Reynolds decomposition

In fluid dynamics and 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 ${\displaystyle u}$ the decomposition would be

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

where ${\displaystyle {\overline {u}}}$ denotes the expectation value of ${\displaystyle u}$, (often called the steady component/time, spatial or ensemble average), and ${\displaystyle u'}$, are the deviations from the expectation value (or fluctuations). The fluctuations are defined as the expectation value subtracted from quantity ${\displaystyle u}$ such that their time average equals zero. [1][2]

The expected value, ${\displaystyle {\overline {u}}}$, is often found from an ensemble average which is an average taken over multiple experiments under identical conditions. The expected value is also sometime denoted ${\displaystyle \langle u\rangle }$, but it is also seen often with the over-bar notation.[3]

Direct Numerical Simulation, or resolving the Navier-Stokes equations completely in ${\displaystyle (x,y,z,t)}$, 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. [4]

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.