高阶谱元区域分解算法及其在流动稳定性中的应用

以无时间分裂误差的区域分解Stokes谱元算法为基础构建整体稳定性分析方法.用Jacobian-free的Inexact-Newton-Krylov算法求解不可压缩Navier-Stokes方程的定常解,将Stokes算法的时间推进步作为Newton迭代的预处理,在此基础上采用Arnoldi方法计算大规模特征值问题,对复杂流动进行稳定性分析,该方法能统一处理定常和非定常计算,没有时间分裂误差,无需显式构造Jacobian矩阵,可以减少内存使用,降低计算量,并加速迭代收敛.对有分析解的Kovasznay流动的计算表明,高阶谱元法具有指数收敛的谱精度.对亚临界方腔对称驱动流的各种定常解的计算及其稳定性分析验证了方法的可行性.

High Order Spectral Element Method and Application in Hydrodynamic Stability

A high order spectral element method with a domain decomposition Stokes solver is presented for hydrodynamic stability analysis. A Jacobian-free Inexact-Newton Kryloy algorithm with a Stokes time-stepping preconditioning technique is introduced to the calculation of steady state of incompressible flows. An Anoldi method is used to calculate the leading eigenvalues and corrosponding eigenvectors, which are responsible for the hydrodynamic instability. The method deals with steady and unsteady simulations in a similar way without time-splitting divergence error and does not need Jacobian matrix. As a result, it reduces memory allocation and computation cost, and speeds up the convergence. Numerical result for Kovasznay flows with an analytic solution shows spectral accuracy with exponentially spacial convergence and superlinear convergence for inexact Newton method. An antisymmetric sinusoidal velocity driven cavity problem is considered at Re = 800. The stable and unstable patterns are analyzed with leading eigenvalues of steady states. The symmetric-breaking Hopf bifurcations are considered in the wake of a circular cylinder limitlessly or between two parallel walls. The onset of instabilities agrees well with experimental and numerical results.