How the elastic axis folds the transonic flutter boundary: shock energy transfer and conformally calibrated active learning
Isogai's case A, a pitch-plunge aerofoil whose elastic axis lies half a chord ahead of the leading edge, has a folded transonic flutter boundary in inviscid flow: near the bottom of the transonic dip several flutter speeds coexist at one Mach number. We map this boundary, and the family obtained by moving the elastic axis, with coupled unsteady Euler simulations, a Gaussian-process surrogate of the damping coefficient trained by batch active learning, and split conformal prediction on independent random simulations, which turns the surrogate into stability calls with a distribution-free error rate. Three failure modes of time-domain damping labels that leave no trace in the residuals are documented, and the half-degree mean incidence left by the common one-degree start is shown to raise the flutter speed at the dip by 2.8%, more than any numerical setting. With 80 simulations the learned case A boundary labels 64 of 65 independent runs correctly, and calibrated calls are wrong in at most 1.6% of cases. Over Mach number, elastic axis and flutter speed index, 193 simulations and 40 calibration runs show that the fold exists only for elastic axes more than about 0.4 chord ahead of the leading edge; it closes between 0.39 and 0.33 chord ahead, on two meshes. Chordwise energy budgets, closed against the solver loads and the growth rate within a few per cent, explain the boundary. The lower branch is crossed through the reduced frequency at unchanged shock phase. On the upper branch the shock is pinned at the trailing edge, and moving the elastic axis aft brings the pivot of the flutter mode towards the section, so that the damping of the supersonic surface falls faster than the residual shock input and the fold closes.