A refined interpretation of the logarithmic structure function law in wall layer turbulence

It has been shown recently that the real-space equivalent of the k−1 law for near-wall turbulence is a logarithmic law for the second-order longitudinal structure function, ⟨(Δu)2⟩(r)=u*2(A+Bln(r∕y)). Here y is the distance from the wall, u* is the shear velocity, A and B are coefficients of order unity and r is measured in the streamwise direction. In this paper we provide theoretical arguments to suggest that, in the limit of large Reynolds number, B is a universal constant while A is of the form A=A′−Bln(P∕ϵ), where A′ is a universal constant and P and ϵ are the rates of production and dissipation of energy, respectively. Hence A is a weak, universal, function of y. Two independent sets of data are examined and it is shown that, to within experimental error, our predictions are consistent with the data.