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In fluid dynamics and the turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations. For example, for a quantity $\scriptstyle u$ the decomposition would be

$u(x,y,z,t)={\overline {u(x,y,z)}}+u'(x,y,z,t)\,$

where $\scriptstyle {\overline {u}}$ denotes the expectation value of $\scriptstyle u\,$ (often called the steady component), and $u'\,$ are the deviations from the expectation value (or fluctuations). The fluctuations are defined as the expectation value subtracted from the quantity u such that their time average equals zero.

This allows us to simplify the Navier–Stokes equations by substituting in the sum of the steady component and perturbations to the velocity profile and taking the mean value. The resulting equation contains a nonlinear term known as the Reynolds stresses which gives rise to turbulence.

See also

• Reynolds-averaged Navier–Stokes equations
• Turbulence

References

Article Evaluation:

The way the 'average' term is defined in the article should be called the expectation value because it does not have to be the mean value, it could be a time averaged, spacially averaged or ensemble average. Several ways of determining this exist.

U' could be more clearly described by saying that the 'fluctuation' are the points that deviate from the expectation value.

It should also be shown that the fluctuation term is the actual quantity subtracted by the expectation value.