Self-similarity in turbulence and its applications

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 M. 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)’.