A Wing That Remembers Its Acceleration ✛

Original: "Unsteady Thin-Airfoil Theory Revisited: An Approximate Analytical Solution and the Self-Similar Wagner Effect in Viscous Flows"

· Tianshu Liu, Jianfeng Lin, Shizhao Wang

arXiv:2609.05839v1 · 2026-09-05 · CC BY 4.0 · 2 min

Fluid Dynamics

A new formula links the growth of wing lift to the viscous wake and shows that Wagner's law holds even with flow separation.

An airplane wing, like a skier on a ski jump, remembers its acceleration. Change the angle of attack sharply, and the lift does not immediately reach its new value. It builds up gradually, as if the wing is not yet sure the maneuver is for real.

This effect was discovered by engineer Herbert Wagner in 1925. He described the growth of lift after an instantaneous change in angle of attack, but his formula remained without an explicit expression. For nearly a century, Wagner's function existed only in tables. Now the authors have derived a formula that links the vortex wake to the quasi-steady circulation of the wing.

The vortex wake consists of the vortices that shed from the trailing edge and carry away some energy. They are like the wing's "memory" of past motions.

The key move was to approximate the kernel of Wagner's integral equation with a function that can be inverted analytically. The error of this approximation is about one percent. This allowed them to obtain a convolution expression for the intensity of the vortex sheet. It turned out that the renormalized vortex lift is self-similar: curves for different circulation growth times collapse onto a single universal curve.

Numerical simulations of flow past a starting plate at angles of attack of 5° and 10° and Reynolds numbers from 126 to 1000 confirmed the self-similarity. Even with moderate flow separation, the data fell onto a single curve. The time to reach the local minimum of lift follows a power law with an exponent of −1/5. The acceleration component of the force is also self-similar, but with different scales, so the total force is not self-similar.

The Reynolds number is the ratio of inertial forces to viscous forces. The smaller it is, the more important viscosity becomes. In the range 126–1000, viscosity cannot be ignored, yet self-similarity persists.

A wing that remembers its acceleration can now be described by a simple formula. This matters for engineers designing flapping wings, helicopter rotors, and wind turbines. Viscous effects are no longer an obstacle to predicting lift in transient regimes.

Whole graph →

The Wagner function, which describes the lift buildup after a sudden change in angle of attack, was first published in 1925 as tables because no explicit formula existed for nearly a century.

In Stanisław Lem's novel The Invincible, mechanical 'flies' are described that fly by rapid wing beats in a dense atmosphere. Understanding unsteady aerodynamics of viscous flows, as achieved in this work, brings us closer to describing such hypothetical flying machines.

\gamma_w(s) = -\frac{c_1}{2\pi} \int_0^s \Gamma_0'(t) S_0(s-t)\,dt - \frac{c_2}{c_3} \int_0^s \Gamma_0'(t) S_1(s-t)\,dt

The wake intensity is expressed as a convolution of the derivative of the quasi-steady circulation with Green's functions S0 and S1

\tilde{C}_{lcir}(\tilde{t}) = \frac{C_{lcir}(t,\tau_{rise}) - C_{lcir,min}(\tau_{rise})}{C_{lcir,max}(\tau_{rise}) - C_{lcir,min}(\tau_{rise})}

Renormalization removes the dependence on circulation rise time, revealing a universal curve

Concepts

Wigner function wind stress Archimedes' principle Wheeler-DeWitt equation empirical orthogonal function Gauss's theorem vorticity velocity distribution viscosity Poiseuille's equation ornstein uhlenbeck process Navier-Stokes equations wald entropy

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A Wing That Remembers Its Acceleration ✛