Extended self-similarity in moment-generating-functions in wall-bounded turbulence at high Reynolds number

In wall-bounded turbulence, the moment generating functions (MGFs) of the streamwise velocity _ _ fluctuations exp(qu+) develop power-law scaling as a function of the wall normal distance z/δ. z Here u is the streamwise velocity fluctuation, + indicates normalization in wall units (averaged fric- tion velocity), z is the distance from the wall, q is an independent variable and δ is the boundary layer thickness. Previous work has shown that this power-law scaling exists in the log-region 3Re0.5 . z+, τ z . 0.15δ where Reτ is the friction velocity-based Reynolds numbers. Here we present empirical evidence that this self-similar scaling can be extended, including bulk and viscosity-affected regions 30 < z+, z < δ, provided the data are interpreted with the Extended-Self-Similarity (ESS), i.e. self-scaling of the MGFs as a function of one reference value, qo. ESS also improves the scaling properties, leading to more precise measurements of the scaling exponents. The analysis is based on hot-wire measurements from boundary layers at Reτ ranging from 2700 to 13000 from the Melbourne High-Reynolds-Number-Turbulent-Boundary-Layer-Wind-Tunnel. Furthermore, we investigate the scalings of the filtered, large-scale velocity fluctuations uzL and of the remaining small-scale com- ponent, uzS = uz − uzL. The scaling of uzL falls within the conventionally defined log region and 1/2 depends on a scale that is proportional to l+ ∼ Reτ ; the scaling of uzS extends over a much wider range from z+ ≈ 30 to z ≈ 0.5δ. Last, we present a theoretical construction of two multiplicative processes for uzL and uzS that reproduce the empirical findings concerning the scalings properties as functions of z+ and in the ESS sense.