Analytic models of heterogenous magnetic fields for liquid metal flow simulations
aa r X i v : . [ phy s i c s . f l u - dyn ] M a y Analytic models of heterogenous magnetic fields for liquidmetal flow simulations
E. V. Votyakov, S. C. Kassinos, X. Albets-Chico
Computational Science Laboratory UCY-CompSci, Department of Mechanical and ManufacturingEngineering, University of Cyprus, 75 Kallipoleos, Nicosia 1678, CyprusCommunicated byReceived date and accepted date
Abstract.
A physically consistent approach is considered for defining an external magnetic fieldas needed in computational fluid dynamics problems involving magnetohydrodynamics (MHD).The approach results in simple analytical formulae that can be used in numerical studies wherean inhomogeneous magnetic field influences a liquid metal flow. The resulting magnetic field isdivergence and curl-free, and contains two components and parameters to vary. As an illustration,the following examples are considered: peakwise, stepwise, shelfwise inhomogeneous magneticfields, and the field induced by a solenoid. Finally, the impact of the streamwise magnetic fieldcomponent is shown qualitatively to be significant for rapidly changing fields.
There are few examples in recent history, when things being obvious to particular specialists, re-mained unexploited by people working in conjugated fields. These include, for instance, the Fast FourierTransform (FFT), which was originally used by Gauss in 1805 and then several times rediscovered byLanczos and Danielson in 1942, and Cooley and Tukey in mid-1960s . Another and more specific ex-ample is the so-called Savitzky-Golay smoothing filter. The least square method being the base of thefilter has been formulated hundred years before experimentalists working with spectra started to useit to treat their data. The paper that popularized this method for the experimentalists is one of themost widely cited papers in the journal Analytical Chemistry .What is considered in this letter is not as far reaching as the two aforementioned examples, never-theless, we believe it will help people studying numerically the flow of liquid metals under the influenceof an inhomogeneous magnetic field. Also, this letter is complementary to the discussion about mag-netic field models given earlier in Votyakov et al. [2008]. The cited work dealt with a 3D distributionof heterogeneous magnetic fields, which are of importance in the case of a magnetic obstacle, whilethe current letter considers simpler 2D distributions, which are needed, for example, for the properdescription of the flow through a fringing magnetic field.Let us recall shortly the effects of magnetic fields on conducting liquid flows. When the inducedmagnetic field is negligible, the external magnetic field interacts with the moving liquid and produces theLorentz force which brakes the flow in the direction of motion, see e.g. Davidson [2001]. This phenomenon Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ”Fast Fourier Transform.” Ch. 12.2 in NumericalRecipes in FORTRAN: The Art of Scientific Computing, 3rd ed. Cambridge, England: Cambridge University Press, Page498, 2007; see also http://mathworld.wolfram.com/FastFourierTransform.html A. Savitzky and Marcel J.E. Golay (1964). Smoothing and Differentiation of Data by Simplified Least Squares Pro-cedures. Analytical Chemistry, 36: 1627-1639; see also http://en.wikipedia.org/wiki/Savitzky-Golay smoothing filter
E. V. Votyakov, S. C. Kassinos, X. Albets-Chico is heavily exploited in many practical applications (Davidson [1999]), such as electromagnetic stirring,electromagnetic brakes, and non-contact flow measurements (Thess et al. [2006]).In reality, one always deals with an inhomogeneous magnetic field because creating a strong and homogeneous magnetic field for experimental needs remains a hard practical challenge. Despite thisfact, starting from the pioneering work of Hartmann & Lazarus [1937] many theoreticians love towork mostly with a homogeneous magnetic field. It is clear, that a constant magnetic field is alreadyresponsible for main phenomena such as the formation of the Hartmann and parallel layers. Nevertheless,a constant field could not produce the well developed M-shaped velocity profile frequently used for theelectromagnetic brake. Due to Kulikovskii [1968], who showed that the flow under the influence of astrong and slowly varying magnetic field can be subdivided in a core and a boundary layer, peoplestarted to exploit theoretically inhomogeneous magnetic fields. It was demonstrated that the flow goesparallel to characteristic surfaces, but all the calculations, including the numerical ones, were carried outeither by neglecting the second magnetic field component, see, e.g. Sterl [1990], Molokov & Reed [2003 b ],Molokov & Reed [2003 a ], Alboussiere [2004], Kumamaru et al. [2004], Kumamaru et al. [2007], Ni et al. [2007] or by employing an especially curvilinear channel Todd [1968] to match boundary conditions.In the case of numerical simulations, there is no particular technical problem preventing the inclusionof the second component of the inhomogeneous magnetic field; nevertheless this has not been done.A probable explanation for this is the absence of suitable and analytically simple models to define aninhomogeneous magnetic field. Therefore, there is the need for convenient formulae that could allowus to do so, and the goal of the paper is to provide a simple method that can be used in numericalstudies so that both components can be varied in a consistent way. At the end of the paper we discussbriefly the legitimacy of omitting the streamwise component of the magnetic field, and show that insome cases this might have been done due to large aspect ratios.It is worth to note that there are examples where correct expressions for a heterogenous magnetic fieldhave been used for MHD flow simulations. These are 2D numerical papers by Cuevas et al. [2006 a , b ]who correctly applied formulae taken from the book of McCaig [1977]. Those 2D simulations hiredonly the transverse component of the magnetic field while the second component played no role. Moresophisticated 3D cases are given in Votyakov et al. [2007, 2008], where all three components of themagnetic field are taken into account.The necessity to have at least two nonzero components of the inhomogeneous magnetic field (hereafterdenoted as MF) follows directly from the requirement that, in the flow region, an externally applied fieldmust be simultaneously divergence-free ∇ · B = and curl-free ∇ × B = . Thus, if the transverse MFcomponent varies along the streamwise coordinate, the streamwise MF component must vary consistentlyalong the transverse coordinate. (The spanwise component can be neglected without violation of thephysical correctness, when the magnetic field is two dimensional.) A vector field that is simultaneouslydivergence-free and curl-free is known as a Laplace vector field and it can be defined in terms of thegradient of any function η which is harmonic in the flow region, B = −∇ η , ∆η = 0. This function η is called the magnetic scalar potential. Although this is the most general approach, see e.g. McCaig[1977], it is not quite convenient because boundary conditions for η must be defined for each specificcase. However, what we need is a magnetic field which either vanishes or goes monotonically onto aconstant level far away from the central point where the intensity of the field is maximal. So, among allthe possible harmonic functions, we may select those that do not vary at far distances, and this formsthe basis for the proposed methodology.A simple, flexible, and physically consistent way to define an inhomogeneous MF for parametricnumerical needs is based on the magnetic field induced by a single magnetic dipole. This field is localin space and its magnetic scalar potential, which below is called ”an elementary potential”, belongs toharmonic functions. The whole magnetic field is the sum of local fields from the single magnetic dipoles,therefore, the whole magnetic field can also be represented as the gradient of a scalar function, i.e. as awhole scalar potential. In other words, the single magnetic dipole can be taken as an elementary unitin the appropriate spatial distribution of magnetic sources. The field B ′ ( r , r ′ ) created at r = ( x, y, z )by a single dipole m = (0 , , m ) located at r ′ = ( x ′ , y ′ , z ′ ) is given by Jackson [1999]: B ′ ( r , r ′ ) = ∇ × (cid:18) ∇ | r − r ′ | × m (cid:19) = ( m · ∇ ) (cid:18) ∇ | r − r ′ | (cid:19) = m ∇ ∂∂z (cid:18) | r − r ′ | (cid:19) , (1) nalytic models of heterogenous magnetic fields where we have used few vector identities and omitted the constant µ / (4 π ). Let the dipoles be distributedin the finite region Ω outside of the flow. Then, it is easy to see that the elementary η ′ and the wholescalar potential η are given by: η ′ ( x, y, z, x ′ , z ′ , z ′ ) = − m ∂∂z (cid:18) | r − r ′ | (cid:19) = m ( z − z ′ ) | r − r ′ | , (2) η ( x, y, z ) = Z Ω η ′ ( x, y, z, x ′ , z ′ , z ′ ) dx ′ dy ′ dz ′ = Z Ω m ( z − z ′ ) | r − r ′ | dx ′ dy ′ dz ′ . (3)This is the simplest model of a permanent magnet occupying the region Ω with the constant magneticdipole distribution m . By taking various Ω , one can define any desired inhomogeneous MF for parametricnumerical needs and mimic roughly various magnetic systems. Below in Fig.1 we give few simple examplesof two-dimensional, i.e. B y ( x, y, z ) = 0, fields constructed in this way to represent typical magnetic fieldconfigurations. Everywhere, x (left to right) is streamwise, y - spanwise (perpendicular to the plane ofFig. 1), and z (down to up) is transverse coordinate.Here it is worth noting that any two-dimensional field can be conveniently expressed through acomplex representation. Introducing the variable ζ = x + iz , and its conjugate ζ ∗ = x − iz , one maydefine a complex function β ( ζ ) = B x + i B z , (4)where B x = Re( β ), and B z = Im( β ). The function β ( ζ ) is given below as well. Linear chain of equal dipoles oriented parallel to z direction and infinitely extended in the y directionat ( x , z ), Ω = { x = x , −∞ ≤ y ≤ ∞ , z = z } : η ( x, y, z ) = Z ∞−∞ Z ∞−∞ Z ∞−∞ m ( z − z ′ ) δ ( x ′ − x ) δ ( z ′ − z ) dx ′ dy ′ dz ′ (( x − x ′ ) + ( y − y ′ ) + ( z − z ′ ) ) / = 2 m ( z − z )( x − x ) + ( z − z ) = 2 m cos ϕl (5) B x ( x, y, z ) = − ∂η ( x, y, z ) ∂x = 4 m ( x − x )( z − z )(( x − x ) + ( z − z ) ) = 2 m sin 2 ϕl , (6) B z ( x, y, z ) = − ∂η ( x, y, z ) ∂z = 2 m (cid:0) ( z − z ) − ( x − x ) (cid:1) (( x − x ) + ( z − z ) ) = − m cos 2 ϕl , (7) β ( ζ ) = 2 m i ( ζ − ζ ) | ζ − ζ | = 2 m i ( ζ ∗ − ζ ∗ ) , (8)here, l and ϕ define the cylindrical coordinate system ( x − x ) = l cos ϕ, ( z − z ) = l sin ϕ , and theangle ϕ is taken as counter-clockwise. Fig.1a shows an example of the magnetic field induced by twosymmetrically located chains of dipoles in the rectangular channel.The sheet of magnetic dipoles, Fig.1 b , is made by stacking the linear chains described above alongthe z axis, Ω = { x = x , −∞ ≤ y ≤ ∞ , z ≤ z ≤ ∞} . Therefore for this case, the magnetic field isobtained by an integration of Eqs. (5-8) over z , from z to infinity: η ( x, y, z ) = − m ln (cid:0) ( x − x ) + ( z − z ) (cid:1) = 2 m ln 1 l (9) B x ( x, y, z ) = 2 m ( x − x )( x − x ) + ( z − z ) = 2 m cos ϕl , (10) B z ( x, y, z ) = 2 m ( z − z )( x − x ) + ( z − z ) = 2 m sin ϕl , (11) β ( ζ ) = 2 m ζ − ζ | ζ − ζ | = 2 m ( ζ ∗ − ζ ∗ ) . (12)This particular magnetic dipole configuration allows the construction of the final magnetic field througha clear geometrical interpretation. Fig. 1 b shows this construction: the vector B is the field created by E. V. Votyakov, S. C. Kassinos, X. Albets-Chico ( a ) ( b )( c ) ( d ) B B B l j j j j== zx B llB ln l l l l l l l l
22 221 1 22 221 1 llB llB zx jj jj --= += j j sinsin coscos llB llB zx jj jj += +=
21 2212 2111 ln jj +== zx B ll llB l j j (x ,z )(x , z )(x, z) (x, z)(x, z) (x , z )(x ,z )(x ,z ) (x , z )(x ,z )(x ,z )(x , z )(x , z ) Figure 1.
Inhomogeneous magnetic field created in the duct by linear chains of identical dipoles( a ), two identical sheetsdipoles( b ), half space ( c ), and two magnets ( d ). Angles ϕ , ϕ , ϕ must be taken as counter-clockwise, m is taken equalone. Also, ( b ) shows a geometrical construction to find the final magnetic field, see Eq. 10-11 the lower sheet and is directed along the ( r − r ), while its length is inverse to l ; on the other hand,vector B is the field created by the upper sheet, it points along the ( r − r ) with a length inverselyproportional to l ; the vector sum B = B + B is the final magnetic field in the duct.The stepwise magnetic field can be obtained by the integration of the sheet fields from the edgepoint x up to infinity, ( Ω = { x ≤ x ≤ ∞ , −∞ ≤ y ≤ ∞ , −∞ ≤ z ≤ z S z ≤ z ≤ ∞} ), seethese coordinates in Fig. 1c). To exclude the divergence for B x at x → ∞ , we add a second magnetichalf-space, then: B x ( x, y, z ) = m ln ( x − x ) + ( z − z ) ( x − x ) + ( z − z ) = 2 m ln l l (13) B z ( x, y, z ) = 2 m (cid:18) arctan z − z x − x + arctan z − z x − x (cid:19) = 2 m ϕ, (14) β ( ζ ) = 2 m ln ζ − ζ ζ − ζ . (15)This, so called fringing, magnetic field is a candidate configuration for the proper study of the trans-formation of the Poiseuille velocity profile to the Hartmann one, and the dependence of a transitionlength as a function of Re and Ha . To our knowledge there are no papers to date that apply a physi-cally correct fringing magnetic field in numerical simulations, even though this field is one of the mostheavily studied in liquid metal flows, see Molokov & Reed [2003 b ], Alboussiere [2004], Kumamaru et al. [2004], Kumamaru et al. [2007], Ni et al. [2007]. Below we derive qualitatively what is the effect of thestreamwise magnetic component of the fringing field on the transverse pressure distribution in the duct. nalytic models of heterogenous magnetic fields The shelfwise magnetic field, created by a homogenous dipole distribution, is obtained if Ω is restrictedin the x direction ( Ω = { x ≤ x ≤ x , −∞ ≤ y ≤ ∞ , −∞ ≤ z ≤ z ∪ z ≤ z ≤ ∞} ). Fig. 1d showsthis configuration: B x ( x, y, z ) = m l l l l = m x − x ) + ( z − z ) )(( x − x ) + ( z − z ) )(( x − x ) + ( z − z ) )(( x − x ) + ( z + z ) ) (16) B z ( x, y, z ) = m ( ϕ + ϕ )= m (cid:18) arctan z − z x − x − arctan z − z x − x + arctan z − zx − x − arctan z − zx − x (cid:19) , (17) β ( ζ ) = m ln ( ζ − ζ )( ζ − ζ )( ζ − ζ )( ζ − ζ ) . (18)By varying the parameters x , x , z , z , one can regulate the width of the central region and the upward(outward) gradient of the magnetic field.In above examples, the z integration is taken over two half-open regions {−∞ ≤ z ≤ z S z ≤ z ≤ ∞} . It is easy to perform the integration up to a finite z-value instead of infinity, however, thiscase is not considered in order to simplify formulae and keep the essence of the method. Moreover, inpractice, a magnetic system is supported by the yoke made of soft iron. This effectively means a closurefor the lines of the magnetic field coming from the external side of the magnet (where the internal sideis adjoined to the duct), therefore, the upper limit of the integration can be safely set as infinity.Up to now, we have learned how to define an external magnetic field by means of magnetic dipoles.Usage of magnetic dipole language dictates the following scaling behavior for the magnetic field far fromthe central point: l − for a single dipole, l − for a linear chain, l − for a sheet, and ln l for a massivebody, where l is a distance from the object, (see Fig. 1). It’s possible to make this slope even moresmooth, i.e. l ln l , if, instead of the magnetic domain, we take a cross section of constant current density j , taken everywhere equal to one. Physically, it means that instead of using the permanent magnetconsidered before, we shall deal with a conducting cable carrying the electric current of homogenousdensity. The easiest way to understand how we could develop mathematically the cable from the magnetis described next.First, we may present the magnetic domain as a non-conducting cross-section enabling surfacecurrents only. Schematically, it is shown in Fig. 2 a by taking six magnetic dipoles together in closecontact: all the internal currents from the adjoining magnetic dipoles are mutually compensated becausethey run in opposite directions, therefore, the only non-compensated electric currents are located onthe bordering surfaces. The next step is to extend the surface current obtained above, in such a waythat it flows through a finite rectangular region, i.e. through cable of rectangular cross-section. Theconfiguration of the rectangular conductor is shown in Fig. 2 b ., where the corners are ( x k , z l ) , k, l = 1 , b denotes electric current running in the direction perpendicular the planeof the figure). Such a cable can be represented as a part of the top (cables II and IV) and bottom (cablesI and III) solenoids assembled around the channel, see Fig. 2 c .To derive the magnetic field produced by the aforesaid primitive solenoids, we note that the shelfwisemagnetic system shown in Fig.1 d , Eqs. (16)-(18), is equivalent to the solenoids having infinitely thincables located at x = x and x = x because all internal currents in this system are mutually compensatedin the way explained above by Fig. 2 a . Therefore, if one integrates β ( ζ ) given by Eq. (18) over thecomplex variable ζ = x + iz confined by cable’s corners shown in Fig. 2 b , then one obtains the followingcomplex function for the individual cable A : β A ( ζ ) = ( ζ A − ζ ) ln ζ A − ζζ A − ζ + ( ζ A − ζ ) ln ζ A − ζζ A − ζ , (19)where A = I, II, III, IV should be taken for different cables, and ζ Ak,l = x Ak + iz Al , k, l = 1 , A , see Fig. 2 c . The final complex function for the top and bottomsolenoids is obtained as a superposition of terms coming from four cables: β ( ζ ) = β I ( ζ ) + β II ( ζ ) − β III ( ζ ) − β IV ( ζ ) , (20) E. V. Votyakov, S. C. Kassinos, X. Albets-Chico ( a) ( b)III IIIIV( c) ( ) ( )
AkAiAik AAAAAAA IVIIIIII zixzix , lnln)( )()()()()( +=+= ---+---= --+= zz zz zzzzzz zzzzzb zbzbzbzbzb II z II z II z II z IV z IV z IV z IV z I z I z I z I z III z III z III z III z (x ,z ) (x ,z )(x ,z )(x ,z ) Figure 2.
Scheme showing an equivalence of the magnetic domain and a body with surface currents ( a ), notations usedto write down magnetic field from a cable ( b ), scheme of two primitive solenoids located under and above duct( c ), thecables are supposed to encircle at y = ±∞ , see text and Eq. (19, 20) where terms from cables I and II are positive and for cables III and IV are negative because the directionof the electric current in cables I and II is opposite to the direction of current in cables III and IV.The final β ( ζ ), Eq. 20, is the linear superposition of the products of logarithmic and rational functionsdepending on a complex variable ζ = x + i z . Then, by means of algebraic transformations, the obtained β ( ζ ) should be presented as a sum of its real and imaginary parts ( cf. Eq. 4), so that B x = Re[ β ( ζ )]and B z = Im[ β ( ζ )]. Although they have been obtained, the mathematical expressions for B x and B z are quite cumbersome and for the sake of brevity they are not reported here.To conclude, we show briefly what is the impact of the streamwise field component on a liquid metalflow in the case of the fringing magnetic field. As it is widely accepted, inertia and viscosity vanish in thecore of a duct MHD flow, therefore, the pressure distribution, p , is governed in the core by the Lorentzforce, ∇ p = j × B , where j are the induced electric currents. This means that ∇ p is perpendicularto B , i.e. the pressure contour lines are matched with magnetic field lines. Thus, if one employs thetransverse field component only, the pressure contour lines are straightened in the transverse direction,while they must be curved in accordance with the curvature of the external magnetic field, see Fig. 3.Curvature of the magnetic field lines depends on the aspect ratio d/h , where d ( h ) is the height ofthe duct (magnetic gap). As d/h decreases, the curvature vanishes, however the inward gradient ofthe magnetic field decreases as well. Hence, in order to study the effects of a rapidly varying fringingmagnetic field, the streamwise field component must be taken into account necessarily.As a paradigmatic example, the well-known nuclear fusion benchmark fringing magnetic field casebased on the experiment of Reed et al. [1987] ( Ha = 6569, N = 10824, c = 0 . nalytic models of heterogenous magnetic fields Figure 3.
Pressure contour lines matching magnetic field lines with ( a ) and without ( b ) streamwise magnetic field component. −15 −10 −5 0 5 1000.010.020.030.040.05 X/R P ( x , , R ) − P ( x , R , ) R σ U b B Ni et al.Albets−Chico et al.Exp. Reed
Figure 4.
Dimensionless transverse pressure difference for the so called Benchmark fringing magnetic field case. Comparisonis given for experimental results (Reed et al. [1987], circles), 3D numerics with a non-consistent fringing magnetic field(dashed, Ni et al. [2007]), and 3D numerics with a consistent magnetic field (solid, Albets-Chico et al. [2009]) The resultsare clearly improved when the consistent magnetic field is considered. studied using a complete numerical resolution of the Navier-Stokes equations by Ni et al. [2007]. Detailsregarding the shape of the fringing magnetic field, geometry and boundary conditions for this case areclearly exposed in both the experimental Reed et al. [1987] and the numerical Ni et al. [2007] works.Ni et al. [2007] presented 3D numerical results in very good agreement with the experimental dataalthough a non negligible under-prediction ( ≈ et al. applied a tanh-based fitting function to approximate the main componentof the experimental magnetic field, while the additional components were neglected. More recently,Albets-Chico et al. [2009] have also addressed this case by means of a complete numerical resolution of thegoverning equations (considering a quasi-static approximation for the induced magnetic field) when usinga nodal-based non-structured finite-volume code. Details regarding the form of the addressed Navier-Stokes equations and the assumed simplifications can be found in Ni et al. [2007], as both works haveemployed essentially the same methodology. Additionally, Albets-Chico et al. have analyzed the effect of E. V. Votyakov, S. C. Kassinos, X. Albets-Chico the consistency of the magnetic field while developing a mathematical technique to generate consistentmagnetic fields from experimental fits. Further, details will be available soon in Albets-Chico et al. [2009].Fig. 4 presents the effect of the consistency of the magnetic field in the transverse pressure differenceresults. When using a consistent magnetic field, the under-prediction for the peak is reduced to approx-imately 9%, which clearly demonstrates how the consistency of the magnetic field plays an importantrole related to the transverse pressure difference in such a case. It is interesting to note that Albets-Chico et al. obtained a perfect agreement with Ni et al. [2007] results when using a non-consistent magneticfield (based on the same tanh-based fitting function as the one used in Ni et al. ). Finally, Albets-Chico et al. explain the still remaining under-prediction in terms of the experimental results in the nature(slope and order of accuracy) of the fitting function.
Acknowledgements.
This work has been performed under the UCY-CompSci project, a Marie Curie Transferof Knowledge (TOK-DEV) grant (contract No. MTKD-CT-2004-014199) funded by the CEC under the 6thFramework Program. Partial support through a Center of Excellence grant from the Norwegian Research Councilto the Center for Biomedical Computing is also greatly acknowledged. E.V.V. is grateful for many fruitfuldiscussions with Oleg Andreev, Yuri Kolesnikov, Andre Thess, and Egbert Zienicke during his time in theIlmenau University of Technology.
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