Visualize 2D potential flow around a cylinder with streamlines and pressure coefficient (Cp) color maps. Experience the Magnus effect and d'Alembert's paradox.
Flow pattern
Parameters
Free-stream velocity U∞1.0 m/s
Source strength Q8.0 m²/s
Circulation Γ0.0 m²/s
Characteristic length a1.00 m
Display
Live Values
1.00
Free-stream velocity U∞ [m/s]
0.0
Circulation Γ [m²/s]
2
Number of stagnation points
2.00
Instantaneous max speed |V| [m/s]
0.00
Lift coefficient CL
0.0
Lift L=ρU∞Γ [N/m]
−3.00
Cp top
−3.00
Cp bottom
Potential Flow — Particle Animation
High speed, low pressure (Cp<0)Low speed, high pressure (Cp→1)
Tracer particles are advected by the velocity field. Colour = pressure coefficient Cp, yellow = stagnation point, green = lift vector. Use the presets to switch between uniform flow, source, cylinder, free vortex and Rankine oval.
When I increase the "Circulation Γ" in this simulator, the flow around the cylinder starts to swirl. What is this useful for?
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Roughly speaking, this is the mechanism that creates "lift". The "Magnus effect", which makes a spinning baseball curve, is caused by the pressure difference produced by this circulation. Increase Γ with the slider above and you can see the flow speed rise above the cylinder (the blue region) and the pressure fall. That produces an upward force.
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Really! But I heard that if the flow is perfectly symmetric, the fluid flowing past the cylinder cancels out front and back and the drag is zero. That's "d'Alembert's paradox", right?
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Exactly! In ideal potential flow, the pressure distribution is symmetric front to back, so the drag is zero. Real fluids have viscosity, so that doesn't happen. Set the "Circulation Γ" back to 0 in the simulator and look at the colour on the cylinder surface (the pressure coefficient $C_p$). The front and back have almost the same colour (pressure), don't they? That is the paradox made visible.
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I see! So what is the "Q" slider for? It looks like it sucks fluid in when it's negative.
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It represents a "source" or a "sink". With the "Source" preset and a positive Q, a source flowing out in all directions from one point combines with the uniform flow to give the flow around a semi-infinite body with a rounded nose (a Rankine half-body). The stagnation point forms upstream of the source at x = −Q/(2πU∞); with Q = 8 m²/s and U∞ = 1 m/s it is about 1.27 m in front. A negative Q makes it a sink. In practice, such simple models are used first to understand the flow near an engine intake or a ventilation fan inlet.
Frequently Asked Questions
It is the theoretical result that, in potential flow of an ideal fluid (inviscid and incompressible), no drag acts on a body in a uniform stream. With the "Cylinder (doublet)" preset and circulation Γ = 0, the pressure coefficient distribution is symmetric front to back (Cp = 1 at the front and rear stagnation points and Cp = −3 at the top and bottom), and you can see that the pressure drag is zero.
Set the circulation strength Γ (gamma) to a positive or negative value. Increasing Γ strengthens the asymmetry of the streamlines around the cylinder and lift is generated. At the same time, the pressure-coefficient colour map shows the pressure difference arising between the top and bottom of the cylinder.
Q is used with the "Source" and "Rankine oval" presets. A positive Q is a "source" from which fluid emerges at one point, and a negative Q a "sink" into which fluid is drawn. Combining a uniform flow with a source gives the flow around a Rankine half-body, and combining a source and a sink gives the flow around a Rankine oval (a closed body); you can see how the body size and the stagnation-point positions are set by Q and U∞.
The inside of the cylinder is treated as having zero velocity, and streamlines, tracer particles and the pressure colour are drawn only outside the cylinder. The complex-potential formula itself has values inside the cylinder too, but they have no physical meaning, so they are not shown.
Real-World Applications
Aeronautical engineering (wing theory):This is the idea underlying thin-aerofoil theory. Adding circulation explains the lift acting on a wing, and the Kutta–Joukowski theorem is at the core of wing design. Watching the lift change as you vary Γ in the simulator is a direct experience of it.
Fluid analysis of sports balls:A banana kick in football, a curveball in baseball and topspin in table tennis are all due to the "Magnus effect" — the generation of circulation by spin. The potential-flow model is the first step to understanding this effect in idealised form.
Early studies for wind power and turbine design:The complex flow around blades is first modelled as inviscid potential flow to get a rough idea of the pressure distribution and streamlines. The concept of circulation is especially important for understanding vortical flows.
Basic studies in architectural and environmental wind engineering:Simple potential-flow models are sometimes used to get a rough picture of wind flow around buildings and structures. Source/sink models help visualise the flow near ventilation openings.
Common Misconceptions and Points to Note
There are several pitfalls that beginners in particular tend to fall into when they start using this simulator. First,don't forget that "potential flow is a model of a 'perfect' fluid". Because it ignores viscosity, it is fundamentally different from real flow. For example, some people see lift generated by adding circulation Γ and jump to the conclusion that "this explains all of a wing's lift!", but on a real wing the "circulation" produced by viscosity and the "trailing-edge condition" are intricately intertwined. Think of this tool asan "idealised first step" that extracts the core physical concept.
Second, pay attention tothe size of the circulation and the position of the stagnation points. While the circulation is small, the cylinder's stagnation points move down (or up) along the surface; when Γ exceeds 4πU∞a they leave the surface and a single stagnation point appears in the flow, with closed streamlines circling the cylinder. For U∞ = 1 m/s and a = 1 m the boundary is Γ ≈ 12.6 m²/s, and at Γ = 15 m²/s the stagnation point moves to about 1.8 m from the centre. This is a correct potential-flow solution, but obtaining this much circulation with a real rotating cylinder requires very fast rotation. Starting with Γ of the same order as U∞a (for example around 5 m²/s for U∞ = 5 m/s and a = 1 m) makes the relation between the movement of the surface stagnation points and the lift easier to see.
Third, understand thatthe "point source/sink (Q)" is a mathematical abstraction. Being told "a sink of 1 m²/s" in practice doesn't mean much, does it? Think of it as representing, by a single "point", the effect of drawing a fixed amount of air through, say, a circular duct 10 cm in diameter. Changing Q changes the streamline pattern dramatically, but this reflects the mathematical strength of the "singularity"; the trick is to interpret it as a"first-order approximation" of a real inlet that has finite size.
Set the free-stream velocity U∞ in the range 0.2–5 m/s
Adjust the characteristic length a (the radius for the cylinder, half the source–sink spacing for the Rankine oval) between 0.4 and 2.0 m
With "Cylinder (doublet)", vary the circulation Γ between −18 and +18 m²/s and observe how the lift changes through the Magnus effect
Switch the flow pattern (uniform flow / source / cylinder (doublet) / free vortex / Rankine oval) and compare how the streamlines and the Cp colour map change
Read the top and bottom pressure coefficients, lift coefficient CL, lift L and number of stagnation points (drag is always zero by d'Alembert's paradox, so it is not shown)
Worked Example
A cylinder of radius R = 1.0 m (diameter D = 2.0 m) in an air stream of U = 5 m/s (air density ρ = 1.225 kg/m³): the Reynolds number is Re = ρUD/μ ≈ 6.8×10⁵. Without circulation (Γ = 0), CD = 0 in line with d'Alembert's paradox for an ideal fluid, but adding Γ = +5 m²/s generates lift L' = ρUΓ = 1.225·5·5 ≈ 30.6 N/m by the Kutta–Joukowski theorem, and CL = Γ/(U·R) = 1.0. At the same time Cp (top) ≈ −3.65 and Cp (bottom) ≈ −2.38, and the streamlines become asymmetric.
Practical Notes
On a real cylinder, Kármán vortex shedding begins above Re ≈ 47, and for Re ≈ 10³–2×10⁵ the drag coefficient is CD ≈ 1.0–1.2, very different from the zero drag of potential flow. Use CFD (RANS/LES) alongside to see the effect of viscosity
On a real rotating cylinder, the circulation arises from boundary-layer viscosity and is smaller than the value estimated from the surface speed. The lift also falls below the Kutta–Joukowski value, so treat the lift from this tool as an upper-bound guide
This tool evaluates the Cp colour map analytically pixel by pixel, so no grid-resolution setting is needed. If it is heavy, mind the rendering load in the browser
The potential-flow pressure distribution is close to reality only on the front half of the cylinder. On the rear half it differs greatly because of boundary-layer separation (around 80° from the front stagnation point for laminar flow)
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