Reynolds-averaged Navier-Stokes equations

The Reynolds-averaged Navier-Stokes (RANS) equations are time-averaged ^[1] equations of motion for Navier-Stokes equations. For an incompressible flow of Newtonian fluid, these equations can be written as

$\rho \frac{\partial \bar{u}_i}{\partial t} + \rho \frac{\partial \bar{u}_j \bar{u}_i }{\partial x_j} = \rho \bar{f}_i + \frac{\partial}{\partial x_j} \left[ - \bar{p}\delta_{ij} + \mu \left( \frac{\partial \bar{u}_i}{\partial x_j} + \frac{\partial \bar{u}_j}{\partial x_i} \right) - \rho \overline{u_i^\prime u_j^\prime} \right ].$