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 are used in fluid dynamics, functional analysis, and invariant theory, and the notation and definitions in these areas differ slightly.
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.
[1] Reynolds operators are often given by projecting onto an invariant subspace of a group action.