Rotating waves of various azimuthal wavenumbers appear in magnetised spherical Couette flows in time-dependent states. The criteria for the flow to enter a time-dependent state are determined by the values of Reynolds and Hartmann numbers for a given geometric configuration of the spherical Couette system. These time-dependent states of the flow have been referred to as the radial jet, the return flow and the shear layer instabilities. In the present work, rotating waves in the flow during radial jet and return flow instabilities are observed.
The rotating waves both have an azimuthal wavenumber of 3, but vary in their spatial and temporal properties.Postprint (published version

García González, Fernando

Gundrum, Thomas

Ogbonna, Jude

Stefani, Frank

English

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   12th PAMIR International Conference - Fundamental and Applied MHD                                                                July 4 – 8, 2022, Krakow, Poland _________________________________________________________________________________________________________________   ROTATING WAVES ARISING FROM THE INSTABILITY OF MAGNETISED SPHERICAL COUETTE FLOWS Jude OGBONNA1*, Ferran GARCIA2, Thomas GUNDRUM1, Frank STEFANI1 1 Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany. 2 Barcelona East School of Engineering, Universitat  Politecnica de Catalunya, Av. D’Eduard Maristany, 16, 08019   Barcelona, Spain. *Corresponding author: j.ogbonna@hzdr.de Abstract: Rotating waves of various azimuthal wavenumbers appear in magnetised spherical Couette flows in time-dependent states. The criteria for the flow to enter a time-dependent state are determined by the values of Reynolds and Hartmann numbers for a given geometric configuration of the spherical Couette system. These time-dependent states of the flow have been referred to as the radial jet, the return flow and the shear layer instabilities. In the present work, rotating waves in the flow during radial jet and return flow instabilities are observed. The rotating waves both have an azimuthal wavenumber of 3, but vary in their spatial and temporal properties. Key words: Experimental fluid dynamics, flow instability, magnetohydrodynamics 1. Introduction The magnetised spherical Couette (MSC) flow occurs when an electrically-conducting fluid is utilised in a spherical Couette geometry and a magnetic field is applied to the flow. The resulting flow is governed by the magnetic Prandtl number Pm = ν/η (where ν and η are the kinematic viscosity and the magnetic diffusivity of the fluid, respectively), the Hartmann number Ha = B0ri/(μ0ρνη)1/2 = B0riσ1/2/(ρν)1/2 (where B0, ri, μ0, ρ and σ are the applied magnetic field, the radius of the inner sphere, the vacuum permeability, the density of the fluid and the electrical conductivity of the fluid, respectively), the ratio of the inner to the outer spheres ri/ro (where ro is the radius of the outer sphere) and the Reynolds number Re = Ωri2/ν. These parameters sufficiently define the standard spherical Couette flow, where the inner sphere rotates while the outer sphere remains stationary. Four different states of the MSC flow have been identified, with one of them being in a steady state while the other three are time-dependent. The steady state of the MSC is its base state, consisting of a flow driven by the rotation of the inner sphere. This rotation flings the fluid outwards by inertia, thus creating a secondary meridional circulation consisting of a narrow jet along the equator towards the outer sphere and a return flow from the rest of the outer sphere towards the origin of the jet [1]. The time-dependent states of the MSC flow are commonly referred to as the radial jet, the return flow and the shear layer instabilities. The physical features of these time-dependent states are described in [1]. The stability curves demarcating the parameter space of Re and Ha into regions representing the base state and the time-dependent states of the MSC flow have been numerically computed for ri/ro = 0.33 [1], 0.5 [2] and 0.6 [2]. The use of bifurcation lines representing a particular property of the flow is also used to graphically represent the MSC flow states. Various bifurcation diagrams for MSC flows of ri/ro =  0.5 have been computed [3, 4]. 12th PAMIR International Conference - Fundamental and Applied MHD July 4 – 8, 2022, Krakow, Poland  The motivation for the present work is to demonstrate the existence of rotating waves in two different time-dependent states of the MSC flow, namely, the radial jet instability and the return flow instability. The work follows an experimental approach by employing the HEDGEHOG (HydromagneticExperiment with Differentially Gyrating sphEres HOlding GaInSn) MSC setup at Helmholtz-Zentrum Dresden-Rossendorf. 2. Equations The axial magnetic field B0 applied to the MSC flow with the spheres rotating about êz, the unit vector representing the z-axis is given in Eq. 1. B0 = B0êz = B0cos (θ) êr – B0sin (θ) êθ   (1) where B0, θ, êr and êθ are the magnetic field, the colatitude and the unit vectors representing the radial distance and the colatitude, respectively. The Navier-Stokes equation ∂tv + v ∙ ∇v = -∇p + Re-1∇2v + Ha2Re-1 (∇ × b) × êz   (2) governs the fluid flow with the no-slip condition applied at the interfaces between the fluid and the spheres. The deviation of the magnetic field induced by the interaction of the flow with B0 is negligible for Rm << 1. This justifies the so-called inductionless approximation of the experiments conducted in the present work since Re = 1000, Pm ~10-6 for GaInSn and Rm = Re Pm. As a result, the induction equation simplifies to 0 = ∇2b + ∇ × (v × êz).   (2) The boundary conditions are assumed to be electrically-insulated in accordance with the experiments. 3. Materials and methods The setup for the HEDGEHOG, which is the main experimental apparatus, and the geometric relationship between a piezoelectric ultrasonic transducer (a component of the ultrasonic Doppler velocimetry system for measuring the fluid velocities) and the HEDGEHOG spheres are shown in Fig 1. 3.1 Experimental apparatus and instrumentation The inner and outer spheres of HEDGEHOG are made from acrylic glass. Their radii are 45 mm and 90 mm, respectively. Thus, ri/ro is fixed at 0.5. The inner sphere is rotated by an electric motor shaft passing through the poles of both spheres. The diameter of the shaft between the inner and outer spheres is only 3 mm, so that the interference caused by its rotation on the major fluid flow is insignificant. GaInSn completely fills the volume between the inner and outer spheres. GaInSn is a eutectic alloy of gallium (67 % by mass), indium (20.5 % by mass) and tin (12.5 % by mass). A lead weight inside the inner sphere counterbalances the buoyancy force exerted by the density of GaInS. The outer surface of the outer sphere contains provisions for mounting the ultrasonic transducers. The surface also contains 168 needle-like copper electrodes for electric potential measurement, rendering it with the distinctive appearance that inspired its name. The axial magnetic field for the flow is generated by current through a Helmholtz coil pair of 80 wire windings in each coil. The radius of each coil and the distance between them are both approximately 300 mm. The associated instrumentation for the HEDGEHOG experimental setup is the ultrasonic Doppler velocimetry system, consisting of piezoelectric ultrasonic transducers (Signal Processing SA, model TR0410LS) and an ultrasonic Doppler velocimeter (Signal Processing SA, model DOP3010). For the present experiments, six transducers were used at colatitudes of 78.9° with respect to the spheres. Each transducer was inclined southward so that the angle between the axis of the transducer and the equator of the spheres was 35°. Each transducer periodically emits an ultrasonic pulse at a frequency of 4 MHz and received the echoes from the particles in the path of the ultrasonic beam. These particles are formed as a result of the oxidation of GaInSn due to its exposure to air. From each transducer channel, the 12th PAMIR International Conference - Fundamental and Applied MHD July 4 – 8, 2022, Krakow, Poland  velocimeter sequentially acquires the echo signals reflected by the particles to determine the velocities of the flow parallel to each transducer axis.   Figure 1. Photographs of the HEDGEHOG setup showing its full view (left) and a close-up view on the outer sphere (right). (b) Meridional section showing the geometric relations for one of the six azimuthally equidistant ultrasonic transducers installed on HEDGEHOG. The dotted line extending from the ultrasonic transducer represents its line of view along which the flow velocity components parallel to the line were measured.  3.2 Procedure Since ultrasonic Doppler velocimetry relies on the reflection of the ultrasonic pulses by the naturally-occurring oxides in GaInSn, the oxides were distributed by circulating the GaInSn from the north pole to the south pole via a peristaltic pump connected to the poles externally. The axial magnetic field B0 was calculated from the current I through the Helmholtz coil pair from the relation B0 = (4/5)3/2 µ0nI/R   (4) where n and R are the number of wire windings in each Helmholtz coil and the radius of each coil (which is equal to the distance between the coils), respectively.  Two sets of experiments were conducted, one at Ha = 5 and and the other at Ha = 35. Both sets of experiments were conducted at Re = 1000. At Re = 1000 and Ha = 5, the flow is in the state of radial jet instability, while the flow is in the state of return flow instability at Re = 1000 and Ha = 35. Thus the MSC flows at two distinct time-dependent states were investigated. The values of Re and Ha were evaluated using the properties of GaInSn given in Table 1. Table 1. Physical properties of GaInSn from [5] ρ (kg m-3) ν (m2 s-1) η = 1/µ0σ (m2 s-1) 6.4 × 103 3.4 × 10-7 0.2428 The results of the experiments were inferred from the velocity data measured by the ultrasonic Doppler velocimetry. The velocity data were sampled at approximately 1.17 seconds per data point. This difference in the measured time between the six transducers were ignored since they are negligible compared to the period of the rotating waves generated in the flow. One hour of the velocity data was analysed for each experiment.   12th PAMIR International Conference - Fundamental and Applied MHD July 4 – 8, 2022, Krakow, Poland  4. Results 4.1 Flow velocities Fig 2 and Fig 3 show the plots of the velocity components of the flow along the axis of one of the six ultrasonic transducers for Ha = 5 and Ha = 35, respectively. The depths indicate the distance away from the transducer along its axis. The sign convention of the velocity is such that flows away from the transducers have positive velocities, while flows towards the transducers have negative velocities.    Figure 2. Plots of the experimentally measured velocity components parallel to the ultrasonic transducer axis at Ha = 5 and Re = 1000. The depth indicates the axial distance from the ultrasonic transducer.  Figure 3. Plots of the experimentally measured velocity components parallel to the ultrasonic transducer axis at Ha = 35 and Re = 1000. The depth indicates the axial distance from the ultrasonic transducer.  4.2 Azimuthal modes The base and time-dependent states of the MSC flow differ by the absence of kinetic energy in the non-axisymmetric azimuthal modes in the former. The kinetic energy correlates with the velocity of the azimuthal modes. The velocities in the azimuthal modes are obtained from the two-dimensional discrete Fourier transform (DFT) of the velocity data in time and space (using the velocity data captured by each of the six ultrasonic Doppler transducers). The azimuthal modes of the MSC flow correspond to the azimuthal wavenumbers of the rotating waves present in the flow. Fig 4 and Fig 5 show the velocities of the various azimuthal modes in the flow for Ha = 5 and Ha = 35, respectively. The dominant azimuthal modes correspond to the azimuthal wavenumber m of the rotating wave in the flow. The DFT for Ha = 5 and Ha = 35 were taken at depths of 25 mm and 40 mm, respectively (refer to Fig 2 and Fig 3 for the depths). The results show that m = 3 for rotating waves in both flows. However, the frequency of the rotating wave at Ha = 5 is approximately 0.011 Hz, whereas it is approximately 0.006 Hz for Ha = 35. Furthermore, the rotating waves in the two flows are dominant at different depths. 12th PAMIR International Conference - Fundamental and Applied MHD July 4 – 8, 2022, Krakow, Poland  This can be inferred from the structure of the velocity plots in Fig 2 and Fig 3, and informed the depth at which the DFT of the velocity data were conducted.   Figure 4. Velocities of the flow decomposed into azimuthal wavenumbers 2, 3 and 4 obtained from the discrete Fourier transform of the velocity data at Ha = 5 and Re = 1000.  Figure 5. Velocities of the flow decomposed into azimuthal wavenumbers 2, 3 and 4 obtained from the discrete Fourier transform of the velocity data at Ha = 35 and Re = 1000.  5. Conclusions  Experiments were conducted at two time-dependent states of the MSC flow. For the radial jet instability, Ha = 5 and Re = 1000, while for the return flow instability, Ha = 35 and Re = 1000. In both cases, rotating waves of m = 3 were observed. However, the rotating waves in the two flows differed in their frequencies and depths at which they were dominant. Acknowledgments This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 787544). REFERENCES [1] R. HOLLERBACH. Non-axisymmetric instabilities in magnetic spherical Couette flow.  Proc. R. Soc. A, vol. 465. [2] V. TRAVNIKOV, K. ECKERT AND S. ODENBACH. Influence of an axial magnetic field on the stability of spherical Couette flows with different gap widths. Acta Mechanica, vol. 219. [3] F. GARCIA AND F. STEFANI. Continuation and stability of rotating waves in the magnetized spherical Couette system: secondary transitions and multistability.  Proc. R. Soc. A, vol. 474. [4] F. GARCIA, M. SEILMAYER, A. GIESECKE AND F. STEFANI. Modulated rotating waves in the magnetised spherical Couette system.  J. Nonlinear Sci., vol. 29. [5] G. RÜDIGER, L. KITCHATINOV AND R. HOLLERBACH. Magnetic Processes in Astrophysics (Wiley-VCH Verlag GmbH & Co. KGaA, 2013). 

Rotating waves arising from the instability of magnetized spherical Couette flows

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Rotating waves arising from the instability of magnetized spherical Couette flows

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