Ahmed M Arslan, Sharma Ati S
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USA.
Department of Aerospace Engineering, University of Southampton, Southampton SO17 1BJ, United Kingdom and Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA.
Phys Rev E. 2020 Jan;101(1-1):012213. doi: 10.1103/PhysRevE.101.012213.
One of the outstanding problems in the dynamical systems approach to turbulence is to find a sufficient number of invariant solutions to characterize the underlying dynamics of turbulence [Annu. Rev. Fluid Mech. 44, 203 (2012)10.1146/annurev-fluid-120710-101228]. As a practical matter, the solutions can be difficult to find. To improve this situation, we show how to find periodic orbits and equilibria in plane Couette flow by projecting pseudorecurrent segments of turbulent trajectories onto the left-singular vectors of the Navier-Stokes equations linearized about the relevant mean flow (resolvent modes). The projections are, subsequently, used to initiate Newton-Krylov-hookstep searches, and new (relative) periodic orbits and equilibria are discovered. We call the process project-then-search and validate the process by first applying it to previously known fixed point and periodic solutions. Along the way, we find new branches of equilibria, which include bifurcations from previously known branches, and new periodic orbits that closely shadow turbulent trajectories in state space.
在湍流的动力学系统方法中，一个突出的问题是找到足够数量的不变解来刻画湍流的潜在动力学[《流体力学年度评论》44, 203 (2012)10.1146/annurev-fluid-120710-101228]。实际上，这些解可能很难找到。为了改善这种情况，我们展示了如何通过将湍流轨迹的拟重复片段投影到关于相关平均流线性化的纳维 - 斯托克斯方程的左奇异向量（预解模态）上，来找到平面库埃特流中的周期轨道和平衡点。随后，这些投影被用于启动牛顿 - 克里洛夫 - 钩子步搜索，并发现新的（相对）周期轨道和平衡点。我们将这个过程称为“先投影后搜索”，并通过首先将其应用于先前已知的固定点和周期解来验证该过程。在此过程中，我们发现了新的平衡点分支，其中包括来自先前已知分支的分岔，以及在状态空间中紧密跟踪湍流轨迹的新周期轨道。
