Editing Reynolds decomposition (section)

==Decomposition==
For example, for a quantity <math>u</math> the decomposition would be
<math display="block">u(x,y,z,t) = \overline{u(x,y,z)} + u'(x,y,z,t) </math>
where <math>\overline{u}</math> denotes the expectation value of <math>u</math>, (often called the steady component/time, spatial or [[ensemble average]]), and <math>u'</math>, are the deviations from the expectation value (or fluctuations). The fluctuations are defined as the expectation value subtracted from quantity <math>u</math> such that their [[time average]] equals zero. <ref>{{cite book |last=Müller |first=Peter |year=2006 |title=The Equations of Oceanic Motions |page=112 }}</ref><ref>{{cite journal| last1=Adrian|first1=R|title=Analysis and Interpretation of instantaneous turbulent velocity fields|journal=Experiments in Fluids| date=2000|volume=29|issue=3|pages=275–290|url=https://www.researchgate.net/publication/227210874| bibcode=2000ExFl...29..275A|doi=10.1007/s003489900087|s2cid=122145330}}</ref>

The expected value, <math>\overline{u}</math>, 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 <math>\langle u\rangle</math>, but it is also seen often with the over-bar notation.<ref>{{cite book|last1=Kundu|first1=Pijush|title=Fluid Mechanics|date=27 March 2015 |publisher=Academic Press|isbn=978-0-12-405935-1|pages=609}}</ref>

[[Direct numerical simulation]], or resolution of the [[Navier–Stokes equations]] completely in <math>(x,y,z,t)</math>, is only possible on extremely fine computational grids and small time steps even when [[Reynolds number]]s are low, and becomes prohibitively computationally expensive at high Reynolds' numbers. 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.<ref>{{Cite thesis|last=Mukerji|first=Sudip|year=1997|title=Turbulence Computations with 3-D Small-Scale Additive Turbulent Decomposition and Data-Fitting Using Chaotic Map Combinations|type=PhD thesis|publisher=University of Kentucky|id={{ProQuest|304354392}} |doi=10.2172/666048 | language=English |osti=666048|doi-access=free}}</ref>

Reynolds decomposition allows the simplification of 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.