$$ \tilde{\omega}_i = \underbrace{\Omega_i}_{\substack{\Omega_i = \varepsilon_{ijk} \tfrac{\partial U_k}{\partial x_j} \\ \text{average vorticity}}} + \underbrace{\omega_i}_{\substack{\omega_i = \varepsilon_{ijk} \tfrac{\partial u_k}{\partial x_j} \\ \text{fluctuating vorticity}}}, \quad \tilde{u}_i = U_i + u_i $$

Substituting equation \(\boxed{\tilde{\omega}_i = \Omega_i + \omega_i, \quad \tilde{u}_i = U_i + u_i}\) into equation \(\boxed{\frac{\partial \tilde{\omega}_i}{\partial t} + \frac{\partial}{\partial x_j} \big( \tilde{u}_j \tilde{\omega}_i \big) = \tilde{\omega}_j \tilde{s}_{ij} + \nu \frac{\partial^2 \tilde{\omega}_i}{\partial x_j \partial x_j} \quad (i, j = 1, 2, 3)}\) and then taking the ensemble average of both sides, the Reynolds-averaged vorticity transport equation is obtained $$ \frac{\partial \Omega_i}{\partial t} + U_j \frac{\partial \Omega_i}{\partial x_j} = \Omega_j S_{ij} + \nu \frac{\partial^2 \Omega_i}{\partial x_j \partial x_j} - \frac{\partial}{\partial x_j} \big( \overline{u_j \omega_i}) + \overline{\omega_j s_{ij}} $$

\[ \frac{\partial \Omega_i}{\partial t} + U_j \frac{\partial \Omega_i}{\partial x_j} = \underbrace{\Omega_j \overbrace{S_{ij}}^{\substack{ \tfrac{1}{2}\left(\tfrac{\partial U_i}{\partial x_j} + \tfrac{\partial U_j}{\partial x_i}\right) \\ \text{average} \\ \text{strain-rate tensor}}}}_{\substack{ \text{average vorticity} \\ \text{amplification} \\ \text{due to average strain-rate}}} + \underbrace{\nu \frac{\partial^2 \Omega_i}{\partial x_j \partial x_j}}_{\substack{ \text{average vorticity} \\ \text{diffusion} \\ \text{due to molecular motion}}} - \underbrace{\frac{\partial}{\partial x_j}\,(\overline{u_j \omega_i})}_{\substack{ \text{average vorticity transport} \\ \text{(or turbulent diffusion)} \\ \text{caused by} \\ \text{fluctuating velocity}}} + \underbrace{\omega_j \overbrace{s_{ij}}^{\substack{ \tfrac{1}{2}\left(\tfrac{\partial u_i}{\partial x_j} + \tfrac{\partial u_j}{\partial x_i}\right) \\ \text{fluctuating} \\ \text{strain-rate tensor}}}}_{\substack{ \text{average vorticity generated by} \\ \text{vortex stretching and tilting} \\ \text{due to fluctuating strain-rate}}} \]