CC BY
15TECHNICAL SCIENCES
DEVELOPMENT OF A NEW DESIGN OF A VORTEX WIND TURBINE
Koshumbayev M.
doctor of technical sciences Department of heat power Engineering JSC "Kazakh agrotechnical University. S. Seifullin»
Nur-Sultan, Kazakhstan Koshumbayev A. master of technical sciences Laboratory of Alternative Energy LLP Almaty, Kazakhstan
Abstract
The relevance of the work consists in the development of a new direction of wind devices with a vertical axis of rotation without mechanisms of pointing to the wind. Based on patent search, a new design of a vortex wind turbine with a flow concentrator has been developed, for which a security document was received. The main purpose of the article was to use a mathematical model to describe the new design of the wind turbine, a distinctive feature of which is the use of the vortex effect and the concentration of the flow at its direct supply to the blades of the wind wheel. The studied design consists of a flow concentrator, inside which is a wind wheel. A vertical pipe mounted on the concentrator is used to extract the flow. Methods of research are numerical calculation and mathematical modeling of air flow within the structure of the wind turbine, which is used to determine the distribution of the velocity field. In the course of the work, a literary review and analysis of mathematical methods and existing designs of wind farms were carried out. A literature review showed that a computational experiment can be used for the method of large-eddy simulation with the solution of the averaged Navier-Stokes equations. The distribution of velocities in the concentrator, inside the structure and the discharge pipe is obtained. As a result of numerical calculations the velocity distribution in the concentrator, inside the structure and the discharge pipe is obtained. The calculation results show that the narrowing channels of the concentrator are curvilinear in plan and are described by logarithmic dependence, which creates a stable vortex motion inside the structure and the vertical pipe. Despite the uneven entrance to the wind device, a uniform vortex motion is created in the hub, which provides constant pressure on the propeller blades. Changing the direction of the external wind flow does not affect the mode of operation of the wind device: the flow enters the hub through narrowing channels and creates a vortex motion in the hub. The results of the research can be used in solving important applied problems related to the numerical simulation of turbulent flows in geometrically complex areas. The methods used to calculate turbulent flows will allow studying aerodynamic processes in wind turbines with a vortex effect. The results of numerical calculations of the flow parameters inside the hub allow us to determine the qualitative picture of the flow. Inaccuracies of the design scheme overestimate the rate of increase in compressed section. Nevertheless, the calculations allow us to find the optimal ratio of the sizes of the concentrator and to improve the parameters of the air flow inside the concentrator. Optimization of the design scheme will be considered in subsequent studies, which provide laboratory experiments on wind station models.
Keywords: Wind power plant; concentrator; vertical axis of rotation; turbulence; vortex flow; wind turbine.
Due to the rapid growth in the use of wind energy, by virtue of accepted international agreements to reduce harmful emissions from the combustion of hydrocarbons and reduce the load on the environment, there is an urgent task to increase the efficiency of wind devices through research, including mathematical modeling, with the construction of new algorithms for solving problems of gas / liquid flow in geometrically complex areas. Despite the increasing use of wind stations, the theory of describing wind devices is constantly evolving and still far from being complete. Nowadays, there are no sufficiently accurate and reliable methods for the aerodynamic calculation of wind turbines.
A distinctive feature of wind devices is their affiliation to renewable energy sources, as ecologically safe energy devices. The most common devices are three-bladed wind turbines with a horizontal axis of rotation. These cars are developed at the beginning of the last
century and were widely adopted, thanks to the simplicity. They can be installed in any remote areas where it is difficult to deliver fuel or other energy sources. Further development of wind energy is impossible without a wide development of scientific research in this direction.
Initial studies of wind devices were conditional and limited to semi-empirical hypotheses. It was assumed that the power of the unit depends on the swept area, which is a circle of radius in length of the blade. The influence of the blade profile on the speed of rotation of the wheel was not considered. One of the main provisions of this approach was that the wind direction should be perpendicular to the plane of rotation of the wheel. In this case, the wind direction is provided along the generator shaft with special mechanisms. Since the wind actually changes its direction, the theoretical description of wind turbines has not had a practical application for a long time. The most well-known analytical
solution to this problem, associated with the flow around a wind wheel, is the study of propellers of ships [1].
Intensive development of wind energy in the past two decades has allowed for detailed research using the basic principles of aerodynamics. In the last quarter of the twentieth century, the problems of wind turbine aerodynamics were summarized and presented in a number of publications [2]. The aging infrastructure of energy production in Kazakhstan and the urgent need to replace it in order to maintain acceptable levels of quality and reliability of electricity supply is an opportunity to use wind energy [3, 9]. Currently, Kazakhstan is one of the largest sources of carbon monoxide emissions in the world per capita. There is a strong dependence (approximately 85%) on the production of electricity using coal. The possibility of switching to a moderate use of coal in the production of electricity is an incentive for the development of wind energy. The very low intensity of carbon dioxide emissions from wind farms attracts investors, since developing financial mechanisms for climate change make such projects commercially feasible [4]. A deterrent factor for wind power is the dependence of wind turbines on changes in wind direction, cantilever blade tension, the destructive effect of precipitation and low temperature on blades and turbine assemblies.
Despite the experience of mathematical modeling of vortex flows, it is necessary to find out their use for calculating the new element of the vortex concentrator of the wind device. Based on a patent search, a new design of a vortex wind device with a flow concentrator has been developed. In order to determine the size and parameters of the unit, it is required a design scheme based on the basic equations of aerodynamics. The relevance of the research in this article is the search for a reliable mathematical model for the calculation of a vortex concentrator for a new wind device design.
The existing classical theories of wind turbine aerodynamics [5, 6] allow us to solve a number of problems of wind devices, including some issues of optimal design. At the same time, the classical theory of an ideal wind turbine describes the process only as a first approximation in the study of a complex physical picture of gas flow in the vicinity of the working elements of a wind unit. The most effective means of further studying the problem is the use of numerical methods for solving and analyzing a system of differential equations and the corresponding initial-boundary value problems reflecting the unsteady nature of gas flow and its interaction with the main working bodies [7]. The technique of mathematical modeling of the vortex motion of the air flow inside the structure is given in the following papers [8, 10]. In these papers, a method of aerodynamic calculation of the air flow in a vortex wind turbine is given (Fig. 1).
Getting into the curvilinear channels of the hub turbulence increases and depends on the geometry of the flow. Turbulence leads to the formation of wavelike structures that can absorb energy from the main flow. As they grow, energy due to nonlinear effects will be transmitted to other forms of disturbances, and thus, disordered pulsations arise, which are usually considered as manifestations of turbulence [11]. In a turbulent flow, large-scale structures absorb the energy of the main flow. They become strongly anisotropic, turbulent, and significantly differ from flow to flow. To the greatest extent, they determine the character of transport processes in turbulent flows [12]. In the six-channel hub the flow enters by one or two curvilinear channels. Moving along the narrowing channels, the flow is concentrated and enters tangentially into the hub. The effect of narrowing the flow allows you to increase its speed and concentrate energy on the blades of the wind wheel, which is located inside the hub. Moving along a curvilinear channel, the flow is turbulent and getting inside the hub forms a vortex turbulent flow.
Fig. 1 Three-dimensional image of the air flow hub.
The characteristic dimensions of small-scale structures are not related to the geometric characteristics of the main flow and are determined by the kinematic viscosity and the total energy flow from large-scale structures, leading to dissipation [13]. According to Kolmogorov, from the analysis of dimensions it follows that the lower limit of the linear scales of turbulence structures is defined as l = (v3/e)1/4, and the energy attributable to vortices with such dimensions (or inversely proportional to their wave numbers) varies according to the equation:
E (k) = k/3. (1)
Here E (k) is the so-called three-dimensional energy density spectrum, the parameter X is the Kolmogo-rov constant.
One of the important features of the cascade process is the transfer of energy from large-scale vortices to small-scale ones. Another distinctive feature of the cascade process is that any spatial orientation of large-scale vortices (small wavenumbers) is lost during the transition to small-scale motion (large wavenumbers), and therefore small-scale turbulence can be considered as locally isotropic [14].
Based on the universal properties of turbulence, it is possible to construct models of turbulence with scales smaller than the computational grid step. The only empirical constant appearing in such models is the Kolmogorov constant X, which characterizes the transfer of momentum. The assumed universal laws are valid only at very high Reynolds numbers, and therefore the applicability of turbulence models based on spectral properties in the inertial sub region with the scale of structures smaller than the grid step decreases with a decrease in the Reynolds number [15].
Among the known methods of numerical simulation of three-dimensional turbulent flows, it is necessary to distinguish direct numerical simulation of turbulence and the solution of the averaged Navier-Stokes equations. To use direct numerical simulation, powerful computational resources are required. On the other hand, the use of the averaged Navier-Stokes equations requires much smaller computational resources, however, the turbulence models used to close the equations do not have acceptable universality and can only be used for a narrow circle of applied problems.
The simulation of the averaged Navier-Stokes equations is complicated by the correct specification of the boundary conditions and the ambiguous determination of the parameters of the boundary layer. Such problems are eliminated by artificially setting the parameters in the boundary layer. This moment is the subject of additional research with laboratory experiments to determine the boundary conditions or dependences of changes in flow parameters in the boundary layer.
In the numerical simulation of external gas flows near the bodies of a real form, it is necessary to construct the geometry of the body being streamlined, a discrete set (grid), and approximate the original system of differential equations with their difference analog. One of the main problems is the construction of a computational grid, which takes into account the geometric
and physical features well, and allows describing the flow under investigation with the required accuracy with a limit on the number of nodes. The large eddy simulation method (LES - Large eddy simulation) is a symbiosis between direct numerical simulation and the solution of the averaged Navier-Stokes equations. In the method of large vortices, the solution of the space-averaged Navier-Stokes equations is carried out and the movement of only large vortices is considered [16], [24], [25]
The method is based on two assumptions. The first is the ability to separate the velocity field into the movement of large and small eddies, and the movement of large eddies can be calculated separately, which is connected with the universality and independence of small-scale structures of turbulent motion. The second assumption is in the possibility of approximating the contribution of nonlinear interactions between large and small eddies to analytical relations, which are called sub grid closure models.
The equations for large vortices are obtained by filtering the Navier - Stokes equations, which is the Fa-vre-type averaging method (averaging the velocity field with weighting). The result of filtering leads to the appearance of non-closed nonlinear terms (Reynolds stresses). For their closure, sub grid models are used [17].
The method of modeling large vortices is widely used in applied fluid dynamics. In 1963, the LES model was developed by Joseph Smagorinsky for solving problems related to atmospheric air flow [18]. The practical implementation of the LES model was carried out in 1970 by the researcher James W. Deardorff [19]. At present, LES models have found wide engineering applications in various industries, for example, in power engineering (combustion process) [20], acoustics [21], and in the study of atmospheric phenomena [22]. As we can see, these studies were associated with the spread of the flow in an unbounded environment. With limited spaces, a direct impact of curved channels and walls of the concentrator on the parameters of the air flow occurs. The question of these studies is how the LES model behaves when studying the movement of air in a hub.
Thus, small-scale motion is eliminated from the original Navier - Stokes equations using a filtering operation and modeled using sub grid models. The most popular and frequently used filtering functions are Gaussian, Fourier, cylindrical. In calculations using the finite volume method, filtering is carried out in a natural way: as a result of integrating differential equations representing conservation laws, by finite volumes. Among the applied sub grid models, one can distinguish the Smagorinsky model, two-point closures, dynamic models, models of one equation [23].
Spatial gas flows are calculated using various mathematical models: non-viscous gas equations, spatial laminar and turbulent boundary layer equations, a "thin" layer containing all terms of the Euler and boundary layer equations, and also based on the complete Navier - Stokes equations.
The basis of the calculation of turbulent flows are the Navier-Stokes equations [26]:
du,. d i \ 1 dp d u,
—L +-[uu, ) =---- + v-
dt dr. V 1 j
p dxt dx dx '
and the continuity equation:
du, —1 = 0, dx.
Through filtering, the averaged Navier-Stokes equations of the following form [27] are obtained:
(2)
(3)
du, d í__\ 1 dp d2u, dr.
dt dx.
(u,uj )=
-H--lu,u, )=-
dul dx,.
= 0,
■ + v-
r = u^j - uflj■
p dxi dxjdxj dxj
(4)
(5)
(6)
where xij - is a sub grid term responsible for small-scale structures that takes into account the influence of sub grid turbulence scales.
The most widely used model of subgrid turbulence is the Smagorinsky model, which is represented as [28] :
rj --
tj 3
(7)
where vT - is the turbulent viscosity, which can be represented as:
vr = Cs A2 (2SS f2 Cs - coefficient, depends on the nature of the
(8)
flow;
A = (a;.A . A^.)13 - width of the mesh filter;
V 2
Í
du
du, ^
i j
dxt dx,.
strain rate tensor.
V j ' J Air flow in the following geometrical area was considered: a six-channel hub with a diameter of 5 m and a height of 2 m. The air flow at an average speed of 5 m / s approaches the hub and enters it through two curvilinear channels into the hub. The tangential inlet ensures a flow swirl inside the hub. The swirling flow exits the concentrator through the upper central opening 1 m in diameter.
Initial and boundary conditions: dU/de = dU/de = dUg/dO = 0, UE = 0, Ug = 0 on the surface of the walls;
Ur = 0, Ug = 0, Ue = const, at the entrance to the
hub.
The characteristic parameters of the problem: Hub channels have a curvature, which is described as follows:
e = - Q lnr / G, where Q is the air flow, G is the circulation.
To solve the system of averaged Navier-Stokes equations, we used the splitting scheme for physical parameters. The increase in running operations affects the results of calculations, the accuracy of which depends on the steps in time and space. At the first stage, it is assumed that the transfer of momentum occurs only through convection and diffusion. The intermediate velocity field is found using the fractional step method when using the sweep method. At the second stage, the pressure field is determined from the found intermediate velocity field. The Poisson equation for the pressure field is solved by the Fourier method using the matrix sweep method to determine the Fourier coefficients. At the third stage, it is assumed that the transfer is carried out only due to the pressure gradient. A mathematical description of turbulent flows associated with the flow around complex shapes, which also include a vortex wind installation with an air flow concentrator, is possible using the averaged Navier-Stokes equations using the large-eddy method.
On the basis of the considered model, the following characteristics were determined: the distribution of the velocity vectors inside the curvilinear channels of the concentrator and the static pressure. The calculation was carried out at an initial air mass velocity of 5 m / s, Re = 10,000, and it should be especially noted that the initial velocity field was set along the entire plane of the inlet of the concentrator channel.
Fig. 2 shows the change in the contour of the air flow rate along the wall of the hub channel at the time of flow stabilization. The flow in the channel stabilizes when t = 0.038 s is reached, and further the flow pattern does not change.
Fig. 2 velocity contour in the channel with the flow velocity of 5 m / s and time t = 0.038 s.
Fig. 3 shows the velocity vector field. As can be seen from the figures, the flow is turbulent in nature, the speed at the site of the narrowing of the channel increases by 6 times, and in the region of the turbulence of the air flow, almost 8 times. In this case, the velocity field in a narrowing channel acquires an established character - in less than 1 second.
Fig. 3. The distribution of velocity vectors. The flow velocity is 5 m / s, the estimated time is t = 0.038 s.
The results of numerical calculations show that the LES model qualitatively describes the flow pattern, the flow is narrowed when it moves in curvilinear channels. The air flow, getting inside the hub, twists and steadily moves up the hub. At the same time, the increase in speed reaches more than eight times, which contradicts the data of practical studies in jet streams. Such errors in the calculations associated with the general assumptions and the accuracy of the design scheme. For a detailed study of the effect of all assumptions on mathematical modeling, it is necessary to present experimental data on models that will be shown in
another article on laboratory studies and optimization of the design of a vortex wind device with a flow concentrator. In this paper, we will consider the results of numerical calculations.
The results of numerical simulation of the vortex turbulent flow in a vortex six channel wind turbine are shown in Figures 4 and 5, in which the dynamics of the air flow in the concentrator are presented in the form of isolines and iso-surfaces. At the same time the wind is directed from one side and the air enters the structure through two channels of the concentrator.
Fig. 4 Speed contours when air is supplied to two channels of the concentrator at a time step of 20 s.
Fig. 5 Iso-surface velocity when air is supplied to two channels of the concentrator at a time step of 20 s.
Fig. 6 and 7 show the distribution of the velocity field in sections. Analysis of the calculated data shows that when air is supplied to two channels of the concentrator, the air flow rate at the outlet increases 10-12 times. And the stream assumes the steady flow pattern in 20 seconds. The data obtained are purely theoretical, since the initial velocity field was given uniform and strictly one direction. At the same time, getting into the channels of the hub, the air flow was directed tangen-tially to the inner surface of the hub. This ensures the vortex movement of air inside the hub. Stable rotational movement creates a favorable mode for the wind wheel. Conducting numerical simulation of the movement of
air flow in a wind turbine shows the effectiveness of the use of air flow concentrators. Due to numerical modeling, it is possible to find not only technologically simple and cheap versions of hub designs, but also it will be possible to increase the efficiency of wind power by reducing the costs of building and maintaining wind power plants by an order of magnitude.
Figures 6 and 7 show the distribution of the velocity field in sections. As can be seen from the figures, a steady vortex flow is formed in the concentrator. Mathematical modeling allows you to create a plausible simulation of the main processes in the hub.
Fig. 6 Distribution of the velocity vector in the transverse and vertical sections when air is supplied to the two
channels of the concentrator at a time step of 20 s.
Fig. 7 Distribution of the velocity vector in a vertical section when air is supplied to two channels of the concentrator at a time step of 20 s.
Analysis of the calculated data shows that when air is supplied to two channels of the concentrator, the air flow rate at the outlet increases by 10-12 times, while the flow assumes the steady flow pattern in 20 seconds. The data obtained are purely theoretical, since the initial velocity field was given uniform and strictly one direction. At the same time, getting into the channels of the hub, the air flow was directed tangentially to the inner surface of the hub. This ensures the vortex movement of air inside the hub. Stable rotational movement creates a favorable mode for the wind wheel. Conducting a numerical simulation of the movement of air flow in a wind turbine shows the effectiveness of the use of air flow concentrators. Thanks to numerical modeling, it is possible to find not only technologically simple and cheap versions of hub designs, but also it will be possible to increase the efficiency of wind power by reducing the cost of building and maintaining wind power plants by an order of magnitude.
Numerical simulation of the vortex motion of air in a turbine is associated with the development of a new design of a wind turbine [29, 30]. The novelty of the design associated with realization of three effects: concentration of the flow on the wind turbine blade, the
formation of a vortex motion inside the hub and air removal using a vent. Calculations showed that the hub allows you to increase the air flow rate by 8-10 times. It did not consider the motion of the wind wheel. Similar studies [31] also show a steady vortex motion created by the guide vanes. The vortex wake control in a turbulent flow is considered in [32]. Management is performed using the guide walls. Thus, the use of special devices for swirling the flow allows you to create a stable and controlled movement of air in special devices. Similar studies were carried out by the German company Turbine Energy AG [33], the Georgia Solar Technology Institute [34] was created at the Georgia Institute of Technology (Atlanta, USA), which uses thermal air flow. In Russia, wind turbines with a vertical axis of rotation are considered, in which the Darya rotor is used in the form of various modifications, such as an orthogonal device, as well as a wind wheel with vertical and helical blades [35]. Despite the existing structures with vortex flow, their use together with the hub in wind aggregates is not observed. This allows us to assert that we are investigating a new technical process that could lead to the creation of new wind stations. Using mathematical modeling, you can determine the size of the hub and the diameter of the discharge pipe.
The calculations allowed us to reveal the optimal ratios of the geometric parameters of the concentrator and the dependence of the flow parameters on them. The effectiveness of the application of this mathematical model can be found out when conducting additional experimental research on the model of a vortex wind device. Additional tasks in conducting experiments include: such issues as increasing the capacity of the structure are not considered. Calculations have shown that it is very important to know the resistance of the concentrator in order to improve the efficiency of the wind generator. The considered mathematical model also does not calculate the optimal profiles of the propeller blades. This task involves the use of a different design scheme for a flow around curved plate.
The objects under study do not address such issues as the capacity of the structure. It is very important to know the resistance of the concentrator in order to improve the efficiency of the wind generator. For vortex wind devices, blade profiles are not optimized.
The problem of used wind turbines with a horizontal axis of rotation and a three-blade wind wheel is associated with its constant winding, variable flow pressure on the blades, noise effects of rotating blades on the surrounding, cantilever stress on the blades, der
B
I
A
structive effects of precipitation and negative temperature on the blade material and turbine nodes, placing them outside the settlements, which leads to additional costs for power lines.
Virtually all of the listed aspects of the problem are resolved with the use of wind turbines with a vertical axis of rotation, which include a hub and a vertical discharge pipe. In this case, there is no need for turbine winding mechanisms to the wind, which reduces the consumption of materials and the number of turbine nodes. Rotating elements are located inside the hub, so there is no noise and precipitation does not fall on the propeller blades. Since there is no cantilever voltage in the mounts of the blades and given that precipitation and temperature do not affect the strength of the material and the operating mode of the rotating elements of the turbine, the blades can be made of light and thin materials, which reduces the weight of the wind wheel. The hub accelerates the air flow and creates a vortex movement inside itself, which ensures a steady flow and constant pressure of the flow on the propeller blades. Exhaust air is removed from the concentrator by means of a discharge pipe due to vertical thrust. The results obtained allowed us to optimize the design [30] (Fig. 8).
BB
A A
12
\
V yO \XJ !■•'
Figure 8 Vortex wind turbine.
Vortex wind turbine, contains a vertical axis of rotation, a wind wheel with a modified Darya rotor, a chimney with an air flow concentrator, represented as a cone with a concave surface placed inside the tent, interconnected by curvilinear baffles forming tapering air channels tangentially directed from the periphery of the tent to the vertical chimney and wind turbine blades. At the same time, the wind wheel (in the form of the Darier rotor) blades are combined with the air flow hub, so that the outlet of the channels is profiled on the wind wheel blade, while the current generator with the vertical axis of rotation is located in the lower part of the hub, on the same axis of rotation of the wind wheel. The outer parts of the exhaust pipe and tent are painted in dark color.
Calculations have shown that the use of a cone from the bottom of the hub improves the process of rotation of the flow and gives it an additional vertical velocity component. In this case, the rotation increases and the flow rushes into the discharge vertical pipe. The mathematical model allows to determine the distribution of speeds inside the hub depending on the value of the speed of the external air flow. Thus, the numerical simulation of the movement of the air flow showed the
qualitative characteristics of the velocity field inside the hub of the vortex wind station. At the same time, calculations show overestimated speed values in a compressed section. This is due to the error of the design scheme and the problem of specifying the boundary conditions in curved channels and walls of the hub.
The developed mathematical model can serve as a program for determining the preliminary geometric dimensions of a concentrator of a vortex wind device with a flow concentrator and air flow parameters inside the concentrator.
References
1. Abramovich G.N., Theory of turbulent jets. Fiz-MatGiz, 1960. 716 p.
2. Gorbunov A.A. The use of simulation in the design of additional aerodynamic surfaces of the wing of the aircraft. Science, Gorbunov Alexander Alekseevich Orenburg State University. 2013. 17 p.
3. Strategy "Kazakhstan - 2050", http://econ-omy.gov.kz/ru/pages/strategiya-kazahstan-2050.
4. Koshumbayev M.B., Kvasov P.A. Energy saving and renewable energy. Republican scientific-theoretical journal "Science and New Technologies", № 2,
2015, Bishkek. P. 21 - 26.
5. Wilson R.E. Wind turbine aerodynamics. J. of Ind. Aerod. 1980, v.5. pp.357-372.
6. Preuss R.O., Sussiu E.O., Morino L. Potential Aerodynamic analysis of horizontal - axis windmills. AIAA Paper. 1977, No. 132. pp. 1132 - 1140.
7. M. Koshumbayev Improving the safety of hydraulic structures in emergency situations by improving the design of the spillways. Avtoref.dis.doct.tech.sci-ence 05.05.2008. Marat Bulatovich Koshumbaev. Kazakh National Technical University named after KI Satpayev. Almaty, 2008. - 37p.
8. Koshumbayev M.B., Myrzakulov B.K., Toleu-khanova A.B. Theoretical and experimental studies on the development of a new design of a wind power plant with a flow concentrator // Joint High School of International Symposium "Combustion and Plasmochemis-try", RSE "Institute of Combustion Problems", September 16-18, 2015, Almaty, pp. 160-165.
9. Koshumbayev M.B. Strategic Goals and Objectives of the Energy Sector. Yearbook of the Institute of Scientific Information of the Russian Academy of Sciences, Moscow. Issue 11, part 2, 2016. pp.369-370.
10. Koshumbayev M.B., Kvasov P.A., Koshumbayev A.M., Eijan A.A. The wind power plant with a flow concentrator. 3rd International Conference on "Innovative Trends in Multidisciplinary Academic Research" (ITMAR - 2016), October 20-21, 2016 Istanbul, Turkey, pp.49-50.
11. Koshumbayev M.B. Development of the fundamental and ramjet turbine and the ramjet turbine, "Fundamental and Applied Physics", 10-11 November
2016, Tashkent, Uzbekistan, pp. 251 - 253.
12. Dolatabadi A., Mohammadi-ivatloo B., Abapour M. Optimal Stochastic Design of Wind Integrated Energy Hub. // IEEE Transactions on Industrial Informatics. 2017. Vol. 13. No. 5. P. 2379 - 2388.
13. Rivera M.K. The Inverse Energy Cascade of Two-Dimensional Turbulence. University of Pittsburgh, 2000. 160 p.
14. Soifer V.A., Korotkova O., Khonina S.N., Shchepakina E.A. Vortex beams in turbulent media: review. Computer Optics, 2016, Vol. 40 (5), pp. 605-624.
15. Bruk M.A., Zhikharev E.N., Rogozhin A.E., Streltsov D.R., Kalnov V.A., Averkin S.N., Spirin A.V. Formation of micro and nanostructures with a well-rounded profile. Microelectronic Eng. 2016. pp. 92-96.
16. Spalart P. R. Detached-eddy simulation. Annual Review of Fluid Mechanics. 2009. pp. 181-202.
17. Fox R. O. Large-eddy-simulation tools for multiphase flows. Annual Review of Fluid Mechanics. 2012. pp. 47-76.
18. Smagorinsky J. General Circulation Experiments with the Primitive Equations. Monthly Weather Review. 1963. pp. 99-164.
19. Deardorff J. A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. Journal of Fluid Mechanics. 1970. pp. 453480.
20. Pitsch H. Large-Eddy Simulation of Turbulent Combustion. Annual Review of Fluid Mechanics.
2006. pp. 453-482.
21. Wagner C., Huttl T., Sagaut P. Large-Eddy simulation foracoustics. Large-Eddy Simulation for Acoustics. 2007. pp. 505-507.
22. Sullivan P.P., McWilliams J.C., Moeng C.H. A subgrid-scale model for large-eddy simulation of planetary boundary-layer flows. Boundary-Layer Meteorology. Kluwer Academic Publishers. 1994. pp. 247-276.
23. Leveque E., Toschi F., Shao L., Bertoglio J.-P. Shear-Improved Smagorinsky Model for Large-Eddy Simulation of Wall-Bounded Turbulent Flows. J. Fluid Mech. No. 6. 2006. pp. 1-11.
24. Ershkov S.V. Non-Stationary Incompressible Flow, Stokes Equations for 3D, International Journal of Fluid Mechanics Research. June 2015. pp. 206-213.
25. Schneiders J. F.G. Time-Supersampling 3D-PIV Measurements by Vortex-in-Cell Simulation. Master of Science Thesis. Aerospace Engineering, Delft University of Technology. 2014. 112 p.
26. Alfonsi G. Reynolds-Averaged Navier-Stokes Equations for Turbulence Modeling. Applied Mechanics Reviews. July 2009. Vol. 62, pp. 040802-1-04080220.
27. Chemin J.-Y., Gallagher, I., Paicu M. Global equations. of math 173 (2011), no. 2. 2011. pp. 9831012.
28. Sawada O. On-site solvability of the Navier -Stokes equations. Adv. Differential Equations. 8 (4) (2003), 2003. pp. 385-412.
29. Koshumbayev M.B. Provisional Patent of Kazakhstan No. 20243 "Vetroagregat", bull. 11 dated 11.17.2008.
30. Koshumbayev M.B., Myrzakulov B.K., Koshumbayev A.M. Patent of Kazakhstan No. 2291 "Vortex wind power unit", bull. № 14 dated 07.31.2017.
31. Zhang Y., Bao W., Du Q. Numerical simulation of the vortex dynamics in the Ginzburg-Landau-Schrodinger equation. Jnl of Applied Mathematics,
2007, vol. 18, pp. 607-630.
33. Eldredge J. D. Numerical simulation of the fluid body with the vortex particle method. J. Comput. Physics. Vol. 221, 2007, pp. 626-648. DOI: 10.1016 / j.jcp.2006.06.038
34. Vertical wind turbines TURBINA Energy / http://syenergy.com.ua/vetrogeneratory/317- wind generator- turbina-te20.html
35. Dansie M. Solar Vortex / https://revolution-green.com/solar-vortex/, Apr 28, 2015.
