**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

### Famous quotes containing the words stress and/or reynolds:

“Anyone who wishes to combine domestic responsibilities and paid employment with the least *stress* and most enjoyment might start by pondering this paradox: the first step to better functioning is to stop blaming herself for not functioning well enough.”

—Faye J. Crosby (20th century)

“When Sir Joshua *Reynolds* died

All Nature was degraded;”

—William Blake (1757–1827)