The use of spectral proper orthogonal decomposition (SPOD) to construct low-order models for broadband turbulent flows is explored. The choice of SPOD modes as basis vectors is motivated by their optimality and space-time coherence properties for statistically stationary flows. This work follows the modeling paradigm that complex nonlinear fluid dynamics can be approximated as stochastically forced linear systems. The proposed stochastic two-level SPOD-Galerkin model governs a compound state consisting of the modal expansion coefficients and forcing coefficients. In the first level, the modal expansion coefficients are advanced by the forced linearized Navier-Stokes operator under the linear time-invariant assumption. The second level governs the forcing coefficients, which compensate for the offset between the linear approximation and the true state. At this level, least squares regression is used to achieve closure by modeling nonlinear interactions between modes. The statistics of the remaining residue are used to construct a dewhitening filter that facilitates the use of white noise to drive the model. If the data residue is used as the sole input, the model accurately recovers the original flow trajectory for all times. If the residue is modeled as stochastic input, then the model generates surrogate data that accurately reproduces the second-order statistics and dynamics of the original data. The stochastic model uncertainty, predictability, and stability are quantified analytically and through Monte Carlo simulations. The model is demonstrated on large eddy simulation data of a turbulent jet at Mach number \(M=0.9\) and Reynolds number \(\mathrm {Re}_D\approx 10^6\).

探索了使用谱适当正交分解 (SPOD) 来构建宽带湍流的低阶模型。选择 SPOD 模式作为基向量的动机是它们的最优性和统计平稳流的时空相干特性。这项工作遵循建模范式，即复杂的非线性流体动力学可以近似为随机受迫线性系统。提出的随机两级 SPOD-Galerkin 模型控制由模态膨胀系数和强迫系数组成的复合状态。在第一级中，模态展开系数由线性时不变假设下的强制线性化 Navier-Stokes 算子推进。第二级控制强迫系数，补偿线性近似值和真实状态之间的偏移。在这个级别，最小二乘回归用于通过对模式之间的非线性相互作用进行建模来实现闭合。剩余残差的统计数据用于构建去白化滤波器，便于使用白噪声来驱动模型。如果将数据残差作为唯一输入，则该模型始终准确地恢复原始流动轨迹。如果将残差建模为随机输入，则该模型会生成替代数据，该数据可准确再现原始数据的二阶统计数据和动态。随机模型的不确定性、可预测性和稳定性通过分析和蒙特卡罗模拟进行量化。\(M=0.9\)和雷诺数\(\mathrm {Re}_D\approx 10^6\)。