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 (\(h\)).