Here, we briefly describe the equations of fully developed turbulent flow in a plane channel. Further details can be found in classical references (see, for example, [9]). The Reynolds hypothesis for turbulent flow assumes that the velocity \( \left( u, v, w \right) \) and pressure \( p \) are decomposed into a mean component and a fluctuating component, i.e.
\[ \left( u , v , w, p \right) = \left( \bar{u}, \bar{v}, \bar{w}, \bar{p} \right) + \left( u^\prime , v^\prime , w^\prime , p^\prime \right) \]where the mean is usually taken over a finite time interval. Consequently, the nonlinear terms in the Navier-Stokes equations generate additional terms called the Reynolds stresses.
In the case of fully developed incompressible two-dimensional flow in a plane channel the Reynolds averaged Navier-Stokes equations reduce to:
\begin{align} \frac{\partial \bar{p}}{\partial x} & = \frac{d}{dy}\left(\mu\frac{d\bar{u}}{dy} - \rho \overline{u^\prime v^\prime}\right) \tag{1A} \\ \frac{\partial \bar{p}}{\partial y} & = -\frac{d}{dy}\left(\rho\overline{v^n}\right) \tag{1B} \end{align}where \( -\rho\overline{u^\prime v^\prime} \) and \( -\rho\overline{v^n} \) are the Reynolds stresses, or second-order moment tensors as they are sometimes called. Unfortunately, to close the problem the transport equations for the second-order moment tensors generate third-order moment tensors and so on. This problem of closure has been the bane of turbulence modeling for over the past hundred years.