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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Reynolds-averaged Navier Stokes (RANS) equations are time-averaged equations of motion for fluid flow. They are primarily used while dealing with turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate averaged solutions to the Navier Stokes equations. For a stationary, incompressible flow of Newtonian fluid, these equations can be written in Einstein notation as: 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 ].