A three-dimensional computational model is developed for simulating the flag motion in a uniform flow. The nonlinear dynamics of the coupled fluid–flag system after setting up of flapping is investigated by a series of numerical tests. At low Reynolds numbers, the flag flaps symmetrically about its centreline when gravity is excluded, and the bending in the spanwise direction is observed near the corners on the trailing edge. As the Reynolds number increases, the spanwise bending is flattened due to the decrease of the positive pressure near the side edges as well as the viscous force of the fluid. At a certain critical Reynolds number, the flag loses its symmetry about the centreline, which is shown to be related to the coupled fluid–flag instability. The three-dimensional vortical structures shed from the flag show a significant difference from the results of two-dimensional simulations. Hairpin or O-shaped vortical structures are formed behind the flag by connecting those generated at the flag side edges and the trailing edge. Such vortical structures have a stabilization effect on the flag by reducing the pressure difference across the flag. Moreover, the positive pressure near the side edges is significantly reduced as compared with that in the center region, causing the spanwise bending. The Strouhal number defined based on the flag length is slightly dependent on the Reynolds number and the flag width, but scales with the density ratio as St ~ ρ^−1/2). On the other hand, the flapping-amplitude-based Strouhal number remains close to 0.2, consistent with the values reported for flying or swimming animals. A flag flapping under gravity is then simulated, which is directed along the negative spanwise direction. The sagging down of the flag and the rolling motion of the upper corner are observed. The dual effects of gravity are demonstrated, i.e. the destabilization effect like the flag inertia and the stabilization effect by increasing the longitudinal tension force.

The immersed boundary method is a methodology for dealing with boundary conditions at fluid–fluid and fluid–solid interfaces. The immersed boundary method has been attracting growing attention in the recent years due to its simplicity in mesh processing. Great effort has been made to develop its new features and promote its applications in new areas. This review is focused on assessing the immersed boundary method fundamentals and the latest progresses especially the strategies to address the challenges and the applications of the immersed boundary method. Various numerical examples are also presented for demonstrating the capability of the immersed boundary method, including blood flow and blood cells, flapping flag, flow around a hoverfly, turbulence flow over a wavy boundary, shock wave-induced vibration, and acoustic waves scattered by a cylinder and a sphere. The major challenges and several open issues in this field are highlighted.

Two tandem flexible flags in viscous flow were modelled by numerical simulation using an improved version of the immersed boundary method. The flexible flapping flag and the vortices produced by an upstream flag were found to interact via either a constructive or destructive mode. These interaction modes gave rise to significant differences in the drag force acting on the downstream flapping flag in viscous flow. The constructive mode increased the drag force, while the destructive mode decreased the drag force. Drag on the downstream flexible body was investigated as a function of the streamwise and spanwise gap distances, and the bending coefficient of the flexible flags at intermediate Reynolds numbers (200 ≤ Re ≤ 400).