基于混合重构高阶间断伽辽金方法的二维层流和湍流数值模拟

Methods for two-dimensional laminar and turbulent numerical simulations

为了提高间断伽辽金方法的计算效率,解决least-squares重构方法无法满足2-exact的缺陷,发展了基于recovery重构和least-squares重构相结合的三阶混合重构方法,用于求解可压缩层流和湍流流动.将Navier-Stokes方程和修正的一方程Negative Spalart-Allmaras模型方程耦合成为系统方程,采用三阶重构间断伽辽金方法进行求解.时间推进采用基于半解析精确Jacobian矩阵的上-下对称高斯赛德尔格式(lower-upper symmetric Gauss-Seidel scheme,LU-SGS)预处理广义极小剩余(generalized minimal residual,GMRES)方法和四阶隐式Runge-Kutta方法;空间对流项离散采用Haten-Lax-van Leer接触(Haten-Lax-van Leer contact,HLLC)格式;黏性项离散采用第二Bassi-Rebay(second Bassi-Rebay,BR2)格式,并对BR2局部和全局提升算子开展三阶重构,达到提高计算精度的目的.通过典型算例验证了发展rDGP_(1)P_(2)方法的准确性和计算效率.研究结果表明:重构的rDGP_(1)P_(2)方法不仅具有较高的计算精度,而且还具有较高的计算效率.

Third-order hybrid reconstructed methods based on the combination of least-squares recovery and reconstruction were developed for solving compressible laminar and turbulent flows to improve the calculation efficiency of the discontinuous Galerkin methods and overcome the deficiency of the least-squares reconstruction in satisfying the two-exact property.Navier-Stokes equations and the modified equation of the negative Spalart-Allmaras model were coupled as the equation system and solved using the developed third-order reconstructed discontinuous Galerkin methods.The lower-upper symmetric Gauss-Seidel preconditioning generalized minimal residual method based on the semianalytical exact Jacobian matrix and the fourth-order implicit Runge-Kutta method were implemented in temporal advance.Harten-Lax-van Leer contact and second Bassi-Rebay(BR2)schemes were adopted to calculate the inviscid and viscous terms,respectively.Third-order reconstruction was applied to solve the local and global lifting operators of the BR2 scheme,improving the calculation accuracy.The benchmarks were selected to verify the accuracy and calculation efficiency of the developed rDGP_(1)P_(2) method.The research results indicate that the developed reconstructed rDGP_(1)P_(2) methods have high computational accuracy and efficiency.