Simultaneous Measurement of Flow Velocity and Electrical Conductivity of a Liquid Metal Using an Eddy Current Flow Meter in Combination with a Look-Up-Table Method.

The Eddy Current Flow Meter (ECFM) is a commonly employed inductive sensor for assessing the local flow rate or flow velocity of liquid metals with temperatures up to 700 ∘C. One limitation of the ECFM lies in its dependency on the magnetic Reynolds number for measured voltage signals. These signals are influenced not only by the flow velocity but also by the electrical conductivity of the liquid metal. In scenarios where temperature fluctuations are significant, leading to corresponding variations in electrical conductivity, it becomes imperative to calibrate the ECFM while concurrently monitoring temperature to discern the respective impacts of flow velocity and electrical conductivity on the acquired signals. This paper introduces a novel approach that enables the concurrent measurement of electrical conductivity and flow velocity, even in the absence of precise knowledge of the liquid metal's conductivity or temperature. This method employs a Look-Up-Table methodology. The feasibility of this measurement technique is substantiated through numerical simulations and further validated through experiments conducted on the liquid metal alloy GaInSn at room temperature.

Contactless Inductive Flow Tomography for Real-Time Control of Electromagnetic Actuators in Metal Casting.

Flow control of liquid metals based on the actual flow condition is important in many metallurgical applications. For instance, the liquid steel flow in the mould of a continuous caster strongly influences the product quality. The flow can be modified by an electromagnetic brake (EMBr). However, due to the lack of appropriate flow measurement techniques, the control of those actuators is usually not based on the actual flow condition. This article describes the recent developments of the Contactless Inductive Flow Tomography (CIFT) towards a real-time monitoring system, which can be used as an input to the control loop for an EMBr. CIFT relies on measuring the flow-induced perturbation of an applied magnetic field and the solution of an underlying linear inverse problem. In order to implement the CIFT reconstructions in combination with EMBr, two issues have to be solved: (i) compensation of the effects of the change in EMBr strength on the CIFT measurement system and (ii) a real-time solution of the inverse problem. We present solutions of both problems for a model of a continuous caster with a ruler-type EMBr. The EMBr introduces offsets of the measured magnetic field that are several orders of magnitude larger than the very flow-induced perturbations. The offset stems from the ferromagnetic hysteresis exhibited by the ferrous parts of the EMBr in the proximity of the measurement coils. Compensation of the offset was successfully achieved by implementing a numerical model of hysteresis to predict the offset. Real-time reconstruction was achieved by precalculating the computationally heavy matrix inverses for a predefined set of regularization parameters and choosing the optimal one in every measurement frame. Finally, we show that this approach does not hinder the reconstruction quality.

Laboratory Investigation of Tomography-Controlled Continuous Steel Casting.

More than 96% of steel in the world is produced via the method of continuous casting. The flow condition in the mould, where the initial solidification occurs, has a significant impact on the quality of steel products. It is important to have timely, and perhaps automated, control of the flow during casting. This work presents a new concept of using contactless inductive flow tomography (CIFT) as a sensor for a novel controller, which alters the strength of an electromagnetic brake (EMBr) of ruler type based on the reconstructed flow structure in the mould. The method was developed for the small-scale Liquid Metal Model for Continuous Casting (mini-LIMMCAST) facility available at the Helmholtz-Zentrum Dresden-Rossendorf. As an example of an undesired flow condition, clogging of the submerged entry nozzle (SEN) was modelled by partly closing one of the side ports of the SEN; in combination with an active EMBr, the jet penetrates deeper into the mould than when the EMBr is switched off. Corresponding flow patterns are detected by extracting the impingement position of the jets at the narrow faces of the mould from the CIFT reconstruction. The controller is designed to detect to undesired flow condition and switch off the EMBr. The temporal resolution of CIFT is 0.5 s.

Experimental Validation of an Inductive System for Magnesium Level Detection in a Titanium Reduction Reactor.

In order to precisely determine the magnesium level in a titanium reduction retort by inductive methods, many interfering influences have to be considered. By using a look-up-table method, the magnesium level can be reliably identified by taking into account the interfering effects of the titanium sponge rings forming at the walls with their unknown geometrical and electrical parameters. This new method uses a combination of numerical simulations and measurements, whereby the simulation model is calibrated so that it represents the experimental setup as closely as possible. Previously, purely theoretical studies on this method were presented. Here, the practical feasibility of that method is demonstrated by performing measurements on a model experiment. The method is not limited to the production of titanium but can also be applied to other applications in metal production and processing.

Chaotic wave dynamics in weakly magnetized spherical Couette flows.

Direct numerical simulations of a liquid metal filling the gap between two concentric spheres are presented. The flow is governed by the interplay between the rotation of the inner sphere (measured by the Reynolds number Re) and a weak externally applied axial magnetic field (measured by the Hartmann number Ha). By varying the latter, a rich variety of flow features, both in terms of spatial symmetry and temporal dependence, is obtained. Flows with two or three independent frequencies describing their time evolution are found as a result of Hopf bifurcations. They are stable on a sufficiently large interval of Hartmann numbers where regions of multistability of two, three, and even four types of these different flows are detected. The temporal character of the solutions is analyzed by means of an accurate frequency analysis and Poincaré sections. An unstable branch of flows undergoing a period doubling cascade and frequency locking of three-frequency solutions is described as well.

Nonlinear Large Scale Flow in a Precessing Cylinder and Its Ability To Drive Dynamo Action.

We have conducted experimental measurements and numerical simulations of a precession-driven flow in a cylindrical cavity. The study is dedicated to the precession dynamo experiment currently under construction at Helmholtz-Zentrum Dresden-Rossendorf and aims at the evaluation of the hydrodynamic flow with respect to its ability to drive a dynamo. We focus on the strongly nonlinear regime in which the flow is essentially composed of the directly forced primary Kelvin mode and higher modes in terms of standing inertial waves arising from nonlinear self-interactions. We obtain an excellent agreement between experiment and simulation with regard to both flow amplitudes and flow geometry. A peculiarity is the resonance-like emergence of an axisymmetric mode that represents a double roll structure in the meridional plane. Kinematic simulations of the magnetic field evolution induced by the time-averaged flow yield dynamo action at critical magnetic Reynolds numbers around Rm^{c}≈430, which is well within the range of the planned liquid sodium experiment.

Continuation and stability of rotating waves in the magnetized spherical Couette system: secondary transitions and multistability.

Rotating waves (RW) bifurcating from the axisymmetric basic magnetized spherical Couette (MSC) flow are computed by means of Newton-Krylov continuation techniques for periodic orbits. In addition, their stability is analysed in the framework of Floquet theory. The inner sphere rotates while the outer is kept at rest and the fluid is subjected to an axial magnetic field. For a moderate Reynolds number Re = 103 (measuring inner rotation), the effect of increasing the magnetic field strength (measured by the Hartmann number Ha) is addressed in the range Ha∈(0, 80) corresponding to the working conditions of the HEDGEHOG experiment at Helmholtz-Zentrum Dresden-Rossendorf. The study reveals several regions of multistability of waves with azimuthal wavenumber m = 2, 3, 4, and several transitions to quasi-periodic flows, i.e modulated rotating waves. These nonlinear flows can be classified as the three different instabilities of the radial jet, the return flow and the shear layer, as found in the previous studies. These two flows are continuously linked, and part of the same branch, as the magnetic forcing is increased. Midway between the two instabilities, at a certain critical Ha, the non-axisymmetric component of the flow is maximum.

A gradiometric version of contactless inductive flow tomography: theory and first applications.

The contactless inductive flow tomography (CIFT) is a measurement technique that allows reconstructing the flow of electrically conducting fluids by measuring the flow-induced perturbations of one or various applied magnetic fields and solving the underlying inverse problem. One of the most promising application fields of CIFT is the continuous casting of steel, for which the online monitoring of the flow in the mould would be highly desirable. In previous experiments at a small-scale model of continuous casting, CIFT has been applied to various industrially relevant problems, including the sudden changes of flow structures in case of argon injection and the influence of a magnetic stirrer at the submerged entry nozzle. The application of CIFT in the presence of electromagnetic brakes, which are widely used to stabilize the flow in the mould, has turned out to be more challenging due to the extreme dynamic range between the strong applied brake field and the weak flow-induced perturbations of the measuring field. In this paper, we present a gradiometric version of CIFT, relying on gradiometric field measurements, that is capable to overcome those problems and which seems, therefore, a promising candidate for applying CIFT in the steel casting industry. This article is part of the themed issue 'Supersensing through industrial process tomography'.

Destabilization of rotating flows with positive shear by azimuthal magnetic fields.

According to Rayleigh's criterion, rotating flows are linearly stable when their specific angular momentum increases radially outward. The celebrated magnetorotational instability opens a way to destabilize those flows, as long as the angular velocity is decreasing outward. Using a local approximation we demonstrate that even flows with very steep positive shear can be destabilized by azimuthal magnetic fields which are current free within the fluid. We illustrate the transition of this instability to a rotationally enhanced kink-type instability in the case of a homogeneous current in the fluid, and discuss the prospects for observing it in a magnetized Taylor-Couette flow.

Experimental evidence for nonaxisymmetric magnetorotational instability in a rotating liquid metal exposed to an azimuthal magnetic field.

The azimuthal version of the magnetorotational instability (MRI) is a nonaxisymmetric instability of a hydrodynamically stable differentially rotating flow under the influence of a purely or predominantly azimuthal magnetic field. It may be of considerable importance for destabilizing accretion disks, and plays a central role in the concept of the MRI dynamo. We report the results of a liquid metal Taylor-Couette experiment that shows the occurrence of an azimuthal MRI in the expected range of Hartmann numbers.

Extending the range of the inductionless magnetorotational instability.

The magnetorotational instability (MRI) can destabilize hydrodynamically stable rotational flows, thereby allowing angular momentum transport in accretion disks. A notorious problem for the MRI is its questionable applicability in regions with low magnetic Reynolds number. Using the WKB method, we extend the range of applicability of the MRI by showing that the inductionless versions of the MRI, such as the helical MRI and the azimuthal MRI, can easily destabilize Keplerian profiles ∝r(-3/2) if the radial profile of the azimuthal magnetic field is only slightly modified from the current-free profile ∝r(-1). This way we further show how the formerly known lower Liu limit of the critical Rossby number Ro≈-0.828 connects naturally with the upper Liu limit Ro≈+4.828.

Experimental evidence for a transient Tayler instability in a cylindrical liquid-metal column.

In the current-driven, kink-type Tayler instability (TI) a sufficiently strong azimuthal magnetic field becomes unstable against nonaxisymmetric perturbations. The TI has been discussed as a possible ingredient of the solar dynamo mechanism and a source of the helical structures in cosmic jets. It is also considered as a size-limiting factor for liquid metal batteries. We report on a liquid metal TI experiment using a cylindrical column of the eutectic alloy GaInSn to which electrical currents of up to 8 kA are applied. We present results of external magnetic field measurements that indicate the transient occurrence of the TI in good agreement with numerical predictions. The interference of TI with the competing large-scale convection, resulting from Joule heating, is also discussed.

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows.

We present numerical simulations of the kinematic induction equation in order to examine the dynamo efficiency of an axisymmetric von Kármán-like flow subject to time-dependent nonaxisymmetric velocity perturbations. The numerical model is based on the setup of the French von Kármán-sodium dynamo (VKS) and on the flow measurements from a water experiment conducted at the University of Navarra in Pamplona, Spain. The principal experimental observations that are modeled in our simulations are nonaxisymmetric vortexlike structures which perform an azimuthal drift motion in the equatorial plane. Our simulations show that the interactions of these periodic flow perturbations with the fundamental drift of the magnetic eigenmode (including the special case of nondrifting fields) essentially determine the temporal behavior of the dynamo state. We find two distinct regimes of dynamo action that depend on the (prescribed) drift frequency of an (m=2) vortexlike flow perturbation. For comparatively slowly drifting vortices we observe a narrow window with enhanced growth rates and a drift of the magnetic eigenmode that is synchronized with the perturbation drift. The resonance-like enhancement of the growth rates takes place when the vortex drift frequency roughly equals the drift frequency of the magnetic eigenmode in the unperturbed system. Outside of this small window, the field generation is hampered compared to the unperturbed case, and the field amplitude of the magnetic eigenmode is modulated with approximately twice the vortex drift frequency. The abrupt transition between the resonant regime and the modulated regime is identified as a spectral exceptional point where eigenvalues (growth rates and frequencies) and eigenfunctions of two previously independent modes collapse. In the actual configuration the drift frequencies of the velocity perturbations that are observed in the water experiment are much larger than the fundamental drift frequency of the magnetic eigenmode that is obtained from our numerical simulations. Hence, we conclude that the fulfillment of the resonance condition might be unlikely in present day dynamo experiments. However, a possibility to increase the dynamo efficiency in the VKS experiment might be realized by an application of holes or fingers on the outer boundary in the equatorial plane. These mechanical distortions provoke an anchorage of the vortices at fixed positions thus allowing an adjustment of the temporal behavior of the nonaxisymmetric flow perturbations.

Paradoxes of magnetorotational instability and their geometrical resolution.

Magnetorotational instability (MRI) triggers turbulence and enables outward transport of angular momentum in hydrodynamically stable accretion discs. By using the WKB approximation and methods of singular function theory, we resolve two different paradoxes of MRI that appear in the limits of infinite and vanishing magnetic Prandtl number. For the latter case, we derive a strict limit of the critical Rossby number. This limit of Ro(c)=-0.802, which appears for a finite Lundquist number of S=0.618, extends the formerly known inductionless Liu limit of Ro(c)=-0.828 valid at S=0.

Role of soft-iron impellers on the mode selection in the von kármán-sodium dynamo experiment.

A crucial point for the understanding of the von Kármán-sodium (VKS) dynamo experiment is the influence of soft-iron impellers. We present numerical simulations of a VKS-like dynamo with a localized permeability distribution that resembles the shape of the flow driving impellers. It is shown that the presence of soft-iron material essentially determines the dynamo process in the VKS experiment. An axisymmetric magnetic field mode can be explained by the combined action of the soft-iron disk and a rather small alpha effect parametrizing the induction effects of unresolved small scale flow fluctuations.

Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos.

Using a homotopic family of boundary eigenvalue problems for the mean-field alpha;{2} dynamo with helical turbulence parameter alpha(r)=alpha_{0}+gammaDeltaalpha(r) and homotopy parameter beta[0,1] , we show that the underlying network of diabolical points for Dirichlet (idealized, beta=0 ) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, beta=1 ) boundary conditions. In the (alpha_{0},beta,gamma) space the Arnold tongues of oscillatory solutions at beta=1 end up at the diabolical points for beta=0 . In the vicinity of the diabolical points the space orientation of the three-dimensional tongues, which are cones in first-order approximation, is determined by the Krein signature of the modes involved in the diabolical crossings at the apexes of the cones. The Krein space-induced geometry of the resonance zones explains the subtleties in finding alpha profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.

Helical magnetorotational instability in a Taylor-Couette flow with strongly reduced Ekman pumping.

The magnetorotational instability (MRI) is thought to play a key role in the formation of stars and black holes by sustaining the turbulence in hydrodynamically stable Keplerian accretion disks. In previous experiments the MRI was observed in a liquid metal Taylor-Couette flow at moderate Reynolds numbers by applying a helical magnetic field. The observation of this helical MRI (HMRI) was interfered with a significant Ekman pumping driven by solid end caps that confined the instability only to a part of the Taylor-Couette cell. This paper describes the observation of the HMRI in an improved Taylor-Couette setup with the Ekman pumping significantly reduced by using split end caps. The HMRI, which now spreads over the whole height of the cell, appears much sharper and in better agreement with numerical predictions. By analyzing various parameter dependencies we conclude that the observed HMRI represents a self-sustained global instability rather than a noise-sustained convective one.

Experimental evidence for magnetorotational instability in a Taylor-Couette flow under the influence of a helical magnetic field.

A recent Letter [R. Hollerbach and G. Rüdiger, Phys. Rev. Lett. 95, 124501 (2005)] has shown that the threshold for the onset of the magnetorotational instability in a Taylor-Couette flow is dramatically reduced if both axial and azimuthal magnetic fields are imposed. In agreement with this prediction, we present results of a Taylor-Couette experiment with the liquid metal alloy GaInSn, showing evidence for the existence of the magnetorotational instability at Reynolds numbers of order 1000 and Hartmann numbers of order 10.

Asymmetric polarity reversals, bimodal field distribution, and coherence resonance in a spherically symmetric mean-field dynamo model.

Using a mean-field dynamo model with a spherically symmetric helical turbulence parameter alpha which is algebraically quenched and disturbed by additional noise, the basic features of geomagnetic polarity reversals are shown to be generic consequences of the dynamo action in the vicinity of exceptional points of the spectrum. This simple paradigmatic model yields long periods of constant polarity which are interrupted by self-accelerating field decays leading to asymmetric polarity reversals. It shows the recently discovered bimodal field distribution, and it gives a natural explanation of the correlation between polarity persistence time and field strength. The dependence of the persistence time on the noise shows typical features of coherence resonance.

Integral equation approach to time-dependent kinematic dynamos in finite domains.

The homogeneous dynamo effect is at the root of cosmic magnetic field generation. With only a very few exceptions, the numerical treatment of homogeneous dynamos is carried out in the framework of the differential equation approach. The present paper tries to facilitate the use of integral equations in dynamo research. Apart from the pedagogical value to illustrate dynamo action within the well-known picture of the Biot-Savart law, the integral equation approach has a number of practical advantages. The first advantage is its proven numerical robustness and stability. The second and perhaps most important advantage is its applicability to dynamos in arbitrary geometries. The third advantage is its intimate connection to inverse problems relevant not only for dynamos but also for technical applications of magnetohydrodynamics. The paper provides the first general formulation and application of the integral equation approach to time-dependent kinematic dynamos, with stationary dynamo sources, in finite domains. The time dependence is restricted to the magnetic field, whereas the velocity or corresponding mean-field sources of dynamo action are supposed to be stationary. For the spherically symmetric alpha2 dynamo model it is shown how the general formulation is reduced to a coupled system of two radial integral equations for the defining scalars of the poloidal and toroidal field components. The integral equation formulation for spherical dynamos with general stationary velocity fields is also derived. Two numerical examples--the alpha2 dynamo model with radially varying alpha and the Bullard-Gellman model--illustrate the equivalence of the approach with the usual differential equation method. The main advantage of the method is exemplified by the treatment of an alpha2 dynamo in rectangular domains.