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

Abstract

AbstractWe 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$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n \leq 6$) in the range of scales$\eta \ll r \ll \delta $, where$\eta $is the Kolmogorov length and$\delta $is the boundary layer thickness. This theory rests on the hypothesis that the turbulence in this intermediate range of scales depends only on the scale$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.