On discontinuity solutions of shallow water equations in channel with cross section jumping

The equations of the first approximation of shallow water theory are widely used  to model the propagation of hydraulic bores generated by total or partial dam breaks. The theory describes such bores as steady discontinuous solutions, which we, following  will name shocks. However, the classical system of the basic shallow water conservation laws, while correctly describing the parameters of hydraulic bores propagating in a regular channel, does not allow describe the water flows on cross section jumping. It is caused by that the equation of full momentum is the exact conservation law only in the case of regular channel. Therefore it cannot be used for obtaining the the Rankine-Hugoniot conditions on the discontinuous arising on cross section jumping. From this it follows that if on such jumping come two characteristics than it is necessary introduction of an additional condition on the discontinuous.

In present paper, following, we will obtain such additional condition from local momentum conservation law which conserve the divergent form in the case of nontrapezoidal channel. Consequence of it is conservation on such discontinuous of full energy of a running stream. As a concrete example it is considered a dam break problem on cross section jumping. Unequivocal resolvability of this problem within the limits of the solutions containing stationary shocks and centered depression waves is investigated.

The work was supported by the Russian Foundation for Basic Research (project no. 09-01-98001 and 10-01-00338) and the Projects on Basic Research of the Presidium of the Russian Academy of Sciences (No. 4.7) and Presidium of the Siberian Division of Russian Academy of Sciences (No. 23).

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