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.
Figures & tables
Figure 1: The typical section of Isogai’s case A, drawn to scale. The elastic axis lies ∣a∣b=c ahead of mid-chord, half a chord ahead of the leading edge, and the centre of gravity xαb aft of it.
Figure 2: Meshes and steady flow. (a) Near field of mesh L1 (upper half) and steady Mach number at M∞=0.85 , zero incidence (lower half; sonic line white, cross: elastic axis of case A). (b–d) Leading-edge region of meshes L0–L2. (e) Upper-surface pressure coefficient at M∞=0.85 and 0.90 on the three meshes.
M∞
L0
L1
L2
p
cd,∞
0.85
0.023233
0.023322
0.023233
–
–
0.90
0.070917
0.072847
0.073184
2.5
0.073255
Table 1: Steady rigid NACA 64A010 at zero incidence: drag coefficient on the three meshes, observed order p and Richardson extrapolation cd,∞ .
Figure 3: Damping identification along a scan of V∗ at M∞=0.85 , a=−2 . (a) Pitch history with the identification window (grey) and the fitted envelope (dashed). (b) Cycle-peak pitch amplitude, normalised by the first peak, for every V∗ . (c) Damping coefficient of the critical mode and the interpolated Vf∗ . (d) Its frequency.
Setting
Variant
Change of Vf∗
Steps per pitch period
128 (production)
+0.02%
(relative to 256)
64
+0.19%
36 (SU2 test case)
+1.2%
32
label reversed
Inner loop
80 iterations, CFL 10, coupling every 3
0.02%
residual stop 10−8
∣Δγ∣≤2.5×10−5
Table 2: Sensitivity of the flutter speed index at the bottom of the dip ( M∞=0.85 , a=−2 ; production: mesh L0, 128 steps per pitch period, 12 periods, zero mean incidence, Vf∗=0.5224 ), from runs at the two flutter speed indices that bracket the boundary of each variant. Time-step rows are relative to 256 steps per period, the others to production.
Figure 4: Three silent failure modes of damping labels. (a) Reported pitch and geometric rotation of the chord line, M∞=0.85 , a=−2 , V∗=1.7 , for the two mesh-deformation paths (they coincide for the volumetric movement); (b) lift coefficient. (c) Growth rate of an early–late envelope estimator against peak pitch for 40 runs of an earlier campaign; colours: label after re-identification, open symbols: wrong labels. (d) Damping coefficient at the two flutter speed indices that bracket the boundary against the number of steps per pitch period; inset: Vf∗ relative to the 256-step value (open symbol: extrapolated).
Figure 5: Effect of the start of the motion at M∞=0.85 , a=−2 . (a) Damping coefficient at V∗=0.53 and 0.56 against the mean incidence (freestream incidence plus mean pitch); filled: impulsive starts, open diamonds: release from a steady solution, crosses: indeterminate; the annotated pair differs eightfold in amplitude. (b) Interpolated Vf∗ against the mean incidence (numbers: pitch amplitude), with the published values of Marques et al. [13] and Méndez [21] ( 1∘ incidence).
Figure 6: Flow along the case A boundary: steady Mach number at 1∘ incidence, the state from which every simulation is released (mesh L0), at five Mach numbers on the lower branch, at the bottom of the dip, in the fold and on the restabilisation branch. White: sonic line; arrows: shock positions on the two surfaces; xs : upper-surface shock.
Figure 7: The case A flutter boundary learned from 80 coupled Euler simulations (mesh L0, zero mean incidence). Colour: posterior mean yˉ of y=tanh(γ/s) ; thick line: learned boundary yˉ=0 ; dashed: calibrated 90% band from 65 independent random simulations ( q=1.59 ). Circles: initial design; diamonds: chosen by the learner; open squares: Marques et al. [13] ; star: Méndez [21] at 1∘ incidence.
Figure 8: Learning efficiency and calibration for case A. (a) Fraction of the domain where the map after n runs disagrees with the map from all runs, for active learning and random sampling from the same 16-run design. (b) Wrongly labelled held-out random runs (mean and standard error, 40 splits); open circles: active learning scored on all 65. (c) Empirical coverage and correct calls outside the band on held-out halves (200 splits) against the nominal level; numbers: share of runs that receive a call.
Figure 9: One flutter cycle at the bottom of the dip: M∞=0.85 , a=−2 , V∗=0.53 (unstable), mesh L1. (a) Pitch over the last cycle and the four instants shown. (b) Mach number with the sonic line (white), the shocks (arrows) and their positions on the upper and lower surfaces. (c) Numerical schlieren ∣∇ρ∣c/ρ∞ . The shock lags the pitch: of the four instants, the upper shock is furthest aft at 2, after the nose-up extreme.
Figure 10: Surface pressure and shock motion at M∞=0.85 , a=−2 (mesh L1), last three cycles. (a) Pitch; (b) space–time map of the upper-surface Cp at V∗=0.53 , centred on Cp∗ , with the shock track; (c) its unsteady part; (d) Cp at four phases and its envelope; (e) shock position against pitch, normalised by the pitch amplitude, below and above the boundary. The weak inflections near x/c=0.1 and 0.4 follow curvature changes of the tabulated section.
Figure 11: Where the flow feeds the flutter mode, M∞=0.85 , a=−2 , mesh L1, last four cycles. (a) Quadrature part of the pressure at the critical pole, Im(p1/α1)/q∞ , at V∗=0.53 , with the mean sonic line (black); the aerofoil contour is coloured by the local work per cycle w/(q∞α^2) done by the flow (red: into the structure). (b) The shock foot on the upper surface. (c) Running integral of w from the leading edge over both surfaces, below and above the boundary, with the net work per cycle.
Figure 12: The elastic-axis family from 193 coupled simulations over (M∞,a,V∗) , mesh L0, zero mean incidence, calibrated by 40 independent random simulations ( q=1.535 at 90%). (a) Learned boundary for seven elastic-axis positions. (b) Flutter speeds along V∗ at each (M∞,a) : none, a single one, a fold not excluded by the calibrated band, and a calibrated fold (a confidently unstable V∗ below a confidently stable one); dots: simulations, crosses: calibration set. (c) Bottom of the dip with its calibrated band, and its Mach number, against the elastic axis.
Figure 13: Restabilisation on the upper branch: V∗=1.2 , a=−2 , mesh L1, at M∞=0.896 (unstable, red) and 0.905 (stable, blue). (a–d) Mach number at the nose-up and nose-down extremes of a cycle (for the decaying stable run, its largest cycle after start-up), with the upper-surface shock position. (e) Shock position against pitch over the last cycle, normalised by the pitch amplitude, with the mean position and lag. (f) Running integral from the leading edge of the work per cycle on both surfaces; end values: net work.
Figure 14: Why the fold closes: one flow, three elastic axes ( M∞=0.905 , V∗=1.2 , mesh L1, shock at the trailing edge). (a–c) Identified flutter mode, drawn at a pitch amplitude of 4∘ at eight phases, with the elastic axis (circle) and the pivot of the mode (cross); titles give the damping over the last four cycles. (d) Running integral of the work per cycle from the leading edge, both surfaces. (e) Work per cycle of the band swept by the shock ( 0.90≤x/c≤1 ), of the surface ahead of it and in total, against the elastic axis.
William Wright Technology Centre (W-Tech) School of Mechanical & Aerospace Engineering Queen’s University Belfast 50 Malone Road, Belfast, BT9 5BS, United Kingdom
Aerial Platforms Department Subdirectorate General for Aeronautical Systems Spanish National Institute for Aerospace Technology (INTA) · Department of Aerospace Engineering Universidad Carlos III de Madrid Madrid, Spain · TUAREG Research Group Universitat Politècnica de Catalunya (UPC) CASE Large-Scale Computational Fluid Dynamics Barcelona Supercomputing Centre (BSC) Barcelona, Spain