4.2.2 Reynolds (Ensemble) Averaging

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:

where $\bar{u}_i$ and $u'_{i}$ are the mean and fluctuating velocity components ( $i = 1, 2, 3$).

Likewise, for pressure and other scalar quantities:

where $\phi$ 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, $\bar{u}$) yields the ensemble-averaged momentum equations. They can be written in Cartesian tensor form as:

Equations  4.2-3 and 4.2-4 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, $- \rho \overline{u'_i u'_j}$, must be modeled in order to close Equation  4.2-4.

For variable-density flows, Equations  4.2-3 and 4.2-4 can be interpreted as Favre-averaged Navier-Stokes equations [ 130], with the velocities representing mass-averaged values. As such, Equations  4.2-3 and 4.2-4 can be applied to density-varying flows.