The following development of the mean-flow equations is for incompressible turbulent flow with constant viscosity where density fluctuations are caused by temperature fluctuations alone

The first step is to separate the dependent-field quantities into components representing the mean (capital letters and those with over bars) and those representing the deviation from the mean (lower case letters and those with primes) $$ \tilde{u}_i = U_i + u_i,\qquad \tilde{p} = P + p,\qquad \tilde{\rho} = \bar{\rho} + \rho',\qquad \tilde{T} = \bar{T} + T' $$ the complete field quantities are denoted by a tilde (~)

First proposed by Reynolds, this decomposition, called Reynolds decomposition or Reynolds averaging, has become the leading method to simulate turbulent flows

However, it leads to a closure problem in the resulting equation set that has still not been resolved without empiricism and modeling