基于Navier-Stokes方程分析飞机抖振问题

A Class of Spectral Method for Solving Unsteady Navier-Stokes Equations

摘要: 针对不定常的Navier-Stokes方程，给出了一种非线性Galerkin-Legendre谱方法(GL方法)，该方法是将勒让德谱方法和非线性Gelerkin方法结合起来，使得速度与压力的混合元空间不需要满足离散的Babuška-Brezzi不等式条件，给出了GL方法的谱格式，并给出该方法的稳定性证明和误差估计。关于飞机抖振问题已经由基于不可压缩N-S方程的方法的研究取代了原来粘、位流分开计算的研究，因此用谱方法研究飞机抖振问题，通过误差分析，减少阻力，达到节约能源的目的。

Abstract: This paper focuses on the issue about using numerical computation to solve unsteady Navier-Stokes equations. We present a scheme called nonlinear Galerkin-Legendre Spectral method, which combines the Legendre spectral method with nonlinear Galerkin-Legendre method. Therefore, it is not necessary to meet Babuska-Brezzi inequality condition in the velocity space and pressure space. In this paper, we give a spectral scheme of nonlinear Galerkin-Legendre methods for solving 2D N-S equations. What’s more, we also provide the error estimation and proof the stability of this scheme. The method based on the separate calculation of viscous flow and potential flow, was used to solve the problem of plane buffeting, and has been replaced by the method based on the incompressible N-S equations. Therefore, it is necessary to use the spectral method to study the chattering problem of aircraft and to reduce the resistance by error analysis and to save energy.