The von Kármán constant for flows over rough mobile beds. Lessons learned from dimensional analysis and similarity

The nature of the von Kármán constant (κκ), i.e. universal, Re  -independent but wall-dependent or, in general, flow dependent, has been subjected to considerable investigation. Contradicting the classic view that κκ should be universal in open-channel flows of clear water with a fully developed boundary layer, irrespectively of the type of boundary, it has recently been proposed that the value of κκ is influenced by the mobility of the bottom boundary.Rather than advocating a particular view on the nature of κκ, the objectives of this work are: (i) to provide a similarity framework for the nature of κκ, highlighting, in the process, the underlying key phenomena at play, and (ii) to explore a theoretical model for κκ as a function of turbulence parameters.Dimensional analysis is performed, including a new key step, following the works of Selim Yalin, in which the shear-rate is interpreted as a two-phase variable, depending on flow as well as sediment parameters. Vaschy–Buckingham’s theorem is applied and similarity arguments leading to the log-law are employed. These introduce appropriate hypothesis concerning complete and incomplete similarity, thus providing a theoretical framework for the discussion of the nature of κκ. A theoretical model is then explored, materialised in a functional relation between κκ and turbulence parameters. The discussion of the nature of κκ becomes a discussion of the nature of those parameters, all susceptible to be measured. Such discussion is based on laboratory results concerning flows over porous immobile and mobile beds. The data reveals changes in the structure of turbulence for flows over weakly mobile gravel-sand beds but is not supportive of a reduction of the value of κκ.