A Kármán vortex street is one of the most recognisable patterns in fluid dynamics: the alternating swirls shed behind a cylinder in crossflow. It is also one of the easiest things to get subtly wrong in a simulation. When the mesh is too coarse to resolve the shed vortices, the street that comes out is not simply blurry. It has a different shape, a different shedding frequency, and a different downstream decay.

The Grid Resolution Problem

Vortex shedding is a resolution-sensitive phenomenon by construction. The street only exists because vorticity concentrates into compact cores that then detach and travel downstream. If those cores are smaller than a few cells, the discretisation cannot hold them together. The pattern that survives is a numerical artefact wearing the costume of a physical result.

Wind-tunnel measurements of cylinder wakes have been the reference standard for decades. Simulations that look qualitatively convincing on a coarse mesh often miss the measured Strouhal number by a margin that matters for any downstream use, such as predicting the forcing frequency on a structure or the acoustic tone of a bluff body.

Practitioners disagree about how much of this is acceptable. Some treat a coarse-mesh vortex street as a useful qualitative indicator. Others refuse to report shedding frequencies unless the mesh has been refined until the result stops moving. Both positions have costs, and neither is obviously wrong.

What Kármán Streets Require

The rule of thumb that circulates in CFD training is that a shed vortex needs somewhere between ten and twenty cells across its core to be tracked with reasonable fidelity. That range is a heuristic, not a derived bound. It varies with the scheme's order of accuracy, the Reynolds number, and how far downstream the quantity of interest sits.

Coarse grids smear vorticity. The discrete analogue of viscosity spreads the vortex core over the available cells, and the peak vorticity drops. This is not a bug in the solver; it is what a truncated Taylor series does to a sharp feature. The practical effect is that the street looks diffused and the alternating pattern weakens.

Numerical diffusion is the term for this, and it behaves like a physical viscosity that nobody specified. On a coarse mesh, the effective viscosity can exceed the molecular value by a large factor. The simulation then reports a flow that is more damped than the real one, and the shedding may weaken or vanish entirely at Reynolds numbers where the experiment shows a clean street.

The heuristic number also hides a directional asymmetry. Streamwise spacing and cross-stream spacing do not carry the same penalty. A mesh that is fine in the flow direction but coarse across it can resolve the advection of a vortex while starving it of the cells it needs to stay round. The core then stretches along the axis of travel, which is one reason coarse-mesh streets come out elongated rather than merely faint. Refining uniformly is the safe move, but it is rarely the cheapest one, and practitioners who refine only where the wake is expected sometimes introduce a spacing jump that generates its own spurious vorticity at the interface.

How the Simulations Were Run

The comparison setup is deliberately plain. One solver, one cylinder geometry, one set of boundary conditions. The only variable is the grid spacing. Reynolds numbers are held in the laminar-to-turbulent transition range where shedding is strong and well documented, so that any discrepancy cannot be blamed on an ambiguous flow regime.

Baseline runs carry no sub-grid model. That choice isolates the effect of the mesh from the effect of a closure. Adding a sub-grid model would change the answer, but it would also confound the comparison, since different closures would need to be tuned separately at each resolution.

The vortex street is detected through velocity spectra taken at probe points in the wake. A clear peak in the spectrum marks the shedding frequency, and the Strouhal number follows from that frequency, the cylinder diameter, and the freestream velocity. The shape of the street is assessed from instantaneous vorticity fields.

Probe placement deserves more care than it usually gets. A probe sitting inside a diffuse vortex core records a weaker, broader spectral peak than one placed in the free stream between successive vortices, and the peak width feeds directly into the uncertainty on the shedding frequency. Two probes at different downstream stations also reveal whether the street is decaying at the rate the experiment shows or collapsing early, which is often the first visible symptom of under-resolution.

Different Streets, Same Parameters

On coarse grids the vortices come out elongated and irregular. The neat alternating pattern gives way to a ragged wake in which successive vortices differ in strength and spacing. The street may persist for only a few diameters downstream before dissolving into a disorganised wake.

Fine grids recover the classic alternating pattern, with compact cores and a regular spacing that matches the experimental record. The transition between these two behaviours is not gradual. There is a resolution band where the result changes quickly, and below it the street is simply a different object.

The Strouhal number shifts by roughly ten to twenty percent across the resolution range. That figure is a range rather than a fixed offset because it depends on the scheme and the Reynolds number. A ten percent error in shedding frequency is enough to move a predicted structural resonance away from the measured one.

Discrepancies persist across solvers. Different codes with different discretisation schemes produce different coarse-mesh streets, which means the coarse-mesh answer is partly a property of the software rather than the flow. The fine-mesh answers converge toward each other and toward the experiment, which is the behaviour a convergence study is designed to expose.

The direction of the drift is not fixed either. First-order upwind schemes add enough dissipation to slow the shedding and lengthen the wake, while some low-dissipation central schemes on the same coarse mesh can over-predict the frequency and produce a street that is too tightly spaced. Two practitioners comparing notes on the same geometry can therefore disagree about whether a coarse mesh runs fast or slow, and both can be right about their own code. That is an uncomfortable position for anyone who wants a single correction factor, and it is the reason the resolution threshold is usually discussed as a property of a solver-plus-mesh pair rather than of the mesh alone.

What the Result Does Not Show

None of this makes coarse grids useless. A coarse simulation that captures the overall wake topology, the recirculation length, and the qualitative shedding behaviour can be entirely fit for purpose if the question is about flow structure rather than frequency. The error matters when the number is used, not when the picture is looked at.

The comparison does not validate any sub-grid model. It says nothing about whether a particular closure improves coarse-mesh predictions, because no closure was tested. A reader who wants that answer needs a separate study with the closure in the loop.

The scope is narrow. The results are limited to a range of Reynolds numbers around the laminar-to-turbulent transition and to two-dimensional simulations. Three-dimensional flows introduce spanwise instability and vortex dislocations that change the resolution requirements, and nothing here should be extrapolated to them. The same caution applies to compressible flows, rough cylinders, and bodies other than a circular cylinder.

There is also a cost side that the comparison deliberately brackets. Refining a wake by a factor of two in each direction multiplies the cell count by roughly four in two dimensions and by roughly eight in three, before the time-step restriction that comes with smaller cells is counted. A convergence study that doubles the resolution three times can consume an order of magnitude more compute than the production run it is meant to validate. That arithmetic is the honest argument for coarse meshes, and it is a better argument than the hope that the frequency will come out right anyway.

Practical Steps for Simulators

Run a grid-convergence study before trusting any vortex prediction. Refine the mesh by a factor of roughly 1.4 to 2 in each direction and check whether the shedding frequency and the street shape stop changing. If they do not, the result is not converged.

Report the cell count per expected vortex length alongside the result. A reader cannot judge a shedding frequency without knowing how many cells spanned the vortex core. State it explicitly rather than leaving it in a supplementary file.

Compare against experimental Strouhal data where such data exist. For a circular cylinder in crossflow, the measured relationship between Reynolds number and shedding frequency is well established, and a quick check against it catches gross resolution errors before they propagate into a design decision.

Document the solver settings and the estimated numerical diffusion. The scheme order, the limiter, and the time-step size all affect how much artificial dissipation enters the solution. A short note in the methods section saves the next reader from repeating the work.

Flag any result that sits near the resolution threshold. If a small change in mesh spacing moves the answer by more than a few percent, say so in the report. The related discussion of how calibration choices shape noise budgets makes a similar point about uncertainty that lives in the setup rather than the physics. The same logic applies to fabrication steps accumulating in qubit coherence: the procedural detail is the result.