The hunt for the Karman '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 Karman '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 times the mean velocity derivative ) 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 in the inner expansion, but moving up across the overlap to the leading in the outer expansion. Here we show that, due to this linear overlap term, values well beyond 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 , and the leading order of the outer expansion. This common part contains a superposition of the log law and a linear term , and corresponds to the linear part of , which, in channel and pipe, is concealed up to by terms of the inner expansion. A new and robust method is devised to simultaneously determine and in pressure-driven flows at currently accessible 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.

Automated summary

The log law of the wall, which connects the inner and outer regions of wall-bounded turbulent flows, has been studied for over a century, but there is still no consensus on its prefactor value or universality. Currently, the most common diagnostic tool to identify the logarithmic parts of the velocity profile is to locate regions where the indicator function is constant. However, in pressure-driven flows like channel and pipe flows, the indicator function is significantly affected by a term proportional to the wall-normal coordinate. By considering the common part of the inner and outer expansions, a new method is proposed to determine the prefactor and the linear term in pressure-driven flows, providing values consistent with those derived from Reynolds number dependence analysis and improving our understanding.