Prediction and understanding of the wall-shear stress modulation by non-linear interactions based on novel machine learning techniques

湍流壁面流体流动对于许多工程和生物医学应用具有重要意义。然而，它们的典型特征是高维、非线性和非定常的动力学，在空间和时间上表现出丰富的多尺度现象。由于这些具有挑战性的特性，我们仍然缺乏对这些复杂流体的全面了解。对于大多数应用来说，一个关键的成分是空间和时间分辨的壁面剪应力。除了提供摩擦阻力的测量-这在运输部门非常重要-它还可以洞察施加在刚性/柔性墙壁上的动态载荷，这在人类医学中可能是至关重要的。尽管它意义重大，但要测量瞬时和空间分辨率良好的壁面剪应力分布仍然异常困难。现有的大多数测量传感器都是单向和/或一维的设备，只能在固定位置检测一个壁面剪应力分量。此外，由于二次电子设备的实验限制，联合部署的传感器的空间分辨率和最大数量通常是有限的。因此，最近绝大多数针对调节壁面剪应力动力学的多尺度现象的研究局限于没有空间分辨率的时间相关数据，因此不能提供复杂物理的全面图景。因此，这一提议的首要目标是发展基于现代深度学习的算法，用于基于容易获得的速度测量来预测壁面剪应力。设想的神经网络是专门设计的，旨在捕捉--而不是限制假设--导致壁面剪应力动态的复杂的非线性和不稳定的相互作用，并使它们对人类来说是可解释的。特别是，该建议的目标是开发一种基于深度学习的体系结构，该体系结构专门设计用于学习从位于湍流壁面边界流动外层的二维速度场到瞬时空间分辨壁面剪应力分布的映射函数。作为补充，通过符号回归从导出的潜在表示中提取与固有学习的传递函数相对应的可解释的数学表达式。为了更深入地了解导致壁面剪应力调制的非线性相互作用，对该表达式进行了进一步研究。在直接的数值模拟数据上进行训练，基于外层的粒子图像测速仪测量和使用微柱剪应力传感器的壁剪应力测量，证明了所设想的学习器的泛化。

Turbulent wall-bounded fluid flows are of significant importance for numerous engineering and biomedical applications. However, they are typically characterized by high-dimensional, non-linear, and unsteady dynamics that exhibit rich multi-scale phenomena in space and time. Due to such challenging characteristics, we still lack a comprehensive understanding of these complex fluids. One key ingredient necessary for a majority of applications is the spatially and temporally resolved wall-shear stress. Besides providing a measure for the friction drag - a quantity of utmost importance in the transportation sector - it also gives insight into the dynamic loads imposed onto rigid/flexible walls, which can be crucial in human medicine. Despite its significance, it is still exceptionally difficult to measure instantaneous and spatially well-resolved wall-shear stress distributions. Most existing measurement sensors are single-direction and/or single-dimension devices that can only detect one wall-shear stress component at a fixed location. Moreover, the spatial resolution and the maximum number of jointly deployed sensors is typically limited due to experimental constraints arising from secondary electronic devices. As a result, the overwhelming majority of recent studies targeting the investigation of multi-scale phenomena that modulate the wall-shear stress dynamics are limited to time-dependent data without spatial resolution and consequently, cannot provide a comprehensive picture of the complex physics. Therefore, the overarching objective of this proposal is the development of modern deep learning based algorithms for the prediction of the wall-shear stress based on easily accessible velocity measurements. The envisioned neural networks are specifically designed to capture - without limiting assumptions - the complex non-linear and unsteady interactions that are responsible for the wall-shear stress dynamics and to make them interpretable for the human kind. In particular, this proposal targets the development of a deep learning based architecture specifically designed to learn a mapping function from two-dimensional velocity fields located in the outer layer of a turbulent wall-bounded flow to the instantaneous spatially resolved wall-shear stress distribution. Complementary, an interpretable mathematical expression, which corresponds to the inherently learnt transfer function, is extracted from the derived latent representation via symbolic regression. This expression is further investigated to gain deeper insight into the non-linear interactions that result in the modulation of the wall-shear stress. Trained on direct numerical simulation data, the generalization of the envisioned learner is evidenced based on simultaneous particle-image velocimetry measurements in the outer layer and wall-shear stress measurements using the Micro-Pillar Shear-Stress Sensor.