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www.nature.com/npjmgrav ARTICLE OPEN Numerical representations for ﬂow velocity and shear rate inside electromagnetically levitated droplets in microgravity 1 2 3 1 Xiao Xiao , Jonghyun Lee , Robert W. Hyers and Douglas M. Matson Electromagnetic levitation techniques are used in a microgravity environment to allow materials research under containerless conditions while limiting the inﬂuence of gravity. The induced advective ﬂow inside a levitated molten alloy droplet is a key factor affecting solidiﬁcation phenomena while potentially inﬂuencing the measurement of thermophysical properties of metallic alloy. It is thus important to predict the ﬂow velocity under various operation conditions during melt processing. In this work, a magnetohydrodynamic model is applied over the range of conditions under which electromagnetically levitated droplets are processed to represent the maximum ﬂow velocity and shear rate as a polynomial function of heating voltage, density, viscosity, and electrical conductivity of molten materials. An example is given for the ternary steel alloy Fe-19Cr-21Ni (at%) to demonstrate how internal advection under different heater settings becomes a strong function of alloy temperature and is a determining factor in the transition from laminar to turbulent ﬂow conditions. The results are directly applicable to a range of other materials with properties in the range considered, including Ni-based superalloys, Ti-6Al-4V, and many other commercially-important alloys. npj Microgravity (2019) 5:7 ; https://doi.org/10.1038/s41526-019-0067-2 INTRODUCTION et al. presented an analysis of heating power and electro- 12–15 magnetically levitated droplet, Szekely et al. developed the Containerless processing techniques involving electromagnetic mutual inductance method to calculate electromagnetic forces in levitation (EML) provide the capability to position and process a 16–18 the spherical droplets, and Lohöfer developed an analytical highly reactive molten metal sample without use of a crucible model for the absorbed power, current distribution and impe- while conducting thermophysical property measurements or dance of an electromagnetically levitated metal sphere. Compared solidiﬁcation studies. For thermophysical property evaluations, to the terrestrial environment, a microgravity environment the viscosity, density, surface tension, resistivity, and heat capacity provides the opportunity to maintain stable EML conditions with of molten metal sample can be measured; for solidiﬁcation studies greatly reduced positioning forces. The levitated molten sample the focus is on nucleation phenomena, growth mechanism, and will form an approximately spherical shape and the induced ﬂow phase selection. In either case, conditions may be signiﬁcantly inside the sample can achieve a wide range of ﬂow velocity from inﬂuenced or controlled by the advective ﬂow inside the levitated 6,19 laminar to turbulent conditions. molten metal droplet. For instance, the viscosity measurement of Due to the difﬁculty of measurement of the ﬂow inside the molten metals could be greatly affected by internal turbulent 2–5 molten sample directly from experiment, numerical methods are ﬂow induced by the electromagnetic forces required to utilized to simulate the advective ﬂow ﬁeld and predict related position, levitate and heat a sample, and well-controlled internal variables such as local ﬂow velocity and shear rate inside the ﬂow conditions are necessary to support the experiments; for levitated molten metal droplets under given experimental phase selection in steels, the transformation of metastable to 6–9 parameters such as the sample’s physical properties and coil stable phases during rapid solidiﬁcation is strongly affected by settings. For magnetohydrodynamic (MHD) simulation, in previous and could be controlled by applied advection inside the molten 12,20 work by Szekely et al. , MHD models for the electromagneti- sample thus inﬂuencing development of the ﬁnal microstructure. cally levitated droplets was developed using a k–ε turbulence For an EML facility, an alternating electromagnetic ﬁeld is model for both terrestrial and microgravity environments. Recent applied to a conductive sample located within a water-cooled coil 1,4 as part of a high frequency oscillating circuit. Eddy currents work by Hyers et al. reported results for laminar ﬂow in spherical induced inside the sample provide heating and positioning droplets in a microgravity EML facility, and extended the results to functions at different frequencies of the oscillating circuits, and turbulent ﬂow of gravitationally-deformed droplets in ground- the temperature of the sample is controlled by adjusting the based EML. Berry et al. surveyed the turbulence models and heating control voltage. Meanwhile, the advective ﬂow inside the stated that RNG k–ε turbulence model (Renormalization Group molten sample is induced by the applied Lorenz force when the method variation) is the most appropriate model for EML droplets. electromagnetic ﬁeld is imposed, and velocity could be high Lee et al. validated the k–ε turbulence model through the under large heater setting, and turbulent ﬂow may result. Okress comparison between the experiments and the predicted ﬂow 1 2 3 Department of Mechanical Engineering, Tufts University, Medford, MA, USA; Department of Mechanical Engineering, Iowa State University, Ames, IA, USA and Department of Mechanical & Industrial Engineering, University of Massachusetts, Amherst, MA, USA Correspondence: Douglas M Matson (douglas.matson@tufts.edu) Received: 4 May 2018 Accepted: 21 December 2018 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA X. Xiao et al. velocity along the surface of an electromagnetically levitated in, the ﬂow velocity and shear rate are predicted and molten copper-cobalt droplet in the terrestrial environment which represented as function of heating control voltage, density, showed excellent agreement between model and experimental viscosity, and electrical conductivities based on around 10,000 observations. The ﬂow is usually characterized by the Reynolds discrete modelling runs for both of laminar and turbulent models. number (Re) as deﬁned in Eq. (1), which represents the ratio of The model is solved in axisymmetric two-dimensional space. u inertial effects to viscous effects and indicates the laminar or and u denote the ﬂow velocity in the angular and radial turbulent condition of the ﬂow. coordinate respectively, u is the velocity magnitude, and u is max the the maximum ﬂow velocity. γ denotes the magnitude of shear ρud rate inside the droplet as deﬁned in Eq. (2), and γ is the Re ¼ (1) max maximum shear rate in the ﬂow ﬁeld. where μ is the viscosity, ρ is the density, u is the velocity, and d is ∂ u 1 ∂u θ r γ_ ¼ r þ (2) the diameter of the sample droplet. For the laminar-turbulent ∂r r r ∂θ transition that is characterized Reynolds number, Hyers et al. At each electrical conductivity value, the maximum velocity u suggested that the transition occurs at Re around 500 to 600, max and maximum shear rate γ_ are ﬁtted into third degree which is experimentally observed from the formation and max polynomials with four variables over a representative range of perturbation of the stagnation line at the equator of the droplet. 23,24 heating control voltage U (i), density ρ (j), natural logarithm of Lee et al. also predicted the ﬂow velocity of electromagne- ctr viscosity ln μ (k), and natural logarithm of electrical conductivity ln tically levitated iron-cobalt droplet in support of the experiments σ (s), as presented in Eq. (3), where the coefﬁcients p are on board the International Space Station (ISS) with characteristic e,l ijks derived using least-squares approach from the raw data. The constraints of temperature and heating current appropriate to test quality of the ﬁts for the interpolated maximum velocity u ^ and conditions and determined the corresponding laminar and max interpolated maximum shear rate γ_ are evaluated using R- turbulent conditions related to the given geometry and realistic max squared metric, where the value closer to 1.0 means a better ﬁt assumptions of the thermophysical properties of the alloy has been obtained. including density, viscosity, and electrical conductivity. Besides 25,26 the k–ε turbulence models, Bojarevics et al. used pseudos- k s b H i j u ^ or γ_ ¼ p U ρ ðln μÞ ðln σ Þ max ijks e;l max ctr pectral methods to solve the Navier–Stokes equations with k–ω i;j;k;s turbulence model, Ai used direct numerical simulation of P 2 P (3) ^ _ _ turbulent ﬂow in EML. ðu u Þ γ γ max max max max R squared ¼ 1 or 1 2 P 2 In the present work, the model development is based on ðu u Þ max max γ_ γ_ max max microgravity EML using a superposition levitation method (the coil To evaluate the contribution of each term to the overall ﬁt, the conﬁguration is called SUPOS for “superposition”) on board ISS; absolute value of Pearson correlation coefﬁcient (PCC), as deﬁned the design speciﬁcations of ISS-EML SUPOS coil are described by 28,29 in Eq. (4), is calculated between simulation results Y = u or γ max Lohöfer. MHD simulations using laminar model and RNG k–ε max k s H i j for each term X ¼ U ρ ðln μÞ ðln σ Þ . ijks e;l turbulence model are conducted to predict the ﬂow velocity and ctr shear rate inside a molten droplet when electromagnetically covðX ; YÞ ijks levitated by the SUPOS coil in a microgravity environment in both ρ ¼ (4) X ;Y ijks σ σ X Y ijks the laminar and turbulent regime, as a function of a series of key experimental parameters. For a given sample size, these The value of ρ is between 0 and 1 for positive correlation, X ;Y ijks parameters include heating control voltage of the coil, density, where a value closer to 1.0 means a signiﬁcation correlation; cov(X , ijks viscosity, and electrical conductivity of the sample material. Y) is the covariance between X and Y,and σ are their standard ijks Finally, the results from MHD simulation are represented as deviation. To select the dominating terms X and reduce the ijks polynomial expressions for convenient reference to be applied to dimension of the regression equation, X is ordered by the value ijks molten materials that requires characterization by MHD methods; ρ ,and the ﬁrst N terms of X are included in the Nth regression ijks X ;Y ijks in practice this involves deﬁning key material properties as a testing until R-squared increases to value closer to 1.0 and function of temperature such that the ﬂow ﬁeld becomes a converges. The regression tests show that the ﬁrst 21 terms were function of applied heating control voltage and sample tempera- signiﬁcant, as displayed in Table 2.The ﬁtted coefﬁcients p and ijks ture, only. overall R-squared values using laminar and turbulent models are displayed separately, and using these values the predicted ^ _ maximum velocity u and predicted maximum shear rate γ max max RESULTS can be readily estimated for any combination of parameters of U , ctr General model ρ, μ,and σ ,byusing Eq. (3)withall thecoefﬁcients p presented e,l ijks The MHD simulation is performed for a 6.5 mm electromagneti- in Table 2 and related indices i, j, k, s applied to each term. Figure 1a cally levitated droplet in microgravity with the ISS-EML SUPOS coil shows an example of the predicted u as function of viscosity μ, max under ﬁxed positioning control voltage U at 5.21 V, and multiple ctr heating control voltage U and density ρ under electrical ctr 5 −1 conditions of heating control voltage, density, viscosity, and conductivity σ = 6.0 × 10 Sm ,and Fig. 1b shows u as e,l max electrical conductivity which are shown in Table 1. For a general function of σ , U ,and ρ under μ= 0.010 Pa s. e,l ctr levitated molten droplet, as expansion plus ﬁtting of monographs DISCUSSION Table 1. Operation conditions for ISS-EML Levitated Droplet In the current settings, the heating ﬁeld produces much stronger ﬂow than the positioner ﬁeld for most of the common operating Parameters Values range. The magnitude of positioner-induced ﬂow and correlated Heating control voltage (V) U ¼ 0:01 6:00ðÞ 8 levels shear rate slightly increases with the positioner voltage U in the ctr ctr −1 −1 −3 P range from 2.0 to 10.0 V, where du ^ =dU is <0.0002 m s V Density (kg m ) ρ = 5000–10,000 (11 levels) max ctr −1 −1 and dγ_ =dU is <0.8 s V . The variance induced from max Viscosity (Pa s) μ = 0.001–0.040 (8 levels) ctr −1 different positioner voltage U ¼ 2:0 V to 10.0 V is <0.001 m s −1 6 ctr Electrical conductivity (S m ) σ = 2.0 × 10–6.0 × 10 (7 levels) e,l −1 for u ^ and <4.0 s for γ_ compared to the results with max max npj Microgravity (2019) 7 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; X. Xiao et al. Table 2. Polynomial coefﬁcients of maximum velocity and shear rate for ISS-EML Levitated Molten Droplet Laminar model Turbulent model −1 −1 −1 −1 Velocity (m s ) Shear rate (s ) Velocity (m s ) Shear rate (s ) −1 2 −1 2 p 2.705 × 10 3.801 × 10 1.025 × 10 2.213 × 10 −2 1 −3 1 p −2.375 × 10 −3.601 × 10 −9.377 × 10 −1.796 × 10 −1 2 −2 2 p 1.481 × 10 1.152 × 10 9.369 × 10 1.432 × 10 −2 1 −2 1 p −1.758 × 10 −1.196 × 10 −1.221 × 10 −1.662 × 10 −4 −2 −4 −1 p 3.930 × 10 3.746 × 10 3.322 × 10 4.002 × 10 −5 −1 −5 −2 p −1.402 × 10 −1.304 × 10 −2.197 × 10 4.452 × 10 −1 3 −1 2 p −9.354 × 10 −1.634 × 10 −4.259 × 10 −7.906 × 10 −1 2 −2 1 p 1.100 × 10 1.881 × 10 5.454 × 10 8.993 × 10 −3 −3 p −3.587 × 10 −5.873 −1.796 × 10 −2.927 −2 1 −3 1 p −3.068 × 10 −8.963 × 10 −1.167 × 10 −1.992 × 10 −3 −2 −3 p −1.638 × 10 1.088 × 10 −1.233 × 10 −2.826 −3 −4 p −2.927 × 10 −5.758 −7.224 × 10 −3.369 −6 −2 −6 −2 p 9.942 × 10 1.262 × 10 4.693 × 10 1.031 × 10 −7 −3 −7 −4 p −8.575 × 10 −1.170 × 10 −4.767 × 10 −9.095 × 10 −6 −3 −7 −3 p 1.135 × 10 1.635 × 10 5.908 × 10 1.245 × 10 −10 −7 −10 −7 p 2.931 × 10 4.580 × 10 1.775 × 10 3.035 × 10 −3 −4 p 3.861 × 10 6.457 −9.398 × 10 1.933 −4 −1 −4 −1 p 7.098 × 10 7.534 × 10 3.206 × 10 6.064 × 10 −3 −4 −1 p 1.269 × 10 1.993 2.329 × 10 9.055 × 10 −8 −5 −8 −6 p −7.639 × 10 −9.026 × 10 −1.077 × 10 3.413 × 10 −4 −1 −4 −1 p −6.038 × 10 −4.668 × 10 −1.623 × 10 −4.845 × 10 R−squared 0.9963 0.9939 0.9982 0.9957 U ¼ 5:21 V, presented up to 10% error when the heater is these tests, MHD modeling was conducted over the range of ctr minimized, and up to 3% error when U is >0.2 V. This variation conditions accessible using the ISS-EML SUPOS coil. Conditions ctr with positioner is negligible for most operational conditions, so would be selected such that the 6.5 mm diameter molten sample positioner voltage U is excluded from the ﬁts. droplet could achieve a wide range of heating rates (up to dT/dt ctr −1 −1 The droplet dimension is an additional factor in the MHD model = 200 K s at T ) or cooling rates (dT/dt = 0–50 K s at T in m m 1,24 −1 which was studied previously that the maximum velocity u ^ vacuum or dT/dt = 0–100 K s at T in helium) and a broad range max and maximum shear rate γ_ increases for larger droplet of thermal hold temperatures T = T ± 200 K such that each is max m diameter d, and gives basis for an extrapolation formula presented characterized by distinct quasistatic ﬂow conditions depending on in Eq. (5) for d = 5.0 mm−7.0 mm based on the predictions under the heating control voltage. The thermophysical propriety values d = 6.5 mm. vary with the temperature as shown in Table 3. For operation conditions heating control voltage U ¼ ctr γ 2 0:01 V 5:7 V with the positioner maintained at U ¼ 5:21 V, max u ^ j 0:253 d 1:887 d 3:393 ctr max d d or (5) and temperatures over the range T = 1515 K–1915 K (T − 200 K 2 m u j b 0:253 d 1:887 d 3:393 max 0 d _ 0 0 γ max to T + 200 K), the MHD model was utilized to predict the advective ﬂow ﬁeld and local shear rate inside the 6.5 mm molten Fe-19Cr-21Ni droplet. Practical application to a speciﬁc case Figure 2 shows the predicted maximum velocity u and max predicted maximum shear rate γ of Fe-19Cr-21Ni under various The general model provides coefﬁcients that are used to predict max heating control voltages U and temperatures T with both of ﬂow at a given heater setting for a given density, viscosity, and ctr laminar and turbulence models, where the dots represent electrical conductivity. In practice, an experimentalist would know the results from the general model extrapolated from Eq. (2) these thermophysical properties for a particular sample material and Table 2, and the curves represent the correlated predicted as a function of temperature, and thus ﬂow can be predicted values as deﬁned in Eq. (6) and Table 4, which are further ﬁtted to given the heater setting and temperature. Then the predictions b H ^ _ obtain expressions of u and γ as function of U and T, max can be used either as a forecasting tool before a test is run or as a max ctr based on the extrapolated values from the general model. characterization tool based on the observed pyrometer tempera- tures after a test is run. The approach is to select a temperature at H i j u ^ or b _ γ ¼ p U T max ij max ctr (6) a given heater setting, evaluate the thermophysical properties, i;j and generate a plot of the ﬂow velocity and shear rate over the available experiment control-space. Based on the Reynolds number calculated using Eq. (1) For an application of the general model, the ternary steel alloy correlated to the predicted maximum velocity, the ﬂow conditions Fe-19Cr-21Ni (atomic %) was selected to represent the family of are determined to be either laminar, transitional, or turbulent. industrially-cast austenitic alloys for phase selection experiments Figure 3a shows the Reynolds number over a range of heating in microgravity on-board the ISS. To quantify advection during control voltage and temperature, utilizing both laminar and Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2019) 7 X. Xiao et al. Above the curve the ﬂow condition is turbulent and below the curve is laminar. This provides a criterion for determination and selection of ﬂow regimes for planning of experimental conditions. In conclusion, the velocity and shear rate inside electromagne- tically levitated droplet in microgravity with the ISS-EML SUPOS coil is numerically predicted and represented using a previously- validated MHD model. For a levitated molten droplet of arbitrary material properties, the ﬂow is represented as function of heating control voltage, density, viscosity, electrical conductivity, and droplet dimensions, for convenient reference over a wide range of possible metallic materials. As an example of how these results may be applied, the ternary steel alloy Fe-19Cr-21Ni system was selected such that the key material properties all become a function of temperature only. The maximum ﬂow velocity is then represented as functions of heating control voltage and tempera- ture; the critical combination of heating voltage and temperature is provided to predict the ﬂow conditions determining the laminar or turbulent condition of the internal advective ﬂow. METHODS ISS-EML SUPOS coil speciﬁcation For the experiment conducted in microgravity onboard the ISS, the sample of 5.0–7.0 mm in diameter was positioned and heated using ISS-EML SUPOS coil in vacuum or in 350 mbar inert helium or argon gas. The ISS- EML SUPOS coil is a single-coil/dual-current type with upper and lower coils wound in one piece such that a single system is used for both heating and positioning. The alternating current through the coil runs at a frequency of 150 kHz for the positioner and generates a quadrupole electromagnetic force ﬁeld to locate the sample near the center of the coil set. The heating current runs at 350 kHz and generates a dipole electromagnetic ﬁeld that controls the sample temperature through a balance between the resistive heating due to the eddy currents and heat Fig. 1 Maximum Velocity as a function of Heating Control Voltage, loss to the environment due to conduction and radiation. The coil currents Density, Viscosity, and Electrical Conductivity (each ﬁgure includes H P and the control voltage has the following linear relations, where I and I 0 0 H P six groups of curves where U is valued at 0.01, 0.20, 0.50, 1.0, 3.0, ctr are the heating and positioning current, U and U are the heating and ctr ctr and 6.0 V, and each group with the same U contains 11 curves positioning control voltage of the facility. ctr −3 where ρ ranges from 5000 to 10,000 kg m for step size of 500). H H 5 −1 I ¼ 19:09 þ 19:00 U a Maximum velocity at σ = 6.0 × 10 Sm , b maximum velocity at 0 ctr e,l (7) P P μ = 0.010 Pa s I ¼ 27:21 þ 27:21 U 0 ctr Table 3. Baseline material properties for Fe-19Cr-21Ni (at.%) MHD modeling techniques MHD of the EML droplet consists interaction between electromagnetic Properties Values (T = 1715 K) ﬁeld through the conductive molten liquid and the internal ﬂow induced from the electromagnetic forces. The electromagnetic forces in the −3 2 Density (kg m ) ρ =−0.71∙T + 8209 molten alloy droplet induced from the EML coil could be calculated 5 18 Viscosity (Pa s) μ = exp(11,980/T − 11.54) through solving a reduced form of quasi-stationary Maxwell’s equations, −1 5 1,31 which is deﬁned in Eq. (8), Electrical conductivity (S m ) σ = 6.63 × 10 + 380(T − T ) e,l m ∇ B ¼ 0 ∂B ∇ ´ E ¼ (8) ∂t ∇ ´ H ¼ J turbulent models. On the ﬁgure, an upper temperature limit is shown representing the heater setting to achieve an isothermal where J is the induced current, H is the magnetic ﬁeld, B is the magnetic hold. This limit is critical for planning of conditions to conduct ﬂux density, and E is the electric ﬁeld. The electromagnetic force which is thermophysical property measurement at a desired temperature also known as Lorentz force is written as, and for identifying the heating control limit for undercooling F ¼ J ´ B (9) experiments. The method of mutual inductances is used to numerically solve The laminar ﬂow starts to become unsteady at Re = 500 and reduced Maxwell’s equations and calculate the electromagnetic force, becomes turbulent above Re = 600. For the accessible range of utilizing a subroutine developed separately. Because the magnetic conditions, the turbulent ﬂow is transitional and not fully- Reynolds number is so small, the coupling between electromagnetism developed nor isotropic, in part due to the constraints on eddy and ﬂow is one-way: the magnetic ﬁeld drives the ﬂow, but is not size imposed by the ﬁnite size of the droplet. It is appropriate to signiﬁcantly perturbed by the ﬂow. use the results from laminar model to calculate the Reynolds The internal ﬂow could be assumed as incompressible and viscous, number that determines the ﬂow conditions. A critical combina- which is governed by the Navier–Stokes equations, tion of the heating control voltage and temperature can be ∇ u ¼ 0 derived such that the correlated Reynolds number is larger than (10) ∂u 1 2 þ u ∇u ¼ ∇p þ μ∇ u þ F ∂t ρ 600 in the range above the critical values. In Fig. 3b, the critical heating control voltage can be seen to vary with the temperature. where u is the velocity vector, p is the pressure, μ and ρ is the viscosity and npj Microgravity (2019) 7 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA X. Xiao et al. Fig. 2 Fe-19Cr-21Ni Maximum Velocity and Maximum Shear Rate as a function of Heating Control Voltage and Temperature (dots represent the results from the general model where U is valued at 0.01, 0.20, 0.40, 0.70, 1.00, 1.20, 1.50, 2.90, 4.40, and 5.70 V, and T ranges from 1515 to ctr 1915 K for step size of 100). a Laminar model: maximum velocity, b Laminar model: maximum shear rate, c Turbulent model: maximum velocity, d Turbulent model: maximum shear rate Table 4. Polynomial coefﬁcients of maximum velocity and maximum shear rate for ISS-EML Levitated Fe-19Cr-21Ni Droplet Laminar model Turbulent model −1 −1 −1 −1 Velocity (m s ) Shear rate (s ) Velocity (m s ) Shear rate (s ) −2 2 −2 1 p −8.675 × 10 −1.376 × 10 −2.943 × 10 −7.501 × 10 −5 −1 −5 −2 p 6.468 × 10 1.025 × 10 2.819 × 10 6.190 × 10 −1 2 −2 +2 p −4.051 × 10 −6.804 × 10 −7.681 × 10 −1.761 × 10 −4 −1 −5 −1 p 4.286 × 10 7.247 × 10 9.342 × 10 1.967 × 10 −8 −4 −8 −5 p −9.704 × 10 −1.686 × 10 −1.965 × 10 −4.102 × 10 −2 +1 −3 p 2.246 × 10 3.418 × 10 1.820 × 10 3.519 −6 −2 −7 −3 p −9.066 × 10 −1.376 × 10 1.730 × 10 1.351 × 10 −4 −1 −4 −1 p −6.403 × 10 −9.437 × 10 −1.308 × 10 −3.905 × 10 R-squared 0.9998 0.9999 0.9999 0.9999 density, and F is the momentum source which corresponds to the where p is the averaged pressure, and u0u0 is the Reynolds stress term electromagnetic force per unit volume for the EML. describing the additional stresses generated from turbulent ﬂuctuations. The boundary conditions are assumed to be a slip wall, where there is no Two additional equations, the turbulent kinetic energy equation and shear stress on the free surface, and no ﬂux across the surface, energy dissipation equation, are included in the k-ε turbulence model, which represent the dissipation rate of the turbulent kinetic energy, τ i j ¼ 0 r¼1 (11) u j ¼ 0 r¼1 ∂k u 2 þ u ∇k ¼ u þ ∇ k þ P ε ∂t σ where τ is shear stress, i is the tangent unit vector, and u is the radial (14) t r μ u ∂ε t 2 ε ε component of u. þ u ∇ε ¼ þ ∇ ε þ C P C 1ε k 2ε ∂t ρ σ k k For simulation of turbulent ﬂow, the RNG k–ε turbulence model is adopted. Adding extra terms, the vector of turbulent velocity u consists of with additional boundary conditions, the time-averaged velocity u and the ﬂuctuation u′, u ¼ u þ u ∂k ¼ 0 ∂r T (12) r¼1 (15) u ¼ lim u dt T 0 ∂ε T!1 ¼ 0 ∂r r¼1 Eq. (10) then becomes the time-averaged Navier–Stokes equations, 1 0 0 The turbulent kinetic energy is deﬁned as k ¼ u u , P ¼ τ ∂u =∂x is k i:j i j 2 i i ∂u 1 0 0 the kinetic energy production, u = C (k /ε) is the kinematic eddy viscosity, t μ þ u ∇u ¼ ∇p þ μ∇ u þ F ∇ u u (13) 0 0 ∂u ∂u i i ∂t ρ and ε ¼ is the dissipation rate. ρ ∂x ∂x j j Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2019) 7 X. Xiao et al. In Eq. (14), the RNG k–ε model uses the following coefﬁcients, In the MHD model, the sample is assumed to be at the center of the coil with limited translational oscillations, is of spherical shape with limited C ¼ 1:42 1ε surface deformation and at thermal pseudo-steady state with constant and C ¼ 1:68 homogeneous thermophysical properties. In practice, the variance due to 2ε oscillation and surface deformation may introduce error <8%, and that of C ¼ 0:085 (16) thermal equilibrium is negligible. The steady-state solver of the prescribed σ ¼ 0:72 MHD model is based on a ﬁnite volume method through the commercial σ ¼ 0:72 ε package ANSYS Fluent. The model includes a mesh consisting of an optimized number of 550 cells and 591 nodes as shown in Fig. 4a, superimposed with the electromagnetic force as the momentum source term in the shape of arrows. For the heater-dominated MHD simulation results, the ﬂow typically consists of two toroidal circulation loops near the stagnation line at the equator of the droplet, turning inward the sphere where the electro- magnetic force archives a maximum around the equator. The predicted ﬂow patterns are displayed as a vector plot of ﬂow velocity and contour of shear rate magnitude as shown in Fig. 4b on right and left side respectively. For the ﬂow with relatively low Reynolds number below 500, the laminar model is appropriate and accurate; for Reynolds numbers much larger than 600 the ﬂow is turbulent and the results from the RNG k– ε turbulence model are more appropriate. Note that the analysis may not be appropriate for application to experimental conditions during rapid heating—for example during melting the sample experiences surface oscillations and inhomogeneous temperatures across sample; during short pulse applications that used to induce surface oscillations for property evaluations, even if deformations are small, the ﬂow is transient and not quasistatic as required by the present model. Future work will extend the model to allow predictions of the shape of deformed samples under either transient or quasi-static conditions. Reporting summary Further information on experimental design is available in the Nature Research Reporting Summary linked to this article. DATA AVAILABILITY The data that support the ﬁndings of this study are available from the corresponding author upon request. ACKNOWLEDGEMENTS Fig. 3 Fe-19Cr-21Ni Reynolds number and ﬂow conditions. a This work was funded by NASA under grant NNX16AB59G at Tufts University and Reynolds Number as function of heating control voltage and NNX16AB40G at University of Massachusetts. The authors wish to thank the staff from temperature, b critical heating control voltage and temperature the Microgravity User Support Center (MUSC) at the German Space Agency (DLR) for Fig. 4 MHD Model for Electromagnetically Levitated Droplet. a Mesh grid with interpolated electromagnetic force density superimposed. b Contour of shear rate magnitude (left side); Vectors of ﬂow velocity (right side) npj Microgravity (2019) 7 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA X. 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Magnetohydrodynamic modeling and experimental validation of for this investigation as well as support of other research projects funded by the convection inside electromagnetically-levitated Co-Cu droplets. Metall. Mater. agency; X.X. is a postdoctoral scholar employed under this grant. R.W.H. is the Trans. B 45, 1018–1023 (2014). recipient of a NASA award funding collaborations for this investigation as well as 22. Hyers, R. W., Trapaga, G. & Abedian, B. Laminar-turbulent transition in an elec- support of other research projects funded by the agency; J.L. was previously partially tromagnetically levitated droplet. Metall. Mater. Trans. B 34,29–36 (2003). funded under both grants. The authors declare no competing interests. 23. Lee, J., Xiao, X., Matson, D. M. & Hyers, R. W. 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Zong, J.-H., Szekely, J. & Schwartz, E. An improved computational technique © The Author(s) 2019 calculating electromagnetic forces and power absorptions generated in spherical Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2019) 7
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