Unsteady aerodynamics · oscillating-foil tunnel

Flapping Wing

How does the timing between a wing's heaving and pitching decide whether it makes drag or thrust? Heaving up and down changes the angle at which the wing meets the oncoming flow. The pitch phase decides whether pitching adds to that change or cancels it, so the same amplitudes can give drag or thrust. By Kelvin's theorem, every change in circulation on the wing is shed into the wake as a vortex, and switching from drag to thrust reverses the arrangement of those wake vortices.

A wing section heaves up and down and pitches back and forth, sinusoidally, inside a transparent wind tunnel. Change the frequency, amplitudes, phase and pivot position, and see how circulatory lift, added mass, the alternating shed vortices and the cycle-averaged thrust and power proxies change together.

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 circulation drift

\epsilon_\Gamma

$0.0\times10^{-8}$

Oscillating-foil controls

Combine heave and pitch, then watch the 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

At every substep the discrete model solves for no flow through the wing and for Kelvin circulation conservation. It includes feedback from the wake and pressure loads on a thin sheet. The drawn thickness is schematic; camber changes the mean line. Leading-edge shedding, suction, friction and stall are not modeled, so the streamwise force and efficiency are only proxies, and a low advance ratio only resembles hovering. Each wake vortex has a fixed core of 6.5% of the chord. To keep at most 192 elements, older vortices of the same sign merge in a way that conserves their circulation and its first moment; their near field is then approximate. Past the downstream cutoff, circulation still counts toward the total but no longer induces velocity. Cycle averages need a complete cycle.

Each phase step advances $15^\circ$ at every flapping frequency, so twenty-four steps make one cycle. Changing a preset or a control leaves the animation paused or playing, as it was.

When the peak effective angle of attack exceeds 15 degrees, the caption warns that the run is outside the range where the linear model works well. The animation keeps running, so you can see where the approximation is being pushed too far.

By default the Lab computes two-dimensional unsteady vortex panels with wake feedback. Switch it off to compare with the Jones–Wagner harmonic reference $C(k)$ . The span is a section drawn in three dimensions; tip vortices, boundary layers, leading-edge separation and dynamic stall are not modeled.

Physics tutorial

Flapping Wing: how phase turns oscillation into propulsion

BackgroundBy default, the Lab splits a thin wing into panels and solves two conditions together: air cannot flow through the wing, and total circulation is conserved (Kelvin’s theorem). The vortices the wing sheds then act back on it. You can switch to a Jones–Wagner harmonic reference that follows Theodorsen’s theory in NACA Report 496. The two are separate models.

Why it mattersHeave or pitch on its own is just back-and-forth motion. The amplitudes, the frequency and the phase between the two motions decide which way the wake points, the timing of the lift and the streamwise force averaged over a cycle.

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: Heaving changes the effective angle of attack, and the velocity induced by the wake changes the circulation on the wing. By Kelvin’s theorem, every change in the wing’s circulation leaves an equal and opposite amount in the wake. Changing the phase $\phi$ therefore rearranges the force and the wake together.

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]

Each panel carries a bound vortex at its quarter point, and air may not cross the panel at its three-quarter point. The pressure term with the rate of change of circulation is the unsteady part; it cannot be split off as a 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, the flow it sees tilts downward. Pitch can add to or cancel the angle this motion creates.

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 sums up what the wake does, a loss of amplitude and a phase lag, in one complex response that is fast enough to evaluate in real time.

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’s circulation theorem

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

Every change in the wing’s bound circulation sheds an equal and opposite increment from the trailing edge. The colored wake lines show those shed increments.

Common difficulties

A reversed vortex street is not a full thrust prediction

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

Better mental modelIn the default mode, forces come from the pressure jump across a thin vortex sheet, with no leading-edge suction, viscosity or stall. The reference mode projects the lift and adds a fixed drag proxy. The efficiency readout is only for comparing phases. It is not a vehicle’s efficiency and need not 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’s theory assumes small disturbances and flow that stays attached to the wing. When the peak effective angle passes the marked limit, the Lab keeps running so you can see where the extrapolation is heading, but it does not treat the result as a credible stall prediction.

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 shed circulation of alternating sign. Wait for a full cycle before comparing the averages. Switching to the harmonic reference changes how the force relates to wake feedback.
02
Set up 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 add to the angle created by heaving. To judge propulsion, compare the full-cycle force and input power; the wake shape alone cannot tell you.
03
Reverse only the phase
Switch to Phase reversed without changing frequency or amplitudes.
What to observe: The wake pattern and the timing of the force change together. Record the full-cycle averages and check whether the streamwise force changes sign.

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

Flapping Wing

Flapping Wing: how phase turns oscillation into propulsion

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’s circulation theorem

Common difficulties

A reversed vortex street is not a full thrust prediction

A linear model does not generate stall automatically

Run the experiment

Separate the two motions

Set up a propulsive phase

Reverse only the phase

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