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.

Panel wall residual $0$

Completed averaging cycles $0$

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

Computed discrete-vortex wakeSolve bound panels and each new shed vortex together; switch off for the Jones–Wagner harmonic comparison.Kelvin wake vorticesShow each circulation increment shed from the trailing edge.Lift and streamwise forceDisplay instantaneous lift and the signed thrust proxy.Unsteady pressure contributionCirculation-rate pressure in discrete mode; added mass in the harmonic reference.Force history panelTrace lift and streamwise-force coefficients through time.Trailing-edge envelopeDraw the closed path swept by the trailing edge each cycle.Bound vortex panels

N

12

Parabolic mean-line camber

m/c

0.000

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

The discrete model solves no-through-flow and Kelvin conservation each substep, with wake feedback and thin-sheet pressure loads. Thickness is schematic; camber changes the mean line. Leading-edge shedding, suction, friction and stall are absent: streamwise force and efficiency are proxies, and low advance is only hover-like. A fixed wake core is 6.5% of chord. To retain at most 192 elements, old like-signed vortices merge conserving circulation and its first moment; this approximates their near field. Beyond the downstream cutoff, circulation remains in the ledger but no longer induces velocity. Cycle means require a complete cycle.

Each phase step advances $15^\circ$ , at every supported frequency. Twenty-four steps complete one cycle. Presets and controls preserve your paused or playing state.

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.

The default computes two-dimensional unsteady vortex panels and wake feedback; switching it off compares the Jones–Wagner harmonic reference $C(k)$ . The spatial span is a section display, with no tip vortices, boundary layers, leading-edge separation or dynamic stall.

Physics tutorial

Flapping Wing: how phase turns oscillation into propulsion

BackgroundThe default solves thin-wing no-through-flow and Kelvin circulation together, feeding shed-vortex velocities back into the wing. The switchable Jones–Wagner harmonic reference follows the Theodorsen NACA 496 framework. These are two distinct models.

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, computed wake velocities change panel circulation, 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

Discrete panels and wake feedback

(\mathbf U_\infty+\mathbf u_b+\mathbf u_w-\mathbf v_{\mathrm{wing}})\cdot\mathbf n_i=0,\qquad \Delta p_i\Delta s_i=\rho\left[U_{t,i}\Gamma_i+\Delta s_i\frac{\partial}{\partial t}\sum_{j\leq i}\Gamma_j\right]

Quarter-panel bound vortices enforce no-through-flow at three-quarter-panel points. The circulation-rate pressure term is unsteady; it is not an independently separated pure added-mass force.

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.

Harmonic reference: 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.

Harmonic reference: 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 modelDefault loads come from thin-sheet pressure jumps but omit leading-edge suction, viscosity and stall; the reference uses lift projection and a fixed drag proxy. Efficiency is a phase-comparison proxy, not vehicle efficiency, and is not guaranteed to lie between zero and one.

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. Wait for a complete cycle before comparing computed means; switching to the harmonic reference changes how force and wake feedback are related.
02
Compose a propulsive phase
Choose Combined phase without changing the displayed amplitudes, then inspect the trailing-edge envelope, thrust proxy, and power proxy.
What to observe: Pitch can cancel or reinforce motion-induced angle. Compare full-cycle force and input power rather than inferring propulsion from wake shape.
03
Reverse only phase
Switch to Phase reversed without changing frequency or amplitudes.
What to observe: Wake organization and force phase change together. Record complete-cycle means and test whether the streamwise force changes sign.

Conceptual inspiration: FluidMotion. Original simulations and graphics; the model limits are described above.

Core mathematical model

Discrete panels and wake feedback

Heave, pitch, and effective angle

Harmonic reference: Theodorsen wake memory

Harmonic reference: 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