Re-understanding the law-of-the-wall for wall-bounded turbulence based on in-depth investigation of DNS data

Based on direct numerical simulation (DNS) data of the straight ducts, namely square and rectangular annular ducts, detailed analyses were conducted for the mean streamwise velocity, relevant velocity scales, and turbulence statistics. It is concluded that turbulent boundary layers (TBL) should be broadly classified into three types (Type-A, -B, and -C) in terms of their distribution patterns of the time-averaged local wall-shear stress (\(\tau _\mathrm{w} )\) or the mean local frictional velocity (\(u_\tau )\). With reference to the Type-A TBL analysis by von Karman in developing the law-of-the-wall using the time-averaged local frictional velocity (\(u_\tau )\) as scale, the current study extended the approach to the Type-B TBL and obtained the analytical expressions for streamwise velocity in the inner-layer using ensemble-averaged frictional velocity (\(\bar{{u}}_\tau )\) as scale. These analytical formulae were formed by introducing the general damping and enhancing functions. Further, the research applied a near-wall DNS-guided integration to the governing equations of Type-B TBL and quantitatively proved the correctness and accuracy of the inner-layer analytical expressions for this type.