Thermodynamically consistent coefficient calibration in nonlinear and anisotropic closure models for turbulence

This work aims to investigate the results of different coefficient calibration strategies in various nonlinear and anisotropic turbulence closure relations and to evaluate the corresponding findings with regard to thermodynamic considerations at the first order level of closure. In particular, an interesting class of recently developed explicit algebraic stress models is subject to a critical assessment based on the work by Sadiki [1] and Sadiki et al., [2], in which the consistency conditions have been derived from the second law of thermodynamics in the form of restrictions upon model-coefficients in closure assumptions of turbulent equations. Such restrictions require turbulence models to provide turbulence quantities which always recover the fundamental physics of turbulence. 1) It is found that many nonlinear and anisotropic turbulent models are thermodynamically inadmissible. 2) It is demonstrated that the so-called "realizability-constraints'' of Schumann [3], du Vachat [4], Lumley [5], Pope [6], Shih [7] or Wang [8] are contained in the second law of thermodynamics as one among other mathematical conditions derived in exploiting the entropy inequality. Therefore they cannot guarantee the thermodynamic realizability of turbulent flows, and thus put unequivocally restrictions upon closure assumptions. 3) It is furthermore shown through the behaviour of model-coefficients and their validity domains that thermodynamically consistent models predict the stability behaviour of the flow well and remain in good agreement with the results of the linear hydrodynamic stability analysis. While the stability has to be adjusted a posteriori in the existing models derived by other methods in the literature, from a thermodynamic point of view one may, a priori, optimize the turbulence models by calibrating the coefficients, so that the thermodynamic consistency conditions are automatically satisfied. At this first order level of closure, the scalars, turbulent kinetic energy and its dissipation rate, emerge as natural basic variables for the turbulence modeling in contrast to the considerations in Wang [8] and others.