The statistical behaviour of attached eddies

Abstract

Townsend's attached eddy hypothesis forms the basis of an established model of the logarithmic layer in wall-bounded turbulent flows in which this inertially dominated region is characterised by a hierarchy of geometrically self-similar eddying motions that scale with their distance to the wall. The hypothesis has gained considerable support from high Reynolds number measurements of the second-order moments of the fluctuating velocities. Recently, Meneveau and Marusic ["Generalized logarithmic law for high-order moments in turbulent boundary layers," J. Fluid Mech. 719, R1 (2013)] presented experimental evidence that all even-ordered moments of the streamwise velocity will exhibit a logarithmic dependence on the distance from the wall. They demonstrated that this was consistent with the attached eddy hypothesis, so long as the velocity distribution is assumed to be Gaussian (which allows the use of the central limit theorem). In this paper, we derive this result from the attached eddy model without assuming a Gaussian velocity distribution, and find that such logarithmic behaviours are valid in the large Reynolds number limit. We also revisit the physical and mathematical basis of the attached eddy hypothesis, in order to increase rigour and minimise the assumptions required to apply the hypothesis. To this end, we have extended the proof of Campbell's theorem to apply to the velocity field corresponding to a forest of variously sized eddies that are randomly placed on the wall. This enables us to derive all moments of the velocity in the logarithmic region, including cross-correlations between different components of the velocity. By contrast, previous studies of the attached eddy hypothesis have considered only the mean velocity and its second order moments. From this, we obtain qualitatively correct skewnesses and flatnesses for the spanwise and wall-normal fluctuations.

Scaling of second- and higher-order structure functions in turbulent boundary layers

“…Using this, Woodcock & Marusic (2015) showed that the log-law for the flow profile follows from Campbell's theorem.…”

“…M. de Silva, I. Marusic, J. D. Woodcock and C. Meneveau The derivation of the structure functions is analogous to that presented by Woodcock & Marusic (2015) for the moments of the velocity. The details of this derivation can be found in appendix A.…”

“…The constant A k is given by which is based on the assumption that h max = δ and h min = 100 following Kline et al (1967). By an analogous derivation to that given by Woodcock & Marusic (2015), it can be shown that (5.18) can be re-expressed as (5.20) where k x is a constant defined such that the expected distance to the next closest eddy in the positive x direction will always be k x h if the height of the next closest eddy were known to be h. (More specifically, k x h represents the distance in a strip of height h to the nearest eddy of height between h and h + dh divided by h and dh.) The constant k y is its spanwise equivalent.…”

“…The eddies are identical once scaled by their heights, and their shapes are unaffected by the presence of nearby eddies. The reasoning behind the attached-eddy hypothesis, as well as the derivation extended here, has been further elaborated upon by Woodcock & Marusic (2015).…”

“…This has allowed the statistical properties of the flow to be derived via Campbell's theorem. This model has been used by Woodcock & Marusic (2015) to derive the moments of the velocity in the logarithmic region. The model treats the flow as a superposition of the velocity fields corresponding to each of a multitude of geometrically equivalent eddies.…”