The Silver Ratio of Turbulence: A Universal Geometric Bound in the Navier–Stokes Equations

February 23, 2026Open Access

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This research aims to establish the silver ratio as a universal lower bound in turbulent flows governed by the Navier-Stokes equations.
Conducted direct numerical simulations (DNS) on a 256^3 periodic grid.
Utilized corrected stochastic forcing schemes for enhanced accuracy.
Implemented four distinct protocols for testing turbulence under varying conditions.
Monitored the angle between velocity and nonlinear terms in regions of high kinetic energy.
The alignment parameter settled at 0.4123 ± 0.0002, closely matching the silver ratio.
Under extreme forcing, the parameter peaked at 0.458 but never fell below 0.39.
Vorticity kurtosis confirmed genuine turbulent behavior with a peak of 5.6.
The bound held even with lower resolutions, indicating its robustness.

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Direct numerical simulations unveil a universal geometric bound for turbulence, suggesting novel insights into fluid dynamics.

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Igor Labadin (2026) studied this question.

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