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. Ft lids 14, 1,73 (2002)] and the RD [Renard and Deck, J. Fluid Mech. 790, 339 (2016)] 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)] through a triple decomposition of energy flux and show how mean (epsilon(M)), coherent (epsilon(C)), and random turbulent (epsilon(R)) 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 Re-tau approximate to 200 via direct numerical simulations (DNS). While all methods yield comparable drag reductions (similar to 20%), OBS yields the maximum net power saving, followed by SOJF, and then SWO. Specifically, for SOJF control, epsilon(C) (induced by the large-scale swirls) is much smaller than epsilon(R) (induced mainly by the small-scale near-wall vortices) and epsilon(M) (due to the spanwise vorticity sheet Omega(z)). In contrast, for SWO control, epsilon(C) (caused by the wall oscillation-induced Omega(x) vortex sheet)-much larger than that of SOJF-is comparable to epsilon(R) and epsilon M. For OBS control, epsilon(R) is notably suppressed without any introduction of epsilon(C) as the energy is injected through the random velocity field. Diagnoses performed at a higher Re-tau (i.e., 2000) for SWO shows that random turbulent dissipation epsilon(R) predominates due to the increasing near-wall vortical structures-hence their suppression should be the target for drag control at high Re-tau. 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.

Automated summary

The proposed framework allows for quantification of the drag reduction efficacy by different control methods in turbulent channel flows, with a focus on the contributions of mean, coherent, and random turbulent dissipations. The study compares three control methods at Re_tau around 200 and suggests a hybrid drag control strategy incorporating both random and coherent controls may be promising for future exploration, especially at higher Reynolds numbers.