Oblique Waves Lift the Flapping Flag

The flapping of the flag is a classical model problem for the understanding of fluid-structure interaction: How does the flat state lose stability? Why do the nonlinear effects induce hysteretic behavior? We show in this Letter that, in contrast with the commonly studied model, the full three-dimensional flag with gravity has no stationary state whose stability can be formally studied: The waves are oblique and must immediately be of large amplitude. The remarkable structure of these waves results from the interplay of weight, geometry, and aerodynamic forces. This pattern is a key element in the force balance which allows the flag to hold and fly in the wind: Large amplitude oblique waves are responsible for lift.

Figure 1

General representation of the flag which is discussed in the Letter. (a) Popular conception of the flapping flag with vertical wave crests. (b) A first pattern of oblique waves. (c) An alternative pattern of accordion waves. (d) Naming of the flag’s parts in analogy to quadrangular sails. (e) Numerical simulation of a flag in the wind by Ref. 13. (f) A flapping flag on a windy day.

Figure 2

Geometrical justification of the oblique waves. (a) Oblique pattern obtained by geometrical deformation of the cloth. (b) Sketch of the oblique surface force applied on the cloth: weight plus fluid drag. (c) Analogy with a tilted cloth and definition of the roach: zone $B$, which would collapse under the cloth weight. (d) Oblique pattern of waves resulting from the collapse of the roach and comparison with a banner aligned with the external oblique force such that no roach is created.

Figure 3

Measurement of the wave angle. (a) Low Reynolds number; the wave angle obtained by a balance of aerodynamic drag and weight for a laminar flow from Eq. (1), compared to numerical simulations from Ref. 13. The flag is colored in blue (red) for positive (negative) deflections of the cloth. (b) High Reynolds number; wave angle estimated from turbulent boundary layer drag using Eq. (2) (dashed line) and with a force magnified to account for extra drag due to large amplitude wave motion (solid line). The flag motion is shown in a movie included as Supplemental Materials [21].

Figure 4

Three causes of lift due to oblique waves. (a) Aerodynamic resistance due to a large wave amplitude. (b) Lift through downpush of the stream. (c) Lift through snapping at the peak.