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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 905379, 11 pages
Research Article

Aerodynamic Performance Prediction of Straight-Bladed Vertical Axis Wind Turbine Based on CFD

College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China

Received 15 October 2012; Revised 9 March 2013; Accepted 10 March 2013

Academic Editor: Toshio Tagawa

Copyright © 2013 L. X. Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Numerical simulation had become an attractive method to carry out researches on structure design and aerodynamic performance prediction of straight-bladed vertical axis wind turbine, while the prediction accuracy was the major concern of CFD. Based on the present two-dimensional CFD model, a series of systematic investigations were conducted to analyze the effects of computational domain, grid number, near-wall grid, and time step on prediction accuracy. And then efforts were devoted into prediction and analysis of the overall flow field, dynamic performance of blades, and its aerodynamic forces. The calculated results agree well with experimental data, and it demonstrates that RNG k- turbulent model is great to predict the tendency of aerodynamic forces but with a high estimate value of turbulence viscosity coefficient. Furthermore, the calculated tangential force is more dependent on near-wall grid and prediction accuracy is poor within the region with serious dynamic stall. In addition, blades experience mild and deep stalls at low tip speed ratio, and thus the leading edge separation vortex and its movement on the airfoil surface have a significant impact on the aerodynamic performance.

1. Introduction

As one of the most promising renewable energy resources, wind energy plays an important role in response to shortage of fossil fuels and climate change [1]. With widely use of small-scale wind turbine, the straight-bladed vertical axis wind turbine (S-VAWT) is attractive for its simple blade design, making it relatively easy to be fabricated and cost saving. There are some distinct advantages over to the commercial horizontal axis wind turbine [25], where no yaw mechanism is required and it responds well to changes of wind direction for its omnidirectional virtue. Whereas, the unsteady flow around blades, Darrieus motion [6], and dynamic changes of angle of attack have a significant impact on aerodynamic loads on blades. Therefore aerodynamic prediction models of VAWT are still immature [7, 8], although concept of VAWT has been proposed by Darrieus [9] as early as 1931. Accurate prediction and effective and efficient optimization tools are urgently needed to realize the goal for an S-VAWT with better performance.

It is commonly believed that streamtube models and vortex models make great contributions to overall aerodynamic performance optimization and local interaction mechanism of blade. Aerodynamic models still cannot meet the demands for various purposes, although the streamtube and vortex models have been improved for many times, including multiple streamtube model [10], double multiple streamtube model [11], FEVDTM [12], and VPM2D method [13]; more details can be found in the literature [1416]. But with the rapid development of computational fluid dynamics (CFD), which has not only accelerated the design process but also has brought down the overall cost of design, the CFD become an attractive solution for aerodynamic performance prediction.

CFD method had been widely used for overall performance prediction. A CFD prediction model based on BE-M theory was presented by Raciti Castelli et al. [17] for evaluating the aerodynamic forces of S-VAWT. Calculation results obtained from large eddy simulation of Iida et al. [18] indicated that the results at high tip speed ratios (TSR) show good agreement with that of momentum theory, but it was not accurate for low TSR cases. Howell et al. provided three-dimensional CFD model based on RNG k- turbulence model [19], and the effects of turbine solidity, roughness, and tip vortices were considered. Numerical investigations on flow curvature effects were conducted by Coiro et al. [20], and he argued the difference between DMS model and VAT-VOR3D code. Hwang et al. proposed a cycloidal water turbine with individual blade pitch control [21], along with optimized cycloidal motion for better performance obtained from CFD simulations.

In addition, CFD simulation is an effective solution for analysis of local flow field around blades, particularly for dynamic stall and wake flow. Two-dimensional computational investigations on dynamic stall with different blade pitching patterns were carried out by Wang et al. [22], and he stated that obtained results agreed well with the experimental data except for high angle of attack. Amet et al. provided a detailed numerical analysis of the physical phenomena that occurs during dynamic stall where compressible RANS k- model and multiblock structured mesh structure were used [23]. Furthermore, studies of dynamic stall show that SST k- turbulence model was better than k- model as reported by Ekaterinaris and Platzer [24] and Qian et al. [25]; Qian et al. said that SST k- turbulence model was an effective model for dynamic stall simulation with high accuracy and the predicted aerodynamic coefficients agreed well with the experimental data.

Due to that CFD method is an attractive solution for performance optimization, thus reliability and accuracy of CFD have become the key factors. The work presented here addressed the issues that the influences of process of numerical simulation on accuracy and efforts were devoted into the effects of mesh structure and time step, expecting for a better performance prediction. Both 2D and 3D CFD simulations of S-VAWT were conducted to assist a better understanding of dynamic stall. In addition, comparison analysis between experimental data from literature [13] and calculated results was done to verify the reliability and accuracy of CFD.

2. Flow Field of Blade

The present work here aims at obtaining the relationship between prediction accuracy and preset parameters of CFD. In order to have an overall understanding of interference terms, it is better to get knowledge of local velocity field around blade.

2.1. Macro Flow Field

Figure 1 is the local velocity field of blade in first quadrant and its transient aerodynamic forces exerting on blade. It indicates that the local relative velocity is dependent on local wind speed and tangential velocity . In addition, Reynolds number and angle of attack are in dynamic change and are functions of azimuth angle . more details can be found in literature [26].

Figure 1: Local velocity field and transient aerodynamic loads of blade.

The aerodynamic forces acting on blade, lift and drag , can be decomposed into tangential force along the rotor and radial force , as shown in Figure 1. where is dynamic pressure coefficient and is equal to . and are tangential force coefficient and radial force coefficient, respectively, and can be derived by

Thus the torque coefficient of S-VAWT is where and are solidity and the number of streamtubes, respectively. Equation (3) suggests that operational capacity of S-VAWT is closely related to velocity field and configuration parameters of wind turbine, and the tangential force states the torque coefficient to some extent.

2.2. Microflow Field

Based on boundary layer theory and mixing length theory, the flow field around blade can be divided into three regions: external potential flow, boundary layer, and wake flow, as shown in Figure 2. The external potential flow is assumed to be ideal fluid, ignoring the effect of viscosity, and it can be solved by potential flow theory. While the boundary layer is sensitive to viscosity and needs more concerns, for that it is the region where vortexes take place, grow up, and shed, which leads to dynamic stall. The wake flow is extremely complex and it affects performance of itself and that of next blades. In this work, attentions were paid to boundary layer, taking local Reynolds number and thickness of boundary layer into account.

Figure 2: Microflow field around blade.

Due to the dependence of characteristics of boundary layer on Reynolds number [27], the thickness of boundary layer will be affected by the local relative velocity . Schlichting and Gersten [27] stated that Reynolds number had a significant impact on thickness of boundary layer which was the cradle of vortexes and turbulence, and thus solving turbulence was more dependent on grids than laminar flow. Also, tangential velocity fluctuation can be reduced by viscous damping of near-wall region and normal velocity fluctuation is arrested by the wall. Thus the predicated accuracy is dependent on the mesh structure of boundary layer.

For the sensitivity of turbulent flow on thickness of boundary layer, the near-wall around blade is divided into three regions: viscous sublayer, blending region, and fully turbulent region, see the literature [28]. In viscous sublayer, dimensionless average velocity is equal to dimensionless distance , while the dimensionless average velocity of fully turbulent region can be derived by (4). In blending region, turbulent stresses and viscous stresses have the same order of magnitude, and the blending region cannot be ignored with the decrease of Reynolds number: where and are Karman parameter and parameter relating to surface roughness, respectively; for the cases of smooth wall, , and and can be expressed as where the wall friction speed , and , respectively, are wall shear stress and kinematic viscosity.

The dimensionless distance is related to flow velocity, viscosity, and shear stress. The wind turbine operates at low speed and the Reynolds number is quite small; in general, the distance from the center of the first layer grid to the wall can be estimated by (6) where is characteristic length.

3. CFD Model

3.1. Theoretical Background

Discretization is one of the critical steps for CFD method, where continuous space will be discretized into finite control volumes, realizing the transformation form differential equations to algebraic equations. However, the discretization error, closing error, and rounding error during transformation, as well as the numerical dissipation, have great impacts on accuracy, reliability, and convergence. Performance prediction of S-VAWT also depends on mesh structure and its solution algorithm, and only the grids structure and solution algorithm are matched, and a higher accuracy and efficiency will be achieved.

The CFD model depends on mass, momentum, and energy conservation equations, and thus the distribution of physical quantities and their changes over time at any position of the complex flow field can be obtained by using CFD method. Flow around wind turbine is assumed to be incompressible, and the energy dissipation is ignored during numerical simulation, and thus the general control equation is where is universal variable, standing for , , , , and so on, and is generalized diffusion coefficient.

For the CFD method, it must be discretized before solving the control equations, where the variables are interpolated into control decent, as shown in (8),

Terms in (8) are successively transient term, convection term, diffusion term, and original term. For time saving and high accuracy, first-order upwind scheme is used in the first place (usually after approximately three to five operation cycles) with a better convergence, and then second-order upwind scheme is employed for higher accuracy.

3.2. Effect of Computational Domain

Computational domain must be determined before meshing, which is significant for accuracy. At a given grid density, there are fewer grids and it is time saving when computational domain is much smaller. But, it will have great impact on accuracy if the domain is too small. It is due to that flow velocity that passed through the S-VAWT does not yet fully be back to inflow speed, thus pressure of wake flow is much smaller than outlet pressure, leading to adverse pressure gradient and recirculation at outlet, see Figure 3. Furthermore, recirculation zone extends with the increase of adverse pressure gradient, leading to meaningless calculated results.

Figure 3: Effects of computational domain.

However, the computation time will be increased with a large computational domain. Taking the accuracy and computing time into account, the distance between rotor axis and inlet (or the two sides) is usually more than five times than radius of S-VAWT, and the distance from outlet to the rotor axis is more than ten times of the radius. In this paper, a computational domain of rectangular shape had been chosen, as shown in Figure 4. The three-bladed wind turbine, using a NACA0015 blade profile with a chord length of 0.4 m, was 4 m in diameter. The computational domain was 62 m in length, 24 m in width, and the distance between rotor axis and inlet was 12 m. The front of domain was defined with velocity inlet, the outlet of domain was pressure outflow, and the two symmetry boundary conditions were defined as nonslip walls.

Figure 4: Mesh structure of the 2D computational domain of S-VAWT.

In order to control the mesh structure for analyzing its influences on prediction accuracy, thus structured meshes and the sliding mesh technology were employed. For turbulence, Reynolds-averaged method had been widely used, where instantaneous N-S equation was replaced by time-averaged one. In addition, the Reynolds number in boundary layer is much smaller and viscous plays a leading role, and thus wall function method or low Reynolds number modeling method will be employed to solve the turbulent flow. Due to that the wall function can reduce the computing time and storage space, and the accuracy can be guaranteed by introducing empirical equations, therefore RNG k- model and wall function were employed in this paper for high accuracy at low Reynolds number.

3.3. Effect of Grid Number

Due to that the density of grids has a significant impact on accuracy, grid number will be considered in this present paper. Effects of grid number on accuracy of CFD simulations were shown in Figures 5 and 6. Figure 5 is the effects of grid independence on the accuracy. It indicates that torque coefficient gradually becomes stable when the grid number is more than 100 thousand, meanwhile the accuracy is acceptable. In addition, the calculated accuracy will not improve significantly with more grids, but with too much computational time. It also shows that obtained results make no sense for too small grid number, and the computational results are incredible when the grid number is less than 100 thousands. (Note: the increase of grid number is for the whole computational domain, the condition of local grid refinement and low density of other domain are nonexistent).

Figure 5: Effects of grid independence on accuracy.
Figure 6: Effects of grid number on accuracy.

Figure 6 shows the relationship between grid number and accuracy. In this thesis, it is regarded as convergent when the error between inlet and outlet flow is less than 0.1% of the total flow, or the errors of the flow field variables in the adjacent cycle are less than 1%. As shown in Figure 6, the torque coefficient becomes stable when it operates more than five cycles, meanwhile the residual for convergence is less than 0.0001, and thus it is concluded that it is convergent and calculations are terminated. The sudden increase of torque coefficient for cases with grid numbers 50 thousands and 70 thousands is caused by conversion of discrete formats. The solutions were initially obtained using first-order discrete format until periodic solutions were achieved, and then second-order discrete format were used to have a higher accuracy. Figure 6 also demonstrates that the more grids are utilized when the grid independence has not been achieved, the higher prediction accuracy can be obtained, while much too less grids will lead to wrong conclusion which is consistent with Figure 5.

3.4. Effects of Near-Wall Grid Size

Dimensionless distance is related to flow velocity, viscosity, and shear stress, and near-wall grid affects the calculated accuracy of turbulence for low wind speed and Reynolds number. Comparison analysis of instantaneous tangential force coefficient of 3D and 2D for S-VAWT was shown in Figure 7.

Figure 7: Comparison analysis of tangential force coefficient of 3D and 2D for S-VAWT.

From curves, the instantaneous tangential force coefficient presented to be periodic change and the number of peak values was equal to blade number. It also indicates that the torque coefficient obtained by 3D simulation is smaller than that of 2D and lags behind, which is due to the effects of cross-flow of wingtip and spanwise flow.

Figure 7 suggests that maximum of instantaneous tangential force occurs at the azimuth angle nearby 75 degrees of upstream, and upstream plays an important role in driving the wind turbine than downstream. It is found that the peak value of blade 2 is obviously higher than that of blade 1 and blade 3. However, the tangential force at upstream should be the same even though it was affected by the curvature effect. With large-scale investigations being done, it was found that the primary cause was the value of near-wall grid of blade 2, and the big difference was eliminated by adjusting the near-wall grid of blade-2. It is clearly seen that the value of near-wall grid has a significant impact on accuracy of CFD method.

3.5. Effect of Time Step

For the unsteady flow around S-VAWT, setting of time step has a significant impact on the calculation accuracy. With a big time step, it will lead to serious false diffusion and stability of CFD becomes poor. Meanwhile the truncation error increases with the time step which cannot well demonstrate the change of flow field over time, failing to obtain expected results. Sometimes the calculation accuracy of a large time step can be improved by increasing the iterative steps in a single time step, but it is not always effective.

Plenty of researches showed that time step was dependant on the thickness of the first layer of near-wall grids. The smaller the thickness of the first near-wall grid, the smaller the time step to achieving convergence, and it was time-consuming. It was proved that the thickness of near-wall grid played an important role in accuracy and stability, and the time step could be determined based on the near-wall grid parameters. Usually, the time step can be calculated by .

The effect of time step on accuracy was shown in Figure 8. It is clearly seen that the accuracy of the case of time step is the best, followed by case of , and case of is the worst, indicating that the calculation accuracy can be guaranteed by a smaller time step.

Figure 8: Curves of instantaneous tangential force coefficient of S-VAWT at various time steps.

4. Performance Prediction of S-VAWT

4.1.  3D Flow Field of S-VAWT

In order to have a better understanding of the tip flow, 3D investigations had been conducted but also for comparison analysis of the difference between 3D and 2D. Based on the computational domain of 2D simulation shown in Figure 4, mesh structure of 3D flow field and detailed mesh structure of rotor blades were shown in Figure 9. The distance from top of the rotor to the ground was 7 m, and length of blade was 2 m. with the purpose to build a realistic flow field for better prediction accuracy, a “sky” (not shown in Figure 9) with 13 m in depth was stacked on the top of the wind turbine, and at the end the total grid of the 3D model was approximately 1.3 million.

Figure 9: Computational domain for the 3D simulation and details of the mesh structure of the rotor blades.

Figure 10 was the three-dimensional flow field of S-VAWT. As shown in Figure 10, it is clearly seen that the velocity from upstream to downstream decreases gradually, leading to a relative low-velocity zone behind the rotor. It is also found that streamlines of leaf blade move down to middle of the blade and streamlines of bottom blade move up to middle of blade. It is due to that pressure is high at pressure surface and low at suction surface, leading to a big accelerating pressure gradient between the two surfaces, and then cross-flows occur both at leaf blade and bottom blade. Also, the pressure of middle of blade is relatively lower than that of the ends of blade, thus fluids at the ends of blade move to the middle of blade. The two strands of fluid mix at wake region and fully develop, producing a free inflow at father wake region.

Figure 10: Diagrams of streamline and slice of 3D flow field.

Due to cross-flow at the ends of blade, the calculation results of 3D were slightly smaller than that of 2D, seen in Figure 7. The difference between 2D and 3D computational results was acceptable by comparison analysis with that of the literature [19], and it basically met the demands for aerodynamic performance prediction of S-VAWT. In order to reduce the computational time, 2D simulations were employed for the following work.

4.2. Aerodynamic Forces of S-VAWT

Two-dimensional investigations had been carried out to obtain the operational capability of the turbine at various TSRs, as shown in Figure 11. Figure 11 suggests that averaged power coefficient at various TSRs is periodic and lags behind each other by 120 degrees. Curves of suggest that fluctuation of flow becomes smaller with the increase of TSR, and the averaged power coefficient becomes more stable. This is due to that Reynolds number increases with the TSR, but the angle of attack becomes smaller, and therefore the region where deep stall might happen becomes smaller, and thus the power generation becomes more stable.

Figure 11: Average power coefficient of three-bladed S-VAWT at various TSRs.

As can be seen from Figure 11, it is obviously seen the averaged power coefficient of cases of TSR ≧ 1.5 beyond the Bates limitation 0.593, which is unlikely to occur according to Bates law. Nevertheless, Jonkman [29] stated that the static pressure on the boundary of the streamtube portion enclosing the rotor disc should be equal to unperturbed ambient static pressure. However, this hypothesis is not verified for the present work but can be explained by the pressure distribution of wind turbine, see Figure 12.

Figure 12: Pressure distribution of S-VAWT ().

Form the pressure distribution of S-VAWT, it is found that the pressure of windward side is positive, but there is a sudden drop of pressure coefficient at suction surface, especially for the region with high-power generation. For example, when the blade comes to azimuth angle 109 degrees, the pressure coefficient of pressure surface is 0.99, while it is −1 at suction surface. Due to the big differential pressure, the thrust exerted to the blade increases sharply, thus the averaged power coefficient exceeds the Bates limit at some region.

As shown in Figure 11, there is a large fluctuation for the power coefficient at , and there is a smaller peak value following the big peak value. The main reasons are the sinusoidal pattern of angle of attack, and more serious dynamic stalls and secondary vortexes; these make great contribution to the formation of aerodynamic forces.

4.3. Dynamic Performance of Blade

The variation of angle of attack with azimuth angle at different TSRs is sinusoidal function [22, 30], and thus we assume that the blades are under sinusoidal oscillatory motion when S-VAWT is under operation. Due to the investigations on effects of reduced frequency, amplitude and original value of sinusoidal function had been extensively conducted [23, 25]; we did an example with NACA0015 blade profile at the sinusoidal pattern of in this paper. The blade executes an oscillatory motion at a fixed pivot (1/4 chord length) with an oscillating period of 2 s; the time step is 0.005 s, and the inflow velocity is 10 m/s. The dynamic performance of blade is shown in Figure 13.

Figure 13: Lift coefficient hysteresis loops of blade under sinusoidal motion.

As seen from Figure 13, it is found that there is a hysteresis loop that the blade cannot immediately return to the flow field before dynamic stall. It shows that the difference between upstroke and downstroke is small at small stage of angle of attack, while the difference cannot be ignored for high angle of attack. Meanwhile, the results suggest that lift coefficient of upstroke at high angle of attack is better than that of downstroke, whereas slightly lower at small angle of attack.

As can be seen in Figure 11, the power coefficient of was relatively good than other cases. In addition, for a better understanding of the flow field of S-VAWT and the effects of angle of attack on aerodynamic performance, changes of flow field and aerodynamic forces of TSR 2.5 during a revolution were shown in Figures 14 and 15.

Figure 14: Flow field around blade in one revolution.
Figure 15: Curves of dynamic changes of lift and drag coefficients of one blade in one revolution.

Figure 14 demonstrates that it is attached flow at azimuth angle and viscosity plays an important role at that region. Effective angle of attack increase with azimuth angle and flow begins to separate at trailing edge, and then the separation point moves to leading edge along with the azimuth angle. Meanwhile, a clockwise circulation occurs at trailing edge at suction surface and the blade is in mild stall. When passes though azimuth angle of 30 degrees, vortex occurs at leading edge with the increase of azimuth angle, and the lift coefficient is proportional to the angle of attack during this process, see Figure 15. The vortex keeps growing in size, meanwhile the interaction with the growing circulation at trailing edge gets stronger with a further increase of azimuth angle, leading to serous energy dissipation, and thus the lift coefficient grows nonlinear.

When it comes to azimuth angle 70 degrees, the vortex rises up and begins to leave the surface of blade, and the blade is in deep stall, leading to a sudden drop in lift coefficient, see Figure 15, and coming into the peak value. With the increase of azimuth angle, the angle of attack begins to decrease, and secondary vortex is formed at trailing edge, which makes significant contributions to the increase of aerodynamic forces during downstroke motion. The induced vortex grows in size but the angle of attack begins to decrease, and therefore the vortex sheds from the suction surface of blade and it becomes to be attached flow when it is at azimuth angle of 180 degrees. At downstream, the flow penetrated from the pressure side of blade into the suction side and the angle of attack began to increase, and therefore a reversed flow occurs at trailing edge, which has a significant impact on the aerodynamic forces.

From Figure 14, it can be concluded that the blade has experienced five stages: attached flow, mild stall, deep stall, stage of secondary vortex, and reattached flow. The separation at trailing edge marks the blade enter into mild stall, and the shedding of leading vortex states that the blade is in deep stall and the highest lift coefficient is obtained.

5. Experimental Verification

The present work here aimed at finding the effects of near-wall grid on prediction accuracy, and then more efforts were devoted to the influences of thickness of first layer near-wall grids on mechanical properties. Investigations by CFD were performed and compared with the experimental data reported by Wang et al. [13]. Based on plenty of attempts, it comes to be convergent at time step of 0.5 s, as shown in Figure 16.

Figure 16: Comparison analysis of tangential and radial force coefficient between CFD and experimental date.

Figure 16 indicates that the thickness of the first-layer near-wall grids has a great impact on accuracy. It is clearly seen that calculation results obtained from RNG k- turbulence model agree well with the experimental data, especially for the radial force coefficient. Furthermore, the thickness of grids has less impact on the accuracy of radial force than that of tangential force. And the vibration frequency of radial force is significantly less than that of tangential force. It is because that the drag makes great contribution to radial force and the drag is quite small compared with the lift, thus the radial force changes smoothly.

From the curves in Figure 16(a), it is also found that curves of 0.8 mm and 1.2 mm are far away from the experimental data. While the result of 0.1 mm is much better, particularly the results at upstream show well agreement with experimental data but are much smaller than experimental data at downstream. It is probably caused by the shedding vortexes form upstream as well as the lower flow velocity, leading to serious dynamic stall and blade flutter. In addition, prediction values of vortexes by using RNG k- model are a bit larger than the real values due to the high estimate value of turbulence viscosity coefficient, and therefore the tangential force at downstream is much smaller than that of upstream and its fluctuation is severe.

In addition, comparison analysis between calculation results and experimental data of literature [19] has been carried out, and the calculated results are shown in Figure 17. Figure 17 states that the results at higher TSR are in good agreement with experimental data, while there is big difference at low TSR, about 3.8%. It is probably due to that the region where dynamic stall takes place expends at lower TSR, thus dynamic stall is severe and effects of vortexes enhanced. Furthermore, prediction results of the employed turbulence models are larger than those of real values and lead to a poor accuracy.

Figure 17: Comparison analysis of power coefficient between CFD and experimental data.

6. Conclusion

The present paper aimed at getting a better aerodynamic performance prediction of S-VAWT, and efforts were devoted to the effects of computational domain, grid independent, near-wall grid, and time step on calculated accuracy. For the present two-dimensional CFD model, the calculated accuracy from RNG k- model is accepted when the total grid number is more than 100,000. In addition, the prediction values of aerodynamic loads agree well with the experimental data, and the near-wall grids have a significant impact on prediction of tangential force than radial force. The calculation results are smaller than those of experimental data especially at downstream; it is because that the employed turbulent model has a larger prediction values than the real values. Thus, a turbulent model should be carefully selected for a higher accuracy.

From the flow field analysis, it is concluded that the blade experiences a complex process during one revolution, including attached flow, mild stall, deep stall, and reattached flow. The highest lift coefficient is obtained with the shedding of leading vortex, and the secondary vortexes also make significant contributions to the aerodynamic performance.


The authors would like to acknowledge the financial support of the Fundamental Research Funds for the Central Universities (HEUCF110707) and the Heilongjiang Province Natural Science Foundation, China (E201216) for this work.


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