Reynolds operator

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In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, that satisfies 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 Template:Harvs and named by Template:Harvs.

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(φ), P(φ), ρ(φ), 〈φ〉, or Template:Overline. Reynolds operators are usually linear operators acting on some algebra of functions, satisfying the identity

R(R(φ)ψ) = R(φ)R(ψ) for all φ, ψ

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(φ)ψ) = R(φ)R(ψ) for all φ, ψ

and

R(1) = 1.

Together these conditions imply that R is idempotent: R2 = 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(φψ) = R(φ)R(ψ) + R((φR(φ))(ψR(ψ))) for all φ, ψ

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

R(R(φ)ψ) = R(φ)R(ψ) for all φ, ψ.

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 ${\displaystyle \phi }$ and ${\displaystyle \psi }$ be two random variables, and ${\displaystyle a}$ be an arbitrary constant. Then the properties satisfied by Reynolds operators, for an operator ${\displaystyle \langle \rangle ,}$ include linearity and the averaging property:

${\displaystyle \langle \phi +\psi \rangle =\langle \phi \rangle +\langle \psi \rangle ,\,}$
${\displaystyle \langle a\phi \rangle =a\langle \phi \rangle ,\,}$
${\displaystyle \langle \langle \phi \rangle \psi \rangle =\langle \phi \rangle \langle \psi \rangle ,\,}$ which implies ${\displaystyle \langle \langle \phi \rangle \rangle =\langle \phi \rangle .\,}$

In addition the Reynolds operator is often assumed to commute with space and time translations:

${\displaystyle \left\langle {\frac {\partial \phi }{\partial t}}\right\rangle ={\frac {\partial \langle \phi \rangle }{\partial t}},\qquad \left\langle {\frac {\partial \phi }{\partial x}}\right\rangle ={\frac {\partial \langle \phi \rangle }{\partial x}},}$
${\displaystyle \left\langle \int \phi ({\boldsymbol {x}},t)\,d{\boldsymbol {x}}\,dt\right\rangle =\int \langle \phi ({\boldsymbol {x}},t)\rangle \,d{\boldsymbol {x}}\,dt.}$

Any operator satisfying these properties is a Reynolds operator.[1]

Examples

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

• The "Reynolds operator" considered by Template:Harvtxt 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 SVG of elements fixed by G.

References

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|CitationClass=citation }} Reprints several of Rota's papers on Reynolds operators, with commentary.

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