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Fluid dynamics · airfoil pressure workbench

Airfoil Circulation Studio

Keep the orbitable three-dimensional tunnel while linking upper/lower surface pressure, a movable pressure probe, a contour-pressure integral, and circulation-derived lift to the same Joukowski solution. Refine the pressure quadrature to audit lift and zero drag; disable Kutta to reveal the trailing-edge singularity instead of hiding it behind clipped pressures.

Interactive modelAirfoil Circulation Studio
Wall / edge residual (relative to inflow)analytic\text{analytic}
Actual samples / panels128128
Net source-flux residual (normalized)analytic\text{analytic}
Circulation Γ\Gamma0 m2 s−10\,\mathrm{m^2\,s^{-1}}
Theory lift CLC_L00
Pressure-integral lift CL,pC_{L,p}00
Force-audit absolute error00
Pressure-integral drag CD,pC_{D,p}00
Edge limit / trailing-panel estimate VTE/U∞V_{\mathrm{TE}}/U_\infty00
Probe pressure difference ΔCp\Delta C_p00
Lift per span L′L'0 N m−10\,\mathrm{N\,m^{-1}}
Model boundary2D potential flow

Physics tutorial

Airfoil Circulation: how a trailing-edge condition selects lift

BackgroundIncompressible, inviscid, irrotational flow can be solved outside a circle and carried to an airfoil by a conformal map. The offset circle controls camber and thickness, while the sharp trailing edge is a critical point of the map.

Why it mattersThe model cannot predict stall, but it does connect geometry, circulation, surface speed, pressure difference, and lift in one complete causal chain.

Start with the essentials

Focus question
Why does the same airfoil admit many circulation solutions mathematically, while attached physical flow selects one?
One-sentence intuition
The Kutta condition requires finite velocity and smooth departure at the trailing edge, selecting one circulation Γ\Gamma; the resulting pressure difference supplies lift.

Core mathematical model

NACA geometry and a solved source/vortex panel system

yt=5t(0.2969x−0.1260x−0.3516x2+0.2843x3−0.1036x4),V⋅n=0,Vt,1+Vt,N=0y_t=5t\left(0.2969\sqrt{x}-0.1260x-0.3516x^2+0.2843x^3-0.1036x^4\right),\quad \mathbf V\cdot\mathbf n=0,\quad V_{t,1}+V_{t,N}=0

Use the NASA TM 4741 four-digit mean line and thickness polynomial, with the final coefficient modified to close the edge. Hess–Smith solves individual source strengths and one uniform vortex-sheet strength from wall tangency and Kutta. Pressure, streamlines, and circulation share that solution. Collocation residual, net source-flux error, and a recomputed half-resolution pressure-force audit are distinct from the analytic Joukowski mode; a small residual does not establish grid convergence.

Finite trailing-edge speed is not zero speed

VTEU∞=bR∣cos⁡(α−β)∣,β=arg⁡(b−ζc)\frac{V_{\mathrm{TE}}}{U_\infty}=\frac{b}{R}\left|\cos(\alpha-\beta)\right|,\qquad \beta=\arg(b-\zeta_c)

The Kutta numerator and map derivative share a trailing-edge zero. Analytic cancellation leaves a finite, generally nonzero mapped speed. If Kutta is disabled and the shared zero disappears, no finite limit exists; the force audit is disabled instead of clipping the singularity.

A separate contour-pressure force calculation

Cx=−∮Cp d(y/c),Cy=∮Cp d(x/c),CL,p=−Cxsin⁡α+Cycos⁡αC_x=-\oint C_p\,\mathrm{d}(y/c),\quad C_y=\oint C_p\,\mathrm{d}(x/c),\quad C_{L,p}=-C_x\sin\alpha+C_y\cos\alpha

Integrate unmodified pressure around the counterclockwise airfoil contour, then project the force perpendicular to the free stream. Compare the result with circulation-derived lift and zero drag. The half-resolution error is recomputed, not guessed. Reference: MIT 18.354J, Lecture 20, contour-pressure forces.

Joukowski conformal map

z=ζ+b2ζz=\zeta+\frac{b^2}{\zeta}

An offset circle in the preimage plane maps into an airfoil with a rounded leading edge and sharp trailing edge.

Complex potential outside the circle

W(ζ)=U∞(e−iαη+eiαR2η)+iΓ2πlog⁡η,η=ζ−ζcW(\zeta)=U_\infty\left(e^{-i\alpha}\eta+e^{i\alpha}\frac{R^2}{\eta}\right)+\frac{i\Gamma}{2\pi}\log\eta,\qquad \eta=\zeta-\zeta_c

Uniform flow, the cylinder image term, and a point-vortex circulation form the analytic circle-plane solution.

Pressure and lift

Cp=1−∣VU∞∣2,L′=ρU∞Γ,CD=0C_p=1-\left\lvert\frac{V}{U_\infty}\right\rvert^2,\qquad L'=\rho U_\infty\Gamma,\qquad C_D=0

Bernoulli converts surface speed into pressure. For a regular trailing-edge solution, Kutta–Joukowski gives lift per span and the closed pressure integral gives zero drag, exposing the missing viscous physics. The closed force audit is not applied to singular solutions.

Common difficulties

The Kutta condition is not hidden thrust

Typical misconceptionTurning on the Kutta condition adds an artificial source of lift.

Better mental modelIt adds no external force; it selects, among mathematically allowed circulation values, the one with finite speed and smooth departure at the trailing edge.

Zero drag is not a real-aircraft conclusion

Typical misconceptionIf the pressure field produces lift but no drag, a real airfoil should also fly without loss.

Better mental modelInviscid potential flow omits boundary layers, wakes, and skin friction, so it necessarily misses drag and stall. That contradiction is D’Alembert’s paradox.

Run the experiment

  1. 01

    Establish the zero-lift baseline

    Choose Symmetric · zero lift while leaving the Kutta condition enabled.

    What to observe: Upper and lower streamlines and pressures are nearly symmetric, with circulation and lift close to zero.
  2. 02

    Increase only angle of attack

    Choose Cruise section and watch the upper-surface pressure taps and lift vector.

    What to observe: The upper flow accelerates, its pressure coefficient falls, and the selected circulation and lift per span rise together.
  3. 03

    Remove the trailing-edge selection

    Choose Kutta off and compare the flow around the unchanged geometry.

    What to observe: Zero circulation gives zero theoretical lift, but the edge speed becomes singular. The pressure plot can exceed its frame while raw data remains intact; the force audit becomes unavailable rather than endorsing a clipped solution.
  4. 04

    Check precision through two force calculations

    Return to Cruise section, refine pressure quadrature from N=32N=32 to N=128N=128, then move the pressure probe.

    What to observe: Integrated lift approaches circulation-derived lift and numerical drag approaches zero. Analytic Joukowski probing evaluates the exact surface; panel probing interpolates the computed panel-midpoint pressures and can change under refinement.

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