Transport equation for the isotropic turbulent energy dissipation rate in the far-wake of a circular cylinder

Abstract

The transport equation for the isotropic turbulent energy dissipation rate $\overline{{\it\epsilon}}_{iso}$ along the centreline in the far-wake of a circular cylinder is derived by applying the limit at small separations to the two-point energy budget equation. It is found that the imbalance between the production and the destruction of $\overline{{\it\epsilon}}_{iso}$, respectively due to vortex stretching and viscosity, is governed by both the streamwise advection and the lateral turbulent diffusion (the former contributes more to the budget than the latter). This imbalance differs intrinsically from that in other flows, e.g. grid turbulence and the flow along the centreline of a fully developed channel, where either the streamwise advection or the lateral turbulent diffusion of $\overline{{\it\epsilon}}_{iso}$ governs the imbalance. More importantly, the different types of imbalance represent different constraints on the relation between the skewness of the longitudinal velocity derivative $S$ and the destruction coefficient of enstrophy $G$. This results in a non-universal approach of $S$ towards a constant value as the Taylor microscale Reynolds number $R_{{\it\lambda}}$ increases. For the present flow, the magnitude of $S$ decreases initially ($R_{{\it\lambda}}\leqslant 40$) before increasing ($R_{{\it\lambda}}>40$) towards this constant value. The constancy of $S$ at large $R_{{\it\lambda}}$ violates the modified similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) but is consistent with the original similarity hypotheses (Kolmogorov, Dokl. Akad. Nauk SSSR, vol. 30, 1941b, pp. 299–303 (see also 1991 Proc. R. Soc. Lond. A, vol. 434, pp. 9–13)) ($K41$), and, more importantly, with the almost completely self-preserving nature of the plane far-wake.