Towards local isotropy of higher-order statistics in the intermediate wake

In this paper, we assess the local isotropy of higher-order statistics in the intermediate wake region. We focus on normalized odd moments of the transverse velocity derivatives, M2n+1(∂u/∂z)=(∂u/∂z)2n+1¯/(∂u/∂z)2¯(2n+1)/2 and N2n+1(∂u/∂y)=(∂u/∂y)2n+1¯/(∂u/∂y)2¯(2n+1)/2, which should be zero if local isotropy is satisfied (n is a positive integer). It is found that the relation M2n+1(∂u/∂z)∼Rλ-1 is supported reasonably well by hot-wire data up to the seventh order (n = 3) on the wake centreline, although it is also dependent on the initial conditions. The present relation N3(∂u/∂y)∼Rλ-1 is obtained more rigorously than that proposed by Lumley (Phys Fluids 10:855–858, 1967) via dimensional arguments. The effect of the mean shear at locations away from the wake centreline on M_{2 n + 1}(∂u/∂z) and N_{2 n + 1}(∂u/∂y) is addressed and reveals that, although the non-dimensional shear parameter is much smaller in wakes than in a homogeneous shear flow, it has a significant effect on the evolution of N_{2 n + 1}(∂u/∂y) in the direction of the mean shear; its effect on M_{2 n + 1}(∂u/∂z) (in the non-shear direction) is negligible.