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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.