An Experimental Investigation of Wind-farm Flows by Blas Muro July 2014 Technical report from Royal Institute of Technology KTH Mechanics SE-100 44 Stockholm, Sweden i Abstract In this thesis, fundamental properties of the turbulent flow above different wind-farm models were determined by means of wind-tunnel measurements. The assessed wind farms consisted in two staggered configurations, and two inline configurations, where two different streamwise spacings were evaluated. The experiment was focused on dense wind farms: the spacing in the spanwise direction was fixed to approximately 3d (where d indicates the rotor diameters) for every case, and two streamwise spacings were used: 2.5d and 5d. Freelyrotating turbines were used to perform this experiment. The wind-turbine models had a diameter of 45 mm and a height of 85 mm from the ground to the top tip. The wind-farm models were placed one at a time inside the test section of the KTH NT2011 wind tunnel, where the inflow was completely flat, i.e there was no simulated atmospheric boundary layer. X-wire anemometry was the measurement technique to measure the streamwise and wall-normal velocities above the wind farms. From the results, it could be observed that close to the turbines, the streamwise mean velocity had variations in the spanwise direction for inline and staggered configurations, even deep downstream on the wind farms. Horizontal averages were applied to the data to account for the inhomogeneity of the properties above the wind farms. A scaling behaviour was found on the flow above all the wind farms studied: in order to have the streamwise mean velocity profile, all that was needed was the boundary layer thickness, the free-stream velocity, and the streamwise velocity close to the top tip of the turbines. Other scaling behaviours were found for the Reynolds stresses. The dispersive stresses above different wind farms did not scale with the friction velocity, since it was seen that these stresses are highly dependent on the wind farm layout. Therefore, it was concluded that Reynolds stresses and dispersive stresses cannot be compared to each other because they come from different scales: the small scales and the large velocity scales, respectively. An equation to estimate the friction velocity above wind farms was derived, requiring measurements of the angular velocity of the turbines, the free-stream velocity, and the mean velocities close to the tip of the turbines. Finally, it was found that the angular velocity of the turbines was 25 % higher for the staggered arrangement, when comparing it with an inline wind farm. ii Acknowledgement First of all, I am thankful to the almighty God and Mary the Virgin for giving me the necessary willingness and determination for completing this thesis. A big thank you to my family, who were always there supporting me, and cheering me up during the bad times. To my cousin Laura Urbani who helped me in the reviewing process.To my cousins Carolina and Ricardo, who are not anymore with us, I love you wherever you are. A really special thank you to my supervisor Antonio Segalini for giving me the golden opportunity of writing my master thesis in KTH. I would also like to thank him for his daily and outstanding help in this project, for having the needed patience when explaining totally new things to me, and for supporting me in my career decisions. It was really fun to work with you. I would like to thank my Alma mater Universidad Simón Bolı́var, that despite the adversities in my country, they gave me the opportunity to go abroad, and experience a fruitful exchange year. I would like to acknowledge Jan Åke Dalberg for lending us his wind-turbine models. To Ramis Örlü, for having the patience of building the X-wire probes for this experiment. To my fellows Tomas Rosén and Marcus Winroth for those good times in the office, and for teaching me Swedish. To all the guys in the lab for the support, afterworks and Monday’s fikas. I will never forget you. To my friends that made easier the way until the end, and also for making special those great weekends in Stockholm, and around Europe. It was such a wonderful time. Thank you all. iii Contents Abstract ii Acknowledgement iii Chapter 1. Introduction 1.1. Wind-farm layouts 1.2. Motivations & aims 1 3 4 Chapter 2. Theoretical background 2.1. Basic fluid mechanic governing equations 2.2. Turbulence 2.3. Wind-turbine aerodynamics 2.4. Force model 7 7 9 17 21 Chapter 3. Experimental apparatuses & setup 3.1. Wind tunnel description 3.2. Prandtl-tube measurement 3.3. Photodiode 3.4. Hot-Wire anemometry 3.5. Traversing system 3.6. Wind-farm model 23 23 23 24 25 32 33 Chapter 4. Experimental procedure 4.1. Base flow measurements 4.2. Inflow-RPM correlation 4.3. Box measurements 4.4. RPM measurements 4.5. Farm measurements 36 36 38 38 44 44 Chapter 5. 45 Results and discussions iv 5.1. Box 2 × 32.5d flow 5.2. Box 3 × 32.5d flow 5.3. Scaled Flow 5.4. Comparison between Reynolds stresses and dispersive stresses 5.5. Farm 2 × 32.5d case 5.6. Farm 3 × 32.5d case 5.7. Comparison between farm 2 × 32.5d and 3 × 32.5d 5.8. Force model validation Chapter 6. Summary and conclusions Bibliography 45 48 51 53 57 62 67 69 72 75 v CHAPTER 1 Introduction Over the years, the study of wind energy has become more and more important. The principal motivation of this trend has been the exponential growth of the wind-energy industry in the last two decades (see Figure 1.1). The reason of this growth is the fact that considerable amounts of electricity are generated using the most clean and cheap fuel, as the wind. Figure 1.1: Global cumulative installed capacity 1996-2013 (from gwec.net) The global prosperity of this type of energy source can be considered as a fact, since many wind farms have been installed in different parts around the world such as Europe, America, and Asia (see Figure 1.2). Furthermore, the trends have shown an increment of the wind-energy industry in developing countries (like those of South America) over the last 3 years, observation that generates an open question: is wind energy an expensive energy source that only highly-developed countries can afford? A wind turbine is a machine that converts the wind’s kinetic energy into electrical energy. This process is obtained by the extraction of momentum from the air (work done by the blades). The rotor is connected by a shaft to a generator which produces electricity and is further distributed to the main electric grid, and then to different places such as houses, companies, etc. 1 Figure 1.2: Annual capacity installed by region 2005-2013 (from gwec.net) The evolution of wind turbines has a long history. There are evidences that two hundred years BC a wind wheel was made by the Persians. But, it was not until the 7th century that a windmill with practical purposes was built in Iran. At that time, the windmills, as the name says, were used for milling different kind of grains to make flour and for drawing up water from wells. Later, in 1887 the first wind turbine was built by James Blyth in Scotland. Afterwards, that same year an American scientist (Charles Brush) built the first automatically operated 12 kW wind turbine. After that moment, it was not until the middle of 1900s that people in the northern countries of Europe (Denmark, Germany, and Sweden for instance) started to build high-power turbines, and started gathering them in the now well known wind farms, with the purpose of producing significant amounts of electrical power from the wind. There are several types of wind turbines, but the most common are the horizontal and vertical axis turbines. As the names show, those turbines only differ in the shape while keeping their functions the same. In this thesis, only the Horizontal Axis Wind Turbine (HAWT) is going to be studied and a brief description of this one is presented. Horizontal turbines consist of a tower on which a set of blades (rotor) and a generator are connected by a shaft, and are installed perpendicular to the tower axis. It is also relevant to comment that horizontal turbines are overall to be seen in wind farms, in contrast to the vertical axis type because the former occupy less surface area. 2 Figure 1.3: An sketch of a Horizontal Axis Wind Turbine (from englishecoenergy.com) 1.1. Wind-farm layouts A wind farm is a group of turbines placed close to each other in a certain space with the purpose of producing energy. Wind farms could be organised in infinite ways, but several applications and studies (Cal et al. (2010), Chamorro et al. (2011), Chamorro & Porté-Agel (2011), etc.) show that the most common layouts in wind farms are the staggered and the inline configurations. It is called inline configuration when the turbines rows are organised one behind the other. This configuration is important for the academic world due to its simplicity to be tested and simulated. For this reason, this arrangement is assessed in this experimental work. A sketch of this layout is shown in Figure 1.4. Alternatively, it in a staggered configuration the turbines are organised so that the wakes of one row do not affect the inflow of the very next row, which makes this arrangement more efficient than its inline counterpart when referring to the power produced by each turbine individually. In fact, it is important to comment that the staggered configuration is the most used configuration 3 Figure 1.4: Inline Configuration of Turbines in installed wind farms around the world, i.e. the wind farms are designed in a way that the most probable wind direction faces a staggered arrangement. For this reason, this arrangement is also assessed in this experimental work. A sketch of this layout is shown in Figure 1.5. Curiously, if the flow direction in an inline configuration is changed of 45◦ , the new wind farm layout would be staggered. Figure 1.5: Staggered Configuration of Turbines 1.2. Motivations & aims Having now defined the most basic concepts, it is important to discuss what other researches have shown in the past. First, it is relevant to comment that wind turbines are designed to operate independently from each other. However, in wind farms, turbines work in the wakes of other turbines. For this reason, many studies about wake models have been done in the fields of layout optimisations and fatigue prediction. One of the most famous wake model was presented by Frandsen (1992). Also, other models have been presented as the one in Barthelmie et al. (2007) and Frandsen et al. (2006). Furthermore, experimental studies of these flows have been performed by different authors, such as Chamorro & Porté-Agel (2009) where they demonstrate that behind an isolated turbine, the near wake deficit is axisymmetric, and that high levels of 4 turbulence intensity are reached in the wake of the turbines. Also, in Medici & Alfredsson (2008) and Medici & Alfredsson (2009), they studied the behaviour of the 3D wake of an isolated turbine, finding that for two and three bladed turbines, there is meandering in the wake at high rotational speeds. The question “does the flow above a wind farm behave as the flow above rough elements?” has been around in the community for some time, and some researches have tried to model the wind farms as roughness applied to the bottom of the Atmospheric Boundary Layer (ABL). The surface is characterised by using a roughness length scale, which translates the effect of the wind farm on the ABL. Some of the studies that have tried to characterise the wind-farm flows using roughness lengths have been performed by Lettau (1969), Frandsen (1992), Frandsen et al. (2006) and Calaf et al. (2010). In many wind-farm studies, horizontal averages are usually used to simplify the analysis of wind-farm flows. For example, Cal et al. (2010) and Calaf et al. (2010) used the horizontal averaging to observe the behaviour of the flow on different wind-farm arrays. However, when using horizontal averages, new terms appear such as the Reynolds stresses; these terms come up when the statistical data are not spatially homogeneous where the average is being done. When studying wind-farm flows, different approaches to characterise the boundary layer are used. For example, Chamorro & Porté-Agel (2011) performed hot-wire velocity measurements within and above a 3 × 3 inline configuration in a wind tunnel. They observed that there are two flow regions: one from the base of the turbines to the top tip, and the other from the tip to the free stream. Furthermore, the first region was fully developed after 4 turbine rows, unlike the other one that never reach developed flow. Additionally, another approach to wind-farm flows was presented by Calaf et al. (2010), where they observed that a better description of the flow would be given, if the flow was decomposed into three different regions: the region above the turbines, the rotor swept region, and the region between the lower tip and the base of the turbine. It is important to comment that the majority of the studies on windfarm boundary layers are performed by measuring the flow velocities on the centreline of the wind farm. For this reason, the first objective of this thesis is to perform velocity measurements above different wind-farm models accounting for the spanwise variation of the flow properties. As done by different authors, the space average of the flow properties is going to be performed for each assessed configuration, and comparisons between these quantities are going to be done, as well as comparisons of the dispersive stresses for different cases. Furthermore, the growth of the space-averaged boundary layer is going to be shown and compared between each of the cases studied. 5 When describing the velocity profile on a wind farm, it is usually done by calculating the roughness length of the model. However, there is a problem with this approach, and that is the difficulty of replicating an atmospheric boundary layer in the wind tunnel. Hence, an expression to estimate the velocity above different wind-farm models without the necessity to have an ABL is going to be derived. 6 CHAPTER 2 Theoretical background In this chapter, the necessary concepts of this work are going to be introduced, such as the equations of motion, basic fluid mechanics equations, an overview of turbulence, and time and space averaging. As in classical dynamics, the Newton’s 2nd law is the governing law for the movement of an object F = m · a. In fluid mechanics the equations of movement are the Navier-Stokes equations, which are the Newton’s 2nd law applied to a control volume, and combined with the continuity equation. 2.1. Basic fluid mechanic governing equations “The movement of fluids is the movement of a great amount of molecules from one point in space to another as a function of time” Munson et al. (2009), but studying each one of these molecules is considered really difficult, hence, the definition of Fluid Particles came up. A fluid particle or parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. The mass of the fluid particle is constant as it moves. This definition states that the motion of a flow is described by particles of fluid interacting with each other in space and time. Keeping this definition in mind, and introducing a frame of reference (x, y, z), the velocity field of a particle can be represented as in Equation 2.1 where u, v and w are the x, y and z velocity components respectively. u(t, x, y, z) = u(t, x, y, z)ı̂ + v(t, x, y, z)̂ + w(t, x, y, z)k̂, (2.1) In order to solve problems regarding fluid physics, questions about the movement of the fluid are assessed in a bounded volume. These discrete volumes are called Control Volumes. The later concept is of high importance in physics, since several fundamental laws of physics are applied on these control volumes, such as Newton’s laws in a fluid, mass conservation, thermodynamic laws, etc. The mass conservation for a fixed, non-deforming control volume (Continuity Equation) is: ∂ ∂t Z Z ρ u · n̂ dA = 0, ρ dV + CV S 7 (2.2) which states that the time rate of change of the mass inside the control volume, plus the net rate of mass flow passing through the surfaces of the volume is equal to 0, since mass cannot be created nor destroyed. Another important governing expression in fluid mechanics is the linear momentum equation or Newton’s second law which states that in an inertial frame of reference, the total force applied to a body is equal to the time rate of change of momentum, namely Fext = ∂Q . ∂t When applying the previous equation to a fluid in a fixed, non-deformable control volume, the following equation is obtained: Fext ∂ = ∂t Z Z u ρ u · n̂ dA ρ u dV + CV (2.3) S which states that the total external force is equal to the time rate of change of the linear momentum inside the control volume, plus the net rate of flow of linear momentum through the control surface. When studying fluids, it is important to talk about energetic quantities. In order to account for these contributions, the 1st law of thermodynamics is used, which states that the time rate of the total energy of a given control volume is equal to the sum of the rate of work done to what is inside the control volume, plus the rate of heat supplied by the universe to the control volume. In symbolic form, this statement is: Q̇received + Ẇdone = ∂ ∂t Z ρ (e+kuk2 /2) dV + CV Z (e+kuk2 /2) ρ u· n̂ dA, (2.4) S where e is the internal energy per unit mass. It is also important to note that, all the governing equations shown before are in Eulerian notation, i.e. the equations are defined for a fixed, nondeformable control volume. The equations mentioned above were focused to describe the properties and behaviour of a fluid within the boundaries of a control volume. Sometimes, this approach is not enough when analysing fluids, because it does not take into account variations of fluid properties such as pressure and velocity inside the control volume. Therefore, if a detailed study of the fluid is required, a differential analysis of the flow has to be done. The basics of this approach are the same as the ones used in the control volume analysis, but in this analysis, smaller (infinitesimal) control volumes are used to describe the fluid properties. The mass conservation law (Continuity equation) in differential form is given by the following expression: 8 ∂ρ ∂ρu ∂ρv ∂ρw + + + = 0. (2.5) ∂t ∂x ∂y ∂z For incompressible flows the density is constant, and Equation 2.5 reduces to: ∂w ∂u ∂v + + =0 (2.6) ∂x ∂y ∂z and the momentum conservation equation in the three coordinates reduce to: ∂u ∂u ∂u ∂u 1 ∂p +u +v +w =− +ν ∂t ∂x ∂y ∂z ρ ∂x ∂v ∂v ∂v ∂v 1 ∂p +u +v +w =− +ν ∂t ∂x ∂y ∂z ρ ∂y ∂w ∂w ∂w 1 ∂p ∂w +u +v +w =− +ν ∂t ∂x ∂y ∂z ρ ∂z ∂2u ∂2u ∂2u + 2 + 2 ∂x2 ∂y ∂z ∂2v ∂2v ∂2v + + ∂x2 ∂y 2 ∂z 2 − fx (2.7) − fy (2.8) ∂2w ∂2w ∂2w + + ∂x2 ∂y 2 ∂z 2 − fz (2.9) where the left hand side of the equations represents the acceleration terms, and the right hand side represents the force terms (pressure, shear, and body force). Another important equation in fluid mechanics is the so called Bernoulli Equation which states that for an incompressible inviscid flow, with conservative volume forces applied, the following expression is constant along a streamline: P kuk2 + + gZ = constant ρ 2 (2.10) It is important to remark that this equation could be seen as an energy 2 balance between kinetic energy kuk 2 , gravitational potential energy gZ, and pressure potency energy Pρ . 2.2. Turbulence Most flows in life are turbulent. For instance, flows at the boundaries of cars, airplanes, ships, wind turbines, etc. Also, most internal flows are turbulent, namely flows inside a pipe, inside internal combustion engines, and so on. It is relevant to comment that this kind of flows are as complicated as they are common. “Turbulent flows are chaotic and unsteady flows with high levels of vorticity distributed along different sizes of eddies, characterised by high difusivity between fluid particles, and by high dissipation of energy into heat” 9 (From Ferro 2012). The study of turbulence began in 1883 when Reynolds was involved in a work related to transition. At that time, he defined a nondimensional number (the Reynolds number) Re = ud ν , and he noticed that when this number reached a certain value in the flow inside a pipe (where u is the velocity, d the pipe’s diameter and ν the fluid’s kinematic viscosity), the flow became irregular. He noticed that, under that regime, the instantaneous flow characteristics were not predictable, but they were for the mean flow quantities. For this reason, he suggested to divide the flow variables into mean and fluctuating contributions. After some years of development (1895), he introduced the Time Averaged Navier-Stokes equations or Reynolds averaged Navier-Stokes (RANS) equations. Before introducing the RANS equations, it is important to introduce some statistical tools for the analysis of turbulence. 2.2.1. Statistical analysis of turbulence In order to understand turbulent flows, it is necessary to introduce a statistical operator, as it is the time average. The time average of {qn (x, y, z, t); n = 1...N } is the time integral of this property (q) in a certain time period, but for steady flows, the time average becomes the arithmetic mean of a large number (N ) of samples, as q=Q= 1 T Z T q(x, y, z, t)dt = 0 N 1 X q(n) N n=1 (2.11) This is the usual form used for steady flow situations, but it can also be used for non-stationary flows if the timescale for the mean flow variation is much larger than the timescale of the turbulence fluctuations. After introducing the time average, the Reynolds decomposition can be written 0 q =Q+q , (2.12) where the time series of a signal (q) is equal to the time-averaged value (Q), 0 plus the fluctuations (q ). After defining the fluctuations, the variance of q is defined as the average of the square of the fluctuation q 0 2 , and the fluctuation intensity or “rms” value (standard deviation) is defined as the square root of the variance σq = (q 0 2 )1/2 . On the other hand, when measuring two time series at the same time, a correlation of the two different signals can be done, this correlation is called covariance. Assuming that two time series of velocities are measured (u and v), the covariance would be the average of the product of their fluctuations (u0 v 0 ). Furthermore, when studying turbulence a correlation coefficient of these two signals is usually computed in the following way: 10 u0 v 0 (2.13) σu σv It is important to comment that the flow under study in this thesis is considered steady, and consequently the time averages are going to be performed by applying Equation 2.11 to the velocities time series. The formulas used for calculating the time averages in this thesis are shown as follows: corr = u=U = N 1 X u(n) N n=1 (2.14) v=V = N 1 X v(n) N n=1 (2.15) u0 2 = N 1 X 02 (u )(n) N − 1 n=1 (2.16) v0 2 = N 1 X 02 (v )(n) N − 1 n=1 (2.17) σu = (u0 2 )1/2 (2.18) σv = (v 0 2 )1/2 (2.19) u0 v 0 = N 1 X 0 0 (u v )(n) N n=1 (2.20) 2.2.2. Reynolds averaged Navier-Stokes equations The complexity and nonlinearity of the Navier-Stokes equations is the reason why fluids are described in such a detailed way. Therefore, a simplification of these equations is usually done by time averaging them. This simplification was done by Reynolds, and these new equations are the so called Reynolds Averaged Navier-Stokes equations or RANS. 2.2.2a. Cartesian Coordinates and Einstein Notation. In order to facilitate the handling of the equations a different coordinates system is defined: u = uı̂ + v̂ + wk̂ = u1 ı̂ + u2 ̂ + u3 k̂ (2.21) r = xı̂ + ŷ + z k̂ = x1 ı̂ + x2 ̂ + x3 k̂ (2.22) 11 With the aim of having a compact way to express the RANS equation, Einstein notation is used. Next, some properties needed to understand this notation are described: ui ui = u1 u1 + u2 u2 + u3 u3 (2.23) ∂u1 ∂u2 ∂u3 ∂ui = + + ∂xi ∂x1 ∂x2 ∂x3 (2.24) ∂ 2 ui ∂ 2 ui ∂ 2 ui ∂ 2 ui + + = ∂xj ∂xj ∂x21 ∂x22 ∂x23 (2.25) Then, the Navier-Stokes equations and the continuity equation for incompressible flows in Einstein Notation become: ∂ui ∂ui 1 ∂p ∂ 2 ui + uj =− +ν − fi ∂t ∂xj ρ ∂xi ∂xj ∂xj (2.26) ∂ui =0 (2.27) ∂xi By using the concept of dividing the flow variables into mean and fluctuating contributions, the so called Reynolds decomposition becomes: 0 0 ui = Ui + ui and p = P + p , (2.28) 0 where ui is the velocity time series, Ui is the time-averaged velocity and ui is the velocity fluctuation of the ith velocity component. 2.2.2b. Reynolds averaged Navier-Stokes equations. The basic step of the derivation is to express the Navier-Stokes equation as a function of fluctuating and mean properties, and then taking the average of the whole equation: 0 0 0 1 ∂(P + p0 ) ∂ 2 (Ui + ui ) ∂(Ui + ui ) 0 ∂(Ui + ui ) + (Uj + uj ) =− +ν ∂t ∂xj ρ ∂xi ∂xj ∂xj and by averages properties, the first term gives: 0 0 0 ∂(Ui + ui ) ∂Ui ∂ui ∂Ui ∂ui ∂Ui = + = + = , ∂t ∂t ∂t ∂t ∂t ∂t since the average of a fluctuating contribution is equal to 0. Following the above properties, the second term becomes: 12 (2.29) 0 0 (Uj + uj ) 0 ∂u 0 0 0 0 ∂uj ui ∂uj ∂(Ui + ui ) ∂Ui ∂Ui 0 ∂u 0 = Uj + uj i = Uj + − ui , ∂xj ∂xj ∂xj ∂xj ∂xj ∂xj 0 0 where ui ∂xjj = 0 because of where it is stated that ∂uj ∂xj ∂uj ∂xj = 0, which comes from the continuity equation, = 0. Since the former relation is linear, it means that the same condition holds for the mean velocity ∂Uj ∂xj = 0, and the fluctuating 0 part ∂uj ∂xj = 0. The third and fourth term follow the same properties applied in the first term. Thus, the final equation has the following form: ∂Ui ∂Ui 1 ∂P ∂ + Uj =− + ∂t ∂xj ρ ∂xi ∂xj 0 ∂Ui 0 0 ν − uj ui ∂xj (2.30) 0 Being −ρ uj ui the so called Reynolds Stresses or turbulent stresses, which have the form of a tensor. It is shown as follows: 0 0 u u 1 1 0 0 −ρuj ui = −ρ u02 u01 0 0 u3 u1 0 0 u1 u2 0 0 u2 u2 0 0 u3 u2 0 0 u1 u3 0 0 u2 u3 . 0 0 u3 u3 The additional terms (Reynolds Stress) that came from averaging the nonlinear terms of the equation provide more unknowns to the equations of motion. Therefore, the Navier-Stokes expression proceeds to get more unknowns than the number of equations. This situation is known as the turbulent closure problem. It is important to comment that this problem is not going to be an issue in this scientific work, because the flow properties are going to be measured. Namely, all the statistical values, such as the mean flow, velocity fluctuations, variances, covariances, etc. are going to be calculated from wind-tunnel data. However, when a flow is simulated, the turbulence closure problem becomes an issue, and different turbulence models have to be applied (See Johansson & Wallin (2013)), but as it was mentioned before it is not a subject to consider in this work. 13 2.2.3. Spatial and time averaged Navier-Stokes equations Figure 2.1: Coordinates System When measuring the velocity profile above a plate, spanwise invariance in the flow is usually assumed. Namely, it is considered that the flow above the whole plate is exactly the same as the flow in the middle of it. However, when measuring the velocity profile above surfaces when there are objects on it, as the case of canopies and wind farms, it is known that the flow is highly three-dimensional, hence, this assumption would be false. But, because of its simplicity, it is still widely used by many authors. In order to avoid this assumption, but at the same time “keeping the definition” of having the same flow characteristics in a certain space, a spatial average of the flow quantities is done. In Figure 2.2, a schematic representation of the space-average decomposition is shown. From the image, it can be seen that the space-average decomposition is similar to the approach used for the Reynolds decomposition, but in this case the fluctuations are in space and not in time. Namely, it can be seen that the flow has different velocity values at different positions, but the flow is divided in a constant part, plus a spacefluctuating one. The measurement location taken into account for doing this spatial average is the one surrounded by the red box shown in Figure 2.3, i.e. between two rows of turbines. It is also important to comment that this average is done at each Y measurement height, as if there was a horizontal slab (see coordinates in Figure 2.1). Raupach & Shaw (1982) presented the spatial and time-averaged NavierStokes equations for flows within a canopy, which in this thesis are being extrapolated for flows within wind farms. Before introducing the final expression for these equations, it is necessary to introduce the averaging operator they 14 Figure 2.2: Spatial-average decomposition: a time averaged signal (U ) is equal to the sum of the space averaged-signal (hU i), plus the fluctuations in space 00 (U ). Figure 2.3: Horizontal Average used for the derivation of the equations. The formal definition of the average operator is hΨi = 1 A Z Z Ψ(x) dx dz (2.31) R where Ψ is a fluid property that is observed inside a region R (see Figure 2.3). Also, it is important to comment that it is assumed that all the properties being spatial averaged are already time averaged. The spatial decomposition of the averaged properties are: 00 Ui = hUi i + Ui and P = hP i + P 00 (2.32) where Ui is the time averaged velocity, hUi i is the spatial average of the time 00 averaged velocity, and Ui is the spatial fluctuation of the velocity. The basic step of the derivation is to express the time averaged NavierStokes equations as a function of fluctuating and mean properties, and then 15 taking the spatial average of the whole equations (Raupach & Shaw 1982). The time and space averaged Navier-Stokes equation in Einstein notation is given by the following expression: 00 00 ∂hUi i ∂ ∂ ∂hUi i 0 0 + hUj i + hU U i + hu u i = ∂t ∂xj ∂xj i j ∂xj j i * 00 + 00 1 ∂hP i 1 ∂P + ν∇2 hUi i + νh∇2 Ui i =− − ρ ∂xi ρ ∂xi (2.33) where: 00 00 • hUi Uj i = hUi Uj i − hUi ihUj i, and it is called the Reynolds dispersive stress, 00 • − ρ1 h ∂P ∂xi i is the form drag imposed by the wind turbines. 00 • νh∇2 Ui i is the viscous drag imposed by the wind turbines. For simplicity, Equation 2.33 is going to be modified by defining 1 − hfi i = νh∇ Ui i − ρ 2 00 * 00 ∂P ∂xi + , (2.34) where −hfi i is an equally distributed force applied by the wind farm to the flow. Finally, the spatial and time averaged Navier-Stokes equations are: 00 00 ∂hUi i ∂ ∂ ∂hUi i 0 0 + hUj i + hU U i + hu u i = ∂t ∂xj ∂xj i j ∂xj j i 1 ∂hP i =− + ν∇2 hUi i − hfi i ρ ∂xi Figure 2.4: Wind farm control volume (side view) 16 (2.35) When applying this equation to the flow inside the red control volume shown in figure 2.4 (inside the wind farm), the following assumptions are done: ii • ∂hU ∂t = 0 since the flow is assumed to be steady. i • hU i ∂hU ∂x ≈ 0 because the flow is assumed not to be accelerating downstream in the wind farm. • ν∇2 hUi i ≈ 0 because the viscous contributions are considered to be small. i ∂hU∞ i by differentiating the Bernoulli equation in the • − ρ1 ∂hP ∂x = hU∞ i ∂x free stream, and by using the thin boundary layer approximation, which states that the pressure does not change in the wall-normal direction. The force to take into consideration is the force in x1 or x, which is: 00 00 00 00 ∂hU∞ i ∂ ∂ 0 0 ∂ ∂ 0 0 hfx i = hU∞ i − hU U i − hu u i − hU V i − hu v i (2.36) ∂x ∂x ∂x ∂y ∂y where the variations of dispersive and Reynolds stresses in the x direction are assumed to be negligible, as well as, the contributions of the spanwise velocity W , leading to the following expression: 00 00 ∂hU∞ i ∂ ∂ 0 0 − hU V i − hu v i (2.37) ∂x ∂y ∂y Integrating Equation 2.37 from the floor to the top tip of the turbines (from G to 0 in Figure 2.4), the final expression of the force applied by the turbines to the fluid is obtained: hfx i = hU∞ i 00 00 ∂hU∞ i hU V i(0) hu0 v 0 i(0) Fx = hfx i = hU∞ i − − ∂x h h (2.38) where: 00 00 • hU V i(0) is the dispersive stress at the turbine-tip height. • hu0 v 0 i(0) is the Reynolds stress at the turbine-tip height. • h is the height of the turbine from the floor to the tip. It is important to comment that the Reynolds stresses and the dispersive stresses are equal to zero close to the floor, and for this reason they do not appear in Equation 2.38. 2.3. Wind-turbine aerodynamics A wind turbine is a machine that transforms kinetic energy from the air into electricity. This process is done by the extraction of momentum with a propeller (turbine) from the incoming air, obtaining rotational energy that is used to generate electricity. This extraction of momentum implies a reduction of flow 17 velocity i.e. the velocity behind the turbine (wake) has to be slower. Physically, the incoming flow perceives the presence of the rotor, and tries to avoid it. The streamlines try to diverge from the rotor but some are forced to go inside the rotor, and experience the extraction of momentum. When the flow pass through the rotor it experiences a continued reduction in velocity, and by Bernoulli’s theorem this monotonic reduction of velocity can be expressed as a pressure jump in the rotor due to the extraction of momentum. This difference in pressure creates a thrust force (T) to the rotor, which could be expressed by the following expression: T = Ad (Pu − Pd ), (2.39) where Ad is the swept area by the rotor, Pu is the pressure upstream the rotor and Pd is the pressure downstream the rotor which is lower than its upstream counterpart. 2.3.1. Actuator disk theory When observing the behaviour of the flow passing through a HAWT, it is common to assume that the rotor behaves like a frictionless porous disk with the same rotor diameter, infinitesimally thin, and uniformly loaded i.e. T is uniformly distributed in the disk area. This disk was defined by Froude and Rankine as the actuator disk. In order to show the theory, the first thing to do is to establish a control volume. This control volume is defined as a stream tube control volume for the sake of simplicity, since the surface-normal vector is perpendicular to the flow direction (U · n = 0) at some boundaries (see Figure 2.5), and as a consequence the surface terms in the conservation Equations 2.2, 2.3 and 2.4 vanish. Formally speaking, the control volume should not have the actuator disk inside it, but since all the properties vary continuously along the rotor except the pressure, this control volume is taken accounting for the contribution of the pressure discontinuity, in terms of an uniform force placed in the rotor, as it is shown in Figure 2.5. It is assumed that the velocity profile in each cross-section is uniform. Moreover, the flow is assumed steady, incompressible and azimuthally homogeneous. The next thing to do to derive the actuator disk theory is to apply the conservation Equations 2.2 (Mass conservation) and 2.3 (Momentum conservation equation). By applying the mass conservation equation to the control volume shown in Figure 2.5: ∂ ∂t Z Z ρ u · n̂ dA = 0, ρ dV + CV S 18 (2.40) Figure 2.5: Actuator Disk Control Volume (From Segalini & Ivanell (2013)) the first term becomes 0 due to the incompressibility condition, and the second term becomes zero at the surface (S) of the stream tube; the remaining terms are associated to the flow passing through the two caps of the control volume, and are equivalent to the mass flow passing through the control volume: Z Z Z ρ Uinf dA − ρ u · n̂ dA = S ρ Uw dA = 0, (2.41) Aw A∞ ṁ = ρUinf A∞ = ρUw Aw (2.42) It is important to remark that the flow needs to expand in the wake due to the reduction of velocity, phenomena that can be explained by looking at Equation 2.42. By recalling the momentum conservation equation: Z Z ∂ ρ u dV + u ρ u · n̂ dA (2.43) ∂t CV S and using a similar analysis to that used previously in the mass conservation equation, the moment conservation becomes: Fext = Z Fviscous + Fpressure + Fvolume − T = Aw ρUw2 dA − Z 2 ρUinf dA, (2.44) A∞ by assuming that the velocity profile is constant on Aw and A∞ Fviscous + Fpressure + Fvolume − T = ṁ(Uw − Uinf ), 19 (2.45) where pressure forces are 0 because the integral of the pressure on the external surface (S) is 0 (see Segalini & Ivanell 2013). It is stated that pw = p∞ , because it is assumed that the pressure is recovered in the far wake, since flows in the wake cannot sustain pressure gradients in steady conditions, and far from solid boundaries. Additionally, volume forces in the horizontal axis are 0, and viscous forces are 0, because the disk was assumed to be porous and frictionless. This leads to: T = ṁ(Uinf − Uw ) (2.46) which is highly important, since the value of the total force applied by the turbine to the fluid is represented on it. A relation between the velocities Uw , Uinf and Ud , is needed because in almost no case all the velocities are known. This relation is achieved by means of Bernoulli’s theorem downstream and upstream the turbine. Using Bernoulli’s equation (Eq 2.10) upstream the turbine, i.e. between points 1 and 2 in figure 2.5, one gets: 2 Uinf pu U2 p∞ + = + d, ρ 2 ρ 2 (2.47) doing the same between points 3 and 4, pd U2 p∞ U2 + d = + w ρ 2 ρ 2 (2.48) subtracting both Bernoulli balances, the following expression is obtained: pu − pd = ρ 2 Uinf U2 − w 2 2 ! . (2.49) By introducing Equations 2.39 and 2.49 in Equation 2.46, the following expression is obtained: Ud = Uinf + Uw 2 (2.50) which indicates that the disk velocity is equal to the arithmetic mean between the free stream velocity and the wake velocity. Therefore, this relation removes one of the variables in the system, and the thrust force can be expressed as: T = Ad ρ Ud (Uinf − Uw ) = ṁ(Uinf − Uw ) = Ad (pu − pd ), 20 (2.51) 2.3.2. Thrust coefficient A normalisation of the disk force introduced in Equation 2.51 is done by defining a characteristic dimensionless value called Thrust coefficient (Ct ), which is given by: Ct = 1 2 T , 2 ρ Ad Uinf (2.52) in this case the ideal normalising factor is a thrust force equivalent to the 2 dynamic pressure of the free-stream velocity ( 21 ρ Uinf ) applied in the disk area (Ad ). Introducing Equation 2.51 in Equation 2.52, the Thrust coefficient gives: Ct = Ad ρ Ud (Uinf − Uw ) 1 2 2 ρ Ad Uinf (2.53) 2.4. Force model A force model is derived in order to get the force distribution along the streamwise direction of the wind farm. The aim of this model is to provide valuable information about the force applied by the turbines to the fluid, by using a simple analysis based on the actuator disk theory. So, the force could be related with flow properties by inserting it in Equation 2.38. Over the years, the development of force models for the Navier-Stokes equations have shown a characteristic feature that is, many of them have been defined as a characteristic velocity squared divided by a length scale i.e −hf i ∼ U2 L , where the length scale in canopy studies is usually the height of the canopy. In this model, the same logic used for the canopies is going to be applied i.e the length scale would be represented by the height of the wind farm. The first step for the derivation of this model is to recall Equation 2.52: 1 2 , T = Ct ρ Ad Uinf 2 where T indicates the uniformly distributed force applied on the disk area Ad . In order to apply a similar approach to wind farms, the expression has to be dimensioned to sm2 , because the integrated Navier-Stokes equations have these units (see Eq 2.38). When thinking about the variables of this force, the following expression comes up to be: hfx i ∼ f (T, Nt , CV ∗, ρ) hmi (2.54) s2 where Nt is the number of turbines, CV* is the mini control volume bounded by the region R used for the spatial average (figure 2.3). After observing the 21 variables dependence of this force, the following reasonable expression of the streamwise force in a wind farm is derived: Fx = 2 Nt T Ct Nt Ad Uinf = Sx Sz h ρ 2 Sx Sz h (2.55) where: • • • • • • Sx = turbines spacing in the streamwise direction. Nt is the number of turbines in this particular row. Ct is the thrust coefficient of the turbines. Uinf is the inflow velocity of the turbines. Ad is the area swept by the turbine rotor. Sz = Sx [(Nt max )Lz ] where Lz is the spacing in the spanwise direction and Nt max is the maximum number of turbines in a wind farm row, i.e. In a staggered arrangement 3 X 2 Nt max = 3. • h = wind farm height. 22 CHAPTER 3 Experimental apparatuses & setup The aim of this thesis is to analyse different fluid quantities in the flow above wind farms, and the way to do this is by performing wind-tunnel experiments of various wind-farm models. The measurements were carried out in the NT2011 wind tunnel located in the Mechanics department at KTH. In the following sections, a description of the facility and the measurement techniques is going to be shown. 3.1. Wind tunnel description The NT2011 is an open circuit wind tunnel, which means that the flow goes from the entrance to the exit of the wind tunnel, without being directly recirculated. However, the facility is located inside a closed room, so the flow gets recirculated anyhow. The principal parts of the facility are the inlet, the contraction, the test section, the diffusor, the fan and the outlet, as they can be seen in the schematic model of the wind tunnel shown in Figure 3.1. The wind tunnel is used at a speed around 8 m/s over the whole experiment. The system is driven by a 15 kW DC fan located after the diffusor. The height of the test section is 500 mm, the width is 400 mm and the length is 1400 mm. In order to have a good flow quality in the test section (low levels of fluctuations), honeycombs and screens are located at the entrance of the wind tunnel. The honeycombs break turbulent eddies, and orient the flow to be parallel to the axis of the test section. 3.2. Prandtl-tube measurement A Prandtl tube is a pressure-based measurement instrument used to measure the velocity of a fluid. It consists of a tube with two holes inside it, one of which is located at the tip of the tube, while the other is located at the pipe’s side. With this layout, both static and total pressure are measured. The total pressure is measured by pointing the Prandtl-tube normal to the flow direction, so that the hole located at the tip of the tube is a stagnation point (U = 0). On the other hand, the static pressure is measured by the pipe’s side holes. A representation of the used Prandtl tube can be observed in Figure 3.2. This device measures the mean velocity of the flow by using Bernoulli’s theorem, which states: 23 Figure 3.1: NT2011 Wind Tunnel Pstatic U2 Pstagnation + = ρ 2 ρ (3.1) obtaining that the velocity is given by s U= 2(Pstagnation − Pstatic ) ρ (3.2) The Prandtl-tube is used for measuring a reference velocity, which is the free-stream velocity at the inlet of the test section. 3.3. Photodiode Photodiodes are devices that convert light into electric current. This device is used to measure the angular velocity of the wind-turbine models. It is done by pointing a red laser to a blade of the turbine with a certain angle, and placing a photodiode behind the model for having contact between the laser and the photodiode when the blade changes position as shown in Figure 3.3. When this happens, a current is generated by the photodiode. Therefore, the rotational speed of the turbines can be calculated from the photodiode output by measuring the time between each pulse, and by relating it with the displacement of the turbine blade. 24 Figure 3.2: Prandtl-tube Since the wind turbine models are two bladed, each time that a pulse is read, the turbine blade has done half a revolution. Thus, the angular velocity ω of the turbine can be calculated from the following equation: πf (3.3) z where z is the distance between two peaks (number of points difference) and f is the sampling frequency. ω= 3.4. Hot-Wire anemometry Hot-wire anemometry is a widely-used technique for measuring turbulent flows with high precision. The principle of this technique is that, a body that is exposed to a flow is mainly cooled by convective heat transfer, which in turn is related to the flow velocity. This apparatus is characterised by its microscopic dimensions. Nowadays, normal hot-wires have diameters between 3-5 µm and length of about 1-2 mm. These devices are composed by the microscopic wire mentioned before. The wire is attached to two supports (prongs) by which current passes through, heating the microscopic wire. A graphic representation of this device can be observed in Figure 3.4. 25 Figure 3.3: Laser-Turbine-Photodiode arrangement Figure 3.4: Single Hot-wire anemometer (From aawe.org) 3.4.1. Hot-Wire operating system Two ways of operating hot-wire probes have prevailed over the years: constant current anemometry (CCA) and constant temperature anemometry (CTA). However, the most common, and the one used in this thesis is CTA. Further information can be found in Perry (1982), Lomas (1986) and Bruun (1995). 26 Constant temperature anemometry (CTA) is when the hot-wire is kept at a constant temperature i.e. when it has a constant resistance (since R is a function of T ). In order keep a constant temperature, it is necessary to apply a variable current to the hot wire. The principle of a CTA circuit is explained in Figure 3.5. The hot wire is inserted in a Wheatstone bridge with an adjustable resistance, and a part of the bridge is connected to an operational amplifier. To cite Bruun (1995) “As the flow conditions vary the error voltage e2 − e1 will be a measure of the corresponding change in the wire resistance. These two voltages form the input of the operational amplifier. The selected amplifier has an output current, i, which is inversely proportional to the resistance change of the hot-wire sensor. Feeding this current back to the top of the bridge will restore the sensor’s resistance to its original value”. An offset value eof f is applied on one side of the amplifier, which gives a constant current to the bridge, making the temperature of the wire under no flow conditions dependent on the value of the variable resistance R3 (Figure 3.5). Figure 3.5: CTA circuit (Electronic testing sub-circuit, Wheatstone bridge and a feedback amplifier (G) (From Bruun 1995) It is important to comment about what is the relation between the obtained signal and the flow velocity, and for this, it is necessary to apply heat-transfer principles. Specifically, it is fundamental to know the behaviour of a heated cylinder located in a stream, which can be seen in many heat transfer books as Incropera & Dewitt (1981), Cengel (2007) or in hot-wire anemometry books as Bruun (1995). It is also important to say that the heat transferred to the flow by a hot-wire is mostly convective, and it is proportional to the power generated by the resistance and the voltage of the hot-wire. This is represented by the following equation: 27 Qh + Qr + Qc = 2 Ew Rw (3.4) where Ew is the voltage difference in the hot-wire, and Rw is the hot-wire resistance. If the heat transferred by radiation Qr , and by conduction to the prongs Qc are considered small in comparison with the heat transferred by natural and forced convection Qh , the expression 3.4 becomes: Qh = π d h L (Tw − Ta ) ≈ 2 Ew Rw (3.5) where: • d = hot-wire diameter. • h = convective heat-transfer coefficient which depends on the stream velocity (different correlations between h and the flow velocity can be found in Incropera & Dewitt (1981)) • L = hot-wire length. • Tw = hot-wire temperature. • Ta = ambient temperature. 3.4.2. X-Wire probes To measure the velocity in the streamwise u and in the traverse (wall-normal) v direction, a single hot-wire is not enough because it can just measure one velocity component. For this reason, an X-wire is used, which is composed by two single hot-wires placed on the same probe. This kind of hot wire allows to measure two velocity components at the same time, making possible the calculation of the covariance between these (Reynolds stresses). The voltage response of a generically oriented hot-wire probe is dependent on the magnitude of velocity kuk and the yaw angle α, so that: 1 E = f (kuk, α) = A + BVen (3.6) where Ve is the effective velocity, which represents an equivalent normal flow velocity that reproduces the same output voltage, and it is defined as: Ve = kukf (α) (3.7) where f (α) is a yaw function (Bruun 1995), which is usually assumed to be: 1 1 Ve = kukf (α) = kuk(cos2 (α) + k 2 sin2 (α)) 2 = (u2n + k 2 vn2 ) 2 with 28 (3.8) un = u cos(α) − v sin(α) (3.9) ut = u sin(α) + v cos(α) (3.10) where it is assumed that the binormal velocity component w is small. The velocities and angles showed in Equations 3.9 and 3.10 are sketched in Figure 3.6. After, introducing Equations 3.9 and 3.10 in Equation 3.8 for both wires, doing the algebra, and assuming that v 2 /u2 is small, the following expressions are obtained: Ve1 Ve2 q q tan(α)(1 − k 2 ) cos2 (α) + k 2 sin2 (α) (3.11) = u cos2 (α) + k 2 sin2 (α) − v 1 + k 2 tan(α) q q tan(α)(1 − k 2 ) cos2 (α) + k 2 sin2 (α) 2 2 2 (3.12) = u cos (α) + k sin (α) + v 1 + k 2 tan(α) by rearranging these equations, it is possible to get: U1 = q U2 = q Ve1 tan(α)(1 − k 2 ) = u − vg(α) 1 + k 2 tan(α) (3.13) tan(α)(1 − k 2 ) = u + vg(α). 1 + k 2 tan(α) (3.14) =u−v cos2 (α) + k 2 sin2 (α) Ve1 =u+v cos2 (α) + k 2 sin2 (α) By isolating both streamwise and vertical (wall-normal) velocity, and by knowing that the wires are inclined with the same angle but in opposite directions, it can be obtained that u= U1 + U2 2 (3.15) and v = (U1 − U2 ) 1 + k 2 tan(α) 2 tan(α)(1 − k 2 ) (3.16) 3.4.3. Calibration The last step before starting the measurements is the calibration of the X-wire probe. It is important to remember at this stage that, in spite of having the Equations 3.15 and 3.16, no information about the velocities could be obtained before doing a correlation between the voltages read by the hot-wire, and the velocity of the flow. Two calibrations have to be performed: a conventional 29 Figure 3.6: Sketch of the velocity components over both hot wires hot-wire calibration and an angular calibration . The first calibration consists in placing the hot-wire in an uniform stream, and measuring the voltages obtained by each wire, meanwhile a reference velocity using a Prandtl-tube is stored. 30 This calibration is done at the beginning and at the end of each measurement day. In Figure 3.7, it can be seen a characteristic curve taken from this first calibration. 10 10 U 2 [ m/s ] 15 U 1 [ m/s ] 15 5 5 0 0 1.4 1.6 E1 [ V ] 1.8 2 1.4 1.6 E2 [ V ] 1.8 2 Figure 3.7: Calibration between the voltages E1 and E2 and the free-stream velocity. The solid and dashed lines represent 3rd degree polynomials that relate E1 and E2 with the free-stream velocity, respectively. Since the conventional hot-wire correlation is done in the empty test section of the wind tunnel, it is expected that the wall-normal velocity is equal to zero (v = 0). For this reason, the following relations can be used: U1 = u = f (E1 ) (3.17) U2 = u = f (E2 ) (3.18) where it is assumed that U1 and U2 are 3rd degree polynomials, which can be seen as the solid and dashed lines in Figure 3.7. It is interesting to comment that the values of the voltage E1 and E2 are almost the same. The second calibration consists on placing the X-wire probe in an angular calibration facility (Figure 3.8) that allows to vary the velocity angle and the velocity magnitude, so that v 6= 0. Therefore, the value of the constant k can be calculated, by comparing the voltage signals with a reference velocity provided by a Prandtl-tube located inside the facility. Moreover, rewriting Equations 3.15 and 3.16 in a easier form to make the calibrations, they give: 31 Figure 3.8: Angular calibration facility. u= U1 + U2 G1 = uest G1 2 (3.19) and v = (U1 − U2 ) U1 − U2 1 + k 2 tan(α) = G2 = vest G2 2 2 tan(α)(1 − k ) 2 (3.20) When comparing vest with vref , the slope G2 could be calculated. The calibration is showed in figure 3.9. Finally, from the slopes of the 1st degree polynomials in Figure 3.9, it was found that G1 = 1 as expected, and G2 = 0.9708. 3.5. Traversing system In order to move the X-wire inside the wind tunnel, it is necessary a traversing system that allows movement in the three directions (streamwise, spanwise and traverse (wall-normal)). This system is composed by an electronic traversing in the y direction, and two manual traversing systems in the streamwise and spanwise directions. It was possible to traverse either in the streamwise and traverse direction, or in the traverse and spanwise direction. For this reason a traversing system capable to be moved in the three directions (Figure 3.10) was built. 32 16 8 6 14 4 12 v r e f [ m/s ] u r e f [ m/s ] 2 10 0 −2 8 −4 6 −6 4 0 5 10 15 −8 −10 20 u e s t [ m/s ] −5 0 5 10 v e s t [ m/s ] 2 Figure 3.9: Angular correlation. vest = U1 −U and uest = 2 between ve st and vr ef , and between ue st and ur ef . U1 +U2 . 2 Comparison Figure 3.10: Traversing System 3.6. Wind-farm model In order to perform the study of flows above wind farms, a model of such systems have to be built. Therefore, different objects are placed in the test 33 section to simulate a wind farm. These objects are showed in Figure 3.11 and described as follows: • A wooden plate of dimensions 1000 × 390 × 15 mm3 with a rounded leading edge attached to the floor of the test-section. • A steel plate of dimensions 990 × 390 × 15 mm3 attached to the wooden plate. • Wind turbine models of φ = 45 mm with a hub height of H = 65 mm. These turbines were provided by Jan-Åke Dahlberg (see Hägglund 2013). Figure 3.11: Wind farm model The used freely-rotating HAWT models were made of steel with a plastic rotor, and a magnet placed in the base for attaching to metal surfaces. The properties of the blades are described by Hägglund (2013). These turbines have a rotor diameter of φ = 45 mm, a total height from base to the top tip of h = 85 mm, and a thrust coefficient described in Figure 3.12. 34 Thrust Coefficient 0.63 0.62 CT 0.61 0.6 0.59 0.58 0.57 500 1000 1500 ω [ r ad /s ] 2000 Figure 3.12: Thrust coefficient vs Angular velocity from Hägglund (2013) It is important to comment that the values of the thrust coefficient at different angular velocities are between 0.55 and 0.62, similar to the values of real-scale wind-turbine thrust coefficients (between 0.8 and 0.5) at velocities around 8 meters per second. 35 CHAPTER 4 Experimental procedure The experiments done in this thesis were divided in two measurement campaigns. The first one was performed in february 2014, and the other one in april 2014. In the first measurement campaign, several wind farm configurations were assessed assuming spanwise homogeneity, i.e it was assumed that the flow in all the spanwise positions had the same behaviour, thus just measurements in the centreline of the wind farms were done. In almost every inline configuration tested in the wind tunnel, a turbine was always placed underneath the measurement point i.e there was always a turbine located at Z = 0 in each row of the wind farm, and for that case the growth of a boundary layer could be clearly seen. However, when there were no turbines present in the measurement plane, it was seen that there was no boundary layer growth, demonstrating that the spanwise homogeneity was not true for all the studied cases. These results led to the realisation of the second measurement campaign where several measurements were done, taking into account the spanwise nonhomogeneity i.e measuring the velocity profiles at different spanwise positions, and at different streamwise positions (as the first measurement campaign). In order to perform these measurements, a special traversing system capable to move in the spanwise direction had to be built. This traversing system was previously shown in Figure 3.10. In order to achieve a complete description of the flow in the study, five different experimental layouts were used in the second measurement campaign, and they are described as follows: • • • • • Base flow measurements. Inflow-RPM correlation. Box measurements. RPM measurements. Farm measurements. 4.1. Base flow measurements In this experiment a characterisation of the flow is done when the wind tunnel has nothing in its test-section but the ground plate. One important factor is that the voltage at which the wind tunnel fan is set up for this experiment, is 36 Y /h the same for all the subsequent experiments. Measurements of the statistics of the flow were taken at x= 500 mm (the middle of the test section) from the free stream to a region close to the plate in the wall-normal direction. Moreover, it is relevant to say that the aim of this experiment is to observe the flow quality of the wind tunnel. These measurements were done during the first measurement campaign. After performing the measurements, profiles of different flow statistics were computed and shown in Figure 4.1. 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 −0.8 −1 0.85 0.9 0.95 1 U /U ∞ 1.05 −1 −1.5 −1 −0.5 0 −3 2 u ′ v ′/U ∞ x 10 −1 0 0.05 σ u /U ∞ 0.1 Figure 4.1: Base flow characteristics. h is the height of the turbine from the floor to the tip of the blade. The dashed lines represent the area swept by the rotor, and the solid lines indicate the turbine’s hub height. What is important to point out in Figure 4.1 is that the flow conditions are optimal in the area of interest: the mean velocity profile is uniform within the turbine rotor, and the shear stresses and turbulent fluctuations are 0 in this region, which means that there is no external perturbation in the flow. However, a small boundary layer due to the ground plate is observed. 37 4.2. Inflow-RPM correlation In this experiment, a correlation between angular velocity and inflow velocity, and between angular velocity and disk velocity is obtained. This was done by placing a turbine in the test section of the wind tunnel at different flow velocities; the angular velocity of the turbine was obtained by a laser-photodiode arrangement, and the disk and inflow velocity were measured using hot-wire anemometry. In Figure 4.2 a relation between the velocities and the angular velocity can be seen. 10 Uinf u [ m/s ] 8 U disk 6 4 2 0 0 500 1000 ω [ r ad /s ] 1500 2000 Figure 4.2: Correlation between ω and Udisk and Uinf . The solid line and the dashed lines are third degree polynomials used to correlate the inflow velocity Uinf and the disk velocity Udisk with the angular velocity ω respectively. The angular velocity ω is used to calculate the force Fx in the streamwise direction using the model proposed in this thesis (Equation 2.55). Also, the value of the angular velocity in each row of turbines is highly important to determine the behaviour of the flow (accelerations, decelerations, peaks, etc). 4.3. Box measurements In this experiment the statistics of the flow are measured between two rows of turbines using a big amount of measurement points in the three directions (spanwise, streamwise and traverse). The goal of this experiment is to do an analysis of the statistics in this domain, so the distributions of these quantities can be known, providing some information about the optimal location of the measurement points for the farm measurements. 38 It is important to comment that the region between two rows of turbines is located deep downstream, therefore, valuable data of the distribution of flow quantities in that region are going to be analysed. This type of experiment was done in the second measurement campaign. Four different box measurements were performed on four different wind farm arrangements. These configurations are as follows: • 2 × 32.5D : this wind farm arrangement consisted on a series of 9 rows by 2-3 columns of wind-turbine models with a spacing of 2.5 diameters in the streamwise direction and a spacing of 2.88 diameters in the spanwise direction. • 3 × 32.5D : this wind farm arrangement consisted on a series of 9 rows by 3-3 columns of wind-turbine models with a spacing of 2.5 diameters in the streamwise direction and a spacing of 2.88 diameters in the spanwise direction. • 2 × 35D : this wind farm arrangement consisted on a series of 5 rows by 2-3 columns of wind-turbine models with a spacing of 2.5 diameters in the streamwise direction and a spacing of 2.88 diameters in the spanwise direction. • 3 × 35D : this wind farm arrangement consisted on a series of 5 rows by 3-3 columns of wind-turbine models with a spacing of 2.5 diameters in the streamwise direction and a spacing of 2.88 diameters in the spanwise direction. A graphic representation of these arrangements is shown in Figure 4.3. Figure 4.3: Graphic representation of all the box measurements done in this thesis. In order to manage the data in terms of spatial-averages, the minimum number of measurement points were needed to be known along with their position in a certain region for doing a good spatial average in terms of time and 39 flow description. That was one of the main reasons why the box measurements were done. The first measured case was the 2 × 32.5d case, where the normal assumption of symmetry in the spanwise direction was assessed i.e. measurements at both positive and negative values of Z were done. The measurement matrix is shown in Figure 4.4, where h = 85 mm is the wind turbine height from the bottom to the top tip, and d = 45 mm is the rotor diameter. An analysis of the flow in this region was done and it was found that the peaks in velocity for this box measurement were located at Z ≈ 0, and at Z ≈ −0.7d in the spanwise direction, and at X ≈ 12.5d and X ≈ 13.7d in the streamwise direction, respectively. Z /d 0 −1 −2 −3 12 12.5 13 13.5 12.5 13 13.5 14 14.5 15 14 14.5 15 1 Y /h 0.5 0 −0.5 −1 12 X/d Figure 4.4: 2 × 32.5d case measurement matrix Since the definition of the spatial average operator (in Equation 2.31) seems to be practically complex, a simplification can be done by defining the spatialaverage operator as the arithmetic average of certain measurement points located inside the region of study, as it was done by Cheng & Castro (2002). In order to confirm that the arithmetic average can be used as the spatialaverage operator, both methods were used and compared. A quadrature scheme was used to numerically solve the integration shown in Equation 2.31. Furthermore, the arithmetic mean was calculated for the points where the peaks in 40 velocity were located. The comparison between each average is presented in Figure 4.5. 80 Y [ mm ] 60 Integration Many points 4 points 40 20 0 6 6.5 7 7.5 U [ m/s ] Figure 4.5: Averaging method comparison for U (staggered case) From this comparison, it could be concluded that the simplification of using an arithmetic average (Many points) instead of the formal definition of spatialaverage can be used, because the variations of averaged velocity between the two methods are small. Moreover, it was observed that by averaging four points where peaks in velocity were found, a good spatial-average can be obtained. For this reason, just 4 points per row were going to be used for the spatial-average in the Farm measurements. The same analysis was done for the variance of the mean velocity (σu2 ) using the same locations for the 4 points average. The comparison between the methods is shown in figure 4.6. Since it was proved that the arithmetic average of certain points over a region is a good approximation for the spatial-average integral operator, the same method was used to calculate the spatial average of an inline 3×32.5d configuration. But, as it is known, flows over inline arrangements have a different behaviour in comparison to the flows over staggered configurations. Therefore, the locations for the 4 points average has to be modified. By assuming spanwise symmetry, measurements are going to be performed on just one side of the plate. The measurement matrix for these measurements is shown in Figure 4.7. An analysis of the flow in this region was performed, and it could be seen that the peaks in velocity were now located at Z ≈ 0, Z ≈ −1.5d in the spanwise direction, X ≈ 12.5d and X ≈ 13.7d in the streamwise direction. The 41 80 Y [ mm ] 60 Integration Many points 4 points 40 20 0 0 0.1 0.2 0.3 σ u2 [ m 2 /s 2 ] 0.4 0.5 Figure 4.6: Averaging method comparison for σu2 1 Z /d 0 −1 −2 −3 −4 12 12.5 13 13.5 12.5 13 13.5 14 14.5 15 15.5 14 14.5 15 15.5 1 Y /h 0.5 0 −0.5 −1 12 X/d Figure 4.7: 3 × 32.5d case measurement matrix comparison between the arithmetic average of all the points, and the arithmetic average of the points where the peaks were located is shown in Figure 4.8. A difference on the velocity values between the two methods can be seen, but it is 42 considered small. Therefore the 4 points average was used for space averaging inline configurations in the Farm measurements. 80 Many points 4 points Y [ mm ] 60 40 20 0 6 6.5 7 7.5 U [ m/s ] Figure 4.8: Averaging method comparison for U (Inline case) The same analysis was done for the variance of the mean velocity (σu2 ) using the same locations for the 4 points average. The comparison between the methods is shown in figure 4.9. 80 Many points 4 points Y [ mm ] 60 40 20 0 0 0.2 0.4 σ u2 [ m 2 /s 2 ] 0.6 0.8 Figure 4.9: Averaging method comparison for σu2 (inline case) 43 4.4. RPM measurements In these measurements the angular velocity of the turbines is calculated by using the system explained in section 3.3. The arrangement showed in Figure 3.3 is used in every row of turbines in order to measure the voltage pulses, which later are transformed in angular velocities using expression 3.3. The angular velocity was measured in just one turbine per row, and it was further assumed that the angular velocity was constant throughout the row. The angular velocity was then related to the inflow velocity of each turbine by the use of the correlation in Figure 4.2, so that the force model showed in equation 2.55 could be used. The cases assessed in this experiments were the 2 × 32.5D , and the 3 × 32.5D cases previously described in Box Measurements. 4.5. Farm measurements One of the aims of this thesis is to observe the velocity profiles above different wind farm arrays, taking into account the spanwise non-homogeneity of the flow, by measuring the velocity time series in specific spanwise, and streamwise positions at different heights. The optimal places for measuring the velocities above the different wind farms were selected from the Box Measurements by observing where the velocity maxima and minima were located. The motivation for measuring in the points where the peaks in velocity were located was to obtain good spatial averages using the minimum amount possible of measurement points, so less time-consuming measurements could be performed. The growth of the boundary layer above the different assessed wind farms is to be measured, together with some turbulent properties of the flow. Furthermore, calculations of the force applied by the turbines to the fluid are to be performed and, as it can be seen in Equation 2.38, space-averaged values of the statistics are required. After, this force is to be compared with the force proposed in Equation 2.55. The assessed cases in this experiment were the 2 × 32.5D , and 3 × 32.5D wind farm configurations. 44 CHAPTER 5 Results and discussions 5.1. Box 2 × 32.5d flow A contour plot of the velocity distribution at Y = 0 (close to the turbine tip) was done (Figure 5.1) with the aim of observing the velocity distribution at the turbines’ tip. 1 0.5 0.95 Z /d 0 0.9 0.85 −0.5 0.8 −1 0.75 −1.5 12.5 0.7 13 13.5 X/d 14 Figure 5.1: U/U∞ at Y = 0 for the 2 × 32.5d case There are 3 important observations regarding Figure 5.1. The first one is that the assumption of symmetry in the spanwise direction is not completely true. Variations smaller than 5 % in velocity could be seen between two opposite spanwise locations (e.g Z = 0.6d and Z = −0.6d) but, the spanwise symmetry assumption can still be used, since these variations are not big. The second important observation is the presence of a Venturi effect between Z = −0.6d and Z = −0.8d, which corresponds to half the spanwise distance that separates two turbines of two different rows. It is called Venturi effect when an acceleration in the flow is observed; in this region this effect is caused by the turbine’s wake interaction 45 The third relevant observation is that the velocity exactly in between two turbines, located on the same row (Z = −1.5d), is approximately the velocity on top of the turbine in the centre, which means that the wake recovery between two rows is quite small. This fact could also be seen when observing that the velocity change in the streamwise direction was almost zero at a fixed spanwise position. Another information from the Box measurement 2 × 32.5d which is interesting to show is the velocity distribution over the 6th (12.5d) row of turbines in the Z − Y plane, as shown in Figure 5.2. From Figures 5.1 and 5.2 large velocity variations in the spanwise direction can be observed (see velocity at Z/d ≈ 0 and at Z/d ≈ −0.7 at Y ≈ 0). It is important to remark that these measurements are done in a place were developed flow is assumed to be reached. The observation of variations on the streamwise mean velocity under developed regime is important, because the assumption of spanwise invariance of the streamwise mean velocity cannot be used for staggered configurations, contradicting what was stated by Chamorro et al. (2011). Also, large differences in the Reynolds stresses at different spanwise positions can be seen in Figure 5.3. 0.5 1 0.4 0.95 0.9 Y/h 0.3 0.85 0.2 0.8 0.1 0.75 0 0.7 0.5 0 −0.5 Z /d −1 −1.5 Figure 5.2: U/U∞ at X = 12.5d for the 2 × 32.5d case It is interesting to show the contributions of the dispersive, and the spaceaveraged Reynolds stresses above the turbines. These are shown in Figure 5.4, where it can be seen that the contribution of the Reynolds stresses is higher than the contribution of the dispersive stresses at low heights, and they both tend to zero after a certain height. In Figure 5.4, the friction velocity is defined 0 0 00 00 as u2∗ = −hu0 v 0 i(0). Also, the term U ∗∗ is equal to U ∗∗ = huuv2 i + hU uV2 i , and ∗ ∗ shows the contribution of the dispersive and Reynolds stresses, where in this case the Reynolds stresses dominates. 46 −3 0.5 x 10 −0.5 0.4 −1 −1.5 Y/h 0.3 −2 0.2 −2.5 0.1 −3 0 −3.5 0.5 0 −0.5 Z /d −1 −1.5 2 at X = 12.5d for the 2 × 32.5d case Figure 5.3: u0 v 0 /U∞ 0.4 Reynolds st Dispersive st U** Y /h 0.3 0.2 0.1 0 −1 −0.5 0 0.5 Figure 5.4: Comparison between the Reynolds stresses (hu0 v 0 i/u∗ ) and the 00 00 dispersive stresses (hU V i/u∗ ) for Box 2 × 32.5d measurements. Also, the sum of both contributions is represented by U ∗∗ . 47 5.2. Box 3 × 32.5d flow A contour of the velocity profile at Y = 0 for the 3 × 32.5d case was done (figure 5.5). It is inferred that the velocity exactly in between two turbines at the same row is approximately the free-stream velocity i.e. the flow is unperturbed at that spanwise location, which means that the wake interaction between the turbines located in the same row is small, despite the fact that this experiment was set up with small spacings in both streamwise and spanwise directions, in comparison with other experiments performed in the past (Chamorro & Porté-Agel 2011). 0 1 0.95 −0.5 Z /d 0.9 0.85 −1 0.8 0.75 −1.5 12.5 0.7 13 13.5 X/d 14 Figure 5.5: U/U∞ at Y = 0 for the 3 × 32.5d case The Venturi effect due to wake interaction observed in Section 5.1 is visible exactly in between two turbines at the same row. However, what can be seen here is that the flow velocity exactly in between two turbines is the free-stream velocity, i.e. the flow is not perturbed by the turbines at this location. Additionally, inside the wake of the turbine a small decrease on the streamwise velocity in the streamwise direction could also be seen. Another information from the Box Measurement 3 × 32.5d which is interesting to show is the velocity distribution over the 6th (12.5d) row of turbines in the Z − Y plane, as shown in Figure 5.6. In this picture, large velocity variations in the spanwise direction can be seen. The velocity discrepancies reach values up to 30 % at different spanwise locations. It is important to remark that these measurements are done in a place were developed flow is assumed to be reached. Furthermore, when comparing the velocity variations in this configuration with the ones in the staggered case, it could be seen that the velocity variations are higher, and the wake interaction is reduced for the inline case. Therefore, it can be stated that a bigger wake interaction is observed in the 48 staggered case, despite the fact that the turbines spacings between the inline and staggered configurations were exactly the same. 0.5 1 0.4 0.95 0.9 Y /h 0.3 0.85 0.2 0.8 0.1 0 −1.5 0.75 0.7 −1 −0.5 0 Z /d Figure 5.6: U/U∞ at X = 12.5d for the 3 × 32.5d case Large differences in the Reynolds stresses in the spanwise direction could be determined in Figure 5.7. When observing this figure and comparing it with Figure 5.3, it is found that higher values of Reynolds stresses are present in the inline case at the position [Y, Z] = [0, 0], but it could also be seen that the Reynolds stresses are more evenly distributed in the spanwise direction for staggered configurations. −3 x 10 0.5 −1 0.4 −2 Y /h 0.3 −3 0.2 −4 0.1 0 −1.5 −5 −1 −0.5 0 Z /d 2 Figure 5.7: u0 v 0 /U∞ at X = 12.5d for the 3 × 32.5d case 49 The contributions of the dispersive and the space-averaged Reynolds stresses above the turbines are shown in Figure 5.8, where it can be seen that the contribution of the dispersive stresses are much higher in comparison with the staggered case shown in Figure 5.4. Therefore, when summing the Reynolds and dispersive stresses contributions up, they practically cancel each other. This can also be observed when comparing the plotted U ∗∗ term between Figures 5.4 and 5.8 . 0.4 Reynolds st Dispersive st U** Y /h 0.3 0.2 0.1 0 −3 −2 −1 0 1 0 0 Figure 5.8: Comparison between the Reynolds stresses ( huuv∗ i ) and the disper00 00 sive stresses ( hU uV∗ i ) for Box 3 × 32.5d measurements. Also, the sum of both contributions is represented by U ∗∗ . 50 5.3. Scaled Flow 5.3.1. Scaled velocity profile After performing all the box measurements mentioned in Chapter 4, and plotting the spatial-averaged streamwise mean velocity (hU i) at different heights, as it is shown in Figure 5.9, it was decided to plot these profiles using a characteristic scaling as it is usually done for boundary layers over flat plates. 80 2 × 32.5d 2×3 Y [ m ] 60 5d 3 × 32.5d 3 × 35d 40 20 0 6 6.5 7 7.5 h U i [ m/s ] 8 Figure 5.9: Space-averaged velocity profiles of the different assessed box measurements. By trying to normalise the velocity profile with a velocity scale, and the wall-normal position with the the boundary layer thickness, the following expression was suggested for flows above wind farms: hU i = hU i(0) + [hU∞ i − hU i(0)]f Y δ∗ (5.1) where δ ∗ is similar to the displacement thickness used in the integral boundary layer equation (Kundu et al. (2012)). This definition is shown as follows: δ∗ = Z ∞ 1− 0 hU i − hU i(0) hU∞ i − hU i(0) dy. (5.2) An important observation is obtained when analysing Figure 5.10. All the assessed wind-farm configurations follow the same profile shape, thus the spaceaveraged velocity profile above a wind farm (at least for developed sections) can 51 be completely described when having the thickness of the boundary layer (δ ∗ ), the space-averaged free-stream velocities, and the space-averaged streamwise velocities close to the turbines (hU∞ i and hU i(0)), as it is shown in Equation 5.1. 12 2 × 32.5d 10 2×3 5d Y /δ ∗ 8 6 3 × 32.5d 3 × 35d 4 2 0 0 0.5 1 1.5 h U i − h U i ( 0) h U ∞ i − h U i ( 0) Figure 5.10: Universality of the normalised velocity profile over wind farms. 5.3.2. Scaled Reynolds stresses After observing that the scaled velocity profile above all the assessed wind farms followed the same trend, a similar approach was attempted with the second order flow statistics, such as the space-averaged Reynolds stresses and velocity variances (hu0 v 0 i, hσu2 i and hσv2 i). For this reason the following expressions were assumed: , hu0 v 0 i Y = f 2 u∗ δ∗ hσu2 i Y =f u2∗ δ∗ (5.3) (5.4) and hσv2 i =f u2∗ where u2∗ is the friction velocity. 52 Y δ∗ (5.5) When observing Figure 5.11, it can be noted that, if the friction velocity and the boundary layer thickness are known, a complete characterisation of the Reynolds stress above a wind farm can be achieved. 12 Y /δ ∗ 2 × 32.5d 10 2 × 35d 8 3 × 32.5d 6 3 × 35d 4 2 0 −1 −0.8 −0.6 −0.4 h u ′ v ′i /u 2∗ −0.2 0 Figure 5.11: Universality of the normalised Reynolds stress over wind farms. The same analysis is done for the variance of the streamwise and vertical velocities, and it is shown in Figures 5.12 and 5.13 respectively. What is relevant to observe from these figures is that they follow the same profile for every windfarm configuration assessed. Moreover, it is also important to comment that other scaling factors were assessed. In particular, the following one 00 00 u2∗ = −hu0 v 0 i(0) − hU V i(0) was tried several times due to the its logical appearance in the space-averaged equations, but no flow generalisation was achieved which such a scaling. It is believed that, the contributions of the Reynolds and dispersive stresses cannot be compared because they come from different scales: the Reynolds stress contribution comes from the small scales (turbulence) , and the dispersive stress comes from the large scales (turbines layout). 5.4. Comparison between Reynolds stresses and dispersive stresses Since horizontal averages are being used in this experiment, additional terms in the equations of motion come up, as they are the dispersive stresses. So, it is important to observe the contributions of such stresses, and compare them with the Reynolds stresses. 53 12 Y /δ ∗ 2 × 32.5d 10 2 × 35d 8 3 × 32.5d 6 3 × 35d 4 2 0 0 1 2 h σ u2 i /u 2∗ 3 4 Figure 5.12: Normalised streamwise-velocity variance over wind farms. 12 2×3 10 Y /δ ∗ 8 6 2.5d 2 × 35d 3×3 2.5d 3 × 35d 4 2 0 0 1 2 h σ u2 i /u 2∗ 3 4 Figure 5.13: Normalised vertical-velocity variance over wind farms. 00 00 Figure 5.14 shows a comparison between hu0 v 0 i and hU V i. It can be seen that the Reynolds stresses collapse on the same curve for the two configurations, but the dispersive stresses do not. The reason of this is that the dispersive stresses come from the inhomogeneity of the turbine spatial distribution. Therefore, using the friction velocity as a scaling for these stresses is 54 not correct because these two contributions do not come from the same scales: The Reynolds stresses come from the small velocity scales, and the dispersive ones come from the mean velocity variation scale. It can also be seen that the magnitude of the dispersive stresses for the staggered configurations is smaller than the contributions for the inline ones. Therefore, it can be concluded that, when the turbines are placed in an inline layout, the flow above the turbines is changing at different positions of the averaged space, i.e. the flow is less homogeneous than the one in staggered arrangements. However, it cannot be said that the flow is perfectly homogeneous above staggered configurations as well, since dispersive stresses were also present for such arrangement. When comparing the contributions of both stresses at different heights, it is seen that the dispersive stresses close to the turbines can be as high as the Reynolds stresses for inline arrangements. The reason why it is interesting to know the contributions of this terms is that, when integrating the Navier-Stokes equations in the streamwise direction (see Equation 2.36), derivatives of these terms in the wall-normal direction come up, and therefore, it is necessary to understand the influence of each contribution. 10 Y /δ ∗ 8 6 4 2 0 −1 −0.5 0 0.5 ′′ ′′ h u ′ v ′i /u 2∗ h U V i /u 2∗ 1 Figure 5.14: Comparison between the contributions of the Reynolds stresses 00 00 (hu0 v 0 i) and dispersive stresses (hU V i). Two wind-farm configurations are shown: an inline () and a staggered (4) with the same streamwise and spanwise spacings. The color blue represents the dispersive stresses, and the red one the Reynolds stresses. The same comparison was done for other dispersive stresses. For exam00 00 ple, Figure 5.15 shows the comparison between hσu2 i and hU U i. It can be seen that close to the turbines, the dispersive stresses were quite higher 55 than the Reynolds stresses for both layouts. However, after integrating the space-averaged Navier-Stokes equations in the streamwise direction (Eq 2.36), derivatives of these terms in the streamwise direction come up and, since it is known that the flow is not changing too much in the streamwise direction, these contributions of the Reynolds and dispersive stresses become less important. 10 Y /δ ∗ 8 6 4 2 0 0 2 4 h σ u2 i /u 2∗ 6 8 ′′ ′′ h U U i /u 2∗ 10 Figure 5.15: Comparison between the contributions of the Reynolds stresses 00 00 (hσu2 i) and dispersive (hU U i) stresses. Two wind-farm configurations are shown: an inline () and a staggered (4) with the same streamwise and spanwise spacings. The color blue represents the dispersive stresses, and the red one the Reynolds stresses. 56 5.5. Farm 2 × 32.5d case From the data reported in Section 5.1, an efficient way of measuring staggered arrangements was adopted, and measurements above a whole wind farm model were performed. This was done for a 2 × 32.5d configuration which had the same wind-farm layout assessed in the box measurements. The measurement matrix shown in Figure 5.16 was applied to this wind farm arrangement. It can be seen that the measurement points are located in the middle of the wind farm, and where the acceleration of the flow produced by the interaction between the wakes of the turbines is expected (as it could be seen in Figure 5.2). Namely, an analysis to determine the most representative 4 points was discussed in the previous chapter. Z /d 0 −1 −2 −3 −4 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 X/d 12 14 16 18 20 1 Y /h 0.5 0 −0.5 −1 Figure 5.16: Farm 2 × 32.5d case measurement matrix The distributions of the streamwise velocity along the farm and close to the tip of the turbines are shown in Figure 5.17. From the plot, it can be seen that the velocity is approximately the free-stream one between the 1st and 2nd row of turbines in both spanwise locations. This effect clearly shows that the turbines are not interacting at the very beginning of the wind farm. Also, when observing the spanwise variability of velocity at different streamwise positions, it can be seen that this difference gets smaller as the number of rows increases. The achieved value for this difference was of approximately 6 % at the last assessed streamwise location. 57 1.05 1 U /U ∞ 0.95 0.9 0.85 0.8 0.75 0.7 0 5 10 X/d 15 20 Figure 5.17: U/U∞ along the farm at Y = 0 for the 2×32.5d case. The black line represents the average of the velocity in each streamwise position, the bottom of the vertical bars represent the velocity values at Z = 0, and the top of them represent the velocities at Z ≈ −0.6d. The same data analysis is done for the Reynolds stresses, and it is shown in Figure 5.18. In the figure, it can be seen a similar effect at the beginning of the wind farm, but here the Reynolds stress is almost zero between the 1st and 2nd row of turbines at the centre of the wind farm, and it is almost zero between the 1st and 4th row of turbines at Z ≈ −0.65d. Another important feature is that the spanwise variability of Reynolds stresses is much higher in comparison with the variability of the streamwise mean velocity. This means that, when performing absorption of momentum calculations (force calculations) for wind farms just by taking into account the flow in the middle of the wind farm, a high overestimation is done due to the fact that this force (See Equation 2.38) is highly dependent on the Reynolds stresses on top of the turbines. After the application of the spatial-average operator to the measurement points at different heights, a distribution of the normalised averaged velocity above the farm was plotted in Figure 5.19. It can be seen that the boundary layer starts to grow between the 2nd and 3rd row. Thus, it is important to comment that, if measurements would have been performed just in the centre of the wind farm, the boundary layer would have started to grow at the same position, but the thickness of it would have been higher at every streamwise position. Therefore, what the space-averaged boundary layer shows is actually a boundary layer which never reaches a parallel condition. 58 −3 x 10 1 0 2 u ′ v ′/U ∞ −1 −2 −3 −4 −5 −6 0 5 10 X/d 15 20 2 along the farm at Y = 0 for the 2 × 32.5d case Figure 5.18: u0 v 0 /U∞ 0.5 1 0.4 0.95 Y /h 0.3 0.9 0.2 0.85 0.1 0 0.8 5 10 X/d 15 Figure 5.19: hU i/hU∞ i. In this figure, the boundary layer growth for the 2 × 32.5d space-averaged case is shown. The same analysis performed for the space-averaged streamwise velocity is done for its standard deviation. This distribution is shown in Figure 5.20, where it can be clearly seen that the turbulence is developing with the downstream distance. By observing Figures 5.19 and 5.20, it is noted that when the velocity reached its lowest value, the turbulent intensity reached its maximum. 59 0.5 0.12 0.4 0.1 0.08 Y /h 0.3 0.06 0.2 0.04 0.1 0.02 0 0 5 10 X/d 15 Figure 5.20: hσu2 i/hU∞ i2 . In this figure, the turbulent boundary layer growth for the 2 × 32.5d space-averaged case is shown. Figure 5.21 shows the turbulence correlation coefficient along the wind farm. It can be seen from this figure that, close to the turbines after the 4th row, the value of this correlation coefficient is equal to −0.4, which is a characteristic value for turbulent flows near walls. This coefficient has been shown in the past by several authors as Priyadarshana & Klewicki (2004) and Monin & Yaglom (1973), and they have concluded that this value can be found above different types of surfaces (smooth and rough). What it is interesting is the fact that above a wind farm the non-slip condition is not true (i.e. there is no wall on top of the turbines), and still this value (−0.4) of the correlation coefficient is found. 5.5.1. Angular velocity of the turbines Another important aspect from a wind farm is of course the rotational speed of the turbines, because this quantity is related to many other ones, like its power generation, inflow velocity, thrust coefficient, torque, etc. Since the turbines in this thesis work are freely-rotating, a relation between rotational speed and power generation cannot be developed, however useful information can be taken from the trend of the rotational speed along the farm. A curve showing the location of the turbines and a normalised angular velocity is shown in Figure 5.22, where it can be clearly seen that the first and second row have more or less the same angular velocity, consistently with the fact that both rows are submerged by the free stream. Another interesting feature of this curve is the sudden decrease of angular velocity after the second row, due to the interaction of these turbines with the wake of the upstream turbines. It can also be seen that, after this decrease in velocity, the velocity increases of a small amount, and then it remains almost constant. 60 0.5 0 0.4 −0.1 Y /h 0.3 −0.2 0.2 −0.3 0.1 0 −0.4 5 10 X/d 15 Figure 5.21: The turbulence correlation coefficient (hu0 v 0 i/(hσu ihσv i)) for the 2 × 32.5d space-averaged case is shown. Since the angular velocity is directly related with the inflow velocity, it can be concluded that after 3 rows, the flow between the top tip of the turbines and the base reaches a developed condition for this staggered arrangement. An gu l ar Ve l o c i ty Y /h , ω /ω ( 0) 1 0.5 0 −0.5 −1 0 5 10 X/d 15 20 Figure 5.22: Angular velocity distribution for the 2 × 32.5d case 61 5.6. Farm 3 × 32.5d case From the data reported in Section 5.1, an efficient way of measuring staggered arrangements was adopted, and measurements above a whole wind farm model were performed. This was done for a 3 × 32.5d configuration which had the same wind-farm layout assessed in the box measurements. The measurement matrix shown in Figure 5.23 was applied to this wind farm arrangement. It is worth to notice that, the measurement points are located in the middle of the wind farm and exactly in between the turbines (as seen in Figure 5.6) at two different streamwise locations where the maximum velocity variations were found. Z /d 0 −1 −2 −3 −4 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 X/d 12 14 16 18 20 1 Y /h 0.5 0 −0.5 −1 Figure 5.23: Farm 3 × 32.5d case measurement matrix A distributions of the streamwise velocity along the farm and close to the tip of the turbines are shown in Figure 5.24. It can be seen that, the velocity is equal to the free-stream velocity in the first row of turbines at Z = 0, and it can also be seen that the velocity at Z = −1.5d is approximately the free-stream velocity along the whole wind farm, which means that the wake interaction in the spanwise direction between the turbines is really weak, despite the high wind farm density. Furthermore, when observing the spanwise variability of velocity at different streamwise positions, it can be seen that it practically remains the same. The velocity difference between Z = 0 and Z = −1.5d is approximately 25% along the whole wind farm. Leading to the conclusion 62 that the streamwise invariance assumption is not valid for inline configurations either. 1 U /U ∞ 0.9 0.8 0.7 0 5 10 X/d 15 20 Figure 5.24: U/U∞ along the farm at Y = 0 for the 3×32.5d case. The black line represents the average of the velocity in each streamwise position, the bottom of the vertical bars represent the velocity values at Z = 0 and the top of them represent the velocities at Z ≈ −1.5d The same analysis is done for the Reynolds stresses as shown in Figure 5.25. In this figure, it can be seen a big variation between the measurements at Z = 0 and Z ≈ −1.5d as well, but it can be clearly seen that, at Z = 0 the magnitude of the Reynolds stress is increasing as the streamwise position increases. Also, it can be observed that the Reynolds stresses are almost zero at Z ≈ −1.5d at all streamwise positions i.e. the variation between Reynolds stresses is almost 100 % between the two spanwise positions. After the application of the spatial-average operator to the measurement points at different heights, a distribution of the averaged streamwise velocity above the farm was plotted (Figure 5.26). It can be seen that the boundary layer starts to grow just after the 1st row of turbines, and it never stops growing. Therefore, what the space-averaged boundary layer shows is actually a boundary layer which never reaches a parallel condition, result that is in agreement with the statement of Chamorro & Porté-Agel (2011) for inline configurations. The same analysis performed for the space-averaged streamwise velocity is done for its standard deviation. This distribution is shown in Figure 5.27, where it can be clearly seen that the turbulence is increasing with the downstream distance. When observing Figures 5.26 and 5.27, it is noted that, when the velocity reached its lowest value, the turbulent intensity reached its maximum. 63 −3 x 10 1 0 2 u ′ v ′/U ∞ −1 −2 −3 −4 −5 −6 −7 0 5 10 X/d 15 20 2 along the farm at Y = 0 for the 3 × 32.5d case Figure 5.25: u0 v 0 /U∞ 0.5 1 0.4 0.95 Y /h 0.3 0.9 0.2 0.85 0.1 0 0.8 5 10 X/d 15 Figure 5.26: hU i/hU∞ i. In this figure, the boundary layer growth for the 3 × 32.5d space-averaged case is shown. Figure 5.28 shows the turbulence correlation coefficient along the wind farm. It can be seen from this figure that close to the turbines, from the 2nd row in the wind farm, the value of this correlation coefficient is again equal to −0.4, which is a characteristic value for turbulent flows near walls, as it was observed for the staggered case. 64 0.5 0.12 0.4 0.1 0.08 Y /h 0.3 0.06 0.2 0.04 0.1 0.02 0 0 5 10 X/d 15 Figure 5.27: hσu2 i/hU∞ i2 . In this figure, the turbulent boundary layer growth for the 3 × 32.5d space-averaged case is shown. 0.5 0 0.4 −0.1 Y /h 0.3 −0.2 0.2 −0.3 0.1 0 −0.4 5 10 X/d 15 Figure 5.28: The turbulence correlation coefficient (hu0 v 0 i/(hσu ihσv i)) for the 3 × 32.5d space-averaged case is shown. 5.6.1. Angular velocity of the turbines A curve showing the location of the turbines and a normalised angular velocity is shown in Figure 5.29, where it can be clearly seen that the first row of turbines has a much higher velocity in comparison with the rest, due to the fact that these turbines are submerged by the free stream directly. Another interesting feature of this curve is the sudden decrease of angular velocity after the first row, due to the interaction of these turbines with the wake of the 65 upstream turbines. It can also be seen that after this velocity decrease, the velocity increases a small amount, and then it remains almost constant. Since the angular velocity is directly related with the inflow velocity, it can be concluded that after 4 rows for this inline arrangement, the flow between the top tip of the turbines and the base reach a developed condition, in agreement with the result of Chamorro & Porté-Agel (2011). An gu l ar Ve l o c i ty Y /h , ω /ω ( 0) 1 0.5 0 −0.5 −1 0 5 10 X/d 15 20 Figure 5.29: Angular velocity distribution for the 3 × 32.5d case 66 5.7. Comparison between farm 2 × 32.5d and 3 × 32.5d In the last sections, values and characteristics of different flow quantities were shown for two different wind-farm configurations. These two layouts only differ in the organisation of the turbines (staggered and inline arrangement), because the streamwise and spanwise spacing were exactly the same for both of them. Therefore, the objective of this section is to observe the differences between flow properties attributed to the wind-farm layouts. The streamwise mean velocity distribution just above the turbines is shown in Figure 5.17 for the staggered case, and in Figure 5.24 for the inline case. The velocity difference between the two spanwise positions tends to converge with the streamwise distance for the staggered case. On the other hand, this difference does not converge at all for the inline case, demonstrating that the wake interaction in the staggered case is much stronger than in the inline case. But, as it was said before, the variations in velocity do not reach yet negligible values, therefore the assumption of spanwise velocity invariance cannot be done for either configuration. Moreover, It can also be seen that for the inline configuration, the streamwise mean velocity becomes almost constant after four rows, unlike the staggered case which does not seem to reach a plateau in its velocity at any location, in fact, it decreases monotonically with the streamwise distance. When comparing the Reynolds stresses (Figures 5.18 and 5.25) just above the turbines, higher differences can be seen. In the staggered case, these differences in Reynolds stress have high values (approximately 50 %), and the values at the centre and at Z ≈ −0.6 do not appear to converge at any location. However, when observing the inline case a divergence in the Reynolds stresses at Z = 0 and Z ≈ −1.5d can be detected i.e. the difference between this covariance between the two spanwise locations increases as the streamwise distance increases. Another difference between the flow above the two layouts is the spaceaveraged boundary layer (Figures 5.19 and 5.26), where it can be seen that in the inline case, the boundary layer starts before than in the staggered case. It can also be seen that lower values in velocity are reached for the boundary layer above the staggered arrangement. Moreover, the streamwise mean velocity just above the turbines for the inline case is almost constant, unlike the one above the staggered configuration which tends to decrease as the streamwise distance increases. However, when comparing the space-averaged boundarylayer thickness at the last row between the two arrangements, it appears to have similar values with a magnitude of approximately 0.35h, where h is the wind-turbine height from the floor to the top tip, leading to the conclusion that for staggered cases the growth of the space-averaged boundary layer is faster, because it starts after, and ends up with the same boundary layer thickness than the inline case. But, in both of them the boundary layer never stops 67 growing, showing that the flow never reaches a developed condition above the wind farms. After the comparison of the growth of the space-averaged boundary layer, it was convenient to study the turbulence above these two wind farms configurations. This was done by comparing Figures 5.20 and 5.27, where two things were important to remark: first, higher values of turbulence were found at the last row for the staggered configuration, and second, the turbulent boundary layer started before in the inline case, which means that the turbulence starts to grow earlier for inline configurations. A similar finding was seen when the turbulence correlation coefficient was plotted for both configurations in Figures 5.21 and 5.28, where it was found that for the inline case the characteristic correlation = −0.4 was reached in the early stages of the wind farm, unlike the staggered case where this correlation coefficient was found after the fourth row of turbines, which means that the turbulence starts to grow first for inline configurations. Furthermore, the fact that this turbulence correlation coefficient reached a value of −0.4 where the boundary layer is, represents an interesting finding, because the flow above these wind farms behaves as the flow above walls. Therefore, this result leads to think that this coefficient is more related to the turbulence in general than to boundary layer flows. A comparison between the angular velocity for both configurations was done. In order to do that, the values shown in Figures 5.22 and 5.29 were replotted in the same figure (Figure 5.30) for the sake of clarity. In this figure, it can be clearly seen that the angular velocity for staggered configurations is higher in overall than the angular velocity for inline configurations: in fact, in the staggered configuration the angular velocity reaches a plateau at a value approximately 25 % lower than the velocity in the 1st row, and in the inline configuration the angular velocity reaches a plateau at a value approximately 50 % lower than the one in the first turbine. In spite of the fact that these are freely rotating turbines, it can be said that there is a direct relation between power generation and angular velocity, which means that in terms of efficiency per turbine, staggered layouts perform better than inline layouts, as it was discussed by Chamorro et al. (2011). However, when there are space restrictions to place the turbines, a specific answer for which configuration is more efficient cannot be determined with the obtained data in this thesis work. Further information on layout efficiency and optimisation can be found in Alfredsson & Dahlberg (1981), Mosetti et al. (1994) and Frandsen & Jørgensen (2009). Another important observation is that developed flow inside the wind farm is first reached for staggered arrangements. This conclusion can be achieved by observing when the angular velocity reaches a plateau for both configurations. 68 An gu l ar Ve l o c i ty Y /h , ω /ω ( 0) 1 0.5 Staggered Inline 0 −0.5 −1 0 5 10 X/d 15 20 Figure 5.30: Angular velocity comparison between the 2 × 32.5d and 3 × 32.5d case 5.8. Force model validation One of the aims of this thesis is to estimate in a simple way the momentum absorption in wind farms with different layouts (staggered and inline), in order to relate it with second-order flow properties, such as the friction velocity. For this reason, a simple force model (Equation 2.55) was developed in Chapter 2. In order to validate this model, the momentum conservation equation was integrated (Equation 2.38) to obtain an expression for the force applied by the turbines to the flow. Before going further, it is important to recall both expressions. First, the force model: 2 Ct Nt Ad Uinf Fx = 2 Sx Sz h where: • • • • • • Sx is the turbines spacing in the streamwise direction. Nt is the number of turbines in this particular row. Ct is the thrust coefficient of the turbines. u∞ is the inflow velocity of the turbines. Ad is the area swept by the turbine rotor. Sz = Sx [(Nt max )Lz ] where Lz is the spacing in the spanwise direction and Nt max is the maximum number of turbines in a wind farm row, i.e. In a staggered arrangement 3 X 2 Nt max = 3. 69 • h is the wind farm height. And second, the integrated space-averaged Reynolds Navier-Stokes equation: 00 Fx = hU∞ i 00 ∂hU∞ i hU V i(0) hu0 v 0 i(0) − − ∂x h h where: 00 00 • hU V i(0) is the dispersive stress at the tip turbine height. • hu0 v 0 i(0) is the Reynolds stress at the tip turbine height. The measurements over the farms 2 × 32.5d and 3 × 32.5d were used for doing the validation, and this one is shown in Figure 5.31. In this figure, it can be seen that this force, as its expressions shows, is constant between rows of turbines. The motivation of defining the force model as a constant force between two rows is to be consistent with the domain of the spatial-average. When observing Figure 5.31, it can be noted that the force model (solid line for both cases) almost matches the value of the integrated force after 5 turbine rows, fact that is interesting since the Equation 2.38 is valid when the flow is completely developed i.e. when there is no acceleration of the flow. As a conclusion, this model can be utilised for estimating the force applied by the turbines to the fluid for staggered or inline developed wind farms with a certain confidence. But, the most relevant result from this model is that, by having the free-stream velocity, the angular velocity of the turbines, and the dispersive stresses close to the turbines, an estimation of the friction velocity can be obtained. This is helpful, since just high-scale measurements are required to obtain micro-scale values such as the friction velocity. 70 F or c e [ N / k g ] 14 Force Model (Stag) Navier−st (Stag) 12 10 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 F or c e [ N / k g ] 20 Force Model (Inli) Navier−st (Inli) 15 10 5 0 0 2 4 6 8 10 X/d 12 14 16 18 Figure 5.31: Force model validation for 2 × 32.5d and 3 × 32.5d cases 71 20 CHAPTER 6 Summary and conclusions The flow above different wind-farm arrangements was analysed experimentally by means of wind-tunnel measurements. As wind-farm models, small freelyrotating turbines were placed above a metallic plate inside the test section of the KTH NT-2011 wind tunnel. The assessed wind-farm arrangements were different in terms of turbine spacings and turbine layout (inline and staggered). Three types of measurements were performed. First, the time series of streamwise and wall-normal velocities were measured between two rows of turbines deep downstream on the wind farm (Box Measurements). Second, the time series of streamwise and wall-normal velocities were measured above the whole wind farm, in this experiment two different wind farm layouts with the same streamwise and spanwise spacing (Farm Measurements) were assessed. Third, the angular velocity of the turbines was measured in each row of turbines (RPM Measurements). To measure the velocities time series, X-wire anemometry was used, while to measure the rotational speed of the turbines, a laser-photodiode device was used. In the Box Measurements 4 different wind farm arrangements were assessed, two inline and staggered configurations with two different streamwise spacings but the same spanwise spacing (2 × 32.5D , 2 × 35D , 3 × 32.5D and 3×35D respectively). However, just two wind farm configurations were assessed in the Farm Measurements and RPM Measurements: an inline and a staggered configuration with the same streamwise and spanwise spacing (2 × 32.5D and 3 × 32.5D ). From the Box Measurements, it was found that close to the turbines, there is a high spanwise variation of streamwise velocity for both staggered and inline configurations, in spite of the fact that the measurements were performed after 6 rows of turbines, and that the wind farms had quite high densities in comparison with the cases studies before in Cal et al. (2010), Chamorro & Porté-Agel (2011) and Chamorro et al. (2011). Moreover, it was found that a good spatial average could be achieved by only measuring in four different points in each plane above the wind farm. It was found that even for these high densities and developed flow conditions, the velocity between two turbines in the same row is the free-stream velocity for inline configurations. Also, It was found that an acceleration of the 72 flow (Venturi effect) takes place at half the spanwise distance that separates two turbines of two different rows for staggered arrangements. A scaling behaviour of the streamwise mean velocity profile and the Reynolds stresses was found. First, it was found that, to know the streamwise velocity profile above a wind farm, it was only necessary to know the free-stream velocity, the streamwise velocity close to the turbines, and the boundary layer thickness (δ ∗ ). Second, it was also found that to know the Reynolds stresses profiles above a wind farm, it was only necessary to know the friction velocity close to the turbines and the thickness of the boundary layer. On the other hand, it was found that the dispersive stresses did not scale with the friction velocity, because they do not come from the same contributions: the friction velocity comes from the small scales (turbulence) while the dispersive stresses come from the large scales (mean flow). Also, it was found that the value of the 00 00 dispersive stresses hU V i are higher for inline configurations, since the turbines in this layout are not distributed as homogeneously as for the staggered counterpart. From the Farm Measurements, it was observed that the boundary layer above the assessed wind farms did not stop growing, which means that developed conditions were not reached for any of the wind farms. Also, it was observed that turbulence starts to grow faster for inline cases, and that in the zones where the streamwise velocity was the smallest, the turbulence was the highest. Moreover, it was found that after 5 rows of turbines, the spaceaveraged velocities close to the wind farm top were lower for staggered configurations. Furthermore, it was found that the turbulence correlation coefficient above the two assessed wind farms was equal to −0.4, which is a characteristic value for flows above walls, leading to think that this value is more related to the turbulence per se, rather than to boundary layer flows. From the RPM Measurements, it was found that the angular velocity of the turbines was around 25 % higher for staggered configurations along the whole wind farm, and as it is intuitive, by having more angular velocity, more power can be produced. Therefore, having a given number of turbines, the best configuration to place them is the staggered one. Plateaus of the angular velocities could be seen for both configurations. However, this plateau was reached faster for the staggered configuration. What is important to conclude is that, since the angular velocity is directly related to the inflow velocity, it can be stated that the condition of developed flow inside the wind farm is reached for both configurations. A model for calculating the absorption of momentum by the wind farm was derived, and then it was compared with the values of the estimated forces, obtained by introducing the measurements on the time and space-averaged integrated Navier-Stokes equations. From the results of this comparison, it could be concluded that the force model was good enough for developed wind 73 farms, which means that by just knowing the angular velocity of the turbines, the thrust coefficient of them and the geometry of the wind farm, the force applied by the turbines to the flow can be estimated. But, the most relevant result is that, by having the free-stream velocity, the properties of the turbines, the angular velocity of the turbines, and the dispersive stresses close to the turbines, an estimation of the friction velocity can be obtained. This is helpful, since just large-scale measurements are required to obtain small-scale values such as the friction velocity. 74 Bibliography Alfredsson, P.-H. & Dahlberg, J.-A. 1981 Measurements of wake interaction effects on the power output from small wind turbine models. The aeronautical research institude of Sweden (FFA HU-2189). Barthelmie, R., Rathmann, O., Frandsen, S., Hansen, K., Politis, E., Prospathopoulos, J., Radon, K., Cabezon, D., Phillips, J., Neubert, A., Schepers, J. & van der Pijl, S. 2007 Modelling and measurements of wakes in large wind farms. J. Phys.: Conf. Ser. 75 (012049). Bruun, H. H. 1995 Hot-Wire Anemometry Principles and Signal Analysis. Oxford Science Publications. Cal, R. B., Lebrón, J., Castillo, L., Kang, H. S. & Meneveau, C. 2010 Experimental study of the horizontally averaged flow structure in a model windturbine array boundary layer. J. Renewable Sustainable Energy 2 (013106). Calaf, M., Meneveau, C. & Meyers, J. 2010 Large eddy simulation study of fully developed wind-turbine array boundary. Phys. Fluids 22 (015110). Cengel, Y. A. 2007 Heat and Mass Transfer . McGraw-Hill. Chamorro, L. & Porté-Agel, F. 2009 A wind-tunnel investigation of wind-turbine wakes: Boundary-layer turbulence effects. Boundary-Layer Meterorol. 132, 129– 149. Chamorro, L. P., Arndt, R. E. A. & Sotiropoulos, F. 2011 Turbulent flow properties around a staggered wind farm. Boundary-Layer Meterorol. 141, 349– 367. Chamorro, L. P. & Porté-Agel, F. 2011 Turbulent flow properties inside and above a wind farm: A wind-tunnel study. Energies 4, 1916–1936. Cheng, H. & Castro, I. P. 2002 Near wall flow over urban-like roughness. Boundary-Layer Meterorol. 104, 229–259. Ferro, M. 2012 Experimental study on turbulent pipe flow. Master’s thesis, KTH. Frandsen, S. 1992 On the wind speed reduction in the center of large clusters of wind turbines. J. Wind Eng. Ind. Aerodyn 39, 251–265. Frandsen, S., Barthelmie, R., Pryor, S., Rathmann, O., Larsen, S., Hjstrup, J. & Thgersen, M. 2006 Analytical modeling of wind speed deficit in large offshore wind farms. Wind Energy 9, 39–53. 75 Frandsen, S. & Jørgensen, H. 2009 The making of a second-generation wind farm efficiency model complex. Wind Energy 12, 445–458. Hägglund, P. B. 2013 An experimental study on global turbine array effects in large wind turbine clusters. Master’s thesis, KTH. Incropera, F. P. & Dewitt, D. P. 1981 Fundamentals of heat transfer . Wiley. Johansson, A. V. & Wallin, S. 2013 An Introduction to Turbulence. KTH Mechanics. Kundu, P. K., Cohen, I. M. & Downling, D. R. 2012 Fluid Mechanics. Elsevier. Lettau, H. 1969 Note on aerodynamic roughness parameter estimations on the basis of roughness-element description. J. Appl. Meteorol. 8 (828). Lomas, C. G. 1986 Fundamentals of Hot Wire Anemometry. Cambridge University Press. Medici, D. & Alfredsson, P.-H. 2008 Measurements behind model wind turbines: Further evidence of wake meandering. Wind Energy 11, 211–217. Medici, D. & Alfredsson, P.-H. 2009 Measurements on a wind turbine wake: 3d effects and bluff body vortex shedding. Wind Energy 9 (219-236). Monin, A. S. & Yaglom, A. M. 1973 Statictical Fluid Mechanics. The MIT press. Mosetti, G., Poloni, C. & Diviacco, B. 1994 Optimisation of wind turbine positioning in large windfarms by means of a genetic algorithm. Wind Engineering & Industrial Aerodynamics 51, 105–116. Munson, B. R., Young, D. F., Okiishi, T. H. & Huebsch, W. W. 2009 Fundamentals of Fluid Mechanics. Wiley. Perry, A. E. 1982 Hot-wire anemometry. Clarendon Press. Priyadarshana, P. J. A. & Klewicki, J. C. 2004 Study of the motions contributing to the reynolds stress in high and low number turbulent boundary layers. Phys. Fluids 16 (4586). Raupach, M. R. & Shaw, R. H. 1982 Averaging procedures for flow within vegetation canopies. Boundary-Layer Meterorol. 22, 79–90. Segalini, A. & Ivanell, S. 2013 Wind Turbine Aerodynamics. KTH Mechanics. 76

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement