3. The Reynolds Number Caveat
In a real physical fluid, the Reynolds number dictates the transition from laminar to turbulent flow and is calculated as:
\[ Re = \frac{U D}{\nu} \]
Where \( U \) is the flow velocity, \( D \) is the cylinder diameter, and \( \nu \) is the kinematic viscosity of the fluid.
Why is this tricky in our webtool?
Our simulation solves the Euler equations, which assume the fluid is perfectly inviscid (\( \nu = 0 \)). Technically, the Reynolds number of an inviscid fluid is infinite. However, because we compute this on a discrete pixel grid, the grid itself introduces errors that act exactly like physical viscosity—a phenomenon called numerical viscosity. Therefore, the fluid behaves as if it has viscosity, allowing vortex shedding to occur.
The \( Re \) displayed in the UI uses the real-world viscosity of water or air for educational context, but the actual fluid behavior you see is dictated by the grid resolution and the Vorticity Confinement slider, not a strictly calculated viscous friction.
4. Mathematical Theory
The Governing Equations
This simulation is built upon the Eulerian approach to fluid dynamics, specifically solving the incompressible Euler equations on a Cartesian MAC grid.
- Conservation of Mass (Incompressibility): The fluid volume cannot compress or expand.
\[ \nabla \cdot \mathbf{u} = 0 \]
- Conservation of Momentum: How the velocity field \( \mathbf{u} \) changes over time, where \( \rho \) is density, \( p \) is pressure, and \( \mathbf{f} \) represents external forces.
\[ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \mathbf{f} \]
The Solver (Gauss-Seidel with SOR)
To enforce the incompressibility condition, the simulation calculates how much fluid is entering and leaving every single cell on the grid. If more fluid enters a cell than leaves (positive divergence), the solver artificially spikes the pressure \( p \) in that cell, which pushes the fluid outward in the next step. This is solved iteratively using Successive Over-Relaxation (SOR) to speed up the computation.
Vorticity Confinement
To counteract the numerical viscosity mentioned earlier, we use a technique called Vorticity Confinement.
First, we calculate the vorticity (the local rotation of the fluid), \( \omega \):
\[ \omega = \nabla \times \mathbf{u} \]
Next, we find the gradient of the vorticity magnitude to determine the direction pointing toward the center of the vortex, and normalize it into a unit vector \( \mathbf{N} \):
\[ \mathbf{N} = \frac{\nabla |\omega|}{|\nabla |\omega||} \]
Finally, we apply a confinement force \( \mathbf{f}_{conf} \) that acts perpendicular to \( \mathbf{N} \), effectively spinning the vortex back up. This force is scaled by the slider parameter \( \epsilon \) and the grid spacing \( h \):
\[ \mathbf{f}_{conf} = \epsilon h (\mathbf{N} \times \omega) \]
This force is added back into the momentum equation as \( \mathbf{f} \), keeping the Von Kármán vortex street crisp and visible over long distances.
Credits & License
Original Developer: Matthias Müller ("Ten Minute Physics")
Modifications & Enhancements: David S. Nobes, University of Alberta (2026)
Modifications tailored for academic cylinder flow visualization, real-time data extraction, and UI enhancements.
Modifications tailored for academic cylinder flow visualization, data extraction, and UI enhancements.
MIT License:
Copyright 2022 Matthias Müller.
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files, to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
