Effect of Differential Rotation of Oscillating Inner Core on Steady Flow Instability in a Rotating Sphere
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ORIGINAL ARTICLE
Effect of Differential Rotation of Oscillating Inner Core on Steady Flow Instability in a Rotating Sphere S.V. Subbotin 1
&
V.G. Kozlov 1
Received: 18 March 2020 / Accepted: 19 May 2020 # Springer Nature B.V. 2020
Abstract A comparative analysis of steady flow excited by an oscillating core in a rotating spherical cavity with a liquid is carried out in different cases: when the core is free and performs differential rotation, and when the core differential rotation is absent. In the cavity reference frame the core, whose density is less than the density of the liquid, oscillates near the cavity center under the action of transverse to the rotation axis external force field. In both cases, the oscillations lead to the appearance of almost twodimensional axisymmetric azimuthal steady flow, with several inflection points in the velocity profile. An increase in the amplitude of core oscillations leads to a loss of stability of the axisymmetric flow. In the supercritical region, there is a series of threshold transitions associated with various instability modes. The instability manifests itself in the development of an azimuthal periodic system of vortices elongated parallel to the rotation axis. The modes differ in an azimuthal wavenumber, the location of the vortices (distance from the axis of rotation), and the azimuthal drift rate of the vortex system relative to the cavity. It is shown that the core differential rotation modifies the azimuthal velocity profile, resulting in a change in the instability thresholds. Due to an additional azimuthal flow, the drift velocity of the same type vortices in the two cases is different. The effect of the core differential rotation on the dispersion relations for various instability modes has been investigated. Keywords Oscillations . Differential rotation . Steady flows . Flow instability . Waves in a rotating fluid . Pattern formation
Introduction It is known that the vibration effect on hydrodynamic systems can generate steady flows, which play an important role in processes that take place under weightlessness conditions (Fernandez et al. 2017). Understanding the mechanisms of steady flows appearance and their stability offers great opportunities for controlling heat and mass transfer in inhomogeneous density systems (Smorodin et al. 2018; Karpunin et al. 2018; Kozlov et al. 2019) and systems with an interface (Pimenova et al. 2018; Aleksandrov et al. 2018). The study of the effect of vibrations on a rotating fluid is not less interesting. The action of inertia forces associated with rotation changes the mechanism of steady flows generation (Kozlov and Ivanova 2009; Kozlov 2015; Le Bars et al. 2015) and ensures the maintenance of internal wave motion (Messio * S.V. Subbotin [email protected] 1
Laboratory of Vibrational Hydromechanics, Perm State Humanitarian Pedagogical University, 614000 Perm, Russia
et al. 2008; Subbotin and Dyakova 2018). In nature, the examples of oscillating rotating systems are planets (or satellites) with liquid and solid c
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