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

  • Z. Hussain (a1), S. J. Garrett (a2), S. O. Stephen (a3) and P. T. Griffiths (a2)

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.

Copyright

Corresponding author

Email address for correspondence: Z.Hussain@mmu.ac.uk

References

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Corke, T. C. & Knasiak, K. F. 1998 Stationary traveling cross-flow mode interactions on a rotating disk. J. Fluid Mech. 355, 285315.
Evans, H. 1968 Laminar Boundary Layer Theory. Addison-Wesley.
Garrett, S. J., Hussain, Z. & Stephen, S. O. 2009 The crossflow instability of the boundary layer on a rotating cone. J. Fluid Mech. 622, 209232.
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), 11841194.
Garrett, S. J. & Peake, N. 2007 The absolute instability of the boundary layer on a rotating cone. Eur. J. Mech. (B/Fluids) 26, 344353.
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, 155199.
Hall, P. 1982 Taylor–Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475494.
Hussain, Z.2010 Stability and transition of three-dimensional rotating boundary layers. PhD thesis, University of Birmingham.
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.
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, 274293.
Hussain, Z., Stephen, S. O. & Garrett, S. J. 2012 The centrifugal instability of a slender rotating cone. J. Algorithms Comput. Technol. 6 (1), 113128.
Kobayashi, R. 1981 Linear stability theory of boundary layer along a cone rotating in axial flow. Bull. Japan Soc. Mech. Engrs. 24, 934940.
Kobayashi, R. 1994 Review: laminar-to-turbulent transition of three-dimensional boundary layers on rotating bodies. Trans. ASME 116, 200211.
Kobayashi, R. & Izumi, H. 1983 Boundary-layer transition on a rotating cone in still fluid. J. Fluid Mech. 127, 353364.
Kobayashi, R., Kohama, Y.  & Kurosawa, M. 1983 Boundary-layer transition on a rotating cone in axial flow. J. Fluid Mech. 127, 341352.
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, 139145.
Kohama, Y. 1985 Flow structures formed by axisymmetric spinning bodies. AIAA J. 23, 14451447.
Reed, H. L. & Saric, W. S. 1989 Stability of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 21, 235284.
Rosenhead, L. 1963 Laminar Boundary Layers. Oxford University Press.
Saric, W. S., Reed, H. L. & White, E. B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413440.
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.
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