Invariant theory

In invariant theory a Reynolds operator R is usually a linear operator satisfying

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

and

${\displaystyle R(1)=1}$

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

${\textstyle R(\phi \psi )=R(\phi )R(\psi )+R\left(\left(\phi -R(\phi )\right)\left(\psi -R(\psi )\right)\right)\quad {\text{ for all }}\phi ,\psi }$

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

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

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 ${\displaystyle \phi }$ 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]