Wall-bounded turbulence with logarithmically scale-local interactions

Turbulence can be envisioned as a collection of interacting coherent structures or eddies, that unlike waves, are simultaneously localized in both physical- and scale-space. In the case of the inertial range of homogeneous isotropic turbulence, the energy cascade mechanism can be thought of as a series of successive logarithmically local-in-scale steps in this simultaneous physical- and scale-space representation. This picture has been modeled using dynamical systems with banded interaction kernels, called shell models (L. Biferale Annu. Rev. Fluid Mech. 2003). Analogously, we represent the wall-parallel dimensions by logarithmically spaced representations with local-in-scale interaction kernels while maintaining full physical space representation in the wall-normal direction. It is shown that if the wall-parallel representation is logarithmically spaced waves, the mean velocity profile has the incorrect Kármán constant. It is only when the relative strengths of the terms in the interaction kernel are weighed to mimic wave packets, à la coherent structures, that this local-in-scale representation achieves the mean velocity profile with the correct Kármán constant. We present the statistical states of this dynamical system at Reynolds numbers inaccessible with modern, traditional DNS, since the number of degrees of freedom grows as Re_τ²Re_τ^3/4 as opposed to our (log_λ⁡Re_τ)²Re_τ^3/4, where λ is our chosen logarithmic spacing.