The present review, which includes some new material, consists of two parts: physical constraints in turbulence modeling and physics-preserving turbulent closure models that preserve the frame indifference and satisfy both the principle of material frame indifference (PMFI) and the second law of thermodynamics.

The former commences with careful definition of the average operation, the Reynolds stress, the turbulent heat flux and the turbulent mass flux. The remainder of this part is on the developments, to date, of three physical constraints imposed by the invariance, the realizability and the PMFI. In particular, two sufficient conditions are discussed for the Reynolds stress tensor and the turbulent heat/mass flux vector to be frame indifferent. The application of the second law of thermodynamics to a thermally isolated system and an irreversible process concludes that realizability inequalities in turbulent flows follow logically from the second law of thermodynamics. How system rotations affect flow fields is critically examined from a basic theoretical standpoint. Also critically reviewed is the literature on the range of validity of the PMFI to the turbulence modeling.

The latter is devoted to the progress, to date, of physics-preserving closure models of the Reynolds stress and the turbulent heat/mass flux. In particular, both necessary and sufficient conditions are developed in a systematic, rigorous way for turbulent closure models to satisfy the three constraints reviewed in the first pan. The results have either confirmed some intuitive arguments or offered new insights into turbulence modeling, and are of significance in clarifying some controversies in the literature, examining how well existing models preserve the physics, and developing new models. Also developed are a linear theory, a quadratic theory and a flow decomposition theorem to simplify and guide the work of developing specific physics-preserving models. This part ends with a further constraint on the physics-preserving models by the disappearance of Reynolds stress and turbulent heat/mass flux at a vanishing value of the mean velocity.

Most methods and results in the present review are also valid for the higher-order correlations and the subgrid-scale (SGS) modeling in the large eddy simulation (LES).