Reynolds number e?ect on drag control via spanwise wall oscillation in turbulent channel ?ows.

The e?ect of Reynolds number (Re$_{\mathrm{\tau }})$ on drag reduction (DR) by spanwise wall oscillation is studied through direct numerical simulation of incompressible turbulent channel ?ows with Re$_{\mathrm{\tau \thinspace }}$ranging from 200 to 2000. For the non-dimensional oscillation period T$^{\mathrm{+}} \quad =$ 100 with maximum velocity amplitude A$^{\mathrm{+}}$ $=$ 12, DR decreases from 35.3{\%} at Re$_{\mathrm{\tau }}=$200 to 22.3{\%} at Re$_{\mathrm{\tau }}=$2000. The oscillation frequency $\omega ^{\mathrm{+}}$ for maximum DR slightly increases with Re$_{\mathrm{\tau }}$, viz., from $\omega^{\mathrm{+}}=$0.06 at Re$_{\mathrm{\tau }}=$200 to 0.08 at Re$_{\mathrm{\tau }}=$ 2000, with DR$_{\mathrm{max}}$ $=$ 23.2{\%}. These results show that DR progressively decreases with increasing Re$_{\mathrm{\tau }}$. Turbulent statistics and coherent structures are examined to explain the degradation of drag control e?ectiveness at high Re$_{\mathrm{\tau }}$. FIK analysis in combination with the spanwise wavenumber spectrum of Reynolds stresses reveals that the decreased DR at higher Re$_{\mathrm{\tau \thinspace }}$is due to the weakened e?ectiveness in suppressing the near wall large-scale turbulence, whose contribution continuously increases due to the enhanced modulation and penetration e?ect of the large-scale and very large-scale motions. Based on the power-law model and the log-law model, we predict more than 10{\%} drag reduction at very high Reynolds numbers, say, Re$_{\mathrm{\tau }}=$ 10$^{\mathrm{5}}$..