Reynolds operator

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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).

Definition[edit]

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

R(R(φ)ψ) = R(φ)R(ψ) for all φ, ψ

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

Invariant theory[edit]

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

R(R(φ)ψ) = R(φ)R(ψ) for all φ, ψ

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[edit]

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

R(φψ) = R(φ)R(ψ) + R((φR(φ))(ψR(ψ))) for all φ, ψ

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

R(R(φ)ψ) = R(φ)R(ψ) for all φ, ψ.

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[edit]

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 \frac{ \partial \phi }{ \partial t } \right\rangle = \frac{ \partial \langle \phi \rangle }{ \partial t }, \qquad 
\left\langle \frac{ \partial \phi }{ \partial x } \right\rangle = \frac{ \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[edit]

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

  • The "Reynolds operator" considered by Reynolds (1895) 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 SVG of elements fixed by G.

References[edit]

  1. ^ Sagaut, Pierre (2006). Large Eddy Simulation for Incompressible Flows (Third Edition ed.). Springer. ISBN 3-540-26344-6.