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

Abstract

Understanding the effects of solid boundaries on turbulent fluctuations remains a long-standing challenge. Available data on mean-square fluctuations in these flows show apparent contradiction with classical scaling. We had earlier proposed an alternative model based on the principle of bounded dissipation. 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 ⟨ u + 2 q ⟩ 1 / q = α q − β q y ∗ 1 / 4 , where the superscript + indicates the wall unit normalization, and brackets stand for averages over time and the homogeneous plane normal to the wall, q is an integer, α q and β q are constants independent of the friction Reynolds number R e τ , and y ∗ = y / δ is the distance away from the wall, normalized by the flow thickness δ . In particular, α q = μ + σ q 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 ≤ q ≤ 5 . For fixed y + = y ∗ R e τ , the present formulation leads to the bounded state ⟨ u + 2 q ⟩ 1 / q = α q as R e τ → ∞ . This work demonstrates the success of the present model in describing the behavior of fluctuations in wall flows.