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The cross-flow instability of the boundary layer on a rotating cone

Published online by Cambridge University Press:  10 March 2009

S. J. GARRETT ,
Z. HUSSAIN  and
S. O. STEPHEN
S. J. GARRETT*
Affiliation:
Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK
Z. HUSSAIN
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
S. O. STEPHEN
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
*
†Email address for correspondence: s.garrett@mcs.le.ac.uk
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Abstract

Experimental studies have shown that the boundary-layer flow over a rotating cone is susceptible to cross-flow and centrifugal instability modes of spiral nature, depending on the cone sharpness. For half-angles (ψ) ranging from propeller nose cones to rotating disks (ψ ≥ 40°), the instability triggers co-rotating vortices, whereas for sharp spinning missiles (ψ < 40°), counter-rotating vortices are observed. In this paper we provide a mathematical description of the onset of co-rotating vortices for a family of cones rotating in quiescent fluid, with a view towards explaining the effect of ψ on the underlying transition of dominant instability. We investigate the stability of inviscid cross-flow modes (type I) as well as modes which arise from a viscous–Coriolis force balance (type II), using numerical and asymptotic methods. The influence of ψ on the number and orientation of the spiral vortices is examined, with comparisons drawn between our two distinct methods as well as with previous experimental studies. Our results indicate that increasing ψ has a stabilizing effect on both the type I and type II modes. Favourable agreement is obtained between the numerical and asymptotic methods presented here and existing experimental results for ψ > 40°. Below this half-angle we suggest that an alternative instability mechanism is at work, which is not amenable to investigation using the formulation presented here.

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 622 , 10 March 2009 , pp. 209 - 232
Copyright
Copyright © Cambridge University Press 2009

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