The classical problem of the hydraulic jump in diverging channels is revisited and reformulated, in terms of linear and angular momentum conservation, at the integral scale. The former balance is used in its well known form, while the latter is expressed in the simplest meaningful formulation, taking into account of the moments exerted by momentum flux, by pressure force in each cross section and on lateral walls, and by vertical stresses. The whole flow is supposed to be ideally divided in a mainstream, conveying the total discharge, and in a roller, exerting stresses on the mainstream. The scheme is the axially-symmetric counterpart of the 2D plane scheme formulated by Valiani [Linear and angular momentum conservation in hydraulic jump, JHR, 35(3), pp.323-352, 1997]. Gravity forces are supposed to be exactly counteracted by bottom reactions. Outside the jump, the inviscid solution is assumed to minimize the degrees of freedom of the problem. The obtained system of two conservation laws, together with the appropriate boundary conditions, is not analytically solvable due to the highly nonlinear relationship between the vertical length scale (depth) and the longitudinal length scale (radius), but a simple numerical solution gives the sequent depths and their positions, as functions of the non-dimensional discharge and of the downstream/upstream energy ratio of the flow. Taking into account the uncertainties in defining the downstream cross section of the jump, the comparison between the numerical solution and a selected set of laboratory data shows the reliability and accuracy of the proposed scheme.

Hydraulic jump in diverging channels

VALIANI, Alessandro;CALEFFI, Valerio
2010

Abstract

The classical problem of the hydraulic jump in diverging channels is revisited and reformulated, in terms of linear and angular momentum conservation, at the integral scale. The former balance is used in its well known form, while the latter is expressed in the simplest meaningful formulation, taking into account of the moments exerted by momentum flux, by pressure force in each cross section and on lateral walls, and by vertical stresses. The whole flow is supposed to be ideally divided in a mainstream, conveying the total discharge, and in a roller, exerting stresses on the mainstream. The scheme is the axially-symmetric counterpart of the 2D plane scheme formulated by Valiani [Linear and angular momentum conservation in hydraulic jump, JHR, 35(3), pp.323-352, 1997]. Gravity forces are supposed to be exactly counteracted by bottom reactions. Outside the jump, the inviscid solution is assumed to minimize the degrees of freedom of the problem. The obtained system of two conservation laws, together with the appropriate boundary conditions, is not analytically solvable due to the highly nonlinear relationship between the vertical length scale (depth) and the longitudinal length scale (radius), but a simple numerical solution gives the sequent depths and their positions, as functions of the non-dimensional discharge and of the downstream/upstream energy ratio of the flow. Taking into account the uncertainties in defining the downstream cross section of the jump, the comparison between the numerical solution and a selected set of laboratory data shows the reliability and accuracy of the proposed scheme.
2010
Hydraulic jump; momentum balaces; angular momentum balances
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/1394498
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