Hierarchical random additive model for the spanwise and wall-normal velocities in wall-bounded flows at high Reynolds numbers

At high Reynolds numbers, the logarithmic range in wall-bounded flows spans many scales. An important conceptual modeling framework of the logarithmic range is Townsend's attached eddy hypothesis [The Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge, 1976)], where high Reynolds number wall-bounded flows are modeled as assemblies of space-filling, self-similar, and wall-attached eddies. Recently, Yang et al. [Phys. Rev. Fluids 1, 024402 (2016)10.1103/PhysRevFluids.1.024402] reinterpreted this hypothesis and developed the "hierarchical random additive process" model (HRAP), which provides further insights into the scaling implications of the attached eddies. For example, in a recent study [Yang, Phys. Rev. Fluids 2, 064602 (2017)10.1103/PhysRevFluids.2.064602], the HRAP model was used for making scaling predictions of the second-order structure function [ui′(x)-ui′(x′)][uj′(x)-uj′(x′)] in the logarithmic range, where ui's are the velocity fluctuations in the ith Cartesian direction. Here, we provide empirical support for this HRAP model using high-fidelity experimental data of all three components of velocity in a high Reynolds number boundary layer flow. We show that the spanwise velocity fluctuation can be modeled as a random additive process, and that the wall-normal velocity fluctuation is dominated by the closest neighboring wall-attached eddy. By accounting for all the three velocities in all the three Cartesian directions, the HRAP model is formally a well rounded model for the momentum-carrying scales in wall-bounded flows at high Reynolds numbers.