Mathematical solution of the problem of roll‐waves in inclined opel channels

Abstract

AbstractThe purpose of this paper is to obtain solutions which are periodic with respect to distance, describing the phenomenon called “roll‐waves,” for water flow along a wide inclined channel, and to discuss the behavior of the mathematical solutions. The basic idea presented in Part I is that discontinuous periodic solutions can be constructed by joining together sections of a continuous solution through shocks (or “bores”). It is shown first that no continuous solutions can be periodic and that only one special continuous solution can be used as the basis for constructing discontinuous periodic solutions. The analysis is based upon the non‐linear partial differential equations of the “shallow water theory,” augmented by the Cheay formula to allow for turbulent resistance. The Bresse profile equation is obtained in a form applicable for progressing wave flows. Shock conditions are derived for the case of an arbitrary continuous channel bed and for a flow subject to a resisting force. The special continuous solution is explicitly obtained and analyzed. Branches of it are then joined together through shocks. It is proved that roll‐waves cannot occur either if the resistance is zero or if the resistance exceeds a certain critical value. As the resistance decreases, the size of the waves decreases also; and if the resistance becomes too large, the profiles reverse their direction and can no longer be joined by shocks. This critical value is reached when the (dimensionless) resistance coefficient equals one‐fourth the value of the channel slope. The presence of a resistance force which varies merely with velocity is not sufficient to permit the construction of periodic solutions; the resistance must also act in such a manner that it decreases as the water depth increases. The analysis proves that the ratio of wave height to wave length of roll‐waves is always independent of the speed of the waves. Explicit expressions for water height and shock height as functions of wave length are derived. The investigation studies the static discharge rate as a function of the wave speed, and asymptotic formulas for the wave speed in terms of the average discharge rate are derived. Twelve sets of curves arc presented, based on the equations obtained here, to illustrate the quantitative behavior of roll‐waves; these may be used to check this theory against observed data. For prescribed values of slope, resistance, and wave speed, there is a one‐parameter family of roll‐wave solutions. If the wave length is also prescribed, the solution will then be unique.