G. van Baalen
Downstream asymptotics in exterior domains: from stationary wakes to time periodic flows
(520K, pdf)
ABSTRACT. In this paper, we consider the time-dependent Navier-Stokes equations
in the half-space $[x_0,\infty)\times{\bf R}\subset{\bf R}^2$, with
boundary data on the line $x=x_0$ assumed to be time-periodic (or
stationary) with a fixed asymptotic velocity ${\bf u}_{\infty}=(1,0)$
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 fuctions. Furthermore, we prove that as $x\to\infty$, the
vorticity decompose itself in a dominant stationary part on the parabolic scale $y\sim\sqrt{x}$ and corrections of order $x^{-\frac{3}{2}+\epsilon}$, while the velocity field decompose
itself in a dominant stationary part in form of an explicit
multiscale expansion on the scales $y\sim\sqrt{x}$ and
$y\sim x$ and corrections decaying at least like
$x^{-\frac{9}{8}+\epsilon}$. 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 ${\bf u}_{\infty}$ 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 $x\to\infty$.
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 ($x=x_0$) 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,
{\em 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
setting.