分数阶湍流问题的机器学习建模及谱元实现

Predicting turbulence is still an open problem, a last frontier in classical physics. We propose to employ fractional operators in conjunction with physics-informed neural networks (PINNs) to discover new governing equations for modeling and simulating wall-bounded turbulent flows at high Reynolds number. Here, we aim to discover non-local functional which is proposed by Prandtl relationships for Reynolds Averaged Navier-Stokes equations (RANS) hydrodynamic turbulence, to model the subgrid stress tensor (SGS) in terms of the resolved field. This is the fundamental “closure” problem in RANS of turbulence. We will use deep learning to target fully-developed turbulent channel flow and developing turbulent boundary layers for which we will use the extensive data base from direct numerical simulations (DNS). In the integer case, we use tensorflow automatic differentiation that leads to high accuracy and avoids any numerical artifacts. However, this is not possible for fractional PDEs, and hence we need to explicitly discretize the fractional operators using appropriate discretization formulas for fractional derivatives. Finally, we will solve the fractional turbulent model by using spectral element method, and we will also extend our model for solving physical problems such as duct flows, wall-bounded turbulent flows with high Reynolds number and turbulent boundary layer problems.

固壁湍流模型问题在经典物理学中仍然是一个悬而未决的开放性问题。大量的实验数据与数值模拟表明，虽然湍流结构十分复杂，但它依旧遵循连续介质的一般动力学规律，即Navier-Stokes方程。本项目主要针对Prandtl提出的非局部湍流模型进行延拓和进一步深入研究，将具有非局部性质的分数阶微分算子引入固壁湍流封闭问题中，并应用物理信息神经网络(PINNs)深度学习机，采用直接数值模拟(DNS)数据学习并创建高雷诺数下固壁湍流的分数阶形式的雷诺平均模型(RANS)。本项目旨在将整数阶PINNs深度学习机推广到分数阶的情形，并用于学习分数阶固壁湍流模型。其主要难点在于分数阶算子的非局部性导致整数阶求导链式法则不再适用，需要应用数值离散格式对分数阶导数进行离散。最后应用谱元法对得到的分数阶模型进行数值离散，并且用于实际问题的数值模拟如：漕道湍流问题、高雷诺数固壁湍流问题、以及边界层问题。