Effect of the tip speed ratio on the wake characteristics of wind turbines using LBM‐LES

In this paper, the wake characteristics of Zell 2000 wind turbine under different tip velocity ratios are studied by using the lattice Boltzmann method and large eddy simulation. The adaptive mesh refinement method is performed to capture the fine flow structure and wake characteristics development. In this paper, we mainly focus on the effect of the tip speed ratio on the flow structure and unsteady characteristics of wind turbine wake. The three‐dimensional flow vorticity structures, the section vorticity diagram, the pressure fluctuation of wake and the lift coefficient of wind turbine wake are utilized to explore the effect of the tip speed ratio on the unsteady physics mechanism of wind turbine wake. With the increase of the tip speed ratio, the distance between two adjacent vortex rings along the axial direction gradually decreases, as the position of the broken vortex circles gradually approaches the center of the blade, separated vortexes are rapidly generated, and the coherent structure appears closer to the wind turbine. A relationship is established between the tip speed ratios and the positions of the broken vortex circles. It is further found that the dominant frequency amplitude gradually increases with the increase of tip speed ratio and the pressure amplitude spectra of vortex increases with the decrease of the distance between the wake and the center of the blade axis. The above series of studies can provide significant physical insight into deep understanding the influence of the tip speed ratio on the wake characteristics of wind turbines.


| INTRODUCTION
Wind energy and other renewable energy sources are expected to grow significantly in the coming decades and play a key role in mitigating climate change and achieving energy sustainability.With the growth of wind energy demand, the wind energy industry is developing toward large-scale design, which is expected to operate reliably under various environmental conditions.2][3][4][5][6] Due to the complex aerodynamic characteristics of wind turbine rotor, the numerical simulation of wind turbines still is a great challenge and a hot topic.
The wake model based on computational fluid dynamics (CFD) solves the Reynolds-averaged Navier-Stokes (RANS) equation 1 or large eddy simulation (LES) 2 controlling the whole flow field.With the rapid growth of computing level, the LES method is widely applied in the CFD of wind energy engineering.Sedaghatizadeh et al. 3 simulated wind turbine wakes using the LES approach and investigated the detailed information about the flow field as well as the wake development.In the above-mentioned studies considered the full wind turbine, one of the most challenging issues is the mesh generation and is to deal with the rotating rotor in the computational domain.In recent years, the adaptive mesh refinement (AMR) method is proposed for numerical simulation of wake flow. 4Zeoli et al. 4 used the AMR method to simulate wind turbine wake.The grid is adaptively refined in a region of strong vortices.The mesh is refined in the region closely related to the vortex structure, so that the wake dynamics can be properly captured.
The wake of wind turbines by changing the operating conditions of wind turbines and their impact on wind turbine power generation are widely studied in engineering application. 5,6Yin et al. 5 studied the method of reducing the wake loss of wind farm through the startup and stop scheduling of wind turbines based on Jensen wake models, and obtained the influence of wake recovery coefficients on unit power generation.Yang et al. 6 conducted the user-defined function modeling for normal and extreme wind conditions and studied the influence of dynamic incoming flow and tower shadow effect on the downstream wake characteristics of wind turbine.Yang also studied and evaluated the torque and power of the tower under the influence of tower shadow effect of wind turbine and the service life of wind turbine.Wen et al. studied the influence of blade number on the near-wake dynamic characteristics of wind turbines and designed an airfoil with high lift coefficient for SHAWT model operating at low Reynolds number. 7he tip speed ratio is a very important parameter to describe the characteristics of wind turbines. 8,9The tip speed ratio is defined as where Ω denotes the angular speed of the rotor and R D = /2, V  is the incoming flow velocity. 8,9The tip speed ratio is an important parameter in the study of wind turbines.When the tip speed ratio is low, the ratio of maximum lift to drag coefficient increases exponentially.It decreases linearly when the tip speed ratio is high.Okulo et al. 10 explored the development characteristics of wind turbine wake by controlling the incoming flow velocity to change the tip speed ratio.In this paper, the tip speed ratio is implemented to control the change of wind turbine angular velocity.][10] However, the tip speed ratio markedly affects the unsteady flow structure and mechanics of wind turbine wake.The unsteady flow structure and mechanics of wake plays a dominant role to study the aerodynamic performance and noise of wind turbine.How do we reveal the effect of tip speed ratios on the wake structure of wind turbine?In this paper, the AMR method is used to encrypt the grid and the lattice Boltzmann method (LBM) and LES (LBM-LES) is implemented to explore the effect of the tip speed ratio on the wake characteristics of wind turbines.The unsteady flow structure and characteristics are accurately captured in the near flow and far field.In this work, we extend the previous work to deeply explore effect of the tip speed ratio on the physical mechanism of the threedimensional (3D) vortex structure of wake in the near and far fields, 2D vorticity characteristics of the tip vortex region, behind wind turbine nacelle and behind the tower in the near and far fields.The effects of the tip speed ratio on the interaction mechanism among the tip vortex region, behind wind turbine nacelle and behind the tower are deeply revealed in the near and far fields.The pressure fluctuation spectrum of wake in the near and far fields and the lift coefficient of wind turbine wake are further explored to deeply understand the unsteady mechanism of wind turbine wake.
The main framework of this paper is as follows.The LBM-LES and the AMR are described in Section 2. Section 3 mainly displays the model of wind turbine and computational domain.In Section 4, the unsteady 3D and 2D flow characteristics of wind turbines wake at different tip speed ratios.Finally, some constructive conclusions are presented.

| LBM-LES
2][13][14] The D3Q19 scheme is used for numerical simulations in this paper.Figure 1 represents the schematic diagram of the D3Q19 model.
The evolution equation of mesoscopic approach is 15 where f i is the particle distribution function, f i eq is the equilibrium distribution function, x is the coordinate of the particle, c i are the direction vectors, and τ is the relaxation time, respectively.The equilibrium distribution function is given as where ρ is the mass density, u is the macroscopic velocity, c s is the speed of sound, and w i is the weight coefficients, respectively.The parameters in D3Q19 are given as 15 The relation between the relaxation time (τ) and the kinematic viscosity (ν) is expressed as The density and momentum can be calculated by the following equation.
Using the Chapman-Enskog expansion, which can obtain the Navier-Stokes equation for incompressible flow as follows 15,16 : 8][19][20] The LBM-LES is implemented in this paper.The idea of LES is mainly added to LBM in the LBM-LES.With the increase of Reynolds number (Re), the viscosity decreases, the maximum velocity gradient increasingly affects the convergence, and the over dense mesh causes the compressibility error.The idea of LES is introduced here, The turbulent viscosity coefficient is calculated by WALL model.The kinematic viscosity is defined as 21 where ν σ is laminar viscosity coefficient and ν t denotes the turbulent viscosity coefficient.
where U denotes the incoming flow velocity and D is the diameter of the wind turbine, respectively.

| Adaptive mesh refinement
Vortex shedding occurs in the wake of wind turbines.
The unsteady flow and characteristics of wind turbine wake can be accurately captured by the AMR method.Compared with fine mesh, the AMR method requires less manual operation, and it can selectively generate rough mesh in areas with low-turbulence.This method also has lower computational expense. 22eoli et al. have proved that it is feasible to encrypt the grid of wind turbine simulation domain by AMR method. 4The AMR method can capture the wake and refine the spiral vortex structure at the blade root and tip. 23The AMR method is implemented to accurately capture the wake characteristics of wind turbine.The work of this paper is to study the development characteristics of wind turbine wake, which effectively assures the accuracy of numerical simulation.
Figure 2 shows the blade and grid distribution in the numerical simulation of wind turbines.It can be seen from Figure 2 that the AMR method encrypts the grid with the development of wake.The grids are fined in the wake area, while the girds are rough in other computational areas.It is not only the multiple resolution mesh, but also the multi-time step integration should be considered.The ∆ ∆ are chosen as the grid spacing of the fine and coarse mesh.The time intervals of multi-time step integration in the fine and coarse mesh, t f ∆ and t c ∆ , are, respectively, determined as 23 The fine mesh is simulated at the time evolution of two subtime steps per a time step of the coarse mesh.
The relaxation parameters ω αβγ should be different along with the multi-time step.It is worth noting that the following equations (Equations 13 and 14) are discussed with the relaxation parameter of BGK collision，but we can apply the same argument also into the collision model.The relaxation parameters satisfying Equation (12) keeps the continuity of the kinetic viscosity (i.e., v v = c f ), and are derived as f c (13)   Moreover, to satisfy the continuity of the shear tensor, the distribution functions are interpolated from coarser mesh to finer mesh and vice versa are modified as 23 ( ) where the superscripts c f  and f c  denote the modification from coarse to fine and vice versa, respectively.The equilibrium function f ijk eq remains unchanged regardless of the mesh resolution, and the macroscopic variables should be conserved within the modifications.
where τ c and τ f in Equations ( 19) and ( 20) should be replaced into ω 1 in Equation (17).The modifications are done on the velocity space ( f ijk ) even when the collision process is modeled by the cumulated model.

| MODEL OF WIND TURBINE AND COMPUTATIONAL DOMAIN
The model of wind turbine mainly includes the rotor (blade and hub), nacelle and wind tower.The height of wind turbine tower is 12 m and the length of rotor blade is 5 m.It is tested in the NASA ames of wind tunnel.The model of wind turbine is one of the most comprehensive experiments, which is carried out on a full-scale wind turbine. 24Figure 3 displays the model of wind turbine and computational domain.As shown in Figure 3, the computational domain is 200 m × 50 m × 50 m corresponding X-direction coordinate, Y-direction coordinate Mesh in the computing domain.
CUI ET AL.
| 1641 and Z-direction coordinate in this paper, respectively.In all numerical simulations, the wind turbine is placed 50 m away from the air inlet.The inlet speed is set to 7 m/s, the maximum size of the grid is set to 1.0 m in the computational domain, and the maximum adaptation level is set to 5, respectively.The adaptive fine grids (1.0 m/25 = 0.03125 m) are distributed on the surface of wind turbine and area of complex wake.The tip speed ratios of λ = 4, 5, 6, 7, 8 are implemented to study the complex flow mechanism of the wind turbine wake.

| NUMERICAL RESULTS AND DISCUSSIONS
In this section, the 3D vortex structure of wake, the 2D vorticity characteristics of the tip vortex region, behind wind turbine nacelle and behind the tower will be explored in the near and far fields.The pressure fluctuation spectrum of wake and the lift coefficient of wind turbine wake will be studied with increasing tip speed ratio in the near and far fields.

| 3D wake vorticity structure at different tip speed ratios
In general, the wind turbine wake is divided into the tip vortex, vortex behind the wind turbine nacelle and vortex behind the wind turbine tower.In this paper, the wake characteristics of wind turbines are studied at different tip speed ratios of wind turbines.Figure 4 illustrates the comparison of 3D wake flow structure around wind turbine between the previous numerical method and present LBM-LES at λ = 5.As illustrated in Figure 4, one can see that the tip vortex breaks up rapidly after five vortex circles in previous work. 25The finer wake vortex is well captured due to implementing the AMR method of LBM-LES in this paper.The previous studies of Hsu et al., 25 is shown in Figure 4C, and the 3D wake flow structure around wind turbines is not well captured due to numerical dissipation.Nevertheless, the very fine flow vortex structure is accurately captured by LBM-LES, which is well consistent with experimental results at the same working condition. 24igure 4A displays the vortex structures of wind turbine by the finite volume method. 24Figure 4B illustrates the comparison of vortex structure between numerical and experimental results at the same inflow condition (v = 7 m/s).As shown in Figure 4C, the vortex structures of wind tunnel by LBM-LES are effectively captured.However, a lot of vortex structures are numerically dissipated by finite volume method.The eight-circle vortex structures of wind tunnel are clearly captured by LBM-LES, which is well consistent with the experimental visualization wake flow of the wind tunnel.The LBM-LES has very low numerical dissipation and the vortex structures of wind tunnel are effectively captured.
In the following subsection, the tip speed ratios of λ = 4, 5, 6, 7, 8 are performed to explore the effect of the tip speed ratio on flow vortex structure of wind turbine.In this paper, the isosurface value of 3D wake flow structure of wind turbine vorticity diagram is 7.4.Figure 5 shows the 3D wake flow structure of wind turbines at different tip speed ratios.Figure 5A displays the 3D flow vortex structure of the wind turbine at λ = 4.As displayed in Figure 5A, it can be seen that 11 relatively whole flow structures of tip vortex circle appear in the near field wake of wind turbine.However, due to the influence of tower, each vortex circle is responsively interrupted along the lower edge region of the vortex and vertical tower.The level of interrupting tip vortex circle increases with the evolution of flow in the near field wake of wind turbine.Moreover, it is also obtained that all flow structures of tip vortex circle are broken and gradually disappeared in the far field of wake of wind turbine.It is surprisingly found that the vorticity inside the tip vortex circle is higher than that of tip vortex circle outside.Figure 5B demonstrates the 3D flow vortex structure of wind turbine at λ = 5.In Figure 5B, the 12 relatively whole flow structures of tip vortex circle can be clearly demonstrated in the near field.And the pitch of tip vortex decreases.With the influence of wind turbine tower, these vortex circles are interrupted along the lower edge region of the vortex and vertical tower.In the near field wake of wind turbine, the influence of tower on the interruption of blade tip vortex gradually is enhanced.It is also demonstrated that all flow structures of tip vortex circle are broken and gradually decomposed in the far field of wake of wind turbine.Figure 5C illustrates the 3D flow vortex structure of wind turbine at λ = 6.In Figure 5C, the 13 relatively whole flow structures of tip vortex circle can be clearly illustrated in the near field.Spiral tail vortex appears behind the nacelle.Due to the influence of tower, the tip vortex is responsively interrupted along the lower edge region of the vortex.In the far field of wake of wind turbine, the tip vortex begins to decompose and disappear.Figure 5D illustrates the 3D flow vortex structure of wind turbine at λ = 7.In Figure 5D, 14 relatively whole flow structures of tip vortex circle can be clearly illustrated in the near field.It is also obtained that spiral tail vortex appears behind the nacelle and disappears quickly.Figure 5E demonstrates the 3D flow vortex structure of wind turbine at λ = 8.In Figure 5E, the 15 relatively whole flow structures of the tip vortex circle can be clearly demonstrated in the near field.By increasing the tip speed ratio, the pitch of tip vortex gradually decreases.It is also obtained that all flow structures of the tip vortex circle begin to mix and gradually broken in the far field of wake of wind turbine.Because the pitch of tip vortex is Comparison of three-dimensional wake flow structure around wind turbine between previous numerical method and present LBM-LES.
short, the mixing phenomenon occurs in the adjacent vortex circles.
In Figure 5, the tip vortex is a spiral continuous curve.And the wake is greatly affected by the blades.In the blade tip area, it is further found that the vortex shedding of the tower is obviously influenced by the blade, which shows that the wake structure of the tower have obvious differences between the upper and lower parts.It is noted in Figure 5 that the wake at the bottom of the tower is not impacted by the blade, and the vortex street is well preserved at the lower part.It is further found that the distance between two adjacent vortex rings along the axial direction gradually decreases with increasing tip speed ratio.In this paper, the L D / is used to study the relationship between tip speed ratios and the positions of the broken vortex circles.L is the distance from wind turbine to the positions of the broken vortex circles and D is the diameter of wind turbine rotor, respectively.
The dimensionless ratio between the position of the broken vortex circle and the diameter of wind turbine rotor is built to quantitatively obtain the influence mechanism of tip speed ratios on separated vortex with increasing tip speed ratio.Figure 6 displays the relationship between tip speed ratios and the positions of the broken vortex circle.A quantitative relation is obtained between the tip speed ratios and the positions of the broken vortex circles.As displayed in Figure 6, it is clearly observed that the vortex circles are broken up at about 4D distances from the wind turbine.Surprisingly, it is further found that the position of the broken vortex circles gradually decreases with the increase of the tip speed ratio, which is helpful to deeply understand the influence mechanism of tip speed ratio on the wake flow characteristics of wind turbine.The secondary vortex generated by the breaking of vortex circle will be further dissipated, which has certain reference significance for the study of wind field layout of wind turbine.
The vortex behind the tower appears a phenomenon similar to Karman vortex street, but the interaction between the rotor and the vortex causes the vortex to disorderly fall off.In the near wake region, the tip vortex and the vortex behind wind turbine nacelle are separated from each other, and stretch and roll up at the action of axial velocity.In the far wake vortex, the rotor interacts with the wake further, which makes the wake vortex further to be unstable.It is obtained that the tip vortex interval behind the wind turbine also increases with the increase of the tip speed ratio.In addition, the fluid vortices are mainly concentrated at the blade tip and the wind turbine nacelle with increasing tip speed ratio.The tip vortex is the main part of the wind turbine wake.The vortex pitch gradually decreases with increasing the tip speed ratio.

| Effect of tip-speed ratio on wake at different sections
To further study the wind turbine wake, the multiple cross-sections are made in the flow field to observe the wake changes and further study the wind turbine wake characteristics.Figure 7 shows the section selected for wind turbine wake vorticity analysis.
Figure 8 displays the vorticity field is made on the section plane at the XOY section to investigate the velocity evolution in the wake region behind the wind turbine.It can be seen from Figure 8 that the wind turbine wake is obviously divided into blade tip vortex, the vortex behind the wind turbine nacelle and vortex behind the tower in the near field.In the laminar flow region, the three vortices are obviously different and developed backward.With the increase of turbulence intensity, the tip vortex intensity and the pitch of tip vortex gradually decreases.The coherent structure gradually becomes unstable and diffuses downstream by strong disturbance.Besides, the trajectory becomes messy and the radial mixing between vortex systems is enhanced.It is surprisingly captured that with the increasing of λ, the tip vortex of the wake begins to fluctuate up and down at 2D, and the vortex behind the wind turbine rotor interacts and integrates at about 4D.
Figure 8A displays the XOY section flow vortex structure of wind turbine at λ = 4.As illustrated in Figure 8A, it is clearly seen that the connected vortex between the tip vortex and the behind nacelle, which is called as attached vortex.In the near field, the tip vortex, vortex behind the nacelle and attached vortex are clear and separated.With the development of the wake, the attached vortex disappears and the vorticity of vortex behind the nacelle decreases rapidly around the 2D distance from the wind turbine.Besides, the vortex behind the wind turbine rotor interacts and integrates at about 4D. Figure 8B illustrates the XOY section flow vortex structure of wind turbine at λ = 5.In the near field, the evolution intensity of the tip vortex, the vortex behind the nacelle and the vortex behind the tower develop are significantly lower behind the wind turbine.The tip vortex is closely related to the separation vortex from the 6D distance from the wind turbine.Figure 8C displays the XOYY section flow vortex structure of wind turbine at λ = 6.As shown in Figure 8C, it is seen that the vortex behind the tower and the vortex behind the nacelle begin to gradually mix at 4D from the wind turbine.Figure 8D illustrates the XOY section flow vortex structure of wind turbine at λ = 7.The tip vortex begins to fluctuate obviously after 4D distance from the wind turbine.More attached vortices are retained in the near field.The vortex behind the tower and the vortex behind the nacelle begin to mix at 4D from the wind turbine and completely develop to be mixed at 6D of the wind turbine.Figure 8E demonstrates the XOY section flow vortex structure of wind turbine at λ = 8. Figure 8E demonstrates the blade tip vortex has obvious fragmentation and dissipation from the 2D distance of the wind turbine.The vortex behind the nacelle and the vortex behind the tower also interact and begin to mix at a close distance.Due to the short pitch of the tip vortex, the tip vortex is closely related to the separation vortex from the 2D distance of the wind turbine.In the far field, the tip vortex, vortex behind the wind turbine nacelle and vortex behind the wind turbine tower mutually mix and gradually develop coherent structures.
Figure 9 illustrates the vorticity field is made on the section plane at the XOZ section.The velocity evolution is clearly captured in the wake region behind the wind | 1645 turbine.It can be seen from the Figure 9 that the blade tip vortex and the vortex behind the wind turbine nacelle tend to be disordered in the laminar flow region and transition region, and the blade tip vortex appears disturbance.However, the interaction between the tip vortex and the vortex behind the wind turbine nacelle is not obvious.It is clearly obtained that the tip vortex and the wake vortex behind the wind turbine nacelle begin to contact and mix with each other, and continue to develop into the coherent structure in the turbulent region.
Figure 9A displays the XOZ section flow vortex structure of the wind turbine at λ = 4.In Figure 9A, the tip vortex and the vortex behind the wind turbine nacelle develop backward independently in the near field.The attached vortices are formed between tip vortices and the vortex behind the nacelle in the near F I G U R E 8 Vorticity field on the section plane at XOY section.field. 26It is clearly observed that the attached vortex rapidly disappears around the 2D distance from the wind turbine with the development of the wake.In the far field, the blade tip vortex is evidently shredded and dissipated.Nevertheless, the tip vortex and the vortex behind the wind turbine nacelle begin to interact and mix. Figure 9B demonstrates the XOZ section flow vortex structure of wind turbine at λ = 5.In Figure 9B, we can see that the tip vortex emerges obviously fluctuation and produces separation vortex at a closer distance than λ = 4 case.The tip vortex begins to obviously fluctuate after the 4D distance of the wind turbine.The blade tip vortex and F I G U R E 9 Vorticity field on the section plane at XOZ section.
the vortex behind the nacelle completely mix at about 6D distances behind the wind turbine.Figure 9C illustrates the XOZ section flow vortex structure of wind turbine at λ = 6.It is obtained that with the increase of λ, the separated vortex is generated rapidly and the coherent structure of wind turbine wake appears earlier behind the wind turbine.The destruction and dissipation of the vortices are increasingly enhanced with the increase of λ. Figure 9D displays the XOZ section flow vortex structure of wind turbine at λ = 7.The increasingly attached vortices are retained in the near field.The vortex behind the tower and the vortex behind the nacelle begin to mix at 4D from the wind turbine and completely mix at 6D from the wind turbine.Figure 9E demonstrates the XOZ section flow vortex structure of wind turbine at λ = 8.Due to the increase of angular velocity, the increasingly attached vortices are retained in the near field.Meanwhile, the decrease of tip vortex pitch makes the adjacent vortex circles to continuously develop towards the back.The blade tip vortex begins to appear separation vortex at about 2D distance, and gradually interacts with the vortex behind the nacelle.
The detailed flow information characteristics of the vortex behind the wind turbine are accurately seized at different heights by the influence of the rotor.In this paper, the vorticity field is made on the section plane at different heights (l R / = 1.02, 1.04, 1.08, 1.16, 1.24) to investigate the wake evolution process behind the tower, where l is the distance from section to rotor center, and R is the radius of wind turbine.
Figure 10A demonstrates the section flow vortex structure of wind turbine at l/R = 1.02.Illustrated in Figure 10A, it is seen that the transverse separation vortices occur around the tower, which is due to the lateral movement of the fluid caused by the rotating and sweeping of the rotor.The separation vortex is generated at each sweep of the rotor and develops backward with the wake evolution, which also enhances the degree of vortex turbulence behind the wind turbine tower.Figure 10B displays the section flow vortex structure of the wind turbine at l/R = 1.04.In Figure 10B, the wake has traces swept by the wind turbine every time, but the lateral separation vortex is dissipated outward less than that in Figure 10A.The transverse separation vortices around the wind turbine tower are significantly decreased.Figure 10C illustrates the section flow vortex structure of wind turbine at l/R = 1.08.It is obtained that the horizontal fluctuation of the vortex around wind turbine tower is obviously weakened in Figure 10C.The vortex structure behind the tower is consistent with that behind the wind turbine nacelle with the development of wake. Figure 10D demonstrates the section flow vortex structure of wind turbine at l/R = 1.16.The effect of the rotor gradually decreases.The transverse separation vortex caused by transverse force obviously decreases.Figure 10E displays the section flow vortex structure of the wind turbine at l/R = 1.24.Compared with other wind turbine wake sections, the transverse separation vortex of the wake is not obvious in Figure 10E.As shown in Figure 10E, it is clearly observed that the wind turbine wake is greatly affected by the tower and the wind turbine wake behind the tower tends to flow around a cylinder.
It can also be seen from Figure 10 that the wake near the section of the blade presents an obvious phenomenon of diffusion to both sides due to the speed in the transverse direction brought by the blade rotation.The vorticity of the flow on both sides of the tail flow behind the tower is increasingly enhanced and the middle vorticity is increasingly suppressed.The wake is gradually dissipated in Figure 10A,B.The vortex behind the tower tends to cylindrical turbulence in Figure 10E.It is further demonstrated that the wake diffusion weakens with increasing distance from the corresponding vertical tip.
As mentioned in the above discussion, we can see that the tip vortex and the vortex behind the nacelle increasingly break up and produce separation vortex.For the vortex behind the tower, the increasing influence of blade sweep on the wake mainly occurs in the increasingly adjacent blade.The lateral force caused by blade sweeping makes the separation vortex and the separation vortex develop to both sides.The wake near the ground tends to flow around a cylinder.The effect of tower exists at 4D downstream of wind turbine, but the effect of tower disappears due to the growth of wake and the mixing of turbulence with the increase of distance. 27

| Development of the vortex system behind the wind turbine
To systematically and accurately capture the wake characteristics of wind turbines, the pressure fluctuations behind the wind turbine are studied at different positions.In this paper, the 180 monitoring points are set behind the wind turbine.As shown in Figure 10, the 60 monitoring points are set every 1 m from y = 1 m along the Y-direction at z = 5.Similarly, 60 monitoring points are set in the same way at z = 0 and z = −5 m.The pressure fluctuation of the tip vortex, the vortex behind the nacelle and vortex behind the tower are studied in this paper.
To comprehensively study the wake characteristics, six points are selected for comparative analysis of wake characteristics.The near field is the flow field before 4.5D       the intense of pressure fluctuation behind the tower is higher than that of the nacelle and the tip vortex.Figure 13B demonstrates the pressure fluctuation of vortex behind the tower at the distance of wind turbine of 40 m and λ = 4, 5, 6, 7, 8.The mean value of pressure decreases with the increase of λ.The separation vortex behind the tower mainly occurs in the near field of the wind turbine wake at λ = 4. Figure 13D     decreased.The maximum amplitude mainly appears at the secondary frequency and λ = 4, which reveals that the separation vortex behind the nacelle appears in transition region of the wind turbine wake.Figure 14C demonstrates the pressure amplitude spectra of vortex behind the nacelle at the radial distance of rotor shaft of 45 m and λ = 4, 5, 6, 7, 8.It can be seen from Figure 14C that the amplitudes obviously decrease with the development of wake.The amplitude of the secondary frequency increases at λ = 4. Figure 14D illustrates the pressure amplitude spectra of vortex behind the nacelle at the radial distance of rotor shaft of 50 m and at λ = 4, 5, 6, 7, 8.The amplitude increasingly decreases with the development of wake.In the turbulent region, the weakening of pressure fluctuation indicates that the large dissipation mainly occurs in the separation vortex.Figure 14E displays the pressure amplitude spectra of vortex behind the nacelle at the radial distance of rotor shaft of 55 m and at λ = 4, 5, 6, 7, 8.It is obtained that the dominant frequency amplitude increases with the increase of λ, which reveals that many separation vortexes emerge behind the wind turbine.Figure 14F demonstrates the pressure amplitude spectra of vortex behind the nacelle at the radial distance of rotor shaft of 60 m at λ = 4, 5, 6, 7, and 8.The pressure fluctuation in the far field is obviously lower than that of near field, which indicates that the distance in the center of the wind turbine decreases, the pressure fluctuation intensity increasingly enhances.The intensity of vortex shedding is gradually weakening along the axial direction.It is further found in Figure 15 that the vortices gradually disappear at the point far away from the wind turbine.This phenomenon mainly results from many separation vortices in the near field, and the vortex has been basically broken up and dissipated.
Figure 16A displays the pressure amplitude spectra of the tip vortex at the distance of the wind turbine of 35 m at λ = 4, 5, 6, 7, and 8.The maximum amplitudes of the characteristic frequencies occur at λ = 5.However, the secondary frequency peak is significantly larger than that of λ = 4. Figure 16B demonstrates the pressure amplitude spectra of the tip vortex at the distance of the wind turbine of 40 m at λ = 4, 5, 6, 7, and 8.The secondary frequency at λ = 4 has the maximum amplitudes, which reveals that the separation vortex behind the wind turbine appears in the near field of the wind turbine wake.Figure 16C displays the pressure amplitude spectra of the tip vortex at the distance of wind turbine of 45 m and λ = 4, 5, 6, 7, 8.It can be seen from Figure 16C that the dominant frequency amplitude gradually increases with the increase of λ. Figure 16E demonstrates the pressure amplitude spectra of the tip vortex at the distance of wind turbine of 55 m and λ = 4, 5, 6, 7, and 8.The amplitudes obviously decrease with the development of wake.
Figure 17A displays the pressure amplitude spectra of vortex behind the tower at the distance of wind turbine tower of 35 m and λ = 4, 5, 6, 7, 8.It can be seen from the Figure 17A that the pressure fluctuation of the vortex behind the tower is violent owing to the impact of the tower.This indicates that a large number of the separation vortexes are produced in this region.Figure 17B demonstrates the pressure amplitude spectra of vortex behind the tower at the distance of wind turbine tower of 40 m and λ = 4, 5, 6, 7, 8.It is obtained that the dominant frequency amplitude increases to a certain extent with the increase of λ.The secondary frequency has the maximum amplitudes at λ = 4, which demonstrates that the separation vortex behind the tower appears in the near field of the wind turbine wake.amplitude decreases in the far field.Figure 17E demonstrates the pressure amplitude spectra of vortex behind the tower at the distance of wind turbine tower of 55 m and λ = 4, 5, 6, 7, 8.In the far field, the weakening of pressure fluctuation indicates that large dissipation mainly occurs in the region of separation vortex.Figure 17F illustrates the pressure amplitude spectra of vortex behind the tower at the distance of wind turbine tower of 60 m and λ = 4, 5, 6, 7, 8.It is obtained that the amplitudes obviously decrease with the development of wake.The pressure fluctuation in the far field is obviously lower than that of the near field.
In the near field, the dominant frequency amplitude has a higher pressure fluctuation peak and many separation vortexes are produced.The highest the dominant frequency amplitude emerges at the λ = 5.It is clearly observed that when the λ is more than 5, the dominant frequency amplitude increases with the increase of λ.In the near field, the pressure fluctuation is weak.The dominant frequency amplitude gradually increases with the increase of λ.At λ = 5, many vortexes are produced in the near field and separation vortexes are well dissipated in the far field.In the wind farm, the incoming flow disturbance caused by the wind turbine in front is weak when λ = 5, which is well consistent with the work of Okulov. 9he probability density function (PDF) of pressure fluctuation is an effective method to study characteristics in the turbulent flow.The probability distribution function represents the probability that the instantaneous amplitude falls within a specified range.And the PDF of random data is where the normal distribution is an important probability distribution.The PDF is: To further study the pressure fluctuation and vorticity dissipation of wind turbines, the PDF of pressure fluctuations is used to study the physical mechanism of wind turbines.Figure 18A   range of pressure fluctuations increase with the increase of λ.

| CONCLUSIONS
In this paper, the wake characteristics of wind turbine are studied by changing the tip speed ratio, and some conclusions are obtained.
In first, 3D vortex structure around the wind turbine and 2D section vorticity field of the wind turbine at different tip speed ratios are investigated.The results show that with the increase of tip speed ratio, the tip vortex increasingly breaks up, dissipates, and mixes with the vortex behind the nacelle.The vortex behind the nacelle develops backward in a spiral shape, finally mixes with the tip vortex and the vortex behind the tower at about 4D distances from the wind turbine.The attached vortex increases with the tip velocity ratio and finally disappears at a distance of about 2D from the wind turbine.Tower effect exists at 4D downstream of wind turbine.However, with the increase of distance, the tower effect is gradually weakening with the increase of wake and turbulent mixing.
In addition, the tip vortex intensity and the pitch of tip vortex are decreased with the increase of λ in the near field.And the position of the broken vortex circles is gradually decreased.The destruction and dissipation of the vortices are increasingly enhanced with the increase of λ.In the far field, the mixing time of tip vortex with the vortex behind the nacelle increasingly increases with the increase of λ, the attached vortex disappears and the vorticity of vortex behind the nacelle rapidly decrease with the development of the wake.The turbulence intensity of wind turbine wake is increasingly enhanced with the increase of λ.The interaction between tip vortex and the vortex behind the nacelle is enhanced with the increase of λ.
A large number of tip-separating vortices appear at a distance of 2D-4D from the wind turbine, where the vortices dissipated rapidly.The vortices behind the nacelle are dissipated behind the wind turbine.The vortex behind the tower produces transverse separation vortices under the lateral influence of blade rotation.In the mixing zone, the separating vortex is quickly broken up and dissipated.In the turbulence zone, the tip vortex, the vortex behind the nacelle and the vortex behind the tower are mixed to a coherent structure.The pressure amplitude spectrum of vortex deceases with the increase of the distance between the wake and the center of the blade axis, which indicates that the strength of vortex shedding gradually decreases along the axial direction.
In the near field, the dominant frequency amplitude has a higher pressure fluctuation peak.Many separating vortexes emerge in this region.The highest amplitude of dominant frequency emerges at the λ = 5.It is found that the dominant frequency amplitude increases with the increase of λ when the λ is higher than 5.In the far field, the dominant frequency peak increases with the increase of λ, which indicates that the separation vortices behind the wind turbine are producing an increase of λ.
The analysis of the PDF of pressure fluctuations leads to the conclusion that in the near-flow field, the wake pressure increases gradually due to the mixing of wake vortices.And in the far field, the pressure fluctuation decreases rapidly due to the large amount of vortex dissipation.The pressure fluctuation is gradually violent with an increase of λ.

F I G U R E 3
Model of wind turbine and computational domain.(A) Wind turbine model.(B) Computational domain.

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I G U R E 5 Three-dimensional wake flow structure at different tip speed ratio.F I G U R E 6 Relationship between tip speed ratios and the positions of the broken vortex circles.

F I G U R E 7
Section of the wind turbine.CUI ET AL.
from the wind turbine.And the far field is the flow field after 4.5D from the wind turbine.Six monitoring points are measured at the radial distance of rotor shaft of 35 m (3.5D), 40 m (4D), 45 m (4.5D), 55 m (5.5D), and 60 m (6D), which accurately captures the characteristics of wind turbine wake in the near and far field.

Figure
11A displays the pressure fluctuation of the vortex behind the nacelle at the distance of wind turbine of 35 m at λ = 4, 5, 6, 7, and 8.It can be seen from the Figure 11A that the pressure fluctuation of the vortex behind the nacelle is violent at λ = 4.A large number of separation vortexes are produced in this region.The mean value of the pressure behind the nacelle is the highest at λ = 5. Figure 11B demonstrates the pressure fluctuation of the vortex behind the nacelle at the distance of wind turbine of 40 m at λ = 4, 5, 6, 7, and 8.It is clearly observed that the mean value of the pressure behind the nacelle gradually increases with the increase of λ.
Figure11Ddisplays the pressure fluctuation of vortex behind the nacelle at the distance of wind turbine of 50 m and λ = 4, 5, 6, 7, 8.The mean value of the pressure behind the nacelle is the lowest at λ = 5.The mean value of the pressure in the far field is lower than that of near field, which indicates that the pressure intensity gradually decreases with the increase of distance in the center of the wind turbine.Figure11E

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I G U R E 10 Vorticity field on the section plane at different heights.demonstrates the pressure fluctuation of vortex behind the nacelle at the distance of wind turbine of 55 m at λ = 4, 5, 6, 7, and 8.It is further demonstrated that the pressure obviously decreases with the development of wake.
Figure 12A displays the pressure fluctuation of the tip vortex at the distance of wind turbine of 35 m at λ = 4, 5, 6, 7, and 8.It can be seen from the Figure 12A that the pressure fluctuation of the tip vortex is violent at λ = 4.The faster angular velocity makes the pressure fluctuation not violent the separation vortex.
Figure 12B demonstrates the pressure fluctuation of the tip vortex at the distance of the wind turbine of 40 m at λ = 4, 5, 6, 7, and 8.The mean value of the pressure at λ = 5 is not F I G U R E 11 Pressure fluctuation of the tip vortex at different monitoring points and the tip speed ratios.(A) monitoring point at 35 m. (B) Monitoring point at 40 m.(C) Monitoring point at 45 m. (D) Monitoring point at 50 m.(E) Monitoring point at 55 m. (F) Monitoring point at 60 m. the highest value.The mean value of the pressure increases with the increase of the λ.
Figure 12D displays the pressure fluctuation of the tip vortex at the distance of the wind turbine of 50 m at λ = 4, 5, 6, 7, and 8.The mean values of the pressure are similar.The pressure fluctuation of the tip vortex is still violent at λ = 4. Figure 12E demonstrates the pressure fluctuation of the tip vortex at the distance of the wind turbine of 55 m at λ = 4, 5, 6, 7, and 8.With the development of wake, the pressure decreases.And there are still a large number of separated vortices causing disturbance here at λ = 4. Figure 13A displays the pressure fluctuation behind the tower at the distance of wind turbine of 35 m at λ = 4, 5, 6, 7, and 8.It can be seen from the Figure 13A that the pressure fluctuation behind the tower is violent at λ = 4. Due to the influence of the tower, F I G U R E 12 Pressure fluctuations behind the tower at different monitoring points and the tip speed ratios.(A) Monitoring point at 35 m. (B) Monitoring point at 40 m.(C) Monitoring point at 45 m. (D) Monitoring point at 50 m.(E) Monitoring point at 55 m. (F) Monitoring point at 60 m.
displays the pressure fluctuation of vortex behind the tower at the distance of wind turbine of 50 m and λ = 4, 5, 6, 7, 8.The pressure decreases with the development of wake. Figure 13E demonstrates the pressure fluctuation of vortex behind the tower at the distance of wind turbine of 55 m, λ = 4, 5, 6, 7, and 8.The mean value of the pressure behind the tower is the lowest at λ = 5.The separation vortex is rapidly dissipated.

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I G U R E 13 Amplitude spectra of vortex behind the nacelle at different monitoring points and the tip speed ratios.(A) Monitoring point at 35 m. (B) Monitoring point at 40 m.(C) Monitoring point at 45 m. (D) Monitoring point at 50 m.(E) monitoring point at 55 m. (F) Monitoring point at 60 m.

Figure
Figure 14A displays the pressure amplitude spectra of vortex behind the nacelle at the radial distance of rotor shaft of 35 m and λ = 4, 5, 6, 7, 8.It is clearly observed that the maximum amplitudes of the characteristic frequencies occur at λ = 5. Surprisingly, we can see that the value of secondary frequency peak is

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I G U R E 14 Pressure fluctuation of vortex behind the nacelle at different monitoring points and the tip speed ratios.(A) Monitoring point at 35 m. (B) Monitoring point at 40 m.(C) Monitoring point at 45 m. (D) monitoring point at 50 m.(E) Monitoring point at 55 m. (F) Monitoring point at 60 m.
Figure 17C displays the pressure amplitude spectra of he vortex behind the tower at the distance of the wind turbine tower of 45 m and λ = 4, 5, 6, 7, 8.The dominant frequency amplitude increases with the increase of λ.Many separation vortices have emerged in the near field.Figure 17D illustrates the pressure amplitude spectra of vortex behind the tower at the distance of wind turbine tower of 50 m at λ = 4, 5, 6, 7, and 8.It is obtained that the F I G U R E 15 Schematic diagram of monitoring point selection.

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I G U R E 16 Amplitude spectra of the tip vortex at different monitoring points and tip speed ratios.(A) Monitoring point at 35 m. (B) Monitoring point at 40 m.(C) Monitoring point at 45 m. (D) Monitoring point at 50 m.(E) Monitoring point at 55 m. (F) Monitoring point at 60 m.

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I G U R E 17 Amplitude spectra of vortex behind the tower at different monitoring points and tip speed ratios.(A) Monitoring point at 35 m. (B) Monitoring point at 40 m.(C) Monitoring point at 45 m. (D) Monitoring point at 50 m.(E) Monitoring point at 55 m. (F) Monitoring point at 60 m.

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demonstrates the PDF of pressure fluctuations behind the nacelle at the distance of wind turbine of 35 m and λ = 4, 5, 6, 7, 8.It can be seen from Figure 18A that the absolute value range of pressure fluctuation increases with the increase of λ.The absolute value range of pressure fluctuation is 0.2 at λ = 4 F I G U R E 18 PDF of pressure behind the nacelle at different monitoring points and the tip.(A) Monitoring point at 35 m. (B) Monitoring point at 40 m.(C) Monitoring point at 45 m. (D) Monitoring point at 50 m.(E) Monitoring point at 55 m. (F) Monitoring point at 60 m.and the absolute value range of pressure fluctuation is 0.3 at λ = 8.It can be seen in Figure18C,D that the pressure fluctuation behind the nacelle mainly occurs around 4D from the wind turbine compared with Figure18A,B.Figure18E,F displays the PDF of pressure behind the nacelle at the distance of wind turbine of 55 and 60 m at λ = 4, 5, 6, 7, and 8.It can be seen from Figure18E,F that the fluctuations of pressure gradually decrease, which reveals that the separation vortex has been dissipated in this area.

Figure
Figure 19A displays the PDF of the tip vortex pressure at the distance of wind turbine of 35 m and λ = 4, 5, 6, 7, 8.It is obtained that the absolute value range of pressure fluctuation at λ = 4 is 0.28 and the absolute value range of pressure fluctuation is 0.3 at λ = 8.The pressure fluctuation of the blade tip vortex is significantly stronger than that of the vortex behind the nacelle.It is found that the sharp pressure fluctuation of blade tip vortex occurs around 4D from the wind turbine.The pressure fluctuation mainly falls on the negative axis.The