Scaling of the Drag on Flapping Flags in Non-Chaotic Flutter Regime

Fluttering flags represent a fluid–structure interaction (FSI) problem that exhibits an interplay between fluid flow and structure. Despite their simple geometry, these systems exhibit complex dynamics influenced by flow speed, structural properties, and boundary conditions. Fluttering flags are widely studied for their relevance to energy harvesting, bio-inspired propulsion, and the design of flexible aerodynamic surfaces. In particular, drag plays a critical role by governing energy dissipation, wake dynamics, and aerodynamic efficiency. Researchers have investigated how unsteady motion, geometry, and structural parameters affect drag to enhance predictions and develop effective control strategies. This study aims to establish a general drag scaling for fluttering flags by introducing a new non-dimensional drag coefficient that incorporates tip velocity, flutter amplitudes, and geometric parameters. Experiments performed at the University of Idaho’s laminar jet facility measured unsteady aerodynamic forces using an ATI nano force/torque transducer and flow fields via particle image velocimetry (PIV). A coupled FSI-POD approach was used to quantify energy exchange between fluid and structure during limit cycle oscillations (LCOs). The low-order models allow for estimating features such as tip velocity and amplitudes (transverse and streamwise). The proposed scaling for the drag coefficient accounts for local unsteady motion, normalized oscillation amplitudes, and geometric parameters not captured by the traditional approach. This scaling of the drag coefficient collapses data across a wide range of Reynolds numbers, mass ratios, and flag geometries into a single curve, providing a compact, generalizable framework for analyzing non-chaotic fluttering flag dynamics.