Turbulence on Time-Averaged Navier-Stokes Equations

TURBULENCE ON TIME-AVERAGED NAVIER-STOKES EQUATIONS  

Equations 10-1 through 10-14 only hold for laminar flow. At values of Reynolds number above a eomplieated situation emerges 

The resulting momentum balanee for the mean veloeity eomponents U, V, and W beeomes  

The extra stress terms result from six additional stresses, three normal stresses, and three shear stresses  

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu, -pv, and -pw are always non-zero beeause they eontain squared veloeity fluetuations. The shear stresses -pu v, -pu w, -pv w and are assoeiated with eorrelations between different veloeity eomponents. If, for instanee, u and v were statistieally independent fluetuations, the time average of their produet u v would be zero. However, the turbulent stresses are also non-zero and are usually large eompared to the viseous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

For a turbulence model to be useful in a general-purpose CFD code, it must be simple, accurate, economical to run, and have a wide range of applicability. Table 10-1 gives the most common turbulence models. The classical models use the Reynolds equations and form the basis of turbulence calculations in currently available commercial CFD codes. Farge eddy simulations are turbulence models where the time-dependent flow equations are solved for the mean flow and the largest eddies and where the effects of the smallest eddies are modeled. 

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