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This is a In Vitro Study study.

August 21, 2025Open Access

The nonlocal nature of the Reynolds stress

EVEmmanuel VillermauxInstitut polytechnique de Grenoble

Key Points

The proposed model achieves consistent predictions for velocity profiles in turbulent flows and boundary layers.
Measurement agreement includes velocity profiles in jets, shear layers, and wakes based on the nonlocal Reynolds stress framework.
This analysis uses a closure based on Prandtl's framework adapting turbulence models for boundary layer behavior.
Understanding nonlocal effects may provide better turbulence modeling, potentially improving physics of fluid dynamics.

Abstract

A nonlocal expression for the turbulent Reynolds stress u ′ v ′ ¯ solves in a unified manner for the velocity profiles in jets, shear layers, wakes, and boundary layers, offering predictions consistent with known measurements in these canonical flows. The framework is the random walk inspired closure of Prandtl (1925) since we write u ′ v ′ ¯=-v ˜ℓ∂ y u with ℓ a mean free path but, by contrast with Prandtl, with v ˜ a nonlocal transfer velocity resulting from an integration over an appropriate portion of space of the mean velocity profile u(y). The status of ℓ is discussed and its value, a fraction (10π) -1 ≈0.031 of the integral scale, is computed in opened shear flows. The closure is adapted to the special case of boundary layers, where the value of the von Kármán constant is found to be κ=6 -1/2 ≈0.41.

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