# 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 Osbourne Reynolds (1895) and named by J. Kampé de Fériet (1934, 1935, 1949).

## 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 ${\overline {\phi }}$. 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: 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{\textstyle 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.

## Examples

Reynolds operators are often given by projecting onto an invariant subspace of a group action.
* The "Reynolds operator" considered by 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 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.