Induction in a von Karman flow driven by ferromagnetic impellers

We study magnetohydrodynamics in a von K\'arm\'an flow driven by the rotation of impellers made of material with varying electrical conductivity and magnetic permeability. Gallium is the working fluid and magnetic Reynolds numbers of order unity are achieved. We find that specific induction effects arise when the impeller's electric and magnetic characteristics differ from that of the fluid. Implications in regards to the VKS dynamo are discussed.


I. INTRODUCTION
The choice of electromagnetic boundary conditions in dynamo studies and its implication on the underlying magnetohydrodynamic processes has long been an open issue. In experiments, a finite flow of an electrically conducting fluid is generally considered, and the outer medium is ultimately electrically insulating (with electromagnetic properties of vacuum).
Studies have considered layers with various properties regarding electrical conductivity σ w and / or magnetic permeability µ w . Motivations for this have varied considerably. In many numerical simulations, it has been computationally convenient to assume an infinite value of µ w , yielding an attachment of magnetic field lines normal to the wall [1]. In some simulations of the geodynamo, a thin wall approach [2] has been implemented in order to account for the rapid changes in electrical conductivity at the Core-Mantle Boundary [3,4]. Experiments have tend to study extensively changes with boundary conditions. The motivations is that the critical magnetic Reynolds number value for the onset of dynamo action must be lowered as much as possible in order to lie within reach of a given experimental facility.
For instance, in the pioneering experiments in Riga and Karlsruhe a volume of stagnant liquid sodium around the flow was found to be very favorable [5,6]. In the same spirit, dynamo action in the von Kármán sodium (VKS) experiment [7] has only been observed in situations where the fluid is driven by the motion of soft iron impellers. When replaced by non-magnetic stainless steel impellers, no dynamo was generated at the highest stirring achievable with the mechanical drives. In addition, the dominant VKS dynamo mode is essentially an axisymmetric dipole, in contradiction with predictions based on the mean flow structure [8,9]. It is the motivation of this work to understand better the influence of the boundary conditions imposed by the ferromagnetic impellers, in particular whether they change significantly the induction processes or simply lower the threshold of a dynamo that would be reached at higher rotation rates of non-magnetic impellers, if that was possible.
To this end, we have performed extensive induction measurements in a gallium von Kármán flow driven by impellers made of stainless steel, copper or soft-iron.
The experimental device is first described. Induction measurements are then presented and detailed in section III. A physical interpretation of these measurements is developed in section IV. In the last section, we present a discussion of the impact of these mechanisms on the VKS experiments.

II. EXPERIMENTAL SETUP
A von Kármán flow is produced by the coaxial rotation of two impellers inside a stainless steel cylindrical vessel filled with liquid galliumc.f. figure 1(a). The cylinder radius R is 97 mm and its length is 323 mm. The impellers have a radius equal to 82.5 mm and are fitted to a set of 8 straight blades with height 10 mm. The impellers are separated by a distance H = 203 mm. They are driven by two AC-motors which provide a constant rotation rate in the interval (F 1 , F 2 ) ∈ [0.5, 25] Hz.
In most of the cases, the flow is driven by symmetric counter-rotation of the impellers at F 1 = F 2 = F . It is structured in two rotating cells in front of each impeller, separated by a large azimuthal shear layer in the mid plane of the cylinder. Driving can also be achieved using the rotation of one single disk, the other being at rest; the flow then consists in a single rotating cell. In both cases, the fluid is also ejected radially outward by the rotating(s) disk(s); this drives an axial flow toward them along the cylinder axis and a recirculation in the opposite direction along the cylinder lateral boundary.
The integral kinematic and magnetic Reynolds numbers are defined as Re = 2πR 2 F /ν and R m = 2πR 2 F/λ. By varying F , values of R m up to 5 are achieved, with corresponding Re values in excess of 10 6 .
Two pairs of induction coils (in a Helmoltz configuration) are aligned parallel to the rotation axis, or perpendicular to it. They create an applied magnetic field B A of a few ten gauss inside the vessel, essentially uniform in the direction of the axis of the coils.
The interaction parameter N = σ(B A ) 2 /2πρF ∼ 10 −5 is small, so that the action of Lorentz forces on the flow can be neglected. Magnetic induction measurements are performed using a Hall sensor probe array (8 probes) inserted into the flow in the mid-plane. Data are recorded using a National Instrument PXI-4472 digitizer at a rate of 1000 Hz with a 23 bits resolution.
The impellers driving the flow are made of a flat disk upon which straight blades are fixed. In the study reported here, both the disk and the blades can be chosen independently   we checked experimentally and numerically -using FEMM 4.2 software [11] -that, with liquid gallium at rest and in the case of soft-iron impellers, the effect of the high permeability of the material produces only a weak distortion of the magnetic field applied by the coils at the probe location as compared to stainless steel impellers.

III. INDUCTION LINKED TO DIFFERENTIAL ROTATION
The swirling flow generated by the exact counter rotation of the impellers (F 1 = F 2 = F ) possesses a large shear layer in the mid-plane of the cylinder [13]. Induction effects are generated by this azimuthal shear. We study the variation of their amplitude and radial profile, as the impellers are counter-rotated with increasing rotation rate F and the impeller material varied.

A. Induction from an axial applied field
We first investigate the case where the applied field is coaxial to the cylinder: B A = B A zẑ . This configuration is sketched in figure 1 (a). When stainless-steel impellers are used, the induced magnetic field along the azimuthal direction, denoted B I θ , is due to the ω-effect, and proportional to the fluid differential rotation ∂ z v θ [14,15,16]. their maximum value in order to compare the induction profiles. We observed that they do not change noticeably when the material of the impellers is varied. The profiles are also independent of the impellers rotation rate. Figure 2(b) shows the evolution of the maximum amplitude of these radial profiles with the rotation rate F . One observes that the impeller material has a very weak influence here. Induction can be understood as the usual ω-effect due to the fluid motions (linear in the flow forcing and proportional to the time-averaged fluid differential rotation ∂ z v θ ) with no noticeable influence of the impellers material. Note that all field amplitudes have been normalized to the applied field measured at the probe location, thus taking into account the weak local distortion of the applied field lines in the case of the soft-iron impellers (the variation is weak -less than 10 %).
Given the very high Reynolds number value of the flow, the induced magnetic field has a turbulent signature, with fluctuations as high as the time-averaged amplitude (see for instance [14]). The evolution of the rms amplitude of the fluctuations as a function of the impellers rotation rate is displayed in figure 3(a). It is linear in F and the dispersion between In this section, we address the issue of induction from a transverse applied field: xx . This configuration is sketched in figure 1(b). The induced magnetic field is probed along the axial direction, B I z . For the case of a flow driven by stainless-steel impellers and enclosed in a stainless-steel vessel, the induction processes at work here have been described in details in [15,16] (in particular, see figure 13 in [16]) and will only be recalled here. Near the impellers, the flow rotational motion advects the imposed magnetic field B A x , so as to induce a perpendicular component B I y . With the rotation opposite on either side of the flow mid-plane, two such contributions (with opposite directions) are generated on each side. These induced fields are associated to a current distribution consisting of two current sheets in the direction of the applied field, one in each flow cell, plus one with opposite direction in the shear layer. Since the outside medium is electrically insulating, these currents loop back inside the flow volume. In doing so, they generate an axial field B I z , maximum in a plane transverse to the applied field. This effect, directly linked to the rotation of the fluid, is linear in F . As in [16] we will refer to this induction process as a BC-effect in the what follows, since it originates in the Boundary Conditions from the steep variation in electrical conductivity at the flow wall. A simplified way to understand this effect is to consider the B I z component in the mid-plane as resulting from the loop-back path of the B I y contributions on either side. As before, we study modifications of this induction process resulting from changing the impellers. For the three materials used, the radial profile of the normalized time-averaged induced axial field B I z is shown in figure 4(a). These normalized profiles are again independent of impellers rotation rate and impellers materials. Since the profiles are unchanged, one may suppose that a similar BC mechanism can be invoked in all cases.
In addition, in the low R m regimes accessible in the device, the BC-effect remains linear with R m whatever the impellers material. However, as seen in figure 4(b), the amplitude of the time-averaged induced field now strongly depends on the impellers materials: a 20% We argue in section IV below that the increase in the mean induced component B I z arises in a process similar to the one described in [16] from a localized ω-effect generated when the magnetic diffusivity of the impellers differs from that of the fluid.

IV. INTERPRETATION
A. An induction mechanism associated to a jump of magnetic diffusivity at the impellers When probing induction effects in the mi-plane, our observations so far can be summarized as follows: -in the case of an axial applied field (induction from differential rotation in the fluid bulk), the induced field seems to be fairly independent of the impeller material.
-in the case of a transverse applied field (induction from differential rotation, coupled with insulating boundary conditions), the time-averaged amplitude of the induced field varies strongly with impeller material, -in both cases, the spatial profile of the induced field is independent of the impellers nature, and the fluctuations of induction are not appreciably changed.
In order to interpret these observations, we need to understand the role of variations of magnetic diffusivity at the interface gallium-impellers on induction effects. We begin with the case of the transverse applied field, which shows the strongest dependence with the impellers nature. Herzenberg and Lowes have studied the case of a finite cylinder rotating at frequency Ω in a uniform field perpendicular to the cylinder axis. To leading order equation 4.12 of [19] for the induced magnetic field writes: where (r, θ, φ) are spherical coordinates with origin at the center of the cylinder of radius a and V cyl the volume of the cylinder. In the above derivation the cylinder and medium are assumed to have the same magnetic diffusivity, and the medium outside the cylinder is at rest. This induced field has approximately the geometry of a dipole with its axis perpendicular both to the axis of rotation and to the applied field. Its amplitude can also be written as B I ∝ B A R cyl m (V cyl /r 3 ), with R cyl m = µσ(2πaΩ)a the magnetic Reynolds number of the rotating cylinder. It corresponds to a (radial) ω-effect, due to the shearing of the magnetic field lines which are 'advected' by the cylinder rotation; it is also the first step of the expulsion of the transverse field from the rotating body [20]. If we now consider a situation where the cylinder is made of a material with a different magnetic diffusivity, a similar induced field will be generated, even if the fluid around the cylinder rotates with the same velocity, i.e. if there is no differential rotation. This is because a change in magnetic permeability or a change in electrical conductivity also contribute to the induction processes.
For instance if the cylinder has a higher permeability, then the higher values of the flux of B in the cylinder will generate a shear of field lines at the fluid boundary. If the cylinder has a better electrical conductivity, then the same flux variation will be able to generate more induced currents. Altogether it is a change in the magnetic diffusivity µσ that matters here. These arguments, supported by recent numerical works [21], have lead us to write the correction to homogenous induction effects, in a first approximation, as for the induction generated by a local jump of either magnetic diffusivity or velocity at the cylinder end.
We then return to our induction measurements, as reported in the previous section.
In the case of an applied field transverse to the flow axis, the rotation of each impeller creates an induced perpendicular dipole (as above). As in the BC-effect, the reconnection of these induced fields and the constraints on currents paths create an axial component in the (yOz) plane. This field will add to the axial field induced by the usual BC mechanism.
This explanation accounts for the very similar induction profile observed (see figure 4(a)) with different materials used for the impellers: the additional effect due to the impellers is produced by a mechanism with a geometry identical to that of the BC-effect. It also explains why the impeller material have so little effect on the fluctuations of induction: the source of induction lies in the rotation of the impellers which is precisely controlled in the experiment to be constant in time, fixed to a prescribed value. Fluctuations are essentially due to changes in the induction generated by the velocity gradients within the flow, which have a quite non-stationary (turbulent) dynamics [18].
We thus write the axial induced field measured in the mid-plane as being the sum of two contributions: (i) the regular BC-effect which originates from the differential rotation of the fluid and (ii) the effect described above, caused by the change in magnetic diffusivity at the impellers: where K f,i are constants taking into account the precise location of the measurement, and the second term has been chosen to emphasize two features: (i) no additional effect occurs when the impellers have a magnetic diffusivity which matches that of the fluid (all is then incorporated in the first term); (ii) the ratio V i /V max i accounts for the fact that at each impeller, disk and blades can be made of materials; V i max is the maximum volume of disk plus blades, and V i is the actual volume of a given material in the impeller (which can be disk and/or blade) so that the effective volume fraction is less than one -V i < V max The combinations used are indicated in the figure, with the first mention corresponding to the disc and the second to the blades. "SS" stands for stainless steel, "Fe" for iron and "Cu" for copper.
The blue circles correspond to the case of a transverse applied field and the red triangles to the axial applied field.
By varying separately blade and disk materials, and performing measurements for a range of rotation rates of the impellers, one may thus probe the above relationship.  there is a non negligible uncertainty in computing the actual volume of the impellers that plays a role in the induction effect. This is because the blades are indeed attached with stainless steel screws that may prevent the optimal development of electrical currents in more conducting materials.
The inhomogeneity of magnetic diffusivity at the impellers is also expected to produce an added induction in the case of an axially applied field -in this case, however, a velocity shear at the fluid-impeller interface is necessary (otherwise no variation of flux can cause induction). Using again the calculation by Herzenberg and Lowes [19] as a reference, we expect a variation of the azimuthal induced field with disks and blades materials. Figure 6 (red triangles) shows that it is indeed the case, but the evolution is much shallower. The explanation lies in the fact that the effect is localized in the vicinity of the impellers -it decays as the inverse third power of the distance to the disk [19](equation 3.21)) and, in this case, there is no global effect of the boundary condition that constrains the currents and help ensure that a sizable magnetic field can be detected in the mid-plane of the flow (where the measurement probes are located).
We thus find that our modelization, which attributes the induction to conventional effects from the shearing motion of the fluid plus an additional contribution due to the inhomogeneity of the magnetic diffusivity of the impellers compared to that of the fluid, gives an adequate interpretation of our experimental data. Our measurements correspond to (F 2 = 0 and F 1 = 0 in the configuration sketched in figure 1(b)). The applied field is transverse, B A = −B A xx . The evolution of the time-averaged induced field B I z measured in the mid-plane is shown in figure 7(a,b) for the most inner probe in the array (r/R = 0.15) and the most outer probe (r/R = 0.87). As can be seen, in the center of the flow, the induction varies quadratically with the impeller rotation rate, in a manner which is independent of its material. On the other hand, nearer the outer wall, the evolution, initially quadratic, becomes linear when soft iron impellers are used.
In all cases, the fluctuations shown in figure 7(c,d) show no dependence with the impeller material. These features are consistent with the interpretation given in the previous section. Finally, as this additional contribution is associated with the (controlled) motion of the impellers, it has much less fluctuations than the induction originating solely from the flow velocity gradients.
Such effects may have a significant impact on dynamo processes in laboratory experiments. They could be quite important in the generation of the VKS dynamo [7]. Expanding on our observations, one may expect that the combination of velocity shear and magnetic diffusivity discontinuity at the impellers generates a quite strong ω-effect in the vicinity of the soft iron impellers. As a result, an axial magnetic field would be efficiently converted into a toroidal field in this region. For the regeneration of the axial field, several types of α-effects have been proposed [8]. In the above scenario, the VKS dynamo generates an axial dipole from α − ω processes which are both localized in the vicinity of the ferromagnetic impellers, which also has the following major contributions: -their large µσ value and associated enhancement of induction effects (as measured in section III B) helps bring a dynamo threshold within the range of R m values accessible to the experiment, -they promote an axial magnetic field. It would help understand why the actual dynamo field generated with soft iron impellers is dominated by an axial dipole, in contrast with numerical simulations made for homogeneous magnetic conditions which predicted a transverse dipole, -the localization of dynamo sources near the impellers may help understand why the evolution of the VKS dynamo field shares many features with low dimensional chaos [23], as would result from non-linear interactions of two weakly coupled dynamos.
Further measurements will of course be needed. They involve the precise measurement of the magnetic fields in the very vicinity of the impellers -an endeavor that requires major changes in the experimental setup and involves several technical challenges. Detailed induction measurements in the sodium (VKS) experiment would also be needed to clarify the evolution with the magnetic Reynolds, as the current gallium studies are restricted to R m values of order unity.