Scale-by-scale energy budget in a turbulent boundary layer over a rough wall

• The friction velocity is measured using the pressure distribution around a roughness element.
• Rough wall boundary layer appears to evolve in a self-similar manner as UτUτ is constant in x.
• Rough wall flows appear to be more isotropic in comparison to smooth wall flows.
• Scale-by-scale budget analysis revealed the role of large-scale inhomogeneities in the flow.

Hot-wire velocity measurements are carried out in a turbulent boundary layer over a rough wall consisting of transverse circular rods, with a ratio of 8 between the spacing (w) of two consecutive rods and the rod height (k  ). The pressure distribution around the roughness element is used to accurately measure the mean friction velocity (UτUτ) and the error in the origin. It is found that UτUτ remained practically constant in the streamwise direction suggesting that the boundary layer over this surface is evolving in a self-similar manner. This is further corroborated by the similarity observed at all scales of motion, in the region 0.2⩽y/δ⩽0.60.2⩽y/δ⩽0.6, as reflected in the constancy of Reynolds number (RλRλ) based on Taylor’s microscale and the collapse of Kolmogorov normalized velocity spectra at all wavenumbers.A scale-by-scale budget for the second-order structure function 〈(δu)2〉〈(δu)2〉 (δu=u(x+r)-u(x)δu=u(x+r)-u(x), where u is the fluctuating streamwise velocity component and r   is the longitudinal separation) is carried out to investigate the energy distribution amongst different scales in the boundary layer. It is found that while the small scales are controlled by the viscosity, intermediate scales over which the transfer of energy (or 〈(δu)3〉〈(δu)3〉) is important are affected by mechanisms induced by the large-scale inhomogeneities in the flow, such as production, advection and turbulent diffusion. For example, there are non-negligible contributions from the large-scale inhomogeneity to the budget at scales of the order of λλ, the Taylor microscale, in the region of the boundary layer extending from y/δ=0.2y/δ=0.2 to 0.6 (δδ is the boundary layer thickness).