Bounded dissipation law and profiles of turbulent velocity moments in wall flows

Abstract

Turbulent wall flows offer the most direct means for understanding the effects of boundaries and viscosity on turbulent fluctuations. Available data on mean-square fluctuations in these flows show apparent contradiction with classical scaling based on the {mean} wall shear stress. We had earlier proposed an alternative model based on the principle of bounded dissipation to describe the data. Despite its putative success, a conclusive outcome requires much higher Reynolds numbers than are available at present, or can be expected to be available in the near future. However, the model can be validated satisfactorily even within the Reynolds number range already available by considering high-order moments and their distributions in the wall-normal direction. Expressions for high-order moments of streamwise velocity fluctuation $u$ are derived in the form $ \langle u^{+2q} \rangle^{1/q}=α_q-β_q y^{\ast1/4}$; here $q$ is an integer, $α_q$ and $β_q$ are constants independent of the friction Reynolds number $Re_τ$, and $y^{\ast} = y/δ$ is the distance away from the wall, normalized by the flow thickness $δ$; in particular, $α_q =μ+σq$ according to the `linear q-norm Gaussian' process, where $μ$ and $σ$ are flow-independent constants. Excellent agreement is found between this formula and the available data in boundary layers, pipes and channels for $1 \leq q \leq 5$. For fixed $y^+ = y^*Re_τ$, the present formulation leads to the bounded state $\langle u^{+2q} \rangle^{1/q}=α_q$ as $Re_τ\rightarrow\infty$. This work demonstrates the success of the present model in describing the behavior of fluctuations in wall flows.