AI 中文总结
本研究开发了无需预先知晓湍流统计特性的低维Galerkin投影模型，可预测方管中湍流二次平均流动，仅需2个模态即可正确预测其形状与方向，模拟结果与直接数值模拟一致性良好。
AI 中文摘要
湍流管道流动中侧壁的存在会引发二次流动结构，具体表现为角落处的反向旋转流向涡（即普朗特第二类二次流动）。本研究开发了降阶Galerkin投影模型，该模型可在无需预先知晓湍流统计特性的情况下，预测方管几何结构中这类湍流平均二次流动的产生与结构。模型通过将纳维-斯托克斯方程投影到线性化系统的特征模态上得到，所得常微分方程组在添加零均值强迫项后进行模拟。研究表明，多数采用主导流向恒定特征模态构建的模型，能正确预测二次平均流动的形状与方向，其中最简模型仅需2个模态。在这些模型中，二次平均流动的贡献源于某一特征模态的非零平均系数，该特征模态具备二次平均流动预期的所有对称特性。模型足够简单，可解析计算该平均系数与其他模态系数联合二阶矩之间的关系。研究证实，采用与降阶模型相同强迫结构的流向恒定直接数值模拟（DNS），会产生相似的二次流动结构；此外，研究还表明，本模型生成的雷诺应力分布与完全解析的直接数值模拟结果在性质上相似，且随着模型维度增加，一致性会提升。
英文摘要
The presence of sidewalls in turbulent duct flows leads to the emergence of secondary flow structures in the form of counter-rotating streamwise vortices near the corners (Prandtl's secondary flow of the second kind). This work develops reduced-order Galerkin projection models that can predict the emergence and structure of these turbulent mean secondary flows in a square duct geometry, without requiring any prior knowledge of the turbulent statistics. The models are obtained by projecting the Navier--Stokes equations onto eigenmodes of the linearised system, with the resulting systems of ordinary differential equations simulated with the addition of zero-mean forcing. We show that most models obtained using leading streamwise-constant eigenmodes predict the correct shape and direction of the secondary mean, with the minimal such model requiring only two modes. In these models, the secondary mean contribution arises due to the nonzero average coefficient of an eigenmode that possesses all of the symmetry properties expected of a secondary mean. The models are sufficiently simple such that the relationship between this mean coefficient and the joint second moment of other mode coefficients can be computed analytically. We confirm that running streamwise-constant direct numerical simulations (DNS) with the same forcing structure as used in the reduced-order models produces similar secondary-flow structures. We additionally demonstrate that our models produce qualitatively similar Reynolds stress distributions to fully resolved direct numerical simulations, with improved agreement as model dimension increases.
