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Internal shear layers from librating objects

Abstract : In this work, we analyse the internal shear layer structures generated by the libration of an axisymmetric object in an unbounded fluid rotating at a rotation rate Ω * using direct numerical simulation and small Ekman number asymptotic analysis. We consider weak libration amplitude and libration frequency ω * within the inertial wave interval (0, 2Ω *) such that the fluid dynamics is mainly described by a linear axisymmetric harmonic solution. The internal shear layer structures appear along the characteristic cones of angle θ c = acos(ω * /(2Ω *)) which are tangent to the librating object at so-called critical latitudes. These layers correspond to thin viscous regions where the singulari-ties of the inviscid solution are smoothed. We assume that the velocity field in these layers is described by the class of similarity solutions introduced by Moore & Saffman [Phil. Trans. R. Soc. A 264, 597-634 (1969)]. These solutions are characterised by two parameters only: a real parameter m, which measures the strength of the underlying singularity, and a complex amplitude coefficient C 0. We first analyse the case of a disk for which a general asymptotic solution for small Ekman numbers is known when the disk is in a plane. We demonstrate that the numerical solutions obtained for a free disk and for a disk in a plane are both well-described by the asymptotic solution and by its similarity form within the internal shear layers. For the disk, we obtain a parameter m = 1 corresponding to a Dirac source at the edge of the disk and a coefficient C 0 ∝ E 1/6 where E is the Ekman number. The case of a smoothed librating object such as a spheroid is found to be different. By asymptotically matching the boundary layer solution to similarity solutions close to a critical latitude on the surface, we show that the adequate parameter m for the similarity solution is m = 5/4, leading to a coefficient C 0 ∝ E 1/12 , that is larger than for the case of a disk for small Ekman numbers. A simple general expression for C 0 valid for any axisymmetric object is obtained as a function of the local curvature radius at the critical latitude in agreement with this change of scaling. This result is tested and validated against direct numerical simulations.
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https://hal-amu.archives-ouvertes.fr/hal-01543666
Contributor : Stéphane Le Dizès <>
Submitted on : Friday, June 23, 2017 - 4:50:40 PM
Last modification on : Monday, July 27, 2020 - 1:00:05 PM
Long-term archiving on: : Saturday, December 16, 2017 - 7:07:14 AM

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Stéphane Le Dizès, Michael Le Bars. Internal shear layers from librating objects. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2017, 826, pp.653-675. ⟨hal-01543666⟩

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