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

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, $${M_{2n + 1}}(\partial u/\partial z) = {{\overline{{{(\partial u/\partial z)}^{2n + 1}}} }}/{{{{\overline{{{(\partial u/\partial z)}^2}} }^{(2n + 1)/2}}}}$$ M 2 n + 1 ( ∂ u / ∂ z ) = ( ∂ u / ∂ z ) 2 n + 1 ¯ / ( ∂ u / ∂ z ) 2 ¯ ( 2 n + 1 ) / 2 and $${N_{2n + 1}}(\partial u/\partial y) = {{\overline{{{(\partial u/\partial y)}^{2n + 1}}} }}/{{{{\overline{{{(\partial u/\partial y)}^2}} }^{(2n + 1)/2}}}}$$ N 2 n + 1 ( ∂ u / ∂ y ) = ( ∂ u / ∂ y ) 2 n + 1 ¯ / ( ∂ u / ∂ y ) 2 ¯ ( 2 n + 1 ) / 2 , which should be zero if local isotropy is satisfied (n is a positive integer). It is found that the relation $$M_{2n+1}(\partial u/\partial z) \sim R_\lambda ^{-1}$$ M 2 n + 1 ( ∂ u / ∂ z ) ∼ R λ - 1 is supported reasonably well by hot-wire data up to the seventh order ( $$n=3$$ n = 3 ) on the wake centreline, although it is also dependent on the initial conditions. The present relation $$N_{3}(\partial u/\partial y) \sim R_\lambda ^{-1}$$ N 3 ( ∂ u / ∂ y ) ∼ R λ - 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 $$M_{2n+1}(\partial u/\partial z)$$ M 2 n + 1 ( ∂ u / ∂ z ) and $$N_{2n+1}(\partial u/\partial y)$$ 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 $$N_{2n+1}(\partial u/\partial y)$$ N 2 n + 1 ( ∂ u / ∂ y ) in the direction of the mean shear; its effect on $$M_{2n+1}(\partial u/\partial z)$$ M 2 n + 1 ( ∂ u / ∂ z ) (in the non-shear direction) is negligible. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Experiments in Fluids Springer Journals

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

, Volume 57 (7) – Jun 14, 2016
8 pages

/lp/springer_journal/towards-local-isotropy-of-higher-order-statistics-in-the-intermediate-CMm8PBhbYp
Publisher
Springer Berlin Heidelberg
Subject
Engineering; Engineering Fluid Dynamics; Fluid- and Aerodynamics; Engineering Thermodynamics, Heat and Mass Transfer
ISSN
0723-4864
eISSN
1432-1114
D.O.I.
10.1007/s00348-016-2198-5
Publisher site
See Article on Publisher Site

### Abstract

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, $${M_{2n + 1}}(\partial u/\partial z) = {{\overline{{{(\partial u/\partial z)}^{2n + 1}}} }}/{{{{\overline{{{(\partial u/\partial z)}^2}} }^{(2n + 1)/2}}}}$$ M 2 n + 1 ( ∂ u / ∂ z ) = ( ∂ u / ∂ z ) 2 n + 1 ¯ / ( ∂ u / ∂ z ) 2 ¯ ( 2 n + 1 ) / 2 and $${N_{2n + 1}}(\partial u/\partial y) = {{\overline{{{(\partial u/\partial y)}^{2n + 1}}} }}/{{{{\overline{{{(\partial u/\partial y)}^2}} }^{(2n + 1)/2}}}}$$ N 2 n + 1 ( ∂ u / ∂ y ) = ( ∂ u / ∂ y ) 2 n + 1 ¯ / ( ∂ u / ∂ y ) 2 ¯ ( 2 n + 1 ) / 2 , which should be zero if local isotropy is satisfied (n is a positive integer). It is found that the relation $$M_{2n+1}(\partial u/\partial z) \sim R_\lambda ^{-1}$$ M 2 n + 1 ( ∂ u / ∂ z ) ∼ R λ - 1 is supported reasonably well by hot-wire data up to the seventh order ( $$n=3$$ n = 3 ) on the wake centreline, although it is also dependent on the initial conditions. The present relation $$N_{3}(\partial u/\partial y) \sim R_\lambda ^{-1}$$ N 3 ( ∂ u / ∂ y ) ∼ R λ - 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 $$M_{2n+1}(\partial u/\partial z)$$ M 2 n + 1 ( ∂ u / ∂ z ) and $$N_{2n+1}(\partial u/\partial y)$$ 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 $$N_{2n+1}(\partial u/\partial y)$$ N 2 n + 1 ( ∂ u / ∂ y ) in the direction of the mean shear; its effect on $$M_{2n+1}(\partial u/\partial z)$$ M 2 n + 1 ( ∂ u / ∂ z ) (in the non-shear direction) is negligible.

### Journal

Experiments in FluidsSpringer Journals

Published: Jun 14, 2016

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