A mixing length model for arbitrary geometry

¹ LadHyX, CNRS, École Polytechnique, Institut Polytechnique de Paris - F-91120 Palaiseau, France
² Institute of Mechanical Sciences and Industrial Applications, ENSTA-Paris, Institut Polytechnique de Paris 828 Bd des Maréchaux, F-91120 Palaiseau, France

Received: 2 May 2025
Accepted: 1 July 2025

Abstract

We present a novel phenomenological model for the mixing length used in turbulence models. It accounts for the nonlocality of the Reynolds stress tensor without introducing transport or integral equations. It has however the advantage of naturally accounting for the object's geometry while satisfying the standard symmetries of the Navier-Stokes equations. We investigate the model for the classical channel and pipe flows to characterize its main findings. We calibrate the three model parameters to recover the damping in the viscous sub-layer, the log-law of the wall, and the outer region behaviors. Our model compares favorably to friction factor measurements in the pipe flow at high Reynolds numbers and provides analytical predictions of the mixing length for several canonical flows.