Self-similar source-type solutions to the three-dimensional Navier-Stokes equations

Abstract

We formalise a systematic method of constructing forward self-similar solutions to the Navier-Stokes equations in order to characterise the late stage of decaying process of turbulent flows. (i) In view of critical scale-invariance of type 2 we exploit the vorticity curl as the dependent variable to derive and analyse the dynamically-scaled Navier-Stokes equations. This formalism offers the viewpoint from which the problem takes the simplest possible form. (ii) Rewriting the scaled Navier-Stokes equations by Duhamel principle as integral equations, we regard the nonlinear term as a perturbation using the Fokker-Planck evolution semigroup. Systematic successive approximations are introduced and the leading-order solution is worked out explicitly as the Gaussian function with a solenoidal projection. (iii) By iterations the second-order approximation is estimated explicitly up to solenoidal projection and is evaluated numerically. (iv) A new characterisation of nonlinear term is introduced on this basis to estimate its strength $N$ quantitatively. We find that $N=O(10^{-2})$ for the 3D Navier-Stokes equations. This should be contrasted with $N=O(10^{-1})$ for the Burgers equations and $N \equiv 0$ for the 2D Navier-Stokes equations. (v) As an illustration we explicitly determine source-type solutions to the multi-dimensional the Burgers equations. Implications and applications of the current results are given.