= Reynolds-averaged momentum equation
{c}
{title2=$\partial_tU_i+U_j\partial_jU_i=-\rho^{-1}\partial_i\bar p+\nu\Delta U_i-\partial_j\overline{u_i'u_j'}$}
= Reynolds-averaged Navier-Stokes equation
{c}
{synonym}
For an <incompressible flow> of constant <mass density> $\rho$ and <kinematic viscosity> $\nu$, <Reynolds averaging> of the <Navier-Stokes equation>, without a <body force>, gives
$$
\partial_tU_i+U_j\partial_jU_i=-\frac1\rho\partial_i\overline p+\nu\Delta U_i-\partial_j\overline{u_i'u_j'},\qquad\partial_iU_i=0.
$$
Here $\mathbf u=\mathbf U+\mathbf u'$ and $\overline{\mathbf u'}=0$, with averaging assumed to commute with the displayed <derivatives>. The mean acceleration $\partial_tU_i$ vanishes only when the averaged <velocity field> is stationary; taking an average does not by itself remove it.
To derive the extra term, average the conservative <momentum flux>: $\overline{u_iu_j}=U_iU_j+\overline{u_i'u_j'}$. Mean <incompressibility> turns the divergence of the first product into $U_j\partial_jU_i$. The second product is the <Reynolds stress> divided by <mass density>, using the positive-covariance convention. It describes transport of mean <momentum> by fluctuating <velocities>. A <turbulence closure> is needed because that covariance is an additional unknown, rather than a specified molecular transport coefficient.
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