Three-sensor machine-learning phase-amplitude reduction speeds airfoil wake synchronization eightfold while curbing lift fluctuations

A data-driven framework integrates sparse identification of nonlinear dynamics (SINDy), adjoint-based phase-amplitude reduction in a low-order sensor space, and a nonlinear decoder to approximate high-dimensional phase and amplitude sensitivity fields from three airfoil-surface sensor measurements. Full-field sensitivities have typically required a numerical adjoint solver; this work shows sensitivity fields can be reconstructed from sparse sensor readings alone, extending phase-amplitude reduction to experimental settings where no numerical solver is available. Evaluated across NACA0006, NACA0012, and NACA0020 airfoils at several post-stall angles of attack, all exhibiting unsteady periodic vortex shedding; the integrated low-order trajectory deviates from and returns to the limit cycle after perturbation, aligning with the high-dimensional flow behavior.

Optimal actuation waveforms are derived analytically to balance synchronization speed against amplitude deviation, with an amplitude penalty term introduced to constrain lift-coefficient fluctuation. Conventional phase-sensitivity-only optimal waveforms pursue fast synchronization alone and can amplify aerodynamic coefficient fluctuations; adding the amplitude penalty changes the optimal waveform from a single peak to two peaks, achieving fast synchronization without excessive amplitude growth. For NACA0012 at a post-stall angle of attack, sinusoidal forcing, optimal forcing without amplitude penalty, and optimal forcing with amplitude penalty are compared; lift-coefficient statistics show the amplitude-penalized waveform has a smaller max-minus-min range than the unpenalized optimal waveform, and lift-element fields show the penalized case introduces a secondary vortex structure that breaks down the large leading-edge vortex at a specific timing.

The spatial reliability of the reconstructed sensitivity fields is quantified, showing the region near the leading edge is reliable while the wake region carries higher uncertainty. By sweeping sensor locations and computing time-ensemble mean and standard deviation fields, the work clarifies where the approximate sensitivity fields can be trusted, informing actuation placement. For NACA0012, two sensors are fixed while the third sensor location is swept, and ensemble mean and standard deviation fields are computed; the standard deviation field shows high uncertainty in the wake region for both phase and amplitude sensitivity, while the near-airfoil leading-edge region shows reasonable reliability with a distinct maximum sensitive point in the ensemble-averaged field.

Sensitivity-field trends with angle of attack and airfoil thickness are characterized. The intensely sensitive region contracts as angle of attack increases and intensifies and expands as airfoil thickness increases, consistent with prior adjoint-based phase-reduction analysis of airfoil wakes. Time-averaged absolute sensitivity fields are evaluated across multiple angles of attack and three airfoil thicknesses; the text attributes contraction to earlier vortex separation and intensified wake unsteadiness at higher angle of attack, and expansion to lower leading-edge curvature of thicker airfoils delaying vortex detachment and mitigating unsteadiness.

The results apply to two-dimensional incompressible laminar post-stall airfoil wakes at a chord-based Reynolds number of 100, for limit-cycle oscillators with periodic vortex shedding where the lift coefficient defines the phase. The method works under the setting of only three airfoil-surface sensors with actuation placed near the leading edge in the reliable sensitive region, suited to practical scenarios where physical actuation is typically applied near the wing. The authors note that because the framework relies primarily on data availability rather than field-specific governing equations, it may be applicable to periodic or rhythmic systems beyond aerodynamics, including chemical or biological systems.

The reconstructed sensitivity fields are approximations through a nonlinear decoder rather than exact adjoint derivations, with reliability concentrated near the leading edge and higher uncertainty in the wake region; the authors note that more actuation points might be needed to obtain sensitivity fields with less uncertainty near the wake. Current validation is limited to two-dimensional low-Reynolds-number airfoil wakes, and extension to higher Reynolds numbers, three-dimensional flows, and flows modeled as multi-frequency oscillators remains an open question. In addition, several equations, parameter values, and figures are not fully rendered in the provided text, so readers needing to reproduce specific settings should consult the original figures and tables.