Click to seed streamlines · Double-click to clear
Theory & Key Formulas
Source: \(\vec F = \dfrac{Q}{2\pi}\dfrac{\hat r}{r}\)
Free vortex: \(\vec F = \dfrac{Q}{2\pi}\dfrac{\hat\phi}{r}\)
Saddle: \(\vec F = (Qx, -Qy)\)
Divergence \(\nabla\cdot\vec F\) and curl \(\nabla\times\vec F\) classify the field topology.
What this visualizer shows
🤩Why are some fields drawn as outward arrows and others as swirls?
🎓A point source has positive divergence and zero curl — flow leaves every neighbourhood. A free vortex has zero divergence and a delta-function curl at the origin — flow rotates without expanding. Switch between "Point source" and "Free vortex" and watch the streamlines rearrange.
🤩What is the saddle case?
🎓A linear hyperbolic field that pulls in along one axis and pushes out along the other. It models the flow near stagnation points and is the bread-and-butter example for stability analysis in dynamical systems.
🤩Where is this useful in CAE?
🎓Potential-flow superposition (uniform + source + dipole = Rankine oval, for example) gives quick analytic insight before launching a CFD case. The same arrow plot machinery is what your CFD post-processor uses for velocity vectors and electric field plots.
Physical model
A 2D vector field assigns a vector $\vec F(x,y)$ to every point. The simulator visualises it three ways: discrete arrows on a grid, integrated streamlines, and an optional magnitude colormap. The streamlines are obtained by integrating $\dot{\vec r}=\vec F/|\vec F|$ with a fixed step.
Two scalar quantities classify the field. Divergence $\nabla\cdot\vec F = \partial_x u + \partial_y v$ measures local outflow; curl $\nabla\times\vec F = \partial_x v - \partial_y u$ measures local rotation. The cursor probe estimates these numerically by central differences.
Real-world applications
Aerodynamics: potential-flow combinations of uniform, source, dipole and vortex elements model lift and bluff-body flow before viscous CFD is run.
Electromagnetism: the same equations describe 2D electrostatic and magnetostatic fields. Source = positive charge, dipole = electric dipole, vortex = magnetic field of a long wire.
Dynamical systems: phase-plane analysis of ODE systems uses exactly this kind of arrow plot to diagnose fixed-point stability.
FAQ
- What is divergence?
- The local outflow per unit area: $\nabla\cdot\vec F=\partial_x u+\partial_y v$. Positive at sources, negative at sinks.
- What is curl?
- The infinitesimal rotation rate: $\nabla\times\vec F=\partial_x v-\partial_y u$. Positive curl rotates a small paddlewheel counter-clockwise.
- Why does the source field blow up at the origin?
- Because $|\vec F|\sim 1/r$ for a 2D point source, the magnitude diverges at $r\to 0$. Real engineering analogues are always smeared over a finite size, but the singular form is a useful mathematical building block.