On the wake and flapping dynamics of different aspect ratio flags

The flapping of flags is a classical problem involving fast and large amplitude deformations of a thin flexible plate and unsteady flow phenomena. We perform systematic time and space-resolved measurements of the deformation and drag acting on flapping flags for various aspect ratios and mass ratios. Bending waves travel from the root to the tip at a speed close to the incoming flow and the typical wavelength of the waves scales with the length of the flag. With smaller aspect ratio, the local dynamic pressure exerted by the fluid on the flag is reduced, lowering the wave propagation speed, and reducing the tip frequency. The effect of aspect ratio on the vortex formation in the near wake is analysed using flow field measurements. We identified two characteristic length scales, the ratio of the flag area over its perimeter $L^*$ and the square root of area $\sqrt{HL}$ that scale the circulation shed during a cycle. Changes in aspect ratio and mass ratio generate a wide scattering of the mean drag coefficient, ranging from 0 to 0.55. We discuss a kinematic-based model for the mean drag coefficient. This model uses the mass ratio and the typical tip speed, which depends linearly on the wave speeds, to predict the mean drag coefficient without any fitting parameter.

💡 Research Summary

This paper presents a comprehensive experimental investigation of the post‑critical flapping dynamics of rectangular flags with varying aspect ratios (AR = H/L) and mass ratios (M*). Using thin paper sheets (80 g m⁻²) cut into 48 different geometries (L = 10–20 cm, H = 4–19.6 cm), the authors explored AR from 0.22 to 1.92, M* from 1.4 to 2.8, and reduced velocities U* = U∞ L ρs e/D between 7 and 31, corresponding to Reynolds numbers of 3 × 10⁴–1.5 × 10⁵.

The experimental set‑up consisted of an open‑section wind tunnel (45 cm × 45 cm) where each flag was clamped vertically at its root. Flow speed was increased stepwise to trigger flutter, then decreased to capture hysteresis and determine an offset velocity (the lowest speed at which significant flapping persists). Flag deformation was recorded with a laser light‑sheet and an event‑based high‑speed camera, providing centre‑line coordinates at 1.5–2 kHz for 50 equally spaced points along the flag. Drag forces were measured simultaneously with a calibrated load cell (2 kHz sampling). For a subset of flags (L = 16 cm, M* = 2.27) particle‑image velocimetry (PIV) was performed near the tip to resolve the near‑wake vortex structures.

Key findings include:

1. Critical and offset velocities – The onset reduced velocity is around U* ≈ 18 for the representative case, after which tip amplitude jumps to ≈ 0.27 L and plateaus at ≈ 0.31 L. The offset reduced velocity varies between 12.8 and 17.2, showing only weak dependence on AR or M*. Experimental points lie between predictions of two‑dimensional potential‑flow models (Argentina & Mahadevan 2005; Shelley et al. 2005) and a three‑dimensional model (Eloy et al. 2007).

2. Bending‑wave propagation – Bending waves travel from root to tip at a speed close to the free‑stream velocity (c_w ≈ U∞) and have a wavelength proportional to the flag length (λ ≈ L). Smaller AR reduces the local dynamic pressure, which in turn lowers c_w and the tip‑frequency f_tip. Consequently, tip speed U_tip = 2π f_tip A_tip decreases with decreasing AR.

3. Near‑wake vortex scaling – The circulation shed per flapping cycle, Γ, scales linearly with two geometric lengths: the ratio of flag area to its perimeter (L* = A/P) and the square‑root of the area (√(HL)). This dual scaling captures three‑dimensional effects that are absent in purely 2‑D analyses. As AR diminishes, vortex tubes become weaker and less coherent.

4. Mean drag coefficient – Measured drag coefficients span a wide range (C_D = 0–0.55). High drag correlates with large AR and high M*, where tip speeds are greatest; low drag occurs for slender, light flags where the dynamic pressure is reduced.

5. Predictive drag model – The authors propose a kinematics‑based model that predicts the mean drag coefficient solely from the mass ratio and the tip speed, which itself depends linearly on the bending‑wave speed. No fitting parameters are required, and the model reproduces the experimental C_D values across all tested geometries.

The study bridges the gap between flag geometry, flapping kinematics, wake dynamics, and aerodynamic loading. By quantifying how AR and M* control wave propagation, vortex shedding, and drag, the work provides practical design guidelines for applications such as energy harvesting (piezoelectric patches placed at high‑bending regions), micro‑scale mixers, and biomedical implants where flutter‑induced shear stresses are critical. The parameter‑free drag model, in particular, offers a fast predictive tool for engineers to assess performance without extensive CFD or wind‑tunnel testing.