Reynolds decomposition

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In fluid dynamics and the theory of turbulence, Reynolds decomposition is a mathematical technique to separate the average and fluctuating parts of a quantity. For example, for a quantity $u$ the decomposition woud be:

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

where $\overline{u}$ denotes the time average of $u\,$ (often called the steady component), and $u'\,$ the fluctuating part (or perturbations). The perturbations are defined 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.