Page 95 - Computational Fluid Dynamics for Engineers

,,M Turbulence Models

3.1 Introduction

The Reynolds-averaged equations of Section 2.3 and their reduced forms in
Section 2.4 cannot be solved without information about the various correlation
terms that make up the stress tensor, and the same is true for the energy
equation. It is well known that these terms, which represent turbulent diffusion,
are much larger than those corresponding to laminar diffusion except in the
immediate vicinity of a wall, and in turbulent wall boundary layers, wakes, jets
and more complex flows, these turbulent diffusion terms are of similar magnitude
to the convective terms.
This chapter presents a brief description of various models to address the
closure problem of turbulence modeling. The subject has been studied exten-
sively in the past three decades and useful reviews have been provided in many
journal articles and in several books, see for example, Wilcox [1], Cebeci [2,3],
Durbin and Reif [4]. For a detailed description of turbulence models and their
accuracy, the reader is referred to these books.
Early approaches to turbulence modeling include the mixing length, £, as-
sumptions of Prandtl [5] and eddy-viscosity, £ m , assumptions of Boussinesq [6]
for wall boundary layers and jets. Kolmogorov [7] and Rotta [8] proposed models
based on partial-differential equations but, in the absence of digital computers,
could not solve them. The early models provided a foundation which is still in
use today. For example, the concepts of a mixing length,

QU'V' = 6f r ^ \ (3.1.1)

and eddy-viscosity expressions of the form
0H
1
- QU'V = ge m— (3.1.2)
dy
have been used in the vast majority of publications concerned with turbulence