Both necessary and sufficient conditions are derived in a systematic, rigorous way for a Reynolds stress model to preserve both frame indifference and positive semi-definiteness of the Reynolds stresses and to satisfy the principle of material frame indifference (PMFI). Also developed are two approximate theories, a linear theory and a quadratic theory, to simplify the work of developing models satisfying such conditions. The results are also valid for the higher-order correlations and the SGS Reynolds stresses. This leads to the results either confirming the previous intuitive arguments or offering new insights into turbulence modelling, and is of significance in clarifying some controversies in the literature, examining how well existing models preserve the physics, and developing new models.

An examination of various existing models with respect to the necessary and sufficient conditions obtained in the present work shows that no one model can simultaneously satisfy the invariance, the realizability and the PMFI. This offers answers to some fundamental questions regarding the existing models, such as the reason why k−ε and k−l models fail to give accurate predictions for the normal Reynolds stresses.