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.

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The article discusses the methods of modeling two-dimensional in terms of turbulent flows. Their difference and advantages are described. The equations of flow motion in the plane of the hodograph are given, as well as the velocity and depth of the flow in the entire region of its flow. Next, a new method is proposed, the purpose of which is to solve a system of differential equations. Using a differential equation connecting the physical cavity of the flow stream and the plane of the travel time curve, a simple model of the flow of a steady, vortex-free water flow in a wide horizontal, smooth outlet channel is proposed. The flow model in the vicinity of the outlet from the pressureless pipe uses the fact that the resistance forces of the flow to the bottom of the outlet channel are small compared with the forces of inertia. The obtained solutions allow us to calculate the flow parameters with an accuracy sufficient for the design of hydraulic structures of the drainage system from the upper pool to the lower one under roads and railways. This model allows you to identify the basis of the laws of the real spreading of the flow and on its basis there is an opportunity for further building models taking into account the resistance forces in order to increase the correspondence of the mathematical model to the real flow.

The paper studies a two-dimensional water flow from a non-pressure rectangular or round pipe placed into a wide horizontal channel. To simplify the problem, the real three-dimensional flow is modeled as a two-dimensional zone by eliminating the liquid particles’ velocities and accelerations in the direction perpendicular to the flow zone. To describe the water flow motion law, L. Euler’s equations for the ideal fluid are used, taking into account the continuity equations and the Bernoulli equation. The two-dimensional flow models in the spreading zone with the adequacy degree sufficient for practice, describe the movement of water flows arising in the lower road drainage systems races, Liman irrigation systems, small bridges, water volley channels, various culverts and water-crossing facilities. The obtained dependences of the velocity distribution, depth and water flow geometry give an accuracy exceeding that known by the previously used methods both by the velocity values and by the boundary current lines geometry.

The algorithm of the theoretical method proposed by the authors using the speed hodograph plane and simple waves for solving the problem of coupling a two-dimensional flow in terms of flow is considered. This technique allows you to determine the boundaries of the free flow of a turbulent flow and its parameters in the flow area. it is revealed that using both planes physical OXYZ and hodograph G (,) it is possible to solve the problem uniquely analytically. All verification calculations were performed in the MathCad package. The calculation results showed a satisfactory convergence of model and experimental parameters in the free flow model, which does not exceed 10 % by error before the flow expansion within  = 7;  = B/b.