The detailed implementation for solving a passive scalar transport equation without diffusion term, i.e.
=S_{y}, \])
with

be the scalar concentration.
(1) If scalar velocity

equals to the "carrier fluid'' (the fluid solved in the Navier-Stokes equation), i.e.,

, how
to implement the scalar source term

in source files?
(2)If

, how to consider this velocity difference between passive scalar and "carrier fluid''?
Thanks!