$$ \underbrace{\frac{\partial \overline{u_i u_j}}{\partial t}}_{\substack{\text{Unsteady} \\ \text{(time) term}}} +\underbrace{C_{ij}}_{\substack{\text{Convection} \\ \text{term}}} = \underbrace{P_{ij}}_{\substack{\text{Production} \\ \text{term}}} +\underbrace{\Phi_{ij}}_{\substack{\text{Pressure--strain-rate} \\ \text{correlation}}} +\underbrace{d^{\nu}_{ij}}_{\substack{\text{Molecular} \\ \text{diffusion}}} +\underbrace{d^{t}_{ij}}_{\substack{\text{Velocity-fluctuation} \\ \text{diffusion}}} +\underbrace{d^{p}_{ij}}_{\substack{\text{Pressure-fluctuation} \\ \text{diffusion}}} -\underbrace{\varepsilon_{ij}}_{\substack{\text{Viscous} \\ \text{dissipation}}} $$

The convection term describes the total rate of change of Reynolds stress in each fluid control volume; it is produced by the imbalance of the terms on the right-hand side of the equation and at the same time reflects the history effect of turbulent transport. The production term describes the rate of change by which turbulence is generated or reduced through the interaction of the mean strain rate with the Reynolds stress. The pressure-strain-rate correlation describes the redistribution among the components of turbulent energy. Because $\phi_{ii}=0$ (as follows from $s_{ii}=0$), this term does not appear in the turbulent kinetic-energy equation; therefore it has a special meaning in the Reynolds-stress transport equation. The molecular diffusion, velocity-fluctuation diffusion, and pressure-fluctuation diffusion terms represent the diffusive nature of turbulence. For example, in a thin shear layer, $\partial U_1/\partial x_2$ is the dominant component; when integrated over any cross-section, the sum of the integrals of these three terms is zero. Thus their role is to promote the spatial distribution of turbulence. In general, the velocity-fluctuation diffusion term is the main part of the diffusion process; the other two terms are relatively significant mainly in the near-wall region. The viscous dissipation term describes the mechanism by which turbulent momentum is dissipated by fluid viscosity and converted into the internal energy of the fluid.

In \(\boxed{\frac{\partial \overline{u_i u_j}}{\partial t}+C_{ij} = P_{ij}+\Phi_{ij}+d^{\nu}_{ij}+d^{t}_{ij}+d^{p}_{ij}-\varepsilon_{ij}}\), if $U_k$ and $\overline{u_i u_j}$ are taken as the basic unknowns to be solved in the computation, then the terms that still need to be modeled are $\Phi_{ij}$, $d^{t}_{ij}$, $d^{p}_{ij}$, and $\varepsilon_{ij}$. The modeling of these terms should be expressed in terms of $U_k$, $\overline{u_i u_j}$, and suitable time- and length-scale variables.

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