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Constructing turbulence models using the kinetic Fokker–Planck equation

Published online by Cambridge University Press:  15 May 2025

Peng Luan Open the ORCID record for Peng Luan [Opens in a new window] ,
Haoyuan Zhang Open the ORCID record for Haoyuan Zhang [Opens in a new window]  and
Jun Zhang Open the ORCID record for Jun Zhang [Opens in a new window]
Peng Luan
Affiliation:
School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, PR China
Haoyuan Zhang
Affiliation:
State Key Laboratory of Aerodynamics, Mianyang 621000, PR China Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, PR China
Jun Zhang*
Affiliation:
School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, PR China
*
Corresponding author: Jun Zhang, jun.zhang@buaa.edu.cn
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Abstract

This study presents a novel approach for constructing turbulence models using the kinetic Fokker–Planck equation. By leveraging the inherent similarities between Brownian motion and turbulent dynamics, we formulate a Fokker–Planck equation tailored for turbulence at the hydrodynamic level. In this model, turbulent energy plays a role analogous to temperature in molecular thermodynamics, and the large-scale structures are characterised by a turbulent relaxation time. This model aligns with the framework of Pope’s generalised Langevin model, with the first moment recovering the Reynolds-averaged Navier–Stokes (RANS) equations, and the second moment yielding a partially modelled Reynolds stress transport equation. Utilising the Chapman–Enskog expansion, we derive asymptotic solutions for this turbulent Fokker–Planck equation. With an appropriate choice of relaxation time, we obtain a linear eddy viscosity model at first order, and a quadratic Reynolds stress constitutive relationship at second order. Comparative analysis of the coefficients of the quadratic expression with typical nonlinear viscosity models reveals qualitative consistency. To further validate this kinetic-based nonlinear viscosity model, we integrate it as a RANS model within computational fluid dynamics codes, and calculate three typical cases. The results demonstrate that this quadratic eddy viscosity model outperforms the linear model and shows comparability to a cubic model for two-dimensional flows, without the introduction of ad hoc parameters in the Reynolds stress constitutive relationship.

JFM classification

Rarefied Gas Flow: Kinetic theory Turbulent Flows: Turbulence modelling

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JFM Papers
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Journal of Fluid Mechanics , Volume 1011 , 25 May 2025 , A44
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© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

†

These two authors contributed equally to this work.

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