In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum. Definition The velocity field of a flow can be split into a mean part and a fluctuating part using Reynolds decomposition. We write with being the flow velocity vector having components in the coordinate direction (with denoting the components of the coordinate vector ). The mean velocities are determined by either time averaging, spatial averaging or ensemble averaging, depending on the flow under study. Further denotes the fluctuating (turbulence) part of the velocity. We consider a homogeneous fluid, whose density ρ is taken to be a constant. For such a fluid, the components τ'''ij of the Reynolds stress tensor are defined as: Another – often used – definition, for constant density, of the Reynolds stress components is: which has the dimensions of velocity squared, instead of stress. Averaging and the Reynolds stress To illustrate, Cartesian vector index notation is used. For simplicity, consider an incompressible fluid: Given the fluid velocity as a function of position and time, write the average fluid velocity as , and the velocity fluctuation is . Then . The conventional ensemble rules of averaging are that One splits the Euler equations (fluid dynamics) or the Navier-Stokes equations into an average and a fluctuating part. One finds that upon averaging the fluid equations, a stress on the right hand side appears of the form . This is the Reynolds stress, conventionally written : The divergence of this stress is the force density on the fluid due to the turbulent fluctuations. Reynolds averaging of the Navier–Stokes equations For instance, for an incompressible, viscous, Newtonian fluid, the continuity and momentum equations—the incompressible Navier–Stokes equations—can be written (in a non-conservative form) as and where is the Lagrangian derivative or the substantial derivative, Defining the flow variables above with a time-averaged component and a fluctuating component, the continuity and momentum equations become and Examining one of the terms on the left hand side of the momentum equation, it is seen that where the last term on the right hand side vanishes as a result of the continuity equation. Accordingly, the momentum equation becomes Now the continuity and momentum equations will be averaged. The ensemble rules of averaging need to be employed, keeping in mind that the average