A Reynolds averaged vorticity transport equation can be derived by taking the curl of the RANS. In many situations, it is necessary to understand the variation of average vorticity in turbulent flows. The relationship between instantaneous vorticity and instantaneous velocity is as follows: $$ \omega = \nabla \times u \quad \text{or} \quad \tilde{\omega}_i = \varepsilon_{ijk} \frac{\partial \tilde{u}_k}{\partial x_j} $$ $\varepsilon_{ijk}$ is the third-order permutation tensor \[ \varepsilon_{ijk} = \begin{cases} 1, & \text{if } ijk = 123, \, 231, \, \text{or } 312 \\ -1, & \text{if } ijk = 132, \, 321, \, \text{or } 213 \\ 0, & \text{if any two indices in } ijk \text{ are repeated} \end{cases} \]

Taking the curl of both sides of equation \(\boxed{\frac{\partial \tilde{u}_i}{\partial t} + \frac{\partial (\tilde{u}_i \tilde{u}_j)}{\partial x_j} = -\,\frac{1}{\rho}\, \frac{\partial \tilde{p}}{\partial x_i} + g_i + \nu \frac{\partial^2 \tilde{u}_i}{\partial x_j \partial x_j} \quad (i,j=1,2,3)}\) yields the instantaneous vorticity transport equation in turbulent flow $$ \frac{\partial \tilde{\omega}_i}{\partial t} + \frac{\partial}{\partial x_j} \big( \tilde{u}_j \tilde{\omega}_i \big) = \underbrace{\tilde{\omega}_j \overbrace{\tilde{s}_{ij}}^{\substack{\tilde{s}_{ij} =\tfrac{1}{2}\left(\tfrac{\partial \tilde{u}_i}{\partial x_j} + \tfrac{\partial \tilde{u}_j}{\partial x_i}\right) \\ \text{strain-rate tensor}}}}_{\substack{\text{vortex stretching and tilting} \\ \text{that causes vorticity amplification}}} + \underbrace{\nu \frac{\partial^2 \tilde{\omega}_i}{\partial x_j \partial x_j}}_{\substack{\text{diffusion of vorticity} \\ \text{due to molecular motion}}} \quad (i, j = 1, 2, 3) $$