Applications of the $k$-$\omega$ model in stellar evolutionary models

The $k$-$\omega$ model for turbulence was first proposed by Kolmogorov (1942). A new $k$-$\omega$ model for stellar convection was developed by Li (2012), which could reasonably describe turbulent convection not only in the convectively unstable zone but also in the overshooting regions. We have revised the $k$-$\omega$ model by improving several model assumptions (including the macro-length of turbulence, convective heat flux, and turbulent mixing diffusivity, etc.), making it applicable for not only convective envelopes but also convective cores. Eight parameters are introduced in the revised $k$-$\omega$ model. It should be noted that the Reynolds stress (turbulent pressure) is neglected in the equation of hydrostatic support. We have applied it into solar models and 5$M_{\odot}$ stellar models, in order to calibrate the eight model parameters, as well as to investigate the effects of the convective overshooting on the sun and intermediate mass stellar models.