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Wind tunnel measurements using particle image velocimetry have been performed around two perforated discs, with varying streamwise distance, in order to simulate the wake interaction between wind turbines. The static pressure footprint (p-f) on ground level associated with the wake behind the disc and wake velocity data for both the streamwise and wall-normal velocity components with the corresponding turbulence intensities are reported. The p-f method shows that the size of the wake regions, behind the wind turbine models, initially drop when a second disc is placed just downstream of the first one. From a mutual distance (Δ χ) of about five disc diameters (5 D), both wake footprints increase as the mutual distance is increased, and for very large mutual distances, approximately Δ χ/ D > 15, the footprint of the downstream disc has recovered and is about the same as for a single disc. At last we conclude that despite very different inlet conditions to the discs, with about 50% of reduced velocity on the centre line upstream of the second disc and an increase of the maximum streamwise fluctuations by 90%, the mean velocities in the wake are proven to scale with the hub height velocity.

Inside a wind power farm, the flow field is strongly modified compared to the undisturbed flow. Behind each turbine there is a velocity deficit, which reduces the power output for the downwind turbines [

The flow around a porous disc differs from that around a solid disc. For the perforated disc, the interaction between the separated shear layers from the edges of the disc is disturbed by the flow through the disc, which gives the perforated disc an increased pressure behind the disc and thus a lower drag as compared to the solid disc. The formation region of the vortices is also larger, which gives the porous disc a higher Strouhal number as compared to the solid disc, as shown by Castro [^{3} and 10^{5}. The Strouhal number was found to be around 0.14 and slightly increasing with the Reynolds number. It was concluded that the anti-phase characteristics of positions located 180˚ apart were preserved, but that the shedding also had a random nature.

Experiments on a disc with approximately the same solidity and thrust coefficient as the ones used in the present experiment have previously been carried out by Medici [

Although a number of numerical and experimental studies have been performed with turbines modelled as porous discs, there is still a need for additional experimental data, both for gaining an increased understanding of wake interaction and for validation of numerical studies.

In the present paper we present static pressure and velocity data obtained with a pressure plate [

The measurements were performed in the atmospheric wind tunnel at the University of Gävle. The wind tunnel facility is a closed loop tunnel with a total length of 28 m and a maximum velocity of 22 m/s. The contraction ratio is 3:1 and the tunnel is operated by a 45 kW motor driving two parallel fans, which are placed in the return circuit. The tunnel is equipped with guiding vanes and honeycombs to improve the flow quality. The size of the test section is 11 × 3 × 1.5 m 3 (length ´ width ´ height).

The hub height (H) and diameter (D) of the discs were both 45 mm. The solidity of the discs was 56% and the thrust coefficient was 0.85, which was measured with a force balance. The present thrust coefficient ( C T ) is in the upper range of real wind turbines and is defined as

C T = T 0.5 ρ U hub 2 A , (1)

where T is the total thrust force, ρ is the air density, U hub is the velocity at hub height and A = π D 2 / 4 is the disc area. A picture of one of the discs is shown in

and the root-mean-square value of the fluctuations with u rms . The corresponding notation for the vertical component is W and w rms .

The pressure plate is a flat square metallic surface with 400 ( 20 × 20 ) uniformly distributed pressure taps. The taps are connected to a pressure transducer via an in-house built multi-pressure scanner system, and the data acquisition was done using LabView via a National Instrument data acquisition card. The pressure at the hub height was measured using a Prandtl tube. The static pressure at the surface was measured and statistically averaged over a period of 25 seconds for each tap. All the results presented from the static-pressure measurements in the present paper were taken at a free-stream velocity of U ∞ = 16 m/s.

Pressure footprints are plotted as contour plots, in non-dimensional form, as the pressure coefficient C p defined as

C p = Δ p q ∞ − B (2)

The dynamic pressure ( q ∞ ) is in turn defined as

q ∞ = p 0 − p ∞ = 1 2 ρ U ∞ 2 , (3)

where p 0 and p ∞ are the total and static pressures from a Prandtl tube mounted upstream of the turbine model in the ceiling and ρ is the air density. The pressure difference Δ p in Equation (2) corresponds to

Δ p = p i − p ∞ , (4)

where p i is the static pressure at the i^{th} pressure tap. B in Equation (2) corresponds to

B = p i clean / ( 1 / 2 ρ ∞ U ∞ 2 ) ,

where p i clean is the clean plate static pressure. This means that the C p shift in terms of B , in Equation (2), is a weak function of U ∞ . As long as U ∞ is constant B will not change.

The actual size of the pressure footprint, i.e. the area enclosed by a specific contour line using the C p distribution, is normalized by D 2 .

For the PIV measurements, a HI Sense Mk II camera was used, with a total number of 1344 × 1024 pixels. The physical size of each image was 0.103 × 0.078 m^{2} for the reference case with one disc and 0.097 × 0.074 m^{2} for the measurements with two discs. A total number of 1000 image pairs were acquired at each measurement position. For the evaluation, an adaptive correlation scheme was used, with 50% overlap and an interrogation area of 32 × 32 pixels. The sampling frequency was 6.1 Hz. The mean velocities and standard deviations are scaled with the velocity at hub height in absence of the discs, which was U hub = 7.0 m / s . The corresponding free-stream velocity for the PIV measurements was U ∞ = 8.8 m / s and the boundary layer thickness ( δ 99 ) was about 0.2 m at the location of the measurements. This gives δ 99 / H ≈ 4.4 , which models a suburb terrain with a typical gradient height of 360 m [

The reason for the free-stream velocity mismatch between the pressure measurements and the velocity measurements is related to the PIV image quality, which improved significantly at lower velocities. This was however realized after the pressure measurements, which was performed before the PIV measurements, and hence not repeated. However, for the present investigation where we only seek qualitative results using this pressure footprint method we believe that the mismatch of the free-stream velocity giving rise to a relatively small change in Reynolds number is of negligible importance.

Three different disc configurations (C1-C3) were investigated, which are summarized in

For the C1 case, measurements were performed in the region − 0.4 ≤ x / D ≤ 5.9 at 14 different spanwise positions located between 0 ≤ y / D ≤ 3.6 . The total covered distance of 6.3D in the streamwise direction corresponded to three different positions for the camera and laser, with a few millimetres of overlap between the measured planes. The vertical extent of each plane was

Case | x domain | y positions ( y / D ) | z domain |
---|---|---|---|

C1 | − 0.4 < x / D < 5.9 | [0 0.2 2.0 2.5 3.0 3.6] | − 0.7 < z / D < 1.0 |

C2 | − 0.3 < x / D < 7.4 | [0 0.3 0.5 0.6 0.7 1.1] | − 0.6 < z / D < 1.1 |

C3 | 1.7 < x / D < 7.9 | [0 0.3 0.5 0.6 0.7 1.1] | − 0.6 < z / D < 1.1 |

− 0.7 ≤ z / D ≤ 1.0 .

For the C2 and C3 cases measurements were performed in the streamwise ranges − 0.3 ≤ x / D ≤ 7.4 and 1.7 ≤ x / D ≤ 7.9 , respectively. Four and three different positions for the camera and laser were applied for C2 and C3, respectively, with a few millimetres of overlap between the measured planes, in order to cover the whole streamwise extents. The vertical extent was − 0.6 ≤ z / D ≤ 1.1 in both cases. Sketches of the setup for all three cases are shown in

An example of the pressure footprint behind a single turbine model located at ( x / D , y / D ) = ( 0 , 0 ) is shown in

m a x { C p } -value of the C1 case with hub height h / D = 1.0 , where max { C p } = 0.058 , i.e. a contour level value of ξ c p = − 0.02 . Here, the minus sign, simply indicates that it is the pressure footprint in the wake of the porous disc, which is of interest.

In

For the analysis of the interaction between two wind turbines using the p-f method, two porous discs were positioned on the same centre line with variable streamwise distance ( Δ x / D ). In Figures 4(a)-(d) contour plots of C p are shown with increasing relative distance ( Δ x / D ) between the upstream and downstream wind turbine model. The white contour lines correspond to the level ξ c p = − 0.02 . When a second disc is added closely behind the first disc one may observe that the wake footprints, both of the upstream and the downstream wind turbine, initially drop in size with respect to a single wind turbine. For a mutual distance in the range Δ x / D ≲ 5 the wake size ratio of the two models is

about two, with the downstream one being larger than the upstream one. Thereafter the wake footprints increase in size as the mutual distance is increased, and for very large mutual distances, approximately Δ x / D > 15 , the footprint of the downstream wind turbine has recovered and is about the same as a single wind turbine. This result is quantified in

The inflow conditions were measured with hot-film anemometry without the presence of discs. The measurements were taken in the middle of the test section, at x / D = 0 . The hot-film data was used to validate the PIV data, which showed good agreement. The results for the mean streamwise velocity and the streamwise turbulence intensity are shown in

Cross-sectional planes of the mean streamwise velocity for the C1 case are shown in

U d = U hub + U w 2 , T = m ˙ ( U hub − U w ) (5)

By combining these two equations and using U hub = 7.0 m / s and thrust coefficient C T = 0.85 , the scaled disc velocity is calculated to U d / U hub = 0.7 and the scaled wake velocity to U w / U hub = 0.4 . This suggests a faster spreading of the wake as compared to the experimental data shown in

to this theory, a wake velocity of U w / U hub = 0.4 would in this case correspond to a wake radius of R w / D = 0.7 .

The wake is also shown for different spanwise positions in

direction, i.e. a sign of the spreading of the wake. At y = 1 D (not shown here) the trace from the wake can no longer be seen.

The region immediately behind the disc at y = 0 is shown in more detail in

The mean streamwise and wall-normal velocity components for the C2 case are shown in

The corresponding turbulence intensities can be seen in

wake at y / D = 0.5 (

The inflow conditions for the first and the second disc in C2 are very different, and is shown in

intensity is generally higher behind the second disc (

A detailed comparison between all three cases (C1, C2 and C3) is shown in

y = 0.7 D and y = 1.1 D . For the inner part of the wake ( y = 0 and y = 0.5 D ), it can be seen that the velocity is lower for C3, especially closer to the second disc, at x = 3.5 D . On the centre line ( y = 0 ) at x / D = 3.5 the minimum velocity in the wake for the C2 and C3 cases are reduced by 21 and 46%, respectively, with respect to the C1 case. This reduction is expected, since it is likely that the upstream influence of the second disc will be stronger approaching the second disc. These differences can however not be seen further out in the wake in the spanwise direction ( y = 0.7 D and y = 1.1 D ). The streamwise turbulence intensity in

Particle image velocimetry and static pressure measurements of the wake flow behind two perforated discs have been performed in a wind tunnel. The discs have a diameter D = 45 mm and a thrust coefficient C T = 0.85 . The discs were positioned in three different configurations: a reference case with one disc (case C1), with the disc 5D apart in the streamwise direction (case C2) and with the discs 4D apart in the streamwise direction (case C3).

For C2, the inlet conditions for the two discs (i.e. the flow 0.25D upstream of

each disc) were significantly different: the mean velocity at the centreline was reduced with about 50% and the maximum streamwise fluctuation was increased by 90% for disc 2, as compared to disc 1. Despite the different inlet conditions, the mean velocity 2D behind each disc remained fairly similar when scaled with the hub height velocity. The turbulence intensities were however in general higher behind the second disc, the difference was around 20% at the upper edge of the disc. Decreasing the distance between the discs from 5D to 4D seemed to have a somewhat larger effect on the mean velocity than on the turbulence intensity. The presence of the second disc was also found to have a significant effect on the wake recovery, already at 1.5D downstream of the most upstream disc.

The pressure distribution at ground level in the wake of a wind turbine, showing the pressure footprint (p-f) of the model turbine, is believed to provide valuable information that can be used for both qualitative and quantitative wake analyses. Detailed information of the pressure and the velocity fields together can be utilized to understand the intricacies of the complex phenomena of wake interaction with good fidelity. In this investigation we show that, after an initial drop in size of the wake footprints when a second disc is added closely behind a first disc, the wake region of the downstream disc is a factor of two greater than the upstream one provided that the mutual distance between the turbine models is in the range Δ x / D ≲ 5 . We can also conclude that, after about Δ x / D = 5 , the wake footprints increase as the mutual distance is increased and for very large mutual distances, approximately Δ x / D > 15 , the footprint of the downstream wind turbine has recovered and is about the same as for single wind turbine.

We believe that the p-f method provides valuable pictures of the static pressure field on ground. The resolution depends on the number of taps and inter-tap spacing i.e. number of taps per unit surface area of the plate. This method is simple, non-intrusive, economical and a fast measuring technique that can be used for field measurements using microbarometers. This technique provides pictures of the wake in the xy-plane (i.e. horizontal plane) and thus regions with strong interaction can be identified immediately.

Future experiments with a higher spatial resolution of the static pressure measurements as well as the PIV measurements are planned, with the main aim to find a correlation relation between the ground pressure and the velocity deficit in the wake behind wind turbine models. In order to develop such a correlation relation it would be favorable to measure in a uniform stream across the wind turbine model and not in shear layer as have been done in the present investigation.

University of Gävle is acknowledged for funding the project. We thank the workshop technicians and research engineer Leif Claesson at the Atmospheric Boundary Layer wind tunnel, University of Gävle, for their valuable help during the experiments.

Khan, M., Odemark, Y., Sandberg, M. and Fransson, J.H.M. (2017) On the Wake Flow Interaction between Model Turbines with Varying Streamwise Distance. Open Journal of Fluid Dynamics, 7, 557-578. https://doi.org/10.4236/ojfd.2017.74038