Reynolds operator

In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, that satisfies 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).

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(φ), P(φ), ρ(φ), 〈φ〉, or φ. Reynolds operators are usually linear operators acting on some algebra of functions, satisfying the identity

and sometimes some other conditions, such as commuting with various group actions.

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

and

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.

In functional analysis a Reynolds operator is a linear operator R acting on some algebra of functions φ, satisfying the Reynolds identity

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

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.

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:

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