For convenience in later applications, the transport equation of turbulent fluctuating velocity \(\boxed{\frac{\partial u_i}{\partial t} + U_j \frac{\partial u_i}{\partial x_j} + u_j \frac{\partial U_i}{\partial x_j} + \frac{\partial}{\partial x_j}\big( u_i u_j - \overline{u_i u_j} \big) = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_j \partial x_j}s}\) is denoted as $$ L(u_i) = 0 $$

Multiplying the transport equation for $u_i$ by $u_j$, and the transport equation for $u_j$ by $u_i$, then adding the two together and taking the average $$ \langle u_j L(u_i) + u_i L(u_j) \rangle = 0 $$

\( \frac{D \, \overline{u_i u_j}}{Dt} = \frac{\partial}{\partial x_k} \left( \nu \frac{\partial \overline{u_i u_j}}{\partial x_k} - \overline{u_i u_j u_k} - \frac{1}{\rho} \overline{u_i p} \, \delta_{jk} + \frac{1}{\rho} \overline{u_j p} \, \delta_{ik} \right) - \left( \overline{u_k u_i} \frac{\partial U_j}{\partial x_k} + \overline{u_k u_j} \frac{\partial U_i}{\partial x_k} \right) + \frac{2}{\rho} \, \overline{p s_{ij}} - 2\nu \, \overline{\frac{\partial u_i}{\partial x_k} \frac{\partial u_j}{\partial x_k}} \)
In the equation, $s_{ij}$ is the fluctuating strain-rate tensor $ s_{ij} = \tfrac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right); $ $\delta_{ik}$ and $\delta_{jk}$ are the Kronecker delta in tensor notation

For convenience, this equation is generally written as $$ c_{ij} = d_{ij} + P_{ij} + \phi_{ij} - \varepsilon_{ij} $$ This equation describes all the mechanisms governing the transport of Reynolds stresses