Control of Near-wall Turbulence by Stabilizing Saddle-type Steady Flow

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In this project we first searched for saddle-type steady flows in a plane Couette system. We obtained quiescent states of the saddle type which appeared transiently in direct numerical simulations of turbulent flow, and then the quiescent states were used as an initial guess for the Newton iteration to numerically compute three-dimensional saddle solutions to the incompressible Navier-Stokes equation. At the lowest Reynolds number for self-sustaining turbulence we have also succeeded in numerically obtaining unstable periodic motion embedded in the Couette turbulence by using the original iterative method with the aid of the direct numerical simulations. This periodic saddle is a three-dimensional periodic solution, and it represents well the regeneration cycle of near-wall coherent structures, i.e., streaks and streamwise vortices. In order to demonstrate similarities of the saddle solutions obtained in the Couette system with those in a plane Poiseuille system we next compared their … More properties with those in a fully turbulent state of these systems. It has been found in the comparison that the mean velocity and the RMS velocities in the buffer region of near-wall turbulence are in excellent agreement with the corresponding velocities of the saddle solutions.On the other hands, we introduced a porous wall into boundary-layer flow to retard a flow separation (or equivalently to enhance momentum transfer), and we observed the significant effects of the wall porosity. We also obtained analytically the unstable eigensolutions to corrugated vortex sheets, which are a model for the near-wall streaky flow, at a long-wavelength limit. In this analysis we have elucidated theoretically the destabilization mechanism of the steaks, and we have found that the saddle-type steady state appears as a nonlinearly saturated state of unstable perturbations. Furthermore, we performed the direct numerical simulations of the flow downstream of a backward-facing step, and we have shown the generation mechanism of three-dimensional vortical structures appearing downstream of the step.We have worked on the stabilization of the saddle (unstable) solutions using the Pyragas' external-force method, and in the numerical experiments of the Couette system we have succeeded in the stabilization of the periodic saddles. In this strategy we can stabilize saddle solutions by small control input. Less