ADD SIMPLE FLOWS. REVEAL A COMPLEX PICTURE.

Potential Flow
Builder

Place a source. Turn a vortex. Discover how a circular boundary and an asymmetric pressure field emerge from the same few equations.

LIVE SETUP B

Shape the field

Uniform speed U∞ · m/s

Uniform direction ° anticlockwise from +x

Cylinder radius R · m · doublet strength updates with U∞R²

Circulation Γ · m²/s · positive is anticlockwise

Drag the cylinder centre. The doublet and vortex move together to preserve the circular boundary.

Centre x · mCentre y · m

Colour the field

Show velocity arrows

B · CIRCULAR CYLINDER−6 m ← x → +6 m

Speed / U∞03+Fixed scale; saturated colours are clipped.

Playback 0.0 s

Click an empty part of the view to place the probe. x points right; y points up. Circles mark exact stagnation points. Tracer spacing is illustrative; each tracer follows the calculated velocity.

Inspect the local flow

Drag the pink probe

x · my · m

Velocity components add; speed and pressure do not.
Contribution at probe	u · m/s	v · m/s

A · no frozen reference

THREE CONNECTED IDEAS

See the mechanism.

Choose a guided A / B comparison.
These change the setup; they are not tests.

Two sources of fluid motion

A source adds fluid per unit span; a sink removes it. Equal strengths make the far-field net source flux zero, while the local velocities remain very different.

A boundary emerges

A doublet aligned with the uniform stream gives zero normal velocity on a circle. An arbitrary collection of elements does not automatically describe a solid object.

Circulation changes pressure

Clockwise circulation speeds up the flow above the cylinder. The integrated pressure force points upwards. The ideal model still predicts zero drag.

THE SAME EQUATIONS, ANOTHER VIEW

Read the response.

B · solid turquoise
A · dashed amber

Pressure around the cylinder

Scroll the plot horizontally to read the axes. Keyboard: focus the plot, then use the left and right arrow keys.

Speed along the probe’s horizontal line

Scroll the plot horizontally to read the axes. Keyboard: focus the plot, then use the left and right arrow keys.

Both setups sampled at the live probe y coordinate. Gaps mark excluded interiors or singularity discs. Curves retain their calculated values.

A / B evidence

Quantity	A · frozen	B · live	Unit

The export includes model version, settings, probe readings and curve samples.

FROM SUPERPOSITION TO PRESSURE

Equations & model limits

Analytically defined · numerically traced

01 · Add velocity vectors

\vec{V} = {\vec{U}}_{\infty} + \sum_{i} {\vec{V}}_{i}

The potential satisfies Laplace’s equation away from singularities. Linear superposition preserves zero divergence and zero curl there.

x, y · m u, v, U∞ · m/s

02 · Source and vortex

V_{r} = \frac{Q}{2 π r} V_{θ} = \frac{Γ}{2 π r}

Q is outward source flux per unit span. Γ is positive anticlockwise circulation. These are separate elements; the first is radial, the second tangential.

Q, Γ · m²/s r · distance from the element

03 · A circle is a streamline

ψ = U_{\infty} r (1 - \frac{R^{2}}{r^{2}}) \sin θ - \frac{Γ}{2 π} \ln \frac{r}{R}

For an aligned doublet with μ = 2πU∞R², r = R has no normal flow. θ is measured anticlockwise from the incoming-flow direction; the surface allows slip.

ψ · m²/s μ · m³/s R · m

04 · Pressure and ideal lift

C_{p} = 1 - \frac{V^{2}}{U_{\infty}^{2}} L^{'} = - ρ U_{\infty} Γ

Steady Bernoulli with one far-field total pressure gives Cp. Positive lift is normal to the uniform stream towards its left-hand side (upwards when the stream points right).

Cp · dimensionless L′ · N/m illustrative ρ = 1.225 kg/m³

What this model can explain

Superposition, stagnation points and dividing streamlines.
Why circulation changes velocity and pressure around an ideal cylinder.
How a boundary condition differs from drawing a shape.

Where it stops

No viscosity, boundary layer, separation, stall, turbulence or compressibility.
Circulation is prescribed. The model does not predict how cylinder rotation generates it.
Changing controls replaces one steady field with another; it is not a transient fluid calculation. Tracers restart after edits.
Arbitrary-element fields are mathematical flows. Only the cylinder presets claim a solid boundary; singularity discs are excluded, not walls.

Numerical display, exact field

Velocities are evaluated analytically. Streamlines use fourth-order Runge–Kutta steps in arclength; tracers use the same method in physical time, with small adaptive substeps. The colour field is sampled for display.

Speed colours saturate at 3U∞ and Cp colours at −3 to +1. Probe values and curves show the actual calculation. Near singularities, even finite values can lie beyond the useful range of an incompressible physical approximation.

Read the model record · Read source provenance

Verification checks analytic identities and integration errors. It is not experimental validation of a real aircraft.

Authoritative foundations & further reading

Equations and original diagrams in this lab were implemented for teaching; no source artwork is reproduced.

CONTINUE THE CONNECTION

Streamlines, Pathlines & StreaklinesSee why steady and unsteady flow pictures differ →2D Wind TunnelExplore shaped bodies with a viscous teaching model →Boundary LayersRestore the near-wall physics omitted here →