---
title: "How Phase Turns Flapping into Propulsion Interactive Lab | Sandbox Physics"
description: "How does the timing between a wing's heaving and pitching decide whether it makes drag or thrust?"
canonical_url: "https://sandboxphysics.com/labs/flapping-wing/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/flapping-wing/"
last_modified: "2026-10-29"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

**Background** By 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 matters** Heave 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 misconception** If the wake looks propulsive, the efficiency readout must be the efficiency of a real vehicle.

**Better mental model** In 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 misconception** Increasing pitch amplitude to its maximum lets the same model study a dynamic-stall vortex.

**Better mental model** Theodorsen’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

1. 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.

2. 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.

3. 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.

Physical conclusion

Flapping faster does not by itself give propulsion. Heave tilts the oncoming flow, pitch sets the effective angle, wake memory delays the circulation, and by Kelvin’s theorem every change in circulation leaves its mark in the wake. The phase between the motions decides the force and power averaged over a cycle.

Conceptual inspiration: [FluidMotion](http://www.bugman123.com/FluidMotion/index.html). Original simulations and graphics; the model limits are described above.

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