# Reynolds operator

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In [fluid dynamics](/source/fluid_dynamics) and [invariant theory](/source/invariant_theory), a '''Reynolds operator''' is a mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules.  In fluid dynamics, Reynolds operators are often encountered in models of [turbulent flows](/source/turbulence), particularly the [Reynolds-averaged Navier–Stokes equations](/source/Reynolds-averaged_Navier%E2%80%93Stokes_equations), where the average is typically taken over the fluid flow under the group of time translations. In invariant theory, the average is often taken over a [compact group](/source/compact_group) or [reductive algebraic group](/source/Reductive_group) acting on a commutative algebra, such as a ring of polynomials.  Reynolds operators were introduced into fluid dynamics by {{harvs|txt|authorlink=Osbourne Reynolds|first=Osbourne|last= Reynolds|year=1895}} and named by {{harvs|txt|authorlink=Kampé de Fériet|last= Kampé de Fériet|first= J. |year1=1934|year2=1935|year3=1949}}.

==Definition==
Reynolds operators are used in fluid dynamics, [functional analysis](/source/functional_analysis), and invariant theory, and the notation and definitions in these areas differ slightly.  A Reynolds operator acting on <math>\phi</math> is sometimes denoted by <math>R(\phi),P(\phi),\rho(\phi),\langle \phi \rangle</math> or <math>\overline{\phi}</math>. 
Reynolds operators are usually linear operators acting on some algebra of functions, satisfying the identity

: <math>R(R(\phi)\psi) = R(\phi)R(\psi) \quad \text{ for all } \phi,\psi</math>

and sometimes some other conditions, such as commuting with various group actions.

===Invariant theory===
In invariant theory a Reynolds operator ''R'' is usually a linear operator satisfying

: <math>R(R(\phi)\psi) = R(\phi)R(\psi) \quad \text{ for all } \phi,\psi</math>

and

:<math>R(1) = 1</math>

Together these conditions imply that ''R'' is [idempotent](/source/Idempotence): ''R''<sup>2</sup> = ''R''. The Reynolds operator will also usually commute with some group action, and project onto the invariant elements of this group action.

===Functional analysis===
In functional analysis a Reynolds operator is a linear operator ''R'' acting on some algebra of functions ''φ'', satisfying the '''Reynolds identity'''
: <math display="inline">R(\phi\psi) = R(\phi)R(\psi) + R\left(\left(\phi-R(\phi)\right)\left(\psi-R(\psi)\right) \right)\quad \text{ for all } \phi,\psi</math>
:

The operator ''R'' is called an '''averaging operator''' if it is linear and satisfies

: <math>R(R(\phi)\psi) = R(\phi)R(\psi) \quad \text{ for all } \phi,\psi</math>

If ''R''(''R''(''φ'')) = ''R''(''φ'') for all φ then ''R'' is an averaging operator [if and only if](/source/if_and_only_if) it is a Reynolds operator. Sometimes the ''R''(''R''(''φ'')) = ''R''(''φ'') condition is added to the definition of Reynolds operators.

=== Fluid dynamics ===
Let <math>\phi</math> and <math>\psi</math> be two random variables, and <math>a</math> be an arbitrary constant.  Then the properties satisfied by Reynolds operators, for an operator <math>\langle \rangle,</math> include linearity and the averaging property:

:<math>
\langle \phi + \psi \rangle = \langle \phi \rangle + \langle \psi \rangle, \,
</math>

:<math>
\langle a \phi \rangle = a \langle \phi \rangle, \,
</math>

:<math>
\langle \langle \phi \rangle \psi \rangle = \langle \phi \rangle \langle \psi \rangle, \,
</math> which implies <math>
\langle \langle \phi \rangle \rangle = \langle \phi \rangle. \,
</math>
In addition the Reynolds operator is often assumed to commute with space and time translations:
:<math>
\left\langle \frac{ \partial \phi }{ \partial t } \right\rangle = \frac{ \partial \langle \phi \rangle }{ \partial t }, \qquad 
\left\langle \frac{ \partial \phi }{ \partial x } \right\rangle = \frac{ \partial \langle \phi \rangle }{ \partial x },
</math>

:<math>
\left\langle \int \phi( \boldsymbol{x}, t ) \, d \boldsymbol{x} \, dt \right\rangle = \int \langle \phi(\boldsymbol{x},t) \rangle \, d \boldsymbol{x} \, dt.
</math>

Any operator satisfying these properties is a Reynolds operator.<ref name="Sagaut_2006">{{cite book
|author=Sagaut, Pierre
|title=Large Eddy Simulation for Incompressible Flows
|publisher=Springer
|year=2006
|edition=Third
|isbn=3-540-26344-6 }}</ref>

==Examples==
Reynolds operators  are often given by projecting onto an [invariant subspace](/source/invariant_subspace) of a group action.

*The "Reynolds operator" considered by {{harvtxt|Reynolds|1895}} was essentially the projection of a fluid flow to the "average" fluid flow, which can be thought of as projection to time-invariant flows. Here the group action is given by the action of the group of time-translations.
*Suppose that ''G'' is a [reductive algebraic group](/source/reductive_algebraic_group) or a [compact](/source/compact_space) group, and ''V'' is a finite-dimensional representation of ''G''. Then ''G'' also acts on the [symmetric algebra](/source/symmetric_algebra) ''SV'' of polynomials.  The Reynolds operator ''R'' is the ''G''-invariant projection from ''SV'' to the [subring](/source/subring) ''SV''<sup>''G''</sup> of elements fixed by ''G''.

== References ==
{{reflist}}
*{{Citation | last1=Kampé de Fériet | first1=J. |journal=La Science Aérienne |volume=3|pages=9–34 |year=1934 | title=L'état actuel du problème de la turbulence I}}
*{{Citation | last1=Kampé de Fériet | first1=J. |journal=La Science Aérienne |volume=4|pages=12–52 |year=1935 | title=L'état actuel du problème de la turbulence II}}
*{{Citation | last1=Kampé de Fériet | first1=J. | title=Sur un problème d'algèbre abstraite posé par la définition de la moyenne dans la théorie de la turbulence | mr=0032718 | year=1949 | journal=Annales de la Société Scientifique de Bruxelles. Série I. Sciences Mathématiques, Astronomiques et Physiques | issn=0037-959X | volume=63 | pages=165–180}}
*{{citation|last=Reynolds|first=O. |title=On the dynamical theory of incompressible viscous fluids and the determination of the criterion|journal=[Philosophical Transactions of the Royal Society A](/source/Philosophical_Transactions_of_the_Royal_Society_A) |volume=186|year=1895|pages=123–164|jstor=90643|doi=10.1098/rsta.1895.0004|bibcode = 1895RSPTA.186..123R |url=https://zenodo.org/record/1432102|doi-access=free}}
*{{Citation | last1=Rota | first1=Gian-Carlo | author1-link=Gian-Carlo Rota | title=Gian-Carlo Rota on analysis and probability | publisher=Birkhäuser Boston | location=Boston, MA | series=Contemporary Mathematicians | isbn=978-0-8176-4275-4 | mr=1944526 | year=2003}} Reprints several of Rota's papers on Reynolds operators, with commentary.
*{{Citation | last1=Rota | first1=Gian-Carlo | author1-link=Gian-Carlo Rota | title=Proc. Sympos. Appl. Math. |volume=XVI | publisher=Amer. Math. Soc. | location=Providence, R.I. | mr=0161140 | year=1964 | chapter=Reynolds operators | pages=70–83}}
*{{Citation | last1=Sturmfels | first1=Bernd | author1-link=Bernd Sturmfels | title=Algorithms in invariant theory | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | series=Texts and Monographs in Symbolic Computation | isbn=978-3-211-82445-0 | mr=1255980 | year=1993| doi=10.1007/978-3-7091-4368-1 }}

Category:Invariant theory
Category:Fluid dynamics
Category:Turbulence

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