Turbulence intermittency and velocity gradients

A central problem in turbulence is understanding small-scale intermittency, which refers to the sporadic generation of intense fluctuations in velocity gradients and increments. These extreme events, strongly non-Gaussian in nature, govern dissipation, mixing and transport processes in virtually all turbulent flows. Yet, despite decades of study, a faithful and predictive characterisation of small scales remains elusive owing to the inherent mathematical intractability of the Navier–Stokes equations and the difficulty in resolving them in both simulations and experiments at high Reynolds numbers. Recent advances in high-resolution simulations and experiments have significantly reshaped this picture, particularly by providing precise data at high Reynolds numbers to probe the full tensorial structure and dynamics at small scales. In this article, we synthesise the current understanding of small-scale intermittency and universality, drawing on modern data from well-resolved simulations and experiments that resolve the full velocity-gradient tensor. The results show that, while prevailing intermittency theories capture several key trends, they fail to describe or account for observed asymmetries between longitudinal and transverse fluctuations or between strain and vorticity amplification. Evidence suggests that intermittency is closely tied to the dynamics and geometry of vorticity and strain fields, with non-locality playing an important role. We argue that a consistent picture has emerged, but a complete theory will require unifying the statistical scaling frameworks with the underlying dynamical mechanisms that govern gradient amplification. Additional implications of these findings are discussed, and several pressing open problems are identified for future work.