The effects of Reynolds and Prandtl numbers on the turbulent transport of passive scalars in rough-wall flows*

Numerical simulations were used to investigate Reynolds- and Prandtl-number effects on scalar transport in channels with rough walls. The Reynolds number based on the friction velocity and the channel half-width ranged from Re_τ=1,050 to 7,500, with the corresponding equivalent sand-grain roughness height k_s⁺ between 70 and 485. Prandtl numbers between 0.2 and 2.0 were considered.

The fully-rough behavior of the scalar log-layer intercept and roughness function are examined as function of the Reynolds and Prandtl numbers. The scalar roughness function may be insensitive to the roughness geometry for Prandtl numbers below unity. For Prandtl above unity there exists a range of Reynolds numbers for which the effect of the roughness on momentum are smaller compared to scalar, and a range where the opposite is true.

The scalar-flux balance is similar to the momentum one. At high enough Reynolds, the normalized Form-Induced (FI) scalar flux approaches a limiting curve that is insensitive to the Prandtl number. This curve is consistently smaller than the normalized total FI drag and corresponds with the solid fraction, suggesting it may only depend on the roughness geometry.

The magnitude of the normalized scalar variance increased with Reynolds for Pr<1 and decreased for Pr≥1, resulting in the lowest- and highest-Prandtl number curves approaching each other as Reynolds increased, implying that the effect of the Prandtl number diminishes at higher Reynolds numbers.

It is possible that the various Pr curves will eventually collapse on each other, resulting in the scalar variance becoming insensitive to Pr altogether.

The budget of the scalar variance shows that the FI production and diffusion terms are relatively insensitive to Reynolds, while the shear production and dissipation are strongly reduced as Reynolds increases. At high Reynolds the FI production dominates the scalar variance budget in the RSL. The effect of Pr on the scalar variance budget remained significant at the highest Reynolds number examined.

Townsend's similarity hypothesis holds on the scalar fluxes, variance and the budgets.