Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-29T02:20:41.406Z Has data issue: false hasContentIssue false
  • English
  • Français

The centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free stream

Published online by Cambridge University Press:  22 December 2015

Z. Hussain ,
S. J. Garrett ,
S. O. Stephen  and
P. T. Griffiths
Z. Hussain*
Affiliation:
School of Computing, Mathematics and Digital Technology, Manchester Metropolitan University, Manchester M1 5GD, UK
S. J. Garrett
Affiliation:
Department of Engineering, University of Leicester, Leicester LE1 7RH, UK
S. O. Stephen
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
P. T. Griffiths
Affiliation:
Department of Engineering, University of Leicester, Leicester LE1 7RH, UK
*
†Email address for correspondence: Z.Hussain@mmu.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

In this study, a new centrifugal instability mode, which dominates within the boundary-layer flow over a slender rotating cone in still fluid, is used for the first time to model the problem within an enforced oncoming axial flow. The resulting problem necessitates an updated similarity solution to represent the basic flow more accurately than previous studies in the literature. The new mean flow field is subsequently perturbed, leading to disturbance equations that are solved via numerical and short-wavelength asymptotic approaches, yielding favourable comparisons with existing experiments. Essentially, the boundary-layer flow undergoes competition between the streamwise flow component, due to the oncoming flow, and the rotational flow component, due to effect of the spinning cone surface, which can be described mathematically in terms of a control parameter, namely the ratio of streamwise to axial flow. For a slender cone rotating in a sufficiently strong axial flow, the instability mode breaks down into Görtler-type counter-rotating spiral vortices, governed by an underlying centrifugal mechanism, which is consistent with experimental and theoretical studies for a slender rotating cone in otherwise still fluid.

JFM classification

Boundary Layers: Boundary layer stability Geophysical and Geological Flows: Rotating flows Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 788 , 10 February 2016 , pp. 70 - 94
Check for updates
Copyright
© 2016 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

Corke, T. C. & Knasiak, K. F. 1998 Stationary traveling cross-flow mode interactions on a rotating disk. J. Fluid Mech. 355, 285–315.Google Scholar
Evans, H. 1968 Laminar Boundary Layer Theory. Addison-Wesley.Google Scholar
Garrett, S. J., Hussain, Z. & Stephen, S. O. 2009 The crossflow instability of the boundary layer on a rotating cone. J. Fluid Mech. 622, 209–232.Google Scholar
Garrett, S. J., Hussain, Z. & Stephen, S. O. 2010 Boundary-layer transition on broad cones rotating in an imposed axial flow. AIAA J. 48 (6), 1184–1194.CrossRefGoogle Scholar
Garrett, S. J. & Peake, N. 2007 The absolute instability of the boundary layer on a rotating cone. Eur. J. Mech. (B/Fluids) 26, 344–353.CrossRefGoogle 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
Hall, P. 1982 Taylor–Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475–494.Google Scholar
Hussain, Z.2010 Stability and transition of three-dimensional rotating boundary layers. PhD thesis, University of Birmingham.Google Scholar
Hussain, Z., Garrett, S. J. & Stephen, S. O. 2011 The convective instability of the boundary layer on a rotating disk in axial flow. Phys. Fluids 23, 1141108.Google Scholar
Hussain, Z., Garrett, S. J. & Stephen, S. O. 2014 The centrifugal instability of the boundary-layer flow over slender rotating cones. J. Fluid Mech. 755, 274–293.CrossRefGoogle Scholar
Hussain, Z., Stephen, S. O. & Garrett, S. J. 2012 The centrifugal instability of a slender rotating cone. J. Algorithms Comput. Technol. 6 (1), 113–128.Google Scholar
Kobayashi, R. 1981 Linear stability theory of boundary layer along a cone rotating in axial flow. Bull. Japan Soc. Mech. Engrs. 24, 934–940.Google Scholar
Kobayashi, R. 1994 Review: laminar-to-turbulent transition of three-dimensional boundary layers on rotating bodies. Trans. ASME 116, 200–211.Google Scholar
Kobayashi, R. & Izumi, H. 1983 Boundary-layer transition on a rotating cone in still fluid. J. Fluid Mech. 127, 353–364.Google Scholar
Kobayashi, R., Kohama, Y.  & Kurosawa, M. 1983 Boundary-layer transition on a rotating cone in axial flow. J. Fluid Mech. 127, 341–352.Google Scholar
Koh, J. C. Y. & Price, J. F. 1967 Non-similar boundary-layer heat transfer on a rotating cone in forced flow. Trans. ASME J. Heat Transfer. 89, 139–145.CrossRefGoogle Scholar
Kohama, Y. 1985 Flow structures formed by axisymmetric spinning bodies. AIAA J. 23, 1445–1447.Google Scholar
Reed, H. L. & Saric, W. S. 1989 Stability of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 21, 235–284.Google Scholar
Rosenhead, L. 1963 Laminar Boundary Layers. Oxford University Press.Google Scholar
Saric, W. S., Reed, H. L. & White, E. B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413–440.Google Scholar
Towers, P. D. & Garrett, S. J. 2014 Similarity solutions of compressible flow over a rotating cone with surface suction. Therm. Sci. Online-First 32, doi:10.2298/TSCI130408032T.Google Scholar