基于保单调全速域DG格式的非定常湍流隐式大涡模拟研究

Revolutionary aircraft design concepts demand high-fidelity simulation for turbulent flows to predict their performance and undoubtedly, high-order numerical methods are starting to find their place in this research community. This project intends to develop a novel discontinuous Galerkin (DG) method with good mathematical property of monotonicity-preserving and with the desired capability of performing all-speed simulation. And we further carry out high order implicit large eddy simulation (ILES) based on the designed DG method. Using the all-speed methodology and the monotonicity-preserving strategy, this research builds high order DG schemes which are aiming at all-speed and scale-resolving simulation. By solving the large system in the discretization fully implicitly, this work develops highly efficient unsteady implicit solver with scalable parallelization for ILES with present DG methods. At last, this project combines the designed DG methods with the physical modeling and establishes the discretization criterion for monotonicity-preserving and all-speed DG used in ILES. The final outcomes of the project will include highly accurate and effective numerical methods and computational models that are suitable for scale-resolving simulation for unsteady turbulent flows. As a result the work guarantees the extended and realistic application of high-fidelity turbulent simulation for practical problems with complex geometries.

当今气动研究领域要求精细多尺度的流动模拟以准确预测气动性能，而高阶格式是该研究领域不可或缺的重要组成部分。本项目拟发展一类新型保单调全速域间断Galerkin（Discontinuous Galerkin, DG）格式，并基于此类格式开展DG高精度隐式大涡模拟。研究将以全速域计算格式和保单调计算方法为关键技术，建立面向全速域、多尺度流动模拟的DG计算方法。通过实现非定常离散系统全隐式并行求解，发展面向DG隐式大涡模拟、高度并行的非定常隐式时间求解方法。最后，本项目将DG格式研究和物理模型结合起来，建立适于隐式大涡模拟的保单调全速域DG离散方案。研究最终要获得DG在非定常多尺度湍流模拟中高精度和高效率的数值方法和计算模型，从而保证精细化湍流模拟技术在实际问题、复杂外形中得到有效推广和切实应用。

本项目研究保单调和全速域计算方法并基于高阶格式开展隐式大涡模拟（ILES）研究。研究利用全速域格式和保单调方法，发展面向多尺度流动模拟的高精度高效率计算方法。通过实现保单调高阶DG方法和高效率隐式求解计算技术，给出高阶格式在隐式大涡模拟中格式分辨率特性。本项目完成三部分研究内容：首先，发展耗散自适应的全速域流动模拟方法，包括基于密度修正的通量计算方法和高分辨率特征重构方法，提高低速流动模拟精度和高速流动强激波干扰计算稳定性。发展了基于改进MOOD（multi-dimensional optimal order detection）的后验计算方法和基于BVD（boundary variation diminishing）重构的DG亚格子分辨激波捕捉方法，实现对Euler方程、Navier-Stokes方程以及包含强非线性源项的Euler方程的间断识别与捕捉，并且完成格式耗散和色散特性分析。其次，开展保单调非定常隐式求解并行算法研究，采用“全隐式”推进策略，完成非定常隐式后验计算方法。发展了图着色分组数值微分方法计算雅可比矩阵，提出了一种高效率、高精度的符号重载自动微分方法，并在典型算例中对比了自动微分方法与数值微分方法的精度、效率和鲁棒性。发展了适用高阶格式的非线性迭代控制策略，模拟了无粘流、层流和湍流等典型流动，对比了方法定常和非定常计算的效率和收敛特性。最后，将上述发展的保单调全速域计算方法应用于可压缩湍流模拟，通过数值实验定量研究了方法对基本湍流物理量模拟精度，为选用合适的数值方法进行隐式大涡模拟提供了指导意见，并且初步实现方法对激波、漩涡、分离及转捩等复杂湍流模拟的应用。本项目发展的保单调和全速域方法，结合亚格子混合计算思想，能够有效提高DG开展可压缩湍流的鲁棒性，为今后进一步开展高阶格式的相关研究打下基础。