Researchers have proposed a novel double time-relaxation kinetic model, designated DtrKM, to address the persistent challenges of modeling compressible turbulence on unresolved computational grids. Published in the journal Advances in Aerodynamics, the study introduces a framework that extends the generalized Bhatnagar-Gross-Krook model to provide a mesoscopic understanding of subgrid-scale turbulence. The work aims to bridge the gap between kinetic theory and macroscopic governing equations, offering a robust alternative to traditional eddy-viscosity models used in large eddy simulations.
The central motivation for this research stems from the limitations of existing large eddy simulation techniques. While large eddy simulation has become a primary tool for high-fidelity turbulence simulation, it requires subgrid-scale models to account for unresolved turbulent structures. Widely used zero-equation models, such as the Smagorinsky model, often suffer from over-dissipative performance near walls and fail to vanish in laminar flow regions. To address these flaws, dynamic Smagorinsky models were developed, allowing coefficients to be computed locally, yet the complexity of compressible flows remains a significant hurdle for accurate modeling.
The proposed DtrKM operates within a double time-relaxation framework, introducing a non-trivial quantity known as unresolved turbulence kinetic energy, or K_utke. This quantity is modeled using a Gamma distribution, which serves as the stationary distribution for turbulence frequency in statistically stationary isotropic turbulence. By incorporating this additional variable, the model captures the dynamic evolution of unresolved turbulent structures more accurately than previous approaches that relied solely on conservative laws for mass, momentum, and total energy.
Through a first-order Chapman-Enskog expansion, the researchers connected the kinetic model to six macroscopic governing equations. The first five equations correspond to the standard conservative laws, while the sixth equation specifically governs the evolution of the unresolved turbulence kinetic energy. This mathematical linkage allows the model to inherit the advantages of well-established gas-kinetic schemes, which are known for their ability to handle discontinuities and multi-scale structures in compressible flows effectively.
To solve the model accurately and robustly, the team developed a high-accuracy gas-kinetic scheme. This numerical method utilizes fifth-order WENO-Z spatial reconstruction and a two-stage fourth-order time discretization. The scheme is designed to maintain the precise and robust performance of high-order gas-kinetic schemes, ensuring that both inviscid and viscous fluxes are recovered from a time-dependent, multi-dimensional gas distribution function at cell interfaces.
The effectiveness of the DtrKM was evaluated through a posteriori numerical tests on two canonical benchmarks: decaying compressible isotropic turbulence and a temporal compressible plane mixing layer. For the decaying turbulence case, the simulation was conducted on unresolved grids with an initial Taylor microscale Reynolds number of 72 and an initial turbulent Mach number of 0.6. The results demonstrated that the dynamic coefficient C_s fluctuated between 0.06 and 0.12, indicating a reasonable dependence on grid resolution and flow evolution.
In the decaying compressible isotropic turbulence simulation, the normalized ensemble unresolved turbulence kinetic energy initially increased before decreasing consecutively. This behavior suggests that the forward resolved energy cascade dominates at the early stage, followed by the dominance of subgrid-scale dissipation terms. The findings confirm that the intrinsic equilibrium assumption used in zero-equation eddy-viscosity models is not valid for this flow regime, highlighting the importance of modeling the dynamic evolution of unresolved turbulence kinetic energy.
The temporal compressible plane mixing layer simulation further assessed the model’s performance in a flow without strong discontinuities. The direct numerical simulation was validated against refereed numerical simulations, showing good agreement in key quantities such as the normalized growth rate and vorticity thickness-based Reynolds number. The large eddy simulation using DtrKM was restarted from the filtered direct numerical simulation solution, allowing for a direct comparison with the dynamic Smagorinsky model on the same unresolved grids.
Results from the mixing layer simulation indicated that the performance of key turbulent quantities, up to second-order statistics, was comparable to the widely used dynamic Smagorinsky model. The normalized growth rate from DtrKM was 0.611, which agreed well with the dynamic Smagorinsky model and was slightly larger than that of the direct numerical simulation. Profiles of normalized turbulent stress showed quite small deviations between the two large eddy simulation models, reinforcing the reliability of the new kinetic approach.
The study highlights the potential of the double time-relaxation framework for simulating multi-scale flow physics. By providing a mesoscopic understanding of the transport equation for compressible subgrid-scale turbulence kinetic energy, the model offers an alternative pathway for compressible turbulence modeling. The researchers noted that the turbulent relaxation time can be significantly larger than the molecular relaxation time on unresolved grids, a feature that is naturally captured by the kinetic formulation.
Limitations of the current work include the use of a Gamma distribution for the unresolved turbulence kinetic energy, which may not fully capture intense events in compressible isotropic turbulence. The authors acknowledge that the optimal physical distribution for this quantity requires further investigation. Additionally, the formal first-order Chapman-Enskog expansion may not be sufficiently solid for all flow regimes, suggesting that future work could benefit from multi-scale kinetic solvers to improve performance and overcome expansion limitations.
Future research directions include implementing the DtrKM for more practical highly compressible turbulent flows, such as compressible wall-bounded turbulent flows and shock-boundary layer interactions. The team also plans to conduct detailed comparisons on accuracy, efficiency, and robustness between the current model and classical subgrid-scale models. These efforts aim to refine the modeling of the source term and the turbulent relaxation time, potentially leading to a complete time-relaxation compressible turbulence model without ad-hoc techniques from macroscopic models.
The development of the DtrKM represents a significant step forward in the field of computational fluid dynamics. By integrating kinetic theory with large eddy simulation, the model addresses key challenges in compressible turbulence modeling. The study demonstrates that a kinetic approach can provide a workable and accurate method for simulating complex turbulent flows, offering valuable insights for both theoretical understanding and practical engineering applications.
The study, authored by Guiyu Cao, Liang Pan, Kun Xu, and Shiyi Chen, was published in the journal Advances in Aerodynamics in December 2025. The research team developed the double time-relaxation kinetic model to address specific gaps in current compressible large eddy simulation techniques. Existing zero-equation models, such as the standard Smagorinsky model, rely on gradient-type assumptions that often result in excessive dissipation near walls and fail to eliminate subgrid-scale effects in laminar regions. While dynamic Smagorinsky models improve upon this by computing coefficients locally, the authors sought a framework that could inherently preserve non-local and historical effects of turbulence through a kinetic perspective. The proposed model extends the Bhatnagar-Gross-Krook model by introducing a second relaxation process, allowing for a more nuanced representation of unresolved turbulent structures on coarse computational grids.
A critical aspect of the model’s construction involves the treatment of the unresolved turbulence kinetic energy, denoted as K_utke. The authors utilized a Gamma distribution to describe the sample-space variable corresponding to this energy. This choice was validated through coarse-graining analysis of direct numerical simulation data, where a Gamma distribution with a shape parameter of 0.39 provided a good fit to the coarsened data, although deviations were observed in the thick-tail region. These deviations indicate that intense events in compressible isotropic turbulence are difficult to capture with this specific distribution. The model assumes that the equilibrium state of the unresolved turbulence follows this Gamma distribution, which allows the kinetic model to self-consistently determine the unknowns through a relaxation process analogous to non-equilibrium kinetic models for multi-temperature flows.
The numerical implementation of the model relies on a finite volume high-accuracy gas-kinetic scheme designed to handle the complexities of compressible flows. This scheme employs fifth-order WENO-Z spatial reconstruction to ensure high-order accuracy and robustness, particularly in the presence of strong discontinuities. For flows without such discontinuities, such as the temporal compressible plane mixing layer, a linear WENO reconstruction based on conservative variables is utilized. The temporal discretization uses a two-stage fourth-order method, which achieves high accuracy within two steps, thereby improving computational efficiency. The scheme recovers both inviscid and viscous fluxes from a time-dependent, multi-dimensional gas distribution function at cell interfaces, a feature that distinguishes it from traditional hydrodynamic solvers that treat these terms separately using Riemann solvers and central differences.
In the decaying compressible isotropic turbulence simulations, the model was tested on unresolved grids with a resolution of 128 cubed, derived from a 512 cubed direct numerical simulation setup. The initial conditions included a Taylor microscale Reynolds number of 72 and a turbulent Mach number of 0.6. The dynamic coefficient C_s, which governs the turbulent relaxation time, was computed locally and fluctuated between 0.06 and 0.12 during the simulation. This variation demonstrates that the dynamic approach allows the model coefficient to adapt to the local flow field and grid resolution, rather than relying on a fixed empirical value. The evolution of the unresolved turbulence kinetic energy showed an initial increase followed by a decrease, reflecting the transition from energy cascade dominance to dissipation dominance.
For the temporal compressible plane mixing layer, the direct numerical simulation was conducted with a convective Mach number of 0.75 and a vorticity thickness-based Reynolds number of 640. The simulation domain was discretized uniformly, with periodic boundary conditions applied in the streamwise and spanwise directions. Non-reflective boundary conditions were used in the transverse direction to handle the outflow of unresolved turbulence kinetic energy. The direct numerical simulation results were validated against previous refereed simulations, showing good agreement in the normalized growth rate of the momentum thickness and the components of the anisotropy stress tensor. The large eddy simulation was restarted from the filtered direct numerical solution at a normalized time of 400, ensuring that the initial conditions for the unresolved variables were consistent with the high-fidelity data.
The comparison between the double time-relaxation kinetic model and the dynamic Smagorinsky model revealed comparable performance in key statistical quantities. In the mixing layer simulation, the normalized growth rate from the kinetic model was 0.611, which was slightly larger than the direct numerical simulation value but agreed well with the dynamic Smagorinsky model. The profiles of normalized turbulent stress components showed small deviations between the two large eddy simulation models, indicating that the kinetic approach provides a reliable alternative for compressible turbulence modeling. The study concludes that the double time-relaxation framework offers a workable approach for simulating compressible turbulence on unresolved grids, with potential applications in more complex flows such as shock-boundary layer interactions.
Subject of Research: Development and validation of a double time-relaxation kinetic model for compressible large eddy simulation
Article Title: Double time-relaxation kinetic model for compressible turbulence modeling
Article References: Cao, G., Pan, L., Xu, K., & Chen, S. (2025). Double time-relaxation kinetic model for compressible turbulence modeling. Advances in Aerodynamics, 7(1), Article 19. https://doi.org/10.1186/s42774-025-00230-7
Image Credits: AI Generated
DOI: 10.1186/s42774-025-00230-7
Keywords: compressible turbulence, large eddy simulation, kinetic model, gas-kinetic scheme, subgrid-scale modeling, computational fluid dynamics, Bhatnagar-Gross-Krook model, turbulence kinetic energy, Double, time-relaxation, kinetic, model
News Source: Audrey Campbell. (November 12, 2026). New Kinetic Model Advances Compressible Turbulence Simulation. Scienmag.
