The inverse of the von Kármán constant $\kappa$ is the leading coefficient in the equation describing the logarithmic mean velocity profile in wall bounded turbulent flows. Klewicki [J. Fluid Mech. 718, 596 (2013)] connects the asymptotic value of $\kappa$ with an emerging condition of dynamic self-similarity on an interior inertial domain that contains a geometrically self-similar hierarchy of scaling layers. A number of properties associated with the asymptotic value of $\kappa$ are revealed. This is accomplished using a framework that retains connection to invariance properties admitted by the mean statement of dynamics. The development leads toward, but terminates short of, analytically determining a value for $\kappa$. It is shown that if adjacent layers on the hierarchy (or their adjacent positions) adhere to the same self-similarity that is analytically shown to exist between any given layer and its position, then $\kappa \equiv {\Phi }^{-2}=0.381966...$, where $\Phi =(1+\sqrt{5})/2$ is the golden ratio. A number of measures, derived specifically from an analysis of the mean momentum equation, are subsequently used to empirically explore the veracity and implications of $\kappa ={\Phi }^{-2}$. Consistent with the differential transformations underlying an invariant form admitted by the governing mean equation, it is demonstrated that the value of $\kappa$ arises from two geometric features associated with the inertial turbulent motions responsible for momentum transport. One nominally pertains to the shape of the relevant motions as quantified by their area coverage in any given wall-parallel plane, and the other pertains to the changing size of these motions in the wall-normal direction. In accord with self-similar mean dynamics, these two features remain invariant across the inertial domain. Data from direct numerical simulations and higher Reynolds number experiments are presented and discussed relative to the self-similar geometric structure indicated by the analysis, and in particular the special form of self-similarity shown to correspond to $\kappa ={\Phi }^{-2}$.

• Received 6 June 2014
• Revised 17 October 2014

DOI:https://doi.org/10.1103/PhysRevE.90.063015