Journal of Mathematical Sciences, Vol. 104 , No. 6 , 2001

WIND-POWER TRANSFORMING SYSTEMS O. V. Kopeika and A. V. Tereshchenko

UDC 533.66:681.3

In this article, we give a brief account of theoretical and experimental investigations, as well as engineering designs, of different types of rotors (propeller-type rotors and Darrieus-type rotors). In numerical studies, we used the vortex lattice method . We obtained instant and averaged-in-time values of the coefficients of the centrifugal, drag, side, and head forces, as well as the value of the relative torque, the wind-power use coefficient, the configuration of the vortex trace, and the velocity field and contour lines. The results of numerical studies agree well with experimental data. Bibliography: 11 titles.

The continuous growth of the cost of power generation, limitations of oil and natural gas resources, and the necessity of taking urgent measures to protect the environment and to ensure a higher safety level of power-production facilities has made it necessary to look for alternative ways of solving the power-generation problem. One way is to use nontraditional and renewable energy sources, in particular, wind energy. At present, there have been significant practical results in wind-power engineering. Countries such as Denmark, the Netherlands, USA, Germany, and others have a developed wind-power industry. The insufficient availability of energy resources in Ukraine makes it topical to use wind energy. At the present time, the creation of wind-power generating facilities (WPGF) assumes an enhancement of existing wind turbines with the use of new technologies and the development of new wind turbines of non-traditional types. In this connection, theoretical and experimental investigations are being conducted, as well as an engineering development of elements of WPGF. One of the main problems in creating an efficient wind-power station is to develop an effective rotor that serves as the main working element. In practice, rotors with axis-symmetric flow of air through the horizontal shaft propeller unit are mainly used. A less studied class of prospective rotors are rotors of the Darrieus type, for which there is a problem of overcoming the starting torque. Below, we give a brief account of theoretical studies of propeller- and Darrieus-type rotors, as well as engineering designs of different types of rotors, air water pumps, accumulators, and energy transformers. At the present time, there exist a number of methods for the aerodynamic calculation of the wind wheel. They can be split into two main groups with regard to their simplicity and precision: impulsive [1– 4] and vortex [5–7] methods. In the impulsive methods, the calculation is based on the relation between the impulse loss of the flow that goes through the area swept by the rotor and the averaged-in-time of the total aerodynamic force that is applied to the blades and calculated from the blades, aerodynamic coefficients. The main drawback of these methods is that it is impossible to perform the calculation for an overloaded rotor, which has zones of air back-flows behind the blades, and to take into account the influence of the blades on the aerodynamic characteristics, which are determined in a stationary approximation. Vortex methods model the (linear, in most cases) nonstationary structure of streamlines for each rotor blade. Here, the solution is found, mainly, in a plane setting. These methods allow one to find instantaneous and averaged-in-time values of both the total aerodynamic characteristics of the rotor as a whole and the aerodynamic loads applied to each blade. The vortex methods give a better description of regions with inhomogeneous distribution of vorticity, which are common for blades with nonstationary movement. Moreover, the vortex methods have the following advantages: the possibility of considering only those parts of the flow where there exists a vorticity, instead of calculating parameters at each node of the grid; the possibility of accounting for a condition at infinity; a big savings in calculation time, which allows one to perform a multiparameter numerical experiment. Applying this group of methods, because of the complexity of the problem, one uses assumptions that do not allow one to account for the influence of the nonlinear vortex structure formed due to rotation of the blades. In such a mathematical model, special attention is paid to overcoming the difficulties mentioned above by using the author’s experience in solving Translated from Obchyslyuval 0 na ta Prykladna Matematyka, No. 82, 1997, pp. 50–54. Original article submitted October 10, 1996. c 2001 Plenum Publishing Corporation 1072-3374/01/1046-1631 $25.00

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Fig. 1 problems on the nonlinear aerodynamic interaction of a system of blades undergoing a nonstationary motion [8, 9]. The mathematical problem is considered in the plane (the Darrieus rotor) and spatial (the rotor with a horizontal revolution axis) settings in the framework of the model of an ideal incompressible fluid. It is assumed that the cutting edges of the blades are well profiled and vorticity gets into the flow only from the rear edge, making free vortex surfaces. The flow is assumed to be potential everywhere except for the blades and free vortex surfaces. The velocity potential in the flow region satisfies the Laplace equation. At the boundaries of the region, the boundary-value condition of the fluid not penetrating the blades is assumed to hold. On the boundary of the region, we have the following boundary-value conditions: the condition that the fluid does not flow through the blades, there is no pressure gradient on the free vortex surfaces, and the perturbation caused by rotation of the blades decreases to zero at infinity. The difficulty of the problem is that it is considered in a nonlinear setting, i.e., the location of free surfaces is not specified but is determined from an additional condition about the motion of points of the free-surface along trajectories of fluid particles. To define initial conditions in the equations of free surface dynamics, we use the Kutta hypothesis that the velocities at rear edges of the blades are finite. A solution of the problem is sought as a potential of the vortex layer that replaces the blades and free surfaces. In finding a numerical solution, the vortex layer is replaced with a family of discrete vortex singularities, and the expressions for velocities, pressures, and forces are replaced with their discrete counterparts as well as continuous processes and functions of time with their discrete analogues [7–10]. The condition that the fluid does not flow through the blades of the rotor is imposed on a family of check points placed according to the conditions of local approximation of the vortex layer. As a result, we obtain a system of linear algebraic equations and solve it for intensities of the discrete vortex singularities on the blades. The conditions on free surfaces imply that the intensities of the free vortices do not change in time and have the values obtained at the time of separation from the rear edge. The equations of motion are integrated by using the Runge–Kutta method. The pressure is found from the Cauchy–Lagrange integral. In solving the problem, we find the instantaneous and averaged-in-time values of the coefficients of the centrifugal, drag, side, and head forces, as well as the value of the relative torque, the wind-power use coefficient, the configuration of the vortex trace, and the velocity field and contour lines. Figures 1 and 2 show some examples of calculating the characteristics of the flow field behind a rotating rotor with vertical rotation axis (Fig. 1) and horizontal rotation axis for a rotor with three blades (Fig. 2). The flow field shown in Fig. 1 for a Darrieus-type rotor with three blades (instant start, initial phase) shows an intensive vortex formation in the trace behind the rotor, which confirms the necessity of using a 1632

Fig. 2 nonlinear mathematical model. Figure 2 shows the vortex picture of the flow (only longitudinal vortex singularities) observed behind a rotor with horizontal rotation axis (the diametric projection). Recently, a number of complex investigations have been carried out in order to use nonstationary aerodynamic effects for improving the operation of turbine rotor in the starting mode and if the velocity of the incoming flow is small. The mathematical model and numerical methods proposed for calculating the aerohydrodynamic characteristics of the wind wheel allow one to solve the following scientific and engineering problems: (1) to perform numerical calculations and experiments for determining instantaneous and averaged-intime values of both the aerohydrodynamic characteristics of the rotor as a whole and aerodynamic pressures applied to each blade separately; (2) to improve the shape of the blades and the layout of the existing rotors for both the traditional propeller-type and the Darrieus-type rotors; (3) to design prospective nontraditional-type rotors; (4) to numerically model a complex flow field located behind working rotors of different types and to effectively calculate statistical characteristics of the adjacent vortex field (mean, pulse, correlation, and spectral characteristics); (5) to theoretically and experimentally determine the mutual influence of operating rotors and to determine their optimal placement when designing a wind mill; (6) to make design calculations. The patterns and effects found in solving these problems can be implemented when designing fundamentally new elements of wind-power generating facilities. References 1. I. Paraschivoiu, “Aerodynamic loads and rotor performance for the Darrieus wind turbines,” AIAA Pap., No. 2582, 9 (1981). 2. M. Fallen and F. Ziegler, “Leistungsberechnung fur einem Windenergirkonverter mit vertikaler Achse,” Brennstoff-Warme-Kraft, 33, No. 2, 54–59 (1981). 3. H. McCoy and J. L. Loth, “Up- and down-wind rotor half interference model for VAWT,” AIAA Pap., No. 2579, 8 (1981). 4. V. V. Samsonov, “An improved calculation procedure for aerodynamic characteristics of wind wheels of vertical axis type, based on the impulse theory,” Promysh. Aerodin., 35, No. 3, 171–182 (1988). 5. P. R. Schatzle and W. R. McKie, “Aerodynamic interference between two Darrieus wind turbines,” in: AIAA/SERI Wind Energy Conference (1980), pp. 1–7. 6. K. P. Vashkevich and V. V. Samsonov, “A calculation procedure of aerodynamic characteristics of wind wheels of vertical axis type with the use of the vortex lattice method,” Promysh. Aerodin., 35, No. 3, 159–170 (1988). 1633

7. T. Sarpkaya, “Computational methods with vortices — the 1988 Freeman scholar lecture,” J. Fluid Eng., No. 1 (1989). 8. S. A. Dovgiˇı and O. V. Kopeika, “The influence of a hard surface on hydrodynamical characteristics of two oscillating wings if they interact nonlinearly,” Bionics, No. 24, 28–33 (1991). 9. S. A. Dovgiˇı and O. V. Kopeika, “A study of hydrodynamical characteristics of two oscillating wings in a system of ‘biplane’ type,” Bionics, No. 26 (1993). 10. S. M. Belotserkovskiˇı, V. A. Vasin, and B. E. Loktev, “On a mathematical nonlinear modeling of a nonstationary flow over a lifting propeller,” Dokl. Akad. Nauk SSSR, 240, No. 6, 1320–1323 (1978). 11. M. Van-Dyke, An Album of Fluid Motion, The Parabolic Press, Stanford, California (1982).

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