Reconciling Nondimensional Forms of the Decaying Taylor Green Vortex
Richard Janutka · 2026-10-16 · Fluid Dynamics · Mathematics · Physics · Engineering
The exact decaying two-dimensional Taylor vortex is a useful reference field, but its numerical interpretation depends on which length defines the trigonometric phase, the computational domain and the Reynolds number. We separate these roles as the vortex scale \ell, square-domain side D and Reynolds reference length L{Re}. Starting from dimensional incompressible flow, we state the complete nondimensional map and derive a translated viscous vortex whose continuity, momentum and pressure-Poisson residuals vanish identically. We then compare two periodic interpretations under an explicitly assumed axis-aligned periodic square: a vortex-scale convention with \nu=1/Re{\ell}, and a domain-scale convention with \nu=2\pi/ReD. At an equal numerical Reynolds label they assign different viscosities and decay rates; at equal physical viscosity they coincide under ReD=2\pi Re{\ell}. We also distinguish physical pressure p from kinematic pressure q=p/\rho, for which the correct Poisson equations differ by a density factor. A literal identification D=\ell=2\pi remains a local solution of the differential equations but is incompatible with the assumed periodic square when the perturbation is nonzero. Full-period invariants, analytical numerical fingerprints and fixed Fourier support provide additional checks. A fresh private reanalysis reproduced the central formulas with 28 executable checks, including deliberately defective controls; archived historical check totals are reported separately as documentary provenance rather than new mathematical evidence. The result is a bounded reconciliation of conventions, not a new vortex solution, numerical-solver validation, turbulence result, industrial-flow study or statement of Airbus intent.
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