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The centrifugal instability of the boundary-layer flow over slender rotating cones

Published online by Cambridge University Press:  14 August 2014

Z. Hussain*
Affiliation:
Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
S. J. Garrett
Affiliation:
Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
S. O. Stephen
Affiliation:
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
*
Email address for correspondence: zh33@le.ac.uk

Abstract

Existing experimental and theoretical studies are discussed which lead to the clear hypothesis of a hitherto unidentified convective instability mode that dominates within the boundary-layer flow over slender rotating cones. The mode manifests as Görtler-type counter-rotating spiral vortices, indicative of a centrifugal mechanism. Although a formulation consistent with the classic rotating-disk problem has been successful in predicting the stability characteristics over broad cones, it is unable to identify such a centrifugal mode as the half-angle is reduced. An alternative formulation is developed and the governing equations solved using both short-wavelength asymptotic and numerical approaches to independently identify the centrifugal mode.

Type
Papers
Copyright
© 2014 Cambridge University Press 

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References

Corke, T. C. & Knasiak, K. F. 1998 Stationary traveling cross-flow mode interactions on a rotating disk. J. Fluid Mech. 355, 285315.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 2003 Global behaviour corresponding to the absolute instability of the rotating-disk boundary layer. J. Fluid Mech. 486, 287329.Google Scholar
Denier, J. P., Hall, P. & Seddougui, S. O. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. Lond. A 335, 5185.Google Scholar
Garrett, S. J. 2010 Linear growth rates of types I and II convective modes within the rotating-cone boundary layer. Fluid Dyn. Res. 42, 025504.CrossRefGoogle 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, 209232.CrossRefGoogle 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), 11841194.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, 344353.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, 155199.Google Scholar
Hall, P. 1982 Taylor–Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475494.Google Scholar
Healey, J. J. 2010 Model for unstable global modes in the rotating-disk boundary layer. J. Fluid Mech. 663, 148159.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 instability of the boundary layer over a disk rotating in an enforced axial flow. Phys. Fluids 23, 114108.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), 113128.Google Scholar
Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2013 An experimental study of edge effects on rotating-disk transition. J. Fluid Mech. 716, 638657.CrossRefGoogle Scholar
Kappesser, R., Greif, R. & Cornet, I. 1973 Mass transfer on rotating cones. Appl. Sci. Res. 28, 442452.Google Scholar
Kobayashi, R. 1981 Linear stability theory of boundary layer along a cone rotating in axial flow. Bull. JSME 24, 934940.Google Scholar
Kobayashi, R. 1994 Review: laminar-to-turbulent transition of three-dimensional boundary layers on rotating bodies. Trans. ASME 116, 200211.Google Scholar
Kobayashi, R. & Izumi, H. 1983 Boundary-layer transition on a rotating cone in still fluid. J. Fluid Mech. 127, 353364.Google Scholar
Kobayashi, R., Kohama, Y. & Kurosawa, M. 1983 Boundary-layer transition on a rotating cone in axial flow. J. Fluid Mech. 127, 341352.CrossRefGoogle Scholar
Kohama, Y. 1985 Flow structures formed by axisymmetric spinning bodies. AIAA J. 23, 14451447.Google Scholar
Kohama, Y. P. 2000 Three-dimensional boundary layer transition study. Curr. Sci. 79 (6), 800807.Google Scholar
Kreith, F., Ellis, D. & Giesing, J. 1962 An experimental investigation of the flow engendered by a rotating cone. Appl. Sci. Res. A 11, 430440.CrossRefGoogle Scholar
Lingwood, R. J. 1995 Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech. 299, 1733.CrossRefGoogle Scholar
Lingwood, R. J. 1996 An experimental study of absolute instability of the rotating-disk boundary layer flow. J. Fluid Mech. 314, 373405.Google Scholar
Malik, M. R. 1986 The neutral curve for stationary disturbances in rotating-disk flow. J. Fluid Mech. 164, 275287.Google Scholar
Pier, B. 2003 Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer. J. Fluid Mech. 487, 315343.Google Scholar
Reed, H. L. & Saric, W. S. 1989 Stability of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 21, 235284.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, 413440.CrossRefGoogle Scholar
Towers, P. D. & Garrett, S. J. 2014a Similarity solutions of compressible flow over a rotating cone with surface suction. Therm. Sci. 00, 3232; doi:10.2298/TSCI130408032T.Google Scholar
Towers, P. D. & Garrett, S. J.2014b The stability of the compressible boundary-layer flows over rotating cones with surface mass flux (in preparation).Google Scholar