Spherical Couette Flow Research Articles

Revisiting visualization of spiral states in a wide-gap spherical Couette flow

Revisiting visualization of spiral states in a wide-gap spherical Couette flow

Magnetically induced gluing bifurcations of three-tori in spherical Couette flows

The global dynamics of three-tori associated with homoclinic/heteroclinic global (infinite period) bifurcations are investigated for the magnetized spherical Couette problem, a paradigmatic model in geo- and astrophysical magnetohydrodynamics (MHD). A novel homoclinic bifurcation, involving collision between three-tori, is described. In addition, a heteroclinic bifurcation connecting two unstable two-tori with a stable three-torus is also analyzed. The role of the flow's spatial symmetries in this bifurcation scenario is also investigated. This bifurcation scenario gives rise to MHD flows that combine small with extremely large time scales.

Numerical simulations of the Tayler–Spruit dynamo in proto-magnetars

ABSTRACT The Tayler–Spruit dynamo is one of the most promising mechanisms proposed to explain angular momentum transport during stellar evolution. Its development in proto-neutron stars spun-up by supernova fallback has also been put forward as a scenario to explain the formation of very magnetized neutron stars called magnetars. Using three-dimensional direct numerical simulations, we model the proto-neutron star interior as a stably stratified spherical Couette flow with the outer sphere that rotates faster than the inner one. We report the existence of two subcritical dynamo branches driven by the Tayler instability. They differ by their equatorial symmetry (dipolar or hemispherical) and the magnetic field scaling, which is in agreement with different theoretical predictions (by Fuller and Spruit, respectively). The magnetic dipole of the dipolar branch is found to reach intensities compatible with observational constraints on magnetars.

Electromagnetically driven anticyclonic rotation in spherical Couette flow

The electromagnetically driven anticyclonic flow in the wide gap of a rotating concentric spheres system is studied experimentally and numerically in the laminar regime. The working fluid is an electrolyte contained in the spherical gap. The outer sphere rotates at constant angular speed, and the inner sphere is at rest. The electromagnetic stirring is generated due to the interaction of a direct current, which is injected radially through ring-shaped electrodes located at the equatorial zone of each sphere, and a dipolar magnetic field produced by a permanent magnet located inside the inner sphere. Experimental velocity fields in the equatorial plane were obtained with particle image velocimetry. Additionally, full three-dimensional numerical simulations were performed. The high shearing at the equatorial region promotes an instability that can be perceived as a triggering mechanism of four tornado-like vortical structures tilted and entangled in the polar direction. The instability structure rotates in either the cyclonic or the anticyclonic direction, depending on the flow parameters, namely, the angular velocity of the outer sphere and the applied electric current. To the best knowledge of the authors, this is the first time the phenomenon under study is reported with electromagnetic forcing. The numerical results agree quantitatively with the experimental observations.

Noise induced effects in the axisymmetric spherical Couette flow.

We study the axisymmetric, wide gap, spherical Couette flow in the presence of noise in numerical simulations and experiments. Such studies are important because most of the flows in nature are subjected to random fluctuations. Noise is introduced into the flow by adding fluctuations to the inner sphere rotation which are random in time with zero mean. Flows of a viscous incompressible fluid are induced either by rotation of the inner sphere only or by the co-rotation of the spheres. Mean flow generation was found to occur under the action of additive noise. A higher relative amplification of meridional kinetic energy compared to the azimuthal component was also observed under certain conditions. Calculated flow velocities were validated by laser Doppler anemometer measurements. A model is proposed to elucidate the rapid growth of meridional kinetic energy for flows induced by varying the co-rotation of the spheres. Our linear stability analysis for flows induced by the rotation of the inner sphere revealed a decrease in the critical Reynolds number, corresponding to the onset of the first instability. Also, in this case, a local minimum of the mean flow generation on approaching the critical Reynolds number was observed, which is consistent with the available theoretical predictions. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (Part 2)'.

Numerical reproduction of the spiral wave visualized experimentally in a wide-gap spherical Couette flow

Spherical Couette flow experiments were conducted according to the work of Egbers and Rath [Acta Mech. 111, 125–140 (1995)]. While the value of the critical Reynolds number obtained by the previous experiments was in good agreement with the numerical prediction, it has remained a question why a spiral wave bifurcating over the critical Reynolds number can be visualized even by a classical flow visualization technique like the mixing of a small amount of aluminum flakes to the working fluid. In the present study, through visualization using aluminum flakes drifting on a horizontal plane illuminated by a laser sheet, the flow was identified as a spiral wave with azimuthal wavenumber m = 3, using the experimentally obtained and numerically deduced comparison between phase velocities. By solving the equation of motion for the infinitesimal planar particles advecting in the flow field of the spiral wave, a visual distribution of reflected light was virtually reproduced, which is in good agreement with the experimentally obtained picture.

Numerical investigation of distinct flow modes in a wide-gap spherical annulus using OpenMP

Spherical Couette flow (SCF) between two rotating concentric spheres has many applications in different fields of Mathematics, Physics, and Engineering. The transitions occurring in this flow phenomena have the relevancy with geophysical motions. In this paper, we use the time-dependent three-dimensional incompressible Navier–Stokes equations (3D-INSEs) to investigate distinct flow modes for a wide-gap clearance ratio (CR) . An Artificial Compressibility method has been used along with three implicit solvers, i.e. alternating direction implicit (ADI), line Gauss–Seidel (LGS), and point Gauss–Seidel (PGS). We apply the basic techniques of OpenMP to our Fortran code. First, we make a performance comparison of the three implicit solvers running on a single central processing unit (CPU) with that of the multi-core CPUs using Open Multi-Processing (OpenMP) to choose the best solver for the simulation of the SCF problem. The comparison shows that the ADI solver is the most efficient solver compared to LGS and PGS solvers. Using the most efficient parallel ADI solver, we compute distinct flow modes for this wide gap in a range of Reynolds number between . For this wide gap, first we obtain the basic 0-vortex flow (0-VF) at . By increasing the the flow becomes symmetric 1-vortex flow (1-VF) and asymmetric spiral 1-VF at and respectively. The flow returns back to super critical basic flow at . Further increasing the the flow becomes spiral 0-VF with m = 4 and m = 5 spiral waves at and respectively. Increasing the further the flow finally becomes turbulent. For this CR, we have first time numerically simulate these flow transitions from basic flow to super critical spiral basic flow with increasing . Abbreviations: ACM: artificial compressibility method; ADI: alternating directionimplicit; CPU: central processing unit; CR: clearance ratio; CVD: circumferentialvelocity distribution; INSEs: incompressible Navier–Stokes equations; LGS: lineGauss–Seidel; OpenMP: open multi-processing; PGS: point Gauss–Seidel; Re: Reynoldsnumber; SCF: spherical Couette flow; SG: spherical-gaps; TG: Taylor–Gortler; TVF:Taylor vortex flow; 0-VF: 0-vortex flow; 1-VF: 1-vortex flow; WENO: weightedessentially non-oscillatory.

The influence of noises with different spectra on viscous incompressible fluid flows

The results of an experimental study of the effects of noises on flows in a rotating spherical layer are presented. Noise is introduced into the flow in the form of random fluctuations of the inner sphere rotational rate relative to a time constant mean value. The flow velocity was measured with a laser Doppler anemometer. It was found that the noises of the equal amplitude, but with different spectra cause the generation of mean flows of different intensity. It is shown that the increase in the average velocities under the action of the noise is larger for the stable flows compared to unstable ones Keywords: noise, flows with rotation, spherical Couette flow.

A thermal convection limit of spiral state in wide-gap spherical Couette flow

Symmetries of flow structures are often prescribed by their mechanical instability and geometry. In this study, as an example, we present the homotopy of a rotating threefold spiral state that is robust in a spherical Couette flow toward a hybrid system with thermal stratification effects. The rotating wave state has not yet been confirmed to smoothly connect to the thermal stratification system. Through continuation, the most dangerous mode at a purely spherical Couette flow of m = 4 modes of spherical harmonics is replaced by l = 4 and m = 3 in a purely thermal convective system. For the state obtained at the limit under only the thermal effect, the residual quantities of both the torque to the outer sphere and meridional circulation are discussed in detail.

SPH Modelling of a Dike Failure with Detection of the Landslide Sliding Surface and Damage Scenarios for an Electricity Pylon

The 3D SPH code SPHERA (RSE SpA) simulates the triggering and runout stages of a full-scale earth-filled dike failure with coastal flood. The code formulation is consistent with the Kinetic Theory of Granular Flow KTGF and requires no tuned input. The main Landslide Sliding Surface LSS within the dike is detected from the field of the quadratic invariant of the strain-rate tensor. The comparison with the experimental LSS position shows Maximum Gross Errors of 0.90m, in terms of depth difference at fixed planimetric point, and of 0.41m, in terms of LSS distance (i.e. 22% and 10% of the initial dike-top height over the excavation bottom). Inter-comparisons consider calibrated FEM results for the triggering stage. The LSS detection criterion is defined and verified via analytical solutions and code validations on spherical Couette flows. The impact on a virtual electricity pylon is assessed. The code improvements and tutorials are available at github.com.

Simulation of different flow regimes in a narrow-gap spherical Couette flow

The spherical Couette flow (SCF) system with the flow between two concentric rotating spheres is one of the most convenient example in exploring the laminar-turbulent transition. The transitions occurred in this flow phenomena has a great relevance with the geophysical motions and has applications in hydrodynamic problems. In the present work, we numerically study different flow instabilities in a narrow-gap SCF between two concentric spheres with only inner sphere rotating and the outer sphere is fixed. The time dependent three-dimensional incompressible Navier-Stokes equations (INSEs) along with a parallelized line Gauss-Seidel(LGS) solver has been used for the simulation of different flow instabilities in a narrow-gap clearance ratios(CR), β=(R2−R1)/R1=0.10. The flow instabilities have been investigated for a range of Reynolds numbers (Re)170≤Re≤12000. For this narrow-gap CR, first we obtain the 0-,1-,2-,3-, and 4-vortex flows at Re=1430,1440,1460,1750 and 1950 respectively by using the Stokes flow (SF) as an initial condition. Further increasing the Re using the same initial condition the flow becomes wavy, spiral wavy, then return back to supercritical basic flow and finally becomes turbulent at higher Re. The computed numerical results have a good agreement with the existing numerical and experimental results. Further more, we have noticed some new flow states which have not been observed in the previous numerical and experimental studies.

Shifting the flow stability limit in the presence of random fluctuation of the rotational velocity

Nonlinear stability of the isothermal three-dimensional flows of viscous incompressible fluid in the rotational spherical layer with the noise presence was studied numerically. Transition between stable non-stationary flow and unstable flow in the form of travelling azimuthal waves is under consideration. Small-amplitude noise inserted to the flow by random broadband disturbances of the inner sphere rotational rate about constant at time averaged value, outer sphere is fixed. An approach is proposed to simplify finding the critical Reynolds number, corresponding to the stability limit in the presence of noise. The results obtained by the proposed and well known methods are compared. Keywords: noise, rotational flows, spherical Couette flow, instability control.

Dynamic transitions of the magnetized spherical Couette flow between its base state and the return flow instability

The transition between the stable base state of the magnetized spherical Couette (MSC) flow and the return flow instability is experimentally investigated. The experiments are conducted using an MSC setup consisting of insulating spheres with the ratio of the inner to the outer radii r i/r o = 0.5, Reynolds number Re = 1000 and Hartmann number Ha ∈ [25, 29]. The transition is characterized by changes in the power spectra of the azimuthal modes in the flow as Ha is dynamically changed. The transition occurs in the interval Ha ∈ [26.5, 27.5]. The evolution of the power spectra of the azimuthal modes exhibits hysteretic effect depending on whether Ha is increased or decreased within the experimental interval. The power spectra in the azimuthal modes m ∈ {3, 4} increases and remains dominant as Ha is increased, while the power spectra in m ∈ {2, 4} are dominant while the flow is time dependent due to return flow instability as Ha is decreased.

Подавление неустойчивости течений с вращением

Experimentally investigated the possibilities of controlling the instability that develops in the form of traveling azimuthal waves in the spherical Couette flow. It was found that the action on the traveling azimuthal waves of periodic modulation of the internal sphere rotation speed can lead to their suppression, when the modulation is switched-off, the instability in the form of traveling azimuthal waves is restored, including with a change in the wave number. It was found that it is possible to maintain a stable stationary flow for a long time after the end of modulation - at least 50 minutes, or 2000 revolutions of the inner sphere, which is four times longer than the transition time in the stationary flow.

Study of the spherical Couette flow with electromagnetic stirring

The flow driven by electromagnetic forcing of an electrolytic fluid in a concentric sphere system with differential rotation is studied experimentally and theoretically in the laminar regime. The electromagnetic force is generated by the interaction of a direct current injected radially through electrodes located at the equatorial zone of the spheres and a dipolar magnetic field produced by a permanent magnet inside the inner sphere. Even though the flows are fully three-dimensional, under some assumptions, they can be mathematically modeled with a general one-dimensional solution from the incompressible Navier–Stokes equations. Additionally, in order to deepen the flow dynamics, a full three-dimensional numerical simulation is also obtained. Experimentally, velocity profiles in the equatorial plane between spheres were obtained with particle image velocimetry. The analytical solution reproduces qualitatively the experimental profiles, whereas the numerical solution reproduces quantitatively the experimental measurements.

Bifurcation aspect of polygonal coherence over transitional Reynolds numbers in wide-gap spherical Couette flow

This study numerically investigates the bifurcation aspect of the wide-gap spherical Couette flow (SCF), with an emphasis on the competition among polygonal coherence with different wave numbers observed over transitional Reynolds numbers. Focusing on a representative case, the half-radius ratio, we confirm that the axisymmetric state becomes unstable over the first transitional Reynolds number at which the 4-fold spiral state bifurcates, using the continuation method based on the Newton-Raphson algorithm. The Galerkin-spectral method was employed to numerically solve the governing equations. It is found that the 3-fold spiral state bifurcates from the axisymmetric state at a slightly higher Reynolds number than the first transitional Reynolds number. The attraction of the 3-fold spiral state expands rapidly with an increase in the Reynolds number, which is determined by verifying the distance of the unstable periodic-like state to both spiral states in the state space. This aspect of the state space explains the experimentally bistable realization of different equilibrium states over the first transitional Reynolds number. This study also found that the periodic-like state is composed of the 3- and 4-fold spiral states, similar to a beat with two different frequencies.

Modulated rotating waves and triadic resonances in spherical fluid systems: The case of magnetized spherical Couette flow

The existence of triadic resonances in the magnetized spherical Couette system is related to the development of modulated rotating waves, which are quasiperiodic flows understood in terms of bifurcation theory in systems with symmetry. In contrast to previous studies in spherical geometry, the resonant modes are not inertial waves but related to the radial jet instability, which is strongly equatorially antisymmetric. We propose a general framework in which triadic resonances are generated through successive Hopf bifurcations from the base state. The study relies on an accurate frequency analysis of different modes of the flow, for solutions belonging to two different bifurcation scenarios. The azimuthal and latitudinal nonlinear coupling among the resonant modes is analyzed and interpreted using spherical harmonics, and the results are compared with previous studies in spherical geometry.

Experimental study of rough spherical Couette flows: Increasing helicity toward a dynamo state

Spherical Couette flows in liquid metals are a suitable candidate for generating magnetic dynamo states in the laboratory. However, results in our 3 meter diameter experiment have shown that an enhancement of the flow's helicity is likely required. Previous works suggested roughening the inner sphere boundary by adding baffles in order to achieve these goals. In the present work, we perform hydrodynamic studies of the effect of different baffle designs in a 40 cm water experiment. Results point to one of these models as a promising upgrade that we will use in our 3 m experiment to effectively increase our chances of obtaining dynamo action.

A comparison of different numerical schemes in spherical Couette flow simulation

We compare the performance of line Gauss–Seidel (LGS), point Gauss–Seidel (PGS), and alternating direction implicit (ADI) linear solvers used in the artificial compressibility method for the numerical solution of the three-dimensional incompressible Navier–Stokes equations. Spatial discretization is carried out using a fifth-order WENO scheme for the convective terms and a second-order central difference scheme for the viscous terms. A comparison is made by simulating the spherical Couette flow problem, with only the inner sphere rotating and the outer one fixed. OpenMP is used for numerical computation in parallel for the three schemes. First, we compare the numerical efficiency of the solvers by computing 0-vortex flow for a medium-gap σ=R2−R1/R1=0.25. Second, we make a residual comparison for a steady-state flow calculation based on the CFL number and artificial compressibility factor. Finally, we compare the three solvers for unsteady flow computations based on the artificial compressibility factor. The results show that the LGS solver is more reliable than the PGS and the ADI solvers. To show the accuracy of the LGS scheme, we compute different flow modes for an intermediate-gap clearance ratio σ=R2−R1/R1=0.14. The computed results have good agreement with the existing numerical results.

Experimental investigation of the return flow instability in magnetized spherical Couette flows

We conduct magnetized spherical Couette (MSC) flow experiments in the return flow instability regime with GaInSn as the working fluid, the ratio of the inner to the outer sphere radii ri/ro = 0.5, the Reynolds number Re = 1000, and the Hartmann number Ha ∈ [27.5, 40]. Rotating waves with different azimuthal wavenumbers m ∈ {2, 3, 4} manifest in certain ranges of Ha in the experiments, depending on whether the values of Ha were fixed or varied from different initial values. These observations demonstrate the multistability of rotating waves, which we attribute to the dynamical system representing the state of the MSC flow tending to move along the same solution branch of the bifurcation diagram when Ha is varied. In experiments with both fixed and varying Ha, the rotation frequencies of the rotating waves are consistent with the results of nonlinear stability analysis. A brief numerical investigation shows that differences in the azimuthal wavenumbers of the rotating waves that develop in the flow also depend on the azimuthal modes that are initially excited.