Elucidation and control of near-wall turbulence-A new approach by saddle solutions

在这个研究项目中，我们重新计算了Couette平面、Poiseuille平面和自治壁系统中最近发现的有吸引力的鞍解（不稳定平衡或周期解），以比较它们的性质。结果表明，尽管驱动系统的方式不同，但在有壁剪切流中存在普遍的鞍形解。这些解可分为两大类，一类是以顺流涡旋运动为主要特征的上分支解，另一类是以条纹结构为主要特征的下分支解。前者表示近壁湍流的井结构和统计数据，而后者显示的壁剪切速率远小于湍流状态。在平面Couette流数值实验中，利用混沌控制理论稳定了壁面剪切速率较低的下支解（不稳定周期解），实现了紊流中更显著的表面摩擦阻力减小。对平面Couette系统的下分支解进行了线性稳定性分析，证明了该解只有一个不稳定特征值。在亚临界雷诺数下，低分支解及其稳定流形形成层流和湍流吸引子之间的盆地边界。研究表明，如果在湍流向下解和盆地边界的过渡过程中施加一个小幅度的控制输入，可以使湍流分层。在平面Couette流中考察了表征近壁湍流的上分支解（不稳定周期解）的性质，表明该解再现了近壁湍流的普遍统计规律，即普朗特壁定律。此外，在具有高对称性的周期盒湍流中发现了同类的上分支解，证实了该解的统计量与湍流态的统计量很好地吻合。研究发现，上分支解还再现了各向同性湍流的普遍统计规律，即Kolmogorov相似定律。少

In this research project, recently found appealing saddle solutions (unstable equilibrium or periodic solutions) were recomputed in plane Couette, plane Poiseuille, and autonomous wall systems to compare their properties. It has been shown that in spite of diffrernce in the way of driving the systems, there exist universal saddle solutions in wall-bounded shear flow. These solutions have been shown to be classified into two families, one of which is an upper-branch solution characterized by dominant streamwise vortical motion and the other of which is a lower-branch solution characterized by dominant streaky structures. The former represents well structures and statistics for near-wall turbulence, while the latter exhibits wall shear rate much less than that for a turbulent state. In numerical experiments of plane Couette flow, the lower-branch solution (unstable periodic solution) with much less wall shear rate has been stabilized by the use of a chaos control theory to accomplish a s … More ignificant reduction of skin friction drag in turbulent flow. The linear stability analysis of the lower-branch solution for the plane Couette system has been performed to demonstrate that the solution has only one unstable eigenvalue. At subcritical Reynolds numbers the lower-branch solution and its stable manifold form the basin boundary between laminar and turbulent attractors. It has been shown that turbulent flow can be laminarized if a small-amplitude control input is imposed on the flow during its transient approach to the lower solution and so to the basin boundary. The properties of the upper-branch solution (unstable periodic solution) representing near-wall turbulence was examined in plane Couette flow to show that the solution reproduces the universal statistical law of near-wall turbulence, i.e., the Prandtl wall law. Furthermore, the same kind of an upper-branch solution has been discovered in periodic-box turbulence with the high-symmetry to confirm that the statistics of the solution is in good agreement with that in a turbulent state. It has been found that the upper-branch solution also reproduces the universal statistical law of isotropic turbulence, i.e., the Kolmogorov similarity law. Less