Implicit and Conservative Mixing-Plane Method for Multistage Turbomachinery Aerodynamic Analysis

A fully implicit Reynolds-averaged Navier-Stokes (RANS) flow solver and a corresponding adjoint solver using a Newton-Krylov algorithm has recently been developed and demonstrated to exhibit good solution robustness and efficiency for flow and adjoint aerodynamic analyses of single-row turbomachines at design and off-design conditions. To extend the algorithm to multistage applications, a robust mixing-plane technique and its fully implicit treatment are crucial. In this work, a mixing-plane approach featuring 1) flux conservation, 2) implicitness, 3) one-dimensional nonreflectiveness, and 4) robustness in handling reserve flow is proposed and incorporated into the existing Newton-Krylov turbomachinery computational fluid dynamics solver. Following a flux-based approach that ensures exact flux conservation, the algorithm of the proposed mixing-plane method is first rigorously derived, and implementation details into the existing single-row RANS solver are then discussed, with emphasis on the fully implicit treatment of the mixing-plane fluxes. Finally, the fully implicit multistage flow solution capability is extended to the corresponding adjoint solver. The resulting algorithms are tested on a subsonic and a transonic compressor to validate the flow solution accuracy, verify the adjoint sensitivity, and demonstrate its robustness in handling reverse flows.