Modern hydraulic structures require highly reliable water-supply channels, free-flow pipes and open spillways. Therefore, they must be built with considering dynamic properties of the affecting flow. The theory of one-dimensional open flows cannot answer many questions raised by the hydraulic engineering practice. Hence, this paper considers two-dimensional graphical open flows, with separate particular models following from the general theory, important for designing couplings, curves in more complex constructions (two different channels) required in hydraulic engineering (Hydraulics technical structures) (HTS). And the more such models are obtained, the more opportunities will appear for scientists and designers to synthesize more complex structures required in HTS. This study identifies three elements, and three simplest models of two-dimensional graphic open water flows similar to flat models: a source, a vortex and a vortex source. The number of such individual models will continue to increase and expand the range of possibilities for designing more complex water technical objects (WTO). The models are obtained analytically from solving the system of two-dimensional graphical open potential water flows in the plane of the velocity hodograph. The elementary construction of each model includes an analytical solution for the potential function and the stream function. The paper considers the problem of determining the parameters of a two-dimensional graphical open water flow at any point in the flow: a source, a vortex, a vortex source. The practical significance of the models lies in the possibility to use the results by the designers of hydraulic structures both at the first stage of solving problems and at the subsequent ones, with flow resistance forces taken into account. The study also represents the transition method from the plane of the flow velocity hodograph to the flow diagram by integrating the models in the plane of the velocity hodograph from the condition of the connection between the considered planes.

The basic system of equations of motion of a non-stationary two-dimensional open water flow is obtained, which does not contradict the previously known system for a stationary flow.

The paper substantiates a general scheme for solving two-dimensional free spreading problem in terms of open water potential flow below the non-pressure pipe. The first link. Since a uniform flow at its outlet from a rectangular pipe with its free spreading is coupled with a non-uniform flow of a general form, then, using a theorem from the general theory of two-dimensional in terms of turbulent water potential flows, the conclusion is given: since a straight-line characteristic is always the boundary between a uniform and non-uniform flow, then only a simple wave can directly adjoin the uniform flow area. Simple waves serve as a transitional form from the uniform flow to the general non-uniform flow. Therefore, it is not entirely correct to pose the free flow spreading boundary problem, trying to satisfy only the boundary conditions and obtain an analytical solution without taking into account the intermediate flow “simple wave”. The second important link in solving the problem is a general flow choice. And it can be selected from the intermediate flow condition found by the authors in the velocity hodograph plane. Let us call this flow the “type A flow”. “Type A flow” satisfies all the requirements of the flow spreading process function. When increasing, i.e. τ tends to 1, flow depth h tends to 0; velocity V tends to the maximum. The flow depth is greater on the symmetry axis than when moving along the equipotential to the flow free boundary. Using these two links, it is possible to apply a flow coupling scheme and solve the boundary problem of determining the entire spectrum of parameters for the potential flow spreading. The results obtained in this work can be used by the designers of hydraulic structures, which length is relatively short and where the flow resistance forces can be neglected.

In this paper, we show in principle a method for solving the problem of a two-dimensional vortex source in terms of flow. The material of the article relates to mathematical modeling of classical two-dimensional problems in terms of potential flows. The resulting solution is similar to the results for a flat vortex source and is an important result for the development of the analytical theory of two-dimensional in terms of potential water flows. Other solutions for real potential two-dimensional flows are considered, which are clarified by the authors, which confirms the relevance of the work.

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