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Unsteady flow in a rotating torus after a sudden change in rotation rate

Published online by Cambridge University Press:  20 October 2011

R. E. Hewitt ,
A. L. Hazel ,
R. J. Clarke  and
J. P. Denier
R. E. Hewitt*
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
A. L. Hazel
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
R. J. Clarke
Affiliation:
Department of Engineering Science, University of Auckland, New Zealand
J. P. Denier
Affiliation:
School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia
*
†Email address for correspondence: richard.e.hewitt@manchester.ac.uk
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Abstract

We consider the temporal evolution of a viscous incompressible fluid in a torus of finite curvature; a problem first investigated by Madden & Mullin (J. Fluid Mech., vol. 265, 1994, pp. 265–217). The system is initially in a state of rigid-body rotation (about the axis of rotational symmetry) and the container’s rotation rate is then changed impulsively. We describe the transient flow that is induced at small values of the Ekman number, over a time scale that is comparable to one complete rotation of the container. We show that (rotationally symmetric) eruptive singularities (of the boundary layer) occur at the inner or outer bend of the pipe for a decrease or an increase in rotation rate respectively. Moreover, on allowing for a change in direction of rotation, there is a (negative) ratio of initial-to-final rotation frequencies for which eruptive singularities can occur at both the inner and outer bend simultaneously. We also demonstrate that the flow is susceptible to a combination of axisymmetric centrifugal and non-axisymmetric inflectional instabilities. The inflectional instability arises as a consequence of the developing eruption and is shown to be in qualitative agreement with the experimental observations of Madden & Mullin (1994). Throughout our work, detailed quantitative comparisons are made between asymptotic predictions and finite- (but small-) Ekman-number Navier–Stokes computations using a finite-element method. We find that the boundary-layer results correctly capture the (finite-Ekman-number) rotationally symmetric flow and its global stability to linearised perturbations.

JFM classification

Boundary Layers: Pipe flow boundary layer

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 688 , 10 December 2011 , pp. 88 - 119
Copyright
Copyright © Cambridge University Press 2011

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References

1. Banks, W. H. H. & Zaturska, M. B. 1979 The collision of unsteady laminar boundary layers. J. Engng Maths 13 (3), 193–212.CrossRefGoogle Scholar
2. Benton, E. R. & Clark, A. 1974 Spin-up. Annu. Rev. Fluid Mech. 6, 257–280.CrossRefGoogle Scholar
3. Berger, S. A., Talbot, L. & Yao, L. S. 1983 Flow in curved pipes. Annu. Rev. Fluid Mech. 15 (1), 461–512.CrossRefGoogle Scholar
4. Cowley, S. J., Van Dommelen, L. L. & Lam, S. T. 1990 On the use of Lagrangian variables in descriptions of unsteady boundary-layer separation. Phil. Trans. R. Soc. Lond. 333 (1631), 343–378.Google Scholar
5. Demkowicz, L., Oden, J. T., Rachowicz, W. & Hardy, O. 1989 Toward a universal h–p adaptive finite element strategy, Part 1. Constrained approximation and data structure. Comput. Meth. Appl. Mech. Engng 77, 79–112.CrossRefGoogle Scholar
6. 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. 335 (1636), 51–85.Google Scholar
7. Elman, H. C., Silvester, D. J. & Wathen, A. J. 2005 Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
8. Hall, P. 1985 The Görtler vortex instability mechanism in three-dimensional boundary layers. Proc. R. Soc. Lond. A 399 (1816), 135–152.Google Scholar
9. Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37, 151–189.CrossRefGoogle Scholar
10. Heil, M. & Hazel, A. L. 2006 oomph-lib – an object-oriented multi-physics finite-element library. In Fluid-Structure Interaction (ed. Schafer, M. & Bungartz, H.-J. ). Lecture Notes on Computational Science and Engineering , pp. 19–49. Springer.CrossRefGoogle Scholar
11. Heroux, M. A., Bartlett, R. A., Howle, V. E., Hoekstra, R. J., Hu, J. J., Kolda, T. G., Lehoucq, R. B., Long, K. R., Pawlowski, R. P., Phipps, E. T., Salinger, A. G., Thornquist, H. K., Tuminaro, R. S., Willenbring, J. M., Williams, A. & Stanley, K. S. 2005 An overview of the trilinos project. ACM Trans. Math. Softw. 31 (3), 397–423.CrossRefGoogle Scholar
12. Mackerrell, O., Blennerhassett, P. J. & Bassom, A 2002 Görtler vortices in the Rayleigh layer on an impulsively started cylinder. Phys. Fluids 14 (9), 2948–2956.CrossRefGoogle Scholar
13. Madden, F. N. & Mullin, T. 1994 The spin-up from rest of a fluid-filled torus. J. Fluid Mech. 265, 217–244.CrossRefGoogle Scholar
14. Otto, S. R. 1993 Stability of the flow around a cylinder: the spin-up problem. IMA J. Appl. Maths 51 (1), 13–26.CrossRefGoogle Scholar
15. del Pino, C., Hewitt, R. E., Clarke, R. J., Mullin, T. & Denier, J. P. 2008 Unsteady fronts in the spin-down of a fluid-filled torus. Phys. Fluids 20 (12), 124104.CrossRefGoogle Scholar
16. Siggers, J. H. & Waters, S. L. 2005 Steady flows in pipes with finite curvature. Phys. Fluids 17, 077102.CrossRefGoogle Scholar
17. Siggers, J. H. & Waters, S. L. 2008 Unsteady flows in pipes with finite curvature. J. Fluid Mech. 600, 133–165.CrossRefGoogle Scholar
18. Simpson, C. J. & Stewartson, K. 1982 A note on a boundary-layer collision on a rotating sphere. Z. Angew. Math. Phys. 33 (3), 370–378.CrossRefGoogle Scholar
19. Stewartson, K., Cebeci, T. & Chang, K. C. 1980 A boundary-layer collision in a curved duct. Q. J. Mech. Appl. Maths 33 (1), 59–75.CrossRefGoogle Scholar
20. Yang, Z.-H. & Keller, H. 1986 Multiple laminar flows through curved pipes. Appl. Numer. Maths 2, 257–271.CrossRefGoogle Scholar
21. Zienkiewicz, O. C & Zhu, J. Z 1992a The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Intl J. Numer. Meth. Engng 33 (7), 1331–1364.CrossRefGoogle Scholar
22. Zienkiewicz, OC & Zhu, JZ 1992b The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity. Intl J. Numer. Meth. Engng 33 (7), 1365–1382.CrossRefGoogle Scholar
23. Zurigat, Y. H. & Malik, M. R. 1995 Effect of cross-flow on Görtler instability in incompressible boundary layers. Phys. Fluids 7, 1616.CrossRefGoogle Scholar