Stewartson-layer instability in a wide-gap spherical Couette experiment: Rossby number dependence. (10th November 2019)

• Record Type: Journal Article
• Authors: Hoff, Michael; Harlander, Uwe
• Is Part Of: Journal of fluid mechanics. Volume 878(2019)
• Page Start: 522
• Page End: 543
• Publication Date: 2019-11-10
• Subjects: rotating flows; Fluid mechanics -- Periodicals; 532.005
• DOI: 10.1017/jfm.2019.636
• Languages: English
• ISSNs: 0022-1120

Abstract

Abstract : Instabilities of a viscous fluid between two fast but differentially rotating concentric spheres, the so-called spherical Couette flow, with a fixed radius ratio of $\unicode[STIX]{x1D702}=r_{i}/r_{o}=1/3$ are studied, where $r_{i}$ is the inner and $r_{o}$ the outer radius of the spherical shell. Of particular interest is the difference between cases where the Rossby number $Ro=(\unicode[STIX]{x1D6FA}_{i}-\unicode[STIX]{x1D6FA}_{o})/\unicode[STIX]{x1D6FA}_{o}>0$ and cases with $Ro<0$, where $\unicode[STIX]{x1D6FA}_{i}$ and $\unicode[STIX]{x1D6FA}_{o}$ are the inner- and outer-sphere angular velocities. The basic state in both situations is an axisymmetric shear flow with a Stewartson layer situated on the tangent cylinder. The tangent cylinder is given by a cylinder that touches the equator of the inner sphere with an axis parallel to the axis of rotation. The experimental results presented fully confirm earlier numerical results obtained by Hollerbach ( J. Fluid Mech., vol. 492, 2003, pp. 289–302) showing that for $Ro>0$ a progression to higher azimuthal wavenumbers $m$ can be seen as the rotation rate $\unicode[STIX]{x1D6FA}_{0}$ increases, but $Ro<0$ gives $m=1$ over a large range of rotation rates. It is further found that in the former case the modes have spiral structures radiating away from Stewartson layer towards the outer shell whereas for $Ro<0$ the modes are trapped in the vicinity of the Stewartson layer. Further, the mean flow excited by inertial mode self-interaction and its correlation with the mode's amplitudes are investigated. The scaling of the critical $Ro$ with Ekman number $E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{o}\, d^{2})$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $d$ the gap width, is well within the bounds that have been established in a number of experimental studies using cylindrical geometries and numerical studies using spherical cavities. However, the present work is the first that experimentally examines Stewartson-layer instabilities as a function of the sign of $Ro$ for the true spherical-shell geometry.