Where, \(U_{i}\) is the velocity component, \(\mu_{t}\) is the turbulent
viscosity, \(G_{k}\) is the generation turbulent kinetic energy due to
mean velocity gradients, \(G_{b}\) is the generation of turbulent
kinetic energy due to buoyancy, \(Y_{m}\) denotes the influence of the
fluctuating dilation in compressible turbulence to total dissipation
rate,\(C_{1\varepsilon}\) , \(C_{2\varepsilon}\) and\(C_{3\varepsilon}\) are the constants, \(\sigma_{k}\) and\(\sigma_{\varepsilon}\) are the Prandtl numbers of k and ɛ respectively\(S_{k}\) and \(S_{\varepsilon}\) are user defined source terms for k
and ɛ respectively.
Solidification modeling
An enthalpy-porosity technique is used to track the liquid-solid
transition with a parameter called liquid fraction, which it indicates
the fraction of each cell of the domain which is in liquid form. The
liquid fraction for is equal to porosity of a cell, which is determined
using following conditions:
\(\beta=0\) if T <\(T_{\text{solidus}}\) (solid)
\(\beta=1\) if \(T\) > \(T_{\text{liquidus}}\) (liquid)
\(\beta=\frac{T-T_{\text{solidus}}}{T_{\text{liquidus}}-T_{\text{solidus}}}\)if \(T_{\text{solidus}}\) < \(T\) <\(T_{\text{liquidus}}\) (mushy zone)
The mushy zone is the region where it is neither completely liquid nor
completely solid. The enthalpy justifies the solidification with the
release or absorption of latent heat (\(\Delta H\)) and sensible heat
(\(h\)).