= Solution
{c}
Write $u_i=U_i+u_i'$ and $p=P+p'$, with $\overline{u_i'}=\overline{p'}=0$. Assume a statistically stationary mean and sufficient regularity for <Reynolds averaging> to commute with spatial <derivatives>. Then
$$
\overline{\partial_tu_i}=\lim_{T\to\infty}\frac{u_i(T)-u_i(0)}T=0
$$
provided the endpoint difference grows sublinearly. Averaging the <continuity equation> gives
$$
\boxed{U_{i,i}=0,\qquad u_{i,i}'=0.}
$$
In <Einstein notation>, the nonlinear term expands as
$$
\begin{aligned}
\overline{u_ju_{i,j}}
&=\overline{(U_j+u_j')(U_{i,j}+u_{i,j}')}\\
&=U_jU_{i,j}+U_j\overline{u_{i,j}'}
+\overline{u_j'}U_{i,j}+\overline{u_j'u_{i,j}'}\\
&=U_jU_{i,j}+\overline{u_j'u_{i,j}'}.
\end{aligned}
$$
Thus the <Reynolds-averaged Navier-Stokes equation> is
$$
\boxed{\rho U_jU_{i,j}=-P_{,i}+\mu U_{i,jj}
-\rho\overline{u_j'u_{i,j}'}.}
$$
The extra term survives because the average of a product of zero-mean fluctuations need not vanish; it represents turbulent transport of mean <linear momentum>.
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