Vortices and Turbulence in Incompressible Fluids: An Introductory Review

Since Reynolds’ work, turbulence has been one of the most important subjects in fluid dynamics. Although its complete understanding seems still out of reach, there is at least one established physical basis that turbulence is a phenomenon of a random but non-trivially correlated assembly of vortices. The knowledge of vortices has thus become a prerequisite for promoting our understanding of the nature of turbulence. In this article, we first review the simple, compact vortex solutions to the Navier–Stokes equations for incompressible viscous fluids and a unified view of a certain type of vortices including Burgers, Sullivan and Bellamy-Knights solutions. The non-equivalence of the inviscid limit of the Navier–Stokes equations and the Euler equations is emphasized. Introducing the notion of observational non-uniqueness, which differs from the non-uniqueness in a certain class of differential equations, of solutions to the Navier–Stokes equations, the observation problem associated with the dense distribution of non-equivalent solutions is argued. The origin of the extreme sensitivity of the solutions to the boundary conditions is clarified. A few examples of vortex phenomena in the real world are also surveyed. We next review the works of constructing turbulence as a random assembly of simple, compact vortices. An attempt to combine the vortex model of turbulence with the Kármán–Howarth equation for the velocity correlation functions of anisotropic turbulence is presented. It is pointed out that the studies in this direction suggested that Kolmogorov’s 2/3 scaling law was generally compatible with anisotropy. A few quantities are proposed as candidates to measure anisotropy in turbulence experiments.