The transport equation of turbulent fluctuating velocity can be obtained by the instantaneous momentum equation \(\boxed{\frac{\partial \tilde{u}_i}{\partial t} + \frac{\partial (\tilde{u}_i \tilde{u}_j)}{\partial x_j} = -\,\frac{1}{\rho}\, \frac{\partial \tilde{p}}{\partial x_i} + g_i + \nu \frac{\partial^2 \tilde{u}_i}{\partial x_j \partial x_j} \quad (i,j=1,2,3)}\) minus the average momentum equation (RANS) \(\boxed{\frac{\partial U_i}{\partial t} +U_j\frac{\partial U_i}{\partial x_j} =-\,\frac{1}{\rho}\frac{\partial P}{\partial x_i} +g_i +\nu\,\frac{\partial^2 U_i}{\partial x_j\partial x_j} -\frac{\partial \overline{u_i u_j}}{\partial x_j}}\) \[ \frac{\partial u_i}{\partial t} + U_j \frac{\partial u_i}{\partial x_j} + u_j \frac{\partial U_i}{\partial x_j} + \frac{\partial}{\partial x_j}\big( u_i u_j - \overline{u_i u_j} \big) = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_j \partial x_j} \]

For incompressible turbulent flow, the fluctuating velocity field also satisfies the continuity condition of zero divergence. From equation \(\boxed{\frac{\partial \tilde{u}_j}{\partial x_j} = 0 }\) minus equation \(\boxed{\frac{\partial U_j}{\partial x_j}=0 }\) \[\frac{\partial u_j}{\partial x_j} = 0 \]

Similarly, from equation \(\boxed{\frac{\partial \tilde{\omega}_i}{\partial t} + \frac{\partial}{\partial x_j} \big( \tilde{u}_j \tilde{\omega}_i \big) = \tilde{\omega}_j \tilde{s}_{ij} + \nu \frac{\partial^2 \tilde{\omega}_i}{\partial x_j \partial x_j}, \quad (i, j = 1, 2, 3)}\) minus equation \(\boxed{\frac{\partial \Omega_i}{\partial t} + U_j \frac{\partial \Omega_i}{\partial x_j} = \Omega_j S_{ij} + \nu \frac{\partial^2 \Omega_i}{\partial x_j \partial x_j} - \frac{\partial}{\partial x_j}\,(\overline{u_j \omega_i}) + \omega_j s_{ij}}\), the transport equation for the vorticity fluctuations can be obtained $$ \frac{\partial \omega_i}{\partial t} + U_j \frac{\partial \omega_i}{\partial x_j} + u_j \frac{\partial \Omega_i}{\partial x_j} + \frac{\partial}{\partial x_j}\left( u_j \omega_i - \overline{u_j \omega_i} \right) = \Omega_j s_{ij} + \omega_j S_{ij} + \big( \omega_j s_{ij} - \overline{\omega_j s_{ij}} \big) + \nu \frac{\partial^2 \omega_i}{\partial x_j \partial x_j} $$

Taking the divergence of both sides of equation \(\boxed{\frac{\partial u_i}{\partial t} + U_j \frac{\partial u_i}{\partial x_j} + u_j \frac{\partial U_i}{\partial x_j} + \frac{\partial}{\partial x_j}\big( u_i u_j - \overline{u_i u_j} \big) = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_j \partial x_j}}\), the Poisson equation for the pressure fluctuations is obtained \[\frac{1}{\rho} \nabla^2 p = - \frac{\partial^2}{\partial x_i \partial x_j} \big( u_i U_j + U_i u_j \big) - \frac{\partial^2}{\partial x_i \partial x_j} \big( u_i u_j - \overline{u_i u_j} \big) \]