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1 Introduction

During their life time, wind turbines are exposed to a large amount of load fluctuations. They are caused for example by atmospheric turbulence [9], the wake of other wind turbines [22], complex terrain [18], yaw misalignment [19] or tower blockage [17]. Those load fluctuations cause fatigue, which can have a negative influence on the life time of the wind turbines. To reduce them, load alleviation systems gain in importance.

In the course of the DFG PAK 780 project, different active and passive load reduction systems are investigated experimentally and numerically. Besides 2d investigations (e.g. [3]), the concepts are also investigated in the rotating environment of a wind turbine. As the implementation of the systems on a real wind turbine is difficult to realise at such an early state of development, the model wind turbine BeRT (Berlin Research Turbine) was designed and built by the Technical University of Berlin in cooperation with the SMART BLADE GmbH [14]. The turbine has a rotor radius of \(R=1.5\) m and exchangeable blades whereby different load reduction systems can be investigated. The turbine rotates with 180 rpm and the wind velocity amounts \(v_{inflow}=8\) m/s, leading to a tip speed ratio of \(\lambda =3.53\). The blades have a single airfoil cross section which is based on the Clark-Y airfoil. This airfoil was chosen because of its ability to provide attached flow for low Reynolds numbers and its good efficiency with leading and trailing edge flaps.

Model wind turbines like BeRT are often investigated in the controlled environment of a wind tunnel whereas full size turbines operate under free stream condition. To avoid wind tunnel wall interference an open jet section, as used in the MEXICO project [16] or in the INNWIND.EU project [10], can be used. In a closed test section, the blockage ratio, which is defined as the rotor swept area divided by the wind tunnel cross-section area, should not exceed a limit of \(10{\%}\) as suggested by Schümann [20], who already experienced an influence of the wind tunnel walls for a blockage ratio of approximately \(14{\%}\). Sørensen et. al. [21] investigated the loads on the NREL PHASE-VI model wind turbine in a wind tunnel with closed test section and a blockage ratio of \(8.8{\%}\). Thereby, only small blockage effects occurred. Krogstad and Lund [11] made experimental and numerical investigations on the performance of a model wind turbine with a blockage ratio of \(11.8{\%}\) and they expected only small blockage effects, too. The BeRT turbine will be investigated in the closed test section of the GroWiKa of the TU Berlin. There, a blockage ratio larger than \(40{\%}\) has to be expected and consequently wall interactions will have a significant impact on the aerodynamic characteristics of the wind turbine.

Within the DFG PAK 780 project, the turbine was already simulated with a large-eddy simulation (LES) approach under free stream condition [5]. However, for the BeRT turbine, the influence of the wind tunnel walls should not be neglected. Therefore, the influence of the limited space in a generic wind tunnel with high blockage ratio on the performance of the BeRT wind turbine is investigated in the present paper. First a description of the numerical setup is given as well as the results of a grid convergence study according to Celik [2] under free stream condition. The CFD based numerical investigations focus for the time being on a pure rotor, which is realised as a \(120^{\circ }\)-model of the turbine with periodic boundary conditions. Loads, axial induction and the angle of attack along the blade of the simulation with free stream condition (hereafter referred to as farfield) are compared to two simulations including a generic wind tunnel, one with walls realised as slip walls and the other as no-slip walls. With these preliminary investigations, a better understanding of the effects caused by turbine wall interaction can be achieved and used for the improvement of future simulations of the real wind tunnel setup.

2 Numerical Setup

In the last few years, a process chain for the numerical investigations of wind turbines was developed at the Institute of Aerodynamics and Gas Dynamics (IAG) [13] which was also used in the present investigations. The simulations have been performed using FLOWer [12], a block structured code developed by the German Aerospace Center (DLR). It was extended for helicopter and wind turbine applications at the IAG. FLOWer uses the finite volume method to solve the compressible unsteady Reynolds-averaged Navier-Stokes equations (URANS). The temporal discretization is realised with an implicit dual-time stepping scheme [6] and the spatial discretization with the second order central discretization JST (Jameson-Schmidt-Turkel method) [7]. In the present fully turbulent approach, turbulence is modelled using the Menter SST turbulence model. The single components of the wind turbine are meshed separately and overlapped using the CHIMERA technique [1] whereby a body fitted mesh can be used for each component. For pure rotor simulations, as used in the present paper, only one blade of the turbine is simulated by applying rotationally periodic boundary conditions. Thereby, computationally efficient studies on the grid resolution and the influence of different parameters under uniform inflow can be performed. For all simulations, a steady simulation was run prior to the unsteady simulation in order to accelerate convergence. The time step for the unsteady simulation corresponds to \(3^\circ \) azimuth and 30–100 inner iterations were used per time step.

Each setup for all three cases (farfield, slip wall, no-slip wall) consists of five individual grids: blade, background, connection, hub and nacelle. The blade mesh is a CH-type grid which was generated automatically through a script developed at the IAG and has a fully resolved boundary layer ensuring \(y^+\)-values below one for the first grid layer. The boundary layer is resolved with 37 cells, 181 cells are used around the airfoil and 101 spanwise. The other meshes are generated manually. As a \(120^{\circ }\)-model is used for these preliminary investigations, a rectangular cross section of the wind tunnel is not feasible. The background grid has to be one third of a cylinder for all three cases. It is approximately 63 m (\(\approx 43 \cdot R_{BeRT}\)) long and the rotor plane is positioned in the middle of the domain and around the turbine, the cells have a maximum size of \(0.25 \times 0.25 \times 0.25\) m. The background grid with farfield condition has a radius of approximately 17 m (\(\approx 11 \cdot R_{BeRT}\)). According to Sayed [15], the size of the computational domain is sufficient. The grids with slip and no-slip walls, representing a generic wind tunnel with high obstruction, have a radius of 2.1 m, leading to a blockage ratio of over \(50\%\) and a constant distance between blade tip and wall of 0.6 m, which corresponds to the minimal distance in the experiment. The setup with slip walls has no refinement towards the wall, whereas the setup with no-slip wall has a refinement for the boundary layer at the wind tunnel walls, leading to \(y^+\)-values below 1.4 for the first grid layer. The nacelle of the turbine is rotating, too. A comparison to simulations with a non-rotating nacelle showed only minor differences between the loads (e.g. \(\varDelta power=0.019{\%}\)). The blade grid consists of approximately 5.5 million cells, the whole setup including the farfield grid consists of approximately 13.1 million cells, the setup with slip walls of approximately 11.4 million cells and the setup with no-slip walls of approximately 13.2 million cells.

Figure 1 shows the grids for the different components of the model wind turbine.

Fig. 1
figure 1

Grids of the \(120^{\circ }\)-model of the BeRT turbine

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3 Grid Convergence Study

For the setup with farfield boundary condition, a grid convergence study according to Celik [2] was performed in order to estimate the dependency of the numerical solution on the grid resolution and to find appropriate grids for upcoming simulations. Three setups of different resolution for all grids have been generated. The grid refinement factor between two setups has been chosen, as recommended by Celik [2], larger than 1.3. Table 1 shows the total number of cells and the refinement factor of the setups used in the grid convergence study.

Table 1 Number of cells for the grid convergence study
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42,000 steady iterations and 25 revolutions were simulated, whereas the last five revolutions were time averaged for the evaluation of the fine grid convergence index (\(GCI_{fine}^{21}\)), which estimates the error caused by the numerical grid, and the extrapolated error \(e_{ext}^{2}\) between the value of the medium grid and the extrapolated value of a theoretical ideal mesh. An extract of the results of the grid convergence study can be found in Table 2.

Table 2 Extract of the results of the grid convergence study
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Power has small values for the grid convergence index (0.38\({\%}\)) and the extrapolated relative error (0.63\({\%}\)), indicating a sufficient resolution of the medium grid. However, the driving force has higher values (\(GCI_{fine}^{21}=13.42{\%}\), \(e_{ext}^{2}=12.76{\%}\)). This is a result of the higher uncertainty of the driving force in the inner part of the blade. As power is computed by multiplying driving force and radius, the influence of the inner part of the blades is low and therefore the \(GCI_{fine}^{21}\) for power is smaller than for driving force. Thrust is the most important value concerning load reduction. Here, the grid convergence index and the extrapolated relative error are small (\(GCI_{fine}^{21}=0.001{\%}\), \(e_{ext}^{2}=0.02{\%}\)). Consequently, the medium grid is sufficient for the evaluation of thrust.

4 Results

Usually, wind turbines operate under free stream conditions. However, in order to investigate them, wind tunnels can be used as they can guarantee defined inflow conditions and consequently enable the investigation of different sets of parameters under identical conditions. Because of the wind tunnel walls, the free stream condition of the flow is restricted, as the walls, depending on the distance between blade tip and wall, can have an influence on the flow around the turbine. In order to estimate this influence, three different setups of the medium grid have been investigated whereby only the background grid was changed between the cases. The turbines have been simulated steady-state for 42,000 iterations prior to the unsteady simulation which was performed with a time step corresponding to 3\(^{\circ }\) azimuth and 30–100 inner iterations. Figure 2 shows the sectional thrust and driving force. The light graphs show the values every 3\(^{\circ }\) for the last five revolutions, whereas the darker curves are averaged over the last five revolutions. The values are normalized with the maximum of the averaged farfield case.

Fig. 2
figure 2

Normalized, sectional driving force (upper) and thrust (lower) for the last five revolutions and averaged over the last five revolutions for all three setups

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Fig. 3
figure 3

Surface solution including stream traces of the suction side for the farfield (left), slip wall (middle) and no-slip wall (right)

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It can be seen that the farfield solution shows significantly smaller loads than the simulations that mimic the wind tunnel environment. The loads of the slip wall case are on average about 60\({\%}\) (driving force) respectively 25\({\%}\) (thrust) higher than the ones of the farfield case. The no-slip wall case is even higher (driving force approximately 70\({\%}\), thrust 30\({\%}\)). Moreover, for both wind tunnel cases, separation occurs not only in the root region, but also in the middle of the blade between \(\mathrm{r/R}=55{\%}\) and \(\mathrm{r/R}=85{\%}\) which can be seen at the thrust distribution and in Fig. 3. For the farfield case, no separation occurs on the suction side which is indicated by the attached stream traces and in the no-slip case the separation is slightly more pronounced than in the slip wall case.

The power of the wind in the present case is approximately 2080 W. The power averaged over the last five revolutions for the farfield case amounts to 755 W, leading to a power coefficient of \(c_P=0.36\), which is considerable smaller than the theoretical maximum value given by Betz [4] as \(c_{pBetz}=0.59\). However, the power coefficient for the wind tunnel cases exceed the one given by Betz (\(c_p\)(slip wall)\(=0.60\) and \(c_p\)(no-slip wall)\(=0.64\)).

The higher loads, power production and the separation are a result of the influence of the wind tunnel walls. Wind turbines extract energy from the wind in form of kinetic energy. As a result, the fluid in the stream tube behind the rotor plane has a lower speed than upstream of the rotor plane. To maintain mass continuity, the flow conduit has to expand behind the rotor. Under free stream condition this can happen without problems. However in the wind tunnel, the walls impede the expansion. On the one hand, the walls are physical boundaries which can not be passed. On the other hand, the fluid which bypasses the turbine does also need space or otherwise blocking would occur. In the converged condition, an equilibrium of the different aspects is achieved. As the fluid conduit has a smaller expansion, the fluid has a higher velocity behind the turbine compared to the farfield case. As the present expansion reduces the available space of the bypass fluid, the corresponding velocity compared to the area in front of the turbine is increased and tip vortices are transported faster. Figure 4 shows the velocity distribution around the turbine for the different cases.

Fig. 4
figure 4

Velocity distribution around the turbine. Top farfield, middle: slip wall, bottom no-slip wall. The flow field of the farfield case is cut off at the top

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In this paper, axial induction and angle of attack are extracted using the reduced axial velocity method according to [8] which can be used to extract the angle of attack and aerodynamic coefficients from steady wind turbine solution. To do so, the velocities in front and behind the rotor are extracted on arcs and the mean velocities in axial and tangential direction for each arc are determined by integration for every radius position. Afterwards the velocities are used to determine the velocity in the rotor plane. However, with the present method, tilt, tower, boundary layer or atmospheric turbulence can not be taken into account.

As the axial velocity around the turbine and in the rotor plane is higher compared to the farfield case, the axial induction (Fig. 5) in the wind tunnel cases is smaller than in the farfield case, leading to larger angle of attack (AoA) over the blade (Fig. 6) and consequently to higher loads and power. All the effects mentioned above are more distinct for the no-slip wind tunnel walls, as the displacement thickness reduces the space even more.

Fig. 5
figure 5

Axial induction over the normalized blade radius

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Fig. 6
figure 6

Angle of attack over the normalized blade radius

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The higher AoA distribution does not only has a direct influence on the loads, it also leads to flow separation on the suction side of the blade (Fig. 3) which has effects on the loads (Fig. 2), too. Due to the proximity between wind tunnel walls and blades, no free stream condition prevails in the wind tunnel as the pressure around the turbine is not constant. Consequently, the Betz limit, which is only applicable for free stream turbines [4], is no longer valid and can therefore be exceeded.

The flow field around the turbine develops in the farfield case, due to the large available space, slowly, leading to a long simulation time until the full expansion of the wake is achieved. In consequence, AoA distribution and axial induction converge slowly, which has in turn influence on the flow around the blades. When the flow is fully developed, the AoA distribution doesn’t decrease further and no more separation occurs in the present case (Fig. 3). Because of the limited space in the wind tunnel, the expansion is less pronounced there and the convergence of the loads is achieved faster. In these cases, the AoA distribution does not decrease until the flow is completely attached. Because of the high axial induction in the root part of the blades, the velocity is slowed down to a high degree there. This effect is stronger for the farfield case, leading even to wake space and backflow behind the nacelle (Fig. 4).

5 Summary and Outlook

A \(120^{\circ }\)-model of a model wind turbine was investigated. A grid convergence study was performed in order to estimate the error caused by the grid and to find an appropriate setup for upcoming simulations. The model wind turbine under investigation, BeRT (Berlin Research Turbine), was designed and built by the technical University of Berlin in cooperation with the SMART BLADE GmbH [14] and has a rotor radius of \(R=1.5\) m. As the experiments will take place in a wind tunnel with high blockage ratio, the influence of wind tunnel walls on the performance of the turbine in a generic wind tunnel with a blockage ratio of approximately \(50\%\) was investigated by means of CFD. For this purpose, three different background grids were created. One representing the farfield condition and two, one with slip walls and the other one with no-slip walls, representing a generic wind tunnel. In the farfield case, the wake of the turbine can fully expand, leading to attached flow over the whole blade and a power coefficient below the Betz limit. In the wind tunnel cases the space is limited, leading to a speed up of the flow around the turbine, flow separation on the blades as well as higher loads and power production. The effects are even more pronounced for the no-slip case, as the displacement thickness of the boundary layer at the wind tunnel walls reduces the available space for the fluid in the wind tunnel even more. To sum up, the wind tunnel walls have a distinct influence on the performance of the wind turbine, as even the Betz limit is exceeded. This leads to the assumption, that the turbine behaves not like a free stream turbine any more and the wind tunnel walls must not be neglected in the simulation.

Full model simulations including tower and the real wind tunnel as well as simulations with yaw misalignment are planned as further steps. The nozzle, which follows the test section of the wind tunnel will also be taken into account, as first measurements have revealed an influence on the turbine performance. Different load alleviation systems (active and passive) will be investigated experimentally and numerically and the results will be compared. Moreover, a velocity deficit, representing a gust, will be realised in the wind tunnel and corresponding CFD simulations are planned too.