Turbulence, Heat and Mass Transfer 6 K. Hanjalić, Y. Nagano and S. Jakirlić (Editors) © 2009 Begell House, Inc.

Solidification of a binary metal alloy in a turbulent melt flow driven by AC magnetic fields

P.A. Nikrityuk1, K. Eckert2 and S. Eckert3 1ZIK VIRTUHCON, Institute of Energy Process Engineering and Chemical Engineering, TU Bergakademie Freiberg, Reiche Zeche, Fuchsmühlenweg 9, 09596 Freiberg, Germany, Petr.Nikrityuk@vtc.tu-freiberg.de 2Chair of Magnetofluiddynamics, Institute of Fluid Dynamics, Technische Universität Dresden, Mommsenstr. 13, 01062 Dresden, Germany, Kerstin.Eckert@tu-dresden.de 3MHD Department, Forschungszentrum Dresden-Rossendorf (FZD), P.O.Box 510119, 01314 Dresden, Germany, S.Eckert@fzd.de

Abstract - In this work we numerically study the impact of Taylor-Görtler vortices inside a melt, driven by a rotating magnetic field, on the macrosegregation and the shape of the mushy zone during unidirectional solidification of Al-7wt%Si alloy. The weakly turbulent flow was modeled by means of direct numerical simulations in an axisymmetric approach, where the transient heat and mass transfer were simulated by means of a standard mixture model. Both types of solidification, columnar and equiaxed, were considered by the application of both a permeability and a hybrid model to treat the fluid flow in the mushy zone. Our results demonstrate that in the case of columnar solidification the Taylor-Görtler vortices cause both a wavy shape of the mushy zone and segregations in the form of a fir tree with a distinct accumulation of silicon along the axis of the cylinder.

1. Introduction Recent experimental studies showed that the application of an RMF during the solidification of a metal alloy is the origin of a new type of macrosegregations [1,2]. These segregations are the result of the secondary flow in the form of a double vortex structure which appears in a rotating flow due to the Ekman pumping effect. Such a secondary flow is responsible for the transport of the solute, rejected by the solidification, towards the axis of rotation. As a result, typical segregation channels with a high concentration of the eutectic phase [1-3] are formed there. Recently Kovacs et al. [2] showed experimentally that strong macrosegregation with a “Christmas-tree” (fir tree) shape developed near the axis of the solidified Al-7wt%Si alloy when the magnetic Taylor number is increased above certain critical values. However, no qualitative analysis of the influence of flow oscillations on the macrosegregation due to the Taylor-Görtler vortices, randomly appearing along the side wall of the cylinder, was performed. Thus the aim of this work is to improve the understanding of the influence of the turbulent flow, driven by an RMF, on the final macrosegregation during unidirectional columnar and equiaxed solidification of binary metal alloys. 2. Problem and Model Formulation

We consider a uniform RMF with an amplitude 0B and an angular frequency, fπω 2= ( 50=f Hz is the frequency of the alternating current), applied to a cylindrical cavity of a radius 025.00 =R m and a height 0625.00 =R .m, filled with a superheated Al-7wt%Si alloy. The interaction of the RMF with the electrical current induced produces an azimuthally

Turbulence, Heat and Mass Transfer 6

dominant Lorentz force, θF , leading to the rotation of the liquid alloy. We use a

low-frequency, low-induction RMF, which means that the skin depth, 0/1 ωσμ , is larger than 0R , where 0μ is the magnetic permeability of the free space. The low-induction condition implies that the angular velocity induced by the RMF does not change the magnetic field applied. Once these conditions are fulfilled, an analytical solution can be written for the time-averaged Lorentz force which is directed in azimuthal direction. It is a function of the position only and has the following form:

( )zrfrBF ,21 2

0 ⋅⋅⋅⋅= ωσθ, (1)

where ),( zrf is a geometrical shape function, for details see Nikrityuk et al. [3,10]. To characterize the regimes of the fluid flow driven by the RMF we use the magnetic Taylor number:

2

40

20

2μωσρ RB

Ta = . (2)

In this work we perform simulations for 51005.4 ⋅=Ta ( 30 101 −⋅=B T), 61053.2 ⋅=Ta

( 30 105.2 −⋅=B T) and 61048.6 ⋅=Ta ( 3

0 104 −⋅=B T). According to Grants and Gerbeth [10] the developed flow for these Ta numbers is unsteady. The scheme of the set-up under consideration is shown in Fig 1a.

We use two different assumptions for the motion of the dendrites in the mushy zone. The first one assumes immobile dendrites (columnar solidification) while the second assumption considers the dendrites as floating within the mushy zone (equiaxed solidification). To investigate both cases we apply the standard continuum model [4] in two different versions: For the columnar solidification the fluid flow in the mushy region is modeled by means of the Darcy law (permeability (K)-model). By contrast, a hybrid model [5] is used for the equiaxed solidification with floating dendrites.. Both models rest on the following assumptions: the shrinkage-driven flow is negligible; the flow of the liquid phase is assumed to be axisymmetric; the thermoelectrical effects (Peltier, Thomson) and Joule effect are neglected; the electrical conductivities of the solid and liquid phased are equal. We note that in the case of non-homogeneous electrical conducting media the Lorentz force has to be calculated numerically, see Nikrityuk et al. [11]. Based on the assumptions made above, the set of equations has the following form:

0=⋅∇ u (3)

( ) ( ) ( ) ( )( ) ,0,0 LllCTlm

m FCCTTguK

upuutu

+−+−+−∇∇+−∇=∇⋅+∂∂ ββρ

μμρρ (4)

( )( ) ( ) ( )

tLTTcu

tTc

pp

∂∂

−∇⋅∇=⋅∇⋅+∂

∂ ερλρρ (5)

( ) ( ) ( )( ) ( )( )( ) ,CCuuCCDCDCutC

lslmm −−⋅∇−−∇⋅∇+∇⋅∇=∇⋅+∂∂ ρρρρρ (6)

( )ελελλ −+= 1sl , ( )εε −+= 1sllm DDD (7) ( )

( ) ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−=

CTCTCTTMAXMINsl

s,0,1ε (8)

where ε refers to the volume fraction of liquid in the mushy zone. The hybrid model uses permeability and viscosity functions, K and μm, to capture the behavior of the flow within the

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P.A. Nikrityuk et al.

mushy zone [5]:

( )2

3

0 1 εε−

= KGK K , ( )[ ]4

100arctan15.0−

⎟⎠⎞

⎜⎝⎛ −+= εε

π crKG , 180

21

0d

K = , (9)

( )

2

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛

−−=

εμμ

μμ

μ

FAA

lm , ( )[ ]εεπμ −−= crF 100arctan15.0 , 4.0=μA , (10)

Here 1d is the dendrite arm spacing. It is assumed that the velocity of the floating dendrites is proportional to the fluid flow velocity in the slurry region of the mushy zone [12], namely

uus ε≈ for crεε > , 0=su for crεε < , 6.0=crε . All material properties of Al-7wt%Si used in the simulations were taken from the work of Willers et al. [6].

a b

c d e Figure 1: Scheme of the set-up – (a), time history of the bottom temperature – (b), next figures are the validation cases: (c) – comparison of radial profiles of the time-averaged, non-dimensional azimuthal velocity with respect to Stiller et al. [7], (d) – the radial profile of the non-dimensional turbulent

kinetic energy at 3

0Hz = , 5103 ⋅=Ta , 3D DNS [9], (e) - the axial profile of the time-averaged

axial velocity following experimental data of Cramer et al. [8].

The weakly turbulent flow was modeled by means of direct numerical simulations in an axisymmetric approach where both types of solidification, columnar and equiaxed, are considered as outlined above. Details about our calculation procedure have been presented elsewhere [3,6]. The basic scheme of the domain and the boundary conditions used are shown in Fig 1a. The boundary condition on the bottom of the cavity for the temperature equation is shown in Fig. 1b. The boundary condition at the bottom of the cavity for the temperature

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Turbulence, Heat and Mass Transfer 6

equation can be approximated as follows ( ) ( )( ) 04.02525 25.08714975)( tttT ⋅−+⋅−= [K], here t is given in sec. Several grid- and time-step-convergence tests were preformed to define the proper grids and time steps leading to grid and time-step independent solutions. The final grid used in the simulations has 120 control volumes in the radial direction and 360 control volumes in the axial direction. The time step was set to 0.1 s.

3. Code Validation To validate the code and to check the ability of axisymmetric numerical simulations (ANS) to predict the correct time-averaged velocity of RMF-driven turbulent flows, we use an experiment by Stiller et al. [7], who measured radial profiles of the time-averaged azimuthal velocity in a liquid metal, driven by an RMF, for three magnetic Taylor numbers 5101 ⋅=Ta ,

5102 ⋅ , 5103 ⋅ , and 3D DNS data calculated for 5103 ⋅=Ta by Frana et al. [9]. The grid resolution used in the validation simulations corresponds to 150 and 450 control volumes in the radial and axial directions, respectively. The nondimensional grid spacing equals to 0.0067. In comparison, the nondimensional grid spacing of 0.018-0.005 was used in 3D DNS [7,9] for the simulation of the flow at 5103 ⋅=Ta . For the calculation of time-averaged

velocities we used the time span scaled with ν

20R equals to 0.1. In comparison, the time span

in 3D DNS [7,9] was set to 0.03. The comparison of experimental data and numerical data (Fig. 1c) shows a very good

agreement. In order to verify the ability of ANS to capture correctly the turbulent parameters, which are responsible for the mixing, we plot the radial profiles of turbulent kinetic energy,

( ) 2/''''''zzrr uuuuuuk ++= θθ in Fig. 2d. It can be seen that in the near side-wall region, where

the azimuthal and meridional velocities reach a maximum, a good agreement can be observed. However, near the axis the ANS results predict higher values of k in comparison to 3D DNS. Thus, it can be seen that ANS overpredicts the turbulent kinetic energy on the axis of rotation, where the meridional and azimuthal velocities are weak, but in spite of that it gives a very good agreement in the turbulent sidewall region where the turbulent mixing takes place, see next Section. In that way, ANS is able to predict adequately the main features of weak turbulent flow driven by RMF.

Next, we present the ANS validation against experimental data for the secondary flow

published recently by Cramer et al. [8] for 6102 ⋅=Ta and 12 0

0 =R

H , corresponding to

6.163

=Ta

Ta Dcr . The values of D

crTa 3 can be found in [10]. All material properties of a liquid

metal used in the validation-simulation were taken from the paper [8]. The grid resolution used in the validation simulations corresponds to 200 and 400 control volumes in the radial and axial directions, respectively. The nondimensional grid spacing equals to 0.005. For the

calculation of the time-averaged velocity we used the time span scaled with ν

20R equals to

0.2. The results of comparison are shown in Fig. 1e. Good agreement is seen. 4. Results Next we discuss the simulations of the Al-7wt%Si-solidification by using both the hybrid and

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P.A. Nikrityuk et al.

the K-model. Figure 2 displays the corresponding snapshots of the azimuthal velocity and the volume fraction of liquid together with the vectors of the meridional velocity.

Independent on the model, the flow field is dominated by the so-called Taylor-Görtler (T-G) vortices occurring at the side walls. The T-G vortices move along the side walls up and down. In the upper part of the mushy zone, the T-G vortices are advected by the strong radial flow resulting from the Ekman pumping. On their way through the mushy zone the T-G vortices are dissipated which results in oscillations of the flow intensity of the radial jet. The feedback of the oscillation is visible in the primary flow, too (see the axial undulations in Fig. 2 a and 2c).

a b c d Figure 2: Comparison between the K-model (a-b) and the hybrid model (c-d) for B=4mT

61048.6 ⋅=Ta . (a) and (c) - contours of the azimuthal velocity at t=70 s; (b) and (d) - meridional velocity vectors plus liquid isolines at t=70 s.

a b c d e f Figure 3: Contour plots of the final relative concentration of Si, (C-C0)100% / C0, calculated by use of the K-model (a-c) and the hybrid model (d-f) for 51005.4 ⋅=Ta , 61053.2 ⋅=Ta and

61048.6 ⋅=Ta , respectively.

Applying the K-model, thus considering columnar dendrites ( 0=su ), the oscillations in the radial flow, which conveys the solute rejected at the solidification front towards the axis of the cylinder, are responsible for a local remelting. This occurs in spots with a particularly

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Turbulence, Heat and Mass Transfer 6

high solute concentration near the axis leading to a wavy shape of the mushy zone. As a result, the final segregation patterns show a distinct accumulation of silicon along the axis of the cylinder in the form of a fir tree, see Fig. 3a,b,c. The increase of Ta leads to an increase of the segregation and to a regularization of fir tree side arms. These findings are in good agreement with the experimental data published recently by Kovac at al. [2].

The analysis of the floating dendrite case ( 6.0>crε ) with the hybrid model shows that the oscillations of the radial flow are damped by the rather high viscosity of the slurry region formed by the mixture of equiaxed dendrites and interdendritic liquid, see Fig. 2d,e,f. The transport of the equiaxed dendrites towards the axis of the cylinder results in a faster cooling of the melt (compare the propagation of the liquid isolines in Fig. 2b and 2d) and to a convex shape of the mushy zone. The latter benefits from the lack of the strong axial silicon accumulation. Consequently the final macrosegregation (Fig. 3d,e,f) is significantly smaller in comparison with the columnar solidification. An increase of the Taylor number leads to a further decrease of the macrosegregation in the middle of the sample (see Fig. 3f) for which we see two reasons. First, the turbulent mixing of the rejected solute and floating dendrites with the bulk liquid, mediated by the T-G vortices, is enhanced. Second, the transport of floating dendrites by the flow reduces the ‘wash’ effect, resulting from the convective transport of the rejected solute from the solid interface, in comparison with non-movable dendrites. 5. Summary This numerical work studies the impact of turbulence on the macrosegregation and the shape of the mushy zone during unidirectional solidification of Al-7wt%Si alloy in an RMF. Our results demonstrate that the Taylor-Görtler vortices cause a local enhancement of the solute transport from the wall to the axis in the upper part of the mushy zone. In the case of columnar solidification this effect leads to a wavy shape of the mushy zone and a segregation in the form of a fir tree with a distinct accumulation of silicon along the axis of the cylinder. In the case of free floating dendrites the increase of RMF intensity leads to a reduction of the macrosegregation on the axis of the cylinder due to the turbulent mixing. 6. Acknowledgements This work was financially supported by Deutsche Forschungsgemeinschaft (DFG) as part of the SFB609. Sächsische Ministerium für Wissenschaft und Kunst of Saxony and Administration of TU Bergakademie Freiberg are gratefully acknowledged for financial support of P.A.N as a part of ZIK ‘VIRTUHCON’. References 1. J.K. Roplekar and J.A. Dantzig. A study of solidification with a rotating magnetic field.

Int. J. Cast. Met. Res., 14: 79-95, 2001. 2. J. Kovacs, et al. Characterization of the structure of Al-7Si-0.6Mg alloy solidified

directionally in a rotating magnetic field. In 5th Int. Conf. on Solidification Proc-'SP07', (ed. H. Jones), Vol.1, Sheffield, The Sheffield University, pp. 235-239; 2007.

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