On the minimal representation of non-linear eddy-viscosity models

This study attempts to provide a minimal explicit non-linear relationship between the Reynolds stresses and the mean strain rate and vorticity. Such relationship is also referred to as the nonlinear eddy-viscosity model (EVM) of the Reynolds stresses. Based on the observation of the independent terms in the Reynolds stress anisotropy tensor, it is proposed here that the explicit Reynolds stress closure can be formulated in a compact vorticity tensor to forma minimal representation. With the application of the generalized Cayley-Hamilton theory, it is shown that the explicit algebraic stress model with 10 integrity bases can be transformed into a five-term model expression. Thus, the present work provides a framework for the development of the explicit Reynolds stress closure in the compact form. Indeed, a cubic EVM is developed here that is capable of revealing the quadratic behaviour of the tangential velocity with respect to the radial distance in the fully developed turbulent rotating pipe flow. This model, hence, removed some major defects in the lower-order EVMs, while the model complexity is kept at minimum.