A stable entropy producing formulation of two-equation turbulence models with particular reference to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo linebreak="goodbreak">−</mml:mo

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). In the context of two-equation turbulence models we introduce space-time averaged variables, the essential concept which enables identification of an appropriate set of conservation variables. This change of variables, compared with the usual formulations of the model, is also key to the ensuing developments. From these, the correct concept of entropy and a set of entropy variables can be defined which leads to a symmetric system of advective-diffusive equations. The equivalence of quasilinear symmetric, advective-diffusive systems with coupled systems having a nonlinear convex entropy function, established by Mock [25] and Godunov [26], provides the theoretical underpinning of our developments. Positivity and symmetry of the equations require certain constraints on the turbulence diffusivity coefficients and the turbulence source terms that we delineate. With these, we are able to design entropy producing formulations of two-equation turbulence models and, in particular, the model, and numerical formulations that inherit these properties. The accuracy of the original model is maintained and we automatically gain computational stability and robustness due to the guaranteed entropy production property. The results suggest possible formulations of other turbulence models that will enhance their behavior in numerical simulation.