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 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.
Read more about this topic: Reynolds Stress
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