# Reynolds operator

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: *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*(*φψ*) =*R*(*φ*)*R*(*ψ*) +*R*((*φ*−*R*(*φ*))(*ψ*−*R*(*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 and be two random variables, and be an arbitrary constant. Then the properties satisfied by Reynolds operators, for an operator include linearity and the averaging property:

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

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*SV*^{G}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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