Turbulence closure problem
Is there a theoretical model that can predict the statistical structure of turbulent flow (e.g. energy distribution across eddy sizes) directly from the Navier-Stokes equations, without ad hoc empirical closures?
History and references
Still open in the literature as of 2026.
Turbulence was called one of the last unsolved problems of classical physics by Richard Feynman and others. Reynolds-averaged and large-eddy approaches give practical engineering closures, and Kolmogorov's 1941 statistical theory captures inertial-range scaling well, but no one has derived the full statistics of turbulence (including intermittency corrections) from first principles. It remains a major open problem bridging fluid dynamics, statistical mechanics, and nonlinear PDE theory.
Turbulence is the physics you can watch swirl past you in a river, yet a century of the world's best mathematicians still cannot fully tame its statistics.
Explanations that circulated
- That Kolmogorov's 1941 -5/3 energy-cascade law is a complete first-principles theory; it is a phenomenological scaling result, not a closure derived from the Navier-Stokes equations
References
- A First Course in Turbulence, Henk Tennekes, John L. Lumley, MIT Press, 1972
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