We propose, analyze, and investigate numerically a novel feedback control strategy for high Reynolds number flows. For both the continuous and the discrete (finite element) settings, we prove that the new strategy yields accurate results for high Reynolds numbers that were not covered by current results. We also show that the new feedback control yields more accurate results than the current control approaches in marginally-resolved numerical simulations of a two-dimensional flow past a circular cylinder at Reynolds numbers $Re=1000$. We note, however, that for realistic control parameters, the stabilizing effect of the new feedback control strategy is not sufficient in the convection-dominated regime. Our second contribution is the development of an adaptive evolve-filter-relax (aEFR) regularization that stabilizes marginally-resolved simulations in the convection-dominated regime and increases the accuracy of the new feedback control in realistic parameter settings. For the finite element setting, we prove that the novel feedback control equipped with the new aEFR method yields accurate results for high Reynolds numbers. Furthermore, our numerical investigation shows that the new strategy yields accurate results for reduced order models that dramatically decrease the size of the feedback control problem.

翻译：本文针对高雷诺数流动提出了一种新颖的反馈控制策略，并对其进行了理论分析与数值研究。在连续和离散（有限元）框架下，我们证明了新策略能够针对当前方法尚未覆盖的高雷诺数区域获得精确结果。同时，通过二维圆柱绕流（雷诺数$Re=1000）数值模拟，表明在弱分辨条件下，新反馈控制方法相较于现有控制方法可获得更高精度。然而需指出，在真实控制参数条件下，该新型反馈控制策略在对流主导区域中的稳定效应仍显不足。为此，我们提出第二个创新点：发展自适应演化-滤波-松弛（aEFR）正则化方法，可稳定对流主导区域的弱分辨数值模拟，并在实际参数设定下提升新反馈控制的精度。在有限元框架中，我们验证了结合aEFR方法的新型反馈控制策略在高雷诺数下仍能保持精确结果。数值实验进一步表明，该新策略可显著缩减反馈控制问题规模，并在降阶模型中获得精确计算结果。