GALCIT Colloquium

The inverse of the von Karman constant, κ, is the leading coefficient in the equation describing the logarithmic mean velocity profile in wall bounded turbulent flows.  Previous research (J. Fluid Mech., 638, 2009, 73, J. Fluid Mech., 718, 2013, 596.) demonstrates that the asymptotic value of κ derives from an emerging condition of dynamic self-similarity on an interior inertial domain, and that these dynamics induce a geometrically self-similar hierarchy of scaling layers.  First-principles based analyses are used to reveal a number of properties associated with the asymptotic value of κ.  The development leads toward, but terminates short of, analytically determining a value for κ.  The analysis does, however, suggest the distinct possibility that κ = Φ-2 = 0.381966..., where, Φ = (1 + √5)/2 is the golden ratio.  Empirical measures derived from the theory are used to explore the veracity and implications of κ = Φ-2.  Consistent with the differential transformations underlying the invariant form admitted by the governing mean equation, it is further demonstrated that the value of κ 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.  Data from direct numerical simulations and higher Reynolds number experiments convincingly support the self-similar geometric structure indicated by the analysis.