Efficient Melt Stirring Using Pulse Sequences of a RotatingMagnetic Field: Part I. Flow Field in a Liquid Metal Column

S. ECKERT, P.A. NIKRITYUK, D. RABIGER, K. ECKERT, and G. GERBETH

The use of a pulsed, rotating magnetic field (RMF) is presented as an auspicious method forobtaining an intensive stirring and mixing in a pool of liquid metal; the RMF pulses within asequence have been applied with a constant or alternating direction. The resulting flow structurein a cylindrical liquid metal column has been explored by numerical simulations and by modelexperiments, using the ternary alloy GaInSn. Ultrasonic Doppler velocimetry (UDV) has beenused to determine profiles of the vertical velocity. Both the numerical results and the velocitymeasurements demonstrate the capability of the proposed stirring regimes for overcoming thelimited mixing character of conventional rotary stirring. The application of a time-modulatedRMF offers considerable potential for providing an optimal flow pattern in a solidifying melt,for reasons of a well-aimed modification of casting properties.

DOI: 10.1007/s11663-007-9096-4� The Minerals, Metals & Materials Society and ASM International 2007

I. INTRODUCTION

ALTERNATING current (AC) magnetic fields arecommonly used in industrial practice for melt stirring.[1–6] The requirements arising from the particular metallur-gical or casting operation are manifold. For example, theelectromagnetic stirring should provide an efficient mix-ing of the melt or counterbalance buoyancy-driven flows.In principle, different magnetic fields (rotating, traveling,pulsating, and combinations of these three) are available,but each field type gives rise to a more or less symptom-atic flow pattern. For the optimization of melt-stirringoperations, a suitable magnetic field design has to bechosen. Furthermore, the magnetic-field parameters(amplitude, frequency, and spatial and temporal struc-ture) have to be adjusted according to the requirements.

In the present article, we consider the standard case ofelectromagnetic stirring by means of a rotating magneticfield (RMF). The rotary stirring is thought to beemployed in steelmaking not only for reasons of ladlemixing or flow control at the mold region in continuouscasting,[2–4] but also during the solidification of metalalloys, where it has been proven a powerful method forachieving a purposeful alteration of the microstructureof casting ingots. The RMF-driven convection promotesa transition from a columnar to an equiaxed dendriticgrowth (CET) and provokes a distinct grain-refiningeffect.[7,8] However, it was also observed that the RMFstirring of a solidifying metal alloy causes macrosegre-gations.[9–12]

In the majority of cases, the RMF stirring is appliedas continuous stirring. For a metal alloy in a cylindricalcolumn, the imposition of an RMF leads basically to aswirling motion of the liquid, usually called a primaryflow. A secondary recirculating flow in the r-z planeresults from the Ekman pumping at the horizontalwalls. Although the amplitude of the primary motion isseveral times higher than the secondary flow, itscontribution to an efficient bulk mixing is marginal,because it consists mainly of an almost rigidly rotatingcore. The convective transport in the vertical and radialdirections is mainly achieved by the secondary flow,which consists of a double-vortex structure for thelaminar case. Therefore, an amplification of the sec-ondary flow causes an intensification of the mixing rate,which can be realized in the simplest way by increasingthe magnetic field. At sufficiently high field intensities,so-called Taylor–Gortler vortices occur, moving alongthe sidewalls of the cylinder and dissipating inside theBodewadt layers. Such vortex structures are veryefficient with respect to melt mixing.[13] On the otherhand, an increase in the magnetic field also acceleratesthe primary, swirling flow, which results in increasingdeflection of the free surface. Instabilities at themeniscus region should be strictly avoided in manyapplications, because they can lead to surface defects orthe entrainment of gas. In addition, specific problemsarise if the rotary stirring is applied during directionalsolidification. For example, for a metal alloy solidifiedin vertical direction, the double-vortex structure of thesecondary flow results in a radial, inward flow along thesolidification front. The flow transports the solutetoward the axis of the ingot and is, therefore, respon-sible for the typical segregation pattern in the form ofvertical channels filled with eutectic alloy.[9–11] Amitigation of the problem could probably be achievedif the flow in the mushy zone is oriented by turns indifferent directions.

S. ECKERT, Senior Research Scientist, D. RABIGER, GraduateStudent, and G. GERBETH, Group Leader, are with the MHDDepartment, Forschungszentrum Dresden-Rossendorf (FZD),D-01314, Dresden, Germany. Contact e-mail: s.eckert@fzd.deP.A. NIKRITYUK, Postdoctoral Fellow, and K. ECKERT, GroupLeader, are with the Institute for Aerospace Engineering, DresdenUniversity of Technology, D-01062, Dresden, Germany.Manuscript submitted May 3, 2007.Article published online November 6, 2007.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 38B, DECEMBER 2007—977

Regardless of the differences between the continuouscasting process and the unidirectional solidification ofmetal alloys, these problems related to rotary stirringimpede the efficient use of an RMF in both cases. Forthis reason, a new concept of electromagnetic stirringhas to be found. A few techniques are known; they havebeen proposed to avoid the disadvantages of rotary,electromagnetic stirring mentioned here. These conceptswill be introduced in Section II of the present paper. Ourapproach is to find a method that makes it possible toamplify the secondary, recirculating flow, but to limitthe amplitude of the primary, swirling flow at the sametime. As a first step toward this goal, we will demon-strate in Part I of this article that a modulated RMFapplication can enforce a higher mixing rate, both at alower energy consumption and at a nearby flat surface.

We study the melt flow excited by the application of atime-modulated RMF. In the present article, twospecific types of RMF stirring have been considered.First, we used a succession of pulses that always havethe same rotational direction, called the RMF pulsesequence of constant direction (RMF-PSCD). Second,we used an RMF pulse sequence of alternating direction(RMF-PSAD); this is related to a periodic inversion ofthe sense of rotation between two consecutive pulses. Anoptimization of the electromagnetic stirring with regardto an efficient mixing or a control of the solidificationprocesses requires a profound knowledge of the flowstructure; therefore, Part I of this article is exclusivelyfocused on obtaining quantitative information on theflow fields in a circular cylinder generated by a singleRMF pulse and a pulse sequence. Here, we consider amodel experiment not exclusively related to a particularindustrial application. In fact, we want to study how aflow field evolves under isothermal conditions if theliquid metal is agitated by a succession of RMF pulses.The knowledge gained is applied in Part II of this article,which is specifically concerned with the effect of pulsedrotary stirring during the directional solidification of Al-Si alloys. Numerical simulations and experimentalinvestigations have been carried out. Velocity measure-ments have been realized using the ultrasound Dopplervelocimetry (UDV). The experimental setup will bedescribed in Section III, whereas the numerical proce-dure is briefly introduced in Section IV. Section V isdevoted to the specific flow pattern arising from animpulsive change in the magnetic-field strength, knownas the spinup or spindown of the flow. Experimental andnumerical results with respect to the velocity field duringelectromagnetic stirring via an RMF pulse sequence ofconstant or alternating direction will be presented inSections VI and VII, respectively. The results will bediscussed in Section VIII and the conclusions can befound in Section IX.

II. ADVANCED CONCEPTS OF ROTARY,ELECTROMAGNETIC STIRRING

Several studies have been devoted to overcoming thehandicaps of rotary stirring, with the specific goal ofproviding a vigorous stirring in the bulk without

considerable deformation of the free surface. Vives[6]

developed a helical induction stirrer, which generatesnot only rotational but also vertical forces. For the samereason, Taniguchi et al.[14] proposed a hybrid stirrer thatcombines the action of a rotating and vertically travelingmagnetic field. An electromagnetic stirring system thatemploys two rotating magnetic fields was considered byChang et al.[4] Whereas the bulk volume of the mold isagitated by a main electromagnetic stirring (M-EMS)system, a second stirrer, a so-called alternating current-stirring modifier (AC-SM), should control the meniscusregion by generating a fluid rotation in the oppositedirection. The opposite stirring is supposed to keep thefree surface quiescent while a vigorous stirring ismaintained in the rest of the pool.All these approaches require a modification of the

stirrer design or the installation of additional inductioncoils. From an economic point of view, it would beattractive to achieve the same benefit with the already-existing coil system. Spitzer et al.[2] proposed a multi-frequency stirring. The authors applied a low-frequency,three-phase current rotating counterclockwise and ahigh-frequency current rotating in a clockwise direction,whereas the high-frequency part only acts in the exteriordomain of the liquid because of the skin effect. It wasshown that, for the relevant time-averaged volume force,the superposition principle is valid with respect to thecontribution of both frequencies. Kojima et al.[15]

adopted a rotary, electromagnetic stirrer to reduce thecenter segregation in the continuous casting of steel.Because previous plant trials showed that continuousstirring does not inhibit such segregation effects, theauthors developed new stirring methods with varyingdirections and intensities of the magnetic field. Onevariant, named the differential-phase-current (DPC)stirring, was implemented by a frequency variation ofa two-phase current. Another technique is based on aperiodic activation of a pulse-shaped RMF, both inintermittent and alternating form. The particular pulsesare separated by dormant periods, while the develop-ment of the fluid flow is left to its own inertia. Theauthors performed tests on Wood�s metal. They carriedout velocity measurements at particular positions in themelt, which revealed an oscillating behavior of the flow.Although a reduction of the center segregation wasachieved using these types of stirring, the methods arenot considered effective enough to meet the qualityrequirements for high-grade steels.An optimized application of an intermittently-applied

RMF for melt stirring requires a better understanding ofthe fluid dynamics in such processes. If a sequence ofRMF pulses is imposed on the melt, the resulting flowexperiences a succession of spinup and spindown cycles.Analytical and numerical studies concerning the spinupof an RMF-driven flow have been published re-cently.[16,17] Nikrityuk et al.[18] investigated the transientbehavior of the flow arising from the application of asingle RMF pulse. Here, the flow is governed by animpulsive spinup from the rest state, followed by aspindown phase resulting from the inertia of the fluid.Under specific conditions determined by the impositionof a pulsed RMF, the secondary, recirculating flow

978—VOLUME 38B, DECEMBER 2007 METALLURGICAL AND MATERIALS TRANSACTIONS B

shows periodical reversals; a considerable strengtheningof the secondary flow with respect to the primary flowcan be noticed as well.[18] These findings give reason toanticipate the possibility of overcoming the disadvan-tages of conventional rotary stirring by means of apulsed RMF.

The present article focuses on a systematic investiga-tion of flow pattern driven by different types of a time-modulated RMF, whereas the temporal structure of themagnetic field is composed of sequential RMF pulses. Astriking method of flow tailoring is in demand forgenerating favorable flow structures featured by anintensive mixing, while at the same time minimizingboth the energy consumption and the interfering per-turbations of the free surface. Moreover, the synthesis ofa secondary flow with periodic inversions of the flowdirection in the vicinity of the solidification front seemsto hold promise for avoiding flow-induced macrosegre-gation during solidification.

III. EXPERIMENTAL SETUP

Model experiments are an important tool for under-standing the details of the flow structure and thetransport properties of flows occurring in real-scalemetallurgical facilities and for validating the multitude ofnumerical codes for the flow simulation. The modelexperiment presented here should help us to understandhow a fluid flow develops under the influence of asequence of RMF pulses, which has to be considered as acomplex fluid dynamical problem. For this reason, wehave chosen a low-melting-point liquid metal in a simplegeometry, namely, a cylindrical cavity. The flow mea-surements were carried out using the ternary eutecticalloy Ga68In20Sn12, which is liquid at room tempera-ture. The cylindrical fluid vessel was made from Perspexwith a radius R0 of 30 mm. The height of the liquid metalcolumn H0 was chosen to be 60 mm, ensuring an aspectratio 2R0/H0 of unity. The experiments were carried outin the magnetic induction system KOMMA* at FZD,

which is composed of two coil systems that allow for thegeneration of both rotating and traveling magnetic fields.A schematic view of the experimental setup is depicted inFigure 1. The bore diameter of the magnetic system is90 mm, wherein the fluid vessel was placed concentri-cally. In order to preclude flow artifacts arising fromsymmetry deviations of the experimental setup (verticalalignment, conformity of both the cylinder, and mag-netic-field axis), special care was necessary to ensure aprecise positioning of the cylinder inside the magneticsystem. The RMF was generated by an arrangement ofsix air-cooled copper coils, which were fed with a three-phase electric current. The magnetic system is able toprovide field intensities of, at most, 25 mT and afrequency in the domain between 10 and 400 Hz. Thehomogeneity of the magnetic field was checked using a

three-axis Gauss meter (Lakeshore model 560, sensortype MMZ2560-UH, Lakeshore). Within a radius of30 mm, which was selected as the radial dimension of thecontainer, the variance of the magnetic-field strength wasfound to be smaller than 5 pct. The 50-hZ frequency inthe magnetic field has been modulated with sequences ofrectangular pulses. The modulation has been carried outby switching the power supply with an external wave-form generator. The modulation frequencies (typicallybelow 1 Hz) are considerably lower than the magnetic-field frequency of 50 Hz. The different variants of themodulation that have been realized in our experimentswill be discussed in the subsequent sections of this article.The UDV, which delivers instantaneous profiles of the

local velocity along the ultrasonic beam, is a veryattractive technique for obtaining experimental datafrom flows of opaque liquids. Details concerning themeasuring principle can be found in Reference 19. Overthe past years, the UDV technique has become anaccepted method for flow investigations in various liquidmetals—for example, gallium and GaInSn.[20,21] Ourmeasurements have been carried out using the DOP2000velocimeter (model 2125, Signal Processing SA, Lau-sanne) equipped with an 8-MHz transducer (TR0805LS,diameter 5 mm, Signal Processing SA, Lausanne). Thetransducer was aligned vertically and attached to thefree surface of the liquid metal. All velocity measure-ments presented in this article have been carried out at aradial position of r/R0 = 0.6. Using this arrangement,profiles of the vertical velocity vz have been recordedbetween the front end of the sensor and the bottom ofthe fluid container. The sensor surface penetrated themelt surface until a depth of about 2 mm. The measur-ing volume can be considered as a series of consecutive

Fig. 1—Schematic view of the experimental setup.

*KOMMA is an experimental magnetic induction system built-onby FZD

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 38B, DECEMBER 2007—979

discs lined up concentrically along the ultrasonic beam.Because of the divergence of the ultrasonic beam, thelateral size of the measuring volume increases with thedistance from the transducer. Hence, the spatial resolu-tion in the lateral direction varies in a range from 5 mmat the sensor to approximately 7.5 mm at the bottom ofthe fluid vessel. In an axial direction, a spatial resolutionof about 1.4 mm was achieved. The velocity data wereacquired with a sampling frequency of 8 Hz. Theresolution of the velocity data is about 0.15 mm/s.

IV. NUMERICAL METHOD

A detailed description of our model and the numericalimplementation can be found in Nikrityuk et al.[18] andwill be briefly summarized here. The simulations con-sider an electrically-conducting liquid in a finite cylinder(radius R0, height H0) exposed to an RMF. The rotationof the magnetic field induces a Lorentz force FL, whichdrives the liquid motion. In general, the Lorentz forceconsists of both a steady axisymmetric part and a time-dependent one oscillating with the double frequency.The latter one becomes relevant only for small frequen-cies[22] and can be neglected for our experimentalconditions. Furthermore, the modeling of the Lorentzforce presumes both the low-frequency and low-induc-tion approximation. The low-induction approximationimplies that both the magnetic Reynolds numberRem = l0ruR0 and the magnetic interaction parameterN = rB0

2/qx are much smaller than unity. The low-frequency approximation requires the skin depth of themagnetic field to be much larger than the radius of thecylinder R0 or, in other words, the shielding parameterS = l0rxR0

2 < 1. If both conditions are fulfilled, theLorentz force has only a nonoscillating part that isunaffected by the fluid motion. In this case, an analyticalsolution can be used for the Lorentz force that has apredominant component in the azimuthal direction:

FL ¼1

2r � x � B2

0 � r

� 1� 2

r0�X1

k¼1

J1 kkr0ð Þ � cosh kk z0 � 12H0� �� �

k2k � 1� �

� J1 kKð Þ � cosh 12 kkH0� �

!½1�

where z¢ = z/R0, r¢ = r/R0, H¢ = H0/R0, J1 and J2 areBessel functions, and the kk terms are the roots ofJ01 xð Þ ¼ 0.

The liquid metal is considered a Newtonian incom-pressible fluid, which is assumed to be axisymmetric. Anopen-source code of a Navier–Stokes solver[23] has beenadopted to solve the transport equations for mass andmomentum:

r � u ¼ 0 ½2�

@u

@tþ urð Þ u ¼ � 1

qrpþ mD uþ FL eh

q½3�

where the cylindrical polar coordinates (r,h,z) are used.Axisymmetry is forced, by cancelling the derivativeswith respect to h. The projection of the momentum

conservation equation in the azimuthal direction has theform

@ uh

@ tþ uz

@ uh

@ zþ ur

@ uh

@ r¼ m

@2 uh

@ z2þ m

r

@

@ rr@ uh

@ r

� �

� ur uh

r� m

uh

r2þ FL

q½4�

No-slip conditions on all walls except the top areapplied. Due to the open-cavity experimental setup, weuse the stress-free condition on the free surface.The SIMPLE algorithm with an arrangement of

collocated variables has been applied, to calculate thepressure and the velocity fields. For the stabilization ofthe pressure-velocity coupling, the Rhie–Chow schemehas been used. The convection terms are discretized by adeferred-correction scheme.[23] The diffusion terms wereapproximated, using the second-order central differenc-ing scheme. The time derivatives are discretized by athree-time-level scheme. The system of linear equationsis solved by using Stone�s strongly-implicit procedure(SIP). Time marching with a fixed time-step was used.The outer iterations for each time-step were stoppedwhen the maximum residual was less than 10–10 inphysical units. Several grid-convergence tests are per-formed. The final grid has 120 and 240 control volumesin the radial and the axial directions, respectively. Thecode was validated in Reference 17.

V. SPINUP AND SPINDOWN OF AN RMF-DRI-VEN FLOW

This section is devoted to the melt flow occurringduring a spinup or spindown of an RMF. The knowl-edge about the respective velocity fields provides a basisfor developing techniques for an efficient melt stirring.Let us consider a stagnant liquid metal in a circular

cylinder suddenly exposed to a RMF characterized bythe amplitude B0 and the frequency x during a timeperiod tp = t1 – t0 (Figure 2(a)). Figure 2(b) showswhat occurs when the sense of rotation of the fieldvector is inverted after the RMF pulse. This situationwill be discussed at the end of this section. At the time t0,the Lorentz force FL starts to accelerate the liquid. Thisprocess is called spinup. The Lorentz force is notuniform along the axial direction, and shows a maxi-mum at the horizontal midplane of the cylinder.Therefore, the fluid rotates faster here compared to theregions at the top and the bottom, where the velocitydeclines to zero at the wall across the developingBodewadt boundary layers. The consequence is animbalance of the centrifugal forces driving a radialoutward flow at the horizontal midplane. For reasons ofcontinuity, a secondary circulation is formed in themeridional plane, which appears as two toroidal vorti-ces. This first period of the spinup, in which viscouseffects play only a negligible role, is also called the initialadjustment phase.[17] The nonlinear interaction betweenthe primary and secondary flow becomes important inthe subsequent inertial phase. The development of the

980—VOLUME 38B, DECEMBER 2007 METALLURGICAL AND MATERIALS TRANSACTIONS B

secondary flow proceeds via inertial oscillations with thetwo-toroidal vortex pattern of the previous initialadjustment phase. As a result, the multilayered structureof the Bodewadt layers at the horizontal endwalls isestablished. For the supercritical values of the magneticinduction, Taylor–Gortler vortices appear near thesidewall layers and move from the horizontal midplanetoward the horizontal endwalls, where they will bedissipated. A detailed analysis of the flow structurearising from a single RMF pulse is reported byNikrityuk et al.[18]

In the present article, we consider the special case onlyif the RMF was switched off before completing thespinup process immediately at the end of the initialadjustment phase. Figures 3(a) and (b) show a transientflow pattern obtained by numerical simulations andUDV measurements in an experimental run, respec-tively, using an RMF pulse with the amplitude B0 of2.7 mT and the pulse length tp of 3.6 seconds, whichcorresponds to the duration of the initial adjustmentphase tia, for the case considered here. The coloredcontour plots display a sequence of the axial profiles ofthe vertical velocity. The amplitude of the velocity isrepresented by the color map. Drawing the particularvelocity profiles along the ordinate and using theabscissa as a timeline, a spatiotemporal structure ofthe flow can be visualized. The comparison between the

numerical and experimental results shows an excellentagreement. At the beginning, a descending flow appearsin the upper part of the container and an ascendingmotion appears in the lower part, which indicates theformation of two toroidal vortices usually appearingduring the initial adjustment phase. The shutdown ofthe RMF at tp = 3.6 s (mark A) does not extinguish thesecondary flow immediately. The double vortex struc-ture undergoes oscillations; specifically, the doublevortex decays temporarily (mark B) and will be dis-placed by another vortex pair emerging at the endwallsand spreading toward the midplane of the container

Fig. 2—Scheme of the applied RMF pulse: (a) single pulse or (b)pulse with subsequent reversal of the direction of the RMF.

(a)

(b)

Fig. 3—Spatiotemporal plots of the vertical velocity showing a flowpattern resulting from an RMF pulse with B0 = 2.7 mT: (a) numeri-cal simulation and (b) velocity measurements recorded atr/R0 = 0.6, where A through E are selected time marks for showingspecific flow structures.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 38B, DECEMBER 2007—981

(mark C). It is important to note that the velocity showsa reversal of its direction. At the midplane, a short-timeerasement of the flow structure seems to occur before

the vortices expand over the other half of the container(mark D). Valentine and Miller[24] studied the spindownof the flow inside a rotating cylinder numerically. Ourresults provide a good coincidence with their calcula-tions, which predict the propagation of a so-called‘‘solitary’’ wave from each endwall toward the midplaneof the cylinder. Whereas the velocities in this periodreach the same magnitude as those observed at the endof the initial adjustment phase, additional verticalstructures appear with decreasing intensity (mark E).Figure 4(a) contains a diagram showing the square

value of the vertical velocity Uz averaged along themeasuring line, giving an indication concerning theintensity of the meridional circulation:

(a)

Fig. 4—Flow pattern resulting from an RMF pulse with B0 = 2.7 mT:(a) Uz

2 averaged over the velocity profile (experimental result) and vectorplots of the meridional velocity (numerical results) for the times (b) 3.6 s(point A), (c) 5 s (point B), (d) 6.3 s (point C), and (e) 9.5 s (point D).

(a)

(b)

Fig. 5—Spatiotemporal plots of the vertical velocity showing a flowpattern resulting from an RMF pulse with B0 = 2.7 mT followed byan inversion of the direction of rotation: (a) numerical simulationand (b) velocity measurements recorded at r/R0 = 0.6.

982—VOLUME 38B, DECEMBER 2007 METALLURGICAL AND MATERIALS TRANSACTIONS B

U2z ¼

1

H0

ZH0

0

u2zdz ½5�

Corresponding snapshots of the meridional flowobtained from the numerical simulations are shown inFigures 4(b) through (e) for the time marks A, B, C, andD, respectively. The maxima of the intensity of themeridional flow (marks A, C, and D) coincide with theappearance of the large vortical structures. The initialadjustment time ti.a. denotes the time period until thefirst maximum of Uz

2 in Figure 4(a) occurs, i.e., ti.a. =t(A).

Following the idea of applying a pulse-shaped RMF,we have performed a similar experiment in which themagnetic field was not turned off but the sense ofrotation of the field vector was inverted after the firstRMF pulse. The modulation scheme is shown inFigure 2(b). Experimental and numerical results reveal-ing the resulting flow pattern can be found in Figures 5and 6. There is a continuous input of energy into thesystem, but if the direction of the driving force wasclockwise during the initial pulse, it changed to coun-

terclockwise afterward. The reorientation of the flowdirection does not occur simultaneously at differentheights of the cylinder, resulting temporarily in strongvertical gradients of the azimuthal velocity. This effectbecomes obvious from the numerical simulations of theazimuthal flow presented in the Figures 6(b) through(d). As already discussed, with respect to the spinupprocess, such gradients in the angular motion drive thesecondary, recirculating flow. Consequently, after theinversion of the driving force, the vertical velocityclearly exceeds the values occurring in the initial doublevortex (Figures 5 and 6(a)). These strong oscillationsdecay quickly and the flow undergoes the spinup processdescribed at the beginning of this section, until thesecondary flow reaches a quasi-steady state.

VI. THE RMF-PSCD

Section V showed the occurrence of strong velocityfluctuations and an amplification of the secondary flowarising from an impulsive change in the amplitude or thedirection of the electromagnetic force, respectively.Moreover, periodic reversals of the direction of thesecondary flow have been observed. It can be assumed

(a)

Fig. 6—Flow pattern resulting from an RMF pulse with B0 = 2.7 mT followed by an inversion of the direction of rotation: (a) Uz2 averaged

over the velocity profile (experimental result), contour plots of the azimuthal velocity (numerical results), and vector plots of the meridionalvelocity (numerical results) for the times (b) 5 s (point B), (c) 7.2 s (point C), and (d) 10 s (point D).

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 38B, DECEMBER 2007—983

that such features of the velocity field could be beneficialfor enforcing the mixing of the melt or for mitigating thesegregation during solidification. To be attractive fortechnological applications, this strong, oscillatingbehavior—especially of the secondary, recirculatingflow—has to be maintained over a longer time period—-for example, for a few minutes. This behavior may lenditself to meeting that goal by a repeated switching onand off of the RMF, i.e., the generation of a successionof RMF pulses, as shown in Figure 7(a). In the sameway, the rotating direction of the RMF can be inverted,as shown in Figure 7(b) and discussed in Section VII.Additional control variables appear, namely, the pulselength tp and the time between the pulses td, also calledthe dormant period.

In the present study, exclusively equal periods ofdriving and dormant phases are used. Thus, we considerthe dependence of the intensity of the secondary flowmerely on the magnetic-field strength B0 and the pulsefrequency fp = 1/(2 Æ tp).

Figure 8 displays spatiotemporal plots of the verticalvelocity obtained for RMF pulses with a magnetic-fieldstrength of 5.8 mT. The variation of the pulse frequencyfp shows a distinct influence on the secondary flow. Inthe case of low pulse frequencies (for example, fp = 0.2Hz in Figure 8(a)), the flow pattern is characterized bya noncontinuous double vortex structure appearing with

a periodicity fp. The interim time is characterized byvelocity oscillations at lower intensities. Reversals of theflow direction can be observed sporadically, but theoriginal direction of the secondary flow obviouslyprevails. In the experiment, the pulse frequency wasincremented in a stepwise manner. A new type of flowpattern has been found for a pulse frequency of0.475 Hz (Figure 8(b)). A pronounced double-vortexstructure appears that shows a periodic inversion of the

Fig. 7—Temporal scheme for the modulation of the electromagneticforce: (a) RMF-PSCD regime (b) RMF-PSAD regime.

(a)

(b)

(c)

Fig. 8—Spatiotemporal plots of the vertical velocity for RMF-PSCDregime, B0 = 5.8 mT: (a) fp = 0.2 Hz, (b) fp = 0.475 Hz, and (c)fp = 0.77 Hz.

984—VOLUME 38B, DECEMBER 2007 METALLURGICAL AND MATERIALS TRANSACTIONS B

sign of vorticity. Note that such a flow pattern can onlybe observed for a very narrow range of frequenciesaround the value of 0.475 Hz, for the particular fieldstrength of 5.8 mT. An additional increase of fp impliesa weakening of the secondary flow. The situation canalso be illustrated by the time series of the local velocitycontained in Figure 9. The time series were recorded at avertical position of 10 mm above the bottom of the fluidcontainer, for the same radial position and RMFparameters as shown in Figure 8. Adjusting the fre-quency of 0.475 Hz provides regular oscillations of thevertical velocity. Moreover, the amplitude of the veloc-ity oscillations exceeds the values obtained for bothsmaller and higher pulse frequencies. Thus, an optimalvalue for the pulse frequency has been identified. Anexplanation of this phenomenon requires an under-standing of the interaction between the primary and thesecondary flow. The energy transfer between the rotarymotion and the meridional flow is governed by inertialwaves. Recently, the importance of these waves wasaddressed in the context of RMF-driven fluids.[17,25]

Preliminary results obtained by a detailed analysis of thefluid mechanical aspects reveal that a characteristic flowpattern, as shown in Figure 8(b), will be achieved if thepulse frequency fp corresponds to the eigenperiod ofinertial waves in a developed regime, as given byGreenspan.[26]

Additional measurements and numerical calculationshave been performed to investigate the dependence ofthe optimal value of the pulse frequency on themagnetic-field induction B0. It has been found that theoptimum of fp,optimum = 1/(2 Æ tp,optimum) correlates withan optimum pulse length of tp,optimum = t(B) - t(A),which means that the duration of the individual pulseshas to coincide with the time period occurring betweenthe first minimum and maximum of Uz

2, as shown inFigure 4(a).

VII. THE RMF-PSAD

Another method of stirring can be realized byapplying an alternately-pulsed RMF, i.e., the sense of

rotation will be periodically inverted. The alternatingcourse of the Lorentz force FL, which drives the fluidmotion, is shown in Figure 7(b). We have already seenin Section V that distinctive oscillations of the secondaryflow can be expected here with higher velocity ampli-tudes, as observed for the RMF-PSCD. It is self-evidentthat the pulse frequency fp = 1/(2 Æ tp) should haveconsiderable impact on the resulting flow structure forthe RMF-PSAD regime, as well.The respective spatiotemporal plots of the vertical

velocity recorded for different frequencies fp at amagnetic-field strength of 2.7 mT are depicted in Fig-ure 10. Figure 10(a) shows a typical situation whenrelatively long cycles (tp = 25 s, in the case underconsideration) have been selected. High intensities of thesecondary flow emerge directly after changing thedirection of the RMF. However, these large velocityvalues are efficiently damped, because of the stabilizingeffect arising from the upcoming rotary motion of theliquid metal. Subsequently, the fluid experiences theusual spinup process until the next reversal of therotational direction will be initiated. The intervalbetween the reversals was reduced until a frequency of0.08 Hz was reached. The corresponding velocity field inFigure 10(b) reveals strong fluctuations and permanentchanges of the sign of the velocity. The flow appearsrather complex; a stationary vortex structure cannot beidentified. A considerable decrease in the flow intensitybecomes apparent for frequencies above 0.1 Hz (as anexample, Figure 10(c), for fp = 0.2 Hz). Due to its owninertia, the fluid can no longer follow the alternatingforcing. To get a measure of the intensity of thesecondary flow, the square value of the vertical velocityuz2 was averaged over both the axial line where the

velocity profiles have been determined and the totalmeasuring time T:

�U2z ¼

1

T

ZT

0

1

H0

ZH0

0

u2zdzdt ½6�

Figure 11 shows a diagram in which the resultingmean values �U2

z are drawn vs the frequency fp. Similar tothe findings for the RMF-PSCD regime, it becomesobvious that pronounced maxima of �U2

z occur atselective frequencies, varying for different values of themagnetic-field strength B0. Note that the location of themaxima corresponds to an optimal pulse length tp =t(A) = ti.a. Whereas the intensity of the secondary flowat lower frequencies shows a level comparable to apermanently-applied RMF, a drastic decay of thesecondary flow has to be noticed at higher frequencies.In the present study, it was not planned to measure

the form of the free surface quantitatively during thestirring. However, the experimental configuration al-lowed for a visual observation in parallel with thevelocity measurements. In a simple, supplementaryexperiment, a continuously-applied RMF was comparedwith the stirring using the RMF-PSAD regime. Amagnetic-field strength of 8.3 mT was chosen for thecontinuous RMF. Here, a distinct deflection of the freesurface could be observed easily. Then, the melt was

time [s]

Fig. 9—Time series of the vertical velocity at z = 10 mm and r/R0 = 0.6 (RMF-PSCD regime, B0 = 5.8 mT).

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 38B, DECEMBER 2007—985

stirred by means of an alternately-pulsed magnetic fieldwith B0 of 4.4 mT and a frequency fp of 0.15 Hz. Forthat case, the velocity measurements showed nearly thesame intensity of the vertical velocity, in terms of thevalue �U2

z . The respective spatiotemporal plots of thevertical velocity can be found in Figure 12. Whereas the

stirring with the continuous RMF causes distinctperturbations of the free surface, an almost planarshape of the surface can be maintained in the case of thealternately-pulsed field (Figure 13).

Fig. 10—Spatiotemporal plots of the vertical velocity: RMF-PSADregime, B0 = 2.7 mT: (a) fp = 0.02 Hz, (b) fp = 0.08 Hz, and (c)fp = 0.2 Hz.

(a)

(b)

Fig. 12—Spatiotemporal plots of the vertical velocity: (a) continu-ously-applied RMF, and B0 = 8.3 mT; and (b) RMF-PSAD regime,B0 = 4.4 mT, and fp = 0.15 Hz.

Fig. 11—Dependence of the intensity of the secondary flow on thefrequency of reversals fp of rotational direction: �U2

z is the averagevalue over the velocity profile and the measuring time.

986—VOLUME 38B, DECEMBER 2007 METALLURGICAL AND MATERIALS TRANSACTIONS B

VIII. DISCUSSION

Our approach to achieving an efficient mixing in themelt is based on a considerable amplification of themeridional, secondary flow, without additional acceler-ation of the primary, rotary motion. In comparison witha continuously-applied RMF, the same or even a highermixing effect can be realized, accompanied by lowerenergy consumption. The flow measurements show thecapability of overcoming the limited mixing character ofconventional rotary stirring, specifically, of providing a

vigorous mixing in the bulk of a liquid metal column,only in connection with a considerable deformation ofthe upper surface. Both stirring regimes studied in thepresent article generate high mixing intensities through adistinct intensification of the secondary flow, whichreduces the mixing time of the process under consider-ation, but leaves the free surface almost unaffected.These features make the electromagnetic stirring bymeans of RMF pulses very attractive for variousindustrial applications in which RMFs are alreadyapplied for melt agitation. For example, specific stirringtechnologies have been proposed for steelmaking pro-cesses using two coil systems.[4,14] The method presentedhere requires only one coil system and consumes a smallamount of electric energy. Furthermore, the secondaryflow can be organized in such a way that periodicreversals of the flow direction occur. This featurebecomes important with respect to an application forunidirectional solidification of metal alloys, because itoffers the potential for preventing macrosegregation.The outcome of the present study demonstrates that a

preferential intensification of the secondary flow re-quires the careful adjustment of the respective frequen-cies fp for the pulsing and the inversion of the rotationaldirection of the RMF. The respective optimum for fpdepends on the magnetic-field strength B0, the materialproperties of the liquid, and the geometry underconsideration. A mismatch of the relevant parametersmay prevent an improvement of the mixing quality;actually, the results may become worse compared to thecase of a continuously-applied RMF. An importantfinding is that the optimum conditions for the stirringcan be achieved if the pulse length tp is chosen to be t(B)- t(A) and t(A) for the RMF-PSCD and the RMF-PSAD regime, respectively. It is worth noting that thecharacteristic time scales t(A) = tia and t(B), knownfrom the fluid mechanical studies of the RMF spinup[17]

or the single RMF pulse,[18] play a significant role in thestirring regimes studied here. This finding can be used asa kind of recipe for discovering the ideal stirringfrequencies fp; however, the characteristic time scaleshave to be specified separately, with respect to theparticular experimental configuration.

IX. CONCLUSIONS

This article presents a tailored method of electromag-netic stirring. An RMF acts upon a column of liquidmetal and the amplitude of the field strength has beenmodulated in two different ways: a sequence of RMF-PSCD pulses (Section V) and a pulse sequence of RMF-PSAD, in which the direction of rotation of the field isinverted periodically (Section VI). Numerical calcula-tions and model experiments showed very good agree-ment with respect to the resulting flow structures.The electromagnetic stirring method that uses a time-

modulated RMF offers considerable potential for a well-aimed modification of casting properties, also. Both theRMF-PSCD and the RMF-PSAD stirring regimesenable us to avoid macrosegregations, by inducingperiodic reversals of the recirculating, secondary flow.

Fig. 13—Photographs of the free surface of the melt during stirring:(a) continuously-applied RMF, and B0 = 8.3 mT; and (b) RMF-PSAD regime, B0 = 4.4 mT, and fp = 0.15 Hz.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 38B, DECEMBER 2007—987

Additional research activities on this subject are underprogress. The application during the directional solid-ification of Al-Si alloys is the subject of Part II of thisarticle.

ACKNOWLEDGMENTS

This work was financially supported by DeutscheForschungsgemeinschaft (DFG) in the form of the col-laborative research center SFB 609, ‘‘ElectromagneticFlow Control in Metallurgy, Crystal Growth and Elec-trochemistry.’’

NOMENCLATURE

B0 magnetic inductionJ Bessel functionU2

z square value of the vertical velocity averagingalong the z axis

�U2z time-averaged value of U2

zfp pulse frequencytp pulse lengthtia duration of the initial adjustment phaseFL Lorentz forceH0 height of cylinderR0 radius of cylinderr,h,z cylindrical coordinate axesu velocityS shielding parameterp pressureN magnetic interaction parameterRem magnetic Reynolds number

GREEK SYMBOLS

x angular frequency of the magnetic fieldr electrical conductivityq densityk root of J¢(x) = 0l0 magnetic permeabilitym kinematic viscosity

REFERENCES1. P.A. Davidson: An Introduction to Magnetohydrodynamics, Cam-

bridge University Press, Cambridge, United Kingdom, 2001, pp.285–300.

2. K.-H. Spitzer, G. Reiter, and K. Schwerdtfeger: ISIJ Int., 1996,vol. 36, pp. 487–92.

3. K.-H. Spitzer: Prog. Cryst. Growth Charact. Mater., 1999, vol. 38,pp. 59–71.

4. F.-C. Chang, J.R. Hull, and L. Beitelman: Metall. Mater. Trans.B, 2004, vol. . 35B, pp. 1129–37.

5. D. Perrier, Y. Fautrelle, and J. Etay: Metall. Mater. Trans. B,2003, vol. 34B, pp. 669–78.

6. C. Vives: Metall. Trans. B, 1993, vol. 24B, pp. 493–510.7. J.K. Roplekar and J.A. Dantzig: Int. J. Cast Met. Res., 2001, vol.

14, pp. 79–95.8. B. Willers, S. Eckert, U. Michel, I. Haase, and G. Zouhar: Mater.

Sci. Eng. A, 2005, vol. 402, pp. 55–65.9. J. Priede and G. Gerbeth: J. Cryst. Growth, 2005, vol. 285, pp.

261–69.10. P.A. Nikrityuk, K. Eckert, and R. Grundmann: Int. J. Heat Mass

Transfer, 2006, vol. 49, pp. 1501–15.11. B. Willers, S. Eckert, P.A. Nikrityuk, and K. Eckert: in Modeling

of Casting, Welding, and Advanced Solidification Processes-XI, C.-A. Gandin and M. Bellot, eds., TMS, Warrendale, PA, 2006, pp.333–40.

12. S. Steinbach and L. Ratke: Mater. Sci. Eng. A, 2005, vols. 413–414, pp. 200–04.

13. P.A. Nikrityuk, K. Eckert, and R. Grundmann: Proc. Conf.Turbulence and Interactions TI2006, ONERA, Toulouse, 2006.CD.

14. S. Taniguchi, K. Mataike, M. Okubo, T. Ando, and K. Ueno:Proc. 4th Int. Conf. on Electromagnetic Processing of Materials(EPM2003), ISIJ, Tokyo, 2003, pp. 339–43.

15. S. Kojima, T. Ohnishi, T. Mori, K. Shiwaku, I. Wakasugi, and M.Ohgarni: 66th Steelmaking Conf., Atlanta, GA, ISS-AIME,Warrendale, 1983, pp. 127–31.

16. M. Ungarish: J. Fluid Mech., 1997, vol. 347, pp. 105–18.17. P.A. Nikrityuk, M. Ungarish, K. Eckert, and R. Grundmann:

Phys. Fluids, 2005, vol. 17, pp. 067101-1–067101-17.18. P.A. Nikrityuk, S. Eckert, and K. Eckert: Eur. J. Mech. B/Fluids,

2007, doi: 10.1016/j.euromechflu.2007.05.004.19. Y. Takeda: Nucl. Eng. Des., 1991, vol. 126, pp. 277–84.20. D. Brito, H.-C. Nataf, P. Cardin, J. Aubert, and J.P. Masson: Exp.

Fluids, 2001, vol. 31, pp. 653–63.21. A. Cramer, C. Zhang, and S. Eckert: Flow Meas. Instrum., 2004,

vol. 15, pp. 145–53.22. L.M. Witkowski, J.S. Walker, and P. Marty: Phys. Fluids, 1999,

vol. 11, pp. 1821–26.23. J.H. Ferziger and M. Peric: Computational Methods for Fluid

Dynamics, Springer, New York, NY, 2002, pp. 196–200.24. D.T. Valentine and K.D. Miller: Phys. Fluids, 1994, vol. 6, pp.

1535–47.25. P.A. Davidson and F. Boysan: Ironmaking and Steelmaking, 1991,

vol. 18, pp. 245–52.26. H.P. Greenspan: The Theory of Rotating Fluids, Breukelen Press,

Brooklyn, NY, 1990, pp. 81–85.

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