Distributed Vortex-Wave Interactions: The Relation Of Self-Similarity To The Attached Eddy Hypothesis

A large-Reynolds-number asymptotic reduction of the Navier-Stokes equations capable of describing a locally periodic vortex-wave array and the associated large-scale variation of the mean-shear velocity field first suggested in Hall (J. Fluid Mech., vol. 850, 2018, pp. 46-82) is extended. The sustaining process of the locally periodic coherent structures is based on the vortex-wave interaction theory of Hall & Smith (J. Fluid Mech., vol. 227, 1991, 641-666), wherein two-dimensional roll-streak fields are supported by localised nonlinear self-interactions of three-dimensional waves that are largest in size within critical layers of the streak field. The variation of the mean velocity is made possible by incorporating a slow change to the mean profile using a Wentzel-Kramers-Brillouin-type approach. As the first extension, we demonstrate that the local structure corresponds to the asymptotic limit of computations in a shearing box. A variety of solutions with different symmetry properties are found via the hybrid numerical asymptotic approach of Blackburn, Hall & Sherwin (J. Fluid Mech. vol. 721, 2013, 58-85). Moreover, some solutions show generic flow features such as uniform momentum zones and spatial intermittency known to occur in near-wall turbulent boundary layers. We extend the vortex-wave interaction array theory to show that, in addition to a Reynolds-averaged-Navier-Stokes-type relationship between the large-scale vertical variation of the mean flow and local roll-streak scale, a higher-order analysis gives a second constraint on the slow-scale dynamics. Those constraints are used for the first time to derive the logarithmic law of the wall through a closed asymptotic analysis of self-similar local coherent structures, consistent with the attached eddy hypothesis.