We study the energy convergence of the Karhunen-Loève decomposition of the turbulent velocity field in a high-Reynolds-number pressure-driven boundary layer as a function of the number of modes. An energy-optimal Karhunen-Loève (KL) decomposition is obtained from wall-modeled large-eddy simulations at “infinite” Reynolds number. By explicitly using Fourier modes for the horizontal homogeneous directions, we are able to construct a basis of full rank, and we demonstrate that our results have reached statistical convergence. The KL dimension, corresponding to the number of modes per unit volume required to capture $90\%$ of the total turbulent kinetic energy, is found to be $2.4\times {10}^5\left|\mathrm{\Omega }\right|/H^3$ (with $\left|\mathrm{\Omega }\right|$ the domain volume and $H$ the boundary layer height). This is significantly higher than current estimates, which are mostly based on the method of snapshots. In our analysis, we carefully correct for the effect of subgrid scales on these estimates.

中文翻译：

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Karhunen-Loeve分解的能量收敛研究在高雷诺数压力驱动边界层大涡模拟中的应用

我们研究了高雷诺数压力驱动边界层中湍流场的Karhunen-Loève分解的能量收敛与模数的函数。能量最佳的Karhunen-Loève（KL）分解是通过“无穷”雷诺数的壁模型大涡模拟获得的。通过在水平齐次方向上显式使用傅里叶模式，我们可以构建满秩的基础，并且证明我们的结果已达到统计收敛。KL尺寸，对应于捕获所需的每单位体积的模式数量$90％$ 总湍动能的 $2.4\times {10}^5\left|\mathrm{\Omega }\right|/H^3$ （与 $\left|\mathrm{\Omega }\right|$ 域数量和 $H$边界层高度）。这大大高于当前的估计，当前的估计主要基于快照方法。在我们的分析中，我们仔细校正了子网格规模对这些估计的影响。