An Improved Model for the Flow in an Electromagnetically Stirred Melt during Solidification

Abstract

A mathematical model for simulating the electromagnetic field and the evolution of the temperature and velocity fields during solidification of a molten metal subjected to a time-varying magnetic field is described. The model is based on the dual suspended particle and fixed particle region representation of the mushy zone. The key feature of the model is that it accounts for turbulent interactions with the solidified crystallites in the suspended particle region. An expression is presented for describing the turbulent damping force in terms of the turbulent kinetic energy, solid fraction, and final grain size. Calculations were performed for solidification of an electromagnetically stirred melt in a bottom chill mold. It was found that the damping force plays an important role in attenuating the intensity of both the flow and turbulent fields at the beginning of solidification, and strongly depends on the final grain size. It was also found that turbulence drops significantly near the solidification front, and the flow becomes laminarized for solid fraction around 0.3.

Appendix A. The Damping Force in the Suspended Particle Region

In the suspended particle region, the fluctuating velocity of the solid particles is zero, and the instantaneous velocity of the two-phase mixture is

$$ {\mathbf{u}} = {\bar{\mathbf{u}}} + {\mathbf{u}}^{\prime } = f_{\text{L}} \left( {{\bar{\mathbf{u}}}_{\text{L}} + {\mathbf{u}}_{\text{L}}^{\prime } } \right) + f_{\text{s}} {\mathbf{u}}_{\text{s}} . $$

(A1)

The instantaneous Navier–Stokes equation in this region can then be written as

$$ \frac{{\partial \left( {\rho \left( {f_{\text{s}} {\bar{\mathbf{u}}}_{\text{s}} + f_{\text{L}} \left( {{\bar{\mathbf{u}}}_{\text{L}} + {\mathbf{u}}_{\text{L}}^{\prime } } \right)} \right)} \right)}}{\partial t} + \nabla \cdot \left( {\rho \left( {f_{\text{s}} {\bar{\mathbf{u}}}_{\text{s}} + f_{\text{L}} \left( {{\bar{\mathbf{u}}}_{\text{L}} + {\mathbf{u}}_{\text{L}}^{\prime } } \right)} \right)\left( {f_{\text{s}} {\bar{\mathbf{u}}}_{\text{s}} + f_{\text{L}} \left( {{\bar{\mathbf{u}}}_{\text{L}} + {\mathbf{u}}_{\text{L}}^{\prime } } \right)} \right)} \right) = - \nabla \left( {\bar{p} + p^{\prime } } \right) + \mu_{\text{l}} \nabla^{2} \left( {f_{\text{s}} {\bar{\mathbf{u}}}_{\text{s}} + f_{\text{L}} \left( {{\bar{\mathbf{u}}}_{\text{L}} + {\mathbf{u}}_{\text{L}}^{\prime } } \right)} \right) + {\mathbf{F}}_{\text{em}} . $$

(A2)

Taking the time average of [A2] yields

$$ \frac{{\partial \left( {\rho {\bar{\mathbf{u}}}} \right)}}{\partial t}{ + }\nabla \cdot \left( {\rho {\mathbf{\bar{u}\bar{u}}}} \right) + \nabla \cdot \left( {f_{\text{L}}^{2} \bar{\tau }_{\text{t}} } \right) = - \nabla \bar{p} + \mu_{\text{l}} \nabla^{2} {\bar{\mathbf{u}}} + {\mathbf{F}}_{\text{em}} , $$

(A3)

where τ _t is the Reynolds stress tensor. From Eqs. [11] and [A3], the damping force is given by

$$ {\mathbf{F}}_{\text{D}} = \left( {1 - f_{\text{L}}^{2} } \right)\nabla \cdot \bar{\tau }_{\text{t}} . $$

(A4)