Fundamentals of Fluid Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 118

The total shear stress is then is given as

μ t ) d

u
dy

¯

τ = ( μ + (5.31)

It should be emphasized that the eddy viscosity is not a fluid property but rather is
a conceptual term that relates the Reynolds stresses to the mean deformation rates
(velocity gradients). The eddy viscosity is strongly flow-dependant, and determining
its value as a function of flow parameters is the central objective of RANS turbulence
models.

5.3.3 Additional Equations for the k-ω Turbulence Model

In this section we present a summary of the k-ω turbulence model because it has
shown promising results for internal flows in transitional and/or low Reynolds number
turbulent flow regimes. In particular, the k-ω-SST model has attracted considerable
attention for flows with a low level of turbulence (Menter et al. 2006), and has
been widely adopted as the turbulence model for respiratory flows (Kleinstreuer and
Zhang 2010; Liu et al. 2010a; Shi et al. 2008). This is because of the presence
of laminar, transitional, and turbulent flow regimes in the respiratory airways. The
family of k - ω models requires solving two additional transport equations for the
kinetic energy of turbulence, k, and the specific dissipation rate, ω (Wilcox 1993),
and in general is classed as a two-equation model. The model provides accurate
predictions for flow separation from smooth surfaces (Bardina et al. 1997). Its variant
models, the k - ω - SST (Shear Stress Transport k - ω model) as described by Menter
(1994) and the LRN- k - ω (Low Reynolds Number k - ω model) and more recently the
k - ω - SST-transistional model (4 equations) have shown to provide good results for
respiratory airway modelling.These model are inherently an LRN model and expands
the solution region all the way to the wall, which means that the mesh near the wall
needs to be fine to provide accurate results.
To describe the model, some preliminary definitions are required first. The tur-
bulence kinetic energy k and the specific dissipation of turbulence energy ω (i.e. the
dissipation rate per unit of turbulence kinetic energy) can be defined and expressed
in Cartesian tensor notation as

1
2 u i u i

ε
k ,

k = and ω =

ν T ∂u i
∂x j

∂u i
∂x j

and i , j

where ε = = 1, 2, 3

The turbulent (eddy) viscosity μ T can be evaluated in terms of the local values of k
and ω as

μ T = α ∗ ρk
ω