Reynolds averaging is quite easy to understand: a quantity that is statistically observable but randomly varying in a single observation is decomposed into a mean (average part) and a fluctuation (random part) $$ f =\underbrace{\overline{f}}_{\text{average part}} +\underbrace{f'}_{\text{fluctuation}} $$ Because the flow is already unstable/turbulent, it cannot be guaranteed that the fluctuation is necessarily much smaller than the average part

\[\tilde{u}_i=U_i+u_i,\qquad \tilde{p}=P+p\]

Substitute \(\boxed{\tilde{u}_i=U_i+u_i, \tilde{p}=P+p}\) into \(\boxed{\underbrace{\frac{\partial \tilde{u}_i}{\partial t}}_{\text{local (unsteady) acceleration}} \;+\; \underbrace{\frac{\partial (\tilde{u}_i \tilde{u}_j)}{\partial x_j}}_{\text{advection}} = -\,\frac{1}{\rho}\, \underbrace{\frac{\partial \tilde{p}}{\partial x_i}}_{\text{pressure gradient}} \;+\; \underbrace{g_i}_{\text{body-force acceleration}} \;+\; \nu\, \underbrace{\frac{\partial^2 \tilde{u}_i}{\partial x_j \partial x_j}}_{\text{viscous diffusion}} (i,j=1,2,3) }\) and \(\boxed{\underbrace{\frac{\partial \tilde{u}_j}{\partial x_j}}_{\text{incompressibility}} = 0}\), then take an ensemble average on both sides to obtain the Reynolds-averaged momentum and continuity equations $$ \frac{\partial U_i}{\partial t} +U_j\frac{\partial U_i}{\partial x_j} =-\,\frac{1}{\rho}\frac{\partial P}{\partial x_i} +g_i +\nu\,\frac{\partial^2 U_i}{\partial x_j\partial x_j} -\frac{\partial \overline{u_i u_j}}{\partial x_j} $$ $$ \frac{\partial U_j}{\partial x_j}=0 $$ The equations that result from Reynolds averaging of any form of the Navier-Stokes equations are commonly known as RANS equations