Reynolds decomposition

A Reynolds decomposition of a field ${\displaystyle \mathbf {u} }$ (e.g., a velocity field) is given by $${\displaystyle \mathbf {u} (\mathbf {x} ,t)={\overline {\mathbf {u} (\mathbf {x} ,t)}}+\mathbf {u} '(\mathbf {x} ,t),}$$ where ${\displaystyle {\overline {\mathbf {u} }}}$ denotes the mean of ${\displaystyle \mathbf {u} }$ (which can be a time, space, or ensemble average), and ${\displaystyle \mathbf {u} '}$ denotes the fluctuations from that mean.^[2] The fluctuating field is defined as $${\displaystyle \mathbf {u} '(\mathbf {x} ,t)\equiv \mathbf {u} (\mathbf {x} ,t)-{\overline {\mathbf {u} (\mathbf {x} ,t)}}}$$ and satisfies^[1][3] $${\displaystyle {\overline {\mathbf {u} '(\mathbf {x} ,t)}}=0.}$$ Note that the mean field ${\displaystyle {\overline {\mathbf {u} }}}$ is also frequently denoted as ${\displaystyle \langle \mathbf {u} \rangle }$.^[4]