The $Re$-number dependence of the longitudinal dispersion in a turbulent channel flow

TL;DR

This paper investigates how the longitudinal dispersion coefficient in turbulent channel flows depends on the Reynolds number, linking it to the turbulent energy spectrum and turbulent cascades, supported by numerical simulations.

Contribution

It predicts the Reynolds number dependence of longitudinal dispersion from turbulence spectra and explores different scaling laws for various turbulent cascades.

Findings

Reynolds number dependence of $K_L$ can be derived from turbulence spectra

Different asymptotic scaling laws for $K_L(Re)$ depending on turbulence type

Numerical simulations confirm theoretical predictions

Abstract

In Taylor's theory, the longitudinal dispersion in turbulent pipe flows approaches, on long timescales, a diffusive behavior with a constant diffusivity $K_L$, that depends \emph{empirically} on the Reynolds number $Re$. We show that the dependence on $Re$ can be determined from the turbulent energy spectrum. By using the intimate connection between the friction factor and longitudinal dispersion in wall-bounded turbulence, we predict different asymptotic scaling laws of $K_L(Re)$ depending on the different turbulent cascades in two-dimensional turbulence. We also explore numerically the $K_L(Re)$ dependence in turbulent channel flows with smooth and rough walls using a lattice Boltzmann method.