Transient Effect of Fluid Flow on Dendrite Growth Direction in Binary FeC Alloys Using Phase Field in OpenFOAM
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Abstract
An opensource fluid flow and phasefield coupled solidification model has been developed in OpenFOAM to investigate the transient nature of the interface growth direction at different degrees of undercooling for an Fe0.15 wt.% C binary alloy under isothermal conditions. Though there are works on melt convection effects in binary alloys, none reported the transient nature of the dendrite growth direction since thermodynamic driving force decreases with time at a particular undercooling. Developing a theoretical relation will be helpful in understanding the competition between the crystallographic growth direction and solute transport. Flow decoupled simulation results have a good quantitative agreement with the literature. The bending angle formulations on the effects of flow velocity and growth speed were separated. At the end, improved theoretical formulations for estimation of the bending angle based on the anisotropy in interface energy were put forward compared with only few available empirical correlations.
Introduction
Steel is one of the most widely used materials in today’s world, manufactured through the continuous casting route. The microstructure that results is important for upstream processes and final product quality. The practice of continuously cast nearnetshape profiles has gained in popularity because of energy savings, but these limit the subsequent inline thermomechanical processing possibilities; therefore, the need to control the cast microstructure becomes vital. The solidified microstructure depends on the alloy chemistry and those conditions in the mould that control the thermal profile and fluid flow. To understand the links and identify the operating conditions that control the evolution of the solidification structure, computational models offer ways to simulate the process and identify sensitivities to distinct conditions.
Phasefield1^{–}3 and cellular automaton4 are currently used to simulate the dependence of the dendritic solidification microstructure on process conditions. Based on the continuum thermodynamic principles, the phasefield method has received much consideration for simulating the complicated interface structure because of its implicit nature of tracking the solid/liquid front via the diffuse interface approach. The real essence of the method stems its origin from the wellknown Cahn–Hilliard equation.5 Wang et al.6 deduced phasefield models that had a strong resemblance to that of Kobayashi7. Starting from a pure metal, Kim et al.1^{,}2^{,}8 and Boettinger et al.9 deduced the phasefield model for alloy solidification in a thermodynamically consistent way. Kim in his model1,2 of binary alloy solidification assumed the interface to be a mixture of solid and liquid phases with different compositions but with same chemical potential. The model1 under the vanishing interface kinetics coefficient condition maintains local equilibrium at the interface.
The thermodynamics of excess solute rejection during the alloy solidification governs the nonuniformity of the chemical composition in the cast product commonly known as segregation. Segregation near a growing front will alter the solidification kinetics and have serious consequences on the final mechanical properties.10 Bulk fluid flow during alloy solidification washes the rejected solute away (thereby changing the local composition) and will thus define the segregation and consequently the dendrite growth direction.11 Ultimately, this influences the evolution of the solidification structure. The solute layer at the dendrite tip, if driven by bulk flow over long distances, may give rise to segregation at the macroscale level.
The phenomenon of dendrite deflection is defined as the biased growth of the solid phase towards the upstream flow direction. Investigations of the modelling of the melt convection effect on the solidification microstructure11^{–}17 have been focussed on the effect of inlet flow magnitude on increased dendrite bending and branching of dendrites in the upstream direction, which is primarily due to the asymmetric solute profile ahead of the deflected dendrite. Not much information is available on the effect of undercooling on dendrite deflection. Most of the research on melt flow has been focussed on nonferrous systems such as NiCu,14^{,}15 AlSi18 and AlCu12^{,}13 while a few exist for Febased alloys such as FeC19 and FeMn,20 which can improve the understanding of the microstructure formation under industrial conditions. To the best of our knowledge, the quantitative dependence of the growth direction on the magnitude of the flow velocity, solute content and solidification speed is limited to two previously reported studies11^{,}16—one showing the empirical dependency of dendrite deflection and in the other study anisotropy in interface energy was not incorporated. Okano et al.,21 also based on limited experiments in a continuously cast steel slab, put forward quantitative relations of the bending angle. For an Fe0.15 wt.% C alloy at relatively high undercooling conditions, using the phasefield method, Natsume et al.19 investigated the singledendrite deflection at three different growth speeds only by considering anisotropy in interface energy. During the progress of casting, levels of undercooling at various positions of the solidification front within the melt change with time depending on the process conditions. This may give rise to different deflection behaviours in combination with the incoming fluid flow. Developing theoretical correlations will help in understanding this behaviour. For pure materials, a few reported22 3D simulations showed that the 3D flow effect is more pronounced than the 2D but did not incorporate the growth direction effect.
In the commercially available CFD software, it is not easy to modify the available numerical models. OpenFOAM23 is an open source computational software, capable of handling a broad spectrum of partial differential equations along with the advantage of having a set of precompiled libraries that can be customised by the user as per the requirements. Despite the wealth of publications24^{,}25 on CFD, a limited amount of literature can be found wherein coupling of OpenFOAM to phasefield methods has been carried out.26^{,}27 Thereby, developing a phasefield method based on an opensource numerical model will not only be useful in understanding the solute advancement behaviour due to fluid convection in FeC binary alloys but also can be helpful for the material research community in the development of new alloys. Moreover, being open source, the fluid flowcoupled model can be extended under different engineering casting conditions to reduce macrosegregation.
The present work involves the development of an opensource phasefield methodbased model in OpenFOAM to understand the transient behaviour of the dendrite growth direction under the influence of maximum flow velocity at the dendrite tip in an FeC alloy at different levels of undercooling. Previous studies have focussed only on the effects of inlet flow velocities. The proposed fit functions are an extension of the Takahashi relation16 and thus can be useful in predicting the solute profile ahead of the growing front under industrial conditions since experimentally it is a challenge to simultaneously measure the flow velocity and growth speed ahead of the interface.
Mathematical Model
Physical Fundamentals
It is to be noted that \( \sigma_{0} \) is the orientationally averaged material interface energy. Thus, since \( \varepsilon \) is related to the material interface energy (Eq. 3), the assumed dependency of \( \varepsilon \) on \( \theta \) also relates the material interface energy to \( \theta \). The material properties have been assumed to be independent of composition.
Physical properties of the FeC alloy
Parameter  Value  Ref. 

Diffusivity in the solid phase (D_{S}), m^{2}/s  6 × 10^{−9}  
Diffusivity in the solid phase (D_{L}), m^{2}/s  2 × 10^{−8}  
Interface energy (σ_{0}), J/m^{2}  0.204  
Kinematic viscosity (ν), m^{2}/s  6.79 × 10^{−7}  
Melting point of pure iron (T_{M}), K  1810  
Molar volume \( (v_{\text{m}} ) \), m^{3}/mol  7.7 × 10^{−6}  
Universal gas constant (R), J/(mol K)  8.314  
Equilibrium partitioning coefficient \( (k_{\text{e}} ) \)  0.178 
Each point within the interface is assumed to be a mixture of solid and liquid phases with different compositions of \( C_{\text{S}} \) and \( C_{\text{L}} \). Local equilibrium has been assumed at the interface.
Numerical Methodology
The code developed to solve the governing equations using the finite volume method in OpenFOAM was written in C++ programming language. The phasefield and solute transport equations were solved using an explicit Euler scheme and the governing equations for fluid flow were solved implicitly with the standard PISO algorithm.23 A random noise of the form \( 16\alpha r\phi^{2} (1  \phi )^{2} \) of 1% amplitude was added to the liquid concentration at the interface to simulate the formation of secondary arms where r is the random number between + 1 and − 1 and \( \alpha \) is the noise amplitude.30 The width of the domain was taken to about 11 times the length of the reported secondary dendrite arm spacing and the domain height was taken to be around 4.5 times the secondary arm spacing. A mesh size of 0.5 µm was used to calculate the dendrite arm spacing. Noflux boundary conditions for both the phasefield and concentration were used. As an initial condition, a uniform thin layer of solid was placed at the bottom of the domain. The effect of cooling rate was implemented in form of the equation \( T(t) = T_{0}  t\Delta T \)3 where \( T_{0} \) is the initial temperature a few degrees below the liquidus temperature of the alloy and \( \Delta T \) is the cooling rate. The initial temperature \( T_{0} \) was constant for all the cases. Growth of secondary dendrite arms was simulated at three different cooling rates, namely—83.33 K/s, 33.33 K/s and 8.33 K/s. Twodimensional simulations were performed for FeC binary alloys under isothermal conditions with the thermodynamic data taken from ThermoCalc software.31 Simulations for solidification coupled with fluid flow were performed under different levels of undercooling. The mesh size was composed of 1000 × 500 cells with an equal distance of 10^{−8} m (= 10 nm). The seed crystal (initial condition) was placed at the centre of the bottom wall of the domain. Liquid melt at a constant inlet velocity along the xaxis enters the left wall (inlet) of the domain and exits at the right wall of the domain with zerogradient velocity boundary condition. The top wall of the domain was assumed to be a slip wall and the bottom wall was assumed to be in contact with stationary mould wall. The uniform value of zero pressure was taken at the right wall and the rest of the walls were taken as zerogradient pressure boundary condition. The boundary conditions for the phasefield and concentration were taken as zero gradient except for the left wall of the domain. For isothermal flowcoupled simulations, the \( k_{\text{e}} \) value was constant up to three significant digits.
Results and Discussion
Model Validation
Fluid Flow Effect
Effect of Undercooling
At a constant maximum flow velocity, it can be seen that with a decrease in the tip growth speed, the bending angle increases.19 The increase in the bending angle is quite steep at lower growth velocities. This is because the fluid passing the dendrite tip has much more time to sweep away the rejected solute from the upstream side of the dendrite tip to the downstream side and thereby contributes to a higher bending angle. However, in this growth speeddominant regime, the solute is not washed away to further distances and instead creates a trail of an asymmetrical solute boundary layer. Thus, this might be the point where diffusion tries to gain importance over bulk convection and may contribute to macrosegregation in the cast product. The authors thus have made an attempt to separate out the two effects—the flow velocity effect and growth speed effect—on the bending angle. This information on the bending angle might be useful for casting operators in finding out the linkage between the dendrite growth direction and casting parameters at various stages of the casting and thereby taking corrective action.
Fit Function Analysis
Conclusion

At a particular level on undercooling, the tip growth speed and flow velocity at the tip changed with time and hence the bending angle. In this way, an array of deflection angle data along with flow velocity at the tip and tip growth speed was obtained for different levels of undercooling.

An attempt to separate out the fluid flow velocity and growth speed effects has been made that can be useful in predicting the interface growth direction under industrial casting conditions by studying at what point of casting and up to what extent both the diffusion and bulk fluid flow interact with the solute layer ahead of the solidification interface.

Anisotropy in interface energybased separate fit functions dependent on flow velocity at the tip and growth speed have been postulated, which are an extension of the empirical correlations proposed by Takahashi.16
In future, they can also be extended to full 3D, which at present is still a formidable challenge.
Notes
Acknowledgements
The present work was carried out as part of a research project fully funded by Tata Steel UK Ltd. The authors gratefully acknowledge the use of the High Performance Computing Cluster available from the Centre for Scientific Computing, University of Warwick.
Conflict of interest
The authors declare that they have no conflict of interest.
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