# Reynolds Stress - Averaging and The Reynolds 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

begin{align} overline{bar a} &= bar a, \ overline{a + b} &= bar a + bar b, \ overline{a bar b} &= bar a bar b. end{align}

One splits the Euler equations 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.