A robust Jacobian-free Newton–Krylov method for turbomachinery simulations

Implicit methods are widely used in computational fluid dynamics numerical simulations of turbomachinery to accelerate convergence speed. However, simplified implicit nonlinear iterative algorithms, such as lower-upper symmetric-Gauss–Seidel method, will suffer from convergence slowdown or even divergence when dealing with off-design operations. In this study, a fast implicit Jacobian-Free Newton–Krylov finite volume method is developed to alleviate convergence difficulties of turbomachinery simulations. Based on the generalized minimal residual method, a matrix free solution for large sparse linear system is implemented, which avoids expensive and complex explicitly assembly of high-order Jacobian matrix. And a low-order Jacobian matrix assembled by graph coloring and finite difference method is adopted as preconditioning matrix. The nonlinear iterations are divided in two stages: startup stage and Newton stage. In the startup stage, an approximate Jacobian linear system is coupled with adaptive Courant–Friedrichs–Lewy number algorithm, solution update strategy and preconditioning lagging method to drive the nonlinear iteration with high computational efficiency. Once the residual is low enough, the Newton stage will be switched with an infinite time step to achieve rapid convergence. The robustness, accuracy and high efficiency of the applied nonlinear solver for different conditions was demonstrated by three test cases: a compressor cascade, a turbine cascade, and an axial compressor rotor. The potentiality of the method to enhance computational robustness of turbomachinery aerodynamic simulation for wide operation range is confirmed.