The RANS characterization starts with the Navier-Stokes equation and the turbulent decomposition
Navier-Stokes Equations
Space time separation
Reynolds Averaging
Conservation of mass
Laminar
Turbulent
Momentum - x
Note the top row has the same structure as the laminar equations. There are 2 additional terms that are shown on the next line.
These terms present a problem. The equation cannot be solved until the fluctuating components are known.
But, there are no additional conservation equations available for their solution.
Therefore have to be substituted in terms of . The dependence is established through various turbulence models
Energy
Laminar
Turbulent
The bottom line in green contains the extra terms
Reynolds Stress
These Reynolds stress and flux term are expressed through the Boussinesq approximation as