Wind tunnel blockage corrections for wind turbine …1078083/...Wind tunnel blockage corrections for wind turbine measurements by Pieter Inghels 880403-T477 August 2013 Technical report

Wind tunnel blockage corrections forwind turbine measurements

by

Pieter Inghels880403-T477

August 2013Technical report from

Royal Institute of TechnologyKTH Mechanics

SE-100 44 Stockholm, Sweden

i

Wind tunnel blockage corrections for wind turbinemeasurementsPieter Inghels880403-T477

Royal Institute of TechnologyKTH MechanicsSE-100 44 Stockholm, Sweden

Abstract:

Wind-tunnel measurements are an important step during the wind-turbine design process. The goal of wind-tunnel tests is to estimate theoperational performance of the wind turbine, for example by measuringthe power and thrust coefficients. Depending on the sizes of both thewind turbine and the test section, the effect of blockage can be sub-stantial. Correction schemes for the power and thrust coefficients havebeen proposed in the literature, but for high blockage and highly loadedrotors these correction schemes become less accurate.A new method is proposed here to calculate the effect a cylindricalwind-tunnel test section has on the performance of the wind turbine.The wind turbine is modeled with a simplified vortex model. Usingvortices of constant circulation to model the wake vortices, the perfor-mance characteristics are estimated. The test section is modeled witha panel method, adapted for this specific situation. It uses irrotationalaxisymmetric source panels to enforce the solid-wall boundary condi-tion. Combining both models in an iterative scheme allows for thesimulation of the effect of the presence of the test-section walls on windturbines performace.Based on the proposed wind-tunnel model, a more general empirical cor-relation scheme is proposed to estimate the performance characteristicsof a wind turbine operating under unconfined conditions by correctingthe performance measured in the confined wind-tunnel configuration.The proposed correction scheme performs better than the existing cor-rection schemes, including cases with high blockage and highly loadedrotors.

Descriptors:Wind tunnel, wind turbine, numerical model, measurements, blockage,empirical correction scheme

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Acknowledgements

I would like to start by thanking Antonio Segalini for giving me theopportunity to perform my master thesis project under his supervisionand for the daily guidance he provided during my stay at the FluidPhysics Laboratory at KTH. During my thesis work, I learned a lotfrom him. I am grateful for many new insights and perspectives hetaught me.

I would also like to thank Henrik Alfredsson for assisting me as well asfor bringing my master thesis to a successful end. I also wish to thankall the people from the Fluid Physics Laboratory. Their welcomingattitude made me feel very welcome in the laboratory.

I want to say special thanks to all my friends I made here during mystay abroad. They made my life during this period so much nicer andthey gave me the opportunity to experience new and exciting culturesin new and exciting places. They supported me during my thesis workand I could rely on them when I needed it.

I would also like to thank my friends in Belgium as well for thecontinuing support from the home front and for visiting me here inStockholm and sharing in my experience.

I wish to convey my sincerest thanks to my family back home inBelgium for the great opportunity they offered me for studying abroadat KTH and their continuing support during the last two fantasticyears I could spend abroad. I will never forget this and I will forevercherish the time I spent here.

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Contents

Abstract ii

Acknowledgements iv

Chapter 1. Introduction 1

1.1. Wind turbine essentials 1

1.2. Wind energy industry 1

1.3. Wind tunnel testing of wind turbines 3

1.4. Why are improved blockage corrections necessary? 4

1.5. Objectives of the thesis 5

Chapter 2. Theoretical background 7

2.1. Governing equations 7

2.2. The panel method 8

2.3. The vortex model 12

2.4. Glauert’s wind tunnel interference model 16

2.5. Mikkelsen & Sørensen blockage model 20

Chapter 3. Wind tunnel model for wind turbines 23

3.1. General model concept 23

3.2. Tunnel geometry 25

3.3. Simulation process 26

3.4. Results of the wind tunnel model 29

Chapter 4. Blockage correction derivation 35

4.1. Simulation cases 35

4.2. Needed correlations 35

4.3. Validation of the proposed correction scheme 41

Chapter 5. Summary and conclusions 47

vi

Appendix A. Glauert model for small blockage ratios 49

A.1. Dimensionless parameters 49

A.2. Derivation of the thrust force 50

A.3. Derivation of the extracted power 54

A.4. Relating to the wake expansion factor 54

A.5. Equivalent free-stream velocity 55

Appendix B. Quadrilateral source panel expressions 57

Appendix C. Ring-shaped source panel expressions 61

C.1. Velocity potential expression 61

C.2. Axial velocity expression 61

C.3. Radial velocity expression 62

Bibliography 65

vii

CHAPTER 1

Introduction

1.1. Wind turbine essentials

In general, a wind turbine is a mechanical device that extracts kinetic energyfrom a flowing stream of air and converts it into mechanical energy (Hansen2008), that is later usually converted to electricity. There are many differ-ent types of wind turbines, but the most common wind-turbine type is theHorizontal-Axis Wind Turbine (HAWT). In this configuration the rotationalaxis of the wind turbine is oriented horizontally. As shown in Figure 1.1, theenveloping power extracting stream-tube expands due to the extraction of ki-netic energy. Extracting kinetic energy from a moving incompressible airflowreduces the velocity and the power extracting stream tube must expand, due tomass conservation. The presence of the wind turbine and the power extractionis already felt upstream of the turbine. This results in a decreased velocityupstream of the turbine blades, compared to the incoming free-stream velocity(Medici et al. 2011). Furthermore, due to the rotation of the wind turbine,an additional rotational velocity component is generated in the wake (Hansen2008).

1.2. Wind energy industry

According to the IPCC (2007), the increased anthropogenic emissions of green-house gases (e.g. CO2 and CH4), partly due to the energy sector, has lead toa substantial effect on the climate balance. In the face of climate change dueto the enhanced greenhouse effect, the interest in finding viable technologiesfor exploiting renewable-energy resources for the production of electricity hasgrown substantially. One of the possibilities to avoid greenhouse gas emissionsis to generate electricity from widely available wind resources. One industrythat has had many successes in developing economically viable technologies isthe wind-energy industry.

Investments in the wind-energy industry are increasing strongly (GWEC2012). Figure 1.2 shows the global annual newly installed wind capacity andit is clear from it that the wind-energy industry is growing fast worldwide.Consequently the global cumulative wind capacity is also increasing and moreand more wind turbines are becoming operational to produce electricity (Fig-ure 1.3). However, there are big discrepancies between geographical regions:

1

2 1. INTRODUCTION

Figure 1.1: Graphical representation of the power extracting stream-tube en-velopping a HAWT. (Burton et al. 2011)

Figure 1.2: Global annual installed wind capacity according to GWEC (2012)

While Europe, North America and Asia are investing heavily in wind energy,Latin America, Africa and the Pacific area are lagging behind. During 2012 inEurope alone, investments in wind energy ranged between 12.9Ge and 17Ge,accounting for almost 26% of the new installed electricity capacity in Europe.

Due to the increased investments in the wind energy sector and the betterenergy conversion technologies, investors start to expect a higher return. Thismeans that total costs in the overall design and manufacturing process mustdecrease. The need for better and more accurate methods in the design processare therefore of paramount importance. Testing of wind turbines is one part ofthis design process and methods used in the evaluation process of wind turbinescan be improved.

1.3. WIND TUNNEL TESTING OF WIND TURBINES 3

Figure 1.3: Global cumulative installed wind capacity according to GWEC(2012)

1.3. Wind tunnel testing of wind turbines

The estimation of the performance of wind turbines can be done in several ways.One possible way is to evaluate the wind turbine using the BEM theory (Burtonet al. 2011). Here, the General Momentum Theory is combined with the BladeElement Method. The blade is represented as a sequence of individual airfoilsegments, combined into a wind turbine blade. Another method often usedin the wind industry is numerical analysis of the fluid using CFD programs(Vermeer et al. 2003). Here the Navier-Stokes equations are solved using adiscretization of the flow field. This is often computationally heavy to performand therefore very expensive, especially if accuracy is pursued. When one needsto evaluate the performance of an existing wind turbine, real-time data can beused. However, in order to have statistically significant data, it is necessary tohave data available spanning a long time period, often up to one or two years.The other option is to evaluate a scaled-down model of the wind turbine andto do the measurements in an enclosed and controlled environment, i.e. a windtunnel (Pope & Rae 1999).

Measuring the performance of a wind turbine in a wind tunnel is a crucialstep during the design process of wind turbines (Manwell & McGowan 2009).Once the preliminary design of a wind turbine is done, experiments are per-formed to assess the actual behavior of the wind turbine and to verify whetheror not the wind turbine operates as designed. In order to control the operatingconditions during the experiments, an enclosed environment is recreated. Dur-ing these wind-tunnel experiments, several wind-turbine operating conditionscan be investigated under different controlled environmental conditions (free-stream velocity, velocity gradient, turbulence, etc.). These test are performedon a scaled model of the designed wind turbine. The scaling down must beperformed ideally in such a way that air flow characteristics (e.g. Reynolds-number, tip-speed ratio and Mach-number) remain the same for the scaleddown model. This leads sometimes to impossible scenarios (e.g. supersonicflows or too high rotational speed of the rotor). Due to the weak dependence

4 1. INTRODUCTION

on the Reynolds number (Adaramola & Krogstad 2011), only matching thetip-speed ratio is pursued during wind-tunnel measurements for wind turbines.

The wind-tunnel test section can be either a closed (enclosed by walls) or anopen (no walls present) section (Pope & Rae 1999). Open test sections have lessinterference from the limited wind-tunnel size, because the air is not boundedby the solid walls. Therefore, the impact of wall effects on the measurementsis limited. According to Glauert (1936), open test sections can handle windturbines up to 60-70% of the test-section area. In a closed section on the otherhand, the presence of walls alters the flow field around the wind turbine insidethe test section compared to the flow field around a wind turbine in unconfinedconditions. The size of the wind turbines must therefore be reduced more formeasurements in closed test sections, given the same wind-tunnel size.

The presence of walls in closed test sections has an impact on the flowfield around the wind turbine (Chen & Liou 2011). The wake behind thewind turbine is not allowed to expand in a similar way as it would do inunconfined conditions. The flow between the wall and the wake will also alter itscharacteristics. In unconfined conditions, this flow will have the same propertiesas the incoming free-stream flow. In the wind tunnel, the flow is funneled inan area with a decreasing cross-section size. This will accelerate the flow andis clearly not the same as in the unconfined case. This accelerating flow willhave an impact on the performance of the wind turbine during measurementsand this phenomenon is called blockage. Blockage correction factors are oftenused to correct for these mismatches. However, for highly loaded rotors, thesecorrections are often not accurate enough (Werle 2009).

1.4. Why are improved blockage corrections necessary?

Blockage correction schemes have been introduced in the past. One of the firstones who proposed a blockage correction scheme was Glauert (1936) and hisscheme was initially derived for propellers. It was later realized that it couldalso be used for wind turbines, although it has severe limitations for windturbines operating under high blockage ratios and heavily loaded conditions.Many years later Mikkelsen & Sørensen (2002) made successful modificationsto this theory to avoid these limitations. Up to today, these blockage correctionschemes have been widely used.

Other blockage correction schemes have been proposed, but are rarelyadopted. Maskell (1963) developed a blockage correction scheme for bluff bod-ies in closed wind tunnels. This theory was the first to address blockage cor-rections for non-streamlined bodies, but it is neither designed for (nor easilyadapted to) wind-turbine measurements. Ashill & Keating (1988) extendedthe work from Maskell who created a correction scheme which does not requiremodeling of flows in the tunnel. They used a two-component method in whichthe knowledge of two components of flow velocity at the outer boundary mustbe known. This method was also not derived with powered propeller or power

1.5. OBJECTIVES OF THE THESIS 5

extracting wind turbines and blockage effects in mind. Hensel (1951) proposeda method, based on the method used by Thom (1943), in which point and linedoublets are used to model bodies of revolution, finite straight wings, and finiteswept wings. Hackett et al. (1979) build off of Hensel’s work: Here source andsink elements are used instead of doublets. During experiments, wall pressureswere measured along the centerline of the tunnel walls or the roof, and on thefloor when necessary. These indicative pressures were then used to determinesource or sink strengths, spans and locations on the tunnel centerline whichdefine a body outline to build up a model equivalent to the test model and itswake. These methods are also not specifically adapted to wind-turbine specificsituations, but it is interesting to point out the use of point and line sourcesor doublets to model solid bodies. Finally, Fitzgerald (2007) proposed a modelwhere propellers and the wind tunnel are modeled with source and sink el-ements or doublet elements: This model is specifically adapted to poweredpropellers and can be extended to wind turbines. Because of the crude pro-peller model, based on a single doublet element, it is not adopted as a generalapproach to determine the effect of blockage for a generic wind turbine duringwind-tunnel measurements.

The correlation between the measured performance data from wind-tunnelmeasurements and the measured performance data of unconfined operatingwind turbines are often not accurate enough. The blockage corrections thathave been used specifically in wind-turbine measurements are derived basedon the Actuator Disk Theory (Glauert 1936). This is however a crude theorybased on assumptions that highly simplify the flow (Werle 2010) and neglecteffects that might have an impact on the performance of the measured windturbine. This results in discrepancies between the corrected measured wind-tunnel data and the data from the unconfined operating wind turbines (Werle2009).

The availability of a new, fast and accurate numerical method for calculat-ing the near-wake structure and the performance of a wind turbine (Segalini &Alfredsson 2013), provides the opportunity to build a numerical model of thewind turbine in a wind tunnel by extending the model with test-section walls.It is indeed possible to simulate the effect of the test-section walls on the near-wake structure and the performance of the wind turbine. The results can becompared with the near-wake structure and the performance of the same windturbine operating under unconfined conditions, resulting in a relation betweenthe two situations.

1.5. Objectives of the thesis

The main objectives of the thesis work are twofold. The first main objectiveis to create a fast numerical model, based on the Simplified Vortex Method(Segalini & Alfredsson 2013), where the effect of the presence of test-sectionwalls is simulated. This vortex method is modeled assuming irrotational flow

6 1. INTRODUCTION

outside the vortex filaments characterizing the wake. The presence of the test-section walls will therefore be modeled using potential flow characteristics. Thewalls will be modeled using source panels, enforcing the numerical boundarycondition that represents the solid wind-tunnel wall at a certain location inspace. The main goal is to be able to compare performance data of a windturbine operating in unconfined conditions with performance data of the samewind turbine operating in a closed test-section configuration. Both cases needto be calculated from the model.

The second main objective is to postulate a new empirical correlationscheme to account for blockage during wind-tunnel measurements of any genericwind turbine. Based on the proposed simulation model, several simulations willbe calculated on a dedicated machine. The results of these simulations will beused to derive a new empirical correlation scheme where the least possibleamount of measurement data is needed to estimate the performance of thesame wind turbine in unconfined operating conditions.

To obtain these main objectives the thesis is built up in several chapters,each describing an important aspect of the content. In chapter 2 an overview ofthe theoretical background and the most important existing blockage correctionschemes is provided. In chapter 3 the implementation of the proposed wind-tunnel model is described together with an example of the calculation process.In chapter 4 a general blockage correction scheme is proposed and validated,based on data collected from numerous simulation cases of the wind-tunnelmodel.

CHAPTER 2

Theoretical background

2.1. Governing equations

The Navier-Stokes equations can describe the behavior of several fluids (in-cluding air and water). These equations are strongly coupled and non-linear innature and therefore very difficult and expensive to solve. Depending on thespecific fluid and flow type, assumptions can be made so that the equationscan be simplified, making them solvable or providing physical insight.

For a steady-state, incompressible, inviscid and irrotational flow some sim-plifications can be made. Steady-state means that all time-dependent termsvanish ( ∂∂t = 0). The incompressibility of the fluid results in a constant fluiddensity (ρ = constant) while the inviscid character of the fluid makes allviscosity-related forces negligible. Irrotationality of the fluid implies that thefluid does not rotate on a macroscopic level, due to shear stresses (∇×u = 0).Since gravity and other body forces are expected to be negligible, these termsare can also be omitted. These simplifications give rise to the Euler equationsin integral form ∫

Σ

(u · n)dΣ = 0 , (2.1)

∫Σ

ui(u · n)dΣ = −∫Σ

pdΣ , (2.2)

while the differential form can be expressed as

∇ · u = 0 , (2.3)

u · ∇u = −∇pρ. (2.4)

Following Anderson (2007), the following identity is valid for any scalarfunction Φ

∇× (∇Φ) = 0 . (2.5)

Therefore, in the case of irrotational flows, the velocity can be expressed as thethe gradient of that scalar function Φ

u = ∇Φ . (2.6)

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8 2. THEORETICAL BACKGROUND

This scalar function is called the velocity potential. Using equation 2.6, thecontinuity equation (eq. 2.3) can be rewritten as

∇ · ∇Φ = ∇2Φ = 0 . (2.7)

This equation is called the Laplace equation for steady state, incompressible,inviscid and irrotational flow. Based on the solution of the velocity potentialof the Laplace equation, the velocity field of a steady state, incompressible,inviscid and irrotational flow can be calculated.

Note that the Laplace equation is a linear differential equation. This im-plies that, if [Φ1,Φ2, ...,Φn] is a set of n solutions of the Laplace equation,then

Φ =

n∑k=1

ckΦk , (2.8)

with ck arbitrary constants, is also a solution of the Laplace equation, namely

∇2Φ =

n∑k=1

ck∇2Φk = 0 . (2.9)

The linearity of the Laplace equation is a very important property. Afterobtaining elementary solutions for either the velocity or velocity potential, thesolution that satisfies the boundary condition can be reduced to searching forthe right linear combination and the solving of a set of algebraic equations.

However, solving the Laplace equation tells nothing about the pressures.This can be done by applying the Bernoulli equation

u2

2+p

ρ= const. (2.10)

Based on the velocities, calculated from the solution of the Laplace equation,the corresponding pressure can be indeed calculated.

2.2. The panel method

2.2.1. Point source

One possible solution of equation 2.7 is a point source/sink element. Thepotential of a point-source element, placed in the origin of a spherical coordinatesystem can be expressed as

Φ(r) = − σ

4πr, (2.11)

where Φ is the velocity potential, r the distance from the point of interest tothe source element and σ the point-source strength expressing a certain amountof volumetric flow. The velocity field, accompanying this velocity potential ispurely radial and can be found by taking the derivative in the radial directionof this velocity potential, as

v = ∇Φ = − σ

4π∇(

1

r

)=

σ

4π

r

r3, (2.12)

2.2. THE PANEL METHOD 9

or in spherical velocity components

(vr, vθ, vφ) =( σ

4πr2, 0, 0

). (2.13)

The velocity field decreases indeed at a rate of r−2 and has a singularity at theposition of the source. If σ > 0, a source is represented and if σ < 0 a sink isrepresented.

If the source element is not located at the origin but at a point r0, theexpressions for the velocity potential and the velocity components need to bechanged. In the general form they become

Φ = − σ

4π |r − r0|, v =

σ

4π

r − r0

|r − r0|3. (2.14)

This basic point-source element principle can be extended to line, surfaceand volume source elements with variable source-strength distributions (Katz& Plotkin 1991). The expression of the velocity potential for a line, surfaceand volume in Cartesian form are, respectively

Φ(x, y, z) = − 1

4π

∫l

σ(x0, y0, z0)dl√(x− x0)2 + (y − y0)2 + (z − z0)2

, (2.15)

Φ(x, y, z) = − 1

4π

∫S

σ(x0, y0, z0)dS√(x− x0)2 + (y − y0)2 + (z − z0)2

, (2.16)

Φ(x, y, z) = − 1

4π

∫V

σ(x0, y0, z0)dV√(x− x0)2 + (y − y0)2 + (z − z0)2

. (2.17)

2.2.2. Ring-shaped source panel

Given a cylindrical tunnel shape and an axisymmetric distribution of the radialinduced velocity, a particular source panel can be used. This panel is ring-shaped and has a constant source-strength distribution along its entire surface.

Because of the axialsymmetry of the ring, it is possible to reduce the num-ber of velocity components. For the specific case, only two induced velocitiesneed to be calculated, i.e. an axial induced velocity and a radial induced ve-locity. For every point in space, the reference frame of the ring can be orientedso that there will only be a radial and axial velocity component. This can bedone because the ring has an axisymmetric source distribution.

The derivation of the analytical expressions for these velocities is performedin Appendix C. The parameters used to calculate the axial and radial inducedvelocities are the source-strength distribution, σ, the geometry of the ring (i.e.its radius R and the axial length 2δ), the axial location, z, of the point in spaceand the radial location, ρ, of the point in space. This is shown in Figure 2.1.

The velocity potential can be calculated from

Φ = −σR4π

∫ 2π

0

[sinh−1

(δ − zG

)− sinh−1

(−δ − zG

)]dθ . (2.18)


Figure 2.1: Schematic representation of the ring parameters.

The axial velocity can be calculated by

uy =σR

π[F1 − F2] . (2.19)

The radial velocity can be calculated using the following formula

uρ =σR

2πρ

[(δ − z)F1 + (δ + z)F2 +

ρ2 −R2

(R+ ρ)2[(δ − z)F3 + (δ + z)F4]

],

(2.20)

where

F1 =1√

(R+ ρ)2 + (δ − z)2K

(√4Rρ

(R+ ρ)2 + (δ − z)2

), (2.21)

F2 =1√

(R+ ρ)2 + (δ + z)2K

(√4Rρ

(R+ ρ)2 + (δ + z)2

), (2.22)

F3 =1√

(R+ ρ)2 + (δ − z)2Π

(4Rρ

(R+ ρ)2,

4Rρ

(R+ ρ)2 + (δ − z)2

), (2.23)

F4 =1√

(R+ ρ)2 + (δ + z)2Π

(4Rρ

(R+ ρ)2,

4Rρ

(R+ ρ)2 + (δ + z)2

), (2.24)

with K(α) the complete elliptic integral of the first kind with argument α andΠ(α, β) the complete elliptic integral of the third kind with arguments α and

β. The values of K(α) and Π(α, β) can be found in tables or in integratedsoftware functions.

The resulting velocity field is shown in Figure 2.2. It is clear that thevelocity field is symmetric, both axially and radially. Close to the ring panel,

2.2. THE PANEL METHOD 11

−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

Axial location

Radiallocation

Figure 2.2: Vector field of a unitary ring-shaped panel. (R = 1, δ = 0.5, σ = 1)

the radial induced velocity is greatest and is pointed inwards. Near the ringaxis the radial induced velocity decreases to zero, due to symmetry. The axialvelocity increases initially when the point of interest moves away from the ringwhile, after some distance, it decreases again.

2.2.3. Wall modeling

To model a solid wall, the wall boundary condition must be satisfied. This con-dition states that the normal velocity at the wall location should be zero. Dueto the linear characteristic of the potential flow, the total velocity vtot(x, y, z)at some point P ≡ (x, y, z) is the sum of all induced velocities, caused by per-turbations in the flow. These induced velocities can be divided into two maingroups. The first represents the induced velocities of all source panels. Thesecond one represents all other external induced velocities.

In order to enforce the wall boundary conditions, the source strength ofeach individual source must be calculated. To do this for n sources, exactlyn control points at the wall are needed. In these control points the boundarycondition of zero normal velocity must be enforced. In all other points ofthe wall, the boundary condition cannot be enforced. Hence, the more panelsrepresenting the wall are used (more points in which the wall condition isenforced) and the better the solid wall is modeled. For each control point P ,


given a constant source-strength distribution σ, the boundary condition can bewritten as

n∑i=1

σi~Vi(P ) · ~n(P ) + ~Vext(P ) · ~n(P ) = 0 , (2.25)

with n the total number of sources, σi the source strength of source i, ~Vi(P )the velocity component induced by the source i with source strength σ = 1

in point P , ~n(P ) the normal direction at point P and ~Vext(P ) the externalinduced velocity at point P .

This equation is valid for all n control points, which leads to n independentequations in n variables (σi) and can be written in matrix form as

VpanΣ = −Vext , (2.26)

where the matrix Σ is the unknown source distributions. This set of equationscan be solved and leads to σ-values for all sources that will enforce the solidwall boundary condition.

2.3. The vortex model

2.3.1. Vortex quantities

The vorticity vector can be calculated from the curl of the velocity field as

ξ = ∇× u . (2.27)

If ξ = 0 at every point in the fluid, the flow is called irrotational, to indicatethat the motion of each fluid particle through space is a pure translation. Ifξ 6= 0 at some points in the fluid, the flow is called rotational and the motionof the fluid has a finite angular velocity in these points. Vortex lines are fieldlines that are parallel to the vorticity vector and are described by

ξ × dl = 0 . (2.28)

Vortex lines that pass through a closed curve in space form a vortex tube.When the cross-sectional area of such a vortex tube is infinitesimally small,it is called a vortex filament. For inviscid flows, it can be proven that thecirculation around a vortex filament is constant

Γvf =

∫ξdS = constant (2.29)

Based on this property, Helmholtz derived his vortex theorems:

1. The strength of a vortex filament is constant along its length.2. A vortex filament cannot start or end in a fluid. It must form a closed

path or extend to infinity.3. The fluid that forms a vortex filament continues to form a vortex fila-

ment and the strength of the vortex filament remains constant as thefilament moves about.

2.3. THE VORTEX MODEL 13

that are equivalent to Kelvin’s theorem, that simply states that the circulationaround any closed curve moving with the fluid remains constant with time.This theorem basically states that if a closed contour is observed at one instantand that closed contour is followed over time, the circulation over the twolocations of this contour are equal.

2.3.2. Biot-Savart’s law

The velocity field may be expressed as the curl of a vector field B

u = ∇×B , (2.30)

and B can be selected in such a way that

∇ ·B = 0 . (2.31)

The vorticity can be rewritten in terms of this vector field, after some vectormanipulation this becomes

ξ = ∇× u = ∇× (∇×B) = ∇(∇ ·B)−∇2B = −∇2B . (2.32)

Using Green’s theorem, the vector field B in a point P, at a distance r0 fromthe origin and at a distance |r0 − r1| from the generic vorticity source, is

B (r0) =1

4π

∫V

ξ(r1)

|r0 − r1|dV , (2.33)

and the induced velocity is

u (r0) =1

4π

∫V

∇× ξ(r1)

|r0 − r1|dV . (2.34)

When the vorticity source is only located in a vortex filament with a crosssection area dS selected so that it is normal to ξ and with the direction of thefilament

dl =ξ

ξdl , (2.35)

and outside this vortex filament there is no vorticity, given the fact the circu-lation is constant for a vortex filament

Γ = ξdS , (2.36)

the induced velocity can be rewritten after executing the curl operation andfor an infinitesimal volume dV = dSdl, resulting in

u (r0) =Γ

4π

∫dl× (r0 − r1)

|r0 − r1|3. (2.37)

This equation is called the Biot-Savart law for inviscid flows. Given the localvorticity distribution of a vortex filament, Equation 2.37 allows for the calcu-lation of the induced velocity in a point in space due to the presence of thatvortex filament.


Figure 2.3: Schematic representation of the vortex model and its referenceframe.

2.3.3. Simplified vortex model for wind turbine wakes

The vortex model is a simplified model based on vortices to simulate inviscidwind turbine wakes, originally proposed by Joukowsky (1912) and extendedby Segalini & Alfredsson (2013). The model assumes a constant circulationalong the blade, from blade tip to blade root, as shown in Figure 2.3. Thisapproximation means that the modeled wind turbine operates at maximumpower production condition for a given set of operating parameters. The modelcalculates the tip-vortex path iteratively, allowing for a wake expansion orcontraction. Based on the tip-vortex path, the induced velocity in the rest ofthe domain can be calculated.

2.3.4. Description

The wind turbine is modeled with Nb line vortices with a constant circulationΓ∗ along the blade location. The line vortex represents a vortex tube witha constant local radius δ∗. According to Helmholtz’s theorem, vortex linescannot begin or end inside the domain. They must either form a closed loop orextend to infinity. The bound vortex at the blade must therefore also extendto infinity, since it does not form a closed loop. The bound vortex will extendfrom the blade root to infinity in a straight line while the bound vortex fromthe blade tip will extend to infinity as a helical line.

To account for the wind-turbine rotation, a rotating reference frame isintroduced, according to Figure 2.3. The x-axis is aligned with the first blade,

2.3. THE VORTEX MODEL 15

the z-axis is oriented normal to the rotating plane of the turbine and the y-axis is chosen normal to both the x- and z-axis so a right-handed axis frameis created. The origin of the reference frame is chosen to be located at thewind-turbine hub. The free-stream velocity is oriented along the z-axis so thatU∞ = U∗∞e3 (with e3 = [0, 0, 1]). The angular velocity of the rotor is alsoaligned with the z-axis, Ω = Ω∗e3

The dimensional quantities are scaled with the density ρ∗, the free-streamvelocity U∗∞ and the rotor radius R∗ in order to make them dimensionless,

λ =Ω∗R∗

U∗∞, Γ =

Γ∗

U∗∞R∗, δ =

δ∗R∗

, x =x∗

R∗, (2.38)

with λ the tip-speed ratio and x the dimensionless vector [x, y, z].

The root-vortex path is assumed to be straight, described by γroot = ze3,while the tip-vortex path is assumed to be helix shaped with a variable helixradius and pitch angle. For the first blade, this vortex path can be describedby

γtip = rW (z)Ψ + ze3 , (2.39)

with

Ψ = [cos(φW (z)), sin(φW (z)), 0] , (2.40)

Ψ = [− sin(φW (z)), cos(φW (z)), 0] . (2.41)

These are orthogonal unitary vectors. Equations 2.39, 2.40 and 2.41 allow forthe description of the path of the tip vortex in a rotating reference frame. Dueto flow symmetry, the tip-vortex paths of the other blades have the same radialfunction rW (z). The angular distribution function φW (z) is shifted for theblade, according to the blade number i

φW,i(z) = φW,1(z) +2π(i− 1)

Nb. (2.42)

The blade is modeled as a line vortex, described mathematically by γblade,i =

µ[cos(αi), sin(αi, 0] where αi = φW,i(0) and µ = (x2 + y2)1/2 ≤ 1 is the nor-malized radial location at the rotor disk.

To compute the induced velocity due to the presence of the vortex lines,the Biot-Savart law is applied (section 2.3.2). At this point, the tip vortexradial function rW (z) and angular function φW (z) are still unknown. Thesecan be determined by means of the Helmholtz theorems. The velocity of the tipvortex system can be derived by adding the two contributions: the free-streamcontribution U∗∞e3 and the velocity induced by the vortex system itself uf .In a rotating reference frame the streamlines are steady state, the followingkinematic equation holds[

uf (γtip) + uext − λe3 × γtip

]× dγtip

dz= 0 , (2.43)


with the first two terms expressing the normalized vortex contribution and anexternally induced velocity (normally only the free-stream velocity), respec-tively, and the third term expresses the change to a rotating reference frame.This equation states that the vortex path is a streamline of the flow field. Given

dγtip

dz=drWdz

Ψ + rWdφWdz

Ψ + e3 , (2.44)

this equation can be rearranged to obtain two ordinary differential equationsfor rW (z) and φW (z)

drWdz

= − ((uf + uext)× e3) · Ψ1 + (uf ×Ψ) · Ψ

, (2.45)

rWdφWdz

=λrW − ((uf + uext)× e3) ·Ψ

(uf × Ψ) ·Ψ− 1. (2.46)

Since solving this set of equations needs information about uf (γtip) itself, aniterative scheme is needed.

2.3.5. Wind turbine performance parameters

The thrust and power coefficient can be calculated by integration over thenormalized blade length µ of the following expressions

dCT =2λNbΓ

πµ(1 + a′b(µ))dµ , (2.47)

dCP =2λNbΓ

πµ(1− ab(µ))dµ , (2.48)

with a′b the tangential induction factor at the blade and ab the axial inductionfactor at the blade. These induction factors can be calculated by using theprojected induced velocity at the blade in the direction of interest, i.e.

ab(µ) = 1− uf,z(µ)

U∗∞, (2.49)

a′b(µ) =uf,θ(µ)

λµU∗∞, (2.50)

where uf,z is the induced velocity from the turbine (namely from the tip vortex,blade vortex and root vortex) in the axial direction, uf,θ the induced velocityfrom the turbine in the azimuthal direction and U∗∞ the magnitude of the free-stream velocity.

2.4. Glauert’s wind tunnel interference model

Glauert’s wind tunnel interference model is based on a similar approach asthe actuator disk theory (Glauert 1936). It also uses a simplified model of awind turbine in a one-dimensional flow. The wind turbine is represented as anideal porous disk extracting power from the flow. The wind tunnel and wind

2.4. GLAUERT’S WIND TUNNEL INTERFERENCE MODEL 17

Figure 2.4: The control volume used in the Glauert wind tunnel interferencemodel approach.

turbine configuration is shown in Figure 2.4. During the analysis, the followingassumptions are made:

• The flow is in steady state, radially homogeneous, incompressible andirrotational.

• There is no frictional drag associated with the turbine disk.• The actuator disk represents a turbine with an infinite number of blades.• There is a uniform distributed thrust force acting on the disk.• The flow field in the wake is non-rotating.

Now consider a control volume around the wind turbine and the windtunnel as shown in Figure 2.4. The power extracting stream-tube around thewind turbine acts as a boundary for two smaller control volumes: the stream-tube control volume (CV1) and the wind tunnel control volume (CV2).

At the inlet, the flow enters the control volume with velocity U∞ andpressure p∞. The cross-area of the stream-tube control volume at the inlet isA∞ and the total cross-area of the whole control volume is C. The fluid flowsthrough the disk with a cross-area Ad at a velocity Ud, exerting a force T on theturbine and extracting a power P from the flow. The pressure just upstream ofthe disk is pu and the pressure just downstream of the disk is pd. At the outlet,the flow inside the stream-tube control volume leaves the control volume at alower velocity UW and a pressure pW . The cross-sectional area of the stream-tube control volume at the outlet is AW and the total cross-sectional area ofthe whole control volume is C. The flow inside the wind-tunnel control volumeleaves the control volume with a velocity U2 and a pressure p2. In the far-wakeregion, this pressure must be equal to the pressure in the stream-tube controlvolume, thus p2 = pW but might be different from the free-stream pressure p∞.

For the quantitative analysis of the Glauert wind-tunnel interference model,the thrust force acting on the wind turbine, the power the wind turbine extractsfrom the flow and the effect of the rotor operation on the local flow field are ofinterest. The full derivation is shown in Appendix A.


The thrust force can be calculated in two ways. The first way is to deter-mine it based on mass and momentum balance. This approach leads to theexpression

T = ρAdU2∞α(1− β)

[1 +

εα

2β2(1 + β)

]+O(ε2) , (2.51)

where α = Ud/U∞, β = UW /U∞ and ε = Ad/C. This expression consistsof two main terms, neglecting the higher order terms of ε. The first term isindependent of ε and can thus be regarded as the unconfined thrust force. Thesecond term depends linearly on ε and for higher blockage ratios ε, this termwill become more important.

The second way to determine the thrust force is based on the pressureacross the actuator disk. This approach leads to a different expression

T =ρAdU

2∞

2(1− β) ((1 + β) + 2εα/β) +O(ε2) , (2.52)

This expression also consists of two main terms. Again, the first term is in-dependent of ε and can thus be regarded as the unconfined thrust force. Thesecond term depends linearly on ε and for higher blockage ratios ε, this termwill become more important.

Equations 2.51 and 2.52 can both be used to determine the thrust force,but they still depend on two unknowns, i.e. α and β that are related to eachother. It is however possible to relate α, ε and β in order to eliminate β fromthe thrust equation. The expression of β in terms of α becomes

β = (2α− 1) + 2εα(1− α)2

(2α− 1)2+O(ε2) . (2.53)

Using this expression for β a new equation for the thrust yields

T = 2ρAdU2∞α(1− α)

[1 + ε

α

2α− 1

]+O(ε2) . (2.54)

This result can be found from both previous expressions for the thrust force.The expression for the extracted power thus becomes

P = Ud T = 2ρAdU3∞α

2(1− α)

[1 + ε

α

2α− 1

]+O(ε2) . (2.55)

The thrust and power coefficients, generally defined as

CT =2T

ρAdU2∞, (2.56)

CP =2P

ρAdU3∞, (2.57)

2.4. GLAUERT’S WIND TUNNEL INTERFERENCE MODEL 19

can then be written as

CT = 4α(1− α)

[1 + ε

α

2α− 1

]+O(ε2) , (2.58)

CP = 4α2(1− α)

[1 + ε

α

2α− 1

]+O(ε2) . (2.59)

These equations state that the impact of the tunnel presence, having a blockageratio ε, can be estimated as the unconfined coefficient plus an extra factordepending on the blockage ratio ε.

2.4.1. The equivalent free-stream velocity

The determination of an equivalent free-stream velocity is a standard way ofdetermining the impact of wind tunnel wall interference. The equivalent free-stream velocity will give the same thrust and disk velocity (and thus power) inunconfined conditions as the wind turbine in confined wind-tunnel conditions.The equivalent free-stream velocity U

′∞ must therefore satisfy the condition

T = 2ρAdUd(U′∞ − Ud) . (2.60)

U′∞ is related to U∞, since the thrust must be the same to equation 2.54

(Appendix A). This yields

U′∞

U∞= 1 + ε

α(1− α)

2α− 1+O(ε2) , (2.61)

or based on the thrust coefficient, in unconfined conditions normally defined asCT = 4α(1− α),

U′∞

U∞= 1 + ε

CT

4√

1− CT+O(ε2) . (2.62)

It can be directly pointed out that for measured thrust coefficients ap-proaching 1, the equivalent free-stream velocity approaches a singularity asCT → 1. For measured thrust coefficients higher than 1, this approach becomesuseless, as visible in Figure 2.5. This is the main drawback of the Glauert modeland this situation needs to be avoided when applying Glauert’s method. Thisusually results in small wind-turbine models in large wind tunnels.

With the equivalent free-stream velocity now calculated, it is possible tocalculate the corrected thrust and power coefficients. They can be calculatedby

C′T = CT

(U∞U ′∞

)2

, (2.63)

C′P = CP

(U∞U ′∞

)3

. (2.64)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

CT (Glauert)

U, ∞

U∞

(Glauert)

Figure 2.5: Results of the Glauert correction on the corrected free-stream ve-locity. Solid line: Blockage ratio of 1%. Dashed line: Blockage ratio of 9%.Dashed-dotted line: Blockage ratio of 16%. Dotted line: Blockage ratio of 25%.

Correcting the thrust and power coefficients can thus be done by simply rescal-ing them with a second or third power of the inverse of the equivalent free-stream velocity. This results in a simple correction scheme, easily implementedonce the measured values and the wind-tunnel geometry is known.

2.5. Mikkelsen & Sørensen blockage model

Mikkelsen & Sørensen (2002) also derived a wind-tunnel blockage model basedon the actuator disk theory, similar to the Glauert wall interference model. Theanalysis starts with the formulation of a set of five equations derived from thecontrol volume analysis. These equations are two mass conservation equations,an expression for the thrust force, an equation for the pressure difference in thepower extracting stream-tube and the global momentum balance on the wholetunnel configuration.

UWAW = UdAd , (2.65)

U2(C −AW ) = U∞C − UdAd , (2.66)

T =ρAd

2(U2

W − U22 ) , (2.67)

∆p = pW − p∞ =ρ

2(U2∞ − U2

2 ) , (2.68)

T −∆pC = ρUWAW (UW − U∞)− ρU2(C −AW )(U∞ − U2) . (2.69)

2.5. MIKKELSEN & SØRENSEN BLOCKAGE MODEL 21

These equations represent a set of 5 equations in 5 variables. Using thedimensionless parameters this set of equations can be rewritten as:

βσ = α , (2.70)

γ(1− εσ) = 1− εα , (2.71)

CT = β2 − γ2 , (2.72)

Cp = 1− γ2 , (2.73)

εCT − Cp = 2βσε(β − 1)− 2γ(1− εσ)(1− γ) . (2.74)

In the Glauert approach, these equations were assumed to behave linearly forsmall blockage ratios ε. Mikkelsen & Sørensen (2002) did not assume this andsolved this set of equations without linearization. They reported the solution inan explicit expression for α in terms of ε and σ (the wake expansion coefficient):

α =σ(εσ2 − 1)

σε(3σ − 2)− 2σ + 1, (2.75)

and the thrust and power coefficients become

CT =(ασ

)2

−(

1− εα1− εσ

)2

, (2.76)

CP = αCT = α

[(ασ

)2

−(

1− εα1− εσ

)2]. (2.77)

The equivalent free-stream velocity, based on the same conditions as the Glauertapproach, can be found by equalizing the thrust coefficient to the equivalentfree-stream thrust coefficient

CT = 4α(α− U ′∞U∞

) , (2.78)

resulting inU′∞

U∞= α− 1

4

CTα. (2.79)

The calculation of the equivalent free-stream is thus performed based on 3parameters that can be measured in wind-tunnel experiments. The α-valuedepends on both the wake expansion coefficient σ and the blockage ratio ε.The third parameter is the measured thrust coefficient. It is directly clear thatthere is no singularity when the thrust coefficient goes to 1 or exceeds it. Thiseffect can be seen in Figure 2.6. It is the main advantage of this approach, eventhough it requires one extra parameter to be measured, i.e. the wake expansioncoefficient.

To estimate the corrected thrust and power coefficients, equation 2.79 canbe applied in the expressions for the corrected thrust and power coefficientsstated in the Glauert model (Eq. 2.63 and 2.64).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

CT (Mikkelsen)

U, ∞

U∞

(Mikkelsen)

Figure 2.6: Results of the Mikkelsen & Sørensen correction on the correctedfree-stream velocity. Solid line: Blockage ratio of 1%. Dashed line: Blockageratio of 9%. Dashed-dotted line: Blockage ratio of 16%. Dotted line: Blockageratio of 25%.

CHAPTER 3

Wind tunnel model for wind turbines

3.1. General model concept

The general concept of the wind tunnel model for wind turbines is based onthe simplified vortex model (see section 2.3) to represent the wind turbine anda panel method (see section 2.2) to model the wind-tunnel walls.

Since the simplified vortex model operates in a rotating reference frame, thetest section is chosen to be cylindrical. To enforce the wall boundary condition,

the method proposed in section 2.2.3 is used, where ~Vext consists in general ofthree main contributions. The first contribution is the induced velocity fromthe wind turbine. The second contribution is the free-stream velocity impact,normal to the wall location. Since this velocity is always located along the windtunnel and wind turbine axis, the free-stream velocity is parallel to the walland no normal component is present. The third contribution is the velocityinduced by the rotating reference frame. The latter two contributions do nothave a component normal to the wind-tunnel wall for a cylindrical test section.

The wind tunnel will introduce an additional externally induced velocityterm in equation 2.43. This means that

uext = ut(γtip) + U∗∞e3 , (3.1)

where

ut(γtip) =

panels∑j=1

(σjU j(γtip)) . (3.2)

This is the induced velocity that all panels exert on the wind-turbine wake.

Since the boundary condition at the wall is of zero normal velocity, theinduced velocity from the wind turbine on the test-section walls needs to beneutralized by the sources. The sources now also induce a certain velocity fieldthroughout the fluid which is felt by the wind-turbine wake too. This newlyinduced velocity will have an impact on the location and shape of the wakeand thus the induced velocity by the wind turbine on the test-section walls. Inturn, the source strength distribution has to be adapted to the new wind turbineinduced velocity. It is clear that this requires an iterative scheme. The generaliterative scheme, used to solve this problem, is represented in Figure 3.1. Themain steps in this iterative scheme are

23

24 3. WIND TUNNEL MODEL FOR WIND TURBINES

Figure 3.1: Schematic representation of the iteration scheme.

1. Initialization: In the initialization step the wind-tunnel geometry isset up, the wind-turbine operating parameters are determined and aninitial wake geometry is initialized. This initial wake geometry is takento be the wake geometry of a wind turbine operating under unconfinedconditions. The initial source strength of the individual panels is set tobe zero.

2. Wind-turbine impact: The wind-turbine wake is calculated, based on thewind-turbine operating parameters, the source-strength distribution ofthe source panels and its induced velocity. When the wind-turbine wakeis calculated, the induced velocity in the test-section wall control pointsis calculated

3. Wind-tunnel impact: The source-strength distribution of the test-sectionsource panels is calculated, based on the induced velocity from the windturbine and the wind-tunnel geometry and the source-panel discretiza-tion.

4. Check convergence: The convergence criterion used to assess the con-vergence of the iterative scheme is based on the source-strength dis-tribution. If the relative difference of the absolute value between theprevious and the current source-strength distribution is smaller than acertain threshold, the iterative scheme is assumed to be converged. Ifthis is not the case, the iteration is started again with the newly calcu-lated source-strength distribution and newly calculated wake geometryas initial values.

5. Generate final results: After the process is converged, the final resultsare calculated and stored for further processing.

3.2. TUNNEL GEOMETRY 25

Table 3.1: Ratio of the radial velocity variation magnitude to the mean radialvelocity magnitude at the tunnel wall at the wind turbine location.

Urad/Urad (%) 10R 8R 6R 5R 4R 3R 2R

1 Blade 0.34 0.31 0.33 0.38 0.48 0.72 1.803 Blades 0.001 0.002 0.004 0.007 0.015 0.042 0.166

3.2. Tunnel geometry

Initially it was proposed to build up the tunnel geometry from quadrilateralsource panels (see Appendix B) covering the test-section wall. When the vari-ation of the local radial velocity is small compared to its average, the inducedradial velocity can be assumed to be axisymmetric and the circumferential vari-ation of the source-strength distribution is also axisymmetric. When this is thecase, ring-shaped sources can be used (section 2.2.2).

Figure 3.2 shows the average of the radial induced velocity for a one bladedturbine operating in unconfined conditions in a test section with a radius of3 times the wind-turbine radius. This means no effect of a wind-tunnel wallpresence is taken into account yet. In the graph, the azimuthal variation isshown with error-bars. At the wind-turbine location, the variation is expectedto be the greatest. With a ratio of standard deviation to average of about0.72%, the effect of variation in the circumferential direction can be neglected.This configuration is the worst case scenario and from Table 3.1 it is clearthat the ratio of standard deviation to mean radial induced velocity decreasesfast for larger wind tunnels and for wind turbines with increasing number ofblades. This table also shows the effect of variation for an extreme blockageratio (R = 2→ ε = 25%). Even for this configuration a maximum variation ofabout 2% is observed and this is still negligible.

The effect of the wind tunnel presence on the asymmetry is difficult toestimate. It is however assumed that the ratio of the radial velocity variationmagnitude to the mean radial velocity magnitude will remain nearly of thesame order of magnitude.

When the panels represent the wall, i.e. the panel center points are thesame as the wall boundary control points, the self-induced velocity of a ring-shaped panel becomes undefined. The problem lies in the complete integral ofthe first kind. Its value goes to infinity if the argument goes to 1. This canbe seen in Figure 3.3. This is clearly the case in equation 2.21 and 2.22 whenρ→ R and y → 0. The argument of the complete elliptic integral is not equalto 1, but it goes towards 1. This can have negative numerical effects and needsto be avoided.


−25 −20 −15 −10 −5 0 5 10 15 20 250

0.01

0.02

0.03

0.04

Axial location

Urad

UU

= 0.72%

Figure 3.2: Mean radial velocity of a single blade wind turbine without windtunnel walls present at a distance of 3R. The error bars indicate the standarddeviation of the radial velocity.

To avoid this problem, the ring panels are placed at a distance farther awayfrom the control points location, so ρ < R. This means that the radius of thecontrol points needs to be at the same radius as the actual modeled tunnel,while the radius of the panels needs to be bigger.

In order to have a smooth transition from a crude mesh to a refined mesh, alinear axial refinement scheme is proposed. This allows the ring length to varyconstantly with a small ring length in the neighborhood of the wind turbine.This is done because the biggest difference in induced velocity is expected inthat neighborhood. A final tunnel geometry is shown in Figure 3.4 and clearlyshows the ring-shaped source panels with an axial refinement in the locationof the wind turbine.

3.3. Simulation process

The simulation process is designed so that the power and thrust coefficientsof a wind turbine operating with or without wind-tunnel walls can be com-pared. Therefore, both cases are calculated with the same turbine operatingparameters.

3.3. SIMULATION PROCESS 27

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.5

2

2.5

3

3.5

4

4.5

5

α

K(α

)

π/2

∞

Figure 3.3: Values for the complete elliptic integral of the first kind.

−100

10

−60−50

−40−30

−20−10

010

2030

4050

60

−10

0

10

y

z

x

Figure 3.4: Wind tunnel geometry of the final model.


The variable input parameters can be subdivided into two main groups:turbine-input parameters and tunnel-input parameters. The turbine-input pa-rameters are the number of blades Nb, the tip-speed-ratio λ and the circulationΓ. The tunnel-input parameter is the test-section radius R.

The output parameters can also be subdivided into two main groups:turbine-output parameters and tunnel-output parameters. The tunnel-outputparameters are the tunnel discretization, the source strength of the panels σ(z)and the mass flow at the tunnel inlet min. The turbine-output parameters arethe wake shape γtip and its components (i.e. rW and φW ), the thrust coefficientdefined with the free-stream velocity Ue, the power coefficient defined with thefree-stream velocity Ue and the mass flow through the turbine disk area mdisk.

3.3.1. Flow field characteristics

One very important characteristic of this wind-tunnel model is the differencebetween the free-stream velocity, here depicted as Ue, and the velocity at theinlet of the tunnel, written as U∞. Due to the presence of the wind turbine,the flow decelerates inside the test section (U∞ < Ue). This is shown schemat-ically in Figure 3.5. The flow velocity at the tunnel inlet can therefore also beregarded as the inlet velocity for an actual wind-tunnel test.

This effect of the numerical model has a consequence on the definition ofthe thrust and power coefficients. This is done based on the free-stream velocityUe, resulting in the thrust coefficient CeT,R and the power coefficient CeP,R. Bycalculating the mass flow at the tunnel inlet min, the flow velocity U∞ can beestimated since min = πR2U∞ρ. This allows for a second definition of thrustand power coefficients based on the tunnel inlet velocity. These values wouldrepresent the measured thrust and power coefficients (C∞T,R and C∞P,R). Thesecoefficients can be related to each other by

CeT,R = (mR)2C∞T,R (3.3)

CeP,R = (mR)3C∞P,R (3.4)

with mR the mass-flow-ratio. This is the ratio of mass flowing through thesame cross section (A = πR2) of the confined case (min) to unconfined case(me):

mR =min

me=ρAU∞ρAUe

=U∞Ue

< 1 (3.5)

Three main types of coefficients are thus calculated from the numericalsimulation model: the coefficients calculated in the unconfined condition case,the coefficients calculated in the confined case scaled with Ue (calculated) andthe coefficients calculated in the confined case scaled with U∞ (measured). Anoverview of the the power and thrust coefficients are given in Table 3.2.

3.4. RESULTS OF THE WIND TUNNEL MODEL 29

Figure 3.5: Schematic representation of the difference between the free-streamflow velocity and the flow velocity at the tunnel inlet, due to the presence of aturbine.

Table 3.2: Overview of the nomenclature for the different thrust and powercoefficients.

Thrust Power

Unconfined CeT,∞ CeP,∞Confined calculated CeT,R CeP,RConfined measured C∞T,R C∞P,R

3.4. Results of the wind tunnel model

The example case input parameters are the following:

• Number of blades Nb = 3• Tip-speed-ratio λ = 5• Circulation Γ = 0.1862• Wind tunnel radius R = 3

This case represents a common wind-turbine type (3 blades) operating at max-imum power production conditions (maximum circulation) and at an averagetip-speed ratio in a wind tunnel with a high blockage ratio.

3.4.1. Source strength distribution

Figure 3.6 shows the radial induced velocity from wind turbine on the test-section wall location in the final iteration. It is clear that the strongest radialinduced velocity occurs in the vicinity of the wind turbine, both before and be-hind. This can obviously be expected for the upstream flow, since the presenceof the wind turbine will divert the incoming axial flow in the radial direction.This effect is stronger close to the wind turbine. In the area just behind the


−60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

z

Urad(z)

Figure 3.6: The induced velocity due to the wind turbine (solid line) and dueto the wind tunnel (dashed line) at the wind tunnel wall location.

wind turbine, the wake is expected to grow and the flow will divert furtheruntil the wake has settled.

To counteract the radial induced velocity from the wind turbine and enforcethe wall boundary condition, the source-strength distribution needs to have adistribution as shown in Figure 3.7. As expected, the highest source strengthsoccur in the vicinity of the wind turbine. The resulting radial induced velocityby the wind tunnel walls is shown in Figure 3.6. It must be the exact negativeof the radial induced velocity from the wind turbine in order to meet the windtunnel wall boundary condition. The radial net effect on the wind tunnel walllocation is therefore 0.

The mass flow, shown in Figure 3.8, is nearly constant (±0.4%) along thetunnel length, clearly demonstrating the quality of the present wall modeling.The mass flow inside the tunnel is m∞ = 26.88 while the mass flow throughthe same cross section of the unconfined case yields me = 28.27. This resultsin a mass-flow ratio of mR = 0.951.

3.4.2. Wake structure

To investigate the wake structure, two wake characteristics are investigated:the wake radius and the wake pitch angle.


−60 −50 −40 −30 −20 −10 0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

z

σ(z)

Figure 3.7: Axial distribution of the source strength for the different sourcepanels in the discretized wind tunnel wall.

It is clear from Figure 3.9 that the radius of the wake decreases due to thepresence of the wind tunnel. This can be expected, since the flow that usuallyflows around the turbine now gets funneled between the wake and the windtunnel walls. It is also clear that the maximum wake radius is reached fasterin the confined case.

Figure 3.10 shows the pitch angle of the wake helix. For the confined casethe wake pitch angle is larger than the one for the unconfined case. This clearlyindicates the elongation of the wake vortex structure. Also, it must be pointedout that the wake pitch angles also settles faster in the confined case than inthe unconfined case.

3.4.3. Thrust and power coefficients

As mentioned before, the numerical model’s output consists of several thrustand power coefficients. All these coefficients are listed in Table 3.2. The val-ues of these thrust and power coefficients for the example case are shown inTable 3.3.

The first observation is that the unconfined thrust coefficient and confinedcalculated thrust coefficient are the same, while the respective power coefficientsare not. Their values are however closer than the unconfined and confinedmeasured power coefficients.


−60 −50 −40 −30 −20 −10 0 10 20 30 40 50 6026

26.2

26.4

26.6

26.8

27

27.2

27.4

27.6

27.8

28

z

m∞

Figure 3.8: The mass flow in the wind tunnel.

Another observation is that the confined measured power coefficient ex-ceeds the Betz-limit (i.e. CP = 0.593). For a wind turbine operating underunconfined conditions this is impossible, but for the same wind turbine oper-ating within a confinement this is possible and is mainly due to the presenceof the wind-tunnel walls and the funneling of the air stream through the windturbine, therefore increasing its power output.

Table 3.3: Overview of the different thrust and power coefficients calculated inthe example model.

Unconfined Confined measured Confined calculated

CT 0.9387 1.0380 0.9387CP 0.5631 0.6727 0.5786


0 5 10 15 20 25 30 35 40 45 50 55 601

1.1

1.2

1.3

1.4

1.5

z

r W(z)

Figure 3.9: Tip vortex radius in the wake structure. Solid line: Unconfinedcase. Dashed line: Confined case.

0 5 10 15 20 25 30 35 40 45 50 55 600.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

z

p(z)

Figure 3.10: Tip vortex pitch angle in the wake structure. Solid line: Uncon-fined case. Dashed line: Confined case.


CHAPTER 4

Blockage correction derivation

The goal of the blockage correlation derivation chapter is to find empirical cor-relations between the measured performance characteristics of a generic windturbine under confined and unconfined conditions. In order to obtain thesecorrelations more simulated cases need to be investigated. Given the modeldescribed in chapter 3, which allows for the fast calculation of a single case, itis possible to simulate many cases in an acceptable time period.

4.1. Simulation cases

The input parameters of the simulation cases are the number of blades Nb,the tip-speed ratio λ, the circulation Γ and the test-section radius Rt. TheNb is chosen to be 1, 3 and 7. The tip-speed ratio λ is investigated between3 and 8. These represent the minimum and maximum values of the typicalrotation speed of a working rotor. Lower values are almost never reached whenwind turbines are in operation and higher tip-speed ratios are usually avoided.The Γ-variable ranges between 10% and 100% of the maximum values of thecirculation Γmax = (8π)/(9λNb) (Segalini & Alfredsson 2013). The test-sectionradius Rt ranges between 10 and 2. These values have been chosen to representa range between a small blockage ratio (Rt = 10→ ε = 1%) and an unusuallylarge blockage ratio (Rt = 2→ ε = 25%).

4.2. Needed correlations

In order to determine a practical correction, it is preferred to use as few mea-sured performance characteristics from the confined case as possible to deter-mine the unconfined performance characteristics. The thrust coefficient, thepower coefficient and the blockage ratio are important measured characteris-tics. The mass-flow ratio is a needed characteristic which is not measurableand needs to be correlated with the measurable characteristics. The thrustand power coefficient of the atmospheric operating wind turbine are the mainperformance characteristics that need to be estimated.

4.2.1. Correlation of the mass-flow-ratio

In the numerical model, the mass-flow-ratio is calculated as one of the modeloutputs. However, in wind-tunnel measurements this value is unknown and

35

36 4. BLOCKAGE CORRECTION DERIVATION

cannot be measured. Therefore a correlations between measurable character-istics, such as the thrust and power coefficients, and the mass-flow-ratio isneeded.

Based on the expected effect both the blockage ratio ε and the measuredthrust coefficient C∞T,R will have on the mass-flow-ratio, the following correla-tion is proposed:

mR,est = 1− f(ε, C∞T,R

), (4.1)

where the function f will have two natural limits:

limε→0

f(ε, C∞T,R

)= 0 , (4.2)

limC∞T,R→0

f(ε, C∞T,R

)= 0 . (4.3)

The first limit states that as the blockage ratio becomes smaller (and thus thetunnel size becomes bigger) the effect of the wall presence decreases and themeasured wind turbine will act more like a wind turbine operating in unconfinedconditions. The second limit states that, for a small thrust coefficient (and thusa small effect of the wind turbine on the flow) the effect on the mass-flow ratiodecreases since a small thrust force does not decelerate the flow too much.

The function f is still unknown. Based on the data it is clear that thefunction should be a smooth function in both variables. As shown in Figure 4.2,it is proposed to use a Taylor expansion of the function f in the two variables,given the two natural limits, of the form

f(ε, C∞T,R

) ∼= εC∞T,R(α1 + α2ε+ α3C

∞T,R + α4ε

2 + α5εC∞T,R + α6

(C∞T,R

)2).

(4.4)

By performing the least squares method (LSM) to determine these coeffi-cients, the general expression for the function f is fitted to the data. For everyset of data, calculated in the simulation cases, mR, C∞T,R and ε are known.It can be expected that higher order expansions will have a better agreementwith the data than lower order expansions, but the higher order expansionsneed more correlating coefficients.

The values of the Taylor expansion are shown in Table 4.1. Comparingthe calculated mass-flow-ratio from the simulation cases with the estimatedmass-flow-ratio, according to the proposed correlation scheme results in a verygood agreement between the two. Figure 4.2 shows the comparison betweenthe calculated mass-flow-ratio and estimated mass-flow-ratio as function of themeasured thrust coefficient for given the blockage ratios.

4.2.2. Correlation of the thrust coefficient

The thrust coefficient calculated in the numerical model CeT,R and the thrustcoefficient measured in the wind tunnel C∞T,R are related to the mass-flow-ratiomR, as stated in section 3.3.1. This is due to the velocity used in the definitions

4.2. NEEDED CORRELATIONS 37

Table 4.1: Empirical coefficients of the Taylor expansion for the estimatedmass-flow-ratio.

a1 a2 a3 a4 a5 a6

0.192 -0.519 0.393 3.029 -1.395 0.010

0.9 0.95 1

0.9

0.95

1

mR (calculated)

mR(estim

ated)

Figure 4.1: Comparison between the simulated mass-flow-ratio mR(calculated)and the estimated mass-flow-ratio mR(estimated) according to the proposedcorrelation scheme, with a margin of error of 0.5%.

of these thrust coefficients. The relation is

CeT,R = (mR)2C∞T,R . (4.5)

As already mentioned in section 3.3.1, the thrust coefficient of the uncon-fined situation and the calculated thrust coefficient from the confined situationfor the example case are the same. Figure 4.3 shows in a more general waythe ratio of the calculated thrust coefficient to the thrust coefficient of the un-confined situation for all the simulated cases. It is clear that the values of thisratio are close to 1 and any difference is negligible (0.02%). It can therefore beassumed that the following relation holds

CeT,∞ ∼= CeT,R . (4.6)


0 0.2 0.4 0.6 0.8 1 1.2 1.40.9

0.92

0.94

0.96

0.98

1

C∞T,R

mR

ε=1/32=0.111

ε=1/62=0.028

ε=1/102=0.010

Figure 4.2: Correlation of the mass-flow-ratio mR with a fourth-order Tay-lor expansion as function of the measured thrust coefficient C∞T,R for differentblockage ratios ε. The individual dots represent the calculated result from thesimulation cases. The solid lines represent the correlation scheme to predictthe mass-flow-ratio.

Combining equation 4.6 and equation 4.5 leads to a correlation between theconfined measured thrust coefficient C∞T,R and the unconfined thrust coefficientCeT,∞, namely

CeT,∞ = (mR)2C∞T,R , (4.7)

where the mass-flow-ratio used in this correlation is the estimated mass-flow-ratio correlation derived in section 4.2.1.

In Figure 4.4 the comparison between the calculated thrust coefficient un-der unconfined condition and the corrected thrust coefficient is shown. Thegreen dots show the values of the thrust coefficient as would be measured ina wind tunnel, the black dots show the values of the measured thrust coeffi-cient corrected with the previously proposed mass-flow-ratio correlation andthe newly proposed thrust coefficient correlation. It is clear that the correctionscheme reduces the over-estimation of the confined measured thrust coefficientto values close to the thrust coefficient under unconfined conditions.

4.2.3. Correlation of the power coefficient

The goal of this section is to relate the confined measured power coefficient(C∞P,R) to the power coefficient under unconfined conditions (CeP,∞). The powercoefficient calculated in the numerical model CeP,R and the power coefficientmeasured in the wind tunnel C∞P,R are related to the mass-flow-ratio mR, as

4.2. NEEDED CORRELATIONS 39

0.94 0.95 0.96 0.97 0.98 0.99 10.998

0.999

1

1.001

1.002

mR

Ce T,R/C

e T,∞

ε=1/32=0.111

ε=1/62=0.028

ε=1/102=0.010

Figure 4.3: Ratio of the calculated thrust coefficient in confined conditions CeT,Rto the calculated thrust coefficient in unconfined conditions CeT,∞ as functionof the mass-flow-ratio mR for different blockage ratios ε.

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

CT (measured-corrected)

CT(unconfined)

Figure 4.4: Comparison of the calculated thrust coefficient in unconfined con-ditions with the confined measured thrust coefficient (green) and the correctedthrust coefficient (black), based on the proposed correlation.


stated in section 3.3.1. This is due to the velocity used in the definitions ofthese thrust coefficients. The relation is

CeP,R = (mR)3C∞P,R . (4.8)

In contrast to the thrust coefficient, the calculated power coefficient fromthe confined situation and the power coefficient from the unconfined situationare not the same. The values of the calculated power coefficient are closer to thepower coefficient under unconfined conditions than to the confined measuredpower coefficient. Figure 4.6 shows the ratio of the calculated power coefficientin confined conditions CeP,R to the calculated power coefficient in unconfinedconditions CeP,∞ as function of the mass-flow-ratio mR for different blockageratios ε. It is clear from this graph that for higher blockage ratios there isa significant difference between the values of the calculated power coefficientand the power coefficient under unconfined conditions, which should not beneglected. It’s however clear that the effect seems to scale with the blockageratio ε. Therefore the following general correlation is proposed

CeP,RCeP,∞

= 1 + εg (ε, mR) , (4.9)

where the function g is an empirical function. With the relation between thecalculated power coefficient and the measured power coefficient (equation 4.8)known, the expression for the correlation between the measured power coeffi-cient and the power coefficient under unconfined condition becomes

CeP,∞ =(mR)

3C∞P,R

1 + εg (ε, mR). (4.10)

This expression has a very important difference with respect to the exist-ing blockage correction schemes. In the Glauert and Mikkelsen & Sørensenschemes, where the equivalent free-stream velocity scaled by the wind tunnelinlet velocity is the inverse of the mass-flow-ratio, the measured power coeffi-cient is scaled with the third power of the mass-flow-ratio to obtain the powercoefficient in unconfined conditions. The additional effect, corrected by thedenominator in equation 4.9, is neglected despite the fact that it has an impactof 2%− 5% and should not be neglected.

At this point, the empirical function g is not yet known and must bederived based on the data calculated in the simulation cases. By introducingthe variable

φ =((1− mR)ε−0.7772

)1/4, (4.11)

the correlation function f is fitted to be the exponent of a third order polyno-mial in the scaled variable φ and can be written as

g (ε, mR) = exp(β3φ

3 + β2φ2 + β1φ+ β0

). (4.12)

4.3. VALIDATION OF THE PROPOSED CORRECTION SCHEME 41

0.3 0.4 0.5 0.6 0.7 0.810

−5

10−4

10−3

10−2

10−1

100

((1 − mR)ε−0.7772i )1/4

((C

e P,R/C

e P,∞

)−

1)ε

−1

i

Figure 4.5: Fitting of the curve after introducing the Φ-variable.

The four empirical constants are [β3 = 34.43;β2 = −52.95;β1 = 47.87;β0 =−21.37] and the fitting is shown in Figure 4.5. The final result of the correlationfunction, using equation 4.11 and equation 4.12, are shown in Figure 4.6.

In Figure 4.7 the comparison between the calculated power coefficient underunconfined condition and the corrected power coefficient is shown. It is clearthat the correction scheme reduces the over-estimation of the measured powercoefficient significantly. The final results of the correction show values close tothe power coefficient under unconfined conditions, with a slight over-estimationof maximum 3%. This over-estimation occurs in situation of very high blockageratios (ε ≥ 3) with a high loading (C∞T,R → 1).

4.3. Validation of the proposed correction scheme

Validation of this newly proposed correction scheme is performed by applyingthe existing Glauert and Mikkelsen & Sørensen correction schemes and theproposed correction scheme to the calculated measured thrust and power co-efficient. Based on the simulation data, where both the measured thrust andpower coefficients and the thrust and power coefficients under unconfined con-ditions are available, the correction schemes are applied and compared. Thecomparison results are shown in Figure 4.8 for a low blockage ratio of 1%, inFigure 4.9 for a high blockage ratio of 11% and Figure 4.10 for an extremeblockage ratio of 25%. In these figures the percentual difference between thecorrected and the calculated thrust or power coefficient are shown in function ofthe measured thrust or power coefficients. The percentual difference in thrust


0.94 0.96 0.98 11

1.02

1.04

1.06

mR

Ce P,R/C

e P,∞

ε=1/32=0.111

ε=1/62=0.028

ε=1/102=0.010

Figure 4.6: Ratio of the calculated power coefficient in wind tunnel conditionsCeP,R to the calculated power coefficient in unconfined conditions CeP,∞ as func-tion of the mass-flow-ratio mR for different blockage ratios ε. Dots: Collecteddata. Solid lines: Results of the final correlation scheme.

coefficient can be defined as

δCeT,∞ =

CeT,∞(corrected)

CeT,∞(calculated)− 1 . (4.13)

In a similar way the percentual difference in power coefficient can be defined.

4.3.1. Results for a low blockage ratio

Figure 4.8 shows the results of the comparison between the blockage correc-tion schemes for a low blockage ratio (i.e. ε = 1%). For the thrust coefficientthe Glauert model seems to be correcting the measured thrust coefficients verywell. The maximum error of this blockage correction scheme, at high turbineloading, is limited to an under-estimation of about 0.5%. The Mikkelsen &Sørensen scheme corrects the measured thrust coefficient less well, with anunder-estimation of about 2%. The proposed model seems to correct the mea-sured thrust coefficient better, with a slight over-estimation of about 0.05%.

For the power coefficient the Glauert model seems to be correcting themeasured power coefficients also very well. The maximum error of this blockagecorrection scheme, at high turbine loading, is limited to an under-estimation ofabout 0.5%. The Mikkelsen & Sørensen scheme corrects the measured power


0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

CP (measured-corrected)

CP(unconfined)

Figure 4.7: Comparison of the calculated power coefficient under unconfinedconditions with the measured thrust coefficient (green) and the corrected powercoefficient (black), based on the proposed correlation.

coefficient less well, with an under-estimation of about 3%. The proposedmodel seems to correct the measured thrust coefficient best, with a slight over-estimation of about 0.05%.

For low blockage ratios, the proposed model seems to be the one that pre-dicts the thrust and power coefficient of a turbine operating under unconfinedconditions best.

4.3.2. Results for a high blockage ratio

Figure 4.9 shows the results of the comparison between the blockage correctionschemes for a high blockage ratio (i.e ε = 11%). For the thrust coefficient theMikkelsen & Sørensen correction scheme seems to be more accurate than theGlauert model for high turbine loading. For low turbine loading, the Glauertmodel seems to be more accurate than the Mikkelsen & Sørensen scheme. Themaximum error of the Mikkelsen & Sørensen scheme is about 4%, while theerror of the Glauert scheme already approaches 10%. Both the Glauert andMikkelsen & Sørensen correction schemes seem to under-estimate the perfor-mance of the actual wind turbine significantly. Note that for measured thrustcoefficients higher than 1, the Glauert model is not able to calculate a correc-tion, since the equivalent free-stream velocity reaches a singularity at CT = 1.


0 0.2 0.4 0.6 0.8 1 1.2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

C∞T,R

δ Ce T,∞

ModelGlauertMikkelsen/Sørensen

0 0.2 0.4 0.6 0.8−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

C∞P,R

δ Ce P,∞


Figure 4.8: Comparison of the thrust (left) and power (right) coefficient errorof the different blockage correction schemes for a blockage ratio of 1%.

The proposed correction scheme seems to correct the measured thrust coeffi-cient better with a slight over-estimation. The maximum error (ca. 0.1%) isvery small compared to the other correction schemes.

For the power coefficient the Mikkelsen & Sørensen correction schemeseems to be more accurate than the Glauert model for high turbine load-ing. For low turbine loading, the Glauert model seems to be more accuratethan the Mikkelsen & Sørensen scheme. The maximum error of the Mikkelsen& Sørensen scheme is about 4%, while the error of the Glauert scheme al-ready approaches 20%. Both the Glauert and Mikkelsen & Sørensen correctionschemes seem to under-estimating the performance of the actual wind turbinesignificantly. The proposed correction scheme seems to correct the measurethrust coefficient best with a slight over-estimation. The maximum error (ca.0.5%) is very small compared to the other correction schemes.

For high blockage ratios, the proposed model seems again to be the onethat predicts the thrust and power coefficient of a turbine operating underunconfined conditions best.

4.3.3. Results for extreme blockage ratio

Figure 4.10 shows the results of the comparison between the blockage correctionschemes for a extreme blockage ratio (i.e ε = 25%). For the thrust coefficientit is clear that the Glauert and Mikkelsen & Sørensen correction schemes be-come less and less useful. The proposed model on the other hand corrects themeasured thrust coefficient very well, even for extreme blockage ratios. Themaximum error is estimated at only 1%, much smaller than the other correctionschemes.


0 0.2 0.4 0.6 0.8 1 1.2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

C∞T,R

δ Ce T,∞


0 0.2 0.4 0.6 0.8−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

C∞P,R

δ Ce P,∞



For the power coefficient there is a similar issue. It seems that the Glauertand Mikkelsen & Sørensen correction schemes have trouble correcting the val-ues, while the proposed correction scheme seems to correct the measured powercoefficient very well. Its maximum error is limited to about a 2% over-estimationand is clearly much lower than the other correction schemes.

Even for extreme blockage ratios, where the other correction schemes havetrouble predicting the thrust and power coefficient, the proposed model seemsagain to be the one that predicts the thrust and power coefficient of a turbineoperating under unconfined conditions best.


0 0.2 0.4 0.6 0.8 1 1.2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

C∞T,R

δ Ce T,∞


0 0.2 0.4 0.6 0.8−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

C∞P,R

δ Ce P,∞



CHAPTER 5

Summary and conclusions

After introducing the necessary theoretical background, the Glauert wind tun-nel interference model and Mikkelsen & Sørensen blockage correction schemesare described. Together with their derivation, their use and limitations arepointed out. The Glauert correction schemes is an easy and straightforwardcorrection scheme, widely used in wind-tunnel measurements. It has a big dis-advantage when the turbine operates in small wind tunnels and when the rotorsare highly loaded, where a singularity is reached for thrust coefficient CT ≈ 1,thus not allowing for accurate calculations in these conditions. The Mikkelsen& Sørensen correction scheme overcomes this problem and it is able to breachthis singularity. However, in this correction scheme one more measured variableis needed, requiring an extra measurement to be done.

Based on the availability of a fast simplified vortex model for modeling thewind turbine, a wind-tunnel model for wind turbines is proposed. Based on theproperties of the simplified vortex model, a cylindrical test section is created.For a wide range of blockage ratios, the radial induced velocity from the windturbine was investigated. The variation of the circumferential distribution ofthe radial induced velocity was found to be very small, even for the worstpossible scenario investigated. The induced velocity from the wind turbine wasthus assumed to be axisymmetric, allowing for the modeling of the wind-tunnelwall with ring-shaped source panels.

The numerical model calculates both a case in unconfined conditions andconfined wind-tunnel conditions. The unconfined case calculates the wakeshape and the performance parameters (i.e. thrust and power coefficient) ofthe wind turbine under unconfined conditions. In the confined wind-tunnelcase, the wake shape and the performance parameters are calculated based onthe same dimensionless unconfined operating conditions. Due to the presenceof the wind turbine and the enclosure of the wind tunnel, the velocity in theinlet area of the wind tunnel is lower than the unconfined free-stream velocity.This velocity is calculated and used to rescale the performance parameters.These rescaled performance parameters are the same as one would measurein the wind tunnel with that inlet velocity. The performance parameters andwake shapes, from the unconfined and confined cases, can be compared. It wasobserved that in the wind-tunnel case the wake shrinks in size and elongates.

47

48 5. SUMMARY AND CONCLUSIONS

Since the simplified vortex model was already validated and used for deter-mining the unconfined performance parameters, a general approach for validat-ing the numerical model is proposed. By deriving a general blockage correctionscheme from many simulation cases and applying this correction scheme andthe existing Glauert and Mikkelsen & Sørensen schemes to the calculated con-fined data, a comparison between their results and the calculated unconfineddata is proposed.

Based on the proposed wind-tunnel model and multiple simulation cases,a blockage correction scheme is thus derived. This process leads to the deriva-tion of three empirical correlation functions where the performance parametersof the wind turbine operating under unconfined conditions can be estimatedonly from performance parameters measured in wind tunnel tests of the samewind turbine. Two of these correlations are needed for the thrust and powercoefficients. The third correlation is needed to estimate the mass-flow ratio, aquantity that cannot be measured in a wind-tunnel test, but is a surrogate ofthe equivalent free-stream velocity. The correction schemes for the mass-flowratio and the thrust coefficient are very accurate. The correction scheme forthe power coefficient is accurate with a small error of ca. 3% for very extremecases, which are almost never reached in wind-tunnel tests.

The newly proposed correlation scheme is compared with the Glauert windtunnel interference model and with the Mikkelsen & Sørensen model to validateit. Here, all models were applied to the measured performance parametersfrom the simulations and compared to the performance parameters of the windturbines in unconfined conditions. The newly proposed model was found tocorrect these values much better than the Glauert or Mikkelsen models.

APPENDIX A

Glauert model for small blockage ratios

Consider a control volume around the wind turbine and the wind tunnel as shownin Figure A.1. The power extracting stream-tube around the wind turbine acts asa boundary for two smaller control volumes: the stream-tube control volume (CV1)and the wind tunnel control volume (CV2).

At the inlet, the flow enters the control volume with a velocity U∞ at a pressurep∞. The cross-area of the stream-tube control volume at the inlet is A∞ and thetotal cross-area of the whole control volume is C. The fluid flows through the diskwith a cross-area Ad at a velocity Ud, exerts a force T on the turbine and extracts apower P from the flow. The pressure just upstream of the disk is pu and the pressurejust downstream of the disk is pd. At the outlet, the flow inside the stream-tubecontrol volume leaves the control volume at a lower velocity UW and a pressure pW .The cross-area of the stream-tube control volume at the outlet is AW and the totalcross-area of the whole control volume is C. The flow inside the wind tunnel controlvolume leaves the control volume with a velocity U2 at a pressure p2. In the far-wakeregion, this pressure must be equal to the pressure in the stream-tube control volume,thus p2 = pW .

A.1. Dimensionless parameters

The dimensionless parameters used in this analysis are:

• The axial induction factor:

a = 1− Ud/U∞ (A.1)

• The α-factor:

α = Ud/U∞ (A.2)

• The wake expansion factor:

σ = AW /Ad = Ud/UW (A.3)

• The wake velocity factor:

β = UW /U∞ (A.4)

• The blockage ratio:

ε = Ad/C (A.5)

• The free-stream acceleration factor:

γ = U2/U∞ =1− εα

1− εα/β (A.6)

49

50 A. GLAUERT MODEL FOR SMALL BLOCKAGE RATIOS

Figure A.1: The control volume used in the Glauert wind tunnel interferencemodel approach.

• The power coefficient:

CP =P

1/2ρAdU3∞

(A.7)

• The thrust coefficient:

CT =T

1/2ρAdU2∞

(A.8)

A.2. Derivation of the thrust force

A.2.1. Thrust force based on mass and momentum balance

For both control volumes, both mass and momentum must be conserved. The massconservation for both control volumes leads to

CV1 : m1 = ρA∞U∞ = ρAdUd = ρAWUW (A.9)

CV2 : m2 = ρ(C −A∞)U∞ = ρ(C −AW )UW (A.10)

while momentum conservation in the axial direction leads to

CV1 : m1(UW − U∞) = −T +X + p∞A∞ − pWA∞ (A.11)

CV2 : m2(U2 − U∞) = −X + p∞(C −A∞)− pW (C −AW ) (A.12)

X is eliminated by summing A.11 and A.12, leading to

X = p∞(C −A∞)− pW (C −AW )− m2(U2 − U∞) (A.13)

Using equations A.10, A.13 and A.11, an expression for the thrust can be derived.

T = C(p∞ − pW )− ρU∞(C −A∞)(U2 − U∞) + ρAdUd(U∞ − UW ) (A.14)

Eliminating (p∞ − pW ) can be done with help from Bernoulli’s equation, applied tothe free-stream control volume

p∞ρ

+U2∞

2=pWρ

+U2

2

2(A.15)

leading to

T =ρC

2(U2

2 − U2∞)− ρU∞(C −A∞)(U2 − U∞) + ρAdUd(U∞ − UW ) (A.16)

A.2. DERIVATION OF THE THRUST FORCE 51

The derivation of the thrust force based on mass and momentum balance, in termsof the dimensionless parameters, is based on the following derivation:

T =ρC

2(U2

2 − U2∞)− ρU∞(C −A∞)(U2 − U∞) + ρAdUd(U∞ − UW )

= ρAdU2∞

[C

2Ad·(U2

2 − U2∞

U2∞

)− C −A∞

Ad·(U2 − U∞U∞

)+

UdU∞·(U∞ − UW

U∞

)]= ρAdU

2∞

[(U2 − U∞U∞

)·[C

2Ad

(U2 + U∞U∞

)− C −A∞

Ad

]+

UdU∞·(U∞ − UW

U∞

)]= ρAdU

2∞

[(1− εα)− (1− εα/β)

(1− εα/β)·[

1

2ε

((1− εα) + (1− εα/β)

(1− εα/β)

)−(

1

ε− α

)]+ α(1− β)

]= ρAdU

2∞

[εα

β· (1− β)

1− εα/β

[2− εα(1 + 1/β)− 2(1− εα)(1− εα/β)

2ε(1− εα/β)

]+ α(1− β)

]= ρAdU

2∞α(1− β)

[εα(1 + 1/β)− 2(εα)2/β

2β(1− εα/β)2+ 1

]= ρAdU

2∞α(1− β)

[εα [(1 + β)− 2εα]

2β2(1− εα/β)2+ 1

](A.17)

For ε << 1, the expression can be simplified. Using the Taylor expansion

[(1 + β)− 2εα] [1− εα/β]−2 = [(1 + β)− 2εα] ·[1 + 2εα/β +O(ε2)

](A.18)

= 1 + β + 2εα/β +O(ε2) (A.19)

and neglecting higher-order terms of ε, the expression for the thrust can be reducedto


[1 +

εα

2β2(1 + β)

]+O(ε2) (A.20)

A.2.2. Thrust force based on the pressure across the actuator disk

Another expression for the thrust force can be based on the pressure difference acrossthe disk area:

T = Ad(pu − pd) (A.21)

The Bernoulli equation is only valid from the far upstream to just in front of the diskand from just behind the disk to far downstream. It is not valid when crossing thedisk. Using Bernoulli’s equation in the stream-tube upstream and downstream of theactuator disk,

puρ

+U2d

2=p∞ρ

+U2∞

2(A.22)

pdρ

+U2d

2=pWρ

+U2W

2(A.23)

Using equations A.15, A.22 and A.23, the expression for the thrust (equation A.21)becomes

T =ρAd

2(U2

2 − U2W ) (A.24)


The derivation of the thrust force based on the pressure across the actuator disk, interms of the dimensionless parameters, is based on the following derivation:

T =ρAd

2(U2

2 − U2W )

=ρAdU

2∞

2

[U2

2

U2∞− U2

W

U2∞

]=ρAdU

2∞

2

[(1− εα

1− εα/β

)2

− β2

]

=ρAdU

2∞

2

[(1− εα)2 − β2(1− εα/β)2

(1− εα/β)2

]=ρAdU

2∞

2

[(1− β2)− 2εα(1− β)

(1− εα/β)2

]=ρAdU

2∞

2(1− β)

[(1 + β)− 2εα

(1− εα/β)2

](A.25)

For ε << 1, the expression can be simplified. Using the Taylor expansion

[(1 + β)− 2εα] [1− εα/β]−2 = [(1 + β)− 2εα] ·[1 + 2εα/β +O(ε2)

](A.26)

= 1 + β + 2εα/β +O(ε2) (A.27)

and neglecting higher-order terms of ε, the expression for the thrust can be reducedto

T =ρAdU

2∞

2(1− β) (1 + β + 2εα/β) +O(ε2) (A.28)

A.2.3. Elimination of wake velocity factor

It’s possible to eliminate β, assuming β scales linearly with ε:

β = β0 + εβ1 (A.29)

Since the thrust is the same, equations A.20 and A.28 must give the same results.Based on these two equations, the elimination of β goes as follows:

ρAdU2∞α(1− β)

[1 +

εα

2β2(1 + β)

]+O(ε2) =

ρAdU2∞

2(1− β) (1 + β + 2εα/β) +O(ε2)

(A.30)

2α

[1 +

εα

2β20

(1 + β0)

]+O(ε2) = 1 + β0 + εβ1 +

2εα

β0+O(ε2) (A.31)

2α+ ε

[α2

β20

+α2

β0

]+O(ε2) = 1 + β0 + ε(β1 + sα/β0) +O(ε2)

(A.32)

To determine β0 and β1, one can see that the first order terms must be equal and thesecond order terms must be equal. The first order equality leads to:

2α = 1 + β0 (A.33)

β0 = 2α− 1 (A.34)

A.2. DERIVATION OF THE THRUST FORCE 53

Second order equality, using the results for β0, leads to an expression for β1:

α2

β20

+α2

β0= β1 +

2α

β0(A.35)

β1 =α2

β20

+α2

β0− 2α

β0(A.36)

=α

β0

(α+ αβ0 − 2β0

β0

)(A.37)

= 2α(1− α)2

(2α− 1)2(A.38)

Finally, this leads to a result for β:

β = β0 + εβ1 (A.39)

β = (2α− 1) + 2εα(1− α)2

(2α− 1)2+O(ε2) (A.40)

A.2.4. Thrust force

Using the expression for β, the thrust coefficients can be written in terms of only αand ε. The elimination of β from equation A.20 goes as follows:


[1 +

εα

2β2(1 + β)

]+O(ε2) (A.41)

= ρAdU2∞α(1− 2α+ 1− 2εα

(1− α)2

(2α− 1)2)

[1 +

εα

2(2α− 1)2(1 + 2α− 1)

]+O(ε2)

(A.42)

= 2ρAdU2∞α

((1− α)− εα (1− α)2

(2α− 1)2

)[1 +

εα2

(2α− 1)2

]+O(ε2) (A.43)

= 2ρAdU2∞α(1− α)

[1 + ε

α2 − α(1− α)

(2α− 1)2

]+O(ε2) (A.44)

= 2ρAdU2∞α(1− α)

[1 + ε

α

2α− 1

]+O(ε2) (A.45)

The elimination of β from equation A.28 goes as follows:

T =ρAdU

2∞

2(1− β) (1 + β + 2εα/β) +O(ε2) (A.46)

=ρAdU

2∞

2

[1− 2α+ 1− 2εα

(1− α)2

(2α− 1)2

] [1 + 2α− 1 + 2εα

(1− α)2

(2α− 1)2+

2εα

2α− 1

]+O(ε2)

(A.47)

= ρAdU2∞2α

[(1− α)− εα (1− α)2

(2α− 1)2

] [1 + ε

(1− α)2 + (2α− 1)

(2α− 1)2

]+O(ε2)

(A.48)

= 2ρAdU2∞α(1− α)

[1 + ε

(1− α)2 + (2α− 1)− α(1− α)

(2α− 1)2

]+O(ε2) (A.49)

= 2ρAdU2∞α(1− α)

[1 + ε

α

2α− 1

]+O(ε2) (A.50)


which is the same equation as equation A.45. The thrust coefficient becomes

CT =T

1/2ρAdU2∞

(A.51)

CT = 4α(1− α)

[1 + ε

α

2α− 1

]+O(ε2) (A.52)

A.3. Derivation of the extracted power

The power extracted from the flow is given by

P = UdT = U∞αT (A.53)

P = 2ρAdU3∞α

2(1− α)

[1 + ε

α

2α− 1

]+O(ε2) (A.54)

The power coefficient becomes

CP =P

1/2ρAdU3∞

(A.55)

CP = 4α2(1− α)

[1 + ε

α

2α− 1

]+O(ε2) (A.56)

A.4. Relating to the wake expansion factor

Starting from the equations A.20 and A.28, given the relation

β =α

σ(A.57)

an expression can be derived to write the α-factor in terms of σ, disregarding theO(ε2)-terms:

2α

[1 +

εα

2β2(1 + β)

]= 1 + β +

2εα

β(A.58)

2α+εα2

β2+εα2

β= 1 + β +

2εα

β(A.59)

2α+ εσ2 + εασ = 1 + α/σ + 2εσ (A.60)

α(2 + εσ − 1/σ) = 1 + 2εσ − εσ2 (A.61)

(A.62)

leading to

α =σ(1 + εσ(2− σ))

(2σ − 1) + εσ2(A.63)

Alternatively, the relation between σ and a becomes

a = 1− α (A.64)

a =(σ − 1)(1 + εσ2)

(2σ − 1) + εσ2(A.65)

A.5. EQUIVALENT FREE-STREAM VELOCITY 55

A.5. Equivalent free-stream velocity

The free-stream velocity must satisfy the condition

T = 2ρAdUd(U′∞ − Ud) (A.66)

Combined with the equation for thrust

T = 2ρAdU2∞α(1− α)

[1 + ε

α

2α− 1

]+O(ε2) (A.67)

the ratio of the equivalent free-stream velocity to the wind tunnel free-stream velocitycan be derived:

2ρAdUd(U′∞ − Ud) = 2ρAdU

2∞α(1− α)

[1 + ε

α

2α− 1

]+O(ε2) (A.68)

UdU∞

(U ′∞U∞− UdU∞

)= α(1− α)

[1 + ε

α

2α− 1

]+O(ε2) (A.69)

α

(U ′∞U∞− α

)= α(1− α)

[1 + ε

α

2α− 1

]+O(ε2) (A.70)

U ′∞U∞

= 1 + εα(1− α)

2α− 1+O(ε2) (A.71)

or based on the expression for CT = 4α(1− α),

U ′∞U∞

= 1 + εCT

4√

1− CT+O(ε2) (A.72)


APPENDIX B

Quadrilateral source panel expressions

A quadrilateral source panel is a flat source distribution surface with four corners. Ingeneral, the four corners are described in the xy-plane, while the z-axis is orientedperpendicular to the panel. The corner point are thus given as (x1, y1, 0), ..., (x4, y4, 0).Following the results obtained by Hess & Smith (1962), the expressions for the velocitypotential and the velocity components, based on the results of the integration processand expressed in the local reference frame of the panel, can be subdivided in far fieldequations and near field equations.

The near-field equations on the other hand are much more complex. The expres-sion for the velocity potential can be written as

Φ(x, y, z) =−σ4π

[(x− x1)(y2 − y1)− ((y − y1)(x2 − x1)

d12lnr1 + r2 + d12r1 + r2 − d12

+(x− x2)(y3 − y2)− ((y − y2)(x3 − x2)

d23lnr2 + r3 + d23r2 + r3 − d23

+(x− x3)(y4 − y3)− ((y − y3)(x4 − x3)

d34lnr3 + r4 + d34r3 + r4 − d34

+(x− x4)(y1 − y4)− ((y − y4)(x1 − x1)

d41lnr4 + r1 + d41r4 + r1 − d41

]+ |z|

[tan−1

(m12e1 − h1

zr1

)− tan−1

(m12e2 − h2

zr2

)+tan−1

(m23e2 − h2

zr2

)− tan−1

(m23e3 − h3

zr3

)+tan−1

(m34e3 − h3

zr3

)− tan−1

(m34e4 − h4

zr4

)+tan−1

(m41e4 − h4

zr4

)− tan−1

(m41e1 − h1

zr1

)](B.1)

with

dij =√

(xj − xi)2 + (yj − yi)2 (B.2)

mij =yj − yixj − xi

(B.3)

ri =√

(x− xi)2 + (y − yi)2 + z2 (B.4)

ei = (x− xi)2 + z2 (B.5)

hi = (x− xi)(y − yi) (B.6)

57

58 B. QUADRILATERAL SOURCE PANEL EXPRESSIONS

−2 −1 0 1 2−2

−1

0

1

2z

x

Figure B.1: Velocity field induced by a source panel.

The velocity components, derived from the velocity-potential equation, are

u(x, y, z) =σ

4π

[y2 − y1d12

lnr1 + r2 + d12r1 + r2 − d12

+y3 − y2d23

lnr2 + r3 + d23r2 + r3 − d23

+y4 − y3d34

lnr3 + r4 + d34r3 + r4 − d34

+y4 − y1d41

lnr1 + r4 + d41r1 + r4 − d41

](B.7)

v(x, y, z) =σ

4π

[x2 − x1d12

lnr1 + r2 + d12r1 + r2 − d12

+x3 − x2d23

lnr2 + r3 + d23r2 + r3 − d23

+x4 − x3d34

lnr3 + r4 + d34r3 + r4 − d34

+x4 − x1d41

lnr1 + r4 + d41r1 + r4 − d41

](B.8)

w(x, y, z) =σ

4π

[tan−1

(m12e1 − h1

zr1

)− tan−1

(m12e2 − h2

zr2

)+tan−1

(m23e2 − h2

zr2

)− tan−1

(m23e3 − h3

zr3

)+tan−1

(m34e3 − h3

zr3

)− tan−1

(m34e4 − h4

zr4

)+tan−1

(m41e4 − h4

zr4

)− tan−1

(m41e1 − h1

zr1

)](B.9)

Figure B.1 shows the vector field representation based on the near field equations.

B. QUADRILATERAL SOURCE PANEL EXPRESSIONS 59

The far-field equations are used when the point of interest is located far awayfrom the source panel, i.e. normally when the distance is 3 to 5 times larger than theaverage panel diameter/diagonal. This will improve the computational efficiency. Inthis case the source panel with area A is perceived as a point source. This results inthe following equations

Φ(x, y, z) = − σA

4π√

(x− x0)2 + (y − y0)2 + (z − z0)2(B.10)

u(x, y, z) =σA(x− x0)

4π√

(x− x0)2 + (y − y0)2 + (z − z0)2(B.11)

v(x, y, z) =σA(y − y0)

4π√

(x− x0)2 + (y − y0)2 + (z − z0)2(B.12)

w(x, y, z) =σA(z − z0)

4π√

(x− x0)2 + (y − y0)2 + (z − z0)2(B.13)

60 B. QUADRILATERAL SOURCE PANEL EXPRESSIONS

APPENDIX C

Ring-shaped source panel expressions

C.1. Velocity potential expression

The starting point for deriving the expression of the velocity potential is the generalexpression for a source/sink panel with a constant source/sink strength distributionin Cartesian coordinates:

Φ = − σ

4π

∫S

dS√(x− x0)2 + (y − y0)2 + (z − z0)2

(C.1)

Any point P (x0, y0, z0) on the surface of the ring, with a constant radius R andan axial distance η, can be described by S = (Rcosθ, η,Rsinθ). This also resultsin dS = Rdηdθ. The ring has a length ranging from (−δ, δ). Since the ring isaxisymmetric, any point in space can be described by only an axial distance y and aradial distance ρ. This results in a new expression for the velocity potential

Φ = −σR4π

∫ 2π

0

dθ

∫ δ

−δ

dη√(ρ−Rcosθ)2 + (y − η)2 + (Rsinθ)2

(C.2)

with

G2 = R2 + ρ2 − 2Rρcosθ (C.3)

and introducing the substitution

(y − η) = Gsinh(ψ) (C.4)

with ∫ b

a

dν√G2 + ν2

=

∫ sinh−1(b/G)

sinh−1(a/G)

dψ (C.5)

= sinh−1

(b

G

)− sinh−1

( aG

)(C.6)

the final expression for the velocity potential becomes

Φ = −σR4π

∫ 2π

0

[sinh−1

(δ − yG

)− sinh−1

(−δ − yG

)]dθ (C.7)

C.2. Axial velocity expression

For deriving the axial velocity, the derivative of the velocity potential must be takenin the axial direction

uy =∂Φ

∂y(C.8)

61

62 C. RING-SHAPED SOURCE PANEL EXPRESSIONS

With

d(sinh−1(x)

)=

dx√1 + x2

(C.9)

this becomes

uy = −σR4π

∫ 2π

0

[1√

G2 + (δ + y)2− 1√

G2 + (δ − y)2

]dθ (C.10)

Using the following transformations

cos(θ) = 2cos2(θ

2

)− 1 (C.11)

β =θ

2, dβ =

dθ

2(C.12)

G2 + ∆2 = R2 + ρ2 + ∆2 − 2Rρcosθ

=[(R+ ρ)2 + ∆2] [1− 4Rρ

(R+ ρ)2 + ∆2cos2(β)

](C.13)

with ∆ = (δ−y) or ∆ = (δ+y) and with the definition of a complete elliptic integralof the first kind

K(α) =

∫ π/2

0

dθ√1− α2cos2(θ)

=1

2

∫ π

0

dθ√1− α2cos2(θ)

(C.14)

the final expression for the axial velocity induced becomes

uy =σR

π

[1√

(R+ ρ)2 + (δ − y)2K

(√4Rρ

(R+ ρ)2 + (δ − y)2

)−

1√(R+ ρ)2 + (δ + y)2

K

(√4Rρ

(R+ ρ)2 + (δ + y)2

)](C.15)

C.3. Radial velocity expression

For deriving the radial velocity, the derivative of the velocity potential must be takenin the radial direction

uρ =∂Φ

∂ρ(C.16)

With

d(sinh−1(x)

)=

dx√1 + x2

(C.17)

this becomes

uρ =σR

4π

∫ 2π

0

ρ−Rcos(θ)

G2

[δ − y√

G2 + (δ − y)2+

δ + y√G2 + (δ + y)2

]dθ (C.18)

C.3. RADIAL VELOCITY EXPRESSION 63

Using the following transformations

cos(θ) = 2cos2(θ

2

)− 1 (C.19)

β =θ

2, dβ =

dθ

2(C.20)

G2 + ∆2 = R2 + ρ2 + ∆2 − 2Rρcosθ

=[(R+ ρ)2 + ∆2] [1− 4Rρ

(R+ ρ)2 + ∆2cos2(β)

](C.21)

ρ−Rcos(θ)

G2=

1

2ρ

(ρ2 − 2ρRcosθ +R2

G2+ρ2 −R2

G2

)=

1

2ρ

(1 +

ρ2 −R2

G2

)(C.22)

with ∆ = (δ − y) or ∆ = (δ + y), the expression reduces to

uρ =σR

2πρ

∫ pi/2

0

(1 +

ρ2 −R2

G2

) δ − y√(R+ ρ)2 + (δ − y)2

1√1− 4Rρ

(R+ρ)2+(δ−y)2 sin2(β)+

δ + y√(R+ ρ)2 + (δ + y)2

1√1− 4Rρ

(R+ρ)2+(δ+y)2sin2(β)

(C.23)

With the definition of a complete elliptic integral of the first kind

K(α) =

∫ π/2

0

dθ√1− α2sin2(θ)

(C.24)

and the definition of a complete elliptic integral of the third kind

Π(α, β) =

∫ π/2

0

dθ

(1− α2sin2(θ))√

1− β2sin2(θ)(C.25)

the final expression for the radial velocity can be written as

uρ =σR

2πρ

δ − y√

(R+ ρ)2 + (δ − y)2K

(√4Rρ

(R+ ρ)2 + (δ − y)2

)

+δ + y√

(R+ ρ)2 + (δ + y)2K

(√4Rρ

(R+ ρ)2 + (δ + y)2

)

+ρ2 −R2

(R+ ρ)2

[δ − y√

(R+ ρ)2 + (δ − y)2Π

(4Rρ

(R+ ρ)2,

4Rρ

(R+ ρ)2 + (δ − y)2

)

+δ + y√

(R+ ρ)2 + (δ + y)2Π

(4Rρ

(R+ ρ)2,

4Rρ

(R+ ρ)2 + (δ + y)2

)](C.26)

64 C. RING-SHAPED SOURCE PANEL EXPRESSIONS

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Wind tunnel blockage corrections for wind turbine …1078083/...Wind tunnel blockage corrections for wind turbine measurements by Pieter Inghels 880403-T477 August 2013 Technical report