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

**Ning Fangfei Xu Liping (Faculty 404, Beijing University of Aeronautics & Astronautics, Beijing 100083, China)**

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.

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