Matrix-Free Resolvent Analysis for Large-Scale Aerodynamic Systems

Abstract

Predicting transonic buffet requires resolving both flow stability and disturbance amplification. Resolvent analysis gives the amplification. However, each gain needs the solution of tens of linear systems with the shifted Jacobian and its adjoint. Sparse direct factorization, the usual route, is extrapolated to need more than $130$ TB for a three-dimensional jet with $4\times 10^{7}$ unknowns. For the stiff Reynolds-averaged operator of a transonic wing, iterative solution has been reported to stall on the adjoint system. To avoid the direct factorization and to converge the adjoint, we present a matrix-free resolvent framework in the Reynolds-averaged Navier-Stokes solver ADflow that combines automatic differentiation, Krylov solution with subspace recycling, and Lanczos bidiagonalization, with backward-error checks on each singular triplet. A three-level study of the NASA Common Research Model wing reaches $1.73\times 10^{8}$ complex unknowns. The coarsest grid predicts an unstable buffet mode, whereas no unstable eigenvalue is found on the finer grids within the shifts searched. The finer-grid gain peaks are consistent with collective amplification by damped modes, whose responses retain the buffet-cell structure and travel outboard at approximately $0.3$ of the free-stream speed. The peak gain remains sensitive to resolution despite these similar structures. These findings motivate future mesh adaptation targeted at convergence of flow stability and disturbance amplification.