This paper presents a systematic derivation of explicit algebraic stress models for three-dimensional turbulent flows in non-inertial frames, based on a hierarchy of second-order closure models. The models are derived using the standard local equilibrium hypothesis and represent the explicit solution to traditional algebraic stress models generalized to include non-inertial effects and a range of pressure-strain models. The explicit models are shown to be equivalent to the equilibrium states predicted by these second-order closures in homogeneous turbulent flows. The paper also discusses the limitations of traditional algebraic stress models, particularly their singularity at large localized strain rates, and proposes a regularization method using a Padé approximation. The regularized explicit algebraic stress model is tested against various two-dimensional and three-dimensional turbulent flow problems, demonstrating its superior performance compared to the standard \(K-\varepsilon\) model and traditional algebraic stress models. The results suggest that the regularized explicit models should replace the traditional models in practical applications, offering a more accurate and robust approach to simulating complex turbulent flows.This paper presents a systematic derivation of explicit algebraic stress models for three-dimensional turbulent flows in non-inertial frames, based on a hierarchy of second-order closure models. The models are derived using the standard local equilibrium hypothesis and represent the explicit solution to traditional algebraic stress models generalized to include non-inertial effects and a range of pressure-strain models. The explicit models are shown to be equivalent to the equilibrium states predicted by these second-order closures in homogeneous turbulent flows. The paper also discusses the limitations of traditional algebraic stress models, particularly their singularity at large localized strain rates, and proposes a regularization method using a Padé approximation. The regularized explicit algebraic stress model is tested against various two-dimensional and three-dimensional turbulent flow problems, demonstrating its superior performance compared to the standard \(K-\varepsilon\) model and traditional algebraic stress models. The results suggest that the regularized explicit models should replace the traditional models in practical applications, offering a more accurate and robust approach to simulating complex turbulent flows.