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

Consistency and stability are two essential ingredients in the design of numerical algorithms for partial differential equations. Robust algorithms can be developed by incorporating nonlinear physical stability principles in their design, such as the entropy production inequality (i.e., the Clausius-Duhem inequality or second law of thermodynamics), rather than by simply adding artificial viscosity (a common approach). This idea is applied to the k-epsilon and two-equation turbulence models by introducing space-time averaging. Then, a set of entropy variables can be defined which leads to a symmetric system of advective-diffusive equations. Positivity and symmetry of the equations require certain constraints on the turbulence diffusivity coefficients and the turbulence source terms. With these, we are able to design entropy producing two-equation turbulence models and, in particular, the k-epsilon model.