基于伴随方程寻找纳维-斯托克斯方程不变解的方法
An adjoint-based approach for finding invariant solutions of Navier-Stokes equations
AI总结：
本文提出一种基于伴随方程的方法，用于寻找纳维-斯托克斯方程的不变解，通过数值计算发现40雷诺数下的21个新稳态和行波解，并揭示了某些解对流体时间间歇性的影响。
摘要：
我们考虑具有周期边界条件和时间不变强迫的不可压缩纳维-斯托克斯方程。对于此类流动，我们推导出伴随方程，其轨迹渐近收敛到纳维-斯托克斯方程的平衡解和行波解。利用伴随方程，任意初始条件会演化到一个(相对)平衡解的附近，随后通过几次牛顿型迭代得到所需的(相对)平衡解。我们将这种方法应用于混沌的二维柯莫哥罗夫流。观察到100%的收敛率，从而在雷诺数Re=40时发现了21个新的稳态和行波解。其中一些新的不变解具有空间局部化结构，此前被认为仅存在于大长宽比域中。我们证明了一个新发现的稳态解支撑了时间间歇性，即流体的高能量耗散事件。更具体地说，显示了通用湍流轨迹的每次间歇性事件都对应于其接近该平衡解的近邻。
英文摘要：
We consider the incompressible Navier--Stokes equations with periodic boundary conditions and time-independent forcing. For this type of flow, we derive adjoint equations whose trajectories converge asymptotically to the equilibrium and traveling wave solutions of the Navier--Stokes equations. Using the adjoint equations, arbitrary initial conditions evolve to the vicinity of a (relative) equilibrium at which point a few Newton-type iterations yield the desired (relative) equilibrium solution. We apply this adjoint-based method to a chaotic two-dimensional Kolmogorov flow. A convergence rate of 100% is observed, leading to the discovery of 21 new steady state and traveling wave solutions at Reynolds number Re=40. Some of the new invariant solutions have spatially localized structures that were previously believed to only exist on domains with large aspect ratios. We show that one of the newly found steady state solutions underpins the temporal intermittencies, i.e., high energy dissipation episodes of the flow. More precisely, it is shown that each intermittent episode of a generic turbulent trajectory corresponds to its close passage to this equilibrium solution.
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