In Reynolds averaging, variables from the solution of the Navier-Stokes equations are decomposed into (1) ensemble averaging or time averaging and (2) fluctuating components, as illustrated in the diagram below for velocity.
Figure 6.4. Averaged fluctuative velocity
For example, for velocity, it can be expressed as the following equation:
(6.1)
Where Ul,bar represents the average component, and U’ represents the fluctuating component (i = 1,2,3).
Similarly, for scalar quantities like pressure, energy, or species concentration, denoted as , the equation can be written as follows:
(6.2)
Substitute equations (6.1) and (6.2) into the momentum and continuity equations at an instantaneous level, then average them over time. This yields equations that can be expressed in Cartesian tensor form as follows:
(6.3)
And, for momentum equation:
(6.4)
Equations (6.3) and (6.4) are referred to as the Reynolds-averaged Navier-Stokes (RANS) equations. Additionally, the form of the term
is also known as Reynolds stresses.
Reference: