Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T07:01:23.441Z Has data issue: false hasContentIssue false
  • English
  • Français

On the non-parallel instability of the rotating-sphere boundary layer

Published online by Cambridge University Press:  29 March 2017

Antonio Segalini Open the ORCID record for Antonio Segalini [Opens in a new window]  and
Stephen J. Garrett Open the ORCID record for Stephen J. Garrett [Opens in a new window]
Antonio Segalini*
Affiliation:
Linné FLOW Centre, KTH Mechanics, 10044 Stockholm, Sweden
Stephen J. Garrett
Affiliation:
Department of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UK
*
†Email address for correspondence: segalini@mech.kth.se
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a new solution for the steady boundary-layer flow over the rotating sphere that also accounts for the eruption of the boundary layer at the equator and other higher-order viscous effects. Non-parallel corrections to the local type I and type II convective instability modes of this flow are also computed as a function of spin rate. Our instability results are associated with the previously observed spiral vortices and remarkable agreement between our predictions of the number of vortices and experimental observations is found. Vortices travelling at 70 %–80 % of the local surface speed are found to be the most amplified for sufficient spin rates, also consistent with prior experimental observations.

JFM classification

Instability: Absolute/convective instability Boundary Layers: Boundary layer stability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 818 , 10 May 2017 , pp. 288 - 318
Check for updates
Copyright
© 2017 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

Appelquist, E., Schlatter, P., Alfredsson, P. H. & Lingwood, R. J. 2015a Global linear instability of the rotating-disk flow investigated through simulations. J. Fluid Mech. 765, 612–631.CrossRefGoogle Scholar
Appelquist, E., Schlatter, P., Alfredsson, P. H. & Lingwood, R. J. 2015b Investigation of the global instability of the rotating-disk boundary layer. Proc. IUTAM 14, 321–328.CrossRefGoogle Scholar
Banks, W. H. H. 1965 The boundary layer on a rotating sphere. Q. J. Mech. Appl. Maths 18, 443–454.CrossRefGoogle Scholar
Banks, W. H. H. & Zaturska, M. B. 1979 The collision of unsteady laminar boundary layers. J. Engng Maths 13, 193–212.CrossRefGoogle Scholar
Barrow, A. & Garrett, S. J. 2013 Convective and absolute instabilities in the boundary layer over rotating spheres with surface mass flux and incident axial flow. Eur. J. Mech. (B/Fluids) 38, 93–100.CrossRefGoogle Scholar
Barrow, A., Garrett, S. J. & Peake, N. 2014 Global linear stability of the boundary-layer flow over a rotating sphere. Eur. J. Mech. (B/Fluids) 49, 301–307.CrossRefGoogle Scholar
Bowden, F. P. & Lord, R. G. 1963 The aerodynamic resistance to a sphere rotating at high speed. Proc. R. Soc. Lond. A 271, 143–153.Google Scholar
Calabretto, S. A. W., Levy, B., Denier, J. P. & Mattner, T. W. 2015 The unsteady flow due to an impulsively rotated sphere. Proc. R. Soc. Lond. A 471 (2181), 20150299.Google Scholar
Cooper, A. J., Harris, J. H., Garrett, S. J., Özkan, M. & Thomas, P. J. 2015 The effect of anisotropic and isotropic roughness on the convective stability of the rotating disk boundary layer. Phys. Fluids 27 (1), 014107.CrossRefGoogle Scholar
Corke, T. C. & Knasiak, K. F. 1998 Stationary travelling crossflow mode interactions on a rotating disk. J. Fluid Mech. 355, 285.CrossRefGoogle Scholar
Corke, T. C., Matlis, E. H. & Othman, H. 2007 Transition to turbulence in rotating-disk boundary layers-convective and absolute instabilities. J. Engng Maths 57, 253.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 2003 Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. J. Fluid Mech. 486, 287–329.CrossRefGoogle Scholar
Dennis, S. C. R. & Duck, P. W. 1988 Unsteady flow due to an impulsively started rotating sphere. Comput. Fluids 16, 291–310.CrossRefGoogle Scholar
Dennis, S. C. R., Singh, S. N. & Ingham, D. B. 1980 The steady flow due to a rotating sphere at low and moderate Reynolds numbers. J. Fluid Mech. 101, 257–279.CrossRefGoogle Scholar
Garrett, S. J.2002 The stability and transition of the boundary layer on rotating bodies. PhD thesis, University of Cambridge.CrossRefGoogle Scholar
Garrett, S. J. 2010 Vortex-speed selection within the boundary-layer flow over a rotating sphere placed in an enforced axial flow. Eur. J. Mech. (B/Fluids) 29, 84–92.CrossRefGoogle Scholar
Garrett, S. J. & Peake, N. 2002 The stability and transition of the boundary layer on a rotating sphere. J. Fluid Mech. 456, 199–218.CrossRefGoogle Scholar
Garrett, S. J. & Peake, N. 2004 The stability of the boundary layer on a sphere rotating in a uniform axial flow. Eur. J. Mech. (B/Fluids) 23 (2), 241–253.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. 248, 155–199.Google Scholar
Hall, P. 1986 An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc. Proc. R. Soc. Lond. A 406, 93–106.Google Scholar
Hollerbach, R., Wiener, R. J., Sullivan, I. S., Donnelly, R. J. & Barenghi, C. F. 2002 The flow around a torsionally oscillating sphere. Phys. Fluids 14, 4192.CrossRefGoogle Scholar
Howarth, L. 1951 Note on the boundary layer on a rotating sphere. Phil. Mag. 42, 1308–1315.CrossRefGoogle Scholar
Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2012 A new way to describe the transition characteristics of a rotating-disk boundary-layer flow. Phys. Fluids 24 (3), 031701.CrossRefGoogle Scholar
Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2013 An experimental study of edge effects on rotating-disk transition. J. Fluid Mech. 716, 638–657.CrossRefGoogle Scholar
Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2014 On the laminar–turbulent transition of the rotating-disk flow: the role of absolute instability. J. Fluid Mech. 745, 132–163.CrossRefGoogle Scholar
Kobayashi, R. & Arai, T. 1990 Spiral vortex behaviour in transition region and separation of three-dimensional boundary layers on spheres rotating in axial flow. In Laminar Turbulent Transition, IUTAM Symposium Toulouse (ed. Arnal, D. & Michel, R.), pp. 551–557. Springer.CrossRefGoogle Scholar
Kohama, O. & Kobayashi, R. 1983 Boundary-layer transition and the behaviour of spiral vortices on rotating spheres. J. Fluid Mech. 137, 153–164.CrossRefGoogle Scholar
Kundu, P. K., Cohen, I. M. & Dowling, D. R. 2016 Fluid Mechanics, 6th edn. Elsevier.Google Scholar
Lingwood, R. J. 1995 Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech. 299, 17–33.CrossRefGoogle Scholar
Malik, M. R. 1986 The neutral curve for stationary disturbances in rotating-disk flow. J. Fluid Mech. 164, 275–287.CrossRefGoogle Scholar
Monkewitz, P. A., Huerre, P. & Chomaz, J.-M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251, 1–20.CrossRefGoogle Scholar
Othman, H. & Corke, T. C. 2006 Experimental investigation of absolute instability of a rotating-disk boundary layer. J. Fluid Mech. 565, 63.CrossRefGoogle Scholar
Phares, D. J., Smedley, G. T. & Flagan, R. C. 2000 The inviscid impingement of a jet with arbitrary velocity profile. Phys. Fluids 12, 2046–2055.CrossRefGoogle Scholar
Pier, B. 2003 Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer. J. Fluid Mech. 487, 315–343.CrossRefGoogle Scholar
Riley, N. 1962 Radial jets with swirl. Part I. Incompressible flow. Q. J. Mech. Appl. Maths 25, 435–458.CrossRefGoogle Scholar
Rubel, A. 1983 Inviscid axisymmetric jet impingement with recirculating stagnation regions. AIAA J. 21, 351–357.CrossRefGoogle Scholar
Sawatzki, O. 1970 Flow field around a rotating sphere. Acta Mechanica 9 (3), 159–214.CrossRefGoogle Scholar
Schwarz, W. H. 1963 The radial free jet. Chem. Engng Sci. 18, 779–786.CrossRefGoogle Scholar
Siddiqui, M. E., Mukund, V., Scott, J. & Pier, B. 2013 Experimental characterization of transition region in rotating-disk boundary layer. Phys. Fluids 25 (3), 034102.CrossRefGoogle Scholar
Simpson, C. J. & Stewartson, K. 1982 A note on a boundary-layer collision on a rotating sphere. Z. Angew. Math. Phys. J. Appl. Math. Phys. 33, 370–378.Google Scholar
Smith, F. T. & Duck, P. W. 1977 Separation of jets or thermal boundary layers from a wall. Q. J. Mech. Appl. Maths 30, 143–156.CrossRefGoogle Scholar
Squire, H. B. 1955 Radial jets. In Fifty Years of Boundary Layer Research (ed. Görtler, H. & Tollmien, W.). Vieweg.Google Scholar
Stewartson, K. 1958 On rotating laminar boundary layers. In Grenzschichtforschung/Boundary Layer Research (ed. Görtler, H.). Springer.Google Scholar
Van Dommelen, L. L. 1990 On the lagrangian description of unsteady boundary-layer separation. Part 2. The spinning sphere. J. Fluid Mech. 210, 627–645.CrossRefGoogle Scholar