Bounded asymptotics for high-order moments in wall turbulence

Turbulent wall-flows are the most important means for understanding the effects of boundary conditions and fluid viscosity on turbulent fluctuations. There has been considerable recent research on mean-square fluctuations. Here, we present expressions for high-order moments of streamwise velocity fluctuation $u$, in the form $ \langle u^{+2q} \rangle^{1/q}=\alpha_q-\beta_q y^{\ast1/4}$; $q$ is an integer, $\alpha_q$ and $\beta_q$ are constants independent of the friction Reynolds number $Re_\tau$, and $y^{\ast} = y/\delta$ is the distance away from the wall, normalized by the flow thickness $\delta$; in particular, $\alpha_q =\mu+\sigma q$ according to the `linear q-norm Gaussian' process, where $\mu$ and $\sigma$ are flow-independent constants. Excellent agreement is found between these formulae and available data in pipes, channels and boundary layers for $1 \leq q \leq 5$. For fixed $y^+ = y^*Re_\tau$, the present formulation leads to the bounded state $\langle u^{+2q} \rangle^{1/q}=\alpha_q$ as $Re_\tau\rightarrow\infty$ while the attached eddy model predicts that the moments continually grow as log Reynolds number.