Decomposition of drag-reducing turbulent events into scale and quadrant contributions using Fukagata-Iwamoto-Kasagi (FIK) identity for a pipe flow with spanwise-oscillated walls

Increased turbulent drag in wall bounded shear flows is associated with an increased velocity gradient in the near wall region resulting from the turbulent momentum transfer. The Fukagata-Iwamoto-Kasagi (FIK) identity allows one to relate the surface drag, or the bulk mean velocity, depending on the problem formulation, with the statistics of the fluid stresses in the interior of the fluid domain. The FIK identity illustrates that, as a function of the distance from the wall, the turbulent Reynolds shear stress contributes to the reduction of the bulk mean velocity or increase in the skin friction from the equivalent laminar flow for a given Reynolds number. The streamwise-wall normal Reynolds shear stress is the product of streamwise and wall normal velocity fluctuations, which can be decomposed into four quadrants based on the sign of the fluctuating pair: outward interactions (Q1), turbulent ejections (Q2), wall-ward interactions (Q3), and turbulent sweeps (Q4). Turbulent ejections and sweeps increase the Reynolds shear stress and thus the near wall velocity gradient. However, where these events occur and what length scales affects the contribution to the mean velocity has not been previously identified. We present the FIK identity recast using the conditionally averaged and filtered streamwise momentum equation to isolate the wall location and length scale at which spanwise wall oscillations in turbulent pipe flow reduce turbulence to affect drag reduction.