The transport of a passive scalar with molecular Prandtl number Pr near unity was studied using direct numerical simulations (DNS) of a fully developed channel with smooth and rough walls. The effects of roughness on the turbulent Prandtl number PrT, turbulent timescale ratio and scalar-dissipation budget were investigated. For the rough-wall case, the turbulent Prandtl number was almost constant (PrT approximate to 0.9) from the roughness centroid up to the roughness crest, matching the common assumption used in modeling. PrT changes rapidly as the base of the roughness is approached, similar to the near-wall behavior in the viscous sublayer of a smooth-wall case. Away from the wall, Townsend's similarity hypothesis holds; the curves collapse when scaled in wall units. An effective turbulent Prandtl number PrT,eff, which also includes the dispersive terms was examined. PrT,eff and PrT have an overall similar behavior, suggesting that they can be interchangeable and that the PrT approximate to 0.9 approximation may be accurate enough when modeling rough-wall flows with constant scalar wall flux. This implies that the ad hoc adjustment of model coefficients is not necessary, and that the scalar dispersive term do not need to be modeled separately. The near-wall behavior of the ratio between turbulent scalar and momentum timescales, R, depends strongly on the molecular Pr for a smooth-wall case; roughness significantly reduces this dependence, tends to lower R, and results in a more uniform behavior. Only below the centroid R rises rapidly as the base of the roughness is approached; as Pr decreases, the location at which this rise occurs approaches the base of the roughness. Townsend's similarity hypothesis also applies to R away from the roughness crest. The budget of the scalar dissipation epsilon theta has a complex near-wall behavior in a smooth-wall case. In a rough-wall case this behavior is significantly simpler, approximating an equilibrium in which several production-of-dissipation terms match the dissipation-of-dissipation. The leading turbulent production-of-dissipation term is only due to the gradients of velocity and scalar fluctuations, which are only indirectly affected by the roughness. Compared with the smooth-wall case, this term is enhanced due to improved alignment between the fluctuating scalar gradient and the principal compressive direction of the strain-rate tensor. The roughness also enhanced the gradient-production of scalar dissipation, which is directly affected by the roughness shape, due to the diffusive nature of the mean scalar, which results in large gradients of the time-averaged scalar field.
