Journal of Physics: Conference Series Numerical analysis of the tip and root vortex position in the wake of a wind turbine To cite this article: S Ivanell et al 2007 J. Phys.: Conf. Ser. 75 012035 View the article online for updates and enhancements. Related content - Numerical investigation of the wake interaction between two model wind turbines with span-wise offset Sasan Sarmast, Hamid Sarlak Chivaee, Stefan Ivanell et al. - Influence of the conservative rotor loads on the near wake of a wind turbinex I. Herráez, D. Micallef and G.A.M. van Kuik - Study of the near wake of a wind turbine in ABL flow using the actuator line method J Nathan, M Bautista, C Masson et al. This content was downloaded from IP address 46.3.197.181 on 20/11/2017 at 03:52
Numerical analysis of the tip and root vortex position in the wake of a wind turbine SIvanell 1,JNSørensen 2, R Mikkelsen 2, D Henningson 3 1 Royal Institute of Technology/Gotland University, Stockholm/Visby, Sweden 2 Technical University of Denmark, Lyngby, Denmark 3 Royal Institute of Technology, Stockholm, Sweden E-mail: stefan.ivanell@hgo.se Abstract. The stability of tip and root vortices are studied numerically in order to analyse the basic mechanism behind the break down of tip and root vortices. The simulations are performed using the CFD program EllipSys3D. In the computations the so-called Actuator Line Method is used, where the blades are represented by lines of body forces representing the loading. The forces on the lines are implemented using tabulated aerodynamic aerofoil data. In this way, computer resources are used more efficiently since the number of mesh points locally around the blade is decreased, and they are instead concentrated in the wake behind the blades. We here present results of computed flow fields and evaluate the flow behaviour in the wake. In particular we compare the position of the root vortices as to the azimuthal position of the tip votices. 1. Introduction A main problem in wind farms is that wake interference may cause severe fatigue problems owing to increased turbulence intensities in the wake of neighboring wind turbines. This effect may be further exaggerated if a wake-affected turbine is exposed to a coherent wake structure consisting of stable tip and /or root vortices. In the present work the stability of tip and root vortices are studied numerically in order to analyse the circumstances under which tip vortices become unstable and breaks down to small-scale turbulence. 2. Simulation Steady simulations were performed to reach a basic steady flow field. This forms the initial condition for the subsequent stability analysis that basically consists of a time-true computation using large eddy simulation. To simplify the analysis only flow cases with constant axial inflow were considered. However, this demands in the order of about 10 million mesh points in order to perform simulations at realistic Reynolds numbers. The point of onset and the growth of the instabilities were evaluated in detail. 2.1. Grid The mesh uses periodic boundary condition which therefore reduces the mesh size to a third, i.e. a 120 degree slice. All simulations are based on steady state calculations as initial condition for the unsteady simulations and assumes constant axial inflow. The mesh consists of 40 blocks c 2007 Ltd 1
that are distributed both in axial and radial direction, allowing the simulations to be run on up to 40 processors. Figure 1 shows how the mesh is designed. Figure 1. The figure shows the design of the 40 block mesh with periodic b.c. The radius of the block closest to the center axis corresponds to 1.25 rotor radii. The actuator line is positioned in the first thin block in the flow direction. The flow direction corresponds to the z-axis. 2.2. Numerical method The main limitation in actuator disc methods is that these methods distributes the forces evenly in the tangential direction of the actuator disc. The influence of the blades is therefore taken as an integrated quantity in the azimuthal direction. An extended three dimensional method, the so-called Actuator Line Method (ACL) introduced by Sørensen and Shen [1], solves this problem by distibuting the body forces along the blades as line sources. In contrast to this method, full CFD simulation would require a great number of mesh points along the blades to resolve the boundary layer. With the actuator lihne method these mesh pints are not needed and the method therefore opens new possibilities for turbine simulations with a well resolved wake. The drawbacks is that the method is still based on tabulated airfoil data from which C L and C D are functions of α, therefore are dependent on the quality of experimental data. The method has been implemented, by Mikkelsen [2] into the EllipSys3D code. EllipSys3D is a general purpose 3D navier-stokes solver developed by N.N.Sørensen and Michelsen [3], [4], [5]. The flow solver is a multi block, finite volume discretization of the Navier-Stokes equations in general curvilinear coordinates. The code is formulated in primitive variables, i.e. in pressure and velocity variables, in a collocated storage arrangement. Rhie/Chow interpolation is used to avoid odd/even pressure decoupling. The presence of the rotor is modeled through body forces, determined from local flow and airfoil data. The Navies-Stokes equations are formulated as: u i t + u u i j = 1 p + f body,i + ν u2 j, x j ρ x i x 2 j u i x i =0 (1) where f body represents the forces acting on the blades. The numerical method uses the third order QUICK (Quadratic Upstream Interpolation for Convection Kinematics) method for the convective terms and fourth order CDS (Central difference schemes) for the diffusive terms. The aerodynamic forces that are distributed along the actuator line cannot be applied discretely along the actuator line because of numerical discontinuities. The forces are therefore distributed among neighboring node points in a Gaussian manner. This is done by taking the convolution of the computed load f rθz and the regularization kernel η ɛ. 2
where the regularization kernel is defined as: f b ɛ = f b rθz η ɛ (2) η ɛ (p) = 1 ɛ 3 π 3/2 e (p/ɛ)2 (3) where p is the distance between cell centered grid points and points at the actuator line. Mikkelsen discovered that by using a 3D Gaussian smoothing results in inconsistencies near the tip region, [2]. Therefore, a 2D Gaussian distribution is used on a 2D-plane orthogonal to the actuator line. This smearing of the forces is therefore done globally, i.e. every node point at a plane orthogonal to the actuator line will be affected, even if the effect is negligible far from the line, because of the Gaussian function. The 3D Gaussian, if applied, would also increase the effective radius of the blade forces. The 2D Gaussian distribution is controlled by the parameter ɛ. The choice of ɛ will also affect the numerical discontinuity at the tip, as a result of a 2D distribution. The choice of the value of ɛ will therefore be critical and may have a great impact on the wake structure. This has been investigated in an earlier report by Ivanell, [6]. Since the aim of this project is to simulate the wake in order to analyse the stability of the flow, the actuator line method has been chosen as it resolves the wake without requiring a mesh of the blades themselves. The Actuator line method captures effects in the ambient flow from the blades with the lowest computational requirements. One can generally say that the performed computation is a large eddy simulation since it solves the Navier-Stokes equations without resolving all scales. 2.3. Numerical setup The simulation has been performed on a linux PC cluster at the Mechanical Department at DTU. The cluster has been explicitly developed for EllipSys3D. It contains 210 PC s with Linux Redhat as an operating system. EllipSys3D is parallelized and uses MPI. EllipSys3D can, however, only handle blocks with the same number of nodes on each block edge or side. The distribution on each side of the block boundaries can, however, be nonlinear. The mesh was created to be able to capture large gradients in the wake. The total number of node points are 10.5 10 6. The number of node points along the actuator line is 70. All simulations have been performed with a 2D Gaussian distribution discussed in section 2.2 and correspond to a wind speed of 10 m/s and a tip speed ratio of 7.07. 2.4. Experimental data Since the actuator line method uses tabulated airfoil data from measurements, data with good quality must be used. Data from the Tjaereborg turbine have been used for all simulations in this project. The Tjaereborg turbine was operational between 1988 and 1998. During these years extensive measuring and testing were performed on the turbine. The turbine was localized 9 km southeast of the city of Esbjerg in the western part of Denmark. The Tjaereborg wind turbine was equipped with a three-bladed upwind horizontal axis rotor. The blade profiles were NACA 4412-43 with a blade length of 29 m giving a rotor diameter of 61 m. The chord length was 0.9 m at the tip, increasing linearly to 3.3 m at hub radius 6m. The blades were twisted 1 per 3m. The tip speed was 70.7 m/s and the rotor solidity was 5,9%. The rated power was 2 MW and the output was controlled bycontinuously varying the pitch angle between 0 and 35 degrees in production mode. The hub height was 60 m [7]. 3
3. Results 3.1. Wake flow field The simulation resulted in a flow field where the velocity field, pressure field, vorticity etc could be evaluated. In figure 2 one example of the steady state flow field is given where the tip and root vortex spiral is identified by an iso surface of the vorticity. The vertical plane also shows the pressure distribution. The horizontal plane shows the stream wise velocity distribution. Figure 2. The vertical plane shows the pressure distribution and the horizontal plane the stream wise velocity. The vortex spiral is identified by an iso-surface with constant vorticity. The axial and radial interference factors, extracted from the flow field, and mean values over azimuth position are plotted in figure 3. The field is evaluated at different axial positions behind the rotor. The rotor is positioned at Z=8, corresponding to 8 rotor radii from the inlet, which means that the extracted data is taken at 1, 2 and 3 rotor radii down stream of the rotor. The mean values over azimuth position of the plotted quantities have then been calculated for each radial position. The mean axial velocity shows the axial breakup and radial wake expansion as expected. The mean azimuthal velocity does show a 1/r relationship between the azimuthal velocity and the radial position, also as expected from classical theory. The mean over azimuth position of axial interference factor, a mean, and the azimuthal interference factor show the same behaviour, only now the sign is of course opposite between a mean and mean axial velocity. 3.2. Vortex positions The aim of the study has been to identify the basic mechanisms behind the breakdown of the tip and root vortex. One starting point has been to study root vortex positions relatively to the tip vortex position in a plane orthogonal to the flow direction. The position of the root and tip vortex are then dependent of the pitch of the vortex spiral. Analytical studies with the same aim has been carried out by Okulov and Sørensen [8]. The aim of this study has therefore been to search for stable and unstable relative azimuthal positions of the root and tip vortex The simulation with a 120 degree slice with boundary conditions resulted in pairing instabilities. However, the simulation stabilizes again about 5000 unsteady iterations because of the influence of the boundary conditions. A 360 degree computation is expected to perform a more realistic simulation when investigating instabilities. In spite of this, the simulations show how the instabilities start and develop. Figures 4 and 5 show how the root vortices start to oscillate in pairs and how the amplitude of the oscillations increase. Notice that the position of the root vortex is in between the tip vortices in azimuthal angle. At the starting point, i.e. the steady state solution, both root and tip vortices are radially aligned at the same azimuthal 4
1 0.8 0.6 0.4 Mean axial velocity 0.2 0 0.5 1 1.5 2 radial position [R] a mean 0.1 0 Mean azimuthal velocity 0.1 z=8 rotor pos z=9 0.2 z=10 z=11 0.3 0 0.5 1 1.5 2 radial position [R] a prim mean 0.6 0.4 0.2 0 0.2 0 0.5 1 1.5 2 radial position [R] 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 radial position [R] Figure 3. The figure shows mean axial and azimuthal velocities and mean axial and azimuthal induction factors at different axial positions at and behind the rotor. angle. In figures 6 and 7 one can see the further development of the instabilities of the root vortex and also the starting point of the oscillation of the tip vortex. Note that there seems to be two root vortices at a azimuthal position between the tip vortices. One may think that there is a second root vortex originating from one of the other blades, but it is more likely that the root vortex has started to break down into smaller scales. The two vortices are however radially aligned at an azimuthal angle between the azimuthal angle of the tip vortices. The development of the instabilities then continues in figures 8 and 9. In figure 9 one can still identify a tendency, that the root vortex position is preferred at an azimuthal angle in between the azimuthal angle of the tip vortex. At a later stage the influence of the boundary condition starts to come into play. After about 6000 unsteady iterations the flow field becomes stable again and corresponds to the steady state solution. The azimuthal position of the root vortex is now back to the original position of the steady state solution, where root and tip vortices are radially aligned at the same azimuthal angle. 3.3. Growth rate of instabilities Figure 10 shows a contour plot of the vorticity when the basic flow, i.e. the converged steady solution, has been removed. The analysis makes it possible to follow the growth rate of the instabilities. Future work will concentrate on evaluating the growth rate and frequencies of the instabilities. The relative position between tip and root vortex, i.e. different pitch, and how that relative position influences the stability will also be studied. 4. Conclusions The basic mechanisms behind the break down of vortices behind a wind turbine has been performed. In real life conditions the wake is also effected by turbulence, shear etc. The basic mechanisms are however still the same and it is therefore important to understand them to be able to base park simulation methods on realistic physical models. From a numerical point one can conclude that the periodic boundary condition effects the stability analysis to the extent that the instability modes found are suppressed. Even though the instability modes are found in the simulation using periodic boundary conditions, they are suppressed by the b.c. s. After a certain number of iterations there is most likely a limiting mechanism holding back certain instability modes that does not have 120 degree symmetry. This will hopefully be resolved in future work where the growth rate and frequencies of the 5
instabilities will be further investigated and the flow field from the periodic simulation will be compared to a full 360 degree mesh simulation. One can from the evaluation of the flow field conclude that there are tendencies pointing in the direction that there are stable positions of the root vortex in relative azimuthal position to the tip vortex. Future work and more extensive analysis is required to quantify this. The work was supported by the Swedish Energy Agency. 5. References [1] Sørensen, J. N., Shen, W. Z., Numerical Modeling of Wind Turbine Wakes, Journal of Fluid Engineering, vol. 124, June 2002. [2] Mikkelsen, R., Actuator Disc Methods Applied to Wind Turbines, MEK-FM-PHD 2003-02, 2003. [3] Sørensen, N. N., General purpose flow solver applied to flow over hills, PhD Dissertation, Ris National Laboratory, Roskilde, Denamark, 1995. [4] Michelsen, J. A., Basis3D - a platform for development of multiblock PDE solvers, Report AFM 92-06, Dept. of Fluid Mechanics, Technical University of Denmark, DTU, 1992. [5] Michelsen, J. A., Block structured multigrid solution of 2D and 3D elliptic PDE s, Report AFM 94-06, Dept. of Fluid Mechanics, Technical University of Denmark, DTU, 1994. [6] Ivanell, S., Numerical Computations of wind turbine wakes, Licentiate thesis, ISSN 0348-467X, KTH, 2005. [7] Tjaereborg Data, http://www.afm.dtu.dk/wind/tjar.html [8] Okulov. V, Sørensen, J. N., Stability of the helical tip vortices in the rotor far wake, Journal of Fluid Mechanics, vol. 576, p 1-25, 2007. Figure 4. The figure shows the solution after 4200 iterations at a position from about 4.25 to 6.25 radii downstream of the ACL(Acuator Line). That is 2200 iterations of unsteady simulation. The unit of the axis are R (turbine radius) Figure 5. The figure shows a slice orthogonal to the flow direction after 4200 iterations at 5R down streams of the ACL. The unit of the axis are R (turbine radius) 6
Figure 6. The figure shows the solution after 4400 iterations at a position from about 4.25 to 6.25 radii downstream of the ACL. That is 2400 iterations of unsteady simulation. Figure 7. The figure shows a slice orthogonal to the flow direction after 4400 iterations at 5R down streams of the ACL. Figure 8. The figure shows the solution after 4800 iterations at a position from about 4.25 to 6.25 radii downstream of the ACL. That is 2800 iterations of unsteady simulation. Figure 9. The figure shows a slice orthogonal to the flow direction after 4800 iterations at 5R down streams of the ACL. 7
Figure 10. The figure shows the vorticity when the basic flow has been removed at 5000 iterations at a position from about 4.25 to 6.25 radii downstream of the ACL. That is 3000 iterations of unsteady simulation. 8