The hunt for the K\'arm\'an "constant'' revisited

Abstract

The logarithmic law of the wall, joining the inner, near-wall mean velocity profile (MVP) 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 pre-factor, the inverse of the K\'arm\'an ``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 $\Xi$ (equal to the wall-normal coordinate $y^+$ times the mean velocity derivative \dd U^+/\dd y^+) is constant. In pressure driven flows however, such as channel and pipe flow, $\Xi$ is significantly affected by a term proportional to the wall-normal coordinate, of order \mathcal{O}(\Reytau^{-1}) in the inner expansion, but moving up across the overlap to the leading $\mathcal{O}(1)$ in the outer expansion. Here we show, that due to this linear overlap term, \Reytau's of the order of $10^6$ and beyond are required to produce one decade of near constant $\Xi$ in channels and pipes. The problem is resolved by considering the common part of the inner asymptotic expansion carried to \mathcal{O}(\Reytau^{-1}), and the leading order of the outer expansion, which is a \textit{superposition} of log law and linear term L_0 \,y^+\Reytau^{-1}. The approach provides a new and robust method to simultaneously determine $\kappa$ and $L_0$ in pressure driven flows at currently accessible \Reytau's, and yields $\kappa$'s which are consistent with the $\kappa$'s deduced from the Reynolds number dependence of centerline velocities. A comparison with the zero-pressure-gradient turbulent boundary layer, henceforth abbreviated ``ZPG TBL'', further clarifies the issues