Self-similarity in turbulence and its applications

TL;DR

This paper explores non-Gaussian self-similar solutions to Navier-Stokes and Burgers equations, identifying new solution types and discussing their implications for understanding turbulence and more general flow solutions.

Contribution

It introduces new self-similar solutions, including kink-type and conjugate vortex solutions, and analyzes their properties and relevance to turbulence modeling.

Findings

Identified a kink-type solution to the 1D Burgers equation.

Derived a conjugate vortex solution for 2D Navier-Stokes.

Discussed implications for 3D Navier-Stokes solutions.

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-Planchon (1996). In order to shed some light on it, we study self-similar solutions to the 1D 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 2D 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 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.