A theory of turbulence
Turbulence statistics cannot be derived from the flow equations; engineers use models tuned to data and simulations that stop far below real Reynolds numbers.
Open in the interactive tree →Turbulent flow, the chaotic swirling of air and water, sets drag on aircraft, mixing in the oceans and the dynamics of storms. Kolmogorov's 1941 theory gave a statistical picture (an energy cascade with a -5/3 spectrum) but is not exactly right: turbulence is not perfectly self-similar, and intermittency breaks its scaling. This is the physics problem of predicting flow statistics; the Clay question about smooth solutions of the equations is a separate problem (see the Navier-Stokes node).
As of October 2026
In February 2026 a Georgia Tech team (Yeung and colleagues) reported the largest direct simulation of turbulence so far: 32,768 cubed (about 35 trillion) grid points on the Frontier supercomputer at a Taylor-scale Reynolds number of about 2,500, which found distinct statistics for energy dissipation and enstrophy, mostly intact classical scalings and stronger intermittency corrections than commonly assumed. Direct simulation needs about Re^(9/4) mesh points and a total cost that grows as Re cubed, so most industrial Reynolds numbers remain beyond the most powerful computers. In January 2026 Ling and Lozano-Durán reported a machine-learned subgrid-stress model that beat the dynamic Smagorinsky model on mean velocity and wall shear stress in adverse-pressure-gradient boundary layers (preprint).
What is missing
- A derivation of turbulence statistics (scaling exponents, intermittency) from the Navier-Stokes equations
- Simulations and measurements at the very high Reynolds numbers of aircraft, oceans and the atmosphere
- An account of rare, extreme local fluctuations, which the 2026 simulation found are often not sufficiently covered by classical theories
- Closure models that stay accurate and stable on flows unlike their training data
Becomes possible once solved
- Aircraft, ships and pipelines designed with less drag and energy loss
- More reliable weather, ocean and combustion models
- Simulations of real flows with quantified error bars
Open steps
- Reliable learned closure models High AI leverageTrain large-eddy and Reynolds-averaged closures on direct-simulation data that stay stable, respect physics and generalize to new geometries and Reynolds numbers.
- Direct simulation at higher Re Medium AI leveragePush direct numerical simulation beyond a Taylor-scale Reynolds number of about 2,500 with efficient exascale codes, hybrid ML-corrected solvers and open datasets.
- Theory of intermittency and extremes Medium AI leverageDerive from the equations why rare intense events in dissipation and vorticity are stronger than Kolmogorov scaling predicts, and quantify the corrections.
- Experiments at high Reynolds number Low AI leverageMeasure turbulence at the Reynolds numbers of pipes, wind tunnels and the atmosphere to test scaling laws and simulation-based theories.
- Link to the Navier-Stokes problem Low AI leverageConnect mathematical results on smoothness and blow-up to statistical turbulence: do singularities matter for real flows?
Where AI could help
Medium AI leverage. AI speeds simulation and learns closure models, but a theory derived from the equations is still missing and high-Reynolds data are scarce.
- Learn subgrid closure models from direct-simulation data and embed them in solvers
- Build fast surrogates to scan flow regimes
- Discover scaling laws and coherent structures in petabytes of simulation data
- Reconstruct flow fields from sparse measurements
Shown so far
- In 2021, Google researchers reported a learned solver for two-dimensional turbulence that matched baseline solvers using 8-10 times finer resolution per dimension, a 40-80x speedup, and stayed stable over long runs (arXiv version). source
- In January 2026, a preprint reported a machine-learned subgrid-stress model for large-eddy simulation that predicted mean velocity and wall shear stress better than the dynamic Smagorinsky model in adverse-pressure-gradient boundary layers. source
Prerequisites
Unlocks
- Predictable Turbulence2040s?