAA Paper 80-0279.Google Scholar
Cobb, E. C. & Saunders, O. A. 1956 Heat transfer from a rotating disk. Proc. R. Soc. Lond. A 236, 343–351.Google Scholar
Davies, C. & Carpenter, P. W. 2003 Global behaviour corresponding to the absolute instability of the rotating-disk boundary layer. J. Fluid Mech. 486, 287–329.CrossRefGoogle Scholar
Davies, C., Thomas, C. & Carpenter, P. W. 2007 Global stability of the rotating-disc boundary layer. J. Engng Math. 57, 219–236.Google Scholar
Federov, B. I., Plavnik, G. Z., Prokhorov, I. V. & Zhukhovitskii, L. G. 1976 Transitional flow conditions on a rotating disk. J. Engng Phys. 31, 1448–1453.Google Scholar
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A 248, 155–199.Google Scholar
Gregory, N. & Walker, W. S. 1960 Experiments on the effect of suction on the flow due to a rotating disk. J. Fluid Mech. 9, 225–234.Google Scholar
Healey, J. J. 2004 On the relation between the viscous and inviscid absolute instabilities of the rotating-disk boundary layer. J. Fluid Mech. 511, 179–199.CrossRefGoogle Scholar
Healey, J. J. 2006 A new convective instability of the rotating-disk boundary layer with growth normal to disk. J. Fluid Mech. 560, 279–310.CrossRefGoogle Scholar
Healey, J. J. 2007 a Instabilities of flows due to rotating disks: preface. J. Engng Math. 57, 199–204.Google Scholar
Healey, J. J. 2007 b Enhancing the absolute instability of a boundary layer by adding a far away plate. J. Fluid Mech. 579, 29–61.Google Scholar
Healey, J. J. 2008 Inviscid axisymmetric absolute instability of swirling jets. J. Fluid Mech. 613, 1–33.Google Scholar
Healey, J. J. 2009 Destabilizing effects of confinement on homogeneous mixing layers. J. Fluid Mech. 623, 241–271.Google Scholar
Huerre, P. 2000 Open shear flow instabilities. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 159–229. Cambridge University Press.Google Scholar
Hunt, R. E. & Crighton, D. G. 1991 Instability of flows in spatially developing media. Proc. R. Soc. Lond. A 435, 109–128.Google Scholar
Kobayashi, R., Kohama, Y. & Takamadate, Ch. 1980 Spiral vortices in boundary layer transition regime on a rotating disk. Acta Mech. 35, 71–82.CrossRefGoogle Scholar
Lingwood, R. J. 1995 Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech. 299, 17–33.CrossRefGoogle Scholar
Lingwood, R. J. 1996 An experimental study of absolute instability of the rotating-disk boundary layer flow. J. Fluid Mech. 314, 373–405.Google Scholar
Malik, M. R., Wilkinson, S. P. & Orszag, S. A. 1981 Instability and transition in rotating disk flow. AIAA J. 19, 1131–1138.Google Scholar
Meunier, N., Proctor, M. R. E., Sokoloff, D. D., Soward, A. M. & Tobias, S. M. 1997 Asymptotic properties of a nonlinear αω-dynamo wave: period, amplitude and latitude dependence. Geophys. Astrophys. Fluid Dyn. 86, 249–285.Google Scholar
Othman, H. & Corke, T. C. 2006 Experimental investigation of absolute instability of a rotating-disk boundary layer. J. Fluid Mech. 565, 63–94.Google Scholar
Pier, B. 2003 Finite-amplitude crossflow vortices, secondary instability in the rotating-disk boundary layer. J. Fluid Mech. 487, 315–343.Google Scholar
Pier, B., Huerre, P., Chomaz, J. M. & Couairon, A. 1998 Steep nonlinear global modes in spatially developing media. Phys. Fluids 10, 2433–2435.Google Scholar
Smith, N. H. 1946 Exploratory investigation of laminar boundary layer oscillations on a rotating disk. NACA Tech. Note 1227.Google Scholar
Soward, A. M. 1977 On the finite amplitude thermal instability of a rapidly rotating fluid sphere. Geophys. Astrophys. Fluid Dyn. 9, 19–74.Google Scholar
Soward, A. M. & Jones, C. A. 1983 The linear stability of the flow in the narrow gap between two concentric rotating spheres. Q. J. Mech. Appl. Math. 36, 19–42.CrossRefGoogle Scholar
Tobias, S. M., Proctor, M. R. E. & Knobloch, E. 1998 Convective and absolute instabilities of fluid flows in finite geometries. Physica D 113, 43–72.Google Scholar
Viaud, B., Serre, E. & Chomaz, J. M. 2008 The elephant mode between two rotating disks. J. Fluid Mech. 598, 451–464.Google Scholar
Wang, S. & Rusak, Z. 1996 On the stability of an axisymmetric rotating flow in a pipe. Phys. Fluids 8, 1007–1016.Google Scholar
Wilkinson, S. P. & Malik, M. R. 1985 Stability experiments in the flow over a rotating disk flow. AIAA J. 23, 588–595.Google Scholar