On the universality of turbulent axisymmetric wakes

Abstract

AbstractDirect numerical simulations (DNS) of two time-dependent, axially homogeneous, axisymmetric turbulent wakes having very different initial conditions are presented in order to assess whether they reach a universal self-similar state as classically hypothesized by Townsend. It is shown that an extensive early-time period exists during which the two wakes are individually self-similar with wake widths growing like$\delta \propto {t}^{1/ 3} $, as predicted by classical dimensional analysis, but have very different growth rates and are thus not universal. Subsequently, however, the turbulence adjusts to yield, eventually, wakes that are structurally identical and have the same growth rate (also with$\delta \propto {t}^{1/ 3} $) so provide clear evidence of a universal, self-similar state. The former non-universal but self-similar state extends, in terms of a spatially equivalent flow behind a spherical body of diameter$d$, to a distance of$O(3000d)$whereas the final universal state does not appear before$O(5000d)$(and exists despite relatively low values of the Reynolds number and no evidence of a spectral${\kappa }^{\ensuremath{-} 5/ 3} $inertial subrange). Universal wake evolution is therefore likely to be rare in practice. Despite its low Reynolds number, the flow does not exhibit the sometime-suggested alternative self-similar behaviour with$\delta \propto {t}^{1/ 2} $(as for the genuinely laminar case) at large times (or, equivalently, distances), since the eddy viscosity remains large compared to the molecular viscosity and its temporal variations are not negligible.