Mathematical model of a vertical-axial wind motor in a viscous gas flow

Abstract

The development of vertical axial wind turbines in Ukraine is in its infancy for many reasons: the lack of systematic theoretical and experimental studies of the aerodynamic characteristics of various schemes of wind turbines, the lack of an appropriate experimental base in technical universities, design organizations, insufficient number of available publications in foreign literature due to high competition between by monopoly firms. At present, various numerical methods are widely used to solve urgent problems of aero hydrodynamics, which are used for the approximate solution of boundary value problems in the form of differential forms of mathematical models. Their common disadvantages are the particularity and laboriousness of solutions, high requirements for computing resources, and, as a consequence, the complexity of solving optimization problems and economic feasibility. These problems can be avoided by using exact or approximate analytical dependences, which allow solving some urgent problems of studying the interaction of a viscous gas with the bearing elements of both aircraft and engineering structures. The existing methods for calculating the aerodynamic characteristics, based on the ideology of the mathematical model of the motion of an ideal medium without viscous interaction, do not correspond to the real processes and demands of practice. The article presents the ideology of determining the aerodynamic characteristics of the interacting system of solid profiles in the configuration of a vertical-axial wind turbine in a viscous gas flow. Based on generalized vector-tensor analysis, contour integral representations of solutions to the main problem of fluid and gas mechanics related to the determination of kinematic and dynamic characteristics of interaction have been constructed. In addition, the existence of a vector potential of the tensor of stresses and deformation velocities has been proved, reducing, in the simplest cases, the process of determining characteristics to integration. The limit values of these integral representations are a system of boundary integral equations, allow for elementary algorithmization, and lead to a system of linear algebraic equations having a single solution.