二维定常湍流计算中的GMRES算法

APPLICATION OF GMRES ALGORITHM IN THE PREDICTIONS OF TWO-DIMENSIONAL TURBULENT STEADY FLOWS

在以前工作的基础上将广义极小残差（Ｇｅｎｅｒａｌｉｚｅｄ Ｍｉｎｉｍｕｍ ＲＥＳｉｄｕａｌ（ＧＭＲＥＳ））算法发展到用于求解二维可压 Ｆａｖｉｅｒ平均 Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程组．控制方程经 Ｎｅｗｔｏｎ线化处理后构成近似的线性系统，然后采用分别耦合了ＬＵＳＧＳ和ＩＬＵ两种预处理矩阵的ＧＭＲＥＳ算法求解．Ｓｐａｌａｒｔ－Ａｌｌｍａｒａｓ湍流模型被用来封闭流体控制方程组，采用与流体控制方程非耦合的方式，使用ＬＵＳＧＳ方法求解．对ＧＭＲＥＳ算法中矩阵－向量的乘积采用了有限插分方法，从而避免了精确的左端系数矩阵的计算和存储．对预处理矩阵的两种使用方法（左预处理和右预处理）进行了分析和讨论．用两个算例对ＬＵＳＧＳ和ＩＬＵ两种预处理矩阵进行了比较，同时探讨了左预处理和右预处理各自的优缺点．通过对Ｓａｊｂｅｎ扩压器和ＮＡＣＡ００１２有攻角流动的计算，表明带有预处理的ＧＭＲＥＳ算法在二维定常跨音黏性流动计算中相比于得到广泛应用的ＤＤＡＤＩ方法具有很大优势，左预处理要优于右预处理．

A GMRES (Generalized Minimun RESidual) algorithm is developed for solving two-dimensional compressible Favier-averaged Navier-Stokes equations based on authors' previous work concentrated on Euler equations. An approximate linear system is constructed resulting from Newton linearization of governing flow equations and then solved by GMRES algorithm coupled respectively with two preconditioning matrices, i.e., LUSGS and ILU. Spalart-Allmaras turbulence model is used to enclose flow equations and solved using LUSGS algorithm in an uncoupled manner with flow equations. The matrix-vector multiplications which emerge in GMRES algorithm is solved using finite-difference approach, thus the calculations of exact left-hand-side coefficient matrix is avoided and computer memory is saved. Some analysis and discussions are made concentrated on two approaches of usage of preconditioning matrix, i.e., left preconditioning and right preconditioning. Based upon the computations of two typical turbulent flow cases, the comparisons of these two types of preconditionings, i.e., LUSGS and ILU, are presented. In addition, the performances of left preconditioning and right preconditioning are discussed. With the predictions of Sajben's diffuser flow and the transonic flow over NACA0012 airfoil, it is shown that GMRES algo- rithm coupled with appropriate preconditioning has superior advantage over widely used so-called DDADI method in convergence rate, left preconditioning is better than right preconditioning.