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Three-dimensional rotating Couette flow via the generalised quasilinear approximation

  • S. M. Tobias (a1) and J. B. Marston (a2)

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.

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Corresponding author

Email address for correspondence: smt@maths.leeds.ac.uk

References

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Bech, K. H. & Andersson, H. I. 1996 Secondary flow in weakly rotating turbulent plane Couette flow. J. Fluid Mech. 317, 195214.
Bech, K. H. & Andersson, H. I. 1997 Turbulent plane Couette flow subject to strong system rotation. J. Fluid Mech. 347, 289314.
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.
Burns, K., Vasil, G., Brown, B., Lecoanet, D. & Oishi, J.2016 Dedalus. http://dedalus-project.org/index.html.
Child, A., Hollerbach, R., Marston, B. & Tobias, S. 2016 Generalised quasilinear approximation of the helical magnetorotational instability. J. Plasma Phys. 82 (03), 905820302-18.
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), 22292253.
Diamond, P. H., Itoh, S.-I., Itoh, K. & Hahm, T. S. 2005 Zonal flows in plasma – a review. Plasma Phys. Control. Fusion 47, R35.
Dickinson, R. E. 1969 Theory of planetary wave-zonal flow interaction. J. Atmos. Sci. 26, 7381.
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.
Faisst, H. & Eckhardt, B. 2000 Transition from the Couette-Taylor system to the plane Couette system. Phys. Rev. E 61, 72277230.
Herring, J. R. 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20 (4), 325338.
Hiwatashi, K., Alfredsson, P. H., Tillmark, N. & Nagata, M. 2007 Experimental observations of instabilities in rotating plane Couette flow. Phys. Fluids 19 (4), 048103.
Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.
Lindzen, R. S. & Holton, J. R. 1968 A theory of the quasi-biennial oscillation. J. Atmos. Sci. 25, 10951107.
Lorenz, E. N. 1967 The Nature and Theory of the General Circulation of the Atmosphere. World Meteorological Organization.
Marston, J. B., Chini, G. P. & Tobias, S. M. 2016 Generalized quasilinear approximation: application to zonal jets. Phys. Rev. Lett. 116 (21), 214501.
Nagata, M. 1998 Tertiary solutions and their stability in rotating plane Couette flow. J. Fluid Mech. 358, 357378.
Salewski, M. & Eckhardt, B. 2015 Turbulent states in plane Couette flow with rotation. Phys. Fluids 27 (4), 045109.
Schmid, P. J. & Henningson, D. S. 2000 Stability and Transition in Shear Flows. Springer.
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.
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289343.
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.
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.
Tillmark, N. & Alfredsson, P. H. 1992 Experiments on transition in plane Couette flow. J. Fluid Mech. 235, 89102.
Tobias, S. M. & Marston, J. B. 2013 Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett. 110 (10), 104502.
Tobias, S. M., Dagon, K. & Marston, J. B. 2011 Astrophysical fluid dynamics via direct statistical simulation. Astrophys. J. 727 (2), 127138.
Tsukahara, T., Tillmark, N. & Alfredsson, P. H. 2010 Flow regimes in a plane Couette flow with system rotation. J. Fluid Mech. 648, 533.
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Three-dimensional rotating Couette flow via the generalised quasilinear approximation

  • S. M. Tobias (a1) and J. B. Marston (a2)

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