Reynolds operator explained

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 and named by .

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(\phi),P(\phi),\rho(\phi),\langle \phi \rangle or \overline{\phi} . Reynolds operators are usually linear operators acting on some algebra of functions, satisfying the identityR(R(\phi)\psi) = R(\phi)R(\psi) forall \phi,\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(\phi)\psi) = R(\phi)R(\psi) forall \phi,\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

$R(\backslash phi\backslash psi)\; =\; R(\backslash phi)R(\backslash psi)\; +\; R\backslash left(\backslash left(\backslash phi-R(\backslash phi)\backslash right)\backslash left(\backslash psi-R(\backslash psi)\backslash right)\; \backslash right)\backslash quad\; \backslash text\; \backslash phi,\backslash psi$

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

R(R(\phi)\psi) = R(\phi)R(\psi) forall \phi,\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

\phi and \psi be two random variables, and a be an arbitrary constant. Then the properties satisfied by Reynolds operators, for an operator \langle \rangle, include linearity and the averaging property:\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, which implies \langle \langle \phi \rangle \rangle = \langle \phi \rangle. In addition the Reynolds operator is often assumed to commute with space and time translations:\left\langle (\partial \phi)/(\partial t) \right\rangle = (\partial \langle \phi \rangle)/(\partial t) , \left\langle (\partial \phi)/(\partial x) \right\rangle = (\partial \langle \phi \rangle)/(\partial x) , \left\langle \int \phi( \boldsymbol{x}, t ) d \boldsymbol{x} dt \right\rangle = \int \langle \phi(\boldsymbol{x},t) \rangle d \boldsymbol{x} dt.

Any operator satisfying these properties is a Reynolds operator.^[1]

Examples

Reynolds operators are often given by projecting onto an invariant subspace of a group action.

• The "Reynolds operator" considered by 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

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

Notes and References

1. Book: Sagaut, Pierre. Large Eddy Simulation for Incompressible Flows. Springer. 2006. Third. 3-540-26344-6 .