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 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 $R(\varphi ),P(\varphi ),\rho (\varphi ),⟨\varphi ⟩$ or $\bar{\varphi }$. Reynolds operators are usually linear operators acting on some algebra of functions, satisfying the identity

$R(R(\varphi )\psi )=R(\varphi )R(\psi )\text{ for all }\varphi ,\psi$

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

$R(R(\varphi )\psi )=R(\varphi )R(\psi )\text{ for all }\varphi ,\psi$

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

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

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

$R(R(\varphi )\psi )=R(\varphi )R(\psi )\text{ for all }\varphi ,\psi$

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 $\varphi$ and $\psi$ be two random variables, and $a$ be an arbitrary constant. Then the properties satisfied by Reynolds operators, for an operator $⟨⟩,$ include linearity and the averaging property:

$⟨\varphi +\psi ⟩=⟨\varphi ⟩+⟨\psi ⟩,$

$⟨a\varphi ⟩=a⟨\varphi ⟩,$

$⟨⟨\varphi ⟩\psi ⟩=⟨\varphi ⟩⟨\psi ⟩,$ which implies $⟨⟨\varphi ⟩⟩=⟨\varphi ⟩.$

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

$⟨\frac{\partial \varphi }{\partial t}⟩=\frac{\partial ⟨\varphi ⟩}{\partial t},⟨\frac{\partial \varphi }{\partial x}⟩=\frac{\partial ⟨\varphi ⟩}{\partial x},$

$⟨\int \varphi (𝒙,t)d𝒙dt⟩=\int ⟨\varphi (𝒙,t)⟩d𝒙dt.$

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