基于小波奇异分析的流场计算方法及应用

Method based on wavelet sigularity analysis for complicated flow and its application

首次将三维小波以及二维小波奇异性分析的思想引进三维以及二维复杂流场的数值计算,发展了一种高效率、高精度、高分辨率的方法.该算法的核心是获取流场中物理量在不同空间点的Hlder指数α,而该指数α的获取又依赖于小波变换以及高维（即二维或三维）小波分析技术.在三维与二维欧氏空间中,为进行小波多分辨分析,需要在尺度空间与小波空间分别引进尺度基与小波基.对二维问题,尺度基与小波基的基底要由1个尺度函数与3个小波函数组成,而三维时要由1个尺度函数和7个小波函数组成.借助于小波奇异分析所找到的流场中奇异点区域与光滑区域,便可分别选用高分辨率、高精度的WENO（weighted essen-tial non-oscillatory）格式与高精度中心差分格式进行流场的离散求解.一系列二维（即,①二维前台阶问题的Euler流;②二维双马赫反射的Euler流;③著名的二维Riemann问题;④跨声速RAE2822翼型二维绕流;⑤跨声速VKI-LS 59二维涡轮叶栅绕流）与三维（即,⑥跨声速轴流压气机转子NASA（National Aeronauticsand Space Administration）Rotor 37三维黏性绕流;⑦跨声速风扇转子NASA Rotor 67三维黏性绕流）算例表明：该方法的计算效率比传统的WENO格式有较大的提高,大部分典型算例能够提高3~5倍,而且可以获取复杂流场中高分辨率的激波结构以及涡系分布,可以得到与有关实验数据较为吻合的数值结果.

Three-dimensional and two-dimensional wavelet sigularity analysis was introduced in complicated flow and a high-order,high-resolution numerical method was developed.The core of this new method is the analysis of Hlder exponent α of complicated flow.The Hlder exponent α depends on wavelet transform and multi-dimensional wavelet analysis.In order to complete the multi-resolution analysis in two-dimensional and three-dimensional Euclidean space,scale basis and wavelet basis have been introduced in scale space and wavelet space.For two-dimensional and three-dimensional cases,scale basis and wavelet basis were constructed by one scale function and by three wavelet functions and seven wavelet functions respectively.High-resolution,high-order WENO（weighted essential non-oscillatory） scheme and high-order central difference scheme can be chosen adaptively according to the results of wavelet sigularity analysis.A series of examples in two-dimensional（（1） forward facing step problem,（2） double Mach reflection of a strong shock problem,（3） two-dimensional Riemann problem,（4） two-dimensional transonic flow past RAE 2822 airfoil problem,（5） two-dimensional transonic VKI-LS 59 turbine cascade flow problem） and three-dimensional（（6） transonic axial compressor,NASA（National Aeronautics and Space Administration） Rotor 37 three-dimensional viscous flow and（7） NASA Rotor 67 three-dimensional viscous flow） were computed.The computation results have been compared with the experiment results in related references and showed that this new numerical method can not only improve the computation efficiency significantly（the efficiency can be increased by 3~5 times for most examples）,but also capture the shock with high-resolution and vortices with an acceptable order.