Theoretical framework for energy flux analysis of channels under drag control

A framework for analyzing energy flux in turbulent channel flows is proposed which enables quantification of the drag reduction efficacy by different control methods. In contrast to the FIK [Fukagata, Iwamoto, and Kasagi, Phys. Fluids 14, L73 (2002)PHFLE61070-663110.1063/1.1516779] and the RD [Renard and Deck, J. Fluid Mech. 790, 339 (2016)JFLSA70022-112010.1017/jfm.2016.12] identities, this framework expresses the skin friction coefficient in terms of the nondimensionalized dissipation rate and the work done by external excitation. We extend the energy-box analysis of Gatti et al. [J. Fluid Mech. 857, 345 (2018)JFLSA70022-112010.1017/jfm.2018.749] through a triple decomposition of energy flux and show how mean (ϵM), coherent (ϵC), and random turbulent (ϵR) dissipations contribute differently to the drag reduction and the net power saving. Three control methods, including our recently developed spanwise opposed wall-jet forcing (SOJF), the spanwise wall oscillation (SWO), and the opposed wall blowing/suction (OBS) controls, are compared at Reτ≈200 via direct numerical simulations (DNS). While all methods yield comparable drag reductions (∼20%), OBS yields the maximum net power saving, followed by SOJF, and then SWO. Specifically, for SOJF control, ϵC (induced by the large-scale swirls) is much smaller than ϵR (induced mainly by the small-scale near-wall vortices) and ϵM (due to the spanwise vorticity sheet ωz). In contrast, for SWO control, ϵC (caused by the wall oscillation-induced ωx vortex sheet) - much larger than that of SOJF - is comparable to ϵR and ϵM. For OBS control, ϵR is notably suppressed without any introduction of ϵC as the energy is injected through the random velocity field. Diagnoses performed at a higher Reτ (i.e., 2000) for SWO shows that random turbulent dissipation ϵR predominates due to the increasing near-wall vortical structures - hence their suppression should be the target for drag control at high Reτ. The analysis also suggests a promising hybrid drag control strategy by incorporating both the random (OBS) and coherent (SOJF or SWO) controls together, an issue for future exploration.