Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T06:01:07.479Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional rotating Couette flow via the generalised quasilinear approximation

Published online by Cambridge University Press:  28 November 2016

S. M. Tobias  and
J. B. Marston
S. M. Tobias*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
J. B. Marston
Affiliation:
Department of Physics, Brown University, Providence, RI 02912-1843, USA
*
†Email address for correspondence: smt@maths.leeds.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the effectiveness of the generalised quasilinear (GQL) approximation introduced by Marston et al. (Phys. Rev. Lett., vol. 116 (21), 2016, 214501). This approximation splits the variables into large and small scales in directions where there is a translational symmetry and removes nonlinear interactions involving only small scales. We utilise as a paradigm problem three-dimensional, turbulent, rotating Couette flow. We compare the results obtained from direct numerical solution of the equations with those from quasilinear (QL) and GQL calculations. In this three-dimensional setting, there is a choice of cutoff wavenumber for the GQL approximation both in the streamwise and in the spanwise directions. We demonstrate that the GQL approximation significantly improves the accuracy of mean flows, spectra and two-point correlation functions over models that are quasilinear in any of the translationally invariant directions, even if only a few streamwise and spanwise modes are included. We argue that this provides significant support for a programme of direct statistical simulation utilising the GQL approximation.

JFM classification

Turbulent Flows: Rotating turbulence Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 810 , 10 January 2017 , pp. 412 - 428
Check for updates
Copyright
© 2016 Cambridge University Press 

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

Bech, K. H. & Andersson, H. I. 1996 Secondary flow in weakly rotating turbulent plane Couette flow. J. Fluid Mech. 317, 195–214.CrossRefGoogle Scholar
Bech, K. H. & Andersson, H. I. 1997 Turbulent plane Couette flow subject to strong system rotation. J. Fluid Mech. 347, 289–314.CrossRefGoogle Scholar
Bretheim, J. U., Meneveau, C. & Gayme, D. F. 2015 Standard logarithmic mean velocity distribution in a band-limited restricted nonlinear model of turbulent flow in a half-channel. Phys. Fluids 27 (1), 011702.CrossRefGoogle Scholar
Burns, K., Vasil, G., Brown, B., Lecoanet, D. & Oishi, J.2016 Dedalus. http://dedalus-project.org/index.html.Google Scholar
Child, A., Hollerbach, R., Marston, B. & Tobias, S. 2016 Generalised quasilinear approximation of the helical magnetorotational instability. J. Plasma Phys. 82 (03), 905820302-18.CrossRefGoogle Scholar
Constantinou, N. C., Farrell, B. F. & Ioannou, P. J. 2016 Statistical state dynamics of jet–wave coexistence in barotropic beta-plane turbulence. J. Atmos. Sci. 73 (5), 2229–2253.CrossRefGoogle Scholar
Diamond, P. H., Itoh, S.-I., Itoh, K. & Hahm, T. S. 2005 Zonal flows in plasma – a review. Plasma Phys. Control. Fusion 47, R35.CrossRefGoogle Scholar
Dickinson, R. E. 1969 Theory of planetary wave-zonal flow interaction. J. Atmos. Sci. 26, 73–81.2.0.CO;2>CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2000 Transition from the Couette-Taylor system to the plane Couette system. Phys. Rev. E 61, 7227–7230.CrossRefGoogle Scholar
Herring, J. R. 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20 (4), 325–338.2.0.CO;2>CrossRefGoogle Scholar
Hiwatashi, K., Alfredsson, P. H., Tillmark, N. & Nagata, M. 2007 Experimental observations of instabilities in rotating plane Couette flow. Phys. Fluids 19 (4), 048103.CrossRefGoogle Scholar
Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.Google Scholar
Lindzen, R. S. & Holton, J. R. 1968 A theory of the quasi-biennial oscillation. J. Atmos. Sci. 25, 1095–1107.2.0.CO;2>CrossRefGoogle Scholar
Lorenz, E. N. 1967 The Nature and Theory of the General Circulation of the Atmosphere. World Meteorological Organization.Google Scholar
Marston, J. B., Chini, G. P. & Tobias, S. M. 2016 Generalized quasilinear approximation: application to zonal jets. Phys. Rev. Lett. 116 (21), 214501.CrossRefGoogle ScholarPubMed
Nagata, M. 1998 Tertiary solutions and their stability in rotating plane Couette flow. J. Fluid Mech. 358, 357–378.CrossRefGoogle Scholar
Salewski, M. & Eckhardt, B. 2015 Turbulent states in plane Couette flow with rotation. Phys. Fluids 27 (4), 045109.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2000 Stability and Transition in Shear Flows. Springer.Google Scholar
Suryadi, A., Segalini, A. & Alfredsson, P. H. 2014 Zero absolute vorticity: insight from experiments in rotating laminar plane Couette flow. Phys. Rev. E 89 (3), 033003.Google ScholarPubMed
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
Thomas, V. L., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27 (10), 105104.CrossRefGoogle Scholar
Thomas, V. L., Lieu, B. K., Jovanović, M. R., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2014 Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow. Phys. Fluids 26 (10), 105112.CrossRefGoogle Scholar
Tillmark, N. & Alfredsson, P. H. 1992 Experiments on transition in plane Couette flow. J. Fluid Mech. 235, 89–102.CrossRefGoogle Scholar
Tobias, S. M. & Marston, J. B. 2013 Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett. 110 (10), 104502.CrossRefGoogle ScholarPubMed
Tobias, S. M., Dagon, K. & Marston, J. B. 2011 Astrophysical fluid dynamics via direct statistical simulation. Astrophys. J. 727 (2), 127–138.CrossRefGoogle Scholar
Tsukahara, T., Tillmark, N. & Alfredsson, P. H. 2010 Flow regimes in a plane Couette flow with system rotation. J. Fluid Mech. 648, 5–33.CrossRefGoogle Scholar