What it does
In an ideal fluid (no viscosity, no compression) flows add up: the flow round a cylinder is just a uniform stream plus a doublet, a spinning cylinder adds a vortex, and a source in a stream makes the nose of a long body. This page lets you build such flows and see them at once. Streamlines are drawn as contours of the stream function ψ, the heavy line is the dividing streamline ψ = 0 (in the presets, the surface of the body), red dots mark stagnation points where the fluid stops, and the shading shows the speed.
How to use it
- Pick a preset: flow past a cylinder, a spinning cylinder with lift, a Rankine half-body, a Rankine oval, a source-sink pair or a vortex pair.
- Set the uniform flow speed and direction (0° is left to right), or 0 for none.
- Edit each element’s strength and position in the list, remove it, or add more. Choose what tapping the picture does: add a source, sink, vortex or doublet there, or measure the velocity, stream function, potential and pressure coefficient at that point.
- Read the force on the elements, worked out with Blasius’s theorem round a circle that encloses them all.
Worked example
The page opens on a 1 m/s stream past a doublet of strength κ = 2π m³/s, which makes a cylinder of radius 1 m. There are 2 stagnation points: (−1, 0), (1, 0), the front and back of the cylinder, and the force is zero in both directions: d’Alembert’s paradox, an ideal fluid exerts no drag on a closed body. Tap just above the cylinder, at about (0, 1.2), and the flow there is about 1.69 m/s with a pressure coefficient of about −1.87; on the surface itself, at (0, 1), the speed reaches 2U and Cp = −3. Choose the spinning cylinder: a clockwise vortex of 2π m²/s moves the stagnation points down to (−0.866, −0.5) and (0.866, −0.5), and the lift is ρUΓ = 7.7 N per metre of span for air at 1.225 kg/m³.
Tips
- For a cylinder of radius a in a stream U, set the doublet strength to 2πUa².
- Increase the vortex strength on the spinning cylinder until Γ = 4πUa: the two stagnation points meet at the bottom, and beyond that one leaves the surface.
- Move the sink of the Rankine oval closer to the source and the body becomes rounder.
FAQ
Why is there no drag? Ideal flow has no viscosity, so there is no boundary layer and no wake. Real bodies have drag; ideal flow is still a good guide to the pressure on the front of a body and to lift.
What about the lines that end at a source? The stream function of a source jumps behind it (a branch cut), and the page leaves out those grid cells so no false line is drawn.
Is it for real design? It is for study and planning, not a substitute for professional judgement.
Where the numbers come from
- Elementary potential flows, superposition, Rankine bodies, flow past a (spinning) cylinder, Blasius's theorem and the Kutta-Joukowski theorem: standard results, for example Anderson, J. D., Fundamentals of Aerodynamics, chapter 3, and Milne-Thomson, L. M., Theoretical Hydrodynamics.
- Checked against SymPy: exact stagnation points (roots of dW/dz) for the presets and the Blasius force from the residue of (dW/dz)².
- Preset strengths and positions are made-up round numbers.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.