15  Laminar Cylinder Vortex Shedding — Strouhal/CD/CL Validation

Validation of unsteady flow, curved meshes, and force integration in the Flux Reconstruction solver (against literature and DBNS)

15.1 Run Provenance and Artifacts

Item	Value
Wake CFD baseline version	0.22.2
Solver	FR
Compute device	GPU (NVIDIA A2000)
Parallel execution	1 MPI rank
Arithmetic precision	FP64
Run date	2026-07-19

Note

Results depend on the solver build that produced them. Read every figure in this report together with the version and date above — re-running the same input on a later build may give different numbers.

15.1.1 Downloadable Files

Not available: standalone input decks and solver result files have not yet been published for download.

15.2 Validation Objective

This case simulates laminar vortex shedding around a circular cylinder with the FR (Flux Reconstruction) solver and compares the results with published values and those from Wake CFD’s FVM-based DBNS solver. Unlike the steady-flow Cases 01 and 02, this is an unsteady flow case that verifies the FR solver implementation in terms of:

• vortex-shedding frequency (Strouhal number)
• mean drag coefficient and lift-coefficient amplitude
• viscous-force integration on curved high-order elements (a Q2 O-grid)
• cross-solver validation between the independently implemented FVM (DBNS) and FR solvers

15.3 Reference Solution and Data

Flow around a circular cylinder at Re_D=100 is a representative unsteady validation case that produces a regular Kármán vortex street and has well-established literature values for Strouhal number, drag, and lift amplitude. A low-Mach (M=0.05) compressible formulation is used to model an essentially incompressible flow.

The same physical condition is also simulated with Wake CFD’s production FVM solver (DBNS), providing cross-validation through two independent numerical paths.

Item	Setting
References	Williamson 1988 (St), Henderson 1995 (C_D), and the Wake CFD DBNS solver for cross-validation

15.4 Simulation Conditions

Item	Setting
Geometry	Circular cylinder, D=1 m, outer boundary R_out=20D
Analysis	Unsteady, compressible laminar Navier–Stokes
Re_D	100
Mach	0.05 (mu=0.2081 is used to match Re_D)
Reference state	ISA sea level — P=101325 Pa, T=288.15 K → rho=1.2218, U=17.03 m/s
Total simulation time	t=24 s (24,000 steps)
FFT window	t=12→24 s (about 34 cycles; initial transient excluded)

15.5 Mesh

Item	Setting
Mesh	Curved Q2 O-grid, 96(θ) × 64(r), first wall-cell height Δr≈0.030

15.6 Boundary and Initial Conditions

Item	Setting
Wall	no-slip, adiabatic
Outer boundary	farfield

Initial field: Not reported.

15.7 Numerical Methods

Item	Setting
Physical time step	dt=1×10^-3 s
Spatial accuracy	FR p2 (third order)
Time integration	Dual time — physical: BDF1; pseudo-time: RK45 + local-PI CFL control + p-multigrid

15.8 Results

Time averaging and FFT analysis over t=12→24 s (about 34 cycles) are compared with literature values and DBNS results at the same condition.

Metric	FR p2 (BDF1, dt=1e-3)	DBNS BDF1 (same scheme and dt)	DBNS BDF2	Literature
St	0.1615	0.1651	0.1670	0.164 (Williamson 1988)
Mean C_D	1.3203 ± 0.0041	1.3430 ± 0.0058	1.3592	1.35 (Henderson 1995)
C_L amplitude	0.2678	0.3088	0.3347	≈0.33
St (drag component, ≈2·St)	0.3229	0.3303	0.3340	0.328

C_L(t) time history — reaches a regular vortex-shedding amplitude after the initial transient

C_D(t) time history — comparison of the time mean with the Henderson reference

C_L spectrum — Strouhal number extracted from the dominant frequency

C_D spectrum — peak at twice the vortex-shedding frequency

St agrees with the literature value within 2%. The drag-spectrum peak occurs at exactly twice the lift-spectrum frequency, reproducing the expected physics: vortices shed alternately from the two sides of the cylinder, so drag oscillates at twice the lift frequency. The C_L amplitude is somewhat lower than the literature and DBNS values. This is consistent with numerical damping from first-order BDF1 time integration; DBNS likewise gives a lower value with BDF1 than with BDF2.

FR and DBNS are two independent numerical paths that do not share a code base. With the same time-integration scheme (BDF1) and dt, they agree broadly, with differences of −2.2% in St, −1.7% in C_D, and −13.3% in C_L amplitude.

15.9 Conclusions and Applicable Scope

• For vortex shedding from a cylinder at Re_D=100, the FR solver agrees with literature within 5% for St and C_D, and within 25% for C_L amplitude, which is the case acceptance criterion.
• Wake CFD’s two solvers (FR and DBNS), which do not share a code base, agree in St and C_D at the same condition, confirming that both implementations reproduce the same physics independently. The comparatively larger discrepancy in C_L amplitude among the solvers and literature requires further study.
• These results were obtained for one Reynolds number (Re_D=100), one mesh, and one time-integration scheme (BDF1). They cannot be generalized to mesh or time-step convergence, higher Reynolds numbers, or three-dimensional effects.
• Follow-on validation should compare polynomial-order (p) convergence, dt convergence, and higher-order time integration (sdirk33) as independent variables.
• The effects of time-integration order (BDF1 versus BDF2) and arithmetic precision (FP32 versus FP64) on this flow are covered separately for the DBNS solver in the time-integration and precision comparison case.

This case shows, through literature comparison and cross-solver validation, that the FR solver’s integrated unsteady viscous-flow, curved high-order-element, and force-integration path functions correctly. It establishes a minimum validation baseline for this range of conditions rather than an absolute accuracy guarantee.

15.10 References

• Williamson, C. H. K., “Defining a Universal and Continuous Strouhal–Reynolds Number Relationship for the Laminar Vortex Shedding of a Circular Cylinder,” Physics of Fluids, 31(10), 1988, pp. 2742–2744.
• Henderson, R. D., “Details of the Drag Curve near the Onset of Vortex Shedding,” Physics of Fluids, 7(9), 1995, pp. 2102–2104.