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 Osbourne Reynolds (1895) and named by J. Kampé de Fériet (1934, 1935, 1949).