Spherical Couette Flow Research Articles

Bi-stability in turbulent, rotating spherical Couette flow

Flow between concentric spheres of radius ratio η=ri/ro=0.35 is studied in a 3 m diameter experiment. We have measured the torques required to maintain constant boundary speeds as well as localized wall shear stress, velocity, and pressure. At low Ekman number E=2.1×10-7 and modest Rossby number 0.07<Ro<3.4, the resulting flow is highly turbulent with a maximum Reynolds number (Re = Ro/E) exceeding 15 million. Several turbulent flow regimes are evident as Ro is varied for fixed E. We focus our attention on one flow transition, in particular, between Ro = 1.8 and Ro = 2.6, where the flow shows bistable behavior. For Ro within this range, the flow undergoes intermittent transitions between the states observed alone at adjacent Ro outside the switching range. The two states are clearly distinguished in all measured flow quantities, including a striking reduction in torque demanded from the inner sphere by the state lying at higher Ro. The reduced angular momentum transport appears to be associated with the development of a fast zonal circulation near the experiment core. The lower torque state exhibits waves, one of which is similar to an inertial mode known for a full sphere and another which appears to be a strongly advected Rossby-type wave. These results represent a new laboratory example of the overlapping existence of distinct flow states in high Reynolds number flow. Turbulent multiple stability and the resilience of transport barriers associated with zonal flows are important topics in geophysical and astrophysical contexts.

Influence of an axial magnetic field on the stability of spherical Couette flows with different gap widths

This paper deals with the linear instability analysis of the spherical Couette flow of an electrically conducting fluid in the presence of an axial magnetic field. The numerical investigations are performed for different ratios, η = 0.5 and η = 0.6, and compared with Hollerbach’s work for η = 0.33 (Hollerbach in Proc R Soc 465:2003, 2009). The corresponding instability diagrams, i.e., the critical values of the Reynolds number Re, the wave number, and the frequency on the Hartmann number Ha, are presented and accompanied by simulation of the transition between three-dimensional flow states of different symmetries. The characteristic subdivision of the linear stability curves into anti-symmetric modes, which is responsible for the instability at small Ha, and symmetric modes occurring at higher Ha is found for larger η, too. However, the extension of the stability corridor between the anti-symmetric and the symmetric modes increases nonlinearly with η. This offers the possibility to stabilize the basic flow up to high Re by appropriately increasing Ha.

Magnetic spherical Couette flow in linear combinations of axial and dipolar fields

We present axisymmetric numerical calculations of the fluid flow induced in a spherical shell with inner sphere rotating and outer sphere stationary. A magnetic field is also imposed, consisting of particular linear combinations of axial and dipolar fields, chosen to make B r = 0 at either the outer sphere, or the inner, or in between. This leads to the formation of Shercliff shear layers at these particular locations. We then consider the effect of increasingly large inertial effects and show that an outer Shercliff layer is eventually destabilized, an inner Shercliff layer appears to remain stable, and an in-between Shercliff layer is almost completely disrupted even before the onset of time-dependence, which does eventually occur though.

Numerical simulations of dynamos generated in spherical Couette flows

We numerically investigate the efficiency of a spherical Couette flow at generating a self-sustained magnetic field. No dynamo action occurs for axisymmetric flow, while we always found a dynamo when non-axisymmetric hydrodynamical instabilities are excited. Without rotation of the outer sphere, typical critical magnetic Reynolds numbers Rm c are of the order of a few thousands. They increase as the mechanical forcing imposed by the inner core on the flow increases (Reynolds number Re). Namely, no dynamo is found if the magnetic Prandtl number Pm = Rm/Re is less than a critical value Pm c ∼ 1. Oscillating quadrupolar dynamos are present in the vicinity of the dynamo onset. Saturated magnetic fields obtained in supercritical regimes (either Re > 2Re c or Pm > 2Pm c ) correspond to the equipartition between magnetic and kinetic energies. A global rotation of the system (Ekman numbers E = 10−3, 10−4) yields to a slight decrease (factor 2) of the critical magnetic Prandtl number, but we find a peculiar regime where dynamo action may be obtained for relatively low magnetic Reynolds numbers (Rm c ∼ 300). In this dynamical regime (Rossby number Ro ∼ −1, spheres in opposite direction) at a moderate Ekman number (E = 10−3), an enhanced shear layer around the inner core might explain the decrease of the dynamo threshold. For lower E (E = 10−4) this internal shear layer becomes unstable, leading to small scale fluctuations, and the favorable dynamo regime is lost. We also model the effect of ferromagnetic boundary conditions. Their presence have only a small impact on the dynamo onset, but clearly enhance the saturated magnetic field in the ferromagnetic parts. Implications for experimental studies are discussed.

Magneto-inertial waves in a rotating sphere

Hydromagnetic oscillations have been observed in recent spherical Couette flow experiments with liquid sodium. In the Derviche Tourneur Sodium (DTS) installation in Grenoble, where the imposed magnetic field is dipolar, they are supposed to be magneto-inertial waves with azimuthal wavenumbers ranging from 1 to 7, and azimuthally propagating in a retrograde way with regard to the rotating fluid. A new numerical simulation code has been developed in order to calculate such magneto-inertial waves in spherical geometry and in the presence of an external magnetic field. A few known examples are first considered in order to validate this code, then preliminary results are obtained in the DTS geometry.

Selection of inertial modes in spherical Couette flow

Spherical Couette flow involves fluid sheared between concentric coaxially rotating spheres. Its scientific relevance lies not only in the simplicity of the system but also in its applicability to astrophysical objects such as atmospheres, oceans, and planetary cores. One common behavior in all rotating flows, including spherical Couette flow, is the presence of inertial modes, which are linear wave modes restored by the Coriolis force. Building on a previous identification of inertial modes in a laboratory spherical Couette cell, here we propose selection mechanisms to explain the presence of the particular modes we have observed. Mode selection depends on both amplification and damping. Our experimental observations are consistent with amplification and selection by over-reflection at a shear layer, and we would expect other spherical Couette devices to behave similarly. Damping effects, due in part to the presence of an inner sphere, add further constraints which are likely to play a role in mode selection in planetary atmospheres and cores, including the core of earth.

Shear-layers in magnetohydrodynamic spherical Couette flow with conducting walls

We consider the steady axisymmetric motion of an electrically conducting fluid contained within a spherical shell and permeated by a centred axial dipole magnetic field, which is strong as measured by the Hartmann number M. Slow axisymmetric motion is driven by rotating the inner boundary relative to the stationary outer boundary. For M ≫ 1, viscous effects are only important in Hartmann boundary layers adjacent to the inner and outer boundaries and a free shear-layer on the magnetic field line that is tangent to the outer boundary on the equatorial plane of symmetry. We measure the ability to leak electric current into the solid boundaries by the size of their relative conductance ɛ. Since the Hartmann layers are sustained by the electric current flow along them, the current inflow from the fluid mainstream needed to feed them increases in concert with the relative conductance, because of the increasing fraction ℒ of the current inflow leaked directly into the solids. Therefore the nature of the flow is sensitive to the relative sizes of ɛ−1 and M.The current work extends an earlier study of the case of a conducting inner boundary and an insulating outer boundary with conductance ɛo = 0 (Dormy, Jault & Soward, J. Fluid Mech., vol. 452, 2002, pp. 263–291) to other values of the outer boundary conductance. Firstly, analytic results are presented for the case of perfectly conducting inner and outer boundaries, which predict super-rotation rates Ωmax of order M1/2 in the free shear-layer. Successful comparisons are made with numerical results for both perfectly and finitely conducting boundaries. Secondly, in the case of a finitely conducting outer boundary our analytic results show that Ωmax is O(M1/2) for ɛo−1 ≪ 1 ≪ M3/4, O(ɛo2/3M1/2) for 1 ≪ ɛo−1 ≪ M3/4 and O(1) for 1 ≪ M3/4 ≪ ɛo−1. On increasing ɛo−1 from zero, substantial electric current leakage into the outer boundary, ℒo ≈ 1, occurs for ɛo−1 ≪ M3/4 with the shear-layer possessing the character appropriate to a perfectly conducting outer boundary. When ɛo−1 = O(M3/4) the current leakage is blocked near the equator, and the nature of the shear-layer changes. So, when M3/4 ≪ ɛo−1, the shear-layer has the character appropriate to an insulating outer boundary. More precisely, over the range M3/4 ≪ ɛo−1 ≪ M the blockage spreads outwards, reaching the pole when ɛo−1 = O(M). For M ≪ ɛo−1 current flow into the outer boundary is completely blocked, ℒo ≪ 1.

Competition of linear modes in a spherical Couette flow after sudden increase of rotational velocity of the inner sphere

Competition of linear modes in a spherical Couette flow after sudden increase of rotational velocity of the inner sphere

AN UNSTABLE SUPERFLUID STEWARTSON LAYER IN A DIFFERENTIALLY ROTATING NEUTRON STAR

Experimental and numerical evidence is reviewed for the existence of a Stewartson layer in spherical Couette flow at small Ekman and Rossby numbers (E ≲ 10−3, Ro ≲ 10−2), the relevant hydrodynamic regime in the superfluid outer core of a neutron star. Numerical simulations of a superfluid Stewartson layer are presented for the first time, showing how the layer is disrupted by nonaxisymmetric instabilities. The unstable ranges of E and Ro are compared with estimates of these quantities in radio pulsars that exhibit glitches. It is found that glitching pulsars lie on the stable side of the instability boundary, allowing differential rotation to build up before a glitch.

Non-axisymmetric instabilities in magnetic spherical Couette flow

We investigate numerically the flow of an electrically conducting fluid confined in a spherical shell, with the outer sphere stationary, the inner sphere rotating and a magnetic field imposed parallel to the rotation axis. We compute both the axisymmetric basic states and their non-axisymmetric instabilities. Two distinct instability classes emerge, one connected to previous non-magnetic results, the other to previous strongly magnetic results. Both instabilities arise from the basic state's meridional circulation, but are otherwise very different from one another, and are separated by a region of stability that persists even for large Reynolds numbers. Finally, we compute the fully three-dimensional nonlinear equilibration of both instabilities. The second class exhibits a rich variety of secondary bifurcations, involving mode transitions between different azimuthal wave numbers.

Superfluid spherical Couette flow

We solve numerically the two-fluid, Hall-Vinen-Bekarevich-Khalatnikov equations for a He-II-like superfluid contained in a differentially rotating, spherical shell, generalizing previous simulations of viscous spherical Couette flow (SCF) and superfluid Taylor-Couette flow. The system tends towards a stationary but unsteady state, where the torque oscillates persistently, with amplitude and period determined by dimensionless gap width δ and rotational shear ΔΩ. In axisymmetric superfluid SCF, the number of meridional circulation cells multiplies as the Reynolds number Re increases. In nonaxisymmetric superfluid SCF, three-dimensional vortex structures are classified according to topological invariants. We find that the mutual friction is "patchy"; that is, it takes different forms in different parts of the vessel, a surprising new result.

Magnetohydrodynamic experiments on cosmic magnetic fields

AbstractIt is widely known that cosmic magnetic fields, i.e. the fields of planets, stars, and galaxies, are produced by the hydromagnetic dynamo effect in moving electrically conducting fluids. It is less well known that cosmic magnetic fields play also an active role in cosmic structure formation by enabling outward transport of angular momentum in accretion disks via the magnetorotational instability (MRI). Considerable theoretical and computational progress has been made in understanding both processes. In addition to this, the last ten years have seen tremendous efforts in studying both effects in liquid metal experiments. In 1999, magnetic field self‐excitation was observed in the large scale liquid sodium facilities in Riga and Karlsruhe. Recently, self‐excitation was also obtained in the French “von Kármán sodium” (VKS) experiment. An MRI‐like mode was found on the background of a turbulent spherical Couette flow at the University of Maryland. Evidence for MRI as the first instability of an hydrodynamically stable flow was obtained in the “Potsdam Rossendorf Magnetic Instability Experiment” (PROMISE). In this review, the history of dynamo and MRI related experiments is delineated, and some directions of future work are discussed.

Transition to chaos in wide gap spherical Couette flow – experiment and DNS

The paper presents the results of the experimental and numerical investigations of transition to chaos in spherical layer δ = 1 with counter rotating boundaries. In experiments the laminar-turbulent transition has been studied near the local maximum at the stability curve at Reo = −810 (−700 < Reo < −950). It was found that transition occurs through four or five bifurcations. Spectra of all supercritical flow states are formed from the frequencies, corresponding to the revealed flow structures. Transition to chaos occurs from one-frequency flow state. Its structure is symmetric with respect to the equator plane and includes three vortex tubes in each hemisphere. This symmetry-breaking bifurcation has a considerable hysteresis. Inner sphere acceleration may change the transition scenario – the last bifurcation leads to spatial-temporal intermittency. For the direct numerical simulation the case with Reo = −900 was chosen. All flow states, including chaotic one, boundaries and hysteresis areas, observed in experiments, were obtained numerically. Furthermore, some details of flow structures were revealed only from numerical simulation.

Instabilities of Shercliffe and Stewartson layers in spherical Couette flow

We explore numerically the flow induced in a spherical shell by differentially rotating the inner and outer spheres. The fluid is also taken to be electrically conducting (in the low magnetic Reynolds number limit), and a magnetic field is imposed parallel to the axis of rotation. If the outer sphere is stationary, the magnetic field induces a Shercliffe layer on the tangent cylinder, the cylinder just touching the inner sphere and parallel to the field. If the magnetic field is absent, but a strong overall rotation is present, Coriolis effects induce a Stewartson layer on the tangent cylinder. The nonaxisymmetric instabilities of both types of layer separately have been studied before; here, we consider the two cases side by side, as well as the mixed case, and investigate how magnetic and rotational effects interact. We find that if the differential rotation and the overall rotation are in the same direction, the overall rotation may have a destabilizing influence, whereas if the differential rotation and the overall rotation are in the opposite direction, the overall rotation always has a stabilizing influence.

Non-isothermal spherical Couette flow of Oldroyd-B fluid

The present paper is concerned with non-isothermal spherical Couette flow of Oldroyd-B fluid in the annular region between two concentric spheres. The inner sphere rotates with a constant angular velocity while the outer sphere is kept at rest. The viscoelasticity of the fluid is assumed to dominate the inertia such that the latter can be neglected in the momentum and energy equations. An approximate analytical solution is obtained through the expansion of the dynamical variable fields in power series of Nahme number. Non-homogeneous, harmonic for axial- velocity and temperature equations and biharmonic for stream function equations, have been solved up to second order approximation. In comparison of the present work with isothermal case; [1,2], two additional terms; a first order velocity and a second order stream function are stem as a result of the interaction between the fluid viscoelasticity and temperature profile. These contributions prove to be the most important results for rheology in this work.

Rapidly rotating spherical Couette flow in a dipolar magnetic field: An experimental study of the mean axisymmetric flow

In order to explore the magnetostrophic regime expected for planetary cores, in which the Lorentz forces balance the Coriolis forces, experiments have been conducted in a rotating sphere filled with liquid sodium, with an imposed dipolar magnetic field (the DTS setup). The field is produced by a permanent magnet enclosed in an inner sphere, which can rotate at a separate rate, producing a spherical Couette flow. The flow properties are investigated by measuring electric potentials on the outer sphere, the induced magnetic field in the laboratory frame just above the rotating outer sphere, and velocity profiles inside the liquid sodium using ultrasonic Doppler velocimetry. This article focuses on the time-averaged axisymmetric part of the flow. The electric potential differences measured at several latitudes can be linked to azimuthal velocities, and are indeed found to be proportional to the azimuthal velocities measured by Doppler velocimetry. The Doppler profiles show that the angular velocity of the fluid is relatively uniform in most of the fluid shell, but rises near the inner sphere, revealing the presence of a “magnetic wind”, and gently drops towards the outer sphere. The transition from a magnetostrophic flow near the inner sphere to a geostrophic flow near the outer sphere is controlled by the local Elsasser number. For Rossby numbers up to order 1, the observed velocity profiles all show a similar shape. Numerical simulations in the linear regime are computed, and synthetic velocity profiles are compared with the measured ones. A good agreement is found for the angular velocity profiles. In the geostrophic region, a torque-balance model provides very good predictions. Radial velocities change sign with the Rossby number, as expected for an Ekman-pumping dominated flow. For a given Rossby number the amplitude of the measured angular velocity is found to vary by as much as a factor of 3. Comparison with numerical simulations suggests that this is due to variations in the electric coupling between liquid sodium and the inner copper sphere, implying an effect equivalent to a reduction of the inner sphere electric conductivity by as much as a factor 100. We show that the measured electric potential difference can be used as a proxy of the actual fluid velocity. Using this proxy in place of the imposed differential velocity, we find that the induced magnetic field varies in a consistent fashion, and displays a peculiar peak in the counter-rotating regime. This happens when the fluid rotation rate is almost equal and opposite to the outer sphere rotation rate. The fluid is then almost at rest in the laboratory frame, and the Proudman–Taylor constraint vanishes, enabling a strong meridional flow. We suggest that dynamo action might be favored in such a situation.

Superfluid spherical Couette flow

We solve numerically for the first time the two-fluid Hall–Vinen–Bekarevich–Khalatnikov (HVBK) equations for an He-II-like superfluid contained in a differentially rotating spherical shell, generalizing previous simulations of viscous spherical Couette flow (SCF) and superfluid Taylor–Couette flow. The simulations are conducted for Reynolds numbers in the range 1 × 102≤Re≤3 × 104, rotational shear 0.1≤ΔΩ/Ω≤0.3, and dimensionless gap widths 0.2≤δ≤0.5. The system tends towards a stationary but unsteady state, where the torque oscillates persistently, with amplitude and period determined by δ and ΔΩ/Ω. In axisymmetric superfluid SCF, the number of meridional circulation cells multiplies as Re increases, and their shapes become more complex, especially in the superfluid component, with multiple secondary cells arising for Re &gt; 103. The torque exerted by the normal component is approximately three times greater in a superfluid with anisotropic Hall–Vinen (HV) mutual friction than in a classical viscous fluid or a superfluid with isotropic Gorter–Mellink (GM) mutual friction. HV mutual friction also tends to ‘pinch’ meridional circulation cells more than GM mutual friction. The boundary condition on the superfluid component, whether no slip or perfect slip, does not affect the large-scale structure of the flow appreciably, but it does alter the cores of the circulation cells, especially at lower Re. As Re increases, and after initial transients die away, the mutual friction force dominates the vortex tension, and the streamlines of the superfluid and normal fluid components increasingly resemble each other. In non-axisymmetric superfluid SCF, three-dimensional vortex structures are classified according to topological invariants. For misaligned spheres, the flow is focal throughout most of its volume, except for thread-like zones where it is strain-dominated near the equator (inviscid component) and poles (viscous component). A wedge-shaped isosurface of vorticity rotates around the equator at roughly the rotation period. For a freely precessing outer sphere, the flow is equally strain- and vorticity-dominated throughout its volume. Unstable focus/contracting points are slightly more common than stable node/saddle/saddle points in the viscous component, but not in the inviscid component. Isosurfaces of positive and negative vorticity form interlocking poloidal ribbons (viscous component) or toroidal tongues (inviscid component) which attach and detach at roughly the rotation period.

Rotating spherical Couette flow in a dipolar magnetic field: experimental study of magneto-inertial waves

The magnetostrophic regime, in which Lorentz and Coriolis forces are in balance, has been investigated in a rapidly rotating spherical Couette flow experiment. The spherical shell is filled with liquid sodium and permeated by a strong imposed dipolar magnetic field. Azimuthally travelling hydromagnetic waves have been put in evidence through a detailed analysis of electric potential differences measured on the outer sphere, and their properties have been determined. Several types of wave have been identified depending on the relative rotation rates of the inner and outer spheres: they differ by their dispersion relation and by their selection of azimuthal wavenumbers. In addition, these waves constitute the largest contribution to the observed fluctuations, and all of them travel in the retrograde direction in the frame of reference bound to the fluid. We identify these waves as magneto-inertial waves by virtue of the close proximity of the magnetic and inertial characteristic time scales of relevance in our experiment.

Inertial waves driven by differential rotation in a planetary geometry

Dynamics occurring in the Earth's outer core involve convection, dynamo action, geomagnetic reversals, and the effects of rapid rotation, among other processes. Inertial waves are known to arise in rotating fluids, and their presence in the core has been previously argued using seismological data (Aldridge and Lumb 1987). They may also be involved in flows affecting the geodynamo. We report experimental observations of inertial wave modes in an Earth-like geometry: laboratory spherical Couette flow with an aspect ratio 0.33, using liquid sodium as the working fluid. Inertial modes are detected via magnetic induction and show good agreement with theoretical predictions in frequency, wavenumber, and magnetic induction structure. Our findings imply that linear wave behavior can dominate the dynamics even in turbulent flows with large Reynolds number Re, where nonlinear behaviors might be expected (here Re ∼ 107). We present evidence that strong differential rotation excites the modes via over-reflection. Earth's inner core may also super-rotate and thereby excite inertial modes in the same way. Zonal flows in the core, likely to have higher speeds than the super-rotation, may be a stronger source for exciting inertial modes in the Earth.

Numerical study of the basic stationary spherical couette flows at low Reynolds numbers

Previously developed iterative numerical methods with splitting of boundary conditions intended for solving an axisymmetric Dirichlet boundary value problem for the stationary Navier-Stokes system in spherical layers are used to study the basic spherical Couette flows (SCFs) of a viscous incompressible fluid in a wide range of outer-to-inner radius ratios R/r (1.1 ≤ R/r ≤ 100) and to classify such SCFs. An important balance regime is found in the case of counter-rotating boundary spheres. The methods converge at low Reynolds numbers (Re), but a comparison with experimental data for SCFs in thin spherical layers show that they converge for Re sufficiently close to Recr. The methods are second-order accurate in the max norm for both velocity and pressure and exhibit high convergence rates when applied to boundary value problems for Stokes systems arising at simple iterations with respect to nonlinearity. Numerical experiments show that the Richardson extrapolation procedure, combined with the methods as applied to solve the nonlinear problem, improves the accuracy up to the fourth and third orders for the stream function and velocity, respectively, while, for the pressure, the accuracy remains of the second order but the error is nevertheless noticeably reduced. This property is used to construct reliable patterns of stream-function level curves for large values of R/r. The possible configurations of fluid-particle trajectories are also discussed.