谱元法应用于涡声传播问题的研究
Investigation of Vortex Sound Propagation Using Spectral Element Method

为了满足计算气动声学对低色散、低耗散高精度数值离散格式的需求,将高精度谱元法结合声比拟理论应用于求解气动声学问题。以伪声压的时间二阶导数作为非齐次波动方程的声源项,空间离散采用谱元法,时间离散应用隐式Newmark法,并在外边界采用C-E-M吸收边界条件,求解了由两个相距为2r0的等环量点涡组成的同向旋转涡对的发声问题。旋转涡对的不可压缩流场通过复位势理论获得,声源由流场量计算得来,并将数值结果与应用多级匹配展开法得到的解析解进行比较,可得数值解与分析解吻合较好。研究结果表明:应用高精度谱元法进行空间离散时,每波长的网格数为11时可达到很高的精度;网格数一定的情况下,时间步长越小得到的数值解与分析解之间的误差就越小;另外,证明了将伪声压对时间的二阶导数作为声源项,能够高精度求解不可压缩流动引起的气动声学问题。
In order to meet the requirements of low dispersive and low dissipative numerical discretization schemes in computational aero？？acoustics, the spectral element method combined with Lighthill’s acoustic analogy theory for the simulation of aeroacoustic problems is investigated. Taking the second temporal derivative of pseudopressure as the source of the inhomogeneous wave equation, and space discretization using spectral element method time discretization using implicit Newmark method, under the C？？E？？M absorbing boundary condition the acoustic field generated by a co？？rotating spinning vortex pair is solved. This co？？rotating vortex pair consists of two point vortices separated by a fixed distance of 2r0 with a circulation intensity Γ. The incompressible flow field is obtained using the complex potential theory, and the acoustic source terms are computed using these hydrodynamic quantities. The acoustic pressure results are evaluated by comparing them with the analytical solution obtained from the matched asymptotic expansion (MAE) method The numerical solutions are in good agreement with the analytical solutions. The results show that the spectral element method can obtain high accuracy using only eleven grids in one wavelength. In the case of the same number of grids, the smaller the time step, the smaller the error between numerical solution and analytical solution. Finally, it is proved that the aero？？acoustic problems induced by incompressible flows can be solved with high accuracy by using the second temporal derivative of the pseudopressure as the acoustic source