Module 3

Locomotion & Flight

Slotted wingtips, silent owl flight, fish undulation, insect clap-and-fling — and how engineers translate these into low-drag aerofoils and micro air vehicles.

3.1 Bird Wing Morphology: Slotted Wingtips and Induced Drag

Soaring birds (eagles, condors, storks) separate their primary feathers into 5–7 slotted wingtips. Each feather acts as an individual winglet, reducing induced drag by spreading wingtip vortices over multiple cores. Birds perfected this design before engineers discovered winglet benefits on commercial aircraft (Whitcomb 1976).

Derivation: Prandtl's lifting-line theory

Induced drag arises from the downwash from wingtip vortices. For a wing of span \(b\)generating lift \(L\) at airspeed \(U\), Prandtl (1918) showed:

\( D_i = \frac{L^2}{\tfrac{1}{2}\rho U^2 \pi b^2 e} \)

where \(e\) is Oswald's span efficiency. For an elliptical loading,\(e = 1\). Slotted wingtips effectively increase the span by distributing the vortex, reducing \(D_i\) by 10–15%.

Engineering application: winglets and spiroid tips

Richard Whitcomb at NASA Langley formalised winglets (1976); now ubiquitous on Boeing and Airbus aircraft (5% fuel savings). Slotted wingtip devices based directly on eagle feathers are under development (Guerrero et al. 2020).

Cross-reference

See our Avian Biophysicscourse for detailed feather aerodynamics and wing-loading scaling.

3.2 Owl Silent Flight

Owls (order Strigiformes) have acoustically stealthy wings allowing nocturnal predation on acute-eared rodents. Three morphological adaptations combine:

Leading-edge comb — 0.5 mm fine serrations on the primary feathers.
Trailing-edge fringe — soft, pliable barbs that break up Kutta-Joukowski trailing vortices.
Velvet wing surface — downy pile that absorbs high-frequency sound and reduces boundary-layer noise.

Derivation: Acoustic scattering at the comb

Lilley (1998) modelled the leading-edge comb as a series of coherent scattering elements. For a comb wavelength \(\Lambda\) and mean flow speed \(U\), the scattered acoustic pressure spectrum is:

\( S_{pp}(f) \propto \frac{1}{f^2}\left|1 - e^{i 2\pi f \Lambda / U}\right|^2 \)

Constructive interference between adjacent comb elements at wavelengths much greater than\(\Lambda\) preserves the low-frequency noise, while short wavelengths are scrambled by phase incoherence — a broadband reduction in the 1–10 kHz range that matches the hearing sensitivity of the owl's prey.

Engineering application: quiet fans and wind turbines

Geyer et al. (2010) applied owl-inspired trailing-edge serrations to subsonic fan blades, measuring 10 dB reductions in broadband noise. Siemens and GE now install serrated blade trailing edges on wind turbines (DinoTail and similar). Sharp & Howe (2014) provide design scaling rules for serration pitch and depth.

Python

script.py83 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Owl silent flight: leading-edge comb breaks up vortex shedding,
# reducing broadband noise. Simulate trailing-edge noise spectrum
# with / without comb using Brooks-Pope-Marcolini (BPM) model style.

freqs = np.logspace(1.5, 4.3, 400)  # 30 Hz - 20 kHz

# Baseline smooth trailing edge: Amiet-type spectrum
def baseline_noise(f, U=10):
    # 1/f^2 low-frequency roll-off times exp high-frequency
    return 60 + 10*np.log10((f/500)**2 / (1 + (f/500)**4))

# Owl feather serrations: broadband reduction of ~10 dB above 1-2 kHz
def owl_noise(f, U=10):
    base = baseline_noise(f, U)
    # Comb cuts high frequencies
    reduction = -10 * (1 - np.exp(-f/2500))
    return base + reduction

# Measured Bachmann & Herweg 2011 owl vs pigeon comparison (approx)
pigeon = baseline_noise(freqs)
owl = owl_noise(freqs)

fig, axes = plt.subplots(1, 2, figsize=(15, 6))
fig.patch.set_facecolor('#030712')

# Panel 1: noise spectra
ax1 = axes[0]
ax1.set_facecolor('#1a0a2e')
ax1.semilogx(freqs, pigeon, color='#f472b6', linewidth=2.2, label='Pigeon (smooth edge)')
ax1.semilogx(freqs, owl, color='#e879f9', linewidth=2.2, label='Owl (serrated comb + velvet)')
ax1.fill_between(freqs, owl, pigeon, alpha=0.2, color='#a78bfa', label='Noise reduction')

ax1.axvspan(500, 5000, alpha=0.1, color='#22d3ee')
ax1.text(1500, 20, 'Prey hearing range', color='#22d3ee', fontsize=10, alpha=0.8)

ax1.set_xlabel('Frequency (Hz)', color='white', fontsize=11)
ax1.set_ylabel('Sound Pressure Level (dB re 20 uPa)', color='white', fontsize=11)
ax1.set_title('Owl vs pigeon: trailing-edge noise spectrum', color='#f0abfc', fontsize=12)
ax1.legend(facecolor='#1a0a2e', edgecolor='#c084fc', labelcolor='white', fontsize=10)
ax1.grid(True, alpha=0.25, color='#a78bfa', which='both'); ax1.tick_params(colors='white')
for sp in ax1.spines.values(): sp.set_color('#c084fc')

# Panel 2: comb geometry and schematic of vortex breakup
ax2 = axes[1]
ax2.set_facecolor('#1a0a2e')

# Draw smooth and comb edges
# Smooth (top):
ax2.plot([0, 10], [3, 3], color='#f472b6', linewidth=3)
ax2.text(11, 3, 'Smooth edge', color='#f472b6', fontsize=11, va='center')
# Comb (bottom):
for i in range(20):
    x0 = 0 + i * 0.5
    ax2.plot([x0, x0 + 0.3, x0 + 0.45, x0 + 0.5], [1, 1.25, 1.0, 1], color='#e879f9', linewidth=1.5)
ax2.plot([0, 10], [1, 1], color='#e879f9', linewidth=1.5)
ax2.text(11, 1, 'Serrated comb', color='#e879f9', fontsize=11, va='center')

# Vortex street above
theta = np.linspace(0, 2*np.pi, 100)
for cx in [1, 3, 5, 7, 9]:
    ax2.plot(cx + 0.3*np.cos(theta), 4 + 0.3*np.sin(theta), color='#f472b6', linewidth=1.3, alpha=0.6)

# Broken up vortices below comb
for cx in np.arange(0.5, 10, 0.5):
    ax2.plot(cx + 0.08*np.cos(theta), 0.5 + 0.08*np.sin(theta), color='#e879f9', linewidth=0.8, alpha=0.45)

ax2.text(5, 4.8, 'Coherent vortex shedding -> tonal noise', color='#f472b6', fontsize=10, ha='center', fontfamily='monospace')
ax2.text(5, 0.1, 'Broken-up small vortices -> broadband, low-level', color='#e879f9', fontsize=10, ha='center', fontfamily='monospace')

ax2.set_xlim(-1, 16)
ax2.set_ylim(-0.3, 5.3)
ax2.axis('off')
ax2.set_title('Mechanism: comb breaks vortex shedding', color='#f0abfc', fontsize=12)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#030712')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

3.3 Fish Swimming: Undulatory to Carangiform

Fish exhibit a continuum of swimming modes (Breder 1926, Webb 1984):

• Anguilliform (eels) — whole-body wave; high manoeuvrability, efficient at low speed.
• Sub-carangiform (trout) — posterior half wave.
• Carangiform (tuna) — posterior third wave.
• Thunniform (marlin, mako shark) — localised to caudal peduncle + stiff lunate tail.
• Ostraciiform (boxfish) — tail-only oscillation.

The progression from eel to marlin corresponds to increasing speed and decreasing manoeuvrability. Reynolds-number regime shifts from\(\text{Re} \sim 10^3\) (larval fish) to \(\text{Re} \sim 10^7\)(sailfish).

Derivation: Lighthill's elongated-body theory

M. J. Lighthill (1960) treated a slender fish as an elongated body with varying cross-section\(S(x)\) undulating with lateral displacement \(h(x, t)\). The far-field mean thrust is:

\( T = \rho S(L)\left[ \frac{1}{2}\left\langle \left(\frac{\partial h}{\partial t}\right)^2\right\rangle - U\left\langle \frac{\partial h}{\partial t}\frac{\partial h}{\partial x}\right\rangle\right]_{x=L} \)

For a travelling wave \(h = A \sin(kx - \omega t)\) with wave speed\(c = \omega/k > U\):

\( T = \tfrac{1}{2}\rho S(L) A^2 \omega^2\, (1 - U/c) \)

Thrust peaks when \(U/c \approx 0.7\) — matching the Strouhal number\(\text{St} = fA/U \approx 0.25\) observed across fish and cetaceans.

Engineering application: biomimetic AUVs

MIT's RoboTuna (Triantafyllou & Triantafyllou 1995) demonstrated carangiform propulsion with \(\eta > 0.75\) — well above propeller efficiency at the same size. Festo AquaPenguin and AquaRay commercial demonstrators use Lighthill-inspired motions. Soft-robotic fish (Marchese, Katzschmann, Rus 2014) use fluid-elastomer actuators for whole-body undulation.

Python

script.py86 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Fish swimming regimes (Webb 1984, Lighthill 1960 elongated-body theory)
# Cost of transport (J / kg / m) as function of swimming speed for three gaits.

U = np.linspace(0.1, 3, 100)    # m/s

# Drag power: P_drag = 0.5 * rho * C_D * A * U^3
rho = 1025
C_D = 0.01
A = 0.005
P_drag = 0.5 * rho * C_D * A * U**3

# Propulsion efficiency varies by gait:
#   Anguilliform (eel): eta ~ 0.4
#   Carangiform (tuna): eta ~ 0.7
#   Thunniform (mackerel): eta ~ 0.85 at design speed
# Also basal metabolism roughly constant
P_basal = 2.0  # W

def cost_of_transport(U, eta):
    P_drag_i = 0.5 * rho * C_D * A * U**3
    P_total = P_basal + P_drag_i / eta
    body_mass = 1.0  # kg
    return P_total / (body_mass * U)

gaits = [
    ('Anguilliform (eel)', 0.40, '#f472b6'),
    ('Carangiform (tuna)', 0.70, '#e879f9'),
    ('Thunniform (marlin)', 0.85, '#22d3ee'),
    ('Squid jet (reference)', 0.20, '#94a3b8'),
]

fig, axes = plt.subplots(1, 2, figsize=(15, 6))
fig.patch.set_facecolor('#030712')

# Panel 1: Cost of transport
ax1 = axes[0]
ax1.set_facecolor('#1a0a2e')
for name, eta, c in gaits:
    cot = cost_of_transport(U, eta)
    ax1.plot(U, cot, color=c, linewidth=2.2, label=f'{name} (eta={eta})')
ax1.set_xlabel('Swimming speed U (m/s)', color='white', fontsize=11)
ax1.set_ylabel('Cost of transport (J / kg / m)', color='white', fontsize=11)
ax1.set_title('Energetics of four swimming modes', color='#f0abfc', fontsize=12)
ax1.legend(facecolor='#1a0a2e', edgecolor='#c084fc', labelcolor='white', fontsize=9)
ax1.grid(True, alpha=0.25, color='#a78bfa'); ax1.tick_params(colors='white')
ax1.set_ylim(0, 50)
for sp in ax1.spines.values(): sp.set_color('#c084fc')

# Panel 2: Lighthill elongated-body thrust vs waveform amplitude & freq
ax2 = axes[1]
ax2.set_facecolor('#1a0a2e')

# Lighthill thrust: T = 0.5 * rho * w(L) * A_tail * U_wave * (U_wave - U)
# For a propulsive wave speed c > swimming speed U
A_tail = 0.001  # m^2
L = 0.5         # body length

freqs = np.linspace(1, 10, 8)  # tail-beat frequency Hz
amps = np.linspace(0.05, 0.3, 8) * L  # amplitude as fraction of body length

AMP, FREQ = np.meshgrid(amps, freqs)
# Wave speed c = wavelength * f, assume wavelength = L
c_wave = L * FREQ
# Assume swim speed U_swim = 0.8 * c (typical Strouhal-optimal)
U_swim = 0.8 * c_wave

# Thrust per unit area ~ pi^2 * rho * f^2 * A^2 (Lighthill)
T = np.pi**2 * rho * FREQ**2 * AMP**2 * A_tail

im = ax2.contourf(AMP*100, FREQ, T, 20, cmap='magma')
plt.colorbar(im, ax=ax2, label='Thrust (N)')
ax2.set_xlabel('Tail-beat amplitude (% body length)', color='white', fontsize=11)
ax2.set_ylabel('Tail-beat frequency (Hz)', color='white', fontsize=11)
ax2.set_title('Lighthill elongated-body thrust map', color='#f0abfc', fontsize=12)
ax2.tick_params(colors='white')
for sp in ax2.spines.values(): sp.set_color('#c084fc')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#030712')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

3.4 Insect Flight: Clap-and-Fling and the Leading-Edge Vortex

For 30 years after Osborne (1951), engineers declared insect flight “impossible” by quasi-steady aerodynamics. The paradox dissolved when Weis-Fogh (1973) identified the clap-and-fling mechanism and, later, the stable leading-edge vortex (LEV) (Ellington et al. 1996).

Leading-edge vortex

When an insect flaps at high angle of attack (\(\alpha \approx 45^{\circ}\)), a bound LEV forms on the upper surface and remains attached to the wing through the entire down-stroke due to span-wise flow draining vortex core fluid. The Polhamus analogy gives the augmented lift:

\( C_L = C_{L,\text{pot}} + K_{\text{vort}} \sin^2\alpha \cos\alpha \)

The LEV contribution raises \(C_{L,\text{max}}\) to 2.5–3, vs 1.5 for a smooth 2D aerofoil. See our Bee Biophysicsand Insect Biophysicscourses for full flapping-flight quantification.

Clap-and-fling

At the top of the up-stroke, small insects (fruit flies, thrips) clap their wings together then peel them apart. The fling generates a strong LEV on each wing immediately, well before the flow would organise quasi-steadily. Weis-Fogh showed this boosts\(C_L\) by \(\sim 50\%\) — crucial for the smallest flyers where the Reynolds number (and therefore LEV attachment time) is marginal.

Engineering application: flapping MAVs

• Festo SmartBird (2011) — 450 g herring-gull analogue, L/D = 14.
• RoboBee (Harvard, Wood 2013) — 80 mg piezo-actuated.
• DelFly (TU Delft, 2008–present) — 20 g flapping MAV.
• AeroVironment Nano Hummingbird (2011) — 19 g flapping, 11-minute flight.

Python

script.py78 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Quasi-steady flapping-wing flight (Sane 2003)
# Lift coefficient vs angle of attack for insect wings (leading-edge vortex)
# reaches CL_max ~ 2.5 - 3 (much higher than smooth airfoil's ~1.5)

alpha_deg = np.linspace(-10, 90, 200)
alpha = np.deg2rad(alpha_deg)

# Smooth airfoil (2D, Kutta-Joukowski):
CL_smooth = 2*np.pi*alpha
CL_smooth[alpha_deg > 15] = 1.0  # stall flat

# Insect LEV-augmented:
# Polhamus leading-edge suction analogy
CL_LEV = 1.8 * np.sin(2*alpha) + 1.95 * np.sin(alpha)**2

# Drag coefficient (insect)
CD_LEV = 1.92 - 1.55 * np.cos(alpha)

fig, axes = plt.subplots(1, 2, figsize=(15, 6))
fig.patch.set_facecolor('#030712')

# Panel 1: CL / CD vs AoA
ax1 = axes[0]
ax1.set_facecolor('#1a0a2e')
ax1.plot(alpha_deg, CL_smooth, color='#94a3b8', linewidth=2, linestyle='--', label='Smooth airfoil (thin-airfoil theory)')
ax1.plot(alpha_deg, CL_LEV, color='#e879f9', linewidth=2.5, label='Insect wing (leading-edge vortex)')
ax1.plot(alpha_deg, CD_LEV, color='#22d3ee', linewidth=2, label='Insect drag CD')
ax1.axvline(45, color='#fbbf24', linestyle=':', linewidth=1, alpha=0.6, label='Typical insect AoA ~ 45 deg')

ax1.set_xlabel('Angle of attack (deg)', color='white', fontsize=11)
ax1.set_ylabel('Force coefficient', color='white', fontsize=11)
ax1.set_title('Insect LEV vs conventional airfoil', color='#f0abfc', fontsize=12)
ax1.legend(facecolor='#1a0a2e', edgecolor='#c084fc', labelcolor='white', fontsize=9)
ax1.grid(True, alpha=0.25, color='#a78bfa'); ax1.tick_params(colors='white')
ax1.set_ylim(-1, 4)
for sp in ax1.spines.values(): sp.set_color('#c084fc')

# Panel 2: Required flapping frequency vs body mass
ax2 = axes[1]
ax2.set_facecolor('#1a0a2e')

# Greenewalt's allometry: f ~ M^(-1/3)
masses = np.logspace(-6, 2, 200)  # 1 ug to 100 kg
f_greene = 6.4 * masses**(-0.273)  # Hz

# Real data
species = [
    ('Honeybee', 0.0001, 230, '#fbbf24'),
    ('Hummingbird', 0.005, 55, '#f472b6'),
    ('Starling', 0.08, 12, '#22d3ee'),
    ('Heron', 2.0, 3.5, '#a78bfa'),
    ('Bumblebee', 0.00015, 150, '#e879f9'),
    ('Gull', 0.5, 4.5, '#86efac'),
    ('Mosquito', 0.0000025, 600, '#fb923c'),
    ('Swan', 10, 2.5, '#c084fc'),
]

ax2.loglog(masses, f_greene, color='#94a3b8', linewidth=2, linestyle='--', label="Greenewalt f ~ M^(-0.27)")
for name, m, f, c in species:
    ax2.scatter(m, f, c=c, s=110, edgecolors='white', linewidth=0.7, zorder=5)
    ax2.annotate(name, (m, f), xytext=(m*1.3, f*1.1), color=c, fontsize=9, fontweight='bold')

ax2.set_xlabel('Body mass (kg)', color='white', fontsize=11)
ax2.set_ylabel('Wingbeat frequency (Hz)', color='white', fontsize=11)
ax2.set_title('Wingbeat frequency across flyers', color='#f0abfc', fontsize=12)
ax2.legend(facecolor='#1a0a2e', edgecolor='#c084fc', labelcolor='white', fontsize=10)
ax2.grid(True, alpha=0.25, color='#a78bfa', which='both'); ax2.tick_params(colors='white')
for sp in ax2.spines.values(): sp.set_color('#c084fc')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#030712')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

3.5 Jumping Robots: Salto and the Galago

The galago (bushbaby) can vertically jump 2.25 m from standing — the highest vertical jump per body size among vertebrates. Peak power density in the leg extensor muscle is\(\sim 1,600\) W/kg, exceeding the \(\sim 350\) W/kg of human muscle by 5x. This is achieved by elastic energy storage + power amplification: slow-contracting muscles gradually load an elastic spring (collagen tendon), which is released explosively.

\( P_{\text{release}} = \frac{U_{\text{elastic}}}{\tau_{\text{release}}}, \quad \tau_{\text{release}} \ll \tau_{\text{load}} \)

Salto (Haldane et al. 2016, Berkeley) is a galago- inspired jumping robot: 100 g, single-leg design, vertical jump agility 1.75 m/s per leap, continuous multi-jumps. Spring-loaded linkage (series-elastic actuator) achieves power amplification via muscle-tendon analogue.

Additional robot exemplars: Boston Dynamics BigDog (rough-terrain quadruped), Ghost Robotics Vision 60 (military reconnaissance), Skydio drones (bird-inspired obstacle avoidance), Festo BionicKangaroo (elastic energy recovery).

3.2a Dragonfly Two-Pair Wings and Gliding

Dragonflies (Odonata) are the only insects that actively beat two wing pairs 90 deg out of phase. This phase-offset interaction between forewing and hindwing allows extraction of energy from the leading vortex shed by the forewing — a passive equivalent of flap-edge vortex recovery in tandem-wing engineering. Combrisson & Thomas (2019) measured a 22% aerodynamic efficiency gain from the two-pair arrangement vs either wing alone.

Dragonflies also glide extensively (Anax species, 15% of forward travel) with a gliding L/D ratio of 6–8, remarkable given their low Reynolds number (\(\text{Re} \sim 10^4\)). This efficiency comes from the corrugated wing cross-section: micro-scale ridges and valleys trap small separation bubbles that act as natural turbulators, delaying full stall. Biomimetic MAVs with corrugated wings (Okamoto et al. 1996, Murphy & Hu 2009) have demonstrated similar performance gains.

3.3a Universal Strouhal Number in Propulsion

Triantafyllou et al. (1991) observed an extraordinary pattern: across fish, cetaceans, birds, insects, and bats, all steady-cruising animals operate in the Strouhal range \(0.25 < \text{St} < 0.35\):

\( \text{St} = \frac{f A}{U} \)

where \(f\) is oscillation frequency, \(A\) tail/wingtip amplitude,\(U\) forward speed. Numerical wake simulations confirm that propulsive efficiency peaks in this range because it corresponds to maximum wake-vortex coherence. Biomimetic underwater vehicles and flapping MAVs now routinely operate in this sweet spot, set as a design constraint before any aerofoil geometry is chosen.

3.4a Hummingbird Hover and Figure-Eight Wing Kinematics

Hummingbirds are unique among birds in their ability to generate lift on both up- and down-strokes by rotating the wing 180 deg at the transitions — a kinematic convergent evolution with insects. The wing traces a figure-eight path in body-fixed coordinates. Warrick et al. (2005, Nature) used PIV to show that the upstroke generates\(\sim 25\%\) of the weight support, vs near-zero in most birds.

\( P_{\text{hover}} = W \sqrt{ \frac{W}{2 \rho A_d} }, \quad A_d = \pi r^2 \)

where \(A_d\) is the wing disc area and \(W\) body weight. This actuator-disc approximation sets the minimum induced power for hover. Hummingbirds approach this ideal by virtue of their very low wing loading (W/A ~ 25 N/m²).

Applications: AeroVironment Nano Hummingbird (2011) is a 19 g MAV capable of stable hover, forward flight, and lateral translation based directly on hummingbird kinematics.

3.5a Kingfisher Beak and the Shinkansen 500

The 500-series Shinkansen bullet train (JR West, 1997) faced a practical aeroacoustic problem: when it exited tunnels at 300 km/h, a compression wave ahead of the nose radiated as asonic boom, shattering house windows up to 200 m from tunnel exits. Chief engineer Eiji Nakatsu — an amateur ornithologist — noticed that the kingfisher Alcedo atthisdives from air into water without splashing, thanks to its long, narrow, dagger-like beak with diamond-shaped cross section. He redesigned the train's 15 m nose to mimic the beak.

The kingfisher-beak nose produces a gradual pressure build-up, preventing the abrupt shock at tunnel exit. Outcome: tunnel booms reduced to inaudibility, 15% lower energy consumption, 10% higher top speed. A textbook example of level-1 (form) biomimicry solving a level-3 (system) problem.

3.5b Humpback Whale Tubercles: WhalePower Blades

Humpback whale (Megaptera novaeangliae) pectoral flippers have distinctive leading-edge tubercles (bumps) that improve manoeuvrability. Frank Fish (West Chester University) showed experimentally that a scaled model flipper with tubercles stalls at higher angle-of-attack than a smooth one (Miklosovic et al. 2004), generating 40% more lift at low speed with 8% less drag. Mechanism: tubercles generate counter-rotating streamwise vortices that delay flow separation — the same mechanism as conventional vortex generators but passive and low-drag.

\( \alpha_{\text{stall,tubercle}} / \alpha_{\text{stall,smooth}} \approx 1.4 \)

Commercial: WhalePower Corporation (Toronto) licenses tubercle-designed fan blades for HVAC and wind turbines. Envira-North's industrial ceiling fans (Altra-Air) achieve 20% greater airflow per watt vs conventional blades.

3.6 Summary and References

Biological locomotion operates across eight orders of Reynolds number and covers walking, running, swimming, flying, jumping, burrowing, climbing, and gliding. Four unifying themes:

• Unsteady aerodynamics beats quasi-steady (LEV, clap-and-fling).
• Elastic energy storage multiplies muscle power density (galago, kangaroo).
• Distributed noise control (owl comb) outperforms broadband mufflers.
• Gait-speed matching (anguilliform → thunniform) minimises cost of transport over the species' ecological range.

References

[1] Lighthill, M.J. (1960). Note on the swimming of slender fish. J. Fluid Mech. 9, 305–317.
[2] Prandtl, L. (1918). Tragflugeltheorie. Nachrichten Ges. Wiss. Gottingen.
[3] Weis-Fogh, T. (1973). Quick estimates of flight fitness in hovering animals. J. Exp. Biol. 59, 169–230.
[4] Ellington, C.P. et al. (1996). Leading-edge vortices in insect flight. Nature 384, 626–630.
[5] Lilley, G.M. (1998). A study of the silent flight of the owl. AIAA Paper 98-2340.
[6] Geyer, T., Sarradj, E., Fritzsche, C. (2010). Measurement of the noise generation at the trailing edge of porous airfoils. Exp. Fluids 48, 291–308.
[7] Triantafyllou, M.S., Triantafyllou, G.S. (1995). An efficient swimming machine. Sci. Am. 272, 64–70.
[8] Sane, S.P. (2003). The aerodynamics of insect flight. J. Exp. Biol. 206, 4191–4208.
[9] Haldane, D.W. et al. (2016). Robotic vertical jumping agility via series-elastic power modulation. Science Robotics 1, eaag2048.
[10] Ma, K.Y., Chirarattananon, P., Fuller, S.B., Wood, R.J. (2013). Controlled flight of a biologically inspired insect-scale robot. Science 340, 603–607.
[11] Webb, P.W. (1984). Form and function in fish swimming. Sci. Am. 251, 72–83.
[12] Alexander, R.McN. (2003). Principles of Animal Locomotion. Princeton Univ. Press.

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