In wall-bounded turbulence, the moment generating functions (MGFs) of the streamwise velocity fluctuations $〈exp\left(q{u}_z^{+}\right)〉$ develop power-law scaling as a function of the wall normal distance $z/\delta$. Here $u$ is the streamwise velocity fluctuation, $+$ indicates normalization in wall units (averaged friction velocity), $z$ is the distance from the wall, $q$ is an independent variable, and $\delta$ is the boundary layer thickness. Previous work has shown that this power-law scaling exists in the log-region $3{\text{Re}}_{\tau }^{0.5}\lesssim z^{+},z\lesssim 0.15\delta$ where ${\text{Re}}_{\tau }$ is the friction velocity-based Reynolds number. Here we present empirical evidence that this self-similar scaling can be extended, including bulk and viscosity-affected regions $30<z^{+},z<\delta$, 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, $q_o$. 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 ${\text{Re}}_{\tau }$ 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 $u_z^L$ and of the remaining small-scale component, $u_z^S=u_z-u_z^L$. The scaling of $u_z^L$ falls within the conventionally defined log region and depends on a scale that is proportional to $l^{+}\sim {\text{Re}}_{\tau }^{1/2}$; the scaling of $u_z^S$ extends over a much wider range from $z^{+}\approx 30$ to $z\approx 0.5\delta$. Last, we present a theoretical construction of two multiplicative processes for $u_z^L$ and $u_z^S$ that reproduce the empirical findings concerning the scalings properties as functions of $z^{+}$ and in the ESS sense.

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• Boundary layers
• Turbulence