Reynolds operator

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 \phi is sometimes denoted by R(\phi),P(\phi),\rho(\phi),\langle \phi \rangle or \overline{\phi}. 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{ for all } \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{ for all } \phi,\psi

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(\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 :

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

: R(R(\phi)\psi) = R(\phi)R(\psi) \quad \text{ for all } \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{ \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 },

: \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.{{cite book

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

• Reprints several of Rota's papers on Reynolds operators, with commentary.