Abstract
The dependence of turbulence statistics in wall-bounded flows on the friction Reynolds number $R\phantom{\rule{0}{0ex}}{e}_{\ensuremath{\tau}}$ is complex. Luchini proposed an empirical law for the mean velocity ${U}^{+}$ which has a dependence proportional to $1/R\phantom{\rule{0}{0ex}}{e}_{\ensuremath{\tau}}$ at fixed ${y}^{+}$, and agrees well with direct numerical simulation for channel (Poiseuille), Couette, and pipe flow. We test similar laws for the Reynolds stresses and their budgets. The figure shows the wall-normal Reynolds stress $\overline{{v}^{+}\phantom{\rule{0}{0ex}}{v}^{+}}$ in channel. The two lower curves are results at two Reynolds numbers, and the upper one the extrapolation to infinite $R\phantom{\rule{0}{0ex}}{e}_{\ensuremath{\tau}}$. The symbols show an experiment in pipe at higher $R\phantom{\rule{0}{0ex}}{e}_{\ensuremath{\tau}}$.
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