Downstream Asymptotics in Exterior Domains: from Stationary Wakes to Time Periodic Flows

Abstract.: In this paper, we consider the time-dependent Navier-Stokes equations in the half-space [x0,∞) × R ⊂ R2, with boundary data on the line x=x0 assumed to be time-periodic (or stationary) with a fixed asymptotic velocity u∞ = (1, 0) at infinity. We show that there exist (locally) unique solutions for all data satisfying a center-stable manifold compatibility condition in a certain class of functions. Furthermore, we prove that as x → ∞, the vorticity decomposes itself in a dominant stationary part on the parabolic scale $$ y \sim \sqrt{x}$$ and corrections of order $$x^{-\frac{{3}}{{2}} + \varepsilon },$$ while the velocity field decomposes itself in a dominant stationary part in form of an explicit multi-scale expansion on the scales $$y \sim \sqrt{x}$$ and y ∼ x and corrections decaying at least like $$ x^{-\frac{{9}}{{8}} + \varepsilon}.$$ The asymptotic fields are made of linear combinations of universal functions with coefficients depending mildly on the boundary data. The asymptotic expansion for the component parallel to u∞ contains ‘non-trivial' terms in the parabolic scale with amplitude ln(x)x−1 and x−1. To first order, our results also imply that time-periodic wakes behave like stationary ones as x → ∞. The class of functions used to prove these results is ‘natural' in the sense that the well known ‘Physically Reasonable' (in the sense of Finn & Smith) stationary solutions of the Navier-Stokes equations around an obstacle fall into that class if the half-space extends in the downstream direction and the boundary (x = x0) is sufficiently far downstream. In that case, the coefficients appearing in the asymptotics can be linearly related to the net force acting on the obstacle. In particular, the asymptotic description holds for ‘Physically Reasonable' stationary solutions in exterior domains, without restrictions on the size of the drag acting on the obstacle. To our knowledge, it is the first time that estimates uncovering the ln(x)x−1 correction are proved in this settin