Fundamentals of Fluid Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 119

and the kinematic turbulent or eddy viscosity is denoted by ν T = μ T / ρ . For the standard
k-ω model α * = 1; but for its variants, such as the LRN and SST versions, α * is a

variable coefficient that accounts for damping of the eddy viscosity.
By substituting the expressions for Reynolds stress given by Eq. (5.28) and the
turbulent heat flux terms in Eq. (5.29) into the governing Eqs (5.24)-(5.27), and for
simplicity removing the over bar for the average quantities, we obtain

∂u
∂x +

∂v
∂y =

0 (5.32)

( ν ( ν

∂u
∂t +

u ∂u v ∂u

1
ρ

∂p
∂x +

∂
∂x

ν T ) ∂u
∂x

∂
∂y

ν T ) ∂u
∂y

∂x + ∂y =−

+ + +

(ν + ν T ) ∂u
∂x

(ν + ν T ) ∂v
∂x

∂
∂x

∂
∂y

+ + (5.33)

(ν (ν

∂v
∂t +

u ∂v v ∂v

1
ρ

∂p
∂y +

∂
∂x

ν T ) ∂v
∂x

∂
∂y

ν T ) ∂v
∂y

∂x + ∂y =−

+ + +

(ν (ν

∂
∂x

ν T ) ∂u
∂y

∂
∂y

ν T ) ∂v
∂y

+ + + + (5.34)

ν
Pr +

∂T
∂x

ν
Pr +

∂T
∂y

∂T
∂t +

u ∂T v ∂T

∂
∂x

ν T
Pr T

∂
∂y

ν T
Pr T ∂x + ∂y =

+

(5.35)

The term ν / Pr appearing in the temperature Eq. (5.35) is obtained from the def-
inition of the laminar Prandtl number, Pr = ν / α, where α = k / ρ C p . Interestingly,
the time-averaged equations above have the same form as those developed for the
laminar equations except for the additional turbulent (eddy) viscosity found in the
diffusion terms for the momentum and the energy equations. Hence, the solution to
turbulent flow in engineering problems entails greater diffusion that is imposed by
the turbulent nature of the fluid flow. The additional differential transport equations
that are required for the standard k-ω model, for the case of a constant fluid property,
are the following:

( ν + σ k ν T ) ∂k
∂x

( ν + σ k ν T ) ∂k
∂y

∂k
∂t +

u ∂k v ∂k

∂
∂x

∂
∂y ∂x + ∂y =

+ + P k − D k

(5.36)

ν ( ν

∂ω
∂t +

u ∂ω v ∂ω

∂
∂x

σ ω ν T ) ∂ω
∂x

∂
∂y

σ ω ν T ) ∂ω
∂y ∂x + ∂y =

+ + + + P ω − D ω

(5.37)

The terms P k and D k represent the production and destruction/dissipation of k and
similarly Pω and Dω are the production and destruction/dissipation of ω . The terms
σ k and σ ω are, respectively, the turbulent Prandtl numbers for k and ω . Eqs (5.35) and