Linear and angular momentum conservation in hydraulic jump in diverging channels

Abstract

This paper addresses the integral conservation of linear and angular momentum in the steady hydraulic jump in a linearly diverging channel. The flow is considered to be divided into a mainstream that conveys the total liquid discharge, and a roller where no average mass transport occurs. It is assumed that no macroscopic rheological relationship holds, so mass, momentum and angular momentum integral balances are independent relationships. Normal stresses are assumed to be hydrostatic on vertical, cylindrical surfaces. Viscous stresses are assumed to be negligible with respect to turbulent stresses. Assuming that the horizontal velocity distribution in the mainstream is uniform and that the horizontal momentum and angular momentum in the roller are negligible with respect to their mainstream counterparts, an analytical solution is obtained for the free surface profile of the flow. This solution is fundamental for finding the sequent depths and their positions. Consequently, it permits solving for the length of the jump, which is assumed to be equal to the length of the roller. Mainstream and roller thicknesses can also be derived from the present solution. This model may also be theoretically used to derive the average shear stresses exerted by the roller on the mainstream and the power losses per unit weight. This second relationship, which returns the well-known classical expression for total power loss in the jump, demonstrates that the strongly idealized mechanical model proposed here is internally consistent.