wall-normal direction, and $385$ and $321$ Fourier modes in the streamwise and spanwise directions. Each simulation has been run for 40,000 time steps with $\Delta t^+ = 0.2$. Throughout the paper, the + superscript indicates quantities scaled by the inner units of the unperturbed turbulent flow. A few calculations with very high $W_m$ have required an even smaller (up to one half) time step, due to stability constraints. A single case at $Re_\tau=200$ has taken 11 seconds for one time step and 5 days for the full simulation with 8 machines. The wall-clock time for the whole set of simulations has been slightly more than two weeks with the full use of the computational power.

For the $Re_\tau=200$ case, we have carried out one simulation for the canonical flow and 28 simulations for different $(T,W_m)$ pairs. The skin-friction coefficient for the fixed-wall case is $C_f = 2 \tau_x / (\rho U_b^2) = 7.94 \times 10^{-3}$, where $\tau_x$ is the time- and space-averaged streamwise wall-shear stress, $\rho$ is the fluid density and $U_b$ is the bulk velocity. This value essentially coincides with the value estimated by the following empirical formula given in @pope-2000 at page 279: $$\label{eq:cf-re-tau}
C_f = 0.0336 Re_\tau^{-0.273}.$$ 