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

TL;DR

This paper introduces a bounded dissipation model for turbulent velocity moments in wall flows, successfully explaining experimental data and predicting high-Reynolds-number behavior of turbulence fluctuations.

Contribution

The study develops and validates a new model based on bounded dissipation, providing explicit formulas for high-order velocity moments in wall turbulence.

Findings

Excellent agreement with experimental data for moments up to q=5.

Model predicts bounded velocity fluctuations at high Reynolds numbers.

Formulas applicable across boundary layers, pipes, and channels.

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}=\alpha_q-\beta_q y^{\ast1/4}$; here $q$ 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 this formula and the available data in boundary layers, pipes and channels 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 $Re_\tau\rightarrow\infty$. This work demonstrates the success of the present model in describing the behavior of fluctuations in wall flows.