复杂几何边界在自然界和工程领域的流动中广泛存在。理解和预测具有复杂几何边界的流动对于解决科学和工程问题至关重要。不同于传统的研究思路，本文运用数据驱动的机器学习方法，从数据科学的角度研究具有复杂几何边界流动的物理机理以及实现流场的高效重构。

首先，我们提出了数据驱动量纲分析方法，可从数据中发现物理系统的主导无量纲量，揭示物理机理。经典的量纲分析将物理问题中众多物理量间的关系归结为若干独立的无量纲量间的关系。其中，部分无量纲量的变化会引起物理问题所关心的因变量的急剧变化，这类无量纲量即为该问题的主导无量纲量，它们代表了物理问题背后的物理机理。根据已有观测数据，我们使用神经网络近似表示各无量纲量间的函数关系。运用活跃子空间方法，发现函数关系变化最快和相对平坦的方向，分别对应主导无量纲量和可忽略的无量纲量，揭示物理系统中主导的物理机理。粗糙圆管流动和柔性体弯曲减阻的流固耦合问题验证了该方法的有效性。我们将该方法应用于兰金体扰流问题的物理机理研究中。兰金体是流体力学势流理论中的基本模型，也是理想的研究复杂几何边界的模型。通过大量的直接数值模拟，本工作深入研究了不同几何和流动状态下兰金体绕流的流体力学特性。依据直接数值模拟的数据，我们运用数据驱动量纲分析方法得到了决定斯特劳哈尔数和摩擦阻力系数的重要无量纲量，并且确定了对应的经验公式。

数据同化方法可以准确且快速地计算大量可供数据驱动量纲分析的流场数据。数据同化结果依赖于观测点位置的选取。面对复杂几何问题，我们提出了优化观测点布置的数据同化方法，能够根据较少的观测点高效重构具有复杂几何边界的流场。为了降低传统数据同化方法对于观测点数量的需求，我们提出了基于卷积神经网络的观测点优化布置方法。该方法根据数据同化的初始样本集，构建基于卷积神经网络的速度涡粘映射，利用卷积神经网络的梯度加权类激活映射识别数据同化中的重要区域。该方法是一种先验方法，在卡尔曼滤波步骤前完全确定观测点位置。相较于均匀布设大量观测点，本方法可通过少量观测点的高可信度数据，辅助集合卡尔曼方法高效重构具有复杂几何边界的平均流场。该观测点优化布置方法应用于具有复杂几何的两个典型流动：周期山状流和轴对称回转体绕流。算例表明基于梯度加权类激活映射的观测点优化布置方法可以有效减少观测点的数量，提高重构流场的精度。

本工作的主要创新点包括：
    1. 提出了数据驱动的机器学习量纲分析方法，可从数据中发现物理系统的主导无量纲量。
    2. 通过直接数值模拟对兰金体绕流展开全面研究，并运用数据驱动量纲分析方法揭示其重要无量纲量，填补了兰金体绕流的研究空白。
    3. 提出了基于类激活映射的观测点优化布置数据同化方法，布置少量观测点就能够实现具有复杂几何边界流动的高精度重构。

Flows with complex geometric boundaries are widely present in nature and engineering fields. Understanding and predicting flows with complex geometric boundaries is crucial for solving scientific and engineering problems. Our manuscript applies data-driven machine learning methods, unlike classical research approaches. We utilized the methods to study the physical mechanisms of flows with complex geometric boundaries from a data science perspective and to achieve effective reconstruction of flow fields.

First, we propose a data-driven dimensional analysis method that can discover the determined dimensionless quantities in physical systems from data, revealing physical mechanisms. Classical dimensional analysis transforms the relationships among numerous physical quantities into relationships among several independent dimensionless quantities in a physical problem. Changes in certain dimensionless quantities can significantly impact the dependent variables of interest in the physical problem. These dimensionless quantities are the determined dimensionless quantities representing the physical mechanisms behind the problem. Using available data, we employ neural networks to approximate the functional relationships among the dimensionless quantities. Using the active subspace method, we identify directions where the functional relationship exhibits the most rapid changes as well as those that remain relatively unchanged, correlating to determined and negligible dimensionless quantities, respectively. This approach reveals the significant physical mechanisms operating within the system. The effectiveness of this method is verified through cases of flows in rough pipes and flow-structure interaction of flexible body. We also apply this method to study the physical mechanism of the Rankine oval flow problem. The Rankine oval is a fundamental model in the potential flow theory of fluid dynamics and an ideal model for studying complex geometric boundaries. Through extensive direct numerical simulations, our manuscript presents a comprehensive study on the hydrodynamic characteristics of flows around various Rankine ovals. Based on the data from direct numerical simulations, we use the data-driven dimensional analysis method to identify the important dimensionless quantities that determine the Strouhal number and the friction drag coefficient. And we fit corresponding empirical functions.

Then, to accurately and rapidly compute a large amount of flow field data for data-driven dimensional analysis, we choose the data assimilation method. The results of data assimilation depend on the selection of observation sensors. When dealing with complex geometries, we proposed an optimized data assimilation method for observation sensors placement, which can efficiently reconstruct flow fields with complex geometric boundaries using fewer observation points. To reduce the demand for the number of observation sensors in traditional data assimilation methods, we introduce an optimal sensor placement method based on convolutional neural networks. This method constructs a velocity-vorticity viscosity mapping based on the initial ensemble samples of data assimilation and uses the gradient-weighted class activation mapping of the convolutional neural network to identify important areas in data assimilation. The method is a priori, determining the positions of the observation sensors before the Kalman filter step. Compared to uniformly using a large number of observation sensors, our method can efficiently reconstruct the mean flow field with complex geometric boundaries using ensemble Kalman filter method, with high-reliability data from a small number of observation sensors. The optimal sensor placement method is applied to two typical flows with complex geometries: periodic hill flow and flow around the axisymmetric body. Cases show that the optimal sensor placement method based on gradient-weighted class activation mapping can effectively reduce the number of observation sensors and improve the accuracy of flow field reconstruction.

The main innovations of this work include:
1. A data-driven machine learning dimensional analysis method is proposed, capable of discovering determined dimensionless quantities in physical systems from data.
2. A comprehensive study of the flow around Rankine ovals is conducted using extensive direct numerical simulations. The data-driven dimensional analysis method is employed to identify important dimensionless quantities, addressing the research gap in Rankine oval flow studies.
3. An optimal sensor placement method based on gradient-weighted class activation mapping for data assimilation is proposed, facilitating high-accuracy reconstruction of flows with complex geometric boundaries using a small number of observation sensors.