Fig 1.
Control volume around a bird and outflow plane in its wake.
The tubes behind the bird give a simplified illustration of the tip vortices of the bird, and the red and blue shapes in the outflow plane illustrate the streamwise component of the vorticity field, with the color corresponding to its sign. The axes of the reference frame used throughout the paper are also represented.
Fig 2.
Various models are used to capture the bird in the simulation framework.
(A): Picture of a real bird while gliding (Original photograph from Peter W. Hills, reproduced with permission). (B) bio-mechanical model of the skeleton and feathers, with the actuated joints being highlighted and two rotations of the wrist being represented as examples. (C) Immersed Lifting and Dragging Line (ILDL) obtained in a particular configuration. (D) final planform represented in the fluid solver. (E) simulated bird and its near wake.
Table 1.
Kinematic parameters used in the simulations for the right wing (the left wing is symmetric).
Fig 3.
Time evolution and spanwise distribution of angle of attack α, lift coefficient CL and vertical and horizontal force coefficients (CZ and CX) through a wingbeat.
Snapshots of the bird during flight are represented above the figure, showing the phase of the flapping motion and the flight direction.
Fig 4.
Scenario 1: Volume rendering of the norm of the vorticity field in the wake of a bird (flying from right to left).
Panels A and B show a top view, C and D an oblique view and E and F a side view. Panels A, C, E represent the instantaneous field at a certain point in the flapping cycle. Panels B, D, F illustrate what can be measured by a single outflow plane by ‘extruding’ the 2D. In panel A, we added a sketch of the main vortical structures of the wake. The red lines of the sketch represent structures shed during the upstroke, and the blue lines during the downstroke.
Fig 5.
Scenario 1: Spanwise component of the vorticity field at various positions behind the bird (on the right) and time derivative of the wing circulation at that spanwise position (on the left), represented by color points at the successive locations of the wing.
In both cases, the bird is flying from right to left, and on the right, x = 0 corresponds to a distance of 10 chords behind the bird.
Fig 6.
Scenario 1: Four snapshots of the streamwise component of the vorticity field in a cross flow plane 10 chords behind the bird.
The observed field corresponds to the wake produced when the wing is in the topmost position (A), mid-downstroke (B), at its downmost position (C), and mid-upstroke (D). The arrows represent the velocity in the plane.
Fig 7.
Scenario 1: Subfigure (a): total vortex lift (adimensionalized with the weight W of the bird) obtained through the ILDL (dashed black) and using Eq (4) (blue), compared to an estimation of the added mass force (green). Subfigure (b): decomposition of the terms of Eq (4) into the corresponding terms of Eq (3).
Fig 8.
Scenario 1: Evaluation of both terms of the lift estimation over time for several downstream positions xout of the outflow plane.
(a) xout/λ = 0.25. (b) xout/λ = 1. (c) xout/λ = 1.75. (d) xout/λ = 2.5.
Fig 9.
Scenario 1: KJ estimations of the instantaneous lift for several positions of the outflow plane xout, shifted in time with Δt = xout/U∞.
Fig 10.
Scenario 1: Average the lift estimation as a function of the downstream distance.
Fig 11.
Scenario 2: Estimation of the lift for non-periodic flapping cycles.
The upper figure shows the estimations of the instantaneous lift and the lower one of the lift averaged over the previous flapping period (), for several distances xout of the outflow plane.