【摘要】 主要利用Jacobian-free的Newton-Krylov方法求解定常不可压缩Navier-Stokes方程,将基于高阶谱元法的区域分解Stokes算法的非定常时间推进步作为Newton迭代的预处理,回避了传统Newton方法Jacobian矩阵的显式装配,节省了程序内存,同时降低了Newton迭代线性系统的条件数,且没有非线性对流项的隐式求解,大大加快了收敛速度。对有分析解的Kovasznay流动的计算结果表明,本高阶谱元法在空间上有指数收敛的谱精度,且对定常解的Newton迭代是二次收敛的。本文模拟了二维方腔顶盖一致速度驱动流,同基准解符合得很好,表明本文方法是准确可靠的。本文还考虑了Re=800时方腔顶盖正弦速度驱动流,除得到已知的一个稳定对称解和一对稳定非对称解外,还获得了一对新的不稳定的非对称解。更多还原

【Abstract】 A Jacobian-Free-Newton-Krylov(JFNK) method with a time-stepping preconditioning technique is presented for the steady incompressible Navier-Stokes equations.The JFNK method combines the Newton method for superlinearly convergent solution of nonlinear equations and Krylov subspace method(such as GMRES) for solving the Newton correction equations.One crucial point for the JFNK method is constructing an effective preconditioner to reduce the number of Krylov iterations.The high order spectral element method with a domain decomposition Stokes solver is introduced as the effective preconditioner for the Newton iteration without forming the Jacobian matrix,which reduces the memory allocation,improves the poor conditioned system,speeds up the convergence rate.This algorithm is more suitable for the preconditioning than the usual time-splitting method and influence-matrix method,because it has no time-splitting divergence error and can be easily extended to the multi-dimensional flows.The numerical result for Kovasznay flow with an analytic solution shows the spectral accuracy with exponentially spatial convergence and superlinear convergence for Newton method.The method is applied to the two-dimensional constant velocity lid-driven problem at Reynolds number 100,400 and 1000.The results are compared with the benchmark data and show excellent agreement.An antisymmetric sinusoidal velocity driven cavity problem is considered for Re=800.Besides the stable patterns of steady symmetric and steady asymmetric solutions,a new pair of unsteady asymmetric solutions are found depending on the different initial conditions.This subcritical flow with multi-state can be served as an effective benchmark for the numerical solution of steady incompressible flows.更多还原