Hierarchical random additive process and logarithmic scaling of generalized high order, two-point correlations in turbulent boundary layer flow

Townsend [Townsend, The Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge, UK, 1976)] hypothesized that the logarithmic region in high-Reynolds-number wall-bounded flows consists of space-filling, self-similar attached eddies. Invoking this hypothesis, we express streamwise velocity fluctuations in the inertial layer in high-Reynolds-number wall-bounded flows as a hierarchical random additive process (HRAP): u(z)(+) = Sigma(Nz)(i=1) a(i). Here u is the streamwise velocity fluctuation, + indicates normalization in wall units, z is the wall normal distance, and a(i)'s are independently, identically distributed random additives, each of which is associated with an attached eddy in the wall-attached hierarchy. The number of random additives is N-z similar to ln(delta/z) where delta is the boundary layer thickness and ln is natural log. Due to its simplified structure, such a process leads to predictions of the scaling behaviors for various turbulence statistics in the logarithmic layer. Besides reproducing known logarithmic scaling of moments, structure functions, and correlation function < u(z)(x)u(z)(x + r)>, new logarithmic laws in two-point statistics such as [3/2 < u(z)(2)(x)u(z)(2)(x + r)> - 1/2 < u(z)(4)(x)>](1/2), [5/2 < u(z)(3)(x)u(z)(3)(x + r)> - 3/2 < u(z)(x)u(z)(5)(x + r)>](1/3), etc. can be derived using the HRAP formalism. Supporting empirical evidence for the logarithmic scaling in such statistics is found from the Melbourne High Reynolds Number Boundary Layer Wind Tunnel measurements. We also show that, at high Reynolds numbers, the above mentioned new logarithmic laws can be derived by assuming the arrival of an attached eddy at a generic point in the flow field to be a Poisson process [Woodcock and Marusic, Phys. Fluids 27, 015104 (2015)]. Taken together, the results provide new evidence supporting the essential ingredients of the attached eddy hypothesis to describe streamwise velocity fluctuations of large, momentum transporting eddies in wall-bounded turbulence, while observed deviations suggest the need for further extensions of the model.