CFD Solution Verification · Code-to-Benchmark · interactive

Laminar Vortex Shedding behind a Circular Cylinder at Re = 100

OpenFOAM pimpleFoam (v2312) · incompressible transient laminar · body-fitted O-grid · digitalmodel CFD capability (issue #1166, epic #1161)

VERIFIED — C̅_d within 0.5% and St within 0.9% of benchmark

SolverOpenFOAM pimpleFoam

Reynolds No.Re = 100

Mesh21,120 cells (O-grid)

Mean C_d1.344

Strouhal St0.1655

C_l amplitude0.325

1 Objective

Verify the digitalmodel transient-CFD pipeline against the canonical laminar Karman vortex street behind a circular cylinder at Reynolds number 100 — the second foundational known-answer case of the CFD validation hierarchy (after the flat-plate boundary layer). The two target quantities have well-established experimental and computational benchmark values:

mean drag C̅_d ≈ 1.33–1.37 Strouhal St = f_sD/U∞ ≈ 0.164

where f_s is the vortex-shedding frequency, D the cylinder diameter and U∞ the free-stream speed.

What this gates in real work — VIV screening

Alternating vortex shedding is the forcing behind vortex-induced vibration of risers, jacket members, mooring lines and pipeline free spans. This case verifies the two numbers VIV screening depends on: the shedding frequency (Strouhal, +0.9%) and the force amplitude (Cd −0.5%, Ĉl in the published band). Try it on a real member:

Member diameter D (m)

Current speed U (m/s)

Structure natural freq fₙ (Hz)

Shedding freq fₛ = St·U/D—

Reduced velocity Uₙ = U/(fₙD)—

Lock-in screen (4 ≤ Uₙ ≤ 8)—

Uses the sub-critical Strouhal St ≈ 0.2 (10³ < Re < 2×10⁵), the flat part of the St–Re curve; this verification case sits at Re = 100 where St = 0.164–0.166, the laminar benchmark point. Lock-in band per DNV-RP-C205 / DNV-RP-F105 screening practice; a hit means fatigue assessment (e.g. the free-span workflow) is required, not that failure occurs.

2 Result — the vortex street

animation frame — Instantaneous spanwise vorticity at t=150: the alternating (red/blue) vortices of the laminar Karman street, shed periodically from the cylinder.

3 Numerical model

Item	Specification
Solver	`pimpleFoam` — transient incompressible PIMPLE, laminar closure
Mesh	Body-fitted O-grid, 21,120 cells; cylinder D=1, circular far-field at 20 D; radial wall-clustering. Non-orthogonality <0.001, max skewness 0.03
Flow	U∞=1, ν=0.01 → Re = U∞D/ν = 100 (laminar shedding regime)
Boundary conditions	cylinder `noSlip`; far-field `freestream`/`freestreamPressure` (entraining)
Discretisation	2nd-order: `backward` time, `Gauss linear` convection; adjustable Δt at Co≤5
Run	t = 0→160; forces sampled over the saturated limit cycle t = 120–160 (5 shedding periods)

4 Triggering the instability (engineering note)

Why the wake first refused to shed. On a perfectly grid-aligned, symmetric setup the discrete solution preserves the up/down symmetry, so the shedding instability — though physically unstable at Re=100 (above the critical Re≈47) — cannot grow from a symmetric state; the lift stayed identically zero to t>40. Because a circle is rotationally symmetric, tilting the free stream by 5° is the same physical flow rotated — it leaves C_d and St unchanged — but it breaks the grid-aligned numerical symmetry and seeds the instability, which then grows exponentially and saturates by t≈120. Forces are measured along the rotated flow/normal axes, so the reported coefficients are unbiased.

5 Force history & quantitative agreement

Left — startup transient: the impulsive start drives C_d to +276 / −136 in the first two output steps — a mesh- and timestep-dependent rendering of the impulsive-start force singularity (Bar-Lev & Yang 1975), not a converged coefficient and excluded from the reported values below. C_d then relaxes to the symmetric-wake plateau ≈ 1.10 with C_l ≈ 0 (no shedding yet). Right — steady-state limit cycle: the saturated Kármán shedding the benchmark values are read from. C_d oscillates about 1.344 (shaded band = 1.33–1.37 benchmark, dashed = mean) at twice the shedding frequency; C_l at ±0.325 at the shedding frequency. Bottom — full history: the complete run — C_d settling to the symmetric-wake plateau ≈ 1.10, the shedding instability growing through t ≈ 55–120 (shaded), and saturation into the limit cycle. The startup spike is clipped off-scale (see panel 1).

Quantity	This CFD	Benchmark	Error	Source
Mean drag C̅_d	1.344	1.33–1.37	-0.5%	Williamson (1996); Henderson (1995)
Strouhal St	0.1655	0.164	+0.9%	Williamson (1996); Roshko (1954)
Lift amplitude Ĉ_l	0.325	~0.32–0.34	in range	Henderson (1995)

5.1 Strouhal extraction

The shedding frequency is taken from the up-crossings of the lift signal over 5 periods (period 6.044 time units), giving St = 0.1655. A raw FFT of the same window returns 0.175, but its frequency-bin width (1/40 ≈ 0.025) is too coarse to resolve St to better than ~15%; the crossing-based estimate is the reliable one. This distinction is itself a verification of the post-processing, not just the solver.

6 Provenance & reproducibility

Solver / version	OpenFOAM ESI v2312, `pimpleFoam`
Mesh	4-block annular O-grid, 21,120 cells, generated programmatically; `checkMesh` OK
Post-processing	OpenFOAM `forceCoeffs` + `vorticity`; PyVista/NumPy/Matplotlib analysis
Acceptance	C̅_d in 1.33–1.37; St within 5% of 0.164
Capability	digitalmodel `solvers/openfoam` (runner + doctor), epic #1161, issue #1166

7 Conclusion

Verified. The digitalmodel transient-CFD pipeline reproduces the laminar vortex street at Re=100 in agreement with the accepted benchmarks: mean drag C̅_d = 1.344 (-0.5%) and Strouhal St = 0.1655 (+0.9%), with the lift amplitude in the literature range. Together with the flat-plate Blasius verification this establishes the capability across both steady boundary-layer and transient bluff-body regimes.

References

Williamson, C. H. K. (1996). "Vortex dynamics in the cylinder wake." Annu. Rev. Fluid Mech. 28, 477–539.
Roshko, A. (1954). "On the development of turbulent wakes from vortex streets." NACA Report 1191.
Henderson, R. D. (1995). "Details of the drag curve near the onset of vortex shedding." Phys. Fluids 7, 2102.
OpenFOAM v2312 User Guide, ESI-OpenCFD (2023).