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

Abstract

In wall-bounded turbulence, the moment generating functions (MGFs) of the streamwise velocity fluctuations $⟨\exp (q{u}_z^{+})⟩$ 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 {\small $3R{e}_{\tau }^{0.5}\lesssim z^{+}$, $z\lesssim 0.15\delta$}, where $R{e}_{\tau }$ 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 $30<z^{+}$, $z<\delta$, 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 $R{e}_{\tau }$ 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 {\small $l^{+}\sim R{e}_{\tau }^{1/2}$}; 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.

References in corpus (1)

• Large-eddy simulation study of the logarithmic law for second and higher-order moments in turbulent wall-bounded flow

Cited by in corpus (1)

• Statistics of turbulence in the energy-containing range of Taylor-Couette compared to canonical wall-bounded flows