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

Abstract

We formalize a systematic method of constructing forward self-similar solutions to the Navier–Stokes equations in order to characterize 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 characterization of nonlinear term is introduced on this basis to estimate its strengthNquantitatively. We find thatN=O(10−2)for the three-dimensional Navier–Stokes equations. This should be contrasted withN=O(10−1)for the Burgers equations andN≡0for the two-dimensional Navier–Stokes equations. (v) As an illustration we explicitly determine source-type solutions to the multi-dimensional Burgers equations. Implications and applications of the current results are given.