Volume I - Section V - CFD Methodology 
Page V ° 13 
To overcome these limitations, variables are split into a mean and fluctuating component, i.e.: 
U = U + u' 
H = H+h' 
(5.5) 
These are then substituted back into the instantaneous momentum equation, producing the 
following: 
a / 
dxj 
(p u,u,) = -^ + 
dx- dx , 
( at/,. 
M- 
ar- pM ^ 
+ 2,(p-Po) 
(5.6) 
This is known as the time averaged momentum equation. A similar equation exists for the 
enthalpy equation: 
d ( \ d 
i \ 
'4 
pit'll' 
(5.7) 
The extra terms produced by this substitution are: 
• Reynolds stress = pu'u'j 
• Reynolds flux = pit'll' 
A turbulent flow is characterized by the dominance of diffusion due to the Reynolds stresses and 
the fluxes over the diffusion due to laminar viscosity or laminar diffusivity of the fluid. The 
spread of contaminants in the animal room, in particular the determination of CCb and NH 3 
levels in both the cages and within the room itself, is controlled strongly by the diffusion of the 
contaminant into the surrounding air volume. The role of turbulence modeling, to calculate the 
Reynolds stresses and fluxes, is therefore of vital importance in the accurate prediction of 
concentration spread in the cages and room. 
The introduction of the Reynolds stresses and fluxes after decomposition of the turbulent 
fluctuating variables means that the equation set is now not closed. Some form of closure is 
required to model these fluxes and stresses. There have been a wide range of methods used to do 
this, varying from the most simple zero-equation models to the much more complex Reynolds 
stress transport equations. Figure 5.04 shows how these turbulence models relate to each other. 
At the center of the zero-, one-, and two-equation models lies the analogy that where a laminar 
stress exists, so can an equivalent turbulent stress (i.e., Reynolds stress). A laminar shear stress is 
defined as: 
