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An Entropy Stable Formulation of Two-equation Turbulence Models with Particular Reference to the k-epsilon Model

This paper presents a robust numerical framework for two-equation turbulence models, particularly the k-epsilon model, by incorporating entropy production principles and space-time averaging to derive symmetric, entropy-stable equations that ensure consistency and positivity without relying on artificial viscosity.

Original authors: Guillermo Hauke, Thomas J. R. Hughes

Published 2026-08-12

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CC BY 4.0

Original authors: Guillermo Hauke, Thomas J. R. Hughes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a river flows around a rock, or how air rushes over a speeding car. In the world of physics, this is the study of fluid dynamics. But fluids are tricky; they don't just flow smoothly like water in a calm stream. They swirl, spin, and crash into themselves in chaotic patterns called turbulence. To understand these wild motions, scientists use a set of rules called the Navier-Stokes equations. Think of these equations as the ultimate instruction manual for how fluids move. However, when fluids get turbulent, the manual becomes so incredibly complex that even the fastest supercomputers in the world can't solve it perfectly for real-world designs.

To get around this, engineers use a shortcut. Instead of tracking every single tiny swirl, they take an "average" of the flow, smoothing out the chaos into a manageable picture. This is called Reynolds-averaging. But here's the catch: when you smooth out the chaos, you lose some of the fundamental rules that keep the universe stable. One of the most important of these rules is the Second Law of Thermodynamics, which basically says that disorder (or "entropy") in a closed system always tends to increase. It's like how a messy room never cleans itself up on its own; energy always dissipates into heat. If a computer simulation ignores this rule, it can become unstable, producing wild, impossible results that crash the program. The big question scientists have been asking is: Can we build a turbulence model that respects this "messiness rule" so our computer simulations stay stable and physically real?

This paper, written by Guillermo Hauke and Thomas J. R. Hughes, tackles that exact problem. The authors propose a new way to write the equations for two-equation turbulence models (specifically the famous $k$ - $\epsilon$ model) so that they naturally obey the Second Law of Thermodynamics. They do this by introducing a clever mathematical trick called "space-time averaging." Imagine looking at a turbulent flow not just through a time-lapse camera, but also through a slightly blurry lens that averages out tiny spatial details. By doing this, they discover a hidden connection between the energy of the swirling eddies and the rate at which that energy disappears (dissipates).

The main finding of the paper is that if you change the variables in your equations to represent these averaged quantities, the entire system becomes "symmetric." In the world of math, symmetry is a superpower; it means the equations are much more stable and less likely to blow up during a computer simulation. The authors show that for this to work, the way turbulence spreads heat and momentum (the diffusivity coefficients) must follow specific constraints. They tested this new, "entropy-stable" version of the $k$ - $\epsilon$ model on a computer by simulating air flowing over a flat plate. The results were promising: the model produced stable, realistic results that matched experimental data well, and it did so without needing the artificial "fixes" (like adding fake viscosity) that older models often require. The authors suggest that this approach provides a more physically consistent way to design turbulence models, ensuring that the computer simulations respect the fundamental laws of nature.

Technical Summary: An Entropy Stable Formulation of Two-equation Turbulence Models with Particular Reference to the k-epsilon Model

Problem Statement
The design of robust numerical algorithms for fluid dynamics relies heavily on consistency and stability. While the second law of thermodynamics (entropy production) provides a fundamental physical stability principle for the Navier-Stokes equations, its application to Reynolds-Averaged Navier-Stokes (RANS) equations is non-trivial. In standard RANS formulations, particularly those using two-equation turbulence models like $k$ - $\epsilon$ , the direct link between the averaged entropy and the averaged dependent variables is lost due to the nonlinearity of the Gibbs relation. Consequently, standard formulations often lack a generalized entropy function, making it difficult to embed physical stability principles directly into the numerical scheme without resorting to artificial viscosity. Previous attempts to restore entropy stability, such as those by Jansen et al., were limited to one-equation models or systems involving six equations for the Reynolds stress tensor, leaving the widely used $k$ - $\epsilon$ model without a rigorous entropy-stable formulation.

Methodology
The authors propose a framework to derive an entropy-stable formulation for two-equation turbulence models, specifically the $k$ - $\epsilon$ model, by introducing space-time averaging.

Space-Time Averaging and Variable Transformation:
The paper introduces a space-time averaging operator to bridge the gap between the dissipation rate ( $\epsilon$ ) and the generalized entropy. By performing a spatial average of the turbulent kinetic energy ( $k$ ) over a small volume, the authors demonstrate that $\epsilon$ accounts for the spatial fluctuations of $k$ . This leads to the definition of a new velocity scale, $q_1 = (\nu \epsilon)^{1/4}$ , in addition to the standard turbulent velocity scale $q_0 = \sqrt{2k}$ .
The system of conservation variables is redefined to include density, momentum, total energy, $q_0$ , and $q_1$ . The total energy is modified to include contributions from both $q_0$ and $q_1$ .
Generalized Entropy Function and Symmetrization:
Using these new variables, the authors construct a generalized entropy function $H(U) = -\rho \hat{s}$ , where $\hat{s}$ is a modified entropy based on the space-time averaged variables. The corresponding entropy variables $V$ are derived as the gradient of $H$ with respect to the conservation variables $U$ .
This change of variables transforms the quasi-linear form of the $k$ - $\epsilon$ equations into a symmetric hyperbolic system (specifically, a symmetric advective-diffusive system). The coefficient matrices for the time derivative and advection terms become symmetric, and the diffusion matrix is shown to be symmetric positive-semidefinite under specific constraints.
Constraints for Entropy Production:
To ensure the system satisfies the Clausius-Duhem inequality (non-negative entropy production), the authors derive specific constraints on the turbulence model coefficients:
- The turbulent diffusivity for the dissipation rate ( $\epsilon$ ) must equal the turbulent diffusivity for the turbulent kinetic energy ( $k$ ). In terms of Prandtl numbers, this requires $Pr_\epsilon = Pr_k$ .
- The source terms in the $q_1$ equation must satisfy specific inequalities to ensure that the production of generalized entropy remains non-negative. This involves modifying the dissipation term in the source function to dominate production terms under certain conditions.
Numerical Implementation:
The formulated equations are implemented using a time-discontinuous Galerkin/least-squares finite element method. The system is solved using a predictor-multicorrector algorithm with a GMRES linear solver. The implementation addresses the stiffness of the $k$ - $\epsilon$ equations, particularly near walls, by utilizing a specific boundary condition for $q_1$ derived from the balance of dissipation and cross-diffusion.

Key Results
The paper presents numerical simulations of a nearly incompressible turbulent flat plate boundary layer using the Lam-Bremhorst $k$ - $\epsilon$ model adapted to this entropy-stable framework.

Validation: The results for the skin friction coefficient and velocity profiles (in wall coordinates) show good agreement with empirical correlations (Karman-Schönherr) and other established computations (Patel et al., Wilcox).
Stability: The numerical scheme demonstrates robustness. Notably, the $q_1$ equation (the transformed dissipation equation) exhibits excellent stability, contrasting with the typical numerical difficulties associated with the standard $\epsilon$ equation.
Grid Independence: A refinement study indicates that the solution is relatively insensitive to the near-wall element size ( $y^+$ ), even for values as high as $y^+ = 5.0$ . This is attributed to the specific boundary condition implementation for $q_1$ , which allows for coarser near-wall meshes compared to standard low-Reynolds number $k$ - $\epsilon$ models that typically require $y^+ < 1$ .
Entropy Production: The simulations confirm that the total entropy production (viscous dissipation plus turbulence contributions) remains non-negative across the boundary layer, validating the theoretical derivation.

Significance and Claims
The authors claim that this work successfully extends the concept of entropy stability to the widely used two-equation $k$ - $\epsilon$ turbulence model. By introducing space-time averaging and the associated variables $q_0$ and $q_1$ , they recover a generalized entropy function that allows the system to be symmetrized.

The primary significance lies in the ability to design numerical algorithms that inherit the physical stability principle of the second law of thermodynamics directly into the discrete formulation. This provides a rigorous alternative to adding artificial viscosity for stability. The paper asserts that the derived constraints (specifically $Pr_\epsilon = Pr_k$ ) are necessary for both entropy production and the positivity of the solution, aligning with previous findings by Mohammadi and Pironneau. The authors conclude that this framework offers a path toward designing more physically coherent turbulence models and stable numerical schemes, as demonstrated by the successful application to the Lam-Bremhorst model.

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