lorentz force transient response at finite …...ieee transactions on magnetics, vol. 52, no. 8,...

IEEE TRANSACTIONS ON MAGNETICS, VOL. 52, NO. 8, AUGUST 2016 6201611

Lorentz Force Transient Response atFinite Magnetic Reynolds Numbers

Vinodh Bandaru, Igor Sokolov, and Thomas Boeck

Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Ilmenau D-98684, Germany

In this paper, we investigate the transient response of Lorentz force at finite magnetic Reynolds numbers Rm on an electricallyconducting rectangular bar that is strongly accelerated in the presence of a localized magnetic field. This is done through numericalsimulations utilizing a coupled finite-difference boundary element approach. The results show good qualitative agreement with existingexperiments with a circular cylinder. The Lorentz force rise time is seen to be a linear function of Rm. The linear dependence ofLorentz force on Rm is found to be valid only for low values of Rm, after which the slope decays leading to an apparent saturationin the Lorentz force at sufficiently large values of Rm. Our results provide important information for the development of Lorentzforce flow meters for transient flow applications.

Index Terms— Lorentz force velocimetry (LFV), magnetic Reynolds number, transient response.

I. INTRODUCTION

ELECTRICALLY conducting flows are often encounteredin various industrial processes such as molten metal flows

in the continuous casting of steel and aluminum and in liquidmetal cooling blankets for fusion reactors [1]. An accuratemeasurement of the flow rates involved may serve to ensureoverall process efficiency and product quality. Among thevarious methods to measure these flows [2], contactless mea-surement techniques are ideal for this purpose and have gaineda lot of attention in the past decade. This is due to thefact that optical methods are unsuitable for flows involvingliquid metals due to their opacity, whereas traditional flowmeasuring probes cannot withstand the environment of hotand chemically aggressive melts. One such recent contactlessmeasurement method that is promising is known as Lorentzforce velocimetry (LFV) [3], [4], the fundamental principleof which is to reconstruct velocity fields in conducting flowsby measuring the Lorentz force that acts on a permanentmagnet (or magnet systems) placed in the vicinity of the flow.Beginning with the measurement of integral quantities likevolume flow rates in channels, this technology has evolvedto a stage where it has recently been demonstrated that3-D velocity vectors fields can be mapped, with the use ofhigh-precision multi-component force and torque sensors [5].However, currently, it is only possible to measure steadyflows through LFV. This is because in a strongly transientflow, a parameter known as the magnetic Reynolds num-ber (Rm) (defined as the ratio of the magnetic diffusiontimescale to the advective timescale) becomes finite. As aconsequence, the Lorentz force will no more be a linearfunction of the velocity and an additional time lag will occurbetween the flow velocity and the measured force signal.Such a behavior is currently not well understood as there

Manuscript received December 30, 2015; revised March 5, 2016; acceptedMarch 20, 2016. Date of publication March 23, 2016; date of cur-rent version July 18, 2016. Corresponding author: V. Bandaru (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2016.2546229

exist no studies that quantify the response of Lorentz force totime-varying/finite Rm flows. This is the main motivation forthis paper.

One of the main features in the case of a finite Rm flowis that the secondary magnetic field (generated by the eddycurrents in the fluid) is significant and hence leads to thedistortion of the originally imposed magnetic field. The studyof magnetic field distortion at finite Rm is nothing new. Ithas been investigated back in the 1960s, first by Parker [6]and Weiss [7] in the context of astrophysical phenomena.They observed magnetic flux expulsion when a conductingsolid/fluid motion is imposed upon a uniform magnetic fieldat high Rm . However, their studies were purely kinematic innature, in the sense that the effect of the field distortion onthe conductor motion is neglected. An attempt to theoreticallyunderstand the dynamic behavior in the finite Rm scenariowas first made by Reitz [8] using a simplified model relevantto applications concerning magnetic levitation and propulsion.This was revived very recently by Weidermann et al. [9]from the viewpoint of LFV, using a canonical 1-D problem oftime-varying motion of conducting rectangular slabs througha uniform magnetic field. They developed an approximate1-D analytical model to determine the Lorentz force responseat finite Rm . Subsequent to this, experiments were conductedby Sokolov et al. [10] in which solid conducting rods wereaccelerated through a localized magnetic field and the timedependence of the resulting Lorentz force was measured.However, a fully consistent theoretical treatment of such aproblem is missing.

In this paper, we investigate the problem of a strongly accel-erated rectangular conducting bar in the presence of a localizedmagnetic field using numerical simulations. This is done witha two-fold purpose. First, this paper complements the existingexperiments allowing a comparison with them and also toextract information of the field variables that are not accessiblewith the experiments. In addition, this enables one to explorethe effects of Rm over a wide range of parameter spaces, whichis not practically possible to achieve through the experiments(experiments with up to Rm ≈ 4 has been performed).

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6201611 IEEE TRANSACTIONS ON MAGNETICS, VOL. 52, NO. 8, AUGUST 2016

This paper is organized as follows. In Section II, the problemsetup is described followed by a description of the governingphysical model and the computational procedure in Section III.This is followed by a discussion of the results from thepresent simulations and their comparison with experimentsin Section IV. A brief summary and outlook to future workfollows in Section V.

II. PROBLEM SETUP

The problem setup consists of a straight conducting barof length 1 m and a square cross section upon which astrong magnetic field (up to 0.2 T) is imposed only on ashort section of the bar. The magnetic field is produced byplacing a set of six equisized permanent magnets of size30 mm × 30 mm × 70 mm (three on either side of the bar)forming a linear Halbach array. The Halbach configuration isused so as to effectively channelize the resultant magnetic fieldfrom all the six magnets normal to the bar [11]. The bar isaccelerated very quickly from rest along its length. Typicalaccelerations considered here range from 0.4 to 4 ms−2. Thesetup is shown in Fig. 1, which is very similar to that of theexperimental setup in [10] except for the difference that rodsof circular cross section were used in the experiments. In theexperiments, rods of different diameters and materials wereused. A brief description of the experimental setup is given inAppendix A. Here, the width L of the bar is chosen so as tohave a cross-sectional area equal to that of the correspondingcircular rod used in the experiments.

The velocity of the bar V = V (t)ex is externallyimposed/prescribed and is known a priori. A typical accel-eration profile is shown in Fig. 2, which shows a rise in barvelocity from rest to ≈136 mm s−1 in a time of ≈0.07 s andthen settles to an approximately constant speed that is close tothe peak velocity. Different acceleration profiles are obtainedby varying only the maximum bar velocity and keeping therise time (≈0.07 s) to be the same. The profile has been chosento be the same as that generated by the motor accelerating therods in the experiments. This explains why the curve is notsmooth. We now turn to the physical modeling of the problemin the next section.

III. PHYSICAL MODEL AND NUMERICAL PROCEDURE

The magnetic field from the Halbach array is completelydiffused into the bar before it is accelerated. Hence, theinitial magnetic field distribution inside the bar is taken tobe the same as it would be without the bar. Furthermore, wedecompose the total magnetic field during the evolution asB0 + B, where B is the secondary magnetic field generateddue to the eddy currents in the conducting bar. It must benoted that the eddy currents considered here consist only ofthe conduction currents that are due to the motion of chargesrelative to the conductor [12]. The advective time scale ischosen to be the typical time that it takes to accelerate the barfrom rest to its peak velocity, which is taken as tadv = 0.067 sin our case. Since there is no inherent velocity scale, we chooseL/tadv, where L is the width of the bar. Besides, we use thescales B0 and B0/μ0 L for the magnetic field and the current

Fig. 1. (a) Schematic of the problem setup showing the strongly accel-erated conducting bar with three magnets placed on either side forming alinear Halbach array configuration. The black arrows indicate the directionof magnetization of the individual magnets. (b) Top view of the sameconfiguration.

densities, respectively, where B0 is the maximum value of themagnitude |B0| and μ0 is the magnetic permeability of freespace. Using these scales, the dimensionless variables can bedefined as

v = V tadv

L, [b, b0] = [B, B0]

B0and j = J

(B0

μ0 L

)−1

. (1)

With this, we obtain the dimensionless form of the magneticinduction equation within the conductor as

∂b∂ t

= −∇ ×[

jRm

− v × (b0 + b)]

(2)

where the current density j = ∇ × b. The magnetic Reynoldsnumber is defined as Rm = L2/(λtadv), where λ = (μ0σ)

−1 is

BANDARU et al.: LORENTZ FORCE TRANSIENT RESPONSE AT FINITE MAGNETIC REYNOLDS NUMBERS 6201611

Fig. 2. Typical velocity of the conducting bar as a function of time.

the magnetic diffusivity, σ being the electrical conductivity ofthe bar. Here, v(t) = v(t)ex is the time-dependent uniformbar velocity known a priori from the experiment as, forexample, in Fig. 2. The bounds of the conducting domain are(0 ≤ x ≤ 3π), (−0.5 ≤ y ≤ 0.5), and (−0.5 ≤ z ≤ 0.5).Periodicity is assumed in the x-direction, which is fullyjustified by the fact that the magnetic field is localized andhas no impact at both ends of the bar. It is important tonote that the boundary conditions for the magnetic field in they- and z-direction are non-trivial. This is due to the fact thatthe secondary magnetic field extends into the non-conductingexterior space where it is additionally curl free and isgoverned by

∇2ψ = 0, b = −∇ψ (3)

where ψ is the magnetic scalar potential in the exterior. Sincewe are interested only in the fields within the conductor,magnetic boundary conditions for the conductor must ensureproper matching with the exterior field at the boundaries inthe y- and z-directions. This is done through the boundaryintegral procedure, which will be explained next along withthe numerical scheme.

A. Numerical Procedure

The solution to the problem is obtained numerically througha coupled finite-difference boundary integral procedure [13],which is briefly explained here. The conducting domain isdiscretized into a rectangular Cartesian grid and the fieldvariables are approximated at the grid points. A non-uniformgrid is used in the y- and z-direction, which is obtained by acoordinate transformation from the uniform-grid coordinates(ζ, η) as

y = Ltanh(Syζ )

tanh(Sy), z = L

tanh(Szη)

tanh(Sz)(4)

where Sy and Sz correspond to the degree of stretching inthe y- and z-direction, respectively. This ensures a finer gridnear the boundaries in order to resolve the steep gradients.However, a uniform grid is chosen is the x-direction to takeadvantage of the periodicity through Fourier transformation.

The governing equation (2) in the strict interior of the baris discretized using second-order finite differences with thesource term treated explicitly as

3bi+1 − 4bi + bi−1

2t= −[2(∇ × ε)i − (∇ × ε)i−1] (5)

where ε represents the electric field given by ε = j − v ×(b0 + b) and the superscripts represent the time levels,i being the current time level. In addition, since the exterioris electrically insulating, the boundary normal component of∇ × ε is continuous across the y and z boundaries [12], [14].Therefore, the discrete form of the induction equation canbe applied only for the normal component of the secondarymagnetic field bn as

3bi+1n − 4bi

n + bi−1n

2t= −[

2(∇ × ε)in − (∇ × ε)i−1n

]. (6)

Here, n represents the outward boundary normal direction.We now introduce a Fourier transformation in the

x-direction as

ψ(x, y, z) = �

⎧⎪⎨⎪⎩

k= Nx2 −1∑

k=0

ψ̂k(y, z) exp(iαk x)

⎫⎪⎬⎪⎭ (7)

where � represents the real part, Nx is the number of gridintervals along the x-direction, and αk is the streamwisewavenumber defined as αk = 2πk/ lx , lx being the length ofthe bar. The governing Laplace equation (3) for the exteriormagnetic potential transforms to the 2-D Helmholtz equationin the k-space as (∇2 − α2

k

)ψ̂k = 0. (8)

Equation (8) along with the far-field condition can be used toderive the boundary integral equation (valid on the rectangularcontour)

β(r ′)ψ̂k(r ′)

= P.V.∮Γ

[Gk(r ′, r)b̂nk(r)+ ψ̂k(r)

∂Gk

∂n(r ′, r)

]dl(r) (9)

where b̂nk(r) = −∂ψ̂k/∂n(r) and β(r ′) is a constant thatdepends on the location of the pole r ′ on the rectangularboundary, and is given by

β(r ′) ={

34 , if r ′ ∈ corner12 , otherwise.

(10)

Here, Gk(r ′, r) represents the Green function or the funda-mental solution of the 2-D Helmholtz operator [15].

Transforming only bn on the lateral face boundaries (inthe y and z directions) obtained from (6) into the Fourierspace provides b̂nk , which is necessary for (9). Using theboundary element method [16], the discrete linear system ofthe boundary integral equation (9) for each wavenumber k canbe written as

Qψ̂ki+1 = mi+1. (11)

Matrix Q is fully occupied due to the non-local nature of theboundary conditions and vector mi+1 contains the first term


on the right-hand side of (9). The tangential components area function of ψ̂ i+1

k as

b̂i+1xk = −iαkψ̂

i+1k , b̂i+1

τk = −∂ψ̂i+1k

∂τ. (12)

Equation (5) is used to compute b in the interior and (6) isused to compute the normal component of b on the boundary.Equations (11) and (12) provide the Fourier coefficients ofthe boundary tangential components, which are transformedback to the real space through an inverse FFT operation. Thiscompletes the computation of b at each time step. We now turnto the modeling of the magnetic field of the Halbach magnetsystem.

B. Magnetic Field of the Halbach Array

In order to model the magnetic field distribution b0 arisingfrom the Halbach array, it is assumed that each of the six mag-nets have unidirectional and constant magnetization. Further-more, it is also assumed that the significant magnetic repulsiveforces that are overcome to form the Halbach configuration donot affect the magnetization in the magnets. In such a case, themagnetic field from a single cuboidal magnet can be expressedin a closed analytical form. For example, the magnetic fieldat a point (x, y, z) outside a magnet of magnetization Ms ez

and with edge coordinates (x1, x2), (y1, y2), and (z1, z2) isgiven by

B0(x, y, z)

(μ0 Ms

4π

)−1

=k=2∑k=1

m=2∑m=1⎡

⎢⎢⎣(−1)k+m ln[P(x, y, z, xm , y1, y2, zk)](−1)k+m ln[H(x, y, z, x1, x2, ym , zk)]

n=2∑n=1

(−1)k+n+m tan−1[(x−xn)(y−ym)

(z−zk)g(x, y, z, xn , ym , zk)

]

⎤⎥⎥⎦

(13)

where the functions P , H , and g are given by

P(x, y, z, xm , y1, y2, zk)

= (y − y1)+ [(x − xm)2 + (y − y1)

2(z − zk)2]1/2

(y − y2)+ [(x − xm)2 + (y − y2)2(z − zk)2]1/2 , (14)

H (x, y, z, x1, x2, ym, zk)

= (x − x1)+ [(x − x1)2 + (y − ym)

2(z − zk)2]1/2

(x − x2)+ [(x − x2)2 + (y − ym)2(z − zk)2]1/2 , (15)

g(x, y, z, xn, ym , zk)

= 1

[(x − xn)2 + (y − ym)2(z − zk)2]1/2 . (16)

The reader is referred to [17] for the derivation of the aboveequations. The imposed magnetic field distribution b0 in thesimulation is the superposition of the fields of the six magnets,each of which is evaluated according to the dimensionlessversions of (13)–(16), with a suitable transformation when themagnetization is not aligned in the z-direction. The magneticfield distribution inside the bar due to the Halbach array isshown in Fig. 3(b). It can be seen that the primary field

Fig. 3. Streamlines of the initial magnetic field b0 in the bar. (a) 3-D linecoloring represents the field magnitude. The red, green, and blue colored axescorrespond to the x-, y-, and z direction, respectively. (b) In the xz-plane, thedirection of motion is from left to right. Only a part of the bar length is shown.The white dotted lines indicate the extent of the magnets in that direction.

component B0z reverses its direction (as one moves along thelength of the bar) through an X-point.

In order to verify that such an analytical description is closeto the field produced in the experiment, we compared themagnetic flux density measurements taken at specific locationsin the vicinity of the magnet system. In the experiment,a Lakeshore 475 DSP gaussmeter and a CYTHS124 Hallsensor array (with seven sensors) have been used to obtainthe magnetic field data.

Fig. 4(b) shows the variation of the primary component ofthe magnetic field B0z at a point midway between the magnets[marked G in Fig. 4(a)] with the distance of separationbetween the magnets. A very close agreement between theexperiment and the analytical model is observed in this case.In fact, the value of the magnetization Ms was determined asthat which leads to this close match of B0z . This is necessarybecause of insufficient information about Ms of the magnetsused in the experiment. Furthermore, Fig. 4(c) and (d) shows,respectively, the variation of B0z along the lines (X ′,Y ′) =(−7 mm, 0 mm) (below the bottom level of the magnets)and (Y ′, Z ′) = (0 mm, 45 mm) (along the symmetry line).The data were obtained in the experiment by sequentiallytraversing the gaussmeter probe along these paths. In thiscase too, the agreement between the analytical model andthe experiment is good with slight differences observed atcertain locations. However, it can be seen from Fig. 4(e)that the values of the out-of-plane component B0y along thecenterline does not match well. For reasons of symmetry, theanalytical model predicts vanishing the y-component in thoselocations contrary to a clear trend seen from the measurements


Fig. 4. (a) Schematic showing the coordinate system and the magnetic field sensor locations in the setup. H1–H7 represent the Hall sensor array placedbetween the rod and the magnet system. Comparison of the imposed magnetic field component B0z (b) at the location of the Gauss sensor located midwaybetween the magnets (marked by the black dot labeled G in the schematic), as a function of the separation distance (Lsep) between the magnets, (c) alongthe line X ′ = −7 mm, and (d) along the line Z ′ = 45 mm. (e) B0y along the line Z ′ = 45 mm and (f) B0z at the locations of the Hall sensors. All the plotscorrespond to the midplane y = 0.

(although the magnitude itself is very low ∼10 mT).In addition, Fig. 4(f) shows the comparison of B0z at thefixed locations of the seven Hall sensors. It is observed thatthe analytical model significantly overpredicts at the locationsof the sensors H4 and H5, although the agreement is fairly

good at the other locations. These differences can be attributedto the shortcomings of the assumptions of uniform (andunidirectional) magnetization and neglecting the alteration thatmight occur in the magnetization when the strong magnetsare brought together. It must be noted that the differences


Fig. 5. Field lines of the magnetic field in the xz midplane near the X-pointat the (a) initial state t = 0 and (b) final steady state when the rod reacheda constant velocity at Rm = 10. Contour coloring is by the 2-D magnitude((b0x + bx )

2 + (b0z + bz)2)

1/2of the total magnetic field.

observed at the sensors H4 and H5 (that are exterior to thebar) are 54% and 73%, respectively, indicating that there is apossibility that the differences of the imposed magnetic fieldmight be of the same order of magnitude at other locationsinside the material of the bar. In summary, the analytical modeldescribes the Halbach magnetic field very well for the mostpart, but also shows significant differences at some locationsthat were examined. With this in mind, we now turn to theresults obtained from the simulations.

IV. RESULTS AND COMPARISON WITH EXPERIMENTS

In order to facilitate comparison with the experiments,simulations were done for square cross-sectional bars of thesame cross-sectional area as those done for the rods used inthe experiments. This would imply that Rm in the simulationswill be slightly lower than that in the experiments of thecorresponding configuration. Although several different barsizes were considered, the primary focus here will be on theconfiguration with a copper bar of 53.26 mm × 53.26 mmcross section (that corresponds to the rod of diameter 60 mmused in the experiments) and a bar-to-magnet surface distanceof 15 mm. This will be the case for most part of this section,unless otherwise stated explicitly. Simulations are carried outduring the same time window, as shown in Fig. 2, i.e., fromthe state of rest until an approximately steady state is reached,with a velocity peak in between. A grid size of 256 × 642

with stretch factors Sy = Sz = 1.5 has been used for all thesimulations.

At first, a qualitative picture of the steady state can beobtained from the streamlines of b at Rm = 10 in themidplane (xz plane) shown in Fig. 5. One can see that thefield lines are advected (the X-point as well) in the direction

Fig. 6. Streamlines of the current density in the steady state at Rm = 3.2.Contour coloring is by the magnitude of current density. The red, green, andblue axes correspond to the x-, y-, and z direction, respectively.

of motion of the bar. This is seen to occur through a seriesof severings and reconnections occurring in the vicinity of theX-point.

The current density streamlines display a three-roll struc-ture, as can be seen from Fig. 6, which corresponds toRm = 3.2. The current density on the surface has thelargest magnitude as displayed by the streamline coloring.In particular, the maximum values of | j | were observed onthe four edges of the bar. The corresponding surface stream-lines are shown in Fig. 7(a), of particular mention being theexistence of critical points (or zero points) of current densityon the y faces (green streamlines).

We now move on to the quantitative results concerning theintegral streamwise Lorentz force over the domain, which isthe quantity of primary interest in this paper. This is computedat regular time intervals in our simulation using

Fl = B20 L2

μ0

∫[ j × (b + b0)]dxdydz. (17)

In the case of a bar acceleration involving a peak velocityVmax = 136.5 mm s−1, the time response of Lorentz forceis shown in Fig. 8(a). One can observe that the Lorentzforce follows a similar profile as the bar velocity, but witha time lag/shift that can be seen from the respective peakvalues. Qualitatively, this is similar to the curve obtained inthe experiment using piezoelectric force sensors. However,the value of the peak Lorentz force is overpredicted in thesimulation by a factor of ≈3.4. This can be clearly attributedto two reasons, both related to the geometry of the problem.The primary geometric difference is in the magnetic fielddistribution of the Halbach array, where there were indications(although in the bar exterior) that the model overpredicts thefield by about 73%, which roughly translates to a factor of1.732 ≈ 3.0 in the Lorentz force (since FL ∼ B2

0 ). Thesecond difference is in the shape of the cross section. For thesereasons, only qualitative comparisons between the experimentand simulations will be discussed. It must also be noted herethat the experimental curve shows a slow decay of Lorentzforce even in the steady state (which has been attributed tocharge leakage effects in the piezoelectric sensors), whereasthe simulations predict a flat profile physically consistent withthe velocity profile.

It is also interesting to see how the predictions of theexisting approximate the 1-D model [9] compared with


Fig. 7. Projected streamlines (a) on the boundary faces and (b) at specific planes of the current density j in the steady state at Rm = 3.2. The red streamlinesare on xy planes.

those of the 3-D simulations performed here. It is recalledhereby that the 1-D model is described by a dimension-less total magnetic field of the form b(z, t)ex + ez , wherethe evolution of the secondary magnetic field b(z, t) isgiven by

∂b

∂ t= ∂2b

∂z2 (18)

b(z, 0) = 0,∂b

∂z(±0.5, t) = −v(t) (19)

and the imposed field is considered to be uniform in thez-direction. The above governing equations of the model aresolved using a finite-difference scheme. For the purpose ofcomparison, a volume-averaged value of the primary compo-nent b0z of the Halbach array in the range 0 mm ≤ X ′ ≤45 mm is used as the uniform value of the imposed magneticfield required for this model. From Fig. 8(a), we can observethat the 1-D model predicts a peak Lorentz force of ≈28 N,which is significantly higher than that of the value (≈12 N)obtained here.

The study of the sensitivity of the result to the grid resolu-tion, grid stretching, and the bar length considered indicates

that the problem is well resolved beyond doubt. Details of thesensitivity study are given in Appendix B.

Furthermore, simulations were also performed with a widerange of peak velocities occurring within the same timeinterval. As expected, the peak Lorentz force shows a lineardependence on the peak bar velocity [see Fig. 8(b)] and witha different slope compared with the measured values. Of keyinterest is the dependence of the peak Lorentz force on Rm .This is done with a fixed configuration and by changing onlythe electrical conductivity σ of the bar to values both muchlower and higher than those of copper. A range of magneticReynolds numbers 0.5 ≤ Rm ≤ 50 000 was considered.Although not relevant for LFV, the large values of Rm areincluded in our study as they are of theoretical interest. Thedependence is shown in Fig. 9, in which the Lorentz forceplotted is normalized by B2

0 L2μ−10 .

The low range of Rm is relevant to LFV, in which asone can observe from Fig. 9, the normalized Lorentz forceincreases linearly with increasing Rm for only low valuesof Rm . After this, the behavior significantly deviates fromlinearity and leads to Lorentz forces that are much lowerthan those predicted by a linear approximation. In quantitative


Fig. 8. (a) Lorentz force as a function of time for the copper rod (experiment)and the bar (simulations) of the same cross section. The additional subscript1-D refers to the result obtained from the 1-D model of [9]. The rod diameteris D = 60 mm and the bar cross section is 53.2 mm. Maximum velocityduring acceleration is Vmax = 136.5mm/s. (b) Maximum Lorentz force as afunction of Vmax.

Fig. 9. Normalized Lorentz force F∗L ,max as a function of Rm . Constant

A = 0.1.

terms, already at Rm = 10, the peak Lorentz force isreduced by approximately 12% when compared to its value atRm = 0. Such information is extremely useful in the

TABLE I

COMPARISON OF t∗rise BETWEEN THE EXPERIMENT AND THE SIMULATION

design and calibration of LFV in transient flow applica-tions. Furthermore, for very low values of Rm , it is pos-sible to normalize the force by σVmax B2

0 lmag L2, which isthe ideal force that would act on a bar length equal tothe length of the magnet array lmag, when a uniform fieldof magnitude B0 is imposed. Using such a normalization,it is interesting to note that the limiting Lorentz forcevalue obtained for low Rm nicely agrees with the Lorentzforce obtained by an independent simulation performed usingthe quasi-static formulation (using the electric potential; seeAppendix C). Further increase in Rm leads to a drasticdrop in the slope of the curve in the range approximately10 ≤ Rm ≤ 500, as shown in Fig. 9. Beyond this, the slopecontinues to decay leading to an apparent saturation of theLorentz force for sufficiently high values of Rm . The structureof the magnetic field in the plane y = 0 at Rm = 1000 isshown in Fig. 10, which is a consequence of no shear insidethe bar although everything is advected in the direction ofmotion. Due to this, the peak magnetic field remains at theboundaries and within the streamwise bounds of the magneticsystem. The same is found to occur for the peak Lorentz forcesat high Rm .

Beside the Lorentz force magnitude, the time lag (comparedwith Rm = 0) that is expected to occur at finite Rm isimportant in transient LFV applications. Information regardingthis is obtained through the dimensionless time t∗rise, which isthe time in advective units that it takes for the Lorentz force toincrease from a value of 2% to 98% of the peak Lorentz force.As can be seen from Fig. 11, the time lag linearly increaseswith Rm . Consistently, at low values of Rm , t∗rise tends tothe baseline, which is the time taken by the velocity itselfto rise from 2% to 98% of its peak value. At Rm = 10, thetime lag is already very significant, being approximately 40%of the time it takes for the corresponding velocity rise. Theslight deviation of the curve from linearity is attributed to thenon-smooth nature of the acceleration profile, due to which asmall ambiguity occurs in determining the exact time instantsat which the 2% and 98% Lorentz force values occur. Further,the time lag is seen to be reasonably close to the only measuredvalue in the experiment that corresponds to this configuration.

In the experiments, several measurements of t∗rise were madewith slightly different configurations and two different rodmaterials, namely, copper and aluminum. The various config-urations differ in geometry due to the various rod diametersused and the variation in the gap d between the rod surfaceand the magnets’ surface. These cases were simulated as well


Fig. 10. Field lines of the magnetic field in the xz midplane in the final steady state when the rod reaches a constant velocity at Rm = 1000. Contourcoloring is by the 2-D magnitude ((b0x + bx )

2 + (b0z + bz)2)

1/2of the total magnetic field. The red dotted lines indicate the extent of the magnets in that

direction.

Fig. 11. Time taken to reach from 2% to 98% of the peak Lorentz force,t∗rise (normalized by the advective time scale), as a function of the magneticReynolds number Rm .

(with an equivalent width L corresponding to D computedby matching the cross section) and a comparison of the risetime is shown in Table I. A maximum difference of ≈10%was observed between the simulation and experimental values.This supports reasonable agreement between the experimentand simulations.

V. CONCLUSION

Fully consistent 3-D simulations of the magnetic fieldevolution in the case of a quickly accelerated rectangular barthrough a Halbach magnetic field were carried out, throughwhich the dependence of Lorentz force and its time responseon Rm was obtained. It is found that the simplifications leadingto the existing 1-D model results in a significant ovepredictionof the peak Lorentz force by a factor of ≈2.3 comparedwith the full 3-D simulations. The Lorentz force rise timeis observed to be a linear function of Rm and is seen to showreasonably good agreement with the experiments that wereconducted on a similar problem but with a different geometry.For values of Rm ∼ 1, the Lorentz force linearly increaseswith Rm . Further increase in Rm shows deviation from the

Fig. 12. Schematic of the experimental setup showing the arrangement ofthe force sensor.

linear behavior. At very high values of Rm , the Lorentz forcesaturates asymptotically.

Future work must be directed toward extending this problemto the case of electrically conducting fluid flows. This helpsin providing a comprehensive theory on finite Rm effects thatcan be used for transient LFV.

APPENDIX AARRANGEMENT OF THE FORCE SENSOR

The experimental setup consists of two 10 mm-thick circu-lar aluminum plates, with a piezoelectric force sensor PCB208C01 mounted in between the plates as shown in Fig. 12.A hole 20 cm in diameter has been made at the center of theplates so that a massive conducting aluminum or copper rod4 cm to 8 cm in diameter can easily go through. A three-phasesynchronous motor is used to rotate a spindle, which througha mechanical arrangement is converted to the linear velocityof the rod. The rotation velocity of the spindle is controlledby a computer with 1 kHz frequency so that the rod can be


Fig. 13. (a) Sensitivity of the Lorentz force response to the length of thebar lx in the simulation. lx = 6π corresponds to the actual length of the rodin the experiment. (b) Sensitivity of the Lorentz force response to the gridstretch factor S = Sy = Sz .

accelerated up to 135 mm s−1 within the advection time of≈70 ms. For a detailed description of the setup and the drivemechanism, see [18]. On the top plate, the magnetic Halbacharray is placed, which creates a time-independent transversemagnetic field in the range from 0 to 1 T depending on thedistance between magnets. Due to the specific arrangement ofthe magnetization vectors, the magnetic field distribution inthe rod has four zones with sharp gradients, which leads toa higher Lorentz force amplitude. Signals are acquired by aneight-channel data logger Graphtec GL 900 with a 50 kHzsampling frequency. The piezoelectric force sensor has a non-linearity of less than 1% and a time response of ∼1 μs.

APPENDIX BSENSITIVITY OF LORENTZ FORCE TIME RESPONSE

The aim of this paper is to ensure that the results ofLorentz force response obtained from the simulations of theaccelerating bar problem are insensitive to the grid parametersand the chosen length lx of the domain. Fig. 13(a) shows thetransient response of the integral Lorentz force in the bar withvarious dimensionless lengths considered.

The actual length of the bar in the experiment wasLx,exp = 1 m, which translates to a dimensionless value of

Fig. 14. Sensitivity of the Lorentz force response to the (a) grid resolutionin the x-direction (direction of motion) and (b) grid resolution in the crosssection.

lx,exp = 18.8 ≈ 6π . It can be seen that already with a lengthlx = 2π , convergent results are obtained, and hence, all thesimulations are performed by modeling a length of 3π insteadof the full length of the bar. This also justifies the usageof periodic boundary conditions in the streamwise direction.Furthermore, Figs. 13(b) and 14(a) and (b) show clearly thatthe grid size of 256 × 642 and grid stretch factor S = 1.5that we use provides sufficient resolution required for thisproblem.

APPENDIX CQUASI-STATIC APPROXIMATION

In the case when the magnetic diffusivity (λ) is high orRm 1, the magnitude of the secondary magnetic field bis small (and can be neglected) although the Lorentz forcesare significant. In the limiting case of Rm → 0, it is possibleto obtain a simplified governing model that is approximatedto the first order and gets rid of dealing with the secondarymagnetic field altogether. This is commonly known as thelow-Rm or the quasi-static approximation. The fact that b isnegligible means that the total magnetic field remains equalto b0 and does not vary with time. This implies that the electricfield ε is curl free and can be expressed as the gradient of ascalar potential φ as ε = −∇φ. With this, the Ohm law and


its divergence will read as

j = −∇φ + v × b0 (20)∇2φ = ∇ · (v × b0). (21)

Here, simple boundary conditions arise, which do not requireconsidering the exterior at all. For example, the condition ofinsulating exterior translates to a Neumann boundary condi-tion, ∂φ/∂n = 0 on the lateral faces of the bar.

ACKNOWLEDGMENT

This work was supported in part by the DeutscheForschungsgemeinschaft (DFG) under the Research TrainingGroup 1567 and in part by the Helmholtz Alliance(Liquid Metal Technologies). The authors would like to thankProf. Y. Kolesnikov at TU Ilmenau, Germany, for providingfruitful discussions and the Computing Center of TU Ilmenaufor providing computer resources.

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