Prandtl's extended mixing length model applied to the two-dimensional turbulent classical far wake

TL;DR

This paper introduces a novel derivation of Prandtl's extended mixing length model for turbulent flows, incorporating kinematic viscosity, and demonstrates its accuracy through numerical solutions matching experimental data.

Contribution

A new derivation of Prandtl's extended model including kinematic viscosity, with similarity solutions and numerical validation against experiments.

Findings

Similarity velocity profiles agree well with experimental data.

Including kinematic viscosity improves model accuracy.

Numerical solutions are obtained using Hermite spectral method.

Abstract

Despite its limitations, Prandtl's mixing length model is widely applied in modelling turbulent free shear flows. Prandtl's extended model addresses many of the shortfalls of the original model, but is not so widely used, in part due to additional mathematical complexities that arise in its derivation and implementation. Furthermore, in both models Prandtl neglects the kinematic viscosity on the basis that it is much smaller in magnitude than the turbulent viscosity. Recent work has shown that including the kinematic viscosity in the original model has both mathematical and physical advantages. In the present work, a novel derivation of the extended model is provided, and it is demonstrated that similar advantages are again obtained when the kinematic viscosity is included. Additionally, through the use of scaling techniques, similarity mean velocity profiles of the extended model are derived, resulting in a single nonlinear ordinary differential equation that is solved numerically with a Hermite spectral method. The computed profiles for the normalised similarity mean velocity and shear stress are compared to experimental observations and shown to be in excellent agreement.