Enhancing convergence and accuracy: A comparative study of preconditioning for steady, laminar, and high gradient flows over a wide range of Mach numbers

Abstract

This paper introduces a novel platform that integrates three preconditioning matrices based on conservative variables: the Turkel, Choi–Merkle and Onur matrices. The platform aims to compare these matrices in terms of accuracy and robustness by investigating their performance in solving three distinct and challenging high-gradient laminar flow problems: (i) Bi-Plane NACA0012 airfoil, (ii) lid-driven flow in a square cavity and (iii) flow in a planar T-junction. These problems serve as new and challenging test cases to accurately determine the abilities of the preconditioning matrices. The preconditioning matrices are applied to evaluate the numerical solutions, and their performance in complex flow fields is assessed in terms of accuracy and efficiency. By solving these flow problems, the effectiveness of the preconditioning matrices is thoroughly analyzed. By integrating these preconditioning matrices into a single platform, this paper significantly contributes to the field. The approach enables a direct and meaningful comparison of the performance of the Turkel, Choi–Merkle and Onur matrices in solving these new and challenging laminar flow problems. While all three matrices demonstrate comparable accuracy in predicting flow characteristics like pressure coefficients, Turkel’s method shows superior convergence acceleration across the almost test cases due to alpha parameter and modifications of the momentum equations. In the external flow case, Turkel converges 22–53% faster than the other matrices by adjusting momentum terms with an alpha parameter. For the internal lid-driven cavity case, Turkel again accelerates convergence up to 38% over Choi–Merkle as Reynolds and Mach numbers increase. However, Onur’s method stalls at high Reynolds/Mach numbers. At low values of Reynolds and Mach numbers, Onur reduces computational cost by 17%. Finally, for the T-junction case, Turkel and Choi–Merkle perform almost identically, decreasing CPU time by 55% vs Onur, which lacks convergence robustness. While the preconditioning matrices have similar accuracy, Turkel offers the best convergence improvement by accounting for momentum effects. Onur works well at low Reynolds/Mach numbers but shows limitations at higher values. The selection of the optimal preconditioning matrix should be based on whether momentum or viscosity dominates in the flow field, as well as the intensity of the viscosity gradient rate.