Elucidation and control of turbulence-driven secondary flow in a rectangular duct using nonlinear traveling-wave solutions

本文采用牛顿法对方形管道中的非线性行波解进行了数值计算。我们成功地得到了与湍流驱动的平均二次流相似的八涡型的定常行波解。该行波的流向平均速度场强度与时向和流向平均湍流速度场强度相当，在低雷诺数范围（Re=1400 ~ 2500），行波不仅再现了湍流驱动平均二次流的定性性质，而且再现了湍流驱动平均二次流的定量性质。我们还研究了矩形管道中非线性行波的数值计算。通过改变管道截面宽高比，对方形管道内的八涡行波进行了跟踪。研究发现，在Re=1200时，只有长径比小于2.4时才存在八涡行波。研究还发现，当宽高比从1开始增大时，行波中的垂直结构集中在大跨度管壁的等分线附近。在更高雷诺数（Re=3500, 5480）下，对方形管道湍流进行了直接数值模拟，利用克莱布希势来识别大尺度的流向滚转。研究表明，在高雷诺数湍流中，不同横流长度尺度的大尺度涡旋对平均二次流的产生起着重要作用。

In this study, numerical computation of nonlinear traveling-wave solutions has been performed using a Newton method in a square duct. We have succeeded in obtaining the steady traveling-wave solutionexhibiting an eight-vortex pattern similar to the turbulence-driven mean secondary flow. The streamwise-averaged velocity field of this traveling wave has a comparable intensity of the time- and streamwise-averaged turbulent velocity field, and in low-Reynolds-number range, Re=1400-2500, the traveling wave reproduces not only qualitative but also quantitative properties of turbulence-driven mean secondary flow. We have also worked on numerical computation of nonlinear-traveling waves in a rectangular duct. The eight-vortex traveling wave found in a square duct has been tracked with changing anaspect ratio of a duct cross-section. It has been found that at Re=1200, the eight-vortex traveling wave exists only for the aspect ratio smaller than 2.4. It has also been observed that when the aspect ratio is increased from unity, vertical structures in the traveling wave are localized around the bisector of the duct wall of wider span. Direct numerical simulations of turbulent square-duct flow have been performed at higher Reynolds numbers, Re=3500, 5480, to identify large-scale streamwise rolls by using the Clebsch potential for cross-stream turbulence motion. It has been suggested that the large-scale rolls of a variety of cross-stream length scales play a significant role in the generation of the mean secondary flow in high-Reynolds-number turbulence.