Semi-Analytic Jacobian-Forming Method for a Newton–Krylov Turbomachinery Flow Solver

Reynolds-averaged Navier–Stokes computational fluid dynamics solvers based on Newton–Krylov algorithms are becoming increasingly popular because of their favorable robustness and efficiency in achieving residual convergence, especially for edge-of-envelope cases. A key ingredient in Newton–Krylov algorithms is the efficient calculation of Jacobian matrices. The widely used graph-coloring method for Jacobian-forming, although straightforward to implement, is computationally expensive, thus impairing the overall efficiency of Newton–Krylov solvers. The semi-analytic approach can significantly reduce computational cost but requires sophisticated manual derivation when being extended to turbomachinery internal flow analysis. This paper introduces an efficient semi-analytic Jacobian-forming method for turbomachinery aerodynamic analysis. Special attention is given to Jacobian correction to account for periodic boundary conditions. The proposed method was compared against the graph-coloring approach on two test cases: a linear compressor cascade and a three-dimensional transonic rotor, with its performance thoroughly evaluated across different flow regimes. Results show that the semi-analytic approach can accelerate the Jacobian-matrix calculations by up to [Formula: see text]. Consequently, nonlinear flow analyses were accelerated by up to [Formula: see text]. The proposed semi-analytic Jacobian-forming method could also be applied to various advanced flow analyses that prefer a Jacobian-explicit approach, such as adjoint and eigenvalue analyses.