title: Reynolds operator
text: # 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).
  
  ## 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 identity
  
      $R(R(\phi )\psi )=R(\phi )R(\psi )\quad {\text{ for all }}\phi ,\psi$
  
  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
  
      $R(R(\phi )\psi )=R(\phi )R(\psi )\quad {\text{ for all }}\phi ,\psi$
  
  and
  
      $R(1)=1$
  
  Together these conditions imply that R is idempotent: $R^2$ = 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(ϕψ)=R(ϕ)R(ψ)+R((ϕ−R(ϕ))(ψ−R(ψ))) for all ϕ, ψ{\textstyle R(\phi \psi )=R(\phi )R(\psi )+R\left(\left(\phi -R(\phi )\right)\left(\psi -R(\psi )\right)\right)\quad {\text{ for all }}\phi , \psi }
  
  The operator R is called an averaging operator if it is linear and satisfies
  
      $R(R(\phi )\psi )=R(\phi )R(\psi )\quad {\text{ for all }}\phi ,\psi$
  
  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:
  
      $\left\langle {\frac {\partial \phi }{\partial t}}\right\rangle ={\frac {\partial \langle \phi \rangle }{\partial t}},\qquad \left\langle {\frac {\partial \phi }{\partial x}}\right\rangle ={\frac {\partial \langle \phi \rangle }{\partial x}},$
      $\left\langle \int \phi ({\boldsymbol {x}},t)\,d{\boldsymbol {x}}\,dt\right\rangle =\int \langle \phi ({\boldsymbol {x}},t)\rangle \,d{\boldsymbol {x}}\,dt.$
  
  Any operator satisfying these properties is a Reynolds operator.
  
  ## Examples
  
  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 $SV^G$ of elements fixed by G.
  