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 two-dimensional 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.
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References
Aleksandrov, V., Kopysov, S., Tonkov, L.: Vortex flows in the liquid layer and droplets on a vibrating flexible plate. Microgravity Sci. Technol. 30, 85–93 (2018)
Barbosa Aguiar, A.C., Read, P.L., Wordsworth, R.D., Salter, T., Yamazaki, Y.H.: A laboratory model of Saturn’s north polar hexagon. Icarus. 206(2), 755–763 (2010)
Calkins, M.A., Noir, J., Eldredge, J.D., Aurnou, J.M.: Axisymmetric simulations of libration-driven fluid dynamics in a spherical shell geometry. Phys. Fluids. 22, 086602 (2010)
Favier, B., Barker, A., Baruteau, C., Ogilvie, G.: Nonlinear evolution of tidally forced inertial waves in rotating fluid bodies. Mon. Not. R. Astron. Soc. 439, 845–860 (2014)
Fernandez, J., Sánchez, P.S., Tinao, I., Porter, J., Ezquerro, J.M.: The CFVib experiment: control of fluids in microgravity with vibrations. Microgravity Sci. Technol. 29, 351–364 (2017)
Hide, R., Titman, C.W.: Detached shear layers in a rotating fluid. J. Fluid Mech. 29, 39–60 (1967)
Hollerbach, R., Futterer, B., More, T., Egbers, C.: Instabilities of the Stewartson layer – part 2. Supercritical mode transitions. Theor. Comput. Fluid Dyn. 18, 197–204 (2004)
Karpunin, I.E., Kozlova, A.N., Kozlov, N.V.: Behavior of a light solid in a rotating horizontal cylinder with liquid under vibration. Microgravity Sci. Technol. 30, 399–409 (2018)
van de Konijnenberg, J.A., Nielsen, A.H., Rasmussen, J.J., Stenum, B.: Shear-flow instability in a rotating fluid. J. Fluid Mech. 387, 177–204 (1999)
Kozlov, N.: Theory of the vibrational hydrodynamic top. Acta Astr. 114, 123–129 (2015)
Kozlov, V.G., Ivanova, A.A.: Dramatic effect of vibrations on dynamics of rotating hydrodynamic systems. Microgravity Sci. Technol. 21, 339–348 (2009)
Kozlov, V.G., Kozlov, N.V.: Vibrational hydrodynamic gyroscope. Dokl. Phys. 52(8), 458–461 (2007)
Kozlov, V.G., Kozlov, N.V., Subbotin, S.V.: Motion of fluid and a solid core in a spherical cavity rotating in an external force field. Dokl. Phys. 59(1), 40–44 (2014a)
Kozlov, V.G., Kozlov, N.V., Subbotin, S.V.: Taylor column instability in the problem of a vibrational hydrodynamic top. Phys. Rev. E. 90(1), 013029 (2014b)
Kozlov, V.G., Kozlov, N.V., Subbotin, S.V.: Influence of an external force field on the dynamics of a free core and fluid in a rotating spherical cavity. Phys. Fluids. 27(7), 074106 (2015)
Kozlov, V.G., Kozlov, N.V., Subbotin, S.V.: Steady flows excited by circular oscillations of free inner core in rotating spherical cavity. Eur. J. Mech. B-Fluid. 58(4), 85–94 (2016)
Kozlov, V.G., Kozlov, N.V., Subbotin, S.V.: Instabilities and pattern formation in rotating spherical cavity with oscillating inner core. Eur. J. Mech. B-Fluid. 63(3), 39–46 (2017)
Kozlov, V., Rysin, K., Vjatkin, A.: Vibroconvective stability of liquid in horizontal plane layer subject to circular translational vibrations. Microgravity Sci. Technol. 31, 759–765 (2019)
Kozlov, V.G., Subbotin, S.V.: Steady flows generated by a core oscillating in a rotating spherical cavity. J. Appl. Mech. Tech. Phys. 59(1), 22–31 (2018)
Kozlov, V., Subbotin, S., Shiryaeva, M.: Instabilities of steady flow in a rotating spherical cavity excited by inner core oscillation. Microgravity Sci. Technol. 31, 775–782 (2019)
Lappa, M.: Assessment of the role of axial vorticity in the formation of particle accumulation structures (PAS) in supercritical Marangoni and hybrid thermocapillary-rotation-driven flows. Phys. Fluids. 25(1), 012101 (2013)
Le Bars, M., Cébron, D., Le Gal, P.: Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163–193 (2015)
Lorenzani, S., Tilgner, A.: Fluid instabilities in precessing spheroidal cavities. J. Fluid Mech. 447, 111–128 (2001)
Makarikhin, I.Y., Smorodin, B.L., Shatrova, E.F.: Drift of spheres in a rotating fluid. Fluid Dyn. 43(4), 506–513 (2008)
Messio, L., Morize, C., Rabaud, M., Moisy, F.: Experimental observation using particle image velocimetry of inertial waves in a rotating fluid. Exp. Fluids. 44, 519–528 (2008)
Morize, C., Le Bars, M., Le Gal, P., Tilgner, A.: Experimental determination of zonal winds driven by tides. Phys. Rev. Lett. 104, 214501 (2010)
Pimenova, A.V., Goldobin, D.S., Lyubimova, T.P.: Comparison of the effect of horizontal vibrations on interfacial waves in a two-layer system of inviscid liquids to effective gravity inversion. Microgravity Sci. Technol. 30, 1–10 (2018)
Pushkin, D.O., Melnikov, D.E., Shevtsova, V.M.: Ordering of small particles in one-dimensional coherent structures by time-periodic flows. Phys. Rev. Lett. 106, 234501 (2011)
Sauret, A., Le Bars, M., Le Gal, P.: Tide-driven shear instability in planetary liquid cores. Geophys. Res. Lett. 41, 6078–6083 (2014)
Schaeffer, N., Cardin, P.: Quasi-geostrophic model of the instabilities of the Stewartson layer in flat and depth varying containers. Phys. Fluids. 17, 104111 (2005)
Smorodin, B.L., Ishutov, S.M., Myznikova, B.I.: On the convection of a binary mixture in a horizontal layer under high-frequency vibrations. Microgravity Sci. Technol. 30, 95–102 (2018)
Subbotin, S., Dyakova, V.: Inertial waves and steady flows in a liquid filled librating cylinder. Microgravity Sci. Technol. 30, 383–392 (2018)
Taylor, G.I.: The motion of a sphere in a rotating liquid. Proc. Roy. Soc. London. Ser. A. 102(102), 180–189 (1923)
Thielicke, W., Stamhuis, E.J.: PIVlab – time-resolved digital particle image velocimetry tool for MATLAB (version: 1.41). J. Open Res. Softw. 2(1), e30 (2014)
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The research was supported by the Russian Science Foundation (project No. 18-71-10053).
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Subbotin, S., Kozlov, V. Effect of Differential Rotation of Oscillating Inner Core on Steady Flow Instability in a Rotating Sphere. Microgravity Sci. Technol. 32, 825–836 (2020). https://doi.org/10.1007/s12217-020-09806-y
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DOI: https://doi.org/10.1007/s12217-020-09806-y