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

在这个项目中，我们首先在平面Couette系统中寻找鞍型稳定流。得到了在湍流直接数值模拟中短暂出现的鞍型静止状态，并将其作为牛顿迭代的初始猜测，数值计算了不可压缩Navier-Stokes方程的三维鞍型解。在最低雷诺数下，我们还利用原始迭代方法，结合直接数值模拟，成功地获得了嵌入在Couette湍流中的不稳定周期运动。这个周期鞍是一个三维周期解，它很好地代表了近壁相干结构（即条纹和流向涡）的再生周期。为了证明在Couette系统中得到的鞍形解与在平面泊泽维尔系统中得到的鞍形解的相似性，我们接下来将它们与这些系统在完全湍流状态下得到的鞍形解进行了比较。在比较中发现，在近壁湍流缓冲区内的平均速度和均方根速度与鞍解的相应速度非常吻合。另一方面，我们在边界层流动中引入多孔壁来延缓流动分离（或等效地增强动量传递），并观察到壁孔隙度的显著影响。在长波长极限下，我们还解析得到了作为近壁条纹流模型的波纹涡片的不稳定本征解。在此分析中，我们从理论上阐明了牛排的失稳机理，并发现鞍型稳态表现为不稳定扰动的非线性饱和状态。在此基础上，对后台阶下游的流动进行了直接数值模拟，揭示了台阶下游出现的三维涡结构的产生机制。我们利用Pyragas的外力法研究了鞍态（不稳定）解的稳定化问题，并在Couette系统的数值实验中成功地实现了周期鞍态的稳定化。在该策略中，我们可以通过小的控制输入来稳定鞍型解。少

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