Theoretical and Computational Fluid Dynamics

, Volume 32, Issue 6, pp 821–845 | Cite as

The transient development of the flow in an impulsively rotated annular container

  • Sophie A. W. Calabretto
  • James P. Denier
  • Trent W. Mattner
Original Article


When a fluid-filled container is spun up from rest to a constant angular velocity the fluid responds in such a way that the fluid–container system is ultimately in a state of rigid-body rotation. The fluid can then be said to have traversed a trajectory in phase space from a simple stable equilibrium state of no motion to another stable equilibrium representing full rigid-body rotation. This simple statement belies the fact that during this process the fluid can undergo a series of transitions, from a laminar through a transient turbulent state, before attaining the stable motion that is rigid-body rotation. Using a combination of analytical and computational methods, we focus on the dynamics resulting from an impulsive change in the rotation rate of a fluid-filled annulus, specifically, the impulsive spin-up of a stationary annulus, or the impulsive spin-down of an annulus already in a state of rigid-body rotation. We explore the initial development of the impulsively generated axisymmetric boundary layer, its subsequent instability, and the larger-scale transient features within this class of flows, allowing us to look at the effect these features have on the time it takes for the system to spin up to a steady state, or spin down to rest.


Transient flow Spin-up Transient turbulence Transition to turbulence 


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  1. 1.
    Benton, E., Clark, A.: Spin-up. Ann. Rev. Fluid Mech. 6, 257–280 (1974)CrossRefGoogle Scholar
  2. 2.
    Blackburn, H.M., Sherwin, S.J.: Formulation of a Galerkin spectral element-Fourier method for three-dimensional incompressible flows in cylindrical geometries. J. Comput. Phys. 197, 759–778 (2004)CrossRefGoogle Scholar
  3. 3.
    Calabretto, S.A.W.: Instability and transition in unsteady rotating flows. PhD thesis, The University of Auckland (2015)Google Scholar
  4. 4.
    Calabretto, S.A.W., Mattner, T.W., Denier, J.P.: The effect of seam imperfections in the spin-up of a fluid-filled torus. J. Fluid Mech. 767, 240–253 (2015)CrossRefGoogle Scholar
  5. 5.
    del Pino, C., Hewitt, R.E., Clarke, R.J., Mullin, T., Denier, J.P. Unsteady fronts in the spin-down of a fluid-filled torus. Phys Fluids 20, 124104 (2008)Google Scholar
  6. 6.
    Do, Y., Lopez, J., Marques, F.: Optimal harmonic response in a confined Bödewadt boundary layer flow. Phys. Rev. E 82, 036301 (2010)CrossRefGoogle Scholar
  7. 7.
    Duck, P.W., Foster, M.R.: Spin-up of homogeneous and stratified fluids. Ann. Rev. Fluid Mech. 33, 231–263 (2001)CrossRefGoogle Scholar
  8. 8.
    Ekman, V.W.: On the influence of the earth’s rotation on ocean-currents. Arkiv för matematik, astronomi och fysik 2(11), 53 (1905)zbMATHGoogle Scholar
  9. 9.
    Görtler, H.: Über eine dreidimensionale Instabilität laminarer Grenzschichten an konkaven Wänden. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1:1–26 (1940) (translated as ‘On the three-dimensional instability of laminar boundary layers on concave walls’, NACA-TM1375)Google Scholar
  10. 10.
    Greenspan, H.P.: The Theory of Rotating Fluids. Cambridge University Press, Cambridge (1968)zbMATHGoogle Scholar
  11. 11.
    Greenspan, H.P., Howard, L.N.: On a time-dependent motion of a rotating fluid. J. Fluid Mech. 17, 385–404 (1963)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hall, P.: Taylor–Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475–494 (1982)CrossRefGoogle Scholar
  13. 13.
    Hall, P.: Centrifugal instabilities of circumferential flows in finite cylinders: the wide gap problem. Proc. R. Soc. Lond. Ser. A 384, 359–379 (1982)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hall, P.: The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58 (1983)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hall, P.: The Görtler vortex instability mechanism in three-dimensional boundary layers. Proc. R. Soc. Lond. Ser. A 399, 135–152 (1985)CrossRefGoogle Scholar
  16. 16.
    Hall, P., Seddougui, S.: On the onset of three-dimensionality and time-dependence in Görtler vortices. J. Fluid Mech. 204, 405–420 (1989)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hewitt, R.E., Hazel, A.L., Clarke, R.J., Denier, J.P.: Unsteady flow in a rotating torus after a sudden change in rotation rate. J. Fluid Mech. 688, 88–119 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Honji, H.: Streaked flow around an oscillating circular cylinder. J. Fluid Mech. 107, 509–520 (1981)CrossRefGoogle Scholar
  19. 19.
    Jewell, N., Denier, J.: The decay of the flow in the end region of a suddenly blocked pipe. J. Fluid Mech. 730, 533–558 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lopez, J., Marques, F., Rubio, A., Avila, M.: Crossflow instability of finite Bödewadt flows: transients and spiral waves. Phys. Fluids 21(114), 107 (2009)zbMATHGoogle Scholar
  21. 21.
    Lopez, J.M., Weidman, P.D.: Stability of stationary endwall boundary layers during spin-down. J. Fluid Mech. 326, 373–398 (1996)CrossRefGoogle Scholar
  22. 22.
    MacKerrell, S., Bassom, A., Blennerhassett, P.: Görtler vortices in the Rayleigh layer on an impulsively started cylinder. Phys. Fluids 14, 2948 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Madden, F.N., Mullin, T.: The spin-up from rest of a fluid-filled torus. J. Fluid Mech. 265, 217–244 (1994)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Otto, S.R.: Stability of the flow around a cylinder: the spin-up problem. IMA J. Appl. Math. 51, 13–26 (1993)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rayleigh, L.: On the motion of solid bodies through viscous liquid. Philos. Mag. 21, 697–711 (1911)CrossRefGoogle Scholar
  26. 26.
    Rayleigh, L.: On the dynamics of revolving fluids. Proc. R. Soc. Lond. Ser. A 93, 148–154 (1917)CrossRefGoogle Scholar
  27. 27.
    Savaş, Ö.: Circular waves on a stationary disk in rotating flow. Phys. Fluids 26, 3445–3448 (1983)CrossRefGoogle Scholar
  28. 28.
    Savaş, Ö.: On flow visualization using reflective flakes. J. Fluid Mech. 152, 235–248 (1985)CrossRefGoogle Scholar
  29. 29.
    Savaş, Ö.: Stability of Bödewadt flow. J. Fluid Mech. 183, 77–94 (1987)CrossRefGoogle Scholar
  30. 30.
    Smirnov, S.A., Boyer, D.L., Baines, P.G.: Nonaxisymmetric effects of stratified spin-up in an axisymmetric annular channel. Phys. Fluids 12, 086601 (2005)CrossRefGoogle Scholar
  31. 31.
    Stewartson, K.: On almost rigid rotations. J. Fluid Mech. 3, 17–26 (1957)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Stokes, G.G.: On the Effect of the Internal Friction of Fluids on the Motion of Pendulums, vol. 9, p. 8. Trans Camb Philos Soc, Cambridge (1856)Google Scholar
  33. 33.
    von Kármán, T.: Über laminare und turbulente Reibung. J. Appl. Math. Mech. (ZAMM) 1:233–252 (1921) (translated as ‘On laminar and turbulent friction’, NACA-TM1092)CrossRefGoogle Scholar
  34. 34.
    Wedemeyer, E.H.: The unsteady flow within a spinning cylinder. J. Fluid Mech. 20, 383–399 (1964)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Weidman, P.D.: On the spin-up and spin-down of a rotating fluid Part 1 Extending the Wedemeyer model. J. Fluid Mech. 77, 685–708 (1976)CrossRefGoogle Scholar
  36. 36.
    Weidman, P.D.: On the spin-up and spin-down of a rotating fluid. part 2. measurements and stability. J. Fluid Mech. 77, 709–735 (1976)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Sophie A. W. Calabretto
    • 1
  • James P. Denier
    • 1
  • Trent W. Mattner
    • 2
  1. 1.Department of Mathematics and StatisticsMacquarie UniversitySydneyAustralia
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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