On the analogy between streamlined magnetic and solid obstacles
aa r X i v : . [ phy s i c s . f l u - dyn ] A ug On the analogy between streamlined magnetic and solid obstacles
E.V. Votyakov ∗ and S.C. Kassinos † Computational Science Laboratory UCY-CompSci,Department of Mechanical and Manufacturing Engineering,University of Cyprus, 75 Kallipoleos, Nicosia 1678, Cyprus
Abstract
Analogies are elaborated in the qualitative description of two systems: the magnetohydrodynamic(MHD) flow moving through a region where an external local magnetic field (magnetic obstacle)is applied, and the ordinary hydrodynamic flow around a solid obstacle. The former problem isof interest both practically and theoretically, and the latter one is a classical problem being wellunderstood in ordinary hydrodynamics. The first analogy is the formation in the MHD flow ofan impenetrable region – core of the magnetic obstacle – as the interaction parameter N , i.e.strength of the applied magnetic field, increases significantly. The core of the magnetic obstacleis streamlined both by the upstream flow and by the induced cross stream electric currents, likea foreign insulated insertion placed inside the ordinary hydrodynamic flow. In the core, closedstreamlines of the mass flow resemble contour lines of electric potential, while closed streamlinesof the electric current resemble contour lines of pressure. The second analogy is the breakingaway of attached vortices from the recirculation pattern produced by the magnetic obstacle whenthe Reynolds number Re , i.e. velocity of the upstream flow, is larger than a critical value. Thisbreaking away of vortices from the magnetic obstacle is similar to that occurring past a solidobstacle. Depending on the inlet and/or initial conditions, the observed vortex shedding can beeither symmetric or asymmetric. ∗ Electronic address: [email protected], [email protected] † Electronic address: [email protected] ntroduction External magnetic fields are heavily exploited in many practical applications [1], suchas electromagnetic stirring, electromagnetic brakes, and non-contact flow measurements [2].The crucial aspect in the above applications is the Lorentz force produced by the interactionof an external magnetic field with induced electric currents. The currents appear becausean electrically conducting fluid moves relative to the external field. The Lorentz force hasa double effect on the flow: it suppresses turbulent fluctuations when the intensity of theexternal field is strong and spatially uniform, but also is able to produce vorticity if theintensity varies spatially. If the external magnetic field is localized in space, i.e. it acts ona finite region of flow, then the flow is decelerated in this region and one can say that thelocal magnetic field produces a virtual obstacle, called a magnetic obstacle. Both a solidand magnetic obstacle have a real physical effect in the sense that they impede the flow.The retarding effect of the external nonuniform magnetic field on the liquid metal flow iswell-known and has been intensively studied in the past, see for instance books [3, 4, 5, 6].The overwhelming majority of works were performed on liquid metal flows in ducts subjectto fringing magnetic fields. The main goal was to study the so-called M-shaped velocityprofile formed by directing the flow into the region of the fringing magnetic field. TheM-shaped profile is characterized by two side jets around a central stagnant region.The flow around a solid obstacle, such as a circular cylinder schematically given inFig. 1( a ), is a classical hydrodynamical problem that is qualitatively well understood. Thestructure of the wake of the cylinder depends on the Reynolds parameter Re = u d/ν , where u is velocity at infinity, d is the cylinder diameter, and ν the kinematic viscosity of the fluid.Physically, Re expresses the ratio of inertial to viscous forces. When the inertia of the flowincreases, two attached vortices appear past the cylinder, Fig. 1( b ). As the inertia of the flowincreases further, the vortices detach from the cylinder and form the von Karman vortexstreet.The flow around a magnetic obstacle like those schematically given in Fig. 1( c ) is a rathernew MHD problem that is not yet completely understood. The first studies devoted to liquidmetal flow around a magnetic obstacle were carried out in the former Soviet Union[7, 8].2D numerical calculations [7] have found two vortices inside the magnetic obstacle, however,especially designed experiments [8] did not confirm the numerical finding. Lately the term2magnetic obstacle’ has been revived for Western readers in 2D numerical works [9, 10, 11],where authors also have found a vortex dipole in a creeping MHD flow [9] and claimed thatvortex generation past a magnetic obstacle is similar to that past a solid obstacle [10].The most recent results for the flow around a magnetic obstacle were obtained by meansof 3D numerics and physical experiments [12, 13]. It turns out that the structure of the wakeof the magnetic obstacle is more complex that that of the solid obstacle. In addition to theReynolds number, Re = u H/ν , an MHD flow is characterized by the magnetic interactionparameter, N = σH B /ρu , where H , u , B are the characteristic length scale, velocity andintensity of the applied magnetic field, and ρ and σ are the density and electric conductivityof the fluid, see e.g. [4, 5, 14, 15]. N represents the ratio of the Lorentz force to the inertialforce. Depending on Re and N , i.e. on the relationship between viscous, Lorentz and inertialforces, the liquid metal flow shows three different regimes: (1) no vortices, when the viscousforce prevails at the small Lorentz force limit, (2) one pair of magnetic vortices when Lorentzforce is high and inertia is small, and (3) three pairs (namely, magnetic, connecting, andattached vortices) when the Lorentz and inertial forces dominate the viscous force. Thelatter case is shown in Fig. 1 d . We believe that this scenario for the wake of the magneticobstacle is the generic one and devote the last Section of the paper to explain in detail whythis is so.An analogy between solid and magnetic obstacles had been suggested from the beginningof MHD liquid metal works in the former USSR [28]. Based on 2D inertialess simulations,Gelfgat et al. [7] remarked that a vortex dipole inside the magnetic obstacle is similar toattached vortices past a solid obstacle. Afterwards, a series of experiments of Gelfgat etal. [8] failed to confirm the numerical results and this let them to question the originalsuggestion. As has been shown recently [12], however, the problem with the suggestedanalogy is that the numerically observed flow structures were magnetic vortices fixed insidethe magnetic obstacle rather than the attached vortices disposed past the magnetic obstacle.The 2D numerical results [7] were correct for creeping flow, while the physical experiments[8] were performed at high Reynolds numbers Re , and so they failed to produce any vorticessince the interaction parameter N was not high enough [13].Then, Cuevas et al. [10, 11], by means of 2D and quasi-2D numerics [29], concludedthat the vortex generation past a magnetic obstacle is similar to that past a solid cylinder.Although this conclusion is correct in general, it is rather obvious: any decelerating force3 IG. 1: Flow around a solid ( a, b ) and magnetic ( c, d ) obstacle. Wake of the solid obstacle ( b ) showstwo attached vortices, wake of the magnetic obstacle ( d ) shows inner magnetic (first),connecting(second) and attached vortices (third pair). generates vorticity that is then translated downstream, exciting in the process the vonKarman vortex street. Moreover, at high Reynolds numbers, 2D numerics is not a suitablemethod to analyze the flow around the magnetic obstacle because it neglects the Hartmannfriction [13], and as a result, fails to describe the stable six-vortex structure shown in Fig. 1 d and discussed later.The possible reason why the previous two analogies were either imprecise [7] or triviallycorrect [10] is because the previous interpretations were not based on full 3D numericalsimulations at high Reynolds numbers. As a result, the previous interpretations sufferedfrom the lack of a concrete and clear demonstration. The basic message of our paper isto report the correct, in our opinion, analogy, and to confirm it by means of concrete 3Dnumerical results.A fruitful way of thinking about the similarity between magnetic and solid obstacles isthat the magnetic and connecting vortices, taken together as one entity, form the body of4 virtual insertion in the MHD flow [12]. One can understand it by recalling the classicalpotential flow theory. In this theory, a real streamlined cylinder is modelled by a virtualimaginable vortex dipole. In the MHD case, we have an opposite picture: a magneticobstacle, that can be understood as a virtual bluff body, manifests itself by means of realphysical vortices. The present paper supports this idea with two new aspects.The first is the impenetrable core of the magnetic obstacle. It originates in the centerof the magnetic gap as the magnetic interaction parameter N increases. When N is verylarge, both mass transfer and electric field vanish in the region between magnetic poles.This region looks as if frozen by the external magnetic field so that the upstream flowand crosswise electric currents can not penetrate inside it. Thus, the core of the magneticobstacle is similar to an insulated solid obstacle inside an ordinary hydrodynamical flowwith crosswise electric currents and without an external magnetic field. In this latter case,because of the absence of a magnetic field, the crosswise electric currents go around theinsulated insertion without affecting the mass flow. Magnetic vortices are located aside thecore and compensate shear stresses, like a ball-bearing between the impenetrable region andupstream flow.At first glance, the appearance of the core of the magnetic obstacle can be admitted asintensively studied before. Indeed, a stagnant region between two side jets is well-knownfor duct flows subject to fringing magnetic fields. However, at a closer examination onefinds that the fringing magnetic field is not the case of the magnetic obstacle. In the formercase, the side jets of the M-shaped velocity profile are caused by a geometrical heterogeneityimposed by the sidewalls of the duct, so the stagnant region tends to spread between thesidewalls. In the latter case, maxima of streamwise velocity appear in an originally freeflow around the region where the magnetic field is of highest intensity, and the core roughlycorresponds to the region where the magnetic field is imposed.To our knowledge, most of numerical studies of fringing magnetic fields were performedwith the 2D assumptions, i.e. the flow was treated as quasi 2D, where only the transversefield component ( B z ) was taken into consideration, while the other components ( B x and B y ) were neglected. As a result, the studied magnetic field was inconsistent, that is, therequirements for the field to be curl- and divergence-free were violated [16]. So, this is oneof the contributions of the present work: a complete systematic 3D numerical study with N changing smoothly from low to high, while maintaining a physically consistent curl- and5ivergence-free external magnetic field.A new physical effect compared to the fringing magnetic fields is a neat demonstrationof the vortices alongside the stagnant core. It has been shown recently that the spanwisehomogeneous fringing magnetic field does not enable any recirculation [13].The second aspect of the paper is the detachment of the vortices from the magneticobstacle when the Reynolds number Re is large enough. Magnetic and connecting vorticesare in rest during the vortex shedding. The shedding can be either symmetric, in which bothattached vortices are coming off simultaneously, or asymmetric, as it usually happens witha solid obstacle when Re exceeds a critical value. The symmetric vortex shedding is alsopossible in an ordinary hydrodynamic flow past an infinitely long cylinder that is speciallyperturbed to provoke the synchronous vortex shedding, see e.g. [17].The presented results are complementary to those published in [12, 13]. They wereunavailable before because there required extensive sets of 3D numerical simulations: aseries of runs for large N to refine the core of the magnetic obstacle, and a series of runsinvolving long time integrations to produce laminar time-periodic vortex shedding at high Re . Each of these sets of runs is discussed below in its own section.The structure of the present paper is as follows. First, we present technical details ofthe simulations: model, equations and 3D numerical solver. Then, we report results for thecore of the magnetic obstacle and demonstrate possible symmetric and asymmetric vortexshedding in the wake past the obstacle. The last Section before the Summary explains thegeneric scenario for the wake of the magnetic obstacle and the similarities with the vortexshedding past the solid obstacle. A summary of the main conclusions ends the paper. Model, equations, numerical method
A schematic of the model is shown in Fig. 1( c ). It is the same as detailed in [13] except forthe fact that, in the present case, we have no side walls. Instead, here we use slip boundaryconditions in the crosswise direction, and therefore, expand the crosswise dimension of thecomputational domain. Also, in vortex shedding simulations, we double the outlet lengthcompared to [13] in order to exclude the outlet influence on the vortex detachment andadvection.The governing equations for an electrically conducting and incompressible fluid, subject6o an external magnetic field, are the Navier-Stokes equations coupled with the Maxwellequations for a moving medium and Ohm’s law. Here, the magnetic Reynolds number R m = µ ∗ σu H is supposed to be much less than one, where, µ ∗ is the magnetic permeability.This corresponds to the so called quasi-static (or inductionless) approximation, where it isassumed that the induced magnetic field is infinitely small in comparison to the externalmagnetic field, see, e.g. [15]. The resulting equations in dimensionless form are: ∂ u ∂t + ( u · ∇ ) u = −∇ p + 1 Re △ u + N ( j × B ) , ∇ · u = 0 , (1) j = −∇ φ + u × B , ∇ · j = 0 , (2)where u denotes velocity field, B is the external magnetic field, j is the electric currentdensity, p is the pressure, and φ is the electric potential. The interaction parameter N and Reynolds number Re , N = Ha /Re , are linked by means of the Hartmann number: Ha = HB ( σ/ρν ) / . The Hartmann number determines the thickness of the Hartmannboundary layers, δ/H ∼ Ha − for flow under constant magnetic field.The origin of the coordinate system is taken in the center of the magnetic gap. Thecomputational domain is: − L x ≤ x ≤ L x , − L y ≤ y ≤ L y , − H ≤ z ≤ H , where x, y, z arerespectively the streamwise, crosswise, and transverse directions, and L x ( L x ), L y , and H are the inlet (outlet), crosswise, and transverse dimensions of the simulation box. L x = 25in runs for the core of the magnetic obstacle, and L x = 50 in runs for vortex shedding; L x = 25, L y = 25, H = 1 in both simulations. Magnetic poles are located at x = 0, y = 0, z = ± h , and the size of the magnet is | x | ≤ M x , | y | ≤ M y , | z | ≥ h . The intensity of theexternal magnetic field B ( r ) is calculated by means of formulae given in [13] with M x = 1 . M y = 2, and h = 1 .
5. Different cuts of the intensity B ( r ) for these parameters are plottedin Fig. 3 and Fig. 4( b ) in paper [13].The characteristic dimensions for the Reynolds number Re , and the interaction param-eter N are the half-height of the duct H , the mean flow rate u , and the magnetic fieldintensity B , taken at the center of the magnetic gap, x = y = z = 0. So, all the distances L x , L x , L x , L y , M y , M x , h are normalized by H ; the velocity u by u ; the magnetic field B by B ; the electric current density j by σu B ; the electric potential by Hu B ; the pressure p by ρνu /H .For a given external field B ( x, y, z ), the unknowns of the partial differential equations (1– 2) are the velocity vector field u ( x, y, z ), and two scalar fields: the pressure p ( x, y, z ) and7he electric potential φ ( x, y, z ). To find the unknowns we use a finite differences method thatwas implemented in a 3D numerical solver as been detailed in[18]. The solver was developedfrom a free hydrodynamic solver created originally in the research group of Prof. M. Griebel([19]). The solver employs the Chorin-type projection algorithm and finite differences onan inhomogeneous staggered regular grid. Time integration is done by the explicit Adams-Bashforth method that has second order accuracy. Convective and diffusive terms are imple-mented by means of the VONOS (variable-order non-oscillatory scheme) method. The 3DPoisson equations are solved for pressure and electric potential at each time step by usingthe bi-conjugate gradient stabilized method (BiCGStab).To complete the numerical model, boundary conditions have to be specified. No slip andinsulating walls were specified in the transverse direction, while slip walls were used in thecrosswise direction. In order to test the effect of boundary conditions, in some of the runscarried out for the core of the magnetic obstacle, the slip conditions in the crosswise directionwere replaced by periodic boundary conditions. However, changing the crosswise boundaryconditions was found to have no effect on the structure of the core. This is because L y = 25is large enough compared to M y = 2 in all the runs for the core.The outlet of the computational domain was treated as a force-free (straight-out) borderfor the velocity. The electric potential at the inlet and outlet boundaries was taken to beequal to zero because the inlet and outlet are sufficiently far from the region of magneticfield. At the inlet, a 2D parabolic (Poiseuille) velocity profile was used that was uniform inthe crossflow (spanwise) direction. In runs for the asymmetric vortex shedding, this profilewas slightly perturbed in the crosswise direction at initial times and then kept symmetricand constant in time. The initial perturbation initiated the asymmetric vortex sheddingand the followed symmetry and constancy assured that the asymmetric vortex shedding isindependent of the inlet conditions.Time integration in the runs for the core of the magnetic obstacles was carried out untila stationary laminar solution has been reached. In all these simulations, we found the samelaminar solution at a given Re and N pair, independently of initial conditions. So, as initialconditions for runs corresponding to new Re and N we used 3D fields of velocity, pressure,and electric potential obtained from the previous runs having the closest Re and N values.Time integration in the runs for the vortex shedding was continued until a time-periodiclaminar solution was reached. These simulations were dependent on the initial conditions.8e found two classes of solutions: symmetric and asymmetric distribution of attached vor-tices in the wake at large times. The details about the initial and inlet conditions for bothcases are given in the beginning of the corresponding sections.The simulation box has been discretized by an inhomogeneous regular 3D grid dependingon the solved problem. Details about the numerical grid are given at the beginning of thecorresponding sections. Core of the magnetic obstacle
In this series of simulations, we focus on the flow around a magnetic obstacle at largeinteraction parameter N . In order to achieve large N = Ha /Re , the simulations werestarted at a small interaction parameter and Ha was smoothly increased, while keeping Re constant. Two values of the Reynolds number were studied, Re = 10 and 100, and noprincipal differences were found at the same N . These low values of Re imply low inertialforces, therefore, only “two-vortex” patterns were produced, without connecting or attachedvortices.The numerical grid was regular and inhomogeneous, N x × N y × N z = 64 . The minimalhorizontal step size in the region of the magnetic gap was ∆ x ≃ ∆ y ≃ .
3, which means thata few dozens points were used for resolving the inner vortices in the core of the magneticobstacle. The minimal vertical step size near the top and bottom (Hartmann) walls was∆ z = 0 . /Ha ) / ∆ z ) points to resolveHartmann layer at Ha = 40 − N . These cuts are shown in Fig. 2a for the streamwise velocity u x ( y ) and in Fig. 2bfor the electric potential φ ( y ). First we discuss how the streamwise velocity changes as N increases.Because N expresses the strength of the retarding Lorentz force relative to the inertialforce, curve 1 in Fig. 2 a ( N = 0 .
1) is only slightly disturbed with respect to a constant.As N increases, the curves u x ( y ) pull further down in the central part u center + u x (0), seefor example curves 2 and 3. At N higher than a critical value N c,m , i.e. for curve 4, thecentral velocities u center are negative. This means that there appears a reverse flow causing9
15 −10 −5 0 5 10 15−1−0.500.51 y φ
5, 643 32 211 (b) −15 −10 −5 0 5 10 1500.511.52 y u (a) FIG. 2: Streamwise velocity ( a ) and electric potential ( b ) along crosswise cuts of middle horizontalplane x = z = 0. Re = 10, N =0.1(solid 1), 1.6(dot-dashed 2), 4.9(solid 3), 40(dashed 4), 250(solid5), and 490(dot-dashed 6). Insertion shows magnified plots for curves 5 and 6. magnetic vortices in the magnetic gap. When N rises even more (see curves 5 and 6) themagnetic vortices become stronger and simultaneously shift away from the center to the sidealong the y direction, see insertion in Fig. 2 a for curves 5 and 6.The fact that the centerline velocity in the center of the magnetic gap goes to zero as N increases is expected and was discussed earlier for fringing magnetic fields. In this respect,the case of the magnetic obstacle is analogous to the fringing magnetic field. What isdifferent in these two cases is that the centerline velocity becomes negative before it goes tozero while this could not be so, and was never actually observed, for the fringing field [13].Fig. 2( b ) shows how the electric potential φ ( y ) varies along the central crosswise cutthrough the magnetic gap. The slope in the central point is the crosswise electric field, E y,center = − dφ/dy | y =0 . One can see that E y,center changes its sign: it is positive at small N and negative at high N . To explain why it is so, one can use the following way ofthinking. Any free flow tends to pass over an obstacle in such a way so as to perform thelowest possible mechanical work, i.e. flow streamlines are the lines of least resistance to thetransfer of mass. The resistance of the flow subject to an external magnetic field is causedby the retarding Lorentz force F x ≈ j y B z , so the flow tends to produce a crosswise electriccurrent, j y , as low as possible while preserving the divergence-free condition ∇ · j = 0. Tosatisfy the latter requirement, an electric field E must appear, which is directed in such a10ay, so as to compensate the currents produced by the electromotive force u × B . Next, weanalyze the crosswise electric current j y = E y + ( u z B x − u x B z ). Due to symmetry in thecenter of the magnetic gap B y = B x = u y = u z = j y = j z = 0 so j y = E y − u x B z . Thismeans that E y tends to have the same sign as u x in order to make j y smaller. At small N ,the streamwise velocity u x is large and positive, so the electric field E y is positive too. Whenthe magnetic vortices appear, there is a reverse flow in the center. Therefore, the centralvelocity is negative now, and the central electric field E y,center is also negative.In [13] the change of the electric field in the magnetic gap is explained in terms of thePoisson equation and the concurrence between external and internal vorticity. This argumentis also valid here, however in contrast to [13], we have no side walls, so the external vorticityin the present case plays only a minor role. As a result, the reversal of the electric fieldappears at a small N (approximately equal to five), which is close to the critical interactionparameter N at κ = 0 . κ is the ratio of the magnet width to theduct width.)The overall data about u center and E y,center in the whole range of studied N are shownin Fig. 3. One can see that both characteristics start from positive values, then, they crossthe zeroth level, reach a minimum, go up again, and finally vanish in the limit of high N .With respect to the streamwise velocity, this means that, at hight N , there is no massflow in the center of the magnetic gap; the other velocity components are equal to zerodue to symmetry. With respect to the crosswise electric field, this means that there are noelectric currents. This occurs because there is no mass flow, therefore, the electromotiveforce vanishes, E y goes to zero, and the other electric field components are equal to zerodue to symmetry. Thus, one can say that the center of the magnetic gap is frozen by thestrong external magnetic field, so that both mass flow and electric currents tend to bypassthe center. In other words, this means that a strong magnetic obstacle has a core, and sucha core is like a solid insulated body, being impenetrable for the external mass and electriccharge flow.In the 2D creeping flow around the magnetic obstacle, the u center and E y,center vanishingbehavior shown in Fig. 3 is impossible because it violates the flow continuity. At very high N and low Re , instead of the frozen core obtained in the 3D case, a 2D flow develops variousrecirculation patterns in the core, because the secondary flow of the 3D magnetic vorticesis forbidden in the 2D case. Paradoxically, the 2D creeping flow discussed in the paper by11
200 400 600−0.200.20.40.60.811.21.4 u c en t e r N(a)N c,m E y , c en t e r N(b)0 u u center φ E y,center = −d φ /dy FIG. 3: Central streamwise velocity u center ( a ) and central spanwise electric field E y,center ( b ) as afunction of the interaction parameter N . N c,m is a critical value where the streamwise velocity isequal to zero. Insertion shows the definition of u center and E y,center . Cuevas et al. [9] is turned out to be more rich than the presented 3D creeping flow betweentwo no-slip endplates. This point is discussed further in the last Section devoted to thegeneric scenario of the wake of the magnetic obstacle.It is convenient to visualize the core of the magnetic obstacle by plotting streamlinesfor the mass flow, (see Fig. 4 a ) and electric charge transfer (see Fig. 4 b ) in the middlehorizontal plane. One can see that the side streamlines envelop the bold dashed rectangle.This rectangle denotes the borders of the external magnet. Alongside the rectangle thereare closed streamlines for mass flow (plot a ), which are magnetic vortices. At high N , thesevortices are located in the region of crosswise gradients of the external magnetic field andcompensate shear stresses between the core of the magnetic obstacle and rest of the flow.Also, the magnetic vortices produce closed electric currents inside the rectangle (plot b ).These internal currents are elongated in the y direction. They are very weak compared tothe external currents enveloping the obstacle.We note that the streamlines of the flow and electric charge resemble contour lines ofthe electric potential (Fig. 4 c ) and pressure (Fig. 4 d ). This happens because inertia andviscosity are vanishing in the core, so equations (1 – 2) become: ∇ p = j × B , ∇ φ = − j + u × B ≈ u × B . In the latter equation, j ≪ ∇ φ and u × B is the dominating term. In the core of the obstacle12 y −5 0 5−6−4−20246 x y (c) −0.100.10 0.01−0.010.000.00 −5 0 5−6−4−20246 x y (b)−5 0 5−6−4−20246 (d)x y FIG. 4: Middle horizontal plane, z = 0: streamlines of the mass ( u x , u y ) ( a ) and electric charge( j x , y y ) ( b ) flow. Contour lines for the electric potential φ ( x, y ) ( c ) and pressure p ( x, y ) ( d ) resemblethe streamlines given above. Re = 10, N = 490. Contours of the electric potential are given withstep 0.01, and contours of the pressure are given with the step 0.4. Dashed bold rectangle showsborders of the external magnet. B = (0 , , B z ) ≈ (0 , , B what obviously not the case shown in Fig. 4 a, b .Magnetic field plus rotation require more sophisticated boundary conditions than justthe Hartmann layer. There is known a solution for the Ekman-Hartman layers, where bothconstant rotation and constant magnetic field are taken jointly into account. This probablydoes not also fit because the vorticity is not constant along the transverse direction, and theshape of vortices is not circular. Moreover, inclusion of the non constant vorticity destroysthe linearity of Kulikovskii’s theory. Therefore, Kulikovskii’s theory could not be used as itstands to predict recirculation a priori . Indeed this explains why the theory has not beenapplied to magnetic vortices, even though it has been known for a while. Nevertheless,Kulikovskii’s theory is useful and must be mentioned because it explains a posteriori theshape of vortices and their matching to electric potential lines. Vortex shedding past a magnetic obstacle
The following simulations were carried out at Re = 900 and N = 9 ( Ha = 90). Thenumerical grid is regular and inhomogeneous, N x × N y × N z = 144 × ×
64. The minimalhorizontal step size is ∆ x ≃ ∆ y ≃ . − .
33 in the region | x | ≤ M x , | y | ≤ M y . Thissupplies few dozens points inside and near the magnetic gap, enough to resolve recirculation.The vertical step size near the top and bottom (Hartmann) walls is ∆ z = 0 . ≤ t ≤ t = 0 in both cases. Theintegration time step varied automatically with the limitation imposed by the viscous layerstability condition. The largest time step was equal to 0.0083.For the unperturbed case, we used the initial ( t = 0) streamwise component velocity u x,s ( x, y, z ) = u P ( z ), where the u P ( z ) = 3 / − z ) is the Poiseuille velocity profile. Thesame inlet velocity profile u x,s (0 , y, z ) was imposed at all times and this explains why wecalled this case unperturbed. Time integration was stopped at t = 612.For the perturbed case we used for the initial ( t = 0) streamwise velocity u x,a ( x, y, z ) = u P ( z ) θ ( y ), where function θ ( y ) = 1 for | y | ≥ λ , and θ ( y ) = (1 + γsin [ π yλ ]) for | y | ≤ λ . Thewavelength of perturbation, λ , was taken to be sufficiently higher than the spanwise size ofthe physical magnet, λ = 2 . M y . The amplitude of the perturbation, γ , was taken to be γ = 0 .
05. This five percent skew was sufficient to avoid the symmetric solution found in theunperturbed case above. However, it also resulted in slightly different flow rates for positiveand negative y . Therefore, the perturbed profile u x,a (0 , y, z ) was imposed at 0 ≤ t ≤ t = 120 the symmetric profile u x,s (0 , y, z ) was prescribed again to restore bythat the equal flow rates. Time integration was stopped at t = 1100.Instantaneous mass flow streamlines for symmetric vortex shedding are shown in Fig. 5.They are plotted for the middle horizontal plane, z = 0, and are symmetric with respectto the centerline y = 0. There are three instances of time with ∆ t = 8. For each, we15 y t=512 A B C −5 0 5 10 15 20 25 30 35 40 45−505 (b)x y t=520 A B C −5 0 5 10 15 20 25 30 35 40 45−505 (c)x y t=528 A B C
FIG. 5: Instantaneous mass flow streamlines for the unperturbed symmetric inlet velocity profile: t =512( a ), 520( b ), 528( c ). Re = 900, N = 9. Dashed bold rectangle shows borders of the externalmagnet. Letters A at x = 5, B at x = 15, and C at x = 35 are points on the centerline, y = 0, fortime histories shown as dashed lines in Fig. 7. see the same configuration of magnetic (first pair) and connecting (second pair) vortices,while attached vortices form a sequence dependent on the time instance. This providesevidence that the attached vortices come off the magnetic obstacle simultaneously and movedownstream slowly. The location of the attached vortices in plot ( a ) ( t = 512) looks similaras in plot ( c ) ( t = 528), therefore one can conclude that vortex breakdown occurs at a timeperiod equal to 16 time units. 16 y t=512 A B C −5 0 5 10 15 20 25 30 35 40 45−505 (b)x y t=520 A B C −5 0 5 10 15 20 25 30 35 40 45−505 (c)x y t=528 A B C
FIG. 6: Instantaneous mass flow streamlines for the initially perturbed symmetric inlet velocityprofile: t =512( a ), 520( b ), 528( c ). Re = 900, N = 9. Dashed bold rectangle shows borders of theexternal magnet. Letters A at x = 5, B at x = 15, and C at x = 35 are points on the centerline, y = 0, for time histories shown as solid lines in Fig. 7. Although symmetric vortex shedding past a bluff body is not typical in ordinary hydro-dynamics, it is possible if one takes special steps, such as artificial forcing, see for instance[17], and references in Table 1 therein. The overwhelming majority of papers devoted tovortex shedding deals with an infinitely long cylinder. The MHD case under considera-tion is three-dimensional, i.e. there are top and bottom walls, so the proper hydrodynamicanalogy is to consider a finite cylinder placed perpendicular between two endplates. There17s evidence in ordinary hydrodynamics that the confinement imposed by the endplates in-creases the stability of the wake, see e.g. [22], [23], [24], [25]. In particular, the range ofthe Reynolds number, Re , where two attached vortices remain symmetric behind a circularcylinder without breaking, is much larger in the presence of no-slip endplates. Therefore, itis also possible that the confinement stabilizes symmetric vortex shedding produced by thesolid cylinder subject to the artificial forcing.If the inlet velocity profile is not symmetric at initial times, then one expects an asym-metric vortex shedding, as shown in Fig. 6. The time instances are the same as in Fig. 5.One can see now that the attached vortices are shifted relative to each other through thecenterline y = 0. Plot a ( t = 512) looks roughly like the mirror image of plot c ( t = 528)giving by that evidence about the half-time period equal approximately to 16 time units.Altogether, the picture is similar to a standard time-periodic laminar vortex shedding past asolid circular cylinder. At the place of the cylinder there is a four-vortex ensemble composedof magnetic and connecting vortices. Because of Hartmann friction, this ensemble is stablein time, and so this represents the body of a virtual solid obstacle imposed by the external,strongly heterogeneous magnetic field.Shown in Fig. 7 are time histories of local instantaneous streamwise ( ∂p/∂x , plot a ) andcrosswise ( ∂p/∂y , plot b ) pressure gradients, for both symmetric (dashed) and asymmetric(solid lines) vortex shedding. The curves are given for points A, B, C located on the center-line y = 0 and denoted in Fig. 5 and Fig. 6. These pressure gradients are selected for timeanalysis because they can be measured experimentally. Moreover, ∂p/∂x and ∂p/∂y can beunderstood as the local drag and lift forces respectively. Time dependencies of drag and liftcoefficients in the case of a solid cylinder are well understood, see e.g. Fig. 7 in [26]. Theyare time-periodic with a single vortex shedding frequency at low Reynolds number.As one can see in Fig. 7, the simulated ∂p/∂x , ∂p/∂y first go through a transitionalregime, which it is then transformed smoothly into a periodic regime. The time period,equal approximately to 16, can be estimated from the zoomed insertions plotted for thetime range 300 ≤ t ≤ St = f L/u . Here, f = 1 /
16 is the frequency of vortexshedding, L = 2 M y = 4 is the crossstream size of the magnet, and u = 1 is the mean flowvelocity, so St = 1 /
4. This value is different from that of 0.1 found in the paper by Cuevas18 ∂ p / ∂ y A (b) −0.0200.02 ∂ p / ∂ x B −0.0200.02 ∂ p / ∂ y B0 200 400 600−0.100.1 time ∂ p / ∂ x C 0 200 400 600−0.100.1 time ∂ p / ∂ y C−0.0200.02 ∂ p / ∂ x A (a) FIG. 7: Streamwise ( ∂p/∂x , plot a ) and crosswise ( ∂p/∂y , plot b ) pressure gradients in points A , B , C given in Fig. 5 (dashed) and 6 (solid lines). Insertions are zoomed for the time range300 ≤ t ≤ et al. in 2D simulations. The difference is obviously explained by the impact of the channelwalls. For the symmetric (dashed) and asymmetric (solid) vortex shedding, the streamwisepressure gradient has similar behavior at locations not far from the magnetic obstacle. Onecan see that ∂p/∂x ’s are very close at point A , slightly disagree at point B and notablydifferent at point C . The crosswise pressure gradient on the centerline for the symmetricshedding is equal to zero. The generic scenario for the wake of the magnetic obstacle
In ordinary hydrodynamics, the generic scenario for the wake past the solid obstacle isthe following when Re smoothly increases [27] before turbulence starts: (i) creeping flow,(ii) two attached vortices, (iii) vortex shedding. If one goes into the details, e.g. considersthe different ways of vortex shedding, then different sub-scenarios can be found depending19n specific conditions, but the generic character of the above classification remains. We em-phasize that the interplay between the viscous and inertial forces is decisive for establishingthis general peculiarities.Analogously, by taking into consideration all possible forces, we derive now a scenario forthe wake of the magnetic obstacle. This was given briefly before in [12, 13] without stressingthat this is generic because of the lack at the time of information about vortex shedding.Now, this gap is filled.In the case of the magnetic obstacle, there are three forces and three corresponding termsin the MHD equations: the viscous force (V), the inertial force (I), and the Lorentz force(L). If we put the forces in order of decreasing intensity, then the total number of all thepossible relationships between forces is six: (1) VIL, (2) IVL, (3) VLI, (4) LVI, (5) LIV, and(6) ILV. (Each capital letter is given for the corresponding term.) The cases (1-2) are ofthe smallest Lorentz term, therefore, they can be treated as outlined above for the ordinaryhydrodynamics. The cases (3-4) are of the smallest inertial term, therefore, there shouldbe no attached vortices past the magnetic obstacle, so the possible scenario are either novortices when the Lorentz force is smaller than the viscous force (case 3) or two alongsidemagnetic vortices when the Lorentz force is larger than the viscous force (case 4). Finally,the cases (5-6) are of the smallest viscous term and the peculiar patterns are either sixvortices (case 5) when the inertial force is so low that the attached vortices are retained orvortex shedding with specific four-vortex pattern (case 6) as shown in previous Section. Inthe latter, the four vortices taken together is an analog of the bluff body as in the ordinaryhydrodynamics.It is important to stress that we discuss a 3D flow between two horizontal no-slip end-plates. This discussion might be projected onto a 2D flow, but carefully. For instance, the2D flow could not produce a 2D region excluded from the flow without violating the conti-nuity requirement, ∇ ⊥ u ⊥ = 0. In a 3D flow, the latter equation is ∇ u = ∇ ⊥ u ⊥ + ∂ z u z = 0,which can be satisfied by the secondary flow in the third direction, i.e. by the ∂ z u z term.This results in the helical streamlines of magnetic vortices, see Fig. 11 in the paper byVotyakov et al. [13]. However, a 3D helix could not be realized in a 2D space, so the u center , E y,center vanishing behavior shown in Fig. 3 becomes impossible. Instead, in the creeping 2Dflow, u center is decreases as N increases. Then, at some high critical N , a u center drop doesstop and makes a flip into positive values causing two additional vortices in the core of the20bstacle. This new resulting flow structure consists of four vortices as shown in Fig. 9 b ofthe paper by Cuevas et al. [9]. If N increases further, then even more intricate recirculationpatterns are produced[30]. It looks paradoxical that a 3D flow has the simpler structure thana 2D flow, however, such simpler behavior is governed by a strong magnetic field, prohibit-ing a penetration into the core, and by the secondary flow in a vertical direction towardsto Hartmann layers. Moreover, it is an issue how to practically realize a 2D creeping flowin order to reveal intricate recirculations in the 2D core without impact of top and bottomflow boundaries.Another question that arises is whether the core of the magnetic obstacle appears fora sufficiently high Re even at high N , that is, whether the upstream flow penetrates thestagnant core made of the magnetic and connecting vortices. Again, the answer dependson whether the flow considered is two- or three-dimensional. We suggest that the magneticvortices must appear in both cases because N is supposed to be sufficiently high to producerecirculation. So the question can be reformulated: whether the vortices and the stagnantcore are stable.In 2D simulations there is no sink for the upstream kinetic energy accumulated by themagnetic vortices. As a result, the rotating magnetic vortices are not fixed in their locationand move freely in the plane by responding to the pulsations of the upstream flow. Thisdestroys the core of the magnetic obstacle so it becomes penetrable. If there are small time-dependent pulsations in the upstream flow, then one can observe different (even exotic)configurations of magnetic vortices, which can be mistakenly taken as sub-scenarios of thegiven 2D simulation.For any 2D approach applied to a realistic system, the main problem is whether 2D as-sumptions are reasonable because the realistic system has always endplates to hold magneticpoles. That is, the Hartmann friction and viscous friction are always present. Of course, tomake 2D middle plane flow, the magnetic poles can be moved far apart while synchronouslyincreasing the magnetic field intensity B to keep the same Ha . But then, the gradient ofthe magnetic field becomes more smooth, therefore, one needs a higher critical N c to observerecirculation [13].In 3D simulations, there is a sink for the kinetic energy because the mass streamlines,that form magnetic vortices, represent helical trajectories into the Hartmann layers as shownrecently in [13]. Then, the pulsations of the upstream kinetic energy are dissipated in the21artmann layers by means of the friction with top/bottom walls. As a result, the rotatingmagnetic vortices are well fixed in their location and they do not move freely. Thus, whenHartmann layers are properly resolved in the 3D simulations and N is enough high to inducealongside recirculation, then the core of the magnetic obstacle must be visible even at high Re .To make more clear the aforesaid statement we consider the following two examples. First,one imagines a flow of moderate Re and such a high Ha that the core of the magneticobstacle is stable. Then, by keeping Ha constant, one increases Re , e.g. by taking a higherflow rate, to find Re c ( Ha ) where the core destabilizes. This happens at a critical value N c, = Ha /Re c . Now, one imagines a flow at moderate Ha and such a high Re thatthe core of the magnetic obstacle does not exist. Then, while keeping Re constant, oneincreases Ha , e.g. by imposing a higher external magnetic field. There exists such a high Ha c ( Re ) where the core stabilizes again. This happens at a critical value N c, = Ha c /Re ,which is supposed to be of the same order of magnitude as N c, . In other words, at anyhigh constant Ha it is possible to find Re destabilizing the core, and vice verse, at any highconstant Re , it is possible to find Ha stabilizing the core.Unfortunately, to confirm the above inference numerically is impossible because it requiresexpensive 3D simulations where the Hartmann layers must be properly resolved. For high Re and N , the thickness of the Hartmann layers is 1 / √ Re ∗ N . Then, e.g. for N = 100 (toguarantee magnetic vortices) and big Re the grid resolution must be around δ/ (10 √ Re ),where 1 /δ = 3 ...
10 is the number of points to resolve Hartmann layers. If Re = 10000, thenaround ( L y /H )( L x /H )10 ∼ numerical nodes is needed for every time step. The totalnumber of time steps must be also very large to be sure that the magnetic vortices do notdestroy the core of the magnetic obstacle. In the previous Section, Re = 900 and N = 9,and the core of the obstacle is shown to be stable.Another interesting issue is whether the vortex shedding past the magnetic obstacle issimilar to that past the solid obstacle. Indeed, the following is valid generally in both cases:(1) the attached vortices are formed from the creeping flow when Re prevails a criticalvalue; (2) in a certain range of Re , the attached vortices are stable; (3) when the inertialforce exceeds the stable threshold, the attached vortices detach from the body. Becausethe inertial force is decisive in both cases and the Lorentz force vanishes past the magneticobstacle, it is expected that the vortex shedding in both cases might be similar as well, at22east for specially selected geometrical conditions. This issue is open currently. Conclusion
In this paper, we attempted to shed light on the peculiarities of the MHD flow passingover a magnetic obstacle when the magnetic interaction parameter N is large, i.e. strengthof the magnetic field is high or when the Reynolds number Re , i.e. inertia of the flow, islarge. The corresponding case for moderate Re and N has been elaborated in [12, 13]. Asit turns out high values of Re and N neatly emphasize analogies between a magnetic anda solid obstacle that have been under discussion from the beginning of MHD in the formerUSSR. In this paper, we have illustrated, by means of 3D numerical simulations, how thecore of the magnetic obstacle is formed when N increases and examined the shedding ofattached vortices when the Re is high enough.With regard to the core of the magnetic obstacle the open issue remains whether ispossible to treat the problem in a simpler way based on the Kulikovskii’s approach [20].This is an old and fruitful idea to subdivide the complicated MHD flow into two parts: coreand periphery and then combine both parts with appropriate boundary conditions. It isshown in this paper that there must be three parts to include recirculation at high N : therest of the flow, immovable core, and transitional region with magnetic vortices.With regard to the shedding of vortices past a magnetic obstacle, it is confirmed thatmagnetic and connected vortices altogether represent a virtual bluff body, which is spatiallyfixed owing to the Hartmann friction [12, 13]. Because the virtual body is fixed, a stagnancyregion is formed. In this region, the intensity of the magnetic field is negligible, hence, theLorentz force vanishes and attached vortices are controlled only by the inertial force. Thesame happens past a solid obstacle. Then, it may appear that the regularities known for theattached vortices past a solid cylinder are valid also for those past a magnetic obstacle. Thisway of thinking is useful provided that one takes into consideration the three-dimensionalityof the problem, namely, the fact that the cylinder is confined between two endplates.23 cknowledgements 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 theCEC under the 6th Framework Program. The simulations were carried out partially on aJUMP supercomputer, access to which was provided by the John von Neumann Institute(NIC) at the Forschungszentrum J¨ulich. E.V.V. is grateful for many fruitful discussions withOleg Andreev, Yuri Kolesnikov, Andre Thess, and Egbert Zienicke during his time in theIlmenau University of Technology. The authors are thankful to the Referee whose commentsled them to write a section for the generic scenario of the wake of the magnetic obstacle. [1] P. Davidson. Magnetohydrodynamics in Materials Processing.
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Lectures on physics. Vol II .Addison-Wesley, 1964.[28] Yuri Kolesnikov, coauthor of [12, 13] used magnetic obstacle as a working term in the 1970sin Riga, MHD center of the former USSR, in order to stress the analogy with a solid obstacle.[29] Fig. 4 in [11] and Fig.4 in [10] are very similar despite the fact that the paper [11] is a 2Dsimulation while the paper [10] is a quasi-2D approach with a friction term. One may concludethat the friction term is of minor significance in 2D models.[30] We performed 2D simulations and found also four vortices shown in the paper by Cuevas etal. [9]. Moreover, other vortex configurations not reported in [9] have been revealed at higher N . The results will be submitted.. The results will be submitted.