The UMIST-N Near-Wall Treatment Applied to Periodic Channel Flow

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 5
1.2 Turbulence Models
RANS models are based upon the idea of filtering turbulence from the gov-
erning equations of a flow so that it may be treated separately. This is due
to Reynolds [59]. In EVMs, the problem of knowing the effect of turbulence
on the mean flow is made tractable by employing the idea of a turbulent
viscosity, based on the work of Boussinesq [6]. Turbulent viscosity may be
calculated in a number of different ways.
The simplest approach is to specify the turbulent viscosity at a given location
based on known local quantities. Models based on this approach are called
‘algebraic models’. Taylor [80] and Prandtl [54] have proposed an algebraic
model in which turbulent viscosity is calculated as a function of a length scale
and a local mean velocity gradient. The length scale is specified as a function
of wall-distance. Unfortunately, an appropriate length scale can be difficult
to obtain in complex geometries. Another limitation of algebraic models is
that the dependence of turbulent viscosity on a local mean velocity gradient
is unrealistic in many types of flow.
To remove the latter limitation of algebraic models, ‘one-equation’ models
track an additional turbulent quantity through the solution of an additional
transport equation. The most popular choice of the additional quantity is
the turbulent kinetic energy per unit mass, k (usually referred to simply as
turbulent kinetic energy). Prandtl [55] proposed this approach. Spalart et
al. [73] have proposed directly solving a transport equation for turbulent
viscosity. A limitation that is common to all one-equation models is that a
length scale is still required to fully specify the modelled flow.