具有粘性逆Lax-Wendroff边界处理和紧凑WENO限制器的自适应网格local discontinuous Galerkin方法

This project aims to apply a new viscous inverse Lax-Wendroff (ILW) method and the local discontinuous Galerkin (LDG) method with new simple and compact viscous WENO limiters on adaptive mesh for numerically simulating the high Reynolds number Navier-Stokes (NS) equations. We technically design a new troubled cell indicator without any artificially adjustable parameters for the LDG method to solve for the high Reynolds number NS equations which contain strong discontinuities, reconstruct a new adaptive algorithm and construct new two and three dimensional adaptive high order accurate finite volume viscous WENO schemes. Then we renovate such new WENO schemes serve as compact limiters for the LDG method. By performing such procedures to solve high Reynolds number NS equations, we can achieve uniform high order accuracy in smooth regions, get sharp, non-oscillatory shock transitions robustly, and can deal with low density and low pressure problems well. Together with the application of associated turbulence model, we could get new schemes for solving hybrid RANS/LES equations. Our project focuses on the developments of structured and unstructured meshes, high order LDG method and associated new adaptive algorithm for the purpose of engineering application requirements.

本项目在自适应网格下采用新型粘性逆Lax-Wendroff(ILW)边界处理方法和带新型紧凑粘性WENO限制器的 Local Discontinuous Galerkin(LDG)方法对高雷诺数Navier-Stokes(NS)方程进行数值模拟。针对LDG方法在求解高雷诺数NS方程包含强间断问题，专门构造一种新型坏单元指示器能准确找到强间断并不含任何可调参数，构造新的自适应加密算法和新型二维，三维网格自适应高精度有限体积粘性WENO方法，将其改造并作为紧凑型限制器应用于LDG方法，使在处理高雷诺数NS问题时数值解在光滑区域具有精度高，强间断区域基本无振荡，并针对一些低密度和低压强的算例能得到较好的数值结果。在耦合湍流模型后构造求解混合RANS/LES方程的新数值方法。项目着重发展结构和非结构网格的新型边界处理方法，高精度LDG方法及与之配套的新的自适应网格处理算法，具有一定工程应用价值。

我们在相关研究工作中能高效生成计算网格、构造了多个具有广泛工程应用背景的高精度计算格式。我们构造了二维和三维粘性复杂物面边界的新型高精度有限体积粘性WENO格式并将其作为Local Discontinuous Galerkin方法限制器求解高雷诺数Navier-Stokes方程。我们设计了自适应高精度粘性数值方法，使绕复杂物形流动的高雷诺数Navier-Stokes方程可使边界层与激波互相干扰问题得到较好解决，使截断误差远小于粘性项作用，计算时内存的占用量降为原先的数分之一，成倍提高程序的计算速度并得到更精确的数值计算结果，使精度算例具有更小的范数误差，达到工程应用要求。