The hunt for the Kármán ‘constant’ revisited

The log law of the wall, joining the inner, near-wall mean velocity profile (MVP) in wall-bounded turbulent flows to the outer region, has been a permanent fixture of turbulence research for over hundred years, but there is still no general agreement on the value of the prefactor, the inverse of the Kármán ‘constant’, or on its universality. The choice diagnostic tool to locate logarithmic parts of the MVP is to look for regions where the indicator function (equal to the wall-normal coordinate y^+ times the mean velocity derivative d U^+/ d y^+) is constant. In pressure-driven flows, however, such as channel and pipe flows, is significantly affected by a term proportional to the wall-normal coordinate, of order O (Re ^-1) in the inner expansion, but moving up across the overlap to the leading O (1) in the outer expansion. Here we show that, due to this linear overlap term, Re values well beyond 10⁵ are required to produce one decade of near constant in channels and pipes. The problem is resolved by considering the common part of the inner asymptotic expansion carried to O (Re ^-1), and the leading order of the outer expansion. This common part contains a superposition of the log law and a linear term S₀ \, y^+Re ^-1, and corresponds to the linear part of, which, in channel and pipe, is concealed up to y^+ 500x20131000 by terms of the inner expansion. A new and robust method is devised to simultaneously determine and S₀ in pressure-driven flows at currently accessible Re values, yielding values which are consistent with the values deduced from the Reynolds number dependence of centreline velocities. A comparison with the zero-pressure-gradient turbulent boundary layer, further clarifies the issues and improves our understanding.