A universal scaling for low-order structure functions in the log-law region of smooth- and rough-wall boundary layers

We consider the log-law layer of both smooth- and rough-wall boundary layers at large Reynolds number. A scaling theory is proposed for low-order structure functions (say $ #1{} #1{ {#1}} ≤ = ≤ = ≥ = ≥ = {Pr} {Fr} {Re}n ≤ 6$ ) in the range of scales $η r δ $ , where $η $ is the Kolmogorov length and $δ $ is the boundary layer thickness. This theory rests on the hypothesis that the turbulence in this intermediate range of scales depends only on the scale $r$ , the local dissipation rate and the shear velocity. Crucially, the structure of the turbulence is assumed to be independent of the distance from the wall, $y$ , except to the extent that $y$ sets the value of the local dissipation rate. A detailed comparison is made between the predictions of the theory and data taken from both smooth- and rough-wall boundary layers. The data support the hypothesis that it is the dissipation rate, and not y , that controls the structure of the turbulence for this range of eddy sizes. Our findings provide the first unified scaling theory for both smooth- and rough-wall turbulence.