Skip to main content

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

  • R. E. Hewitt (a1), A. L. Hazel (a1), R. J. Clarke (a2) and J. P. Denier (a3)

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.

Corresponding author
Email address for correspondence:
Hide All
1. Banks, W. H. H. & Zaturska, M. B. 1979 The collision of unsteady laminar boundary layers. J. Engng Maths 13 (3), 193212.
2. Benton, E. R. & Clark, A. 1974 Spin-up. Annu. Rev. Fluid Mech. 6, 257280.
3. Berger, S. A., Talbot, L. & Yao, L. S. 1983 Flow in curved pipes. Annu. Rev. Fluid Mech. 15 (1), 461512.
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), 343378.
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, 79112.
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), 5185.
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.
8. Hall, P. 1985 The Görtler vortex instability mechanism in three-dimensional boundary layers. Proc. R. Soc. Lond. A 399 (1816), 135152.
9. Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37, 151189.
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. 1949. Springer.
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), 397423.
12. Mackerrell, O., Blennerhassett, P. J. & Bassom, A 2002 Görtler vortices in the Rayleigh layer on an impulsively started cylinder. Phys. Fluids 14 (9), 29482956.
13. Madden, F. N. & Mullin, T. 1994 The spin-up from rest of a fluid-filled torus. J. Fluid Mech. 265, 217244.
14. Otto, S. R. 1993 Stability of the flow around a cylinder: the spin-up problem. IMA J. Appl. Maths 51 (1), 1326.
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.
16. Siggers, J. H. & Waters, S. L. 2005 Steady flows in pipes with finite curvature. Phys. Fluids 17, 077102.
17. Siggers, J. H. & Waters, S. L. 2008 Unsteady flows in pipes with finite curvature. J. Fluid Mech. 600, 133165.
18. Simpson, C. J. & Stewartson, K. 1982 A note on a boundary-layer collision on a rotating sphere. Z. Angew. Math. Phys. 33 (3), 370378.
19. Stewartson, K., Cebeci, T. & Chang, K. C. 1980 A boundary-layer collision in a curved duct. Q. J. Mech. Appl. Maths 33 (1), 5975.
20. Yang, Z.-H. & Keller, H. 1986 Multiple laminar flows through curved pipes. Appl. Numer. Maths 2, 257271.
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), 13311364.
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), 13651382.
23. Zurigat, Y. H. & Malik, M. R. 1995 Effect of cross-flow on Görtler instability in incompressible boundary layers. Phys. Fluids 7, 1616.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 30 *
Loading metrics...

Abstract views

Total abstract views: 200 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th July 2018. This data will be updated every 24 hours.