User:Charlesreid1/Reynolds operator

A Reynolds operator is a mathematical operator that satisfies a set of properties called Reynolds rules. Reynolds operators are often encountered in models of turbulent flows, particularly the Reynolds-averaged Navier-Stokes equations. Reynolds operators are named after Osbourne Reynolds.

Properties
Let $$\phi$$ and $$\psi$$ be two random variables, and $$a$$ be an arbitrary constant. Then the properties satisfied by by Reynolds operators, for an operator $$\langle \rangle$$, are:

$$ \langle \phi + \psi \rangle = \langle \phi \rangle + \langle \psi \rangle $$

$$ \langle a \phi \rangle = a \langle \phi \rangle $$

$$ \langle \langle \phi \rangle \psi \rangle = \langle \phi \rangle \langle \psi \rangle $$

$$ \langle \frac{ \partial \phi }{ \partial s } \rangle = \frac{ \partial \langle \phi \rangle }{ \partial s } \qquad s = \boldsymbol{x}, t $$

$$ \displaystyle{ \langle \int \phi( \boldsymbol{x}, t ) d \boldsymbol{x} dt \rangle } = \int \langle \phi(\boldsymbol{x},t) \rangle d \boldsymbol{x} dt $$

Any operator satisfying these properties is a Reynolds operator.

Reynolds operators also satisfy several additional relations, which are consequences of the above properties.

One of these relations is

$$ \langle \langle \phi \rangle \rangle = \langle \phi \rangle. $$

An operator satisfying this property is called a projector. Let $$P$$ denote the operator $$\langle \rangle$$. Then the above property can be written

$$ P \circ P \phi = P \phi , $$

which can be generalized to

$$ P \circ P \circ \dots \circ P \phi = P^{n} \phi = P \phi. $$

Additionally, for a quantity $$\phi^{\prime}$$ defined as

$$ \phi^{\prime} = \phi - \langle \phi \rangle, $$

Reynolds operators also satisfy the property

$$ \langle \phi^{\prime} \rangle = 0. $$