Reynolds decomposition: Difference between revisions

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{{Refimprove|date=April 2016}}
{{Refimprove|date=April 2016}}
In [[fluid dynamics]] and [[turbulence]] theory, '''Reynolds decomposition''' is a mathematical technique used to separate the [[expectation value]] of a quantity from its [[statistical fluctuations|fluctuations]].
In [[fluid dynamics]] and [[turbulence]] theory, a '''Reynolds decomposition''' is a mathematical technique used to separate a field into its [[expected value|mean]] and [[statistical fluctuations|fluctuating]] components.<ref name="Pope" >{{cite book |last=Pope |first=Stephen |year=2001 |title=Turbulent Flows |page=83 }}</ref>


==Decomposition==
==Decomposition==
For example, for a quantity <math>u</math> the decomposition would be
A Reynolds decomposition of a field <math>\mathbf{u}</math> (''e.g.,'' a velocity field) is given by
<math display="block">u(x,y,z,t) = \overline{u(x,y,z)} + u'(x,y,z,t) </math>
<math display="block">\mathbf{u}(\mathbf{x},t) = \overline{\mathbf{u}(\mathbf{x},t)} + \mathbf{u}'(\mathbf{x},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>
where <math>\overline{\mathbf{u}}</math> denotes the mean of <math>\mathbf{u}</math> (which can be a time, space, or [[ensemble average]]), and <math>\mathbf{u}'</math> denotes the fluctuations from that mean.<ref>{{cite journal|last1=Alfonsi|first1=G.|title=Reynolds-Averaged Navier–Stokes Equations for Turbulence Modeling|journal=Applied Mechanics Reviews|date=July 2009|volume=62|issue=4|pages=040802|doi=10.1115/1.3124648}}</ref> The fluctuating field is defined as
<math display="block">\mathbf{u}'(\mathbf{x},t) \equiv  \mathbf{u}(\mathbf{x},t) - \overline{\mathbf{u}(\mathbf{x},t)} </math>
and satisfies<ref name="Pope"/><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>
<math display="block">\overline{\mathbf{u}'(\mathbf{x},t)} =0. </math>
Note that the mean field <math>\overline{\mathbf{u}}</math> is also frequently denoted as <math>\langle \mathbf{u}\rangle</math>.<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>


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>
==Application==
[[Direct numerical simulation]], or resolution of the [[Navier–Stokes equations]] (nearly) completely in both space and time, is only possible on extremely fine computational grids using small time steps even for low [[Reynolds number]]s. Running direct numerical simulations often 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>


[[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, to obtain the [[Reynolds-averaged Navier–Stokes equations]]. The resulting equation contains a nonlinear term known as the [[Reynolds stresses]], representing effects of turbulence.
 
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.


==See also==
==See also==

Latest revision as of 19:33, 20 August 2025

Template:Refimprove In fluid dynamics and turbulence theory, a Reynolds decomposition is a mathematical technique used to separate a field into its mean and fluctuating components.[1]

Decomposition

A Reynolds decomposition of a field 𝐮 (e.g., a velocity field) is given by 𝐮(𝐱,t)=𝐮(𝐱,t)‾+𝐮′(𝐱,t), where 𝐮‾ denotes the mean of 𝐮 (which can be a time, space, or ensemble average), and 𝐮′ denotes the fluctuations from that mean.[2] The fluctuating field is defined as 𝐮′(𝐱,t)≡𝐮(𝐱,t)−𝐮(𝐱,t)‾ and satisfies[1][3] 𝐮′(𝐱,t)‾=0. Note that the mean field 𝐮‾ is also frequently denoted as ⟨𝐮⟩.[4]

Application

Direct numerical simulation, or resolution of the Navier–Stokes equations (nearly) completely in both space and time, is only possible on extremely fine computational grids using small time steps even for low Reynolds numbers. Running direct numerical simulations often 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.[5]

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, to obtain the Reynolds-averaged Navier–Stokes equations. The resulting equation contains a nonlinear term known as the Reynolds stresses, representing effects of turbulence.

See also

References

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