Self-similarity in turbulence and its applications.

Abstract

First, we discuss the non-Gaussian type of self-similar solutions to the Navier-Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne et al. (1996 Commun. Partial. Differ. Equ. 21, 179-193). In order to shed some light on it, we study self-similar solutions to the one-dimensional Burgers equation in detail, completing the most general form of similarity profiles that it can possibly possess. In particular, on top of the well-known source-type solution, we identify a kink-type solution. It is represented by one of the confluent hypergeometric functions, viz. Kummer's function [Formula: see text]. For the two-dimensional Navier-Stokes equations, on top of the celebrated Burgers vortex, we derive yet another solution to the associated Fokker-Planck equation. This can be regarded as a 'conjugate' to the Burgers vortex, just like the kink-type solution above. Some asymptotic properties of this kind of solution have been worked out. Implications for the three-dimensional (3D) Navier-Stokes equations are suggested. Second, we address an application of self-similar solutions to explore more general kind of solutions. In particular, based on the source-type self-similar solution to the 3D Navier-Stokes equations, we consider what we could tell about more general solutions. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.