Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-07-01T01:54:48.498Z Has data issue: false hasContentIssue false
  • English
  • Français

Instabilities and inertial waves generated in a librating cylinder

Published online by Cambridge University Press:  07 November 2011

J. M. Lopez  and
F. Marques
J. M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
F. Marques
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A librating cylinder consists of a rotating cylinder whose rate of rotation is modulated. When the mean rotation rate is large compared with the viscous damping rate, the flow may support inertial waves, depending on the frequency of the modulation. The modulation also produces time-dependent boundary layers on the cylinder endwalls and sidewall, and the sidewall boundary layer flow in particular is susceptible to instabilities which can introduce additional forcing on the interior flow with time scales different from the modulation period. These instabilities may also drive and/or modify the inertial waves. In this paper, we explore such flows numerically using a spectral-collocation code solving the Navier–Stokes equations in order to capture the dynamics involved in the interactions between the inertial waves and the viscous boundary layer flows.

JFM classification

Boundary Layers: Boundary layer stability Instability: Nonlinear instability Turbulent Flows: Rotating turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 687 , 25 November 2011 , pp. 171 - 193
Copyright
Copyright © Cambridge University Press 2011

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

1. Aldridge, K. D. & Lumb, L. I. 1987 Inertial waves identified in the Earth’s fluid outer core. Nature 325, 421–423.CrossRefGoogle Scholar
2. Aldridge, K. D. & Toomre, A. 1969 Axisymmetric inertial oscillations of a fluid in a rotating spherical container. J. Fluid Mech. 37, 307–323.CrossRefGoogle Scholar
3. Barcilon, V. 1968 Stewartson layers in transient rotating fluid flows. J. Fluid Mech. 33, 815–825.CrossRefGoogle Scholar
4. Barrett, K. E. 1969 Resonant torsional oscillations of a pair of discs in a rotating fluid. Z. Angew. Math. Phys. 20, 721–729.CrossRefGoogle Scholar
5. Benney, D. J. 1965 The flow induced by a disk oscillating about a state of steady rotation. Q. J. Mech. Appl. Maths 18, 333–345.CrossRefGoogle Scholar
6. Bewley, G. P., Lathrop, D. P., Maas, L. R. M. & Sreenivasan, K. R. 2007 Inertial waves in rotating grid turbulence. Phys. Fluids 19, 071701.CrossRefGoogle Scholar
7. Bhattacharjee, J. K. 1989 Rotating Rayleigh–Bénard convection with modulation. J. Phys. A 22, L1135–L1139.CrossRefGoogle Scholar
8. Bhattacharjee, J. K. 1990 Convective instability in a rotating fluid layer under modulation of the rotating rate. Phys. Rev. A 41, 5491–5494.CrossRefGoogle Scholar
9. Busse, F. H. 2010a Mean zonal flows generated by librations of a rotating spherical cavity. J. Fluid Mech. 650, 505–512.CrossRefGoogle Scholar
10. Busse, F. H. 2010b Zonal flow induced by longitudinal librations of a rotating cylindrical cavity. Physica D 240, 208–211.CrossRefGoogle Scholar
11. Busse, F. H. & Simitev, R. 2004 Inertial convection in rotating fluid spheres. J. Fluid Mech. 498, 23–30.CrossRefGoogle Scholar
12. Calkins, M. A., Noir, J., Eldredge, J. D. & Aurnou, J. M. 2010 Axisymmetric simulations of libration-driven fluid dynamics in a spherical shell geometry. Phys. Fluids 22, 086602.CrossRefGoogle Scholar
13. Do, Y., Lopez, J. M. & Marques, F. 2010 Optimal harmonic response in a confined Bödewadt boundary layer flow. Phys. Rev. E 82, 036301.CrossRefGoogle Scholar
14. Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
15. Greenspan, H. & Howard, L. 1963 On a time-dependent motion of a rotating fluid. J. Fluid Mech. 17, 385–404.CrossRefGoogle Scholar
16. Hart, J. E. & Mundt, M. D. 1996 Instability of oscillatory Stokes–Stewartson layers in a rotating fluid. J. Fluid Mech. 311, 119–140.CrossRefGoogle Scholar
17. Jones, A. F. 1969 The resonance effect of a disk oscillating about a state of steady rotation. J. Fluid Mech. 39, 269–281.CrossRefGoogle Scholar
18. Kelley, D. H., Triana, S. A., Zimmerman, D. S., Braun, B., Lathrop, D. P. & Martin, D. H. 2006 Driven inertial waves in spherical Couette flow. Chaos 16, 041105.CrossRefGoogle ScholarPubMed
19. Kelley, D. H., Triana, S. A., Zimmerman, D. S. & Lathrop, D. P. 2010 Selection of inertial modes in spherical Couette flow. Phys. Rev. E 81, 026311.CrossRefGoogle ScholarPubMed
20. Kelley, D. H., Triana, S. A., Zimmerman, D. S., Tilgner, A. & Lathrop, D. P. 2007 Inertial waves driven by differential rotation in a planetary geometry. Geophys. Astrophys. Fluid Dyn. 101, 469–487.CrossRefGoogle Scholar
21. Landau, L. D. & Lifshitz, E. M. 1984 Fluid Mechanics, 2nd edn. Pergamon Press.Google Scholar
22. Liu, Y. & Ecke, R. E. 1999 Nonlinear travelling waves in rotating Rayleigh–Bénard convection: stability boundaries and phase diffusion. Phys. Rev. E 59, 4091–4105.CrossRefGoogle Scholar
23. Lopez, J. M. & Marques, F. 2009 Centrifugal effects in rotating convection: nonlinear dynamics. J. Fluid Mech. 628, 269–297.CrossRefGoogle Scholar
24. Lopez, J. M. & Marques, F. 2010 Sidewall boundary layer instabilities in a rapidly rotating cylinder driven by a differentially co-rotating lid. Phys. Fluids 22, 114109.CrossRefGoogle Scholar
25. Lopez, J. M., Marques, F., Mercader, I. & Batiste, O. 2007 Onset of convection in a moderate aspect-ratio rotating cylinder: Eckhaus–Benjamin–Feir instability. J. Fluid Mech. 590, 187–208.CrossRefGoogle Scholar
26. Lopez, J. M., Marques, F., Rubio, A. M. & Avila, M. 2009 Crossflow instability of finite Bödewadt flows: transients and spiral waves. Phys. Fluids 21, 114107.CrossRefGoogle Scholar
27. Lorenzani, S. & Tilgner, A. 2001 Fluid instabilities in precessing spheroidal cavities. J. Fluid Mech. 447, 111–128.CrossRefGoogle Scholar
28. Marques, F., Mercader, I., Batiste, O. & Lopez, J. M. 2007 Centrifugal effects in rotating convection: axisymmetric states and three-dimensional instabilities. J. Fluid Mech. 580, 303–318.CrossRefGoogle Scholar
29. Mercader, I., Batiste, O. & Alonso, 2010 An efficient spectral code for incompressible flows in cylindrical geometries. Comput. Fluids 39, 215–224.CrossRefGoogle Scholar
30. Niemela, J. J., Smith, M. R. & Donnelly, R. J. 1991 Convective instability with time-varying rotation. Phys. Rev. A. 44, 8406–8409.CrossRefGoogle ScholarPubMed
31. Noir, J., Brito, D., Aldridge, K. & Cardin, P. 2001 Experimental evidence of inertial waves in a precessing spherical cavity. Geophys. Res. Lett. 28, 3785–3788.CrossRefGoogle Scholar
32. Noir, J., Calkins, M. A., Cantwell, J. & Aurnou, J. M. 2010 Experimental study of libration-driven zonal flows in a straight cylinder. Phys. Earth Planet. Inter. 182, 98–106.CrossRefGoogle Scholar
33. Noir, J., Hemmerlin, F., Wicht, J., Baca, S. M. & Aurnou, J. M. 2009 An experimental and numerical study of librationally driven flow in planetary cores and subsurface oceans. Phys. Earth Planet. Inter. 173, 141–152.CrossRefGoogle Scholar
34. Noir, J., Jault, D. & Cardin, P. 2001 Numerical study of the motions within a slowly precessing sphere at low Ekman number. J. Fluid Mech. 437, 283–299.CrossRefGoogle Scholar
35. Roxin, A. & Riecke, H. 2002 Rotating convection in an anisotropic system. Phys. Rev. E 65, 046219.CrossRefGoogle Scholar
36. Rubio, A., Lopez, J. M. & Marques, F. 2008 Modulated rotating convection: radially travelling concentric rolls. J. Fluid Mech. 608, 357–378.CrossRefGoogle Scholar
37. Rubio, A., Lopez, J. M. & Marques, F. 2009 Interacting oscillatory boundary layers and wall modes in modulated rotating convection. J. Fluid Mech. 625, 75–96.CrossRefGoogle Scholar
38. Sauret, A., Cébron, D., Morize, C. & Le Bars, M. 2010 Experimental and numerical study of mean zonal flows generated by librations of a rotating spherical cavity. J. Fluid Mech. 662, 260–268.CrossRefGoogle Scholar
39. Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory, eighth edn. McGraw-Hill.CrossRefGoogle Scholar
40. Thompson, K. L., Bajaj, K. M. S. & Ahlers, G. 2002 Travelling concentric-roll patterns in Rayleigh–Bénard convection with modulated rotation. Phys. Rev. E 65, 04618.CrossRefGoogle ScholarPubMed
41. Wang, C.-Y. 1970 Cylindrical tank of fluid oscillating about a state of steady rotation. J. Fluid Mech. 41, 581–592.CrossRefGoogle Scholar
42. Yih, C.-S. 1977 Fluid Mechanics. West River Press.Google Scholar
43. Zatman, S. & Bloxham, J. 1997 Torsional oscillations and the magnetic field within the Earth’s core. Nature 388, 760–763.CrossRefGoogle Scholar
44. Zhang, K. 1994 On coupling between the Poincaré equation and the heat equation. J. Fluid Mech. 268, 211–229.CrossRefGoogle Scholar
45. Zhang, K. 1995 On coupling between the Poincaré equation and the heat equation: non-slip boundary condition. J. Fluid Mech. 284, 239–256.CrossRefGoogle Scholar
46. Zhang, K., Liao, Z. & Busse, F. H. 2007 Asymptotic theory of inertial convection in a rotating cylinder. J. Fluid Mech. 575, 449–471.CrossRefGoogle Scholar

Lopez and Marques supplementary movie

Movie 1. Azimuthal vorticity on the sidewall over one period at $Re=10^4$, $\gamma=1$,$\omega=0.8\pi$ and $\alpha=0.82$. (Movie corresponds to figure 8c in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 3.9 MB

Lopez and Marques supplementary movie

Movie 1. Azimuthal vorticity on the sidewall over one period at $Re=10^4$, $\gamma=1$,$\omega=0.8\pi$ and $\alpha=0.82$. (Movie corresponds to figure 8c in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 6.5 MB

Lopez and Marques supplementary movie

Movie 2. Azimuthal vorticity on the sidewall over one period at $Re=10^4$, $\gamma=1$,$\omega=0.8\pi$ and $\alpha=0.90$. (Movie corresponds to figure 8d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 4.2 MB

Lopez and Marques supplementary movie

Movie 2. Azimuthal vorticity on the sidewall over one period at $Re=10^4$, $\gamma=1$,$\omega=0.8\pi$ and $\alpha=0.90$. (Movie corresponds to figure 8d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 7.8 MB

Lopez and Marques supplementary movie

Movie 3. Azimuthal vorticity in a meridional plane $\theta=0$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.9\pi$ and $\alpha=0.8$. (Movie corresponds to figure 9c in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 3.8 MB

Lopez and Marques supplementary movie

Movie 3. Azimuthal vorticity in a meridional plane $\theta=0$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.9\pi$ and $\alpha=0.8$. (Movie corresponds to figure 9c in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 5.8 MB

Lopez and Marques supplementary movie

Movie 4. Azimuthal vorticity in a horizontal plane $z=0.25$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.9\pi$ and $\alpha=0.8$. (Movie corresponds to figure 9d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 4.1 MB

Lopez and Marques supplementary movie

Movie 4. Azimuthal vorticity in a horizontal plane $z=0.25$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.9\pi$ and $\alpha=0.8$. (Movie corresponds to figure 9d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 7.6 MB

Lopez and Marques supplementary movie

Movie 5. Azimuthal vorticity on the sidewall over one period at $Re=10^4$, $\gamma=1$, $\omega=0.6\pi$ and $\alpha=0.7$. (Movie corresponds to figure 10d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 4.1 MB

Lopez and Marques supplementary movie

Movie 5. Azimuthal vorticity on the sidewall over one period at $Re=10^4$, $\gamma=1$, $\omega=0.6\pi$ and $\alpha=0.7$. (Movie corresponds to figure 10d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 7.1 MB

Lopez and Marques supplementary movie

Movie 6. Azimuthal vorticity in a meridional plane $\thata=0$ at $Re=10^4$, $\gamma=1$, $\omega=0.6\pi$ and $\alpha=0.7$. (Movie corresponds to figure 11d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 3.8 MB

Lopez and Marques supplementary movie

Movie 6. Azimuthal vorticity in a meridional plane $\thata=0$ at $Re=10^4$, $\gamma=1$, $\omega=0.6\pi$ and $\alpha=0.7$. (Movie corresponds to figure 11d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 5.5 MB

Lopez and Marques supplementary movie

Movie 7. Azimuthal vorticity in a horizontal plane $z=0.25$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.6\pi$ and $\alpha=0.3$ (Movie corresponds to figure 12b in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 3.6 MB

Lopez and Marques supplementary movie

Movie 7. Azimuthal vorticity in a horizontal plane $z=0.25$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.6\pi$ and $\alpha=0.3$ (Movie corresponds to figure 12b in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 4.7 MB

Lopez and Marques supplementary movie

Movie 8. Azimuthal vorticity in a horizontal plane $z=0.25$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.6\pi$ and $\alpha=0.7$. (Movie corresponds to figure 12d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 4.2 MB

Lopez and Marques supplementary movie

Movie 8. Azimuthal vorticity in a horizontal plane $z=0.25$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.6\pi$ and $\alpha=0.7$. (Movie corresponds to figure 12d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 7.5 MB

Lopez and Marques supplementary movie

Movie 9. Azimuthal vorticity in a meridional plane $\thata=0$ at $Re=10^4$, $\gamma=1$, $\omega=0.4\pi$ and $\alpha=0.7$. (Movie corresponds to figure 16d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 3.7 MB

Lopez and Marques supplementary movie

Movie 9. Azimuthal vorticity in a meridional plane $\thata=0$ at $Re=10^4$, $\gamma=1$, $\omega=0.4\pi$ and $\alpha=0.7$. (Movie corresponds to figure 16d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 5.3 MB

Lopez and Marques supplementary movie

Movie 10. Azimuthal vorticity in a horizontal plane $z=0.25$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.4\pi$ and $\alpha=0.7$. (Movie corresponds to figure 17d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 3.8 MB

Lopez and Marques supplementary movie

Movie 10. Azimuthal vorticity in a horizontal plane $z=0.25$ over one period at $Re=10^4$, $\gamma=1$, $\omega=0.4\pi$ and $\alpha=0.7$. (Movie corresponds to figure 17d in the paper).

Download Lopez and Marques supplementary movie(Video)
Video 6.4 MB