  We adopt a set of second-order differential equations ($k-\omega$ model) to
handle core convective overshooting in massive stars, simulate the evolution of
WNL stars with different metallicities and initial masses, both rotating and
non-rotating models, and compare the results with the classical overshooting
model. The results indicate that under the same initial conditions, the
$k-\omega$ model generally produces larger convective cores and wider
overshooting regions, thereby increasing the mass ranges and extending the
lifetimes of WNL stars, as well as the likelihood of forming WNL stars. The
masses and lifetimes of WNL stars both increase with higher metallicities and
initial masses. Under higher-metallicity conditions, the two overshooting
schemes significantly differ in their impacts on lifetimes of the WNL stars,
but insignificant in the mass ranges of the WNL stars. Rotation may drive the
formation of WNL stars in low-mass, metal-poor counterparts, with this effect
being more pronounced in the OV model. The surface nitrogen of metal-rich WNL
stars formed during the MS phase is likely primarily from the CN-cycle, while
it may come from both the CN- and NO-cycles for relatively metal-poor
counterparts. Our model can effectively explain the distribution of WNL stars
in the Milky Way, but appears to have inadequacies in explaining the WNL stars
in the LMC.
