Temporal waveforms of the considered wall oscillation. The following symbols are used throughout the paper for the different waveforms: (a) (○), (b) (♦), (c) (*), (d) (+), (e) (■), (f) (●), (g) (◀), (h) (▶), (i) (▲), and (j) (▼).
Power consumption rate P in (top), reduction of pumping power R (center), and net energy saving rate S as a function of for T + = 125 (left) and of T + for (right) for five selected waveforms, see Figure 1 for reference. The sinusoidal case is shown with the empty symbol. The continuous lines in the top plots are .
Maximum values of net energy saving, S, for each pair of (T, W m )-values investigated. Full circles correspond to S >, open circles to S < 0 while the diameter of the circles reflects the obtained absolute value of S. The letter above each circle indicates the waveform yielding the best S.
(Left) Power consumption data P in for all the simulations versus the analytical prediction based on the Stokes layer assumption, Eq. (5) . The inset plot shows the percentage error. (Right) Turbulent drag reduction R as a function of (ℓ3/2 T −1/2)+, with ℓ computed from Eq. (7) using a threshold corresponding to W th = 1.2. Gray symbols are for T + = 62.5, black symbols for T + = 125, and open symbols for T + = 250.
(Left) Turbulent drag reduction rate, R, as a function of for T + = 125 (black symbols) and T + = 62.5 (gray symbols). The solid lines are the prediction (8) , with ℓ computed via the first mode, Eq. (10) . (Right) Bi-harmonic waveform (solid line) designed to increase the sinusoidal net energy savings at low W m values.
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