高精度时空间断谱元方法求解复杂区域跨音速流动及声传播问题

A significant challenge against the numerical simulation of the flow and sound field in the complex computational domain is introduced by the shock wave in transonic flow. It is necessary for the numerical method to be capable of solving the numerical discontinuity or large-gradient problems with high-order accuracy. Therefore, the high-order accurate time-space discontinuous Galerkin spectral element method and new triangular spectral element method are proposed in this program, and they are applied to solve the flow and sound propagation problems in transonic cascade, aiming to establish the high-order accurate numerical method with low dispersion and low dissipation for the transonic flow and acoustic problems in the complex computational domain. Based on the characteristic analysis of the algorithm, the compressible flow in the transonic cascade will be solved accurately, especially for the shock wave capture and the flow field details near the shock wave. And then the blasting test for the transonic cascade is carried out to testify the validity of the numerical method. Furthermore, it is essential to improve the high-order absorbing boundary condition. After setting up the special sound source, apply the time-space discontinuous Galerkin triangular spectral element method to numerically stimulate the sound propagation in the transonic cascade, focus on the change of the acoustic variables while passing through the shock wave, and analyze the influence mechanism of shock wave on sound propagation. The research produces an important academic value for promoting the high-order accurate numerical method in computational fluid dynamics and computational aero-acoustics, and it also has practical significance for designing the method of noise control.

跨音速流动条件下复杂计算区域内由于激波的存在，对流场及声场数值模拟提出了挑战，要求能够在复杂计算区域内实现对数值间断或大梯度问题的高精度求解。为此，本项目拟构建高精度时空间断Galerkin谱元方法及新型三角谱元方法，以跨音速叶栅的流动及声传播问题为求解对象，建立跨音速流动条件下复杂区域流场及声场的低色散、低耗散、高精度求解方法。在解析算法求解特性的基础上，求解跨音速叶栅流道内可压缩流动问题，特别是实现激波的精确捕捉及激波前后流场的精确计算，并开展跨音速叶栅吹风实验，验证数值方法的正确性。进一步，改进高阶吸收边界条件，构建声源分布，采用时空间断Galerkin三角谱元方法求解跨音速叶栅流场的声传播过程，着重研究激波前后声学量的变化，解析激波对声波传播的影响机制。研究内容对发展计算流体动力学及计算气动声学高精度数值方法具有重要学术价值，同时对指导设计噪声控制方法具有重要实际意义。

跨音速流动条件下复杂计算区域内由于激波的存在，对流场及声场数值模拟提出了挑战，要求能够在复杂计算区域内实现对数值间断或大梯度问题的高精度求解。本项目围绕复杂区域内跨音速气动声学数值模拟方法，针对其在数值间断、时间格式、无反射边界条件、复杂几何离散等方面的难点开展了研究。本项目主要研究结论如下：以谱元方法为基础构建了DG谱元方法，检验了其求解跨音速激波问题的能力，并发展了参数可调的人工粘性项以获得兼具稳定性和精度的数值间断问题求解算法；针对谱元方法低耗散、低色散特性导致的计算稳定性问题，一方面构造了适用于谱元方法的SIMPLE算法，另一方面将时间分裂法与最小二乘变分相结合形成改进的时间分裂法，这两种稳定可靠的新时间离散格式有效拓宽了可计算的雷诺数范围，提高了计算精度与求解效率；针对复杂计算区域，构造了Gauss-Fekete三角谱元方法，并通过等参映射获得曲边单元，在数值精度基本保持不变的情况下实现了对任意复杂边界的精确刻画；针对声场模拟的无反射边界条件，将分裂式完全匹配边界层引入声传播方程，实现对出射声波的精确刻画；针对气动声学的流声耦合问题，一方面结合膨胀理论发展了以不可压缩流动压力的时间二阶导数项作为声源的流声分离算法，并利用构建的方法对平面叶栅气动噪声问题进行了求解，另一方面初步探索了流声统一求解算法，并将其应用于开缝圆柱降噪效果探究。本项目为复杂气动声学问题的求解提供了一套完整的解决方案，对发展计算流体动力学及计算气动声学高精度数值方法具有重要学术价值，同时对指导设计噪声控制方法具有重要实际意义。