AttachedEddyHypothese

AttachedEddyHypothese, commonly referred to in English as the attached-eddy hypothesis, is a theoretical framework for understanding wall-bounded turbulence. It was developed by A. A. Townsend and subsequent collaborators to explain the structure of the turbulent boundary layer, especially in the logarithmic region near a wall. The central idea is that the turbulent field can be described as a hierarchy of self-similar eddies that are attached to the wall.

In this view, each eddy has a size l that scales with its distance y from the wall, so larger eddies extend farther from the surface. The eddies form a population across a range of sizes from the near-wall region to scales comparable with the boundary-layer thickness. The distribution of eddy sizes is approximately logarithmic: the number of eddies per size interval scales roughly as 1/l, leading to a continuous, self-similar spectrum when eddies of all sizes are superposed. The superposition of many attached eddies produces the observed statistics of the turbulent field, including the mean shear and velocity fluctuations.

Predicted consequences of the hypothesis include the presence of a kx^{-1} range in the streamwise energy spectrum and a logarithmic dependence of certain velocity statistics on the Reynolds number, particularly in the logarithmic layer. The attached-eddy framework provides a coherent explanation for the origin of the logarithmic mean velocity profile and the scaling of velocity fluctuations with Reynolds number.

The hypothesis has been supported by laboratory and atmospheric experiments and by direct numerical simulations, and it remains a foundational concept in the interpretation of wall-bounded turbulence. It is often used as a organizing principle in turbulence modeling and in the analysis of spectral and two-point statistics, though real flows can show deviations and require refinements.