Structure function tensor scaling in the logarithmic region derived from the attached eddy model of wall-bounded turbulent flows

We investigate the scaling of the velocity structure function tensor Dij(r,z) in high Reynolds number wall-bounded turbulent flows, within the framework provided by the Townsend attached eddy hypothesis. Here i,j=1,2,3 denote velocity components in the three Cartesian directions, and r is a general spatial displacement vector. We consider spatial homogeneous conditions in wall-parallel planes and dependence on wall-normal distance is denoted by z. At small scales (r=|r|z) where turbulence approaches local isotropy, Dij(r,z) can be fully characterized as a function of r and the height-dependent dissipation rate ϵ(z), using the classical Kolmogorov scalings. At larger distances in the logarithmic range, existing previous studies have focused mostly on the scaling of Dij for r in the streamwise direction and for the streamwise velocity component (i=j=1) only. No complete description is available for Dij(r,z) for all i,j, and r directions. In this paper we show that the hierarchical random additive process model for turbulent fluctuations in the logarithmic range (a model based on the Townsend's attached eddy hypothesis) may be used to make new predictions on the scaling of Dij(r,z) for all velocity components and in all two-point displacement directions. Some of the generalized scaling relations of Dij(r,z) in the logarithmic region are then compared to available data. Nevertheless, a number of predictions cannot yet be tested in detail, due to a lack of simultaneous two-point measurements with arbitrary cross-plane displacements, calling for further experiments to be conducted at high Reynolds numbers.