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12.2.2 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:

 u_i = \bar{u}_i + u'_{i} (12.2-1)

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:
 \phi = \bar{\phi} + \phi' (12.2-2)

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:
 \frac{\partial \rho}{\partial t} + \frac{\partial} {\partial x_i} (\rho u_i) = 0 (12.2-3)

 \frac{\partial}{\partial t} (\rho u_i) + \frac{\partial}{\p... ...\frac{\partial}{\partial x_j} (- \rho \overline{u'_i u'_j}) (12.2-4)

Equations  12.2-3 and 12.2-4 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  12.2-4.

For variable-density flows, Equations  12.2-3 and 12.2-4 can be interpreted as Favre-averaged Navier-Stokes equations [ 142], with the velocities representing mass-averaged values. As such, Equations  12.2-3 and 12.2-4 can be applied to density-varying flows.

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