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, \({M_{2n + 1}}(\partial u/\partial z) = {{\overline{{{(\partial u/\partial z)}^{2n + 1}}} }}/{{{{\overline{{{(\partial u/\partial z)}^2}} }^{(2n + 1)/2}}}}\) and \({N_{2n + 1}}(\partial u/\partial y) = {{\overline{{{(\partial u/\partial y)}^{2n + 1}}} }}/{{{{\overline{{{(\partial u/\partial y)}^2}} }^{(2n + 1)/2}}}}\), which should be zero if local isotropy is satisfied (n is a positive integer). It is found that the relation \(M_{2n+1}(\partial u/\partial z) \sim R_\lambda ^{-1}\) 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 \(N_{3}(\partial u/\partial y) \sim R_\lambda ^{-1}\) 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 \(M_{2n+1}(\partial u/\partial z)\) and \(N_{2n+1}(\partial u/\partial y)\) 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 \(N_{2n+1}(\partial u/\partial y)\) in the direction of the mean shear; its effect on \(M_{2n+1}(\partial u/\partial z)\) (in the non-shear direction) is negligible.