$$ f =\underbrace{\overline{f}}_{\text{average part}} +\underbrace{f'}_{\text{fluctuation}} $$

The Navier–Stokes equation for incompressible flow can be written as $$ \underbrace{\frac{\partial \vec{u}}{\partial t}}_{\text{Local acceleration}} + \underbrace{(\vec{u} \cdot \nabla)\vec{u}}_{\text{Advection}} = - \underbrace{\frac{1}{\rho} \nabla P}_{\text{Pressure force}} + \underbrace{g\vec{k}}_{\text{Gravity}} + \underbrace{\nu \nabla^2 \vec{u}}_{\text{Viscous diffusion}} $$ $g\vec{k}$ with typically $ \vec{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} $ defines gravity as a vector field acting downward (if you use $-g\vec{k}$, then $\vec{k}$ points upward)

For any field variable (velocity $\vec{u}$, pressure $P$, or scalar tracer $T$), we decompose it into a mean and a fluctuating part $$ \vec{u} = \overline{\vec{u}} + \vec{u}' \quad \text{and} \quad P = \overline{P} + P' \quad \text{and} \quad T = \overline{T} + T' $$

Apply the time-average operator to the NS equation. The nonlinear advection term becomes $$ (\vec{u} \cdot \nabla)\vec{u} = (\overline{\vec{u}} \cdot \nabla)\overline{\vec{u}} + \nabla \cdot \overline{\vec{u}'\vec{u}'} $$ $$ \overline{u_i' u_j'} \;\;\Rightarrow\;\; \text{new unknowns!} $$ This introduces a Reynolds stress tensor

So the Reynolds-averaged Navier-Stokes (RANS) equation becomes $$ \frac{\partial \overline{\vec{u}}}{\partial t} + (\overline{\vec{u}} \cdot \nabla)\overline{\vec{u}} = - \frac{1}{\rho} \nabla \overline{P} + g\vec{k} + \nu \nabla^2 \overline{\vec{u}} - \nabla \cdot \overline{\vec{u}' \vec{u}'} $$ The term $- \nabla \cdot \overline{\vec{u}'\vec{u}'}$ is the turbulent momentum transport, and it leads directly to the turbulence closure problem