Study on Resistance Law in Mountainous Rivers with Large Roughness

Abstract

This paper is concerned with the resistance law of mountainous rivers whose bed surfaces are composed of sediments having a very wide range of grain diameter.
A new relationship between the cross-sectional mean velocity and the geometric parameters of bed roughness is derived through hydrodynamical consideration of both the mechanism of the resisting force and the characteristic of the vertical velocity distribution profile. The longitudinal component of the gravity force is equated with sum of the drag force due to large gravels or cobbles projecting considerably high above the mean bed level and the shear force due to the remaining rough surface. As the velocity disribution equation, Prandtl-von Karman's logarithmic law is appropriated only for the upper layer of the flow, while an empirical one is proposed for the lower layer.
Most of the geometric parameters mentioned above cannot be directly measured in case of actual problems because those are introduced from a hydraulical standpoint. Therefore, a set of equations to calculate them from river surveying data has been derived in consideration of the stochastic properties in the arrangement of bed materials; however, only a brief summary is given in this paper.
The relationship is quantitatively verified by the results of flume experiments performed by the authors, where they use sediment mixture whose range of grain diameter is 0.5 mm to 200 mm. Furthermore, some numerical examples of the relationship are compared with other experimental data and field data obtained in gravel bed rivers.