The purpose of studies on the critical tractive force which have been carried out by many researchers since du Boys is mostly to create empirical formula by finding experimentally the relation between the diameter of sand grain and the critical velocity or the critical shearing stress on channel bottom. On the other hand, the theories on the mechanism of initiation of sand movement have been studied by only a few researchers, such as C. M. White (1940) and M. Kurihara (1948), in recent years.
C. M. White expressed the effect of turbulence by a factor named as the turbulence factor, and M. Kurihara explained by a skilful application of turbulence theory that the turbulence factor is a function of Reynolds Number with respect to roughness. Since Kurihara's theory, however, is difficult to understand, another theoretical analysis is attempted by the author in this paper, based on the idea and treatment different from Kurihara's theory.
The basic idea of the author's theory is to construct the equilibrium condition using the forces which are acting on a spherical sand grain, such as the gravity force, the fluid resistance and the resistance resulting from pressure gradient, and to evaluate these resistance considering the velocity fluctuation. In this process, the concepts of the mixing length of turbulence and the minimum scale of eddies in turbulence theory are utilized.
A dimensionless function of critical tractive force like as the function by Shields is derived by the analysis based on the idea described above, and an experimental constant named as the sheltering coefficient is introduced to this function so as to agree with the experimental results obtained with a small tunnel having a uniform and square cross-section.
On the basis of the experimental results and theoretical relationships, a new formula of critical tractive force is created and moreover compared with the empirical formulas proposed by many authorities.