Mathematical models and analysis of turbulent, wall-bounded, complex flows.

In the classical wall bounded turbulent flow a fundamental statement is the existence of a layer, called overlap layer, in which every flow behaves the same and the mean streamwise velocity of each system can be described with only the wall normal coordinate with a logarithmic profile, characterized by the von Kármán constant. This law has been at first derived on data on parallel flows and boundary layer, that are model flows for wall turbulence, but indeed have a much simpler flow than complex shape geometries. The formulation of Millikan has much more general requirement on the flow and it is based on the asymptotic expansion of the velocity field; this theory of the logarithmic behavior of the overlap layer is an asymptotic approximation, and so holds for very high Reynolds numbers, Re_τ → ∞. For this reason much of the research effort has been directed at increasing the Reynolds number. However, due to the limits in resources, and so in the possibility of reaching the highest possible value, every similarity theory is still incomplete; but like all asymptotic approximations, it can be improved with the addition of higher-order terms. We develop a correction of the classical von Kármán logarithmic law for a turbulent Taylor-Couette (TC) flow, the fluid flow developing between two coaxial, independently rotating cylinders, when the curvature of the system is small, i.e. with an inner to outer radius ratio η = r_i /r_o ≥ 0.9, when both the cylinder rotates with the same magnitude but in opposite directions. While in straight geometries like channel or pipe, the deviation from the law can be ascribed to the effect of pressure gradient, in small gap TC flow this effect can be accounted to the conserved transverse current of azimuthal motion. We show that, when the correction is applied, the logarithmic law is restored even when varying the curvature, and that the parameters founded here for TC flow converge to the ones founded in [P. Luchini. European Journal of Mechanics B Fluids, 71, 2018.] for plane Couette flow, in the limit of vanishing curvature η → 1.