Subject of inquiry is a single vane profile, and cascade of vane profiles in vane machinery such as pumps or turbines in a viscous incompressible fluid. The objective is to improve the quality of the object by changing the geometry — build a vane profile with minimal loss and thus improving efficiency of vane machine. The paper presents the implementation of three mathematical methods, the results are compared with each other and with experimental data: — the method of potential flow over profile and the method of flow over profile based on Euler equations using the theory of the boundary layer, — numerical method for calculating the flow over profile based on Navier— Stokes equations for viscous turbulent flows using turbulence model. The first two methods allow to carry out multiple calculations to optimize the profile for a short time, the third, more accurate method is used to verify the results. The problems of optimizing the profile of van Dyk are solved by all the proposed mathematical methods. The comparison is made of a flow over the initial profile with the experimental data resulted in the atlas of van Dyk. Reliability and efficiency of the proposed method is demonstrated by the example of vane profiles' cascade creating with minimal losses.
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