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Instabilities in electrically driven rotating MHD layers

Flows of electrically conducting fluids exposed to intense magnetic fields exhibit a common feature i.e. the formation of uniform cores in which electromagnetic forces are dominant. Cores are separated from each other by thin layers that extend along magnetic field lines. Across these parallel layers strong gradients of flow variables are present, which can lead to the onset of instabilities and non-linear flow transitions. In this work we investigate dynamics and stability issues of rotating parallel layers driven by electromagnetic forces caused by the interaction of injected electric currents with an applied magnetic field. The geometry considered consists of two coaxial circular electrodes used for current injection. They are placed in parallel electrically insulating planes perpendicular to a uniform magnetic field. The basic axisymmetric steady state flow, characterized by a rotating velocity jet confined in a parallel layer that connects the rims of the electrodes, is rather well understood. By increasing the driving current above a critical value the basic flow becomes unstable and undergoes a sequence of supercritical bifurcations.


Introduction
Electrically driven magnetohydrodynamic (MHD) flows have been investigated numerically for the experimental configuration used in [1] and a test-section has been manufactured and instrumented to study MHD phenomena observed in the simulations. Electric current is injected between two coaxial, parallel circular electrodes, which are perpendicular to an externally applied uniform magnetic field. As a result of the MHD interaction of current and magnetic field, an azimuthal flow originates that is mainly confined in an annular fluid layer, which connects the rims of the electrodes. A similar type of flow can appear also in large scale phenomena typical of geo-and astrophysics [2]. The aim of the present study is getting an overview of unstable flow patterns that occur when the current injected across the liquid metal layer is increased above a critical value [3] [4].
In the past experiments and analytical studies have been carried out to investigate MHD flows, where the fluid is set in motion by electromagnetic forces generated by current injected between a pair of concentric electrodes on the bottom of an electrically insulating container [5] [6] [7]. These studies can be regarded as the electrical analogon of those performed in [8]. The established flow consists of a rotating annular core and two shear layers, aligned with the magnetic field. Here the current flows in axial direction before closing through the Hartmann layers that form along surfaces perpendicular to the magnetic field. The width  of the internal layers reduces by increasing the magnetic field strength as δ∼Ha -1/2 . The Hartmann number Ha gives a non-dimensional measure for the strength of the imposed magnetic field. This flow distribution differs from the one described in this work, where cores are stagnant and the induced stationary rotating motion consists of jets confined in an internal parallel layer. Here the maximum jet-velocity increases with Ha and the flow in the layer can become unstable. MHD flows created by injecting current through circular electrodes of finite diameter, as considered in the present study, are discussed in [1]. An approximate steady state asymptotic solution is developed and compared with experiments. For high Hartmann numbers the current is confined to the cylinder of fluid between the electrodes and when reducing the magnetic field the electric resistance of the fluid decreases due to the spread of current streamlines across the enlarged internal layer. The established azimuthal velocity distribution results from the balance between viscous and electromagnetic Lorentz forces in the layer. Discrepancies between experiments and asymptotic solution are explained by suggesting the occurrence of an axial-radial secondary flow. The contribution of the latter is not taken into account in [1]. A more rigorous asymptotic solution for the description of a steady, electrically driven shear layer is given in [9]. Experiments in [1] were carried out in a range of parameter where the flow is expected to be stable. Local electric potential was measured by using a potential probe and velocities by a Pitot tube. An attempt to investigate experimentally instabilities that occur in this type of electrically driven flows can be found in [10] [11], where instantaneous velocities are measured by means of hot-film anemometry. Experiments were reproducible only for sufficiently intense magnetic fields and data could be obtained for a limited range of parameters, 300 < Ha < 650. Considerable deviations between experiments and theoretical studies [9] were observed and explained by a deterioration of measurements due to the influence of magnetic field on heat transfer properties of the hot-film sensor and the occurrence of contributions from the radial velocity. Malcolm [11] finds that for large magnetic fields the critical current for the onset of instabilities and the corresponding Reynolds number vary as I c ∼Ha 4/3 and Re c ∼Ha 1/3 , respectively. Empirical observations point out that instability sets in as wave-like flow with steady amplitude.
Available theoretical studies of electrically driven MHD flows neglect the contribution of radial and axial velocities. In our analysis we consider this additional superimposed secondary circulation, the magnitude of which can be used to estimate the degree of anisotropy of the established MHD flow. For the described MHD flows only very few experimental results are available. Hence an experimental program has been proposed to thoroughly investigate instabilities in electrically driven rotating layers. B is applied aligned with the axis of the container. The electrodes are kept at constant electric potential so that a potential difference  is present between them. Half of the distance a between the electrodes has been chosen as characteristic size of the problem. The electrode rim is located at b/a = 0.5, where b is the electrode radius. The imposed potential difference  drives a current between the electrodes and the resulting electromagnetic forces induce a primary azimuthal flow that is practically located inside a thin parallel layer originating from the electrode rims. The shear layer is caused by the discontinuity in the electrical boundary conditions on the top and bottom of the cylinder. If Δ is smaller than a critical value a steady state solution establishes and the layer takes the shape of a velocity jet travelling in circumferential direction. The 3D velocity distribution for a stable MHD flow is depicted in Figure 1b.

Formulation of the problem
The established MHD flow can be mathematically described by equations for conservation of momentum, mass and charge that in non-dimensional form can be written as ( The variables v, p, j, B and  indicate velocity, pressure, current density, magnetic field and electric potential normalized by u₀, σu₀B²a, σu₀B, B, and u₀Ba, respectively. Here B is the magnitude of the magnetic field, and u₀ a typical velocity. The electrical conductivity σ, the density ρ and the kinematic viscosity ν of the fluid remain constant. It is assumed that the magnetic field induced by the flow is negligible compared to the imposed field. This assumption is called inductionless approximation and it is valid for small magnetic Reynolds numbers, R m = au 0 << 1, where  is the magnetic permeability [12]. The dimensionless groups in (1) are the interaction parameter N and the Hartmann number Ha: . , The former one describes the ratio between electromagnetic and inertia forces, the latter gives a non-dimensional measure for the strength of the magnetic field. Its square quantifies the ratio of electromagnetic to viscous forces. The magnitude of the potential difference Δ between the electrodes is expressed in dimensionless form by the Reynolds number Re As electric boundary conditions fixed potential values are set at the electrodes, while the external surface of the cylinder is electrically insulated, i.e.  n  = 0. The no slip condition, v = 0, is applied to all walls.

Numerical results
Data from numerical simulations are discussed for a constant Hartmann number, Ha = 588, and various Reynolds numbers. The selected Hartmann number has been chosen since for this magnetic field strength experimental and theoretical data for stationary conditions are available in literature [9]. The code used for the numerical simulations has been obtained by introducing the Lorentz force term and the equations determining electric current density and potential into the finite volume open source code OpenFOAM.

Steady solution
For Δ < 140μV (Re < 500) a steady state solution is found, which is characterized by two counter rotating layers (see Figure 1b) mainly confined in an annular region that separates an internal and an external core as schematically displayed in Figure 1a. The fluid is stagnant both in the internal and in the external cores and it moves with a jet-like profile inside the parallel layer. The 3D velocity distribution for a stable laminar MHD flow at Ha = 588 and Re = 358 is depicted in Figure 1b. In Figure 2 the scaled circumferential velocity V θ = v θ Ha 1/2 bl(σρν) 1/2 /I is plotted along the normalized radial direction at various axial positions x/a for a stable MHD flow at Ha = 588. The total driving current density is named I. The velocity is scaled according to [9]. The predicted velocity profile at x/a = 0.905 reproduces well the one from the asymptotic solution indicated by dots, as given in [9]. This suggests that, for the potential difference considered, inertia effects are negligible compared to electromagnetic and viscous forces, and the secondary flow in axial direction is much weaker than the circumferential flow, as assumed in the cited theoretical study. The magnitude of the circumferential velocity increases linearly from zero at the symmetry plane to a maximum value when approaching the Hartmann walls perpendicular to B. In the Hartmann layers the velocity reduces with a strong gradient over a distance of O(Ha -1 ) to satisfy the no slip condition at the wall.
The layer takes the shape of a velocity jet travelling in circumferential direction, whose thickness δ remains constants when it rotates, as visualized in Figure 3. Here the potential  and the axial vorticity component ω x are plotted on a plane perpendicular to the magnetic field, at an axial coordinate x/a = 0.5, at half distance between electrode and symmetry plane. Across the internal layer, in radial direction, the electric potential drops from the inner core value  c1 , which varies linearly along x, to  c2 = 0 in the outer core. The strong radial potential gradient primarily balances the induced electric field, and the potential difference  c1 - c2 through the layer is responsible for the formation of the rotating jet.
The vorticity profile along the dashed line in the middle of the geometry at z = 0 (see Figure 3b) exhibits maxima and minima as marked by green stars. The occurrence of instabilities of the flow in the layers is associated with the presence of extrema in the vorticity field that correspond to inflection points in the velocity profile.

Unstable flow regime
Simulations have been performed by increasing the applied potential difference, i.e. the Reynolds number, while holding the magnetic field constant, Ha = 588.
In Figure 4a the normalized time-averaged circumferential velocity is plotted along the scaled radial direction at x/a = 0.905 for various Reynolds numbers. By increasing Re the maximum of the azimuthal velocity reduces and shifts to higher radial positions. The negative velocity at the interface between inner core and layer progressively disappears, and a minimum occurs in the outer core. The time-averaged thickness δ of the internal layer becomes larger and the mean velocity in the jet reduces due to the increased transfer of momentum related to the occurrence of time-dependent fluctuations. This behavior is analogous to the one of unstable MHD flows in field-aligned boundary layers in ducts, although the physics behind the phenomena is different [13] [14] [15]. In Figure 4b the thickness δ of the internal layer, estimated as the ratio between the integral of the velocity profile V θ and its maximum V θmax , is plotted as a function of the Reynolds number. For 100μV<Δ<140μV, i.e. 350<Re<500, the flow is still steady and one could expect that δ remains constant. However, the internal layer thickens and flow structures extend to larger radial positions due to the occurrence of an increasing radial velocity. In the inertialess limit (Re0) the thickness of the internal layers parallel to the magnetic field should scale as Ha -1/2 . In the present problem, when secondary flow and time dependent flow structures appear, the thickness δ of the layer can depend on the interaction parameter N, or a combination of N and Ha. This is due to the fact that radial-axial flow and unstable patterns are caused by inertial forces.
For Re>500 the flow becomes unstable in agreement with experiments in [11]. The instability sets in as sinusoidal oscillations and vortices develop, which have a main vorticity component ω x in magnetic field direction.  An interesting flow feature that did not receive adequate attention in the past is the occurrence of a radial-axial flow superposed to the main azimuthal circulation. The occurrence of this secondary flow was already suggested in [1] as an explanation for discrepancies between experiments and asymptotic solution, but an exhaustive description of the phenomenon was not given. Preliminary numerical calculations [3] show that the secondary flow consists of a motion towards the electrodes in the internal part of the layer and towards the symmetry plane in the external portion. This results in a centripetal flow near the symmetry plane and a centrifugal flow at the interface between core and Hartmann layer. This secondary motion is caused by a balance of pressure, electromagnetic and inertial centrifugal forces. The pressure force points towards the inner core as visible in Figure 6 on the right, where pressure contours are depicted on a plane perpendicular to B at x/a = 0.5. The electromagnetic Lorentz force slows down the secondary flow and balances the viscous forces in layers.

Experimental set up
The conceptual design of the fabricated test-section is shown in Figure 7a. It consists of a vessel filled with the eutectic alloy GaInSn. On the electrically insulating top and bottom walls that are perpendicular to the imposed magnetic field (Hartmann walls) two circular coaxial copper electrodes with radius b are inserted flush with the internal surface of the plates. The axial length of the electrodes has been selected long enough to achieve sufficiently homogeneous current density at the fluid-solid interface. According to the numerical results, the diameter D of the outer cylinder has been defined such that the lateral wall does not interact with the layer or with unstable vortex patterns. Another constraint for the size of the vessel is represented by the dimension of the bore of the superconducting magnet at TUI (376 mm), where experiments will be performed. A reasonable choice is D = 9b. influence of the axial distance between the electrodes on onset and development of instabilities in the rotating fluid layer. This design flexibility allows also modifying the set-up of the instrumentation.
In order to preclude flow artifacts arising from symmetry deviations of the experimental configuration special care is required to ensure a precise positioning inside the magnet. The fluid vessel has to be placed concentrically inside the bore of the magnetic system.

Instrumentation
During the experimental campaign, values of local electric potential and potential differences are recorded by means of sensors located on Hartmann walls and by movable probes in the liquid metal. These data can be directly compared with numerical results. Spatial-temporal information about velocity field can be obtained by using ultrasound Doppler velocimetry (UDV) techniques. Temperature has also to be monitored at different locations on the test-section in order to calculate the physical properties of the liquid metal and to verify that experiments are carried out under isothermal operating conditions so that thermo-electric effects can be neglected. Figure 7b shows a picture of the fully instrumented test section and in Figure 8 the position of the various instruments is displayed.

Electric potential probe
In order to determine local velocities in liquid metal flows electric potential differences, induced by the motion of an electrically conducting fluid in the imposed magnetic field, are measured by means of potential probes. Their operating principle is based on Ohm's law (3), through which the velocity components perpendicular to x B can be expressed in non-dimensional form as The local velocity is determined by potential gradients and by currents induced in the fluid. Therefore, the potential difference measured at the sensor tips can be directly interpreted in terms of velocity components only if the current density is very small or known [16].
In the planned experiments it is foreseen to employ a movable potential probe with four sensing tips in order to measure the intensity of time-dependent fluctuations of potential and potential gradients. A principle sketch of the probe is shown in Figure 9 the movable electric potential probe consists of a carbon rod the diameter of which should be minimized in order to reduce the obstruction of the flow by the sensor (Figure 9(b)). Due to circumferential symmetry of the flow it is not possible to record average profiles of radial velocity, since the mean potential is constant along a circumference. However, fluctuations of the radial velocity can be measured. Detection of the onset of these oscillations, compatibly with the accuracy of measuring instruments, could be an efficient method to identify the stability limit as a function of Reynolds and Hartmann numbers. Measurements of fluctuations of the much larger circumferential velocity require the average value to be filtered out before amplifying it.

Electric potential sensors on Hartmann wall
The distribution of potential gradient on walls perpendicular to the applied magnetic field can be directly related to the flow velocity. It gives information about flow structure at the interface between Hartmann layer and core of the flow if two conditions are satisfied. First there should be no variation of the electric potential across the thin Hartmann layer, i.e. the potential at the wall coincides with the value at the core-layer interface, and second the current density at the core-layer edge is negligible. Under these assumptions measurements of electric potential on the Hartmann walls allow obtaining an estimate of velocity components on a plane orthogonal to the magnetic field. Comparison with numerical simulations can support the proper physical interpretation of recorded quantities.
In order to measure the two dimensional distribution of the electric potential on a plane perpendicular to the magnetic field, an array of sensors (copper pins) has been installed on the Hartmann wall as shown in Figure 8b. They are in electrical contact with the liquid metal. This array permits to reconstruct an image of vortex patterns, i.e. to map the electric potential distribution at the core-Hartmann layer interface. The sensors are located along radii and regularly spaced along concentric circles. The spacing between the pins is selected so that higher spatial resolution is guaranteed. The proper distance is chosen according to numerical results that allow identifying small scale recirculating flow patterns. Signal correlations between points on the same circumference allow determining the travelling speed of the waves, the wave number and therefore the number of vortices forming the fluctuating patterns.
Electric potential sensors on the Hartmann wall are also located along a circle that corresponds to the path of the movable probe (see Figure 8b). In this way it is possible to define correlations between pairs of measured electric potential fluctuations recorded simultaneously at corresponding points along the same magnetic field line.

Conclusions and future work
Numerical simulations have been performed to study the stability of rotating liquid metal layers that are driven by MHD interaction of injected currents with an imposed magnetic field. Two coaxial electrodes are inserted in insulating parallel plates, which are perpendicular to an imposed magnetic field. The basic steady state flow consists of two counter rotating layers whose velocity vanishes at the symmetry plane. The maximum velocity is reached near the Hartmann layers. The radial profile of the azimuthal velocity exhibits inflection points and for sufficiently large driving current, the flow becomes unstable.
At the onset of instability the flow pattern is characterized by a chain of periodic small vortices. By increasing the Reynolds number the vortices merge together to give rise to larger structures.
An experimental test-section has been fabricated and it will be inserted in the superconducting magnet at the technical university of Ilmenau (TUI). The aim is both complementing the numerical results and comparing the measured data with the simulations. A stability map will be defined to identify the different flow regimes depending on Reynolds and Hartmann numbers.
During the experiments electric potential will be recorded on the Hartmann walls of the cylinder as well as inside the liquid metal by means of a movable probe.