22 Sep 2026
Oxford researchers use machine learning to simplify a longstanding turbulence problem
A compact mathematical formula describes how turbulent energy is produced and dissipated near a wall - a tool researchers can inspect, test and reuse
One formula across three types of turbulent flow. Black curves show the Oxford team’s formula for the ratio of turbulent energy production to dissipation. Coloured curves show all 19 external benchmark profiles used in the paper’s main comparison.
Researchers in the University of Oxford’s Department of Engineering Science have combined machine learning with turbulence physics to develop a simple formula for a difficult part of fluid-flow modelling. The study, by Jianou Jiang and Professor Budimir Rosic, was published in Physics of Fluids on 9 September 2026 and selected as a Featured Article.
When air or water flows along a surface, turbulent motion continually gains energy from the main flow and loses it through dissipation into heat. Understanding this balance matters for modelling flows in engineering. Detailed simulations, including work by Mansour, Kim and Moin in 1988, have shown that production and dissipation can differ substantially close to a wall.
The Oxford researchers used symbolic regression, a machine-learning technique that searches for mathematical expressions in data, alongside analysis of the underlying physics. The result is a formula with just four constants that describes how the ratio of energy production to dissipation changes with distance from a smooth wall.
After fitting those constants to channel-flow simulations, the team compared the formula with 19 external benchmark profiles from four different simulation codes, without refitting. It closely matched the tested near-wall regions in channels, pipes and boundary layers—the thin regions of flow next to a surface.
For researchers, the immediate benefit is a compact reference for analysing energy budgets and comparing turbulence models. The published Python code lets them evaluate the formula without repeating the machine-learning search. A five-minute browser-based tutorial demonstrates its use with simulation data, making the method easier to try and its results easier to check.
“A short equation gives other researchers something concrete to inspect, test and challenge. We have shared the code and a short tutorial so they can try the method directly and explore how it behaves on their own data.”
Co-author Jianou Jiang
In a separate test using detailed simulation data, the formula reduced the root mean square error in estimating energy dissipation by 88–95%, compared with assuming that production and dissipation are equal. This is evidence for analysing the tested simulation budgets; whether it improves the speed or accuracy of complete engineering flow simulations remains to be established.
Full Paper
J. Jiang and B. Rosic, “Combining symbolic regression and turbulence physics to develop a compact inner-layer production–dissipation closure”, Physics of Fluids 38, 095140 (2026). DOI: 10.1063/5.0347368.
Simulation Data Acknowledgement
The underlying DNS profiles are from Hoyas and Jiménez (channel), El Khoury and colleagues and Yao and colleagues (pipe), and Schlatter and Örlü (boundary layer), as cited in the paper. The graphic uses the original budget data and the paper’s published evaluation masks. The original data providers retain their rights to the source datasets.
Main image
Caption: One formula across three types of turbulent flow. Black curves show the Oxford team’s formula for the ratio of turbulent energy production to dissipation. Coloured curves show all 19 external benchmark profiles used in the paper’s main comparison: four channel profiles, five pipe profiles and ten boundary-layer profiles. The dashed line marks equal production and dissipation. Each simulation profile is shown only over its tested near-wall range; distance is expressed in wall units on a logarithmic scale. The same four formula constants are used throughout, without refitting. The boundary-layer cases have zero pressure gradient.
Credit: Adapted from J. Jiang and B. Rosic, Physics of Fluids 38, 095140 (2026), https://doi.org/10.1063/5.0347368, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). This graphic replots the published comparison with new layout and plain-language labels; it is not an unchanged journal figure.
Technical note
Only intended-domain external comparisons are shown. These are external re-evaluations, not untouched blind tests: most profiles had been seen in earlier model selection. The plot excludes the training profiles and the adverse-pressure-gradient and Couette stress tests. The published formula is intended for 1 < y+ ≤ min(150, 0.3 Re_tau), with the paper’s positivity and numerical-quality masks. It does not establish accuracy in complete engineering flow solvers. Source-file hashes and all 19 per-profile scores were checked before this graphic was generated.
