Unsteady aerodynamics · oscillating-foil tunnel

Flapping Wing

A spatial wing section heaves and pitches sinusoidally inside a transparent wind tunnel. Change frequency, amplitudes, phase, and pivot position to connect circulatory lift, added mass, alternating shed vortices, and cycle-averaged thrust and power proxies.

Motion phase $0^\circ$

Reduced frequency

k

$0.18$

Strouhal number

St

$0.08$

Instantaneous lift

C_L

$0.000$

Mean thrust proxy

\overline{C_T^{\ast}}

$0.000$

Mean power proxy

\overline{C_P^{\ast}}

$0.000$

Propulsive proxy

\eta^{\ast}

$0\%$

Kelvin ledger drift

\epsilon_\Gamma

$0.0\times10^{-8}$

Oscillating-foil controls

Compose heave, pitch, and wake

Kelvin wake vorticesShow each circulation increment shed from the trailing edge.Lift and streamwise forceDisplay instantaneous lift and the signed thrust proxy.Added-mass contributionSeparate the non-circulatory acceleration force.Force history panelTrace lift and streamwise-force coefficients through time.Trailing-edge envelopeDraw the closed path swept by the trailing edge each cycle.Free-stream speed

U_\infty

12\,\mathrm{m\,s^{-1}}

Flapping frequency

f

1.4\,\mathrm{Hz}

Heave amplitude

h_0/c

0.22\,c

Pitch amplitude

\theta_0

10^\circ

Pitch phase relative to heave

\phi

270^\circ

Pitch-axis position

x_p/c

0.00\,c

When peak effective angle exceeds 15 degrees, the caption marks the run outside the linear model’s comfort zone; the animation continues so you can see where the approximation is being stretched.

This is a two-dimensional, small-disturbance, incompressible, inviscid Theodorsen thin-airfoil model using the Jones–Wagner rational approximation for $C(k)$ wake memory. The spatial span and wake lines extrude a two-dimensional section; streamwise force uses lift projection minus a fixed profile-drag proxy. It does not resolve boundary layers, stall, dynamic stall, tip vortices, or a viscous wake, and it is not CFD.

Physics tutorial

Flapping Wing: how phase turns oscillation into propulsion

BackgroundTheodorsen developed linear unsteady aerodynamics for an oscillating thin airfoil in NACA 496, and Garrick then applied the same framework to flapping propulsion in NACA 567. This Lab retains circulatory force, added mass, and wake memory, while recording every bound-circulation change in a visible trailing-edge wake.

Why it mattersHeave or pitch alone is merely reciprocal motion. Their amplitudes, frequency, and relative phase determine wake direction, lift phase, and the cycle-averaged streamwise force.

Start with the essentials

Focus question: Why can the same wing, using the same amplitudes and frequency, switch from drag production to propulsion when only pitch phase changes?
One-sentence intuition: Heave velocity changes effective angle of attack, the Theodorsen function delays circulatory lift relative to motion, and Kelvin circulation requires each bound-circulation change to enter the wake with opposite sign; phase $\phi$ therefore reorganizes both force and wake.

Core mathematical model

Heave, pitch, and effective angle

h(t)=h_0\sin\omega t,\qquad \theta(t)=\theta_0\sin(\omega t+\phi),\qquad \alpha_{\mathrm{eff}}=\theta-\tan^{-1}\!\left(\frac{\dot h}{U_\infty}\right)

When the section moves upward, its relative flow tilts downward; pitch can reinforce or cancel that motion-induced angle.

Theodorsen wake memory

C(k)\approx1-\frac{0.165\,ik}{0.0455+ik}-\frac{0.335\,ik}{0.3+ik},\qquad k=\frac{\omega c}{2U_\infty}

The Jones–Wagner rational approximation compresses wake-induced amplitude loss and phase lag into a complex response suitable for real-time evaluation.

Circulatory and added-mass lift

L'=\pi\rho b^2\!\left(-U_\infty\dot\theta-\ddot h+ab\ddot\theta\right)+2\pi\rho U_\infty b\,C(k)\!\left[U_\infty\theta-\dot h+b\left(\frac12-a\right)\dot\theta\right]

The first term is an acceleration force that needs no steady circulation; the second is circulatory force with wake memory. Here b is the semi-chord and a locates the pitch axis.

Kelvin circulation ledger

\Gamma_b+\sum_j\Gamma_{w,j}=\mathrm{constant},\qquad \Delta\Gamma_w=-\Delta\Gamma_b

Every change in bound circulation releases an opposite increment at the trailing edge; the colored wake lines are that discrete ledger made visible.

Common difficulties

A reversed wake is not a complete propulsion prediction

Typical misconceptionIf the wake looks propulsive, the efficiency readout must be the efficiency of a real vehicle.

Better mental modelStreamwise force here combines lift projection with a fixed profile-drag proxy. Viscous wake loss, three-dimensional tip loss, and dynamic stall are absent, so the starred efficiency is only a phase-comparison metric.

A linear model does not generate stall automatically

Typical misconceptionIncreasing pitch amplitude to its maximum lets the same model study a dynamic-stall vortex.

Better mental modelTheodorsen theory assumes small disturbance and attached potential flow. When peak effective angle exceeds the marked boundary, the Lab keeps running to expose the extrapolation trend but does not call it a credible stall result.

Run the experiment

01
Separate the two motions
Compare Pure heave and Pure pitch, watching wake circulation, instantaneous lift, and mean streamwise force.
What to observe: Both motions create alternating circulation, but in the current lift-projection model only a case with heave velocity directly produces a mean propulsive contribution.
02
Compose a propulsive phase
Choose Propulsive phase without changing the displayed amplitudes, then inspect the trailing-edge envelope, thrust proxy, and power proxy.
What to observe: Pitch limits excessive motion-induced angle while pairing lift with heave velocity favorably through the cycle, making mean streamwise force positive.
03
Reverse only phase
Switch to Phase reversed without changing frequency or amplitudes.
What to observe: Wake organization and force phase reverse together, and mean streamwise force changes from propulsion to drag. Phase is not a decorative parameter.

Core mathematical model

Heave, pitch, and effective angle

Theodorsen wake memory

Circulatory and added-mass lift

Kelvin circulation ledger

Common difficulties

A reversed wake is not a complete propulsion prediction

A linear model does not generate stall automatically

Run the experiment

Separate the two motions

Compose a propulsive phase

Reverse only phase