TITLE:
Theoretical Study of Fully Developed Turbulent Flow in a Channel, Using Prandtl’s Mixing Length Model
AUTHORS:
Luigi A. Antonialli, Aristeu Silveira-Neto
KEYWORDS:
Mixing Length, Channel, Pipe, Turbulent Flow
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.6 No.4,
April
12,
2018
ABSTRACT: This research used the common decomposition of the
velocity and pressure in an average part and a fluctuating part, for high
Reynolds number, of the Navier-Stokes equation, which leads to the classic
problem of turbulent closure. The Prandtl’s mixing length model, based on the
Boussinesq hypothesis and traditionally used for free shear flows, was chosen
and adapted for internal flows to solve the closure problem. For channel flows,
Johann Nikuradse proposed a model for the Prandtl mixing length. In the present
paper, which has an academic character, the authors made a return to the model
of the mixing length of Prandtl and the model of Nikuradse, to solve turbulent
flows inside a plane channel. It was possible to develop an ordinary
differential model for the velocity in the direction of the flow whose solution
occurs computationally in a simple but extremely accurate way when compared
with Direct Numerical Simulation databases. For the viscous stress on the wall, it was possible to determine the exact mathematical solution of the
ordinary differential equation. It is a model of great academic value and even
to be used as reference for verification of computational codes destined to the
solution of complete numerical and computational models.