Reynolds stress

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. 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 u ( x , t ) {displaystyle mathbf {u} (mathbf {x} ,t)} being the flow velocity vector having components u i {displaystyle u_{i}} in the x i {displaystyle x_{i}} coordinate direction (with x i {displaystyle x_{i}} denoting the components of the coordinate vector x {displaystyle mathbf {x} } ). The mean velocities u i ¯ {displaystyle {overline {u_{i}}}} are determined by either time averaging, spatial averaging or ensemble averaging, depending on the flow under study. Further u i ′ {displaystyle u'_{i}} denotes the fluctuating (turbulence) part of the velocity. We consider a homogeneous fluid, whose density ρ is taken to be a constant. For such a fluid, the components τ'ij of the Reynolds stress tensor are defined as: