Linear and angular momentum conservation in hydraulic jump

The present paper deals with the integral conservation of linear momentum and angular momentum in the stationary hydraulic jump in a wide rectangular channel. The flow is considered to be divided into a mainstream, that conveys the total liquid discharge, and a roller, in which no average mass transport occurs. Referring to the infinitely large case, a purely two dimensional motion is considered. The interface between the two flow regions is a streamline, corresponding to a stream function value equal to the total discharge per unit width. The present approach consists in satisfying the mechanical balances of mass, momentum and angular momentum, while no (large scale) constitutive relation is assumed for the turbulent motion of the liquid. Regarding the stress tensor, hydrostatic normal pressure distribution is assumed, while nothing is assumed regarding shear stresses, except that viscous stresses are negligible with respect to turbulent stresses. A paradox is put in evidence, that in the classical hydraulic jump (specific force conserving solution) angular momentum conservation is apparently not satisfied. Taking into account of integral balances not only in terms of linear horizontal momentum but also in linear vertical momentum and angular momentum the paradox is overcome. Under some simplified assumptions regarding uniform horizontal velocity distribution in the mainstream, and negligibility of horizontal momentum and angular momentum in the roller with respect to other terms, an analytical solution is obtained in terms of free surface profile, mainstream thickness and roller thickness. Average shear stresses acting on the mainstream by the roller and power losses for unit weight may be theoretically derived. Assuming as known the growth rate of the mainstream at the beginning of the jump, also the length of jump, here assumed identical to the length of the roller, may be determined, together with the volume of the roller, the volume of the mainstream and the volume of the whole stream between the sequent depths