Fundamentals of Fluid Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 120

(5.36) form the statement: The rate of change and the convective transport of k or ω
equal the diffusion transport combined with the rate of production and destruction
ofkorω . Improvements to the standard k-ω model have been made by Menter
(1994) who created the k-ω-SST model. The improved model uses the original k-ω
by Wilcox (1993) in the inner region of the boundary layer but then switches to the
standard k-ε model in the outer region. This modification improves the prediction of
the flow with adverse pressure gradients ( ∂p / ∂x > 0) where flow separation tends to
occur. This is common in the laryngeal region (Fig. 5.6 ). It is important to note that
the use of the k-ω model in the inner region of the boundary layer means that the
model directly resolves the flow behaviour all the way down to the wall. This means
that a very fine mesh is needed in the vicinity of the wall (see later Sect. 5.3.5). Some
disadvantages of the model include its overprediction of k in regions of accelerating
and stagnating flows.

5.3.4 Other Turbulence Models

k-ε Models One of the most widely used turbulence models in a wide range of
engineering applications is the k - ε model . The model is also classed as a two-equation
model as it has two additional transport equations, solving for the turbulence kinetic
energy k and the turbulence dissipation rate ε . The eddy viscosity is then given by

C μ ρ k 2
ε

μ t = (5.38)

where C μ is a model constant. The standard k-ε model was derived empirically
mainly for fully turbulent flows. It can be used with the so-called wall function
boundary conditions that connect the core turbulent flows to the wall and therefore
obviate the need for a fine mesh close to the wall. The model's performance has
been assessed against a number of practical flows. It has achieved notable successes
in predicting thin shear layers, boundary layers and duct flows without the need for
case-by-case adjustment of the model constants. It has also been shown to perform
quite well when the Reynolds shear stresses are dominant in confined flows. This
accommodates a wide range of flows with industrial applications, which explains the
model's popularity among CFD users. Sometimes the k - ε model is used as an initial
run to establish a turbulent flow field before embarking on more complex models.
Some variations of the k - ε model that are commonly found in commercial CFD codes
include the Low-Reynolds-Number (LRN), Realizable, and the ReNormalisation
Group (RNG) k - ε models. The LRN 1 model allows full resolution of the near-wall
boundary layer but requires a fine mesh. Both the Realizable and RNG models
improve to an extent the performance of the standard model by better predicting
flows involving moderate swirl, vortices, and locally transitional flows.

1 It should be noted that the term LRN does not refer to the bulk flow Reynolds number, but rather
the turbulent Reynolds number, which is low in the viscous sublayer (see Sect. 5.3.5 for viscous
sublayer description). In fact the LRN models can be applied for a fluid flow that has a high flow
Reynolds number.