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Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system
Flow Measurement and Instrumentation ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system Sven Franke n, Dirk Räbiger, Vladimir Galindo, Yunhu Zhang, Sven Eckert Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 21 March 2015 Received in revised form 17 September 2015 Accepted 18 September 2015
This paper presents a combined experimental and numerical study of the properties of a liquid metal flow inside a cylinder driven by the application of a strong electrical current. The interaction between the electric current running through the melt and the corresponding induced magnetic field produces socalled electro-vortex flows. We consider here a configuration of two parallel pencil electrodes immersed at the free surface. Velocity measurements were performed by means of the Ultrasound Doppler method. A linear array of 25 singular transducers was used to determine the two-dimensional pattern of the vertical flow component. Numerical simulations of the magnetohydrodynamic (MHD) problem were conducted to calculate the Lorentz force, the Joule heating and the induced melt flow. Experimental and numerical results reveal a complex three-dimensional flow structure of the liquid metal flow. In particular, two pronounced downward jets are formed below both electrodes. The flow structure appears to be symmetrical with respect to two vertical cross sections being perpendicular to each other and one of the two planes contains the electrodes. The comparison between the experimental data and the numerical results shows a very good agreement. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Liquid metal flows Electrically driven flows Velocity measurements Ultrasound Doppler method Ultrasonic transducer array
1. Introduction Liquid metal technologies gain in importance for the metal industry, for example, the production of cast parts, the continuous casting of steel and the crystal growth at the semiconductor industry. As well the role of liquid metal technologies grows steadily for the branch of energy industry. Present and future examples may be found at the nuclear power industry, more precisely at fast breeder power plants and fusion reactors. However, also renewable energies may benefit from liquid metal technologies, for example with respect to the application of liquid metal coolants at concentrated solar power plants. Applications in the energy sector typically utilize liquid metals as coolant, however, future prospects consider liquid metals also for energy storage systems. Safe and reliable operation of such liquid metal systems, optimization of industrial processes and the guarantee for best product quality require a comprehensive knowledge of fluid flow phenomena and related transport processes. Numerical simulations could provide a better understanding of complex flow behavior, but, experimental data are indispensable with respect to a validation of the respective CFD codes. The determination of flow n
Corresponding author. E-mail addresses: [email protected] (S. Franke), [email protected] (D. Räbiger), [email protected] (V. Galindo), [email protected] (Y. Zhang), [email protected] (S. Eckert).
quantities in liquid metals is considerably impeded by the special material properties. Powerful optical methods as used for measurements in transparent liquids are obviously not applicable in molten metals. For the last two decades the ultrasound Doppler method became a very powerful tool to investigate the velocity structure in liquid metal flows as reported for various metallic fluids [1–3]. Originating from the medical branch the pulse-wave ultrasound Doppler method has been established for fluid flow measurements in physics and engineering by the pioneering work of Takeda [4]. The measuring principle is based on the pulsed echo technique. Ultrasonic pulses of a few cycles emitted from an acoustic transducer propagate into the fluid along a measuring line which is identical to the continuation of the transducer axis. A part of the ultrasonic pulse energy is scattered by microparticles suspended in the liquid. Their echo signal is received by the same transducer within the time period between two emissions. A short sequence of such echo signals contains the entire information of the velocity profile along the ultrasonic beam. Knowing the sound velocity of the liquid, the axial position of the scattering particles along the measuring line is determined from the measured time span between the burst emission and the reception of the respective echo signals. The movement of the scattering particles inside the measuring volume between two consecutive bursts will result in a small time shift of the echo signal. A correlation analysis between the echo signals of consecutive bursts reveals the flow velocity
http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004 0955-5986/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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S. Franke et al. / Flow Measurement and Instrumentation ∎ (∎∎∎∎) ∎∎∎–∎∎∎
profile for all positions along the measuring line. Owing to the Nyquist theorem, the product of measurable maximum velocity and penetration depth is limited by the sound velocity and the ultrasonic frequency. Ultrasonic methods are non-invasive, but not contactless since a continuous acoustic path from the ultrasonic transducer to the fluid under investigation is required. For a more detailed description of the basics of the measuring principle the reader is referred to Takeda [5]. The standard ultrasound Doppler systems measure linear velocity profiles, however, the enhancement of the capabilities towards a multidimensional flow mapping with high frame acquisition rates and spatial resolutions would be exceedingly desirable for examinations of highly turbulent, three-dimensional flows such as occurring during electromagnetic stirring of metals. Detailed investigations of such kind of complex flows and a validation of respective numerical simulations require a multidimensional acquisition of the flow field with a reasonable temporal and spatial resolution. Imaging techniques becomes more and more important for detailed explorations of three-dimensional turbulent flows, in particular with respect to the generation of a suitable experimental data base for an efficient validation of respective numerical simulations. First approaches of a flow mapping with a multiline ultrasound Doppler system were performed by Takeda [6] who measured the mercury flow in a liquid metal target of a spallation neutron source. Mapping rates about 2 Hz were realized while applying up to 12 measuring lines. More recently, measurements of a submerged liquid metal jet have been reported by Timmel et al [7]. A horizontal adjustment of 10 transducers provided a two-dimensional visualization of the horizontal velocity component utilizing a 10-channel commercial ultrasonic Doppler device. However, the use of a finite number of singular sensors by sequential sampling restricts the spatial and temporal resolution of the measurements. New measuring systems are under development using specific combinations of linear sensor arrays and have recently been demonstrated in liquid metal flows driven by a rotating magnetic field [8]. In the present paper we demonstrate the capabilities of an array measuring system using the example of a liquid metal flow driven by a strong electric current. The application of electric currents during the solidification of metal alloys was found to be beneficial for achieving superior mechanical properties of the solidified material. The improvement of the product quality is obtained by a distinct grain refinement [9,10]. However, the physical mechanism of the grain refinement has not been understood so far. Various effects are under discussion, such as the fragmentation of dendrites induced by the electric current [11], the reduction of the nucleation activation energy [12], or the break out and the transport of little grains from the boundary by the periodic Lorentz force [13]. Most of the previous studies did not consider the possibility that significant melt flows can be created by an intense Lorentz forces which results from the interaction between the strong electrical current and the self-induced magnetic field. The effect of such kind of electro-vortex flows on the solidification of Al–Si alloys was investigated by a recent study [14]. This paper presents an extension of this experimental and numerical study focusing on the characteristics of the forced melt flow induced by the strong electric currents in an isothermal melt without solidification. A set of experiments was conducted using the eutectic alloy GaInSn at room temperature to obtain quantitative information about the isothermal flow field. The mean flow structure as well as the time-varying flow field are measured and compared to predictions obtained by corresponding numerical simulations. The paper is structured as follows: The flow problem under consideration and respective numerical simulations are briefly described in the following section. A description of the experimental
setup and the measuring system is given in Section 3. Section 4 contains the presentation and discussion of the measuring results. Some experimental findings are compared with corresponding numerical predictions. The concluding remarks can be found in Section 5.
2. Formulation of the problem and numerical scheme We consider a finite cylinder with radius R0 and height H0 filled with a liquid metal. Two parallel pencil electrodes with electrically insulated lateral surfaces are immersed into the melt through the free surface and arranged symmetrically with respect to the cylinder axis. The application of an electric potential between the electrodes produces an electric current which closes through the liquid metal between the electrodes. Any electric current through an incompressible, viscous and electrically conducting liquid is accompanied by corresponding magnetic field. In general, the interaction between the applied electric current and the induced magnetic field creates a Lorentz force J × B which may drive a fluid motion. The simplest configuration of a homogeneous current distribution through a cylindrical column becomes unstable when the axial current exceeds a critical value being specific for a given fluid and geometry. Recently, this phenomenon known as Tayler instability [15] was experimentally observed and numerically analyzed [16,17]. The experimental assembly of two pencil electrodes as considered within this study implies a non-homogeneous distribution of the electric current (see Fig. 1(a)). Such a configuration shows a significant rotational component of the Lorentz force and does not feature a threshold value of applied current which has to be exceeded before a flow sets in. Convection starts as early as a weak electric current is flowing through the liquid metal. Numerical computations have been performed to calculate the actual distributions of electric current and magnetic field as well as the resulting Lorentz force. Furthermore, these simulations provide predictions of the flow field in the liquid metal column. The finite element code OPERA (Cobham plc.) was used to compute the electric current J and the magnetic induction B distributions in the melt and all electrically conducting parts of the facility including the immersed electrodes. In case of a direct current imposed on the electrodes we solve the Laplace equation for the electric potential ∇2φ = 0 taking into account the charge conservation ∇⋅J = 0 and the continuity of the current density J at the interface between two regions with different electric conductivities: Jn = − σ1n⋅∇φ1 ¼ −σ 2 n⋅∇φ2. The magnetic induction B was calculated from the current density J = σ ( − ∇φ) using the Biot–Savart law:
B (r ) =
μ0 4π
∫ d3r′ J (r′)|r ×− (rr′|−3 r′)
(1)
The boundary conditions for the electric potential are defined using the amplitude of the imposed electric current at the end of the electrodes: I = ∫ ds⋅J , where A is the area of the electrode’s A
cross section. The flow in the volume containing the melt was simulated numerically by means of the open source library OpenFOAM [18] solving the Navier–Stokes equation together with the incompressibility condition ∇⋅u = 0 and including an electromagnetic force density term:
⎛∂u ⎞ ρ⎜ + (u⋅∇) u⎟ = − ∇p + η∇2u + J × B ⎝ ∂t ⎠
(2)
The boundary conditions for the flow field are the no-slip condition u = 0 at the solid container walls. For the melt surface
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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B
F J
F [m/s²] 0.5 0.4 0.3 0.2 0.1 0
Fig. 1. Flow field development in given setup: (a) simplified model of the induced Lorentz forces, (b) computed Lorentz force distribution and (c) snapshot of corresponding three-dimensional flow structure from the numerical simulation (predicting a downward-directed jet flow below the electrodes and a upward-directed bulk flow).
either u = 0 or the conditions for a stress-free, non-deformable surface un =0 and ∂ ut / ∂ z = 0 are applied depending on whether the melt flow is evaluated in an open or an enclosed container. A computational grid with 650000 volume elements was used. Fig. 1(b) shows the distribution of the Lorentz force for an electric current of I ¼48 A. It becomes obvious that a relevant Lorentz force only occurs just below the electrodes. This phenomenon can be explained by the distribution of the electric current. The current distribution diverges just beneath the electrodes almost in radial direction with respect to the axis of the electrode. The interplay with the azimuthal magnetic field around the electrodes results in a significant downward component of the electromagnetic force. The consequence is the formation of downward jets below the electrodes as shown in Fig. 1(c). The numerical simulations also reveal the three-dimensional structure of the global flow field. Besides the two downward jets a recirculating flow can be observed covering the lateral area close to the cylinder walls on both sides of the electrodes. The needed experimental validation of the numerical results is the essential motivation for the flow measurements presented below.
3. Experimental setup 3.1. Ultrasonic Doppler flow mapping system The fundamental principle of the ultrasound Doppler method provides spatial information about the flow velocity along the propagation direction of the ultrasonic beam. For a detailed and more reliable study of transient flow structures, however, the mapping of entire instantaneous flow fields is considerably more beneficial than the measurement of a single velocity profile. Recently, a two-dimensional flow mapping system based on the application of linear ultrasonic transducer arrays was reported which achieves a better spatial and temporal resolution by the applying advanced pulsing and driving strategies [8,19]. This flow mapping system was used within the present study. The sensor configuration consists of a plane linear array composed of 25 transducer elements (transmission center frequency 8 MHz) each of a size of 2.3 5 mm2 with an element pitch of 2.7 mm (Fig. 2). Hence, an array of this configuration facilitates a flow mapping of the velocity component perpendicular to the
Fig. 2. Dimensions of ultrasonic transducer array.
transducer surface over a field width of 67 mm. Multiple array arrangements are feasible allowing, for example, the measurement of both in-plane velocity vectors or the mapping of several planes. An extended time-division multiplex scheme for the excitation of the transducers elements guarantees a flow mapping with high frame rates while retaining a proper spatial resolution. The multiplex scheme includes a specific segmentation technique at which two adjacent array elements are grouped in operation to work as one transducer of approx. 5 5 mm2. This effective transducer aperture may be traversed by a step width of one pitch length / half transducer aperture providing a dense scanning of the flow field while retaining a low beam divergence (as a result of the large transducer aperture). In this manner intermediate measuring lines are obtained taking into account that the self-focusing nature of ultrasonic beams constricts the effective beam diameter to a size smaller than the transducer aperture over a significant penetration depth of the beam. The high temporal resolution for measurements of multiple measuring lines is achieved by a multi-beam approach in combination with a refined pulsing and signal acquisition strategy. The aspiration is the simultaneous acquisition of the maximum possible number of measuring lines, whereas an acoustic crosstalk between neighboring transducers has to be avoided. This multibeam approach is extended by a special pulsing strategy considering that the repetition rate of the ultrasonic pulses determines the maximum measurable flow velocity and measuring depth [20]. Since most laboratory scale experiments with moderate measuring depths exhibit low flow velocities the acquisition time of the echo signals (defining the measuring depth) is distinctly shorter than the required pulse repetition time (defining the measurable velocity range). The resulting idle time period between the recordings of the echo signals resulting from
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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Fig. 3. Pulsing scheme of array operation.
successive ultrasonic pulses for a given measuring line is used for data acquisition along other measuring lines. This feature is a novelty in comparison to previous multiline approaches. The final pulsing scheme as applied within this study is shown in Fig. 3. The measurement electronics for one transducer array channel is shown in Fig. 4. Its main component is the custom-built multiplex unit which implements the pulsing scheme from Fig. 3 and facilitates a versatile range of pulsing patterns. The multiplex unit is separated into transmit and receive multiplexer in order to enhance the signal-to-noise ratio since the transmission of highlevel pulse signals and the reception of low-level echo signals actually require different circuitry-wise implementations. The pulse signal for the ultrasonic burst is generated by an arbitrary function generator, intensified by a RF-amplifier and directed by the transmit multiplexer to the active transducer elements (according to the current pulsing scheme). The echo signals received by the active transducer elements are amplified and feed the analog-to-digital converter of the data acquisition system (National Instruments FlexRIO system with components NI PXIe-1073, NI 5761R and NI PXIe-7961R). The digitized echo signals are preprocessed by the FPGA (Field Programmable Gate Array) unit of the acquisition system and transferred to a high-performance computer. Software components based on LabView and Matlab routines manage the data storage and the final signal processing. The timing and triggering of the burst generator and the data acquisition is controlled by the multiplex unit. Multi-plane measurements as well as the measurement of multiple velocity components are managed by adding further transducer array channels which requires a duplication of the burst generator and the multiplex unit and an extension of the
Fig. 5. Electro-vortex experimental setup.
number of input channels of the data acquisition unit (Fig. 3). 3.2. Experimental configuration for flow measurements The experiments were performed in a vessel with the inner geometrical shape of a circular cylinder. The experiments were carried out at room temperature using the eutectic liquid metal alloy Ga68In20Sn12 (melting point 10.5 °C). The thermophysical properties of GaInSn are reported by Plevachuk [21]. The cylinder with an inner radius of R0 ¼ 25 mm was made of Perspex owing to its preferable acoustic properties. The filling level of the liquid metal column amounts to H0 ¼ 60 mm yielding an aspect ratio of 1.2. Two circular rod electrodes with a diameter of 8 mm are mounted at opposing radial positions of RE ¼18 mm. The electrodes are aligned parallel to the cylinder axis and immersed into the liquid up to a depth of 10 mm below the free liquid metal surface. The lateral surface of the electrodes is coated with an electrically insulating material, so that the electric current enters the liquid metal only through the front surface of the electrodes. The experimental setup is shown in Fig. 5. The electrical feed line between the electrodes and the power supply were fixed at a considerable distance to the liquid metal
Fig. 4. Measurement system for a single array transducer channel.
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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Fig. 6. Comparison of the mean vertical velocity field of numerical simulations (left column) with experimental results (right column) for the (a), (b) 0°-plane, (c), (d) 30°plane, (e), (f) 60°-plane, (g), (h) 90°-plane, (i), (j) 120°-plane and (k), (l) 150°-plane (I ¼ 48 A).
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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Fig. 6. (continued)
column in order to minimize the interaction between the induced magnetic field around the feed lines and the electric current flow in the liquid metal. The direct current was generated by the power
supply pe86CWD from plating electronic GmbH. The experiments were conducted using current amplitudes up to about 136 A corresponding to voltages up to 2 V. The electrodes and feed lines had
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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a distinctly higher electrical resistance compared to the liquid metal leading to the result that only 1% of the voltage drop occurred in the metal melt. The heat input into the melt by Joule heating was below 3 W in the applied parameter range complying sufficiently the isothermal condition. As presented in Section 2 the numerical simulation predicts a complex three-dimensional flow structure. Within this paper we follow the approach to provide an appropriate depiction of the velocity field by a two-dimensional mapping of the vertical velocity component in various vertical cross sections of the cylinder. Distinguished cross sections are the planes containing the electrodes and its perpendicular counterpart. Consequently, the transducer array is installed radially at the base of the cylindrical vessel. The vertical alignment of the ultrasonic beam lines enables the detection of the axial flow component in the radial–meridional plane of the liquid metal column. The lid of the vessel fixing the parallel electrodes is pivoted with respect to the cylinder and the transducer array. Using this configuration we performed a twodimensional flow mapping of the axial velocity component in various radial–meridional planes. The angle in Fig. 5 denotes the orientation of the measuring plane with respect to the plane containing the parallel electrodes (hereafter indicated as 0°plane). The transducer array divides the mapping plane into 18 singular velocity measuring lines (with a pitch of 2.7 mm). The measurements were performed with a burst length of 8 cycles (sound velocity in GaInSn: 2740 m/s; transmission frequency: 8 MHz) which leads to an axial resolution (length of measurement volume) of about 1.4 mm. The lateral resolution of about 2.5 mm is achieved in the near field length of the beam and 6 mm at the maximum measuring depth of 60 mm (in reference to the -20 dB lateral decrease in the pressure amplitude of the pulse-echo acoustic field of the ultrasonic beam). The temporal resolution ranges from 50 ms to 200 ms. The mean intensity of the bulk flow was used as an evaluation criterion of the effect of different current configurations. In this study the level of bulk flow intensity is determined by the root mean square of the axial velocity over the volume of the liquid metal column by
uz, rms (t ) =
1 V
∫ d3r uz2 (r, t )
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upwards flow occurs along the cylinder side walls, which is less strong as the downward jets. The area around the cylinder axis shows a descending flow in the upper part and an ascending flow in the bottom part. This indicates the occurrence of a vortex pair in the central region between the electrodes. The flow structures found for the 120°- (Fig. 6(i), (j)) and the 150°-plane (Fig. 6(k), (l)) are very similar to those measured in the 60°- and the 30°-plane, respectively. This observation reveals the symmetry of the flow with respect to the 90°-plane and the 0°-plane as well. The agreement between the experimental data and the numerical calculations is very good. The flow pattern and the velocity amplitude are well reproduced. A minor difference can be noticed for the 90°-plane shown in Fig. 6(g) and (h). The velocity measurements show a slight asymmetry in the flow structure, which does not appear in the numerical results. The reason for this deviation does not become obvious instantly. It stands to reason that the velocity field could be influenced by a non-symmetric configuration of the power lines around the experimental facility. This assumption is supported by the fact, that the measured flow field shows a deflection in the other direction if the electric current is applied with reversed polarity. The respective flow measurement is displayed in Fig. 7. The feed lines of the electrodes lead to the power supply in a distance of 540 mm to the liquid metal column. Obviously, this is not far enough to avoid any contribution of the magnetic field of the feeding lines on the Lorentz force in the liquid metal column. The mean velocity in the jet reaches maximum values up to about 100 mm/s corresponding to a Reynolds number Re¼ UL/ν (U – characteristic velocity in the jet, L – diameter of the jet, ν – kinematic viscosity of the liquid) of about 2800 (Fig. 6(a)). A submerged jet at such a Re number is supposed to produce turbulent velocity fluctuations. The spatio-temporal plot in Fig. 8 illustrates the development of the vertical flow recorded by a singular transducer just below an electrode for an electrical current of I¼ 48 A. The measurement was started at that moment when the electrical current was switched on. The flow intensity increases gradually during an initial period of a few seconds ( 10 s) before a quasi-steady state with an almost constant velocity is reached. Distinct fluctuations of the jet intensity become obvious after
(3)
where V is the volume of the liquid metal column.
4. Experimental and numerical results The numerical simulations predict a distinct three-dimensional structure of the flow field. Therefore, two-dimensional patterns of the axial velocity component were measured at different radial– meridional planes. Six positions of the ultrasonic array were realized allowing access to the 0°-, 30°-, 60°-, 90°-, 120°- and 150°plane, respectively (see also Fig. 5). The experiments were started from the state of rest. Turning-on the electric current drives a flow which develops during an initial state. The data acquisition was started after reaching a steady state when the time-averaged flow becomes constant. The right column of Fig. 6 presents this timeaveraged flow structure measured over a time-period of 100 s. Corresponding results from the numerical computations are shown in the left column of Fig. 6. Figs. 6(a) and (b) (0°-plane containing the electrodes) exhibit the intense downwards-directed jets below the electrodes whereas an upwards back flow can be detected at the axis. The flow in the 30°- and 60°-plane (Fig. 6 (c)–(f)) is already dominated by the recirculating flow which reaches the maximum in the 90°-plane (Fig. 6(g), (h)). Here, the
Fig. 7. A reversal of the electrode polarity induces a vertical flipping of the flow structure in the plane perpendicular to the electrode plane compared to the original polarity from Fig. 6(h).
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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Fig. 8. Spatio-temporal map of the vertical velocity profile at the axis of one electrode (r ¼ 17 mm) from the experimental measurements at I ¼ 48 A.
about 20 s. These perturbations are especially pronounced in the lower part of the vessel at larger distances from the electrode. The reason can be found in a spreading and breakdown of the jet. Moreover, an oscillation of the jet position can occur. Fig. 9 shows respective snapshots of the flow field. A fully developed jet structure can be observed after 9 s. This velocity field almost corresponds to the time-averaged flow pattern shown in Fig. 6(a). In the course of the experiment the jet structure starts to fluctuate. Emergent turbulent structures can be observed in Fig. 9(b). The jet structure becomes decomposed into smaller vortices. Moreover, the jet in its entirety undergoes distinct oscillations (see Fig. 9(c)). Fig. 10 displays the spatio-temporal structure of the jet flow for I ¼48 A obtained by direct numerical simulations (DNS). The pattern shown here is rather similar compared to the experimental findings in Fig. 8 with respect to the initial development time of the flow and the upcoming perturbations. Fig. 11 contains a comparison of the time series of the local velocity uz at a radial position of r ¼17 mm (below the jet) and a height of z¼ 20 mm. The diagram demonstrates a very good accordance of the qualitative behavior of the velocity fluctuations with respect to their amplitudes. The temporal behavior of the flow is dominated by low-frequent oscillations which can be ascribed to the oscillations of the jets. Fig. 12 shows the development of the volume-averaged flow from the state of rest whereas the experimental curve represents an average resulting from measurements in the six planes as illustrated in Fig. 5. The flow develops almost immediately achieving a quasi-steady value already after about 5 s. Characteristic
Fig. 10. Spatio-temporal map of the vertical velocity profile at the axis of one electrode (r ¼ 17 mm) from the numerical simulation at I ¼ 48 A.
Fig. 11. Comparison of experimental and numerical results of the velocity evolution at an exemplary point below one electrode (r ¼ 17 mm, z ¼20 mm) at I ¼ 4 A.
fluctuations of this quasi-steady exhibit a periodicity of a few seconds and amplitudes up to about 10% of the mean value. Measurements of the developing flow averaged over the 0°- and 90° plane are presented in Fig. 13 for different values of the applied electric current. An increase of the electric current intensifies both the mean flow and the related fluctuations. It is interesting to note
Fig. 9. Snapshots of the flow structure at I ¼ 48 A in the electrode plane corresponding the spatio-temporal from Fig. 8: (a) at t ¼ 9.0 s (symmetrical flow structure with almost stationary jets during the initial state), (b) at t ¼43.1 s and (c) at t ¼ 121.9 s (asymmetrical oscillation of jets at random points for the fully-developed flow).
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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Fig. 12. Comparison of the evolution of the volume-averaged vertical velocity for the measurement (averaged over the measuring planes at 0°, 30°, 60°, 90°, 120°, 150°) and for the simulation (averaged over the entire volume).
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in the plane spanned by height and diameter of the liquid metal column. Numerical simulations of the magnetohydrodynamic (MHD) problem were conducted in parallel. Such calculations enable an understanding of the distributions of the electric field in the melt, the corresponding induced magnetic field and the resulting Lorentz force. In this study we used the numerical simulations to predict the flow structure. Experimental and numerical results reveal a complex threedimensional structure of the liquid metal flow. In particular, two pronounced downward jets are formed below both electrodes. The flow structure appears to be symmetrical with respect to two distinguished vertical cross sections, namely the 0°-plane (containing the electrodes) and the 90°-plane. The comparison between the experimental data and the numerical results shows a very good agreement. The detailed investigation of the flow structure in this isothermal experiment delivers essential knowledge for a better understanding of the flow field during solidification and the development of optimal stirring techniques to obtain solidified materials with superior mechanical properties.
Acknowledgements The authors acknowledge the financial support from the German Helmholtz Association in the framework of the HelmholtzAlliance “LIMTECH”.
References
Fig. 13. Comparison of the evolution of the volume-averaged vertical velocities excited by different currents (averaged over the measuring planes at 0° and 90°).
that maximum flow intensities can be observed during an initial phase up to 20 s after the actuation of the electric current.
5. Conclusions We performed flow measurements in a liquid metal column using the ultrasound Doppler method. The melt flow was driven by the application of a strong electric current. Two parallel pencil electrodes were positioned just beneath the free surface of the liquid metal. The superposition of the radially diverging electric current at the electrode surface and the azimuthal component of the induced magnetic field generate a vertical Lorentz force directed downwards. This configuration results in two downward jets just below the electrodes and a recirculating flow covering the lateral area close to the cylinder walls on both sides of the electrodes. Such flows are under consideration for flow control during solidification of metal alloys. In this study we determined the two-dimensional velocity field based on the application of a linear ultrasonic array in combination with specific array driving techniques which allows for a twodimensional flow mapping. The linear array, which comprises singular 25 ultrasonic transducers, was located radially at the bottom of the cylinder vessel to measure the axial flow component
[1] Y. Takeda, Measurement of velocity profile of mercury flow by ultrasound Doppler shift method, Nuclear Technol. 79 (1) (1987) 120–124. [2] S. Eckert, G. Gerbeth, Velocity measurements in liquid sodium by means of ultrasound Doppler velocimetry, Exp. Fluids 32 (5) (2002) 542–546, http://dx. doi.org/10.1007/s00348-001-0380-9. [3] A. Cramer, C. Zhang, S. Eckert, Local flow structures in liquid metals measured by ultrasonic Doppler velocimetry, Flow Meas. Instrum. 15 (3) (2004) 145–153, http://dx.doi.org/10.1016/j.flowmeasinst.2003.12.006. [4] Y. Takeda, Velocity profile measurement by ultrasound Doppler shift method, Int. J. Heat Fluid Flow 7 (4) (1986) 313–318, http://dx.doi.org/10.1016/ 0142-727X(86)90011-1. [5] Y. Takeda, Development of an ultrasound velocity profile monitor, Nucl. Eng. Des. 126 (2) (1991) 277–284, http://dx.doi.org/10.1016/0029-5493(91)90117-Z. [6] Y. Takeda, H. Kikura, Flow mapping of mercury flow, Exp. Fluids 32 (2) (2002) 161–169, http://dx.doi.org/10.1007/s003480100296. [7] K. Timmel, S. Eckert, G. Gerbeth, F. Stefani, T. Wondrak, Experimental modeling of the continuous casting process of steel using low melting point metal alloys – the LIMMCAST program, ISIJ Int. 50 (8) (2010) 1134–1141, http://dx.doi.org/ 10.2355/isijinternational.50.1134. [8] S. Franke, H. Lieske, A. Fischer, L. Büttner, J. Czarske, D. Räbiger, S. Eckert, Twodimensional ultrasound Doppler velocimeter for flow mapping of unsteady liquid metal flows, Ultrasonics 53 (3) (2013) 691–700, http://dx.doi.org/ 10.1016/j.ultras.2012.10.009. [9] E.O. Hall, The Deformation and Ageing of Mild Steel: III Discussion of Results, Proc. Phys. Soc. Lond. 64B (1951) 747–753, http://dx.doi.org/10.1088/ 0370-1301/64/9/303. [10] J.C.M. Li, Y.T. Chou, The Role of Dislocations in the Flow Stress Grain Size Relationships, Met. Trans. 1 (1970) 1145–1159, http://dx.doi.org/10.1007/ BF02900225. [11] M. Nakada, Y. Shiohara, M.C. Flemings, Modification of solidification structures by pulse electric discharging, ISIJ Int. 30 (1) (1990) 27–33, http://dx.doi.org/ 10.2355/isijinternational.30.27. [12] Y. Dolinsky, T. Elperin, Thermodynamics of phase transitions in current-carrying conductors, Phys. Rev. B 47 (1993) 14778–14785, http://dx.doi.org/ 10.1103/PhysRevB.47.14778. [13] J. Li, J. Ma, Y. Gao, Q. Zhai, Research on solidification structure refinement of pure aluminum by electric current pulse with parallel electrodes, Mater. Sci. Eng. A 490 (1) (2008) 452–456, http://dx.doi.org/10.1016/j.msea.2008.01.052. [14] D. Räbiger, Y. Zhang, V. Galindo, S. Franke, B. Willers, S. Eckert, The relevance of melt convection to grain refinement in Al–Si alloys solidified under the impact of electric currents, Acta Mater. 79 (2014) 327–338, http://dx.doi.org/ 10.1016/j.actamat.2014.07.037. [15] R.J. Tayler, Stability of Twisted Magnetic Fields in a Fluid of Finite Electrical Conductivity, Rev. Mod. Phys. 32 (1960) 907–913, http://dx.doi.org/10.1103/ RevModPhys.32.907.
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[16] M. Seilmayer, F. Stefani, T. Gundrum, T. Weier, G. Gerbeth, M. Gellert, G. Rüdiger, Experimental Evidence for a Transient Tayler Instability in a Cylindrical Liquid-Metal Column, Phys. Rev. Lett. 108 (2012) 244501, http://dx. doi.org/10.1103/PhysRevLett.108.244501. [17] N. Weber, V. Galindo, F. Stefani, T. Weier, T. Wondrak, Numerical simulation of the Tayler instability in liquid metals, T. New J. Phys. 15 (2013) 043034, http: //dx.doi.org/10.1088/1367-2630/15/4/043034. [18] 〈http://www.openfoam.org〉. [19] S. Franke, L. Büttner, J. Czarske, D. Räbiger, S. Eckert, Ultrasound Doppler
system for two-dimensional flow mapping in liquid metals, Flow Meas. Instrum. 21 (3) (2010) 402–409, http://dx.doi.org/10.1016/j. flowmeasinst.2010.05.001. [20] Y. Takeda, Ultrasonic Doppler Velocity Profiler for Fluid Flow, Springer, Japan (2012) http://dx.doi.org/10.1007/978-4-431-54026-7 (Ultrasonic Doppler Method, Murakawa). [21] Y. Plevachuk, V. Sklyarchuk, S. Eckert, G. Gerbeth, R. Novakovic, Thermophysical Properties of the Liquid Ga–In–Sn Eutectic Alloy, J. Chem. Eng. Data 59 (3) (2014) 757–763, http://dx.doi.org/10.1021/je400882q.
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Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system Sven Franke n, Dirk Räbiger, Vladimir Galindo, Yunhu Zhang, Sven Eckert Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 21 March 2015 Received in revised form 17 September 2015 Accepted 18 September 2015
This paper presents a combined experimental and numerical study of the properties of a liquid metal flow inside a cylinder driven by the application of a strong electrical current. The interaction between the electric current running through the melt and the corresponding induced magnetic field produces socalled electro-vortex flows. We consider here a configuration of two parallel pencil electrodes immersed at the free surface. Velocity measurements were performed by means of the Ultrasound Doppler method. A linear array of 25 singular transducers was used to determine the two-dimensional pattern of the vertical flow component. Numerical simulations of the magnetohydrodynamic (MHD) problem were conducted to calculate the Lorentz force, the Joule heating and the induced melt flow. Experimental and numerical results reveal a complex three-dimensional flow structure of the liquid metal flow. In particular, two pronounced downward jets are formed below both electrodes. The flow structure appears to be symmetrical with respect to two vertical cross sections being perpendicular to each other and one of the two planes contains the electrodes. The comparison between the experimental data and the numerical results shows a very good agreement. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Liquid metal flows Electrically driven flows Velocity measurements Ultrasound Doppler method Ultrasonic transducer array
1. Introduction Liquid metal technologies gain in importance for the metal industry, for example, the production of cast parts, the continuous casting of steel and the crystal growth at the semiconductor industry. As well the role of liquid metal technologies grows steadily for the branch of energy industry. Present and future examples may be found at the nuclear power industry, more precisely at fast breeder power plants and fusion reactors. However, also renewable energies may benefit from liquid metal technologies, for example with respect to the application of liquid metal coolants at concentrated solar power plants. Applications in the energy sector typically utilize liquid metals as coolant, however, future prospects consider liquid metals also for energy storage systems. Safe and reliable operation of such liquid metal systems, optimization of industrial processes and the guarantee for best product quality require a comprehensive knowledge of fluid flow phenomena and related transport processes. Numerical simulations could provide a better understanding of complex flow behavior, but, experimental data are indispensable with respect to a validation of the respective CFD codes. The determination of flow n
Corresponding author. E-mail addresses: [email protected] (S. Franke), [email protected] (D. Räbiger), [email protected] (V. Galindo), [email protected] (Y. Zhang), [email protected] (S. Eckert).
quantities in liquid metals is considerably impeded by the special material properties. Powerful optical methods as used for measurements in transparent liquids are obviously not applicable in molten metals. For the last two decades the ultrasound Doppler method became a very powerful tool to investigate the velocity structure in liquid metal flows as reported for various metallic fluids [1–3]. Originating from the medical branch the pulse-wave ultrasound Doppler method has been established for fluid flow measurements in physics and engineering by the pioneering work of Takeda [4]. The measuring principle is based on the pulsed echo technique. Ultrasonic pulses of a few cycles emitted from an acoustic transducer propagate into the fluid along a measuring line which is identical to the continuation of the transducer axis. A part of the ultrasonic pulse energy is scattered by microparticles suspended in the liquid. Their echo signal is received by the same transducer within the time period between two emissions. A short sequence of such echo signals contains the entire information of the velocity profile along the ultrasonic beam. Knowing the sound velocity of the liquid, the axial position of the scattering particles along the measuring line is determined from the measured time span between the burst emission and the reception of the respective echo signals. The movement of the scattering particles inside the measuring volume between two consecutive bursts will result in a small time shift of the echo signal. A correlation analysis between the echo signals of consecutive bursts reveals the flow velocity
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Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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profile for all positions along the measuring line. Owing to the Nyquist theorem, the product of measurable maximum velocity and penetration depth is limited by the sound velocity and the ultrasonic frequency. Ultrasonic methods are non-invasive, but not contactless since a continuous acoustic path from the ultrasonic transducer to the fluid under investigation is required. For a more detailed description of the basics of the measuring principle the reader is referred to Takeda [5]. The standard ultrasound Doppler systems measure linear velocity profiles, however, the enhancement of the capabilities towards a multidimensional flow mapping with high frame acquisition rates and spatial resolutions would be exceedingly desirable for examinations of highly turbulent, three-dimensional flows such as occurring during electromagnetic stirring of metals. Detailed investigations of such kind of complex flows and a validation of respective numerical simulations require a multidimensional acquisition of the flow field with a reasonable temporal and spatial resolution. Imaging techniques becomes more and more important for detailed explorations of three-dimensional turbulent flows, in particular with respect to the generation of a suitable experimental data base for an efficient validation of respective numerical simulations. First approaches of a flow mapping with a multiline ultrasound Doppler system were performed by Takeda [6] who measured the mercury flow in a liquid metal target of a spallation neutron source. Mapping rates about 2 Hz were realized while applying up to 12 measuring lines. More recently, measurements of a submerged liquid metal jet have been reported by Timmel et al [7]. A horizontal adjustment of 10 transducers provided a two-dimensional visualization of the horizontal velocity component utilizing a 10-channel commercial ultrasonic Doppler device. However, the use of a finite number of singular sensors by sequential sampling restricts the spatial and temporal resolution of the measurements. New measuring systems are under development using specific combinations of linear sensor arrays and have recently been demonstrated in liquid metal flows driven by a rotating magnetic field [8]. In the present paper we demonstrate the capabilities of an array measuring system using the example of a liquid metal flow driven by a strong electric current. The application of electric currents during the solidification of metal alloys was found to be beneficial for achieving superior mechanical properties of the solidified material. The improvement of the product quality is obtained by a distinct grain refinement [9,10]. However, the physical mechanism of the grain refinement has not been understood so far. Various effects are under discussion, such as the fragmentation of dendrites induced by the electric current [11], the reduction of the nucleation activation energy [12], or the break out and the transport of little grains from the boundary by the periodic Lorentz force [13]. Most of the previous studies did not consider the possibility that significant melt flows can be created by an intense Lorentz forces which results from the interaction between the strong electrical current and the self-induced magnetic field. The effect of such kind of electro-vortex flows on the solidification of Al–Si alloys was investigated by a recent study [14]. This paper presents an extension of this experimental and numerical study focusing on the characteristics of the forced melt flow induced by the strong electric currents in an isothermal melt without solidification. A set of experiments was conducted using the eutectic alloy GaInSn at room temperature to obtain quantitative information about the isothermal flow field. The mean flow structure as well as the time-varying flow field are measured and compared to predictions obtained by corresponding numerical simulations. The paper is structured as follows: The flow problem under consideration and respective numerical simulations are briefly described in the following section. A description of the experimental
setup and the measuring system is given in Section 3. Section 4 contains the presentation and discussion of the measuring results. Some experimental findings are compared with corresponding numerical predictions. The concluding remarks can be found in Section 5.
2. Formulation of the problem and numerical scheme We consider a finite cylinder with radius R0 and height H0 filled with a liquid metal. Two parallel pencil electrodes with electrically insulated lateral surfaces are immersed into the melt through the free surface and arranged symmetrically with respect to the cylinder axis. The application of an electric potential between the electrodes produces an electric current which closes through the liquid metal between the electrodes. Any electric current through an incompressible, viscous and electrically conducting liquid is accompanied by corresponding magnetic field. In general, the interaction between the applied electric current and the induced magnetic field creates a Lorentz force J × B which may drive a fluid motion. The simplest configuration of a homogeneous current distribution through a cylindrical column becomes unstable when the axial current exceeds a critical value being specific for a given fluid and geometry. Recently, this phenomenon known as Tayler instability [15] was experimentally observed and numerically analyzed [16,17]. The experimental assembly of two pencil electrodes as considered within this study implies a non-homogeneous distribution of the electric current (see Fig. 1(a)). Such a configuration shows a significant rotational component of the Lorentz force and does not feature a threshold value of applied current which has to be exceeded before a flow sets in. Convection starts as early as a weak electric current is flowing through the liquid metal. Numerical computations have been performed to calculate the actual distributions of electric current and magnetic field as well as the resulting Lorentz force. Furthermore, these simulations provide predictions of the flow field in the liquid metal column. The finite element code OPERA (Cobham plc.) was used to compute the electric current J and the magnetic induction B distributions in the melt and all electrically conducting parts of the facility including the immersed electrodes. In case of a direct current imposed on the electrodes we solve the Laplace equation for the electric potential ∇2φ = 0 taking into account the charge conservation ∇⋅J = 0 and the continuity of the current density J at the interface between two regions with different electric conductivities: Jn = − σ1n⋅∇φ1 ¼ −σ 2 n⋅∇φ2. The magnetic induction B was calculated from the current density J = σ ( − ∇φ) using the Biot–Savart law:
B (r ) =
μ0 4π
∫ d3r′ J (r′)|r ×− (rr′|−3 r′)
(1)
The boundary conditions for the electric potential are defined using the amplitude of the imposed electric current at the end of the electrodes: I = ∫ ds⋅J , where A is the area of the electrode’s A
cross section. The flow in the volume containing the melt was simulated numerically by means of the open source library OpenFOAM [18] solving the Navier–Stokes equation together with the incompressibility condition ∇⋅u = 0 and including an electromagnetic force density term:
⎛∂u ⎞ ρ⎜ + (u⋅∇) u⎟ = − ∇p + η∇2u + J × B ⎝ ∂t ⎠
(2)
The boundary conditions for the flow field are the no-slip condition u = 0 at the solid container walls. For the melt surface
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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B
F J
F [m/s²] 0.5 0.4 0.3 0.2 0.1 0
Fig. 1. Flow field development in given setup: (a) simplified model of the induced Lorentz forces, (b) computed Lorentz force distribution and (c) snapshot of corresponding three-dimensional flow structure from the numerical simulation (predicting a downward-directed jet flow below the electrodes and a upward-directed bulk flow).
either u = 0 or the conditions for a stress-free, non-deformable surface un =0 and ∂ ut / ∂ z = 0 are applied depending on whether the melt flow is evaluated in an open or an enclosed container. A computational grid with 650000 volume elements was used. Fig. 1(b) shows the distribution of the Lorentz force for an electric current of I ¼48 A. It becomes obvious that a relevant Lorentz force only occurs just below the electrodes. This phenomenon can be explained by the distribution of the electric current. The current distribution diverges just beneath the electrodes almost in radial direction with respect to the axis of the electrode. The interplay with the azimuthal magnetic field around the electrodes results in a significant downward component of the electromagnetic force. The consequence is the formation of downward jets below the electrodes as shown in Fig. 1(c). The numerical simulations also reveal the three-dimensional structure of the global flow field. Besides the two downward jets a recirculating flow can be observed covering the lateral area close to the cylinder walls on both sides of the electrodes. The needed experimental validation of the numerical results is the essential motivation for the flow measurements presented below.
3. Experimental setup 3.1. Ultrasonic Doppler flow mapping system The fundamental principle of the ultrasound Doppler method provides spatial information about the flow velocity along the propagation direction of the ultrasonic beam. For a detailed and more reliable study of transient flow structures, however, the mapping of entire instantaneous flow fields is considerably more beneficial than the measurement of a single velocity profile. Recently, a two-dimensional flow mapping system based on the application of linear ultrasonic transducer arrays was reported which achieves a better spatial and temporal resolution by the applying advanced pulsing and driving strategies [8,19]. This flow mapping system was used within the present study. The sensor configuration consists of a plane linear array composed of 25 transducer elements (transmission center frequency 8 MHz) each of a size of 2.3 5 mm2 with an element pitch of 2.7 mm (Fig. 2). Hence, an array of this configuration facilitates a flow mapping of the velocity component perpendicular to the
Fig. 2. Dimensions of ultrasonic transducer array.
transducer surface over a field width of 67 mm. Multiple array arrangements are feasible allowing, for example, the measurement of both in-plane velocity vectors or the mapping of several planes. An extended time-division multiplex scheme for the excitation of the transducers elements guarantees a flow mapping with high frame rates while retaining a proper spatial resolution. The multiplex scheme includes a specific segmentation technique at which two adjacent array elements are grouped in operation to work as one transducer of approx. 5 5 mm2. This effective transducer aperture may be traversed by a step width of one pitch length / half transducer aperture providing a dense scanning of the flow field while retaining a low beam divergence (as a result of the large transducer aperture). In this manner intermediate measuring lines are obtained taking into account that the self-focusing nature of ultrasonic beams constricts the effective beam diameter to a size smaller than the transducer aperture over a significant penetration depth of the beam. The high temporal resolution for measurements of multiple measuring lines is achieved by a multi-beam approach in combination with a refined pulsing and signal acquisition strategy. The aspiration is the simultaneous acquisition of the maximum possible number of measuring lines, whereas an acoustic crosstalk between neighboring transducers has to be avoided. This multibeam approach is extended by a special pulsing strategy considering that the repetition rate of the ultrasonic pulses determines the maximum measurable flow velocity and measuring depth [20]. Since most laboratory scale experiments with moderate measuring depths exhibit low flow velocities the acquisition time of the echo signals (defining the measuring depth) is distinctly shorter than the required pulse repetition time (defining the measurable velocity range). The resulting idle time period between the recordings of the echo signals resulting from
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Fig. 3. Pulsing scheme of array operation.
successive ultrasonic pulses for a given measuring line is used for data acquisition along other measuring lines. This feature is a novelty in comparison to previous multiline approaches. The final pulsing scheme as applied within this study is shown in Fig. 3. The measurement electronics for one transducer array channel is shown in Fig. 4. Its main component is the custom-built multiplex unit which implements the pulsing scheme from Fig. 3 and facilitates a versatile range of pulsing patterns. The multiplex unit is separated into transmit and receive multiplexer in order to enhance the signal-to-noise ratio since the transmission of highlevel pulse signals and the reception of low-level echo signals actually require different circuitry-wise implementations. The pulse signal for the ultrasonic burst is generated by an arbitrary function generator, intensified by a RF-amplifier and directed by the transmit multiplexer to the active transducer elements (according to the current pulsing scheme). The echo signals received by the active transducer elements are amplified and feed the analog-to-digital converter of the data acquisition system (National Instruments FlexRIO system with components NI PXIe-1073, NI 5761R and NI PXIe-7961R). The digitized echo signals are preprocessed by the FPGA (Field Programmable Gate Array) unit of the acquisition system and transferred to a high-performance computer. Software components based on LabView and Matlab routines manage the data storage and the final signal processing. The timing and triggering of the burst generator and the data acquisition is controlled by the multiplex unit. Multi-plane measurements as well as the measurement of multiple velocity components are managed by adding further transducer array channels which requires a duplication of the burst generator and the multiplex unit and an extension of the
Fig. 5. Electro-vortex experimental setup.
number of input channels of the data acquisition unit (Fig. 3). 3.2. Experimental configuration for flow measurements The experiments were performed in a vessel with the inner geometrical shape of a circular cylinder. The experiments were carried out at room temperature using the eutectic liquid metal alloy Ga68In20Sn12 (melting point 10.5 °C). The thermophysical properties of GaInSn are reported by Plevachuk [21]. The cylinder with an inner radius of R0 ¼ 25 mm was made of Perspex owing to its preferable acoustic properties. The filling level of the liquid metal column amounts to H0 ¼ 60 mm yielding an aspect ratio of 1.2. Two circular rod electrodes with a diameter of 8 mm are mounted at opposing radial positions of RE ¼18 mm. The electrodes are aligned parallel to the cylinder axis and immersed into the liquid up to a depth of 10 mm below the free liquid metal surface. The lateral surface of the electrodes is coated with an electrically insulating material, so that the electric current enters the liquid metal only through the front surface of the electrodes. The experimental setup is shown in Fig. 5. The electrical feed line between the electrodes and the power supply were fixed at a considerable distance to the liquid metal
Fig. 4. Measurement system for a single array transducer channel.
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Fig. 6. Comparison of the mean vertical velocity field of numerical simulations (left column) with experimental results (right column) for the (a), (b) 0°-plane, (c), (d) 30°plane, (e), (f) 60°-plane, (g), (h) 90°-plane, (i), (j) 120°-plane and (k), (l) 150°-plane (I ¼ 48 A).
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Fig. 6. (continued)
column in order to minimize the interaction between the induced magnetic field around the feed lines and the electric current flow in the liquid metal. The direct current was generated by the power
supply pe86CWD from plating electronic GmbH. The experiments were conducted using current amplitudes up to about 136 A corresponding to voltages up to 2 V. The electrodes and feed lines had
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a distinctly higher electrical resistance compared to the liquid metal leading to the result that only 1% of the voltage drop occurred in the metal melt. The heat input into the melt by Joule heating was below 3 W in the applied parameter range complying sufficiently the isothermal condition. As presented in Section 2 the numerical simulation predicts a complex three-dimensional flow structure. Within this paper we follow the approach to provide an appropriate depiction of the velocity field by a two-dimensional mapping of the vertical velocity component in various vertical cross sections of the cylinder. Distinguished cross sections are the planes containing the electrodes and its perpendicular counterpart. Consequently, the transducer array is installed radially at the base of the cylindrical vessel. The vertical alignment of the ultrasonic beam lines enables the detection of the axial flow component in the radial–meridional plane of the liquid metal column. The lid of the vessel fixing the parallel electrodes is pivoted with respect to the cylinder and the transducer array. Using this configuration we performed a twodimensional flow mapping of the axial velocity component in various radial–meridional planes. The angle in Fig. 5 denotes the orientation of the measuring plane with respect to the plane containing the parallel electrodes (hereafter indicated as 0°plane). The transducer array divides the mapping plane into 18 singular velocity measuring lines (with a pitch of 2.7 mm). The measurements were performed with a burst length of 8 cycles (sound velocity in GaInSn: 2740 m/s; transmission frequency: 8 MHz) which leads to an axial resolution (length of measurement volume) of about 1.4 mm. The lateral resolution of about 2.5 mm is achieved in the near field length of the beam and 6 mm at the maximum measuring depth of 60 mm (in reference to the -20 dB lateral decrease in the pressure amplitude of the pulse-echo acoustic field of the ultrasonic beam). The temporal resolution ranges from 50 ms to 200 ms. The mean intensity of the bulk flow was used as an evaluation criterion of the effect of different current configurations. In this study the level of bulk flow intensity is determined by the root mean square of the axial velocity over the volume of the liquid metal column by
uz, rms (t ) =
1 V
∫ d3r uz2 (r, t )
7
upwards flow occurs along the cylinder side walls, which is less strong as the downward jets. The area around the cylinder axis shows a descending flow in the upper part and an ascending flow in the bottom part. This indicates the occurrence of a vortex pair in the central region between the electrodes. The flow structures found for the 120°- (Fig. 6(i), (j)) and the 150°-plane (Fig. 6(k), (l)) are very similar to those measured in the 60°- and the 30°-plane, respectively. This observation reveals the symmetry of the flow with respect to the 90°-plane and the 0°-plane as well. The agreement between the experimental data and the numerical calculations is very good. The flow pattern and the velocity amplitude are well reproduced. A minor difference can be noticed for the 90°-plane shown in Fig. 6(g) and (h). The velocity measurements show a slight asymmetry in the flow structure, which does not appear in the numerical results. The reason for this deviation does not become obvious instantly. It stands to reason that the velocity field could be influenced by a non-symmetric configuration of the power lines around the experimental facility. This assumption is supported by the fact, that the measured flow field shows a deflection in the other direction if the electric current is applied with reversed polarity. The respective flow measurement is displayed in Fig. 7. The feed lines of the electrodes lead to the power supply in a distance of 540 mm to the liquid metal column. Obviously, this is not far enough to avoid any contribution of the magnetic field of the feeding lines on the Lorentz force in the liquid metal column. The mean velocity in the jet reaches maximum values up to about 100 mm/s corresponding to a Reynolds number Re¼ UL/ν (U – characteristic velocity in the jet, L – diameter of the jet, ν – kinematic viscosity of the liquid) of about 2800 (Fig. 6(a)). A submerged jet at such a Re number is supposed to produce turbulent velocity fluctuations. The spatio-temporal plot in Fig. 8 illustrates the development of the vertical flow recorded by a singular transducer just below an electrode for an electrical current of I¼ 48 A. The measurement was started at that moment when the electrical current was switched on. The flow intensity increases gradually during an initial period of a few seconds ( 10 s) before a quasi-steady state with an almost constant velocity is reached. Distinct fluctuations of the jet intensity become obvious after
(3)
where V is the volume of the liquid metal column.
4. Experimental and numerical results The numerical simulations predict a distinct three-dimensional structure of the flow field. Therefore, two-dimensional patterns of the axial velocity component were measured at different radial– meridional planes. Six positions of the ultrasonic array were realized allowing access to the 0°-, 30°-, 60°-, 90°-, 120°- and 150°plane, respectively (see also Fig. 5). The experiments were started from the state of rest. Turning-on the electric current drives a flow which develops during an initial state. The data acquisition was started after reaching a steady state when the time-averaged flow becomes constant. The right column of Fig. 6 presents this timeaveraged flow structure measured over a time-period of 100 s. Corresponding results from the numerical computations are shown in the left column of Fig. 6. Figs. 6(a) and (b) (0°-plane containing the electrodes) exhibit the intense downwards-directed jets below the electrodes whereas an upwards back flow can be detected at the axis. The flow in the 30°- and 60°-plane (Fig. 6 (c)–(f)) is already dominated by the recirculating flow which reaches the maximum in the 90°-plane (Fig. 6(g), (h)). Here, the
Fig. 7. A reversal of the electrode polarity induces a vertical flipping of the flow structure in the plane perpendicular to the electrode plane compared to the original polarity from Fig. 6(h).
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Fig. 8. Spatio-temporal map of the vertical velocity profile at the axis of one electrode (r ¼ 17 mm) from the experimental measurements at I ¼ 48 A.
about 20 s. These perturbations are especially pronounced in the lower part of the vessel at larger distances from the electrode. The reason can be found in a spreading and breakdown of the jet. Moreover, an oscillation of the jet position can occur. Fig. 9 shows respective snapshots of the flow field. A fully developed jet structure can be observed after 9 s. This velocity field almost corresponds to the time-averaged flow pattern shown in Fig. 6(a). In the course of the experiment the jet structure starts to fluctuate. Emergent turbulent structures can be observed in Fig. 9(b). The jet structure becomes decomposed into smaller vortices. Moreover, the jet in its entirety undergoes distinct oscillations (see Fig. 9(c)). Fig. 10 displays the spatio-temporal structure of the jet flow for I ¼48 A obtained by direct numerical simulations (DNS). The pattern shown here is rather similar compared to the experimental findings in Fig. 8 with respect to the initial development time of the flow and the upcoming perturbations. Fig. 11 contains a comparison of the time series of the local velocity uz at a radial position of r ¼17 mm (below the jet) and a height of z¼ 20 mm. The diagram demonstrates a very good accordance of the qualitative behavior of the velocity fluctuations with respect to their amplitudes. The temporal behavior of the flow is dominated by low-frequent oscillations which can be ascribed to the oscillations of the jets. Fig. 12 shows the development of the volume-averaged flow from the state of rest whereas the experimental curve represents an average resulting from measurements in the six planes as illustrated in Fig. 5. The flow develops almost immediately achieving a quasi-steady value already after about 5 s. Characteristic
Fig. 10. Spatio-temporal map of the vertical velocity profile at the axis of one electrode (r ¼ 17 mm) from the numerical simulation at I ¼ 48 A.
Fig. 11. Comparison of experimental and numerical results of the velocity evolution at an exemplary point below one electrode (r ¼ 17 mm, z ¼20 mm) at I ¼ 4 A.
fluctuations of this quasi-steady exhibit a periodicity of a few seconds and amplitudes up to about 10% of the mean value. Measurements of the developing flow averaged over the 0°- and 90° plane are presented in Fig. 13 for different values of the applied electric current. An increase of the electric current intensifies both the mean flow and the related fluctuations. It is interesting to note
Fig. 9. Snapshots of the flow structure at I ¼ 48 A in the electrode plane corresponding the spatio-temporal from Fig. 8: (a) at t ¼ 9.0 s (symmetrical flow structure with almost stationary jets during the initial state), (b) at t ¼43.1 s and (c) at t ¼ 121.9 s (asymmetrical oscillation of jets at random points for the fully-developed flow).
Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i
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Fig. 12. Comparison of the evolution of the volume-averaged vertical velocity for the measurement (averaged over the measuring planes at 0°, 30°, 60°, 90°, 120°, 150°) and for the simulation (averaged over the entire volume).
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in the plane spanned by height and diameter of the liquid metal column. Numerical simulations of the magnetohydrodynamic (MHD) problem were conducted in parallel. Such calculations enable an understanding of the distributions of the electric field in the melt, the corresponding induced magnetic field and the resulting Lorentz force. In this study we used the numerical simulations to predict the flow structure. Experimental and numerical results reveal a complex threedimensional structure of the liquid metal flow. In particular, two pronounced downward jets are formed below both electrodes. The flow structure appears to be symmetrical with respect to two distinguished vertical cross sections, namely the 0°-plane (containing the electrodes) and the 90°-plane. The comparison between the experimental data and the numerical results shows a very good agreement. The detailed investigation of the flow structure in this isothermal experiment delivers essential knowledge for a better understanding of the flow field during solidification and the development of optimal stirring techniques to obtain solidified materials with superior mechanical properties.
Acknowledgements The authors acknowledge the financial support from the German Helmholtz Association in the framework of the HelmholtzAlliance “LIMTECH”.
References
Fig. 13. Comparison of the evolution of the volume-averaged vertical velocities excited by different currents (averaged over the measuring planes at 0° and 90°).
that maximum flow intensities can be observed during an initial phase up to 20 s after the actuation of the electric current.
5. Conclusions We performed flow measurements in a liquid metal column using the ultrasound Doppler method. The melt flow was driven by the application of a strong electric current. Two parallel pencil electrodes were positioned just beneath the free surface of the liquid metal. The superposition of the radially diverging electric current at the electrode surface and the azimuthal component of the induced magnetic field generate a vertical Lorentz force directed downwards. This configuration results in two downward jets just below the electrodes and a recirculating flow covering the lateral area close to the cylinder walls on both sides of the electrodes. Such flows are under consideration for flow control during solidification of metal alloys. In this study we determined the two-dimensional velocity field based on the application of a linear ultrasonic array in combination with specific array driving techniques which allows for a twodimensional flow mapping. The linear array, which comprises singular 25 ultrasonic transducers, was located radially at the bottom of the cylinder vessel to measure the axial flow component
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Please cite this article as: S. Franke, et al., Investigations of electrically driven liquid metal flows using an ultrasound Doppler flow mapping system, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.09.004i