It has previously been shown that by increasing the Reynolds number across a channel by spatially varying the viscosity does not cause an immediate change in the size of turbulent structures and a delay is in fact observed in both wall shear and friction Reynolds number (Coppo Leite, V, & Merzari, E., Proceedings of the ASME 2020 FEDSM, p. V003T05A019). Furthermore, it is also shown that depending on the length in which the flow condition changes, turbulence bursts are observed in the turbulence field. For the present work we propose a new version of the standard Reynolds Averaged Navier Stokes (RANS) k-τ model that includes some modifications in the production term in order to account for these effects. The new proposed model may be useful for many engineering applications as turbulent flows featuring temperature gradients and high heat transfer rates are often seen in heat exchangers, combustion chambers and nuclear reactors. In these applications, thermal and viscous properties of the working fluid are important design parameters that depend on temperature; hence it is likely to observe strong gradients on these scalars' fields. To accomplish our goal, the modifications for the k-τ model are implemented and tested for a channel flow with spatial varying viscosity in the streamwise direction. The numerical simulations are performed using Nek5000, a spectralelement code developed at Argonne National Laboratory (ANL). Finally, the results considering a turbulence channel using the proposed model are compared against data obtained using Direct Numerical Simulations from the earlier work.