Open-channel waves generated by propagation of a discontinuous wave over a bottom step

Abstract

This paper presents the results of theoretical and experimental studies of open-channel waves generated by the propagation of a discontinuous dam-break wave over a bottom step. The cases where the initial tailwater level is higher than the step height (the step is under water) and where this value is smaller than the step height (at the initial time, water is absent on the step) are considered. Exact solutions are constructed using modified first-approximation equations of shallow-water theory, which admit the propagation of discontinuous waves in a dry channel. On the stationary hydraulic jump formed above the bottom step, the total free-stream energy is assumed to be conserved. These solutions agree with experimental data on various parameters (types of waves, wave propagation velocity, asymptotic depths behind the wave fronts).Introduction. The first-approximation equations of shallow-water theory [1-3] are widely used in modeling the propagation of discontinuous waves [4-7] (hydraulic bores [8,9]) resulting from complete or partial break of hydraulic dams or the impact of large sea tsunami-type waves [10] on shallow water. However, the classical system of the basic conservation laws of shallow-water theory (consisting of the laws of conservation of mass and total momentum [3]), while correctly describing the parameters of discontinuous waves propagating in a liquid of finite depth above an even bottom [1], is not suitable for describing wave flows above various bottom-relief features, in particular, discontinuous-wave propagation over a bottom step or a drop. This is due to the fact that the totalmomentum equation is an exact conservation law only in the case of a horizontal bottom and it cannot be used to derive Hugoniot conditions for the discontinuities arising from bed level changes.A method of deriving relations for stationary discontinuities above bed level changes using the shallowwater equations was proposed in [11] and validated in [12]. This method is based on the assumption that if two characteristics arrive at such a discontinuity, then, along with the continuity of the discharge, which follows from the mass conservation law, it is necessary to require the continuity of the Bernoulli function, which follows from the local-momentum conservation law and the conservation law for the total free-stream energy. If three characteristics arrive at a discontinuity above a bed level change, the discharge continuity is sufficient to determine all flow parameters at this discontinuity. The total energy at such a discontinuity is lost, which serves as a criterion for its stability [3]. These assumptions and a generalized method of adiabats [13] were used to study the unique solvability of dam-break problems above a bottom step [14] and a drop [15]. The obtained self-similar solutions are in fairly good agreement with experimental data [16-18] on various parameters (types of waves, wave propagation velocity, asymptotic depths behind the wave fronts).