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

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