Measurement of velocity in swirling flows of liquid metals in the presence of a magnetic field

Studies of the dynamics of a round submerged jet of liquid metal in a transverse magnetic field. The experiments were performed in the range of Reynolds numbers up to 1000 and Hartmann numbers up to 1300 using a probe technique for measuring velocity using potential sensors. Detailed three-dimensional flow structures are visualized using a direct numerical simulation method independent of the experiment. The paper presents data on the averaged velocity field and statistical characteristics of the flow, reproducing the complex nature of a substantially unsteady flow, in which zones of reverse fluid flows are detected and a significant heterogeneity of the velocity field is observed.


Introduction
Electrically conductive liquids interacting with a magnetic field are an essential element of many industrial processes, for example, liquid metal circuits in thermonuclear reactors [1], control in crystal growing technologies [2], continuous casting of steel in metallurgical processes [3], liquid metal batteries [4], etc. Ensuring the feasibility of such processes requires a fundamental understanding of the influence of the magnetic field on the flow of an electrically conductive liquid.
Recent studies show that the magnetic field completely changes the properties of the flow [5]. At high values of similarity criteria, such as the Reynolds or Rayleigh number, flows become unstable even in very strong magnetic fields, which leads to the development of large-scale, quasi-two-dimensional disturbances dominating flows, such as, for example, shear layers or jets. The behavior of such structures is often unsteady and leads to high-amplitude pulsations of velocity, pressure and temperature. These phenomena are poorly studied and have both fundamental and practical significance. Quasi-twodimensional structures completely change the mode of heat and mass transfer, and can lead to the formation of stagnant zones, reverse flow areas, concentrated hot or cold jets, abnormal temperature fluctuations, cyclic mechanical stresses in the walls, etc.
Of particular interest are jet streams that occur, for example, in streams with sudden expansions or during mixing, heating, or cooling of liquid. There are factors that limit the use of experimental methods for studying such tasks. The opacity of liquid metals practically excludes the use of optical measurement and control methods in the flow. Therefore, contactless methods of measuring velocity and temperature  [6]) are considered the most preferable, however, at the current level of development, these methods do not make it possible to obtain three-dimensional velocity and temperature fields. An alternative is contact methods, when various miniature sensors are immersed into the flow (microthermocouples; correlation, electromagnetic and fiber-optic velocity sensors; thinfilm thermoanemometers) and probes with coordinate mechanisms for local measurements, adapted, as a rule, for low-temperature or isothermal three-dimensional measurements.
This paper discusses the features of using a four-electrode conduction anemometer (potential sensor) to measure the flow characteristics of a circular submerged jet in a transverse magnetic field. This technique, which was developed back in the 70s of the last century [8], is based on the generalized Ohm's law, according to which an electric field arises in the flow of an electrically conductive medium moving in a magnetic field Therefore, under these conditions, the local components of the velocity vector field V{Vx,Vy,Vz} can be calculated if the gradient of the electric field potential φ and the current density j are measured locally. In the external applied transverse magnetic field B{0,By,0} (figure 1a), then the following relations will be valid for the components of the electric potential gradient: Further, as it was shown in [8], it is possible to obtain an approximation away from the walls and in a sufficiently strong magnetic field, according to which, in equations (2) and (4), the second terms in the right part can be neglected and considered: from where, by estimating the gradient of the electric potential, it is possible to determine the components of the flow velocity that are perpendicular to the induction of the magnetic field. Subsequent studies, however, have shown that to ensure acceptable accuracy, this method requires special calibration and compliance with a number of conditions in the experiment [9].  Note that directly determining the velocity component , coinciding in the direction with the induction of the magnetic field, is problematic, nevertheless, statistical characteristics associated with this component can be obtained using current fluctuations measurement data . Moreover, as shown in the work [9], using the data of measurements of the potential gradient, it is possible to estimate the statistical parameters of turbulence in the flow.
A conduction anemometer can have different design options that implement (with some variations) the same idea. Figure 1b, for example, shows one of these options, a four-electrode sensor, involving the measurement of two components of the velocity vector transverse to the externally applied magnetic field: The influence of induced currents and on the error of determining the velocity components when measuring the electric potential is determined by the coefficients с1 and c2. With forced flow in the channel, away from the walls and at small values of the external applied magnetic field B (Ha=Ba(σ/η) 1/2 <100, where a is the half-height of the channel, σ is the electrical conductivity, η is the dynamic viscosity coefficient), the values of the coefficients can be determined by calibration, and at large values B (На>100), their values can be taken equal to one, while the expected uncertainty of velocity measurements will not exceed 10%. Nevertheless, the issues of applicability of this technique for measurements in substantially unsteady flows, for example, during flow in jets or sudden channel expansions, as well as the influence of channel walls on measurement accuracy, remain not fully investigated, what have motivated this study.

Problem formulation
The scheme of the flow under consideration is shown in figure.2. The submerged jet is formed when liquid metal flows out of a tube with a diameter of 6 mm in a channel 56x56 mm located in the region of a electromagnet that creates a magnetic field transverse to the flow. The probe, with a conduction anemometer fixed at its end (see figure. 3), is inserted into the flow against the flow, fixed in the selected cross-section and, using coordinate mechanism, can move in a plane perpendicular to the magnetic field, thus measuring the flow velocity profile. To measure the readings of the conduction anemometer, 3xNI-4071 modules are used, which record rapidly alternating pairwise signals from the electrodes in a time-synchronous mode within the framework of an automated measuring PXI system. One of the stages of optimization of the method of velocity measurements using a conduction anemometer can be a comparison of the experimental results with the results of direct numerical simulation (DNS) of the flow in the configuration under consideration. DNS is performed using a technique calculated for rectangular and cylindrical geometries, which has shown its effectiveness in modeling MHD flows at high Reynolds and Hartmann numbers [10], as well as submerged jets in a transverse magnetic field [11]. In a strong magnetic field (Ha = 1000), the effective length of the jet decreases, it loses stability at a sufficiently small distance from the entrance (figure 5) and tends to "stick" to one of the walls of the channel with the formation of a return circulation movement.

Problem formulation
The structure of the swirling flow of a submerged jet is successfully reproduced in an experiment using a conduction technique for measuring the flow velocity. For example, figure 6 shows a dimensionless comparison of the measured fields of the averaged longitudinal and transverse velocities in the vertical section x/a=1.6 with the calculation data. Their good correspondence can be seen,

Conclusions
The results of velocity measurements using a conduction anemometer are compared with the results of direct numerical simulation of submerged jet flow and give a good correspondence both in terms of averaged and statistical characteristics of the flow, reproducing the complex nature of a substantially