Reynolds stresses

In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier-Stokes equations to account for turbulent fluctuations in fluid momentum.

The velocity field of a flow can be split into a mean part and a fluctuating part using Reynolds decomposition. We write

with ${\displaystyle \mathbf {u} (\mathbf {x} ,t)}$ being the flow velocity vector having components ${\displaystyle u_{i}}$ in the ${\displaystyle x_{i}}$ coordinate direction (with ${\displaystyle x_{i}}$ denoting the components of the coordinate vector ${\displaystyle \mathbf {x} }$). The mean velocities ${\displaystyle {\overline {u_{i}}}}$ are determined by either time averaging, spatial averaging or ensemble averaging, depending on the flow under study. Further ${\displaystyle u'_{i}}$ denotes the fluctuating (turbulence) part of the velocity.

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