The phase-locked average images of the vortex wakes reveal a boundary between a region of periodic (crisp images) and chaotic (blurred images) wakes as a function of flapping kinematics; (A
* = 1, 2) and a
* = 3) (see the electronic supplementary material, figure S1 for A
* = 0, 4). Several crisp images of common periodic wake types can be found at relatively low wavelengths for A
* = 1, such as ‘two vortex pairs’ (λ
* = 5, α0
= 15°, 30°) and ‘two single vortices’ (λ
* = 5, α0
= 0°) per flap period (). Such periodic vortex configurations are both found in the wakes of fishes as a function of swimming kinematics (e.g. Müller et al. 2001
; Borazjani & Sotiropoulos 2008
). At shorter wavelengths and higher amplitudes, the wakes are chaotic resulting in blurred images of the wake, the border between periodic and chaotic vortex wakes shifts to higher wavelengths for higher amplitudes.
Figure 2. (a) Phase-locked averages wake images for A* = 3 (). (b) Pooled normalized standard deviation of I0, I1,r and I2,r (with respect to maximum value) as a function of αind at constant α0 shows approximately exponential growth. (more ...)
We find that the normalized standard deviation in I0, I1,r and I2,r is similar valued (see the electronic supplementary material, figure S2 for evidence) and grow roughly exponentially as a function of αind at constant α0, b. This illustrates the dramatic growth in variance in the moment of area integrals owing to chaos. We define the chaos boundary at 15 per cent of the maximum standard deviation in I0, I1,r and I2,r in b and capture it through fitting the combined normalized standard deviations of I0, I1,r and I2,r with an exponential function as a function of αind at constant α0 (0°, 15°, 30°, 45°, 60°; we excluded 75°, because data lacked for a proper fit) and calculating the intersection.
To determine how relevant chaos could be for our understanding of animal propulsion through fluids, we plot our proxy for chaos, the standard deviations of I0, I1,r and I2,r (§2), as a function of effective and induced angle of attack, c. This condensed plot for A* = 1–4 features our linear approximated borderline between chaotic and periodic wakes, which we calculated as follows. We linearly fitted the intersection values of αind at 15 per cent of the maximum standard deviation in b and the corresponding values of αeff (plotted in c). The linear fit αeff = 247° − 3.15αind describes the 15 per cent maximum standard deviation boundary between chaotic and periodic flows well (r2 = 0.99). This boundary yields similar chaos boundaries in figures a,b and a: λ* = 2π × A*/tan(50° + α0/4) (rounded coefficients) that match well with the transition between sharp (periodic) and blurry (chaotic) wake images.