激波精确判别方法及高精度低耗散数值格式的研究与应用

In recent decades, the development of numerical methods in computational fluid dynamics (CFD) is focused on how to capture shock wave accurately and to satisfy the requirement of the high order accuracy in smooth complex regions. For solving the fluid dynamics equations, the shock capturing schemes suppress the oscillation near shock waves by using some kind of schemes' numerical characteristics. However, the deficiency is that these schemes will reduce the high resolution in smooth regions due to their intrinsic numerical diffusion. In order to simulate accurately the multi-scale complex flow fields with the shock wave/boundary layer interaction or shock wave/turbulence interaction, it is necessary to develop the shock capturing methods with high order accuracy and low diffusion. With the development of various numerical methods, how to detect the shock wave accurately becomes a very important problem. In this project, we propose to develop the accurate shock detecting methods, in which no artificial or problem-dependent parameter is used. Based on the accurate shock detecting method, a series of high order accuracy low diffusion schemes will be developed and applied to study the complex flows with the interaction of shock wave/boundary layer, shock wave/turbulence. The accurate shock detecting methods can efficiently reduce the numerical diffusion in smooth regions, which may be detected as shock wave regions by using a shock detecting method with an artificial parameter, and can avoid the spurious oscillation, which may be caused by the unsuitable parameter regards the shock wave as a smooth solution. This is very important for studying the separated flows caused by shock waves and resolving the small-scale structures in the shock wave/turbulence interaction flows. This project is expected to provide a new approach to construct the high performance numerical methods and to study the compressible turbulence.

计算流体力学的数值方法主要是围绕怎样才能准确捕捉激波而又能在光滑区域满足高精度的要求而不断得到发展的。激波捕捉方法是在求解过程中，利用格式的某种性质来抑制在激波附近产生的非物理振荡，其缺点也正是因为格式带来的数值耗散降低了其它区域的计算品质。为了能精确模拟带有激波/边界层、激波/湍流干扰等复杂结构的多尺度流场，必须发展具有高精度低耗散性质的激波捕捉方法，而如何准确判别激波成为一个极其重要的关键性问题。本项目拟综合研究激波捕捉格式中的数值机制，发展激波准确（无人为、问题相关参数）判别方法及理论，在此基础上发展一系列的高精度低耗散数值格式，并应用于激波/边界层干扰、激波/湍流干扰等复杂流动的研究。激波准确判别能有效避免人为参数误判激波带来的数值耗散或非物理振荡，这对精确研究激波引起的分离、激波/湍流干扰中的小尺度结构是非常重要的。本研究将为发展高性能计算方法、数值研究可压缩湍流提供一条新途径。

计算流体力学的数值方法主要是围绕怎样才能准确捕捉激波而又能在光滑区域满足高精度的要求而不断得到发展的。激波捕捉方法是在求解过程中，利用格式的某种性质来抑制在激波附近产生的非物理振荡，其缺点也正是因为格式带来的数值耗散降低了其它区域的计算品质。为了能精确模拟带有激波/边界层、激波/湍流干扰等复杂结构的多尺度流场，必须发展具有高精度低耗散性质的激波捕捉方法。本项目的研究内容主要包括发展高精度激波模板判别方法、 高精度低耗散加权基本无振荡格式、 高精度自适应激波捕捉格式以及高超声速复杂流动的数值模拟研究。在本项目的支持下，课题组构造了高阶整体光滑因子，利用高阶整体光滑因子与子模板光滑因子之间的量级关系，发展了有效鲁棒的高精度激波模板判别方法；针对当前在计算流体力学中得到广泛应用的加权基本无振荡（WENO）格式存在的一些问题和不足之处，提出了新的模板光滑衡量因子及加权函数，所构造的格式在高精度与低耗散方面获得了较大的改进；提出和发展了多步加权基本无振荡格式（Multistep-WENO）、紧致重构多步加权（Multistep-CRWENO）格式；提出了一种不含人为、问题相关参数的混合开关函数，发展了有效的流场自适应混合格式；通过高性能并行的大规模数值模拟研究和分析了压缩性、粗糙元等因素对超声速流动转捩的影响。本项目的研究成果为激波/复杂流动干扰（如激波/边界层干扰引起的分离点附近流场、激波/湍流干扰等）的数值模拟研究提供了系列有效的高精度方法。