In Reynolds averaging, the solution variables in the instantaneous (exact) Navier-Stokes equations are decomposed into the mean (ensemble-averaged or time-averaged) and fluctuating components. For the velocity components:

(4–1)

where and are the mean and fluctuating velocity components ().

Likewise, for pressure and other scalar quantities:

(4–2)

where denotes a scalar such as pressure, energy, or species concentration.

Substituting expressions of this form for the flow variables into the instantaneous continuity and momentum equations and taking a time (or ensemble) average (and dropping the overbar on the mean velocity, ) yields the ensemble-averaged momentum equations. They can be written in Cartesian tensor form as:

(4–3)

(4–4)

Equation 4–3 and Equation 4–4 are called Reynolds-averaged Navier-Stokes (RANS) equations. They have the same general form as the instantaneous Navier-Stokes equations, with the velocities and other solution variables now representing ensemble-averaged (or time-averaged) values. Additional terms now appear that represent the effects of turbulence. These Reynolds stresses, , must be modeled in order to close Equation 4–4.

For variable-density flows, Equation 4–3 and Equation 4–4 can be interpreted as Favre-averaged Navier-Stokes equations  [247], with the velocities representing mass-averaged values. As such, Equation 4–3 and Equation 4–4 can be applied to variable-density flows.