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An analytical framework for studying the logarithmic region of turbulentchannels is formulated. We build on recent findings (Moarref et al., J. FluidMech., 734, 2013) that the velocity fluctuations in the logarithmic region canbe decomposed into a weighted sum of geometrically self-similar resolventmodes. The resolvent modes and the weights represent the linear amplificationmechanisms and the scaling influence of the nonlinear interactions in theNavier-Stokes equations (NSE), respectively (McKeon & Sharma, J. Fluid Mech.,658, 2010). Originating from the NSE, this framework provides an analyticalsupport for Townsend's attached-eddy model. Our main result is thatself-similarity enables order reduction in modeling the logarithmic region byestablishing a quantitative link between the self-similar structures and thevelocity spectra. Specifically, the energy intensities, the Reynolds stresses,and the energy budget are expressed in terms of the resolvent modes with speedscorresponding to the top of the logarithmic region. The weights of the triadmodes -the modes that directly interact via the quadratic nonlinearity in theNSE- are coupled via the interaction coefficients that depend solely on theresolvent modes (McKeon et al., Phys. Fluids, 25, 2013). We use the hierarchiesof self-similar modes in the logarithmic region to extend the notion of triadmodes to triad hierarchies. It is shown that the interaction coefficients forthe triad modes that belong to a triad hierarchy follow an exponentialfunction. The combination of these findings can be used to better understandthe dynamics and interaction of flow structures in the logarithmic region. Thecompatibility of the proposed model with theoretical and experimental resultsis further discussed.

A foundation for analytical developments in the logarithmic region of turbulent channels