Flapping dynamics of a flexible flag clamped vertically in a viscous uniform flow

A numerical simulation of a single flexible flag clamped vertically (here, ‘vertical flag’) in a viscous uniform flow is performed to explore the flapping dynamics of the flag. Four typical flapping modes of the vertical flag are identified as the bending rigidity (γ) and mass ratio (μ) vary at a fixed Reynolds number Re = 200, i.e., biased, flapping, deflected and deflected-flapping modes. As μ increases, the flapping state of the vertical flag changes from the deflected mode to the flapping mode and to the deflected-flapping mode, whereas it is reversed with an increase in γ. For a vertical flag in the flapping mode, a continuous increase of the mean amplitude is observed, and the drag coefficient is proportional to the streamwise amplitude of the flag. In addition, when the vertical flag is in the flapping and deflected modes, the streamwise amplitude of the trailing edge is characterized by two peaks with different amplitudes due to the process of forming the negative leading edge vortex. A comparison of the deflected-flapping mode of the vertical flag with the flapping mode of a single flexible flag where the leading edge of the flag is fixed at an upstream location with a free trailing edge (here, ‘conventional flag’) indicates that the amplitude and frequency of the conventional flag are greater than those of the vertical flag in the irregular flapping state with accompanying vortex dynamics. An inspection of snapping events with rapid changes in the drag and acceleration of the flags shows that the number of snapping events for the conventional flag is higher than that for the vertical flag, indicating the presence of more intermittent and violent irregular flapping for the conventional flag. Finally, a comparison of the hysteresis phenomena between the vertical and conventional flags suggests that the loop width of the vertical flag is larger than that of the conventional flag due to the high elastic restoration energy.