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The hybrid vortex method, in which vortex panel method is combined with the viscous-vortex particle method (HPVP), was established to model the wind turbine aerodynamic and relevant numerical procedure program was developed to solve flow equations. The panel method was used to calculate the blade surface vortex sheets and the vortex particle method was employed to simulate the blade wake vortices. As a result of numerical calculations on the flow over a wind turbine, the HPVP method shows significant advantages in accuracy and less computation resource consuming. The validation of the aerodynamic parameters against Phase VI wind turbine experimental data is performed, which shows reasonable agreement.

Among various aerodynamic theories of the wind turbine rotor aerodynamic models [

The flow field around wind turbine blade is characterized with high Reynolds number and low Mach number. The viscous diffusions of vortices are mainly reflected in the blade surface and wake region. According to the principle of the vortex method, an accurate solution to Navier-Stokes equations will be obtained in the condition of the full-scale vortex particle simulation combined with the vortices field of the blades and wakes. It is hard to perform this simulation because the blade surface eddy scale is very small. The simulation of this surface eddy requires huge of vortex particles and the calculation equals the direct numerical simulation. Actually, when the flow separation is small, the viscous diffusion of the blade surface is concentrated in a thin viscosity layer, which can be regarded as a potential flow vortex sheet and the vortex sheet strength can be quickly obtained by applying the theory of the potential flow. When vortex sheet releases from the blade trailing edge to nonboundary wake zone, the vortex diffusion becomes clear and the vortex scale becomes bigger and bigger. Therefore, the discrete vortex particles can be used to simulate the flow in viscosity eddy which is far from the wake region. The idea of hybrid vortex method was derived from the vortex sheet method by Chorin [

The work presented here targets establishing a hybrid panel and viscous-vortex particle method (HPVP) and developing corresponding computation procedure for the aerodynamic analysis of wind turbine. The vortex surface based on potential flow theory is employed to solve the flow around the blade surface and the region near trailing edge. Meanwhile, the turbulent boundary layer surface friction formula is used to correct the blade surface viscous force. In the zones far from the blade wake, the viscous-vortex particle method is adopted to solve the flow equations. The coupling transfer between the vortex particle and vortex panel is built based on the equivalent principle of the wake panel and the vortices strength. This calculation program is used to conduct a computation case of NREL Phase VI rotor and verified by experimental measure data.

In this section, the governing equations for the hybrid vortex method and its formulations for numerical calculations are presented.

In the rotating coordinate system (

The schematic flow field of the wind wheel.

According to the Green function, any point

On infinity of the outside interface, the disturbance velocity of the wind turbine tends to zero:

On the wheel surface

It assumes that the wind turbine wake is zero thickness plane and parallel to the flow line. On both sides of the trailing vortex surface, the normal velocity difference and pressure difference should be zero:

where

where

The calculation grids of the blade surfaces and the wake regions are generated, respectively; each mesh surface is called the surface element. It is assumed that the perturbation potential

Firstly, the body-fitted coordinate of the blade surface is established. Then, the approximate basic functions are selected which are the three adjacent velocity potentials on each direction of the blade surface. Then, each coefficient of the model function is solved by the method of undetermined coefficients and the derivative of each panel control point can be obtained, namely, the tangential disturbance velocity. Finally, the general tangential velocity

When the surface velocity of the blade is determined by the velocity potential, the pressure distribution of the blade surface can be calculated according to the unsteady Bernoulli equations. The dimensionless pressure coefficient can be expressed as follows:

The rotor thrust and torque can be calculated in numerical integration method based on the solved pressure

It is assumed that the flow field of wind impeller is incompressible; thus the blade vorticity field is described by the three-dimensional incompressible viscous flow. According to the vortex principle, the Navier-Stokes equations can be expressed as velocity-vorticity (

In order to avoid the solution error caused by the numerical dissipation in Euler coordinate system, the Lagrangian system is used to describe the vorticity field, and the vorticity field is discreted into

The

According to the Biot-Savart law, the particle velocity is expressed as

For the vorticity dissipation term, many methods can be used to simulate it, for example, random-walk method [

To accurately calculate the viscous diffusion effect of the wake, the viscous discrete vortex is adopted to replace the dipole wake surface element, and the relationship between the unsteady panel method and viscous-vortex wake method is established, as shown in Figure

Schematic of the panel and particle wake vortex method.

According to Hess equivalence principle, the induced velocity of dipole bin with the strength

At time step

When the upstream wake flows through downstream wind turbine blades, the interference of wake vortices and blade vortex occurs; therefore the impact of wakes on the blade surface element should be considered. In this paper, the surface element distributions are determined by the Neumann object plane boundary conditions and calculated by the induced velocity of vortex particle on blade surface element. The intensity of the panel element is defined as follows:

The calculation programs are developed based on the hybrid panel and viscous-vortex particle method (HPVP); the process diagram is shown in Figure

Flow diagram of HPVP method.

The NREL Phase VI wind turbine is selected as a validation numerical example. The aerodynamic performance results of the NREL Phase VI wind turbine is gotten by the measurement in NASA Ames wind tunnel of 24.4 m

In calculation, the 7 m/s inflow wind speed condition is adopted. In this condition, most of the flow area is attached to the blade surface apart from a small area flow separation near the blade root, which accords with the flow field characteristics of variable pitch wind turbine. The numerical validation shows that 50 bins arranged along blade spanwise, 70 bins arranged along blade circumferential, and 120 vortex particles divided at wake region are enough for a grid-independent solution. The calculation program can be running in normal quad-core processor and finished with less than 300 seconds computer CPU time.

Figure

Wake vortex comparison in different calculation methods.

HPVP method

CFD method

Figure

Blade suction surface limit streamline comparison in different methods.

HPVP method

CFD method

For this comparison, the conditions of the measurement and the calculation are set by the same conditions which ensure the comparability. Figure

Cross pressure coefficients comparisons between the test and HPVP method.

By comparing the pressure distributions at various blade cross-sections in Figure

Figure

Tangential and normal force coefficients comparisons.

Tangential direction

Normal direction

Table

The turbine torque and axial thrust comparisons.

Test data | Calculated value | Error | |
---|---|---|---|

Torque | 805 | 837 | 4% |

Axial thrust | 1130 | 1092 | 3.4% |

In this paper, a hybrid panel and viscous-vortex particle method (HPVP) has been established and corresponding computation procedure for the aerodynamic analysis of wind turbine has been developed. Phase VI wind turbine was selected as a simulation example to validate the accuracy of the HPVP method with the measured data. It showed the good capability of the HPVP method in the performance prediction of the wind turbine.

The HPVP method has the advantages of less empirical correction, less computing resource consuming, and accuracy. Compared to the CFD method, the calculation in HPVP method can quickly obtain accurate results and take up only one-thousandth of CFD computing time. The simulation in HPVP method considers the viscous effect of wake vortex, which can be closer to the exact solutions to the

Some other detailed flow parameters of wind turbine are also solved in HPVP method besides the surface pressure distributions and the limit streamlines of the blade. It is helpful for conducting the investigation of the aerodynamic and aeroelastic issue of the wind turbine complex, which has great significance on the optimization design process of the wind turbine.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was funded by Henan Provincial Key Science and Technology Projects (142102210059). This support is most gratefully acknowledged.