A minimal wake-vortex model explains formation flight of flapping birds

Abstract

Collective patterns of motion emerge across biological taxa: insects swarm, fish school, and birds flock. In particular, large migratory birds form strikingly ordered V-shaped formations, which experiments and direct numerical simulations have demonstrated provide substantial energetic benefits during long-distance flight. However, the precise aerodynamic and morphological mechanisms underlying these benefits remain unclear. In this work, we develop a reduced-order model of the wake-vortex interactions between two flapping birds flying in tandem. The model retains essential unsteady flapping dynamics while remaining computationally tractable. By optimizing over a six-dimensional state space, which comprises the follower's three-dimensional relative position and three independent flapping parameters, we identify the energetically optimal leader-follower configuration of northern bald ibises. The predicted optimum agrees quantitatively with live-bird measurements. Because of its simplicity, the model allows for direct interrogation of the physical mechanisms responsible for this optimum. In particular, it isolates precisely how the follower's wing kinematics interact with the leader's wake to enhance aerodynamic efficiency. The model predicts an 11% reduction in total mechanical power for a follower in formation flight -- consistent with experimental estimates -- and shows that this saving arises from reductions in both induced and profile power, dominated by decreased profile power enabled primarily through reduced flapping amplitude and, secondarily, reduced upstroke flexion. These results provide a mechanistic explanation for the structure of V-formations and offer new insight into the aerodynamic principles governing collective flight.

Figures (5)

• Figure 1: Constant-circulation vortical wake of a flapping bird at the end of its upstroke, adapted directly from Spedding et al. spedding2003family. (a) Approximation of true wake structure: wings sweep out on the downstroke, forming curved filaments (green), and contract during the upstroke, forming straight filaments (blue). This contraction is modulated by the upstroke flexion ratio $R\in(0,1)$. Cross-stream vortices in the spanwise direction $y$, denoted by dashed lines, have zero circulation. (b) A simplified version of (a), again adapted directly from spedding2003family, constructed to succinctly express upstroke and downstroke impulses mathematically. The wake is modeled as a series of vortex filaments, which form closed rectangles and ellipses. The resulting impulse $\vec{I}$ always points upward in the $z$ direction, but due to wingspan contraction on the upstroke, the projected area of the elliptical downstroke region is larger, which generates a net propulsive impulse in the $-x$ (upstream) direction.
• Figure 2: Available leader-follower configurations: up-up (a), up-down (b), down-up (c), and down-down (d). Rankine vortex filaments, which comprise an approximation of the leader's unsteady vortical wake, are indicated by straight green (downstroke) and blue (upstroke) lines. The leader's wake is a simplified version of that in Fig. \ref{['fig:leader_impulse']}(b): here, the elliptical downstroke wake portion is reduced to a rectangle, akin to the upstroke portion, for simplicity. Both the upstroke and downstroke rectangle widths are modified by a wingtip vortex roll-up factor of $\pi/4$hummel1983aerodynamichainsworth1988induced. Vortices are endowed with signed constant circulation $\pm\Gamma$ (purple). The follower is modeled as a lifting line directed along the spanwise $y$-axis, with symmetric circulation distribution such the circulation at each endpoint is equal to $\Gamma$. The net wake-induced force on the follower ($\vec{F}_{uu}$, $\vec{F}_{ud}$, and so on) is marked by a black arrow in each panel (arrow not to scale). The follower "flaps" by heaving up and down in the $z$ direction with amplitude $h_F/2$.
• Figure 3: Results of optimization procedure. (a) Optimal position of the follower (magenta) relative to its leader (cyan) in the dimensionless $(x/\lambda, y/b)$ plane. For comparison, the average experimental observation of Portugal et al. portugal2014upwash is shown by a green dot. The dashed vertical line has equation $x/\lambda=0.5$, and the dashed horizontal line has equation $y/b=\pi/4 + (1+R_F)/2$. Bird silhouettes are meant to orient the reader and are not to scale. (b) Optimal position in the dimensionless $(x/\lambda, z/h_L)$ plane. The extrapolated wakes of the leader (cyan) and follower (magenta) align closely, demonstrating leader-follower wingtip path coherence in the optimal solution. (c,d) Wake-induced force components, computed at the optimal solution, which are used in Eq. (\ref{['eqn:total_induced_force']}) to determine the total force on the follower. Panel (d) is an inset of panel (c).
• Figure 4: Planar sections of normalized vertical and horizontal net forces $F_z^F/W$ (a,c,e) and $F_x^F/D$ (b,d,f) on the follower. The follower's optimal dimensionless location in state space is shown by a magenta dot in each section. This location precisely satisfies $F_z^F/W = F_x^F/D=1$. The leader's location is shown by a cyan dot at $(x_L,y_L,z_L)$. Force balance is more closely satisfied in regions of red, whereas the follower produces insufficient force in non-red regions. The variation of $F_x^F$ is much larger than that of $F_z^F$, even dropping as low as 50% of the total thrust requirement -- while the vertical force only drops as low as about 90% of the total lift requirement. Bird silhouettes are meant to orient the reader and are not to scale.
• Figure 5: Schematic of the follower's (magenta) optimal spanwise positioning relative to the leader (cyan) in the down-up (a) and up-down (b) model configurations. This optimal position is such that the follower's wingtips lie just outside of the nearest streamwise vortex filament. Downstroke and upstroke wake vortex filaments are shown respectively in green and blue. The cycle-averaged optimal distance is then the average of these two distances. (a) Down-up configuration. The optimal wingtip-to-vortex spanwise distance is $y = \pi b/4 + R_F b$. (b) Up-down configuration. The optimal wingtip-to-vortex spanwise distance is $y=\pi b/4 + b$.