The dynamic mechanism for turbulent drag reduction using rigid fibers based on Lagrangian conditional statistics

Abstract

The orientation moments and stresses for a suspension of rigid fibers are calculated along Lagrangian pathlines in a drag-reduced turbulent channel flow. The turbulent flow fields are calculated using the methods developed by Paschkewitz et al. [“Numerical simulation of turbulent drag reduction using rigid fibers,” J. Fluid Mech. 518, 281 (2004)]. These authors investigated turbulent drag reduction using rigid fibers with the Eulerian frame of reference direct numerical simulations, and demonstrated a correlation of drag reduction with fluctuations or “bursts” of fiber stress in intervortex extensional flow regions. These bursts are defined as events where certain components of the fiber stress exceed a specified level. Using probability density functions (PDFs) of the fiber stress contribution to the dissipation of turbulent kinetic energy weighted by the probability of occurrence of a particular stress level, we demonstrate that less than 0.001% of the dissipation is created by stress fluctuations of magnitude ∣τ13∣>5 and ∣τ22∣>7. We therefore define a “small” burst or fluctuation as one of this magnitude or less and a “large” burst to be one having a magnitude greater than these thresholds. Using conditional statistics in the Lagrangian frame, the detailed dynamics responsible for the generation of these stress bursts are quantitatively characterized. As a precursor to the burst, the fibers move into extensional flow regions and align with the wall-normal or spanwise directional axis in a process that takes approximately two local strain units, where the strain rate is defined along the direction of the fiber orientation. After the stress increases past a chosen large value and the stress burst begins, the fibers continue to align and generate increasing stress until the burst is roughly half complete; at this time, the extensional character of the surrounding flow is reduced to a level such that the fibers begin to realign in the flow direction and the fiber stresses decrease. These stress bursts also have a total duration of approximately two local strain units. By considering the flow kinematics in regions near the particle, as well as the time autocorrelation of fiber stress and the second flow invariant Q, we demonstrate that the bursts end because the surrounding vortices are strongly weakened or destroyed by the gradients in fiber stresses. Both frequently observed small stress fluctuations and rare large stress fluctuations exhibit this behavior, with the primary difference being the strength of the nearby vortices and the resulting extensional flow region that the fiber resides in during the burst. However, the more commonly observed small stress fluctuations appear to make the largest contribution to fiber dissipation of turbulent kinetic energy and thus are responsible for the majority of the drag reduction effect. The largest contribution to fiber dissipation of turbulent kinetic energy is made by small fluctuations in the spanwise shear stress component τ13, which result from fibers primarily confined to the x-z plane weakly rotating towards the z axis.