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copied from Reynold's decomposition

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 $\scriptstyle u$ 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 time average of $\scriptstyle u\,$ (often called the steady component), and $u'\,$ the fluctuating part (or perturbations). The perturbations are defined such that their time average equals zero.

Direct Numerical Simulation, or resolving the Navier-Stokes equations completely in (x,y,z,t), is only possible on small computational grids and small time steps when Reynolds numbers are low. Due to computational constraints, simplifications of the Navier-Stokes equations are useful to parameterize turbulence that are smaller than the computational grid, allowing larger computational domains.

Reynolds decomposition allows the simplification 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.