Self-sustaining process of minimal attached eddies in turbulent channel flow

Abstract

It has been recently shown that the energy-containing motions (i.e. coherent structures) in turbulent channel flow exist in the form of Townsend’s attached eddies by a numerical experiment which simulates the energy-containing motions only at a prescribed spanwise length scale using their self-sustaining nature (Hwang, J. Fluid Mech., vol. 767, 2015, pp. 254–289). In the present study, a detailed investigation of the self-sustaining process of the energy-containing motions at each spanwise length scale (i.e. the attached eddies) in the logarithmic and outer regions is carried out with an emphasis on its relevance to ‘bursting’, which refers to an energetic temporal oscillation of the motions (Flores & Jiménez, Phys. Fluids, vol. 22, 2010, 071704). It is shown that the attached eddies in the logarithmic and outer regions, composed of streaks and quasi-streamwise vortical structures, bear the self-sustaining process remarkably similar to that in the near-wall region: i.e. the streaks are significantly amplified by the quasi-streamwise vortices via the lift-up effect; the amplified streaks subsequently undergo a ‘rapid streamwise meandering motion’, reminiscent of streak instability or transient growth, which eventually results in breakdown of the streaks and regeneration of new quasi-streamwise vortices. For the attached eddies at a given spanwise length scale ${\it\lambda}_{z}$ between ${\it\lambda}_{z}^{+}\simeq 100$ and ${\it\lambda}_{z}\simeq 1.5h$, the single turnover time period of the self-sustaining process is found to be $Tu_{{\it\tau}}/{\it\lambda}_{z}\simeq 2$ ($u_{{\it\tau}}$ is the friction velocity), which corresponds well to the time scale of the bursting. Two additional numerical experiments, designed to artificially suppress the lift-up effect and the streak meandering motions, respectively, reveal that these processes are essential ingredients of the self-sustaining process of the attached eddies in the logarithmic and outer regions, consistent with several previous theoretical studies. It is also shown that the artificial suppression of the lift-up effect of the attached eddies in the logarithmic and outer regions leads to substantial amounts of turbulent skin-friction reduction.