计算气动声学中的高阶Nodal-DG方法研究

气动噪声的直接模拟对数值格式的色散、耗散特性提出了严格的要求。基于描述声波的线性双曲方程，运用本征值方法分析了高阶Nodal-DG方法的色散、耗散特性。结果发现，对于任意给定的m阶多项式基函数，数值波解有m+1个值，但仅有一个能够表示对应微分方程的物理波传播方式，其余的都是寄生波，且两种波型的传播方向相反。通过与Tam的DRP格式和Lele的六阶紧致格式进行比较，发现在相同的计算精度下，Nodal-DG方法的有效求解波数范围介于DRP格式和六阶紧致格式之间。通过对初始扰动为高斯波形的计算比较发现，在较少的网格数下，Nodal-DG方法的计算结果可以与紧致格式的计算结果相比，但优于DRP格式的计算结果，非常适合于气动声学的数值模拟，为气动声场的直接计算提供了一种新的方法

Higher-order nodal discontinuous galerkin method in computational aeroacoustics

Numerical schemes that result in minimal dispersion and dissipation errors are generally preferred for direct simulation of noise propagation.Dispersion and dissipation properties of nodal discontinuous Galerkin(DG) method for the linearized Euler equation were investigated by utilizing an eigenvalue analysis technique.It was found that for any given mth order of basis functions,there are m+1 modes of numerical waves.But among them only one represents propagating mode of physical wave corresponding to the partial differential equation,the rest belongs to numerical parasite modes,and moreover the propagation directions of these two kinds of numerical modes are opposite.The comparisons of dispersion properties among the nodal discontinuous Galerkin method,DRP schemes and compact finite difference schemes with the same order show the solvable wavenumber range of nodal DG method lies between those of DRP schemes and compact finite difference schemes.A test problem of wave propagation with initial disturbance consisting of a Gaussian profile was solved.The quality of solution obtained by nodal DG method with less grid number is analogous to that by compact finite difference schemes,but better than that by DRP schemes,which indicates the method is appropriate to direct numerical simulation of aeroacoustics.