= Reynolds-averaged momentum equation
{c}
{title2=$U_j\partial_jU_i=-\rho^{-1}\partial_i\bar p+\nu\Delta U_i-\partial_j\overline{u_i'u_j'}$}
= Reynolds-averaged Navier-Stokes equation
{c}
{synonym}
An <Eulerian time average> of an incompressible constant-density <Navier-Stokes equation> gives the displayed mean momentum equation. Decompose velocity into its mean and a zero-mean fluctuation. In conservative form, the averaged momentum flux is $\overline{u_iu_j}=U_iU_j+\overline{u_i'u_j'}$; mean <incompressibility> then turns its first term into mean advection. The fluctuation covariance is the <Reynolds stress> divided by density, using the positive covariance convention. Closing this extra unknown requires a turbulence model.
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