  In Taylor's theory, the longitudinal dispersion in turbulent pipe flows
approaches, on long timescales, a diffusive behavior with a constant
diffusivity $K_L$, that depends \emph{empirically} on the Reynolds number $Re$.
We show that the dependence on $Re$ can be determined from the turbulent energy
spectrum. By using the intimate connection between the friction factor and
longitudinal dispersion in wall-bounded turbulence, we predict different
asymptotic scaling laws of $K_L(Re)$ depending on the different turbulent
cascades in two-dimensional turbulence. We also explore numerically the
$K_L(Re)$ dependence in turbulent channel flows with smooth and rough walls
using a lattice Boltzmann method.
