Reynolds operator

In fluid dynamics and 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, particularly the Reynolds-averaged Navier–Stokes 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 or reductive algebraic group acting on a commutative algebra, such as a ring of polynomials. Reynolds operators were introduced into fluid dynamics by and named by .

Definition

Reynolds operators are used in fluid dynamics, functional analysis, and invariant theory, and the notation and definitions in these areas differ slightly. A Reynolds operator acting on φ is sometimes denoted by $R(\phi),P(\phi),\rho(\phi),\langle \phi \rangle$ or $\overline$.
Reynolds operators are usually linear operators acting on some algebra of functions, satisfying the identity

: $R(R(\phi)\psi) = R(\phi)R(\psi) \quad \text \phi,\psi$

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

: $R(R(\phi)\psi) = R(\phi)R(\psi) \quad \text \phi,\psi$

and

:$R(1) = 1$

Together these conditions imply that ''R'' is idempotent: ''R''² = ''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
: $R(\phi\psi) = R(\phi)R(\psi) + R\left(\left(\phi-R(\phi)\right)\left(\psi-R(\psi)\right) \right)\quad \text \phi,\psi$
:

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

: $R(R(\phi)\psi) = R(\phi)R(\psi) \quad \text \phi,\psi$

If ''R''(''R''(''φ'')) = ''R''(''φ'') for all φ then ''R'' is an averaging operator 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 $\phi$ and $\psi$ be two random variables, and $a$ be an arbitrary constant. Then the properties satisfied by Reynolds operators, for an operator $\langle \rangle,$ include linearity and the averaging property:

:$\langle \phi + \psi \rangle = \langle \phi \rangle + \langle \psi \rangle, \,$

:$\langle a \phi \rangle = a \langle \phi \rangle, \,$

:$\langle \langle \phi \rangle \psi \rangle = \langle \phi \rangle \langle \psi \rangle, \,$ which implies $\langle \langle \phi \rangle \rangle = \langle \phi \rangle. \,$
In addition the Reynolds operator is often assumed to commute with space and time translations:
:$\left\langle \frac \right\rangle = \frac, \qquad \left\langle \frac \right\rangle = \frac,$

:$\left\langle \int \phi( \boldsymbol, t ) \, d \boldsymbol \, dt \right\rangle = \int \langle \phi(\boldsymbol,t) \rangle \, d \boldsymbol \, dt.$

Any operator satisfying these properties is a Reynolds operator.

Examples

Reynolds operators are often given by projecting onto an invariant subspace of a group action.

*The "Reynolds operator" considered by 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 or a compact group, and ''V'' is a finite-dimensional representation of ''G''. Then ''G'' also acts on the symmetric algebra ''SV'' of polynomials. The Reynolds operator ''R'' is the ''G''-invariant projection from ''SV'' to the subring ''SV''^''G'' of elements fixed by ''G''.

References

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* Reprints several of Rota's papers on Reynolds operators, with commentary.
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*{{Citation , last1=Sturmfels , first1=Bernd , author1-link=Bernd Sturmfels , title=Algorithms in invariant theory , publisher=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

Invariant theory
Fluid dynamics
Turbulence