Turbulent Transition Criteria: From Reynolds Experiment to Modern Numerical Methods

ABSTRACT: Turbulent transition refers to the complex flow phenomenon of laminar flow evolving into turbulence, and the criteria for judging transition constitute a core research topic in fluid mechanics, computational fluid dynamics, flow experiments, and engineering applications. This study systematically reviews the evolutionary framework of transition criteria over the past century, ranging from the empirical Reynolds number rule, linear stability theory, and the engineering eⁿ method, to semi-empirical correlations, the intermittency factor γ model, energy gradient theory, and state-of-the-art numerical criteria based on direct numerical simulation (DNS) and large eddy simulation (LES). Specialized stability criteria for non-parallel flows such as Taylor–Couette flow and Dean flow are sorted out, and the evolutionary paths, theoretical foundations, and applicable working conditions of low-turbulence natural transition and high-disturbance bypass transition are clarified. The progress of existing research is summarized, with critical bottlenecks identified, including high-Reynolds-number transition, three-dimensional coupled complex flows, and poor generalization of data-driven models. The applicable scopes and inherent defects of diverse transition theories are compared: energy gradient theory explains the instability mechanism of finite-amplitude perturbations; the intermittency factor model dominates industrial numerical simulations; and high-fidelity numerical simulation serves as the core tool for uncovering micro transition mechanisms. Four major future research directions are proposed: unified multi-scale theoretical frameworks, efficient computational fluid dynamics (CFD) algorithms, data-driven prediction models, and engineering applications for novel fluid machinery, delivering theoretical support for transition prediction and flow control.