In Taylor's theory, the longitudinal dispersion in turbulent pipe flows approaches, on long time scales, a diffusive behavior with a constant diffusivity K_{L}, which depends 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 the 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.