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Effect of enclosure height on the structure and stability of shear layers induced by differential rotation

  • Tony Vo (a1), Luca Montabone (a2) and Gregory J. Sheard (a1)


The structure and stability of Stewartson shear layers with different heights are investigated numerically via axisymmetric simulation and linear stability analysis, and a validation of the quasi-two-dimensional model is performed. The shear layers are generated in a rotating cylindrical tank with circular disks located at the lid and base imposing a differential rotation. The axisymmetric model captures both the thick and thin nested Stewartson layers, which are scaled by the Ekman number ( $\mathit{E}\,$ ) as $\mathit{E}\,^{1/4}$ and $\mathit{E}\,^{1/3}$ respectively. In contrast, the quasi-two-dimensional model only captures the $\mathit{E}\,^{1/4}$ layer as the axial velocity required to invoke the $\mathit{E}\,^{1/3}$ layer is excluded. A direct comparison between the axisymmetric base flows and their linear stability in these two models is examined here for the first time. The base flows of the two models exhibit similar flow features at low Rossby numbers ( $\mathit{Ro}$ ), with differences evident at larger $\mathit{Ro}$ where depth-dependent features are revealed by the axisymmetric model. Despite this, the quasi-two-dimensional model demonstrates excellent agreement with the axisymmetric model in terms of the shear-layer thickness and predicted stability. A study of various aspect ratios reveals that a Reynolds number based on the theoretical Ekman layer thickness is able to describe the transition of a base flow that is reflectively symmetric about the mid-plane to a symmetry-broken state. Additionally, the shear-layer thicknesses scale closely to the expected ${\it\delta}_{vel}\propto A\mathit{E}\,^{1/4}$ and ${\it\delta}_{vort}\propto A\mathit{E}\,^{1/3}$ for shear layers that are not affected by the confinement ( $A\mathit{E}\,^{1/4}\lesssim 0.34$ in this system, the ratio of tank height to shear-layer radius). The linear stability analysis reveals that the ratio of Stewartson layer radius to thickness should be greater than $45$ for the stability of the flow to be independent of aspect ratio. Thus, for sufficiently small $A\mathit{E}\,^{1/4}$ and $A\mathit{E}\,^{1/3}$ , the flow characteristics remain similar and the linear stability of the flow can be described universally when the azimuthal wavelength is scaled against $A$ . The analysis also recovers an asymptotic scaling for the normalized azimuthal wavelength which suggests that ${\it\lambda}_{{\it\theta},c}^{\ast }\propto (|\mathit{Ro}|/\mathit{E}\,^{2})^{-1/5}$ for geometry-independent shear layers at marginal stability.


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Present address: Space Science Institute, Boulder, CO 80301, USA.



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Aguiar, A. C. B.2008 Instabilities of a shear layer in a barotropic rotating fluid, PhD Thesis, University of Oxford.
Aguiar, A. C. B., Read, P. L., Wordsworth, R. D., Salter, T. & Hiro Yamazaki, Y. 2010 A laboratory model of Saturn’s north polar hexagon. Icarus 206 (2), 755763.
Avila, M. 2012 Stability and angular-momentum transport of fluid flows between corotating cylinders. Phys. Rev. Lett. 108 (12), 124501.
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215241.
Bergeron, K., Coutsias, E. A., Lynov, J. P. & Nielsen, A. H. 2000 Dynamical properties of forced shear layers in an annular geometry. J. Fluid Mech. 402 (1), 255289.
Blackburn, H. M. & Sherwin, S. J. 2004 Formulation of a Galerkin spectral element – Fourier method for three-dimensional incompressible flows in cylindrical geometries. J. Comput. Phys. 197 (2), 759778.
Busse, F. H. 1968 Shear flow instabilities in rotating systems. J. Fluid Mech. 33 (3), 577589.
Chomaz, J. M., Rabaud, M., Basdevant, C. & Couder, Y. 1988 Experimental and numerical investigation of a forced circular shear layer. J. Fluid Mech. 187, 115140.
Cogan, S. J., Ryan, K. & Sheard, G. J. 2011 Symmetry breaking and instability mechanisms in medium depth torsionally driven open cylinder flows. J. Fluid Mech. 672, 521544.
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic Stability of Parallel Flow of Inviscid Fluid Vol. 9. Academic.
Fletcher, L. N., Irwin, P. G. J., Orton, G. S., Teanby, N. A., Achterberg, R. K., Bjoraker, G. L., Read, P. L., Simon-Miller, A. A., Howett, C., de Kok, R., Bowles, N., Calcutt, S. B., Hesman, B. & Flasar, F. M. 2008 Temperature and composition of Saturn’s polar hotspots and hexagon. Science 319 (5859), 7981.
Früh, W. G. & Nielsen, A. H. 2003 On the origin of time-dependent behaviour in a barotropically unstable shear layer. Nonlinear Process. Geophys. 10 (3), 289302.
Früh, W. G. & Read, P. L. 1999 Experiments on a barotropic rotating shear layer. Part 1. Instability and steady vortices. J. Fluid Mech. 383, 143173.
Gissinger, C., Goodman, J. & Ji, H. 2012 The role of boundaries in the magnetorotational instability. Phys. Fluids 24 (7), 074109.
Godfrey, D. A. 1988 A hexagonal feature around Saturn’s north pole. Icarus 76, 335356.
Godfrey, D. A. & Moore, V. 1986 The Saturnian ribbon feature – a baroclinically unstable model. Icarus 68 (2), 313343.
Gombosi, T. I. & Ingersoll, A. P. 2010 Saturn: atmosphere, ionosphere, and magnetosphere. Science 327 (5972), 14761479.
Hollerbach, R. & Fournier, A. 2004 End-effects in rapidly rotating cylindrical Taylor–Couette flow. In MHD Couette Flows: Experiments and Models (ed. Rosner, R., Rüdiger, G. & Bonanno, A.), AIP Conference Proceedings, vol. 733, pp. 114121.
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97, 414443.
van de Konijnenberg, J. A., Nielsen, A. H., Juul Rasmussen, J. & Stenum, B. 1999 Shear-flow instability in a rotating fluid. J. Fluid Mech. 387, 177204.
Kuo, H. 1949 Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere. J. Atmos. Sci. 6, 105122.
Lian, Y. & Showman, A. P. 2008 Deep jets on gas-giant planets. Icarus 194 (2), 597615.
Liu, W. 2008 Magnetized Ekman layer and Stewartson layer in a magnetized Taylor–Couette flow. Phys. Rev. E 77 (5), 056314.
Luz, D., Berry, D. L., Piccioni, G., Drossart, P., Politi, R., Wilson, C. F., Erard, S. & Nuccilli, F. 2011 Venus’s southern polar vortex reveals precessing circulation. Science 332 (6029), 577580.
Montabone, L., Wordsworth, R., Aguiar, A. C. B., Jacoby, T., Manfrin, M., Read, P. L., Castrejon-Pita, A., Gostiaux, L., Sommeria, J., Viboud, S. & Didelle, H. 2010 Barotropic instability of planetary polar vortices: CIV analysis of specific multi-lobed structures. In Proceedings of the HYDRALAB III Joint Transnational Access User Meeting, Hannover, p. 191.
Niino, H. & Misawa, N. 1984 An experimental and theoretical study of barotropic instability. J. Atmos. Sci. 41 (12), 19922011.
Paoletti, M. S., van Gils, D. P. M., Dubrulle, B., Sun, C., Lohse, D. & Lathrop, D. P. 2012 Angular momentum transport and turbulence in laboratory models of Keplerian flows. Astronom. Astrophys. 547, A64.
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.
Piccioni, G., Drossart, P., Sanchez-Lavega, A., Hueso, R., Taylor, F. W., Wilson, C. F., Grassi, D., Zasova, L., Moriconi, M. & Adriani, A. et al. 2007 South-polar features on Venus similar to those near the north pole. Nature 450 (7170), 637640.
Rabaud, M. & Couder, Y. 1983 Shear-flow instability in a circular geometry. J. Fluid Mech. 136, 291319.
Rayleigh, L. 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 11, 5772.
Read, P. L. 1988 On the scale of baroclinic instability in deep, compressible atmospheres. Q. J. R. Meteorol. Soc. 114 (480), 421437.
Sánchez-Lavega, A., Río-Gaztelurrutia, T., Hueso, R., Pérez-Hoyos, S., García-Melendo, E., Antuñano, A., Mendikoa, I., Rojas, J. F., Lillo, J. & Barrado-Navascués, D. et al. 2014 The long-term steady motion of Saturn’s hexagon and the stability of its enclosed jet stream under seasonal changes. Geophys. Res. Lett. 41 (5), 14251431.
Schaeffer, N. & Cardin, P. 2005 Quasigeostrophic model of the instabilities of the Stewartson layer in flat and depth-varying containers. Phys. Fluids 17, 104111.
Schartman, E., Ji, H., Burin, M. J. & Goodman, J. 2012 Stability of quasi-Keplerian shear flow in a laboratory experiment. Astronom. Astrophys. 543 (A94).
Sheard, G. J. 2009 Flow dynamics and wall shear-stress variation in a fusiform aneurysm. J. Engng Math. 64 (4), 379390.
Sheard, G. J. 2011 Wake stability features behind a square cylinder: focus on small incidence angles. J. Fluids Struct. 27, 734742.
Sheard, G. J. & Ryan, K. 2007 Pressure-driven flow past spheres moving in a circular tube. J. Fluid Mech. 592, 233262.
Smith, S. H. 1984 The development of nonlinearities in the $E^{1/3}$ Stewartson layer. Q. J. Mech. Appl. Maths 37 (1), 7585.
Sommeria, J., Meyers, S. D. & Swinney, H. L. 1991 Experiments on vortices and Rossby waves in eastward and westward jets. Nonlinear Topics Ocean Phys. 109, 227269.
Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 1726.
Szklarski, J. & Rüdiger, G. 2007 Ekman–Hartmann layer in a magnetohydrodynamic Taylor–Couette flow. Phys. Rev. E 76 (6), 066308.
Taylor, F. W., Beer, R., Chahine, M. T., Diner, D. J., Elson, L. S., Haskins, R. D., McCleese, D. J., Martonchik, J. V., Reichley, P. E. & Bradley, S. P. 1980 Structure and meteorology of the middle atmosphere of Venus: infrared remote sensing from the Pioneer Orbiter. J. Geophys. Res. 85 (A13), 79638006.
Vo, T., Montabone, L. & Sheard, G. J. 2014 Linear stability analysis of a shear layer induced by differential coaxial rotation within a cylindrical enclosure. J. Fluid Mech. 738, 299334.
Vooren, A. I. 1992 The Stewartson layer of a rotating disk of finite radius. J. Engng Maths 26 (1), 131152.
Williams, G. P. 2003 Jovian dynamics. Part III: Multiple, migrating, and equatorial jets. J. Atmos. Sci. 60 (10), 12701296.
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Effect of enclosure height on the structure and stability of shear layers induced by differential rotation

  • Tony Vo (a1), Luca Montabone (a2) and Gregory J. Sheard (a1)


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