Modeling of the UAE Wind 

Turbine for Refinement of 

FAST_AD 

December 2003 • NREL/TP-500-34755 

J.M. Jonkman 

National Renewable Energy Laboratory 

1617 Cole Boulevard 

Golden, Colorado 80401-3393 

NREL is a U.S. Department of Energy Laboratory 

Operated by Midwest Research Institute • Battelle 

Contract No. DE-AC36-99-GO10337 
National Renewable Energy Laboratory 

1617 Cole Boulevard 

Golden, Colorado 80401-3393 

NREL is a U.S. Department of Energy Laboratory 

Operated by Midwest Research Institute • Battelle 

Contract No. DE-AC36-99-GO10337 

December 2003 • NREL/TP-500-34755 

Modeling of the UAE Wind 

Turbine for Refinement of 

FAST_AD 

J.M. Jonkman 

Prepared under Task No. WER3 2010 
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 iii

Abstract

The Unsteady Aerodynamics Experiment (UAE) research wind turbine was modeled both 

aerodynamically and structurally in the FAST_AD wind turbine design code, and its response to 

wind-inflows was simulated for a sample of test cases. A study was conducted to determine why 

there are wind turbine load magnitude discrepancies—inconsistencies in aerodynamic force 

coefficients, rotor shaft torque, and out-of-plane bending moments at the blade root across a 

range of operating conditions—between load predictions made by FAST_AD and other 

modeling tools and measured loads taken from the actual UAE wind turbine during the NASAAmes wind tunnel tests. The NASA-Ames wind tunnel had a highly controlled testing 

environment and an enormous 24.4 m (80 ft) by 36.6 m (120 ft) test section. The acquired 

experimental test data represent the finest, most accurate set of wind turbine aerodynamic and 

induced flow field data available today. A sample of the FAST_AD model input parameters 

most critical to the aerodynamics computations was also systematically perturbed to determine 

their effect on load and performance predictions. Attention was focused on the simpler upwind 

rotor configuration, zero yaw error test cases. Inconsistencies in input file parameters, such as 

aerodynamic performance characteristics, explain a noteworthy fraction of the load prediction 

discrepancies of the various modeling tools. Additionally, discrepancies between modeling tool 

load predictions and physically measured load values highlighted weaknesses in tip loss, stall 

delay, post stall, and other frequently used aerodynamic models. Furthermore, unexplainable 

FAST_AD load prediction and measured load discrepancies uncovered flaws and limitations in 

the structural models employed by FAST_AD and suggested ways to improve models and codes. 

The ways these weaknesses and flaws affect the design of wind energy systems were noted. 

Additional development and refinement of rotary wing aerodynamics and structural models 

should lead to more optimal wind turbine designs and decreases in wind-generated electricity 

costs. 
 iv
 v

Contents 

1. INTRODUCTION.............................................................................................. 1 

2. WIND TURBINE AERODYNAMICS THEORY................................................. 5 

2.1 NATURAL WIND CHARACTERISTICS ................................................................. 5 

2.2 RANKINE-FROUDE ACTUATOR DISC THEORY AND BETZ LIMIT ........................... 6 

2.3 EFFECTS OF WAKE ROTATION ON BETZ LIMIT ................................................ 10 

2.4 BLADE ELEMENT THEORY ............................................................................ 16 

2.5 BLADE ELEMENT MOMENTUM THEORY AND PERFORMANCE PREDICTION ......... 23 

2.6 ADVANCED AERODYNAMICS TOPICS.............................................................. 27 

3. STRUCTURAL MODELING THEORY........................................................... 29 

3.1 GEOMETRY, COORDINATE SYSTEMS, AND DEGREES OF FREEDOM.................. 29 

3.2 BLADE AND TOWER DEFLECTIONS ................................................................ 41 

3.3 KINEMATICS ................................................................................................ 50 

3.4 KINETICS .................................................................................................... 54 

3.5 KANE’S EQUATIONS OF MOTION.................................................................... 58 

4. FAST_AD DESIGN CODE OVERVIEW AND LIMITATIONS ........................ 59 

4.1 FAST_AD DESIGN CODE OVERVIEW ........................................................... 59 

4.2 FAST_AD DESIGN CODE LIMITATIONS......................................................... 61 

5. UAE AND BLIND COMPARISON OVERVIEW ............................................. 64 

5.1 UAE WIND TURBINE SPECIFICATIONS........................................................... 65 

5.2 NASA-AMES WIND TUNNEL TESTING ........................................................... 76 

5.3 BLIND COMPARISON OVERVIEW.................................................................... 77 

6. BLIND COMPARISON INPUT PARAMETERS AND RESULTS................... 81 

6.1 FAST_AD INPUT FILES DEVELOPMENT FOR THE BLIND COMPARISON STUDY.. 81 

6.2 BLIND COMPARISON RESULTS AND DISCUSSION ............................................ 98 

7. SENSITIVITY STUDY INPUT PARAMETERS AND RESULTS .................. 118 

7.1 SENSITIVITY STUDIES AND FAST_AD INPUT PARAMETER ALTERATIONS ....... 118 

7.2 SENSITIVITY STUDY RESULTS AND DISCUSSION ........................................... 132 

8. CONCLUSIONS........................................................................................... 149 

REFERENCES................................................................................................. 151 

APPENDIX I. FAST_AD INPUT FILE FOR TEST CASE S0700000 ............... 153 

APPENDIX II. MODES INPUT FILES FOR USE IN FAST_AD ....................... 160 

APPENDIX III. WIND DATA FILE FOR TEST CASE S0700000 ..................... 162 

APPENDIX IV. AIRFOIL DATA TABLE FILES OF THE S809 AIRFOIL......... 163 

APPENDIX V. CRUNCH DATA FILE FOR TEST CASE S0700000................ 173 
 1

1. Introduction 

Nonrenewable resources such as coal and natural gas are the main sources of energy for many 

parts of the world. However, these energy resources are harmful to the environment and supplies 

are limited. Resources such as wind possess great potential because they are renewable and 

nonpolluting. 

Wind-driven power systems that can produce large amounts of power are relatively new 

technologies. Many earlier models failed catastrophically because blades struck their support 

structure or tore off the hubs, or towers collapsed. Such failures have diminished in recent years 

because of improved designs, but other failures such as unacceptable material fatigue and 

component malfunction are common. In general, failures occur because the effects of 

unanticipated loads and unknown load levels and load paths are substantial. To combat such 

effects, wind turbine designers rely heavily on time-tested safety factors that account for this lack 

of knowledge. Such a design methodology greatly limits the cost effectiveness of new wind 

turbine designs, which is critical if wind-generated electricity is to be economically competitive 

with traditional energy sources. 

The U.S. Department of Energy’s (DOE) National Renewable Energy Laboratory (NREL) has 

sponsored the development and validation of comprehensive system dynamics codes that can 

accurately predict, quantify, and understand the complex physical interactions that characterize 

component loads and overall wind turbine performance. Researchers and designers use such 

codes to efficiently, safely, and cost effectively design and analyze wind energy systems. These 

design codes enable the user to (1) model a wind turbine aerodynamically and structurally given 

the turbine layout and aerodynamic and mechanical properties of its members, and (2) simulate 

the wind turbine’s aerodynamic and structural response by imposing complex virtual windinflow conditions. Outputs of these simulations include time-series data on the loads and 

responses of the structural members of the wind turbine. Post-processing codes are used to 

analyze these data. 

In many respects, design codes (also labeled as modeling tools) bridge the gap between theorized 

predictions and experimental or observable measurements. Design codes essentially perform 

virtual experiments that can yield load analysis results quickly and cheaply. In many situations, 

virtual experimentation offers the only practical method of research and testing. 

A streamlined design code titled FAST_AD was recently developed through a partnership 

between NREL and Oregon State University. FAST stands for Fatigue, Aerodynamics, 

Structures, and Turbulence modeling; the AD postscript indicates that the wind-inflow data used 

to excite the model during simulation routines are developed with the AeroDyn aerodynamics 

subroutine package, developed by engineering faculty and staff at the University of Utah. 

Design codes must be validated to be effective in the design and analysis environment. 

Validation involves comparing simulated load predictions with load predictions made by other 

modeling tools and with measurements taken experimentally, then assessing the degree to which 
 2

they correspond. FAST_AD has yet to be validated. For accurate and reliable experimental 

measurements to improve the model and validate the code, NREL planned and conducted the 

comprehensive test program known as the Unsteady Aerodynamics Experiment (UAE). In the 

most recent phase of this program (Phase VI), a heavily instrumented, 10-m diameter, 20-kW 

wind turbine, known as the UAE research wind turbine, was tested in the world’s largest wind 

tunnel located at the NASA-Ames Research Center. These tests were completed in May 2000, 

following three weeks of exhaustive testing. The data acquired from the tests represent the 

finest, most accurate set of wind turbine aerodynamic and induced flow field data available. 

To test the validity and range of applicability of the models used in design codes, NREL solicited 

wind turbine modeling experts from around the world to predict loads and performance of the 

UAE wind turbine at selected operating conditions. These predictions were done “blind” in that 

the only information given to the participants concerned basic machine parameters of the UAE 

wind turbine and essential wind-inflow data, not measured wind tunnel experimental data. 

During a later comparison phase, the predictions from all the Blind Comparison modelers were 

presented and compared with the experimentally measured data. 

The objective of this report is to model the aerodynamic and structural aspects of the UAE 

research wind turbine in the FAST_AD design code and to run simulations by exciting the model 

with the same conditions seen in the NASA-Ames wind tunnel for several UAE test cases. 

Furthermore, to facilitate validation of FAST_AD, the load predictions are compared with load 

predictions made by similar modeling tools employed by other modelers during the Blind 

Comparison exercise and with experimental wind tunnel test measurements. 

Load magnitude discrepancies, such as inconsistencies in aerodynamic force coefficients, rotor 

shaft torque, and out-of-plane bending moments at the blade root, appear between these various 

load predictions and measurements across a range of operating conditions. Some, but not all, are 

explainable by inconsistencies in the input file parameters of the modeling tools. Discrepancies 

among load predictions and measured load values highlight weaknesses in commonly used 

aerodynamic models and uncover flaws and limitations in the structural models employed by 

FAST_AD. Consequently, the aerodynamic and structural models engaged by FAST_AD 

should be refined before validation concludes. Moreover, the code should be progressively 

updated and revalidated as our understanding of the physics involved in wind turbine 

aerodynamics improves. Doing so will lead to more optimal wind turbine designs that are 

cheaper to manufacture and can produce cheaper wind-generated electricity. 

Treating the FAST_AD design code as a “black box” would be careless if a reasonable study is 

to explain the discrepancies between code predictions and experimental test data, or even when 

simply assembling the FAST_AD input file. The assumptions and approximations used to 

develop the models employed by the code must be fully understood before any results can be 

characterized or any conclusions drawn. Thus, an exhaustive development of the aerodynamic 

and structural modeling theories of FAST_AD is presented here. Aerodynamic models are 

presented in Chapter 2 and structural models are presented in Chapter 3. During the 

development of these models, several flaws in the structural models implemented by FAST_AD 

are uncovered. These flaws are separate from any bugs in the coding. An exhaustive overview 

of the coding is not developed. The structural models derived in Chapter 3 are presented in their 
 3

corrected form. Discrepancies between these models and those implemented in the FAST_AD 

design code are acknowledged using footnotes where appropriate. Code improvement 

recommendations for code designers of FAST_AD and the AeroDyn aerodynamics subroutines 

are italicized in these footnotes and in the main body of the report where appropriate. 

An overview of the FAST_AD design code and descriptions of many of its limitations is given in 

Chapter 4. Descriptions of the UAE wind turbine, UAE NASA-Ames wind tunnel tests, and 

Blind Comparison exercise are presented in Chapter 5. A summary of the work performed to run 

simulations in FAST_AD for the Blind Comparison exercise and the associated results and 

discussion is given in Chapter 6. In this chapter, reasons for inconsistencies in the load 

predictions of the various modeling tools are explained, though an explanation of every load 

prediction discrepancy is not submitted. Additionally, weaknesses in the commonly used 

aerodynamic models are highlighted. Attention is focused on the descriptions of load prediction 

discrepancies and inaccuracies for the simpler upwind rotor configuration, zero yaw error test 

cases. To further account for and explain the load prediction and measured load discrepancies 

observed in the Blind Comparison results, a sensitivity analysis is performed as documented in 

Chapter 7. In the sensitivity study, a sample of FAST_AD model inputs is systematically 

perturbed to determine effects on load and performance predictions. A summary of conclusions 

is given in Chapter 8. 

To familiarize readers who are unfamiliar with wind turbine technology on the subject matter, 

various wind turbine classes and configurations are summarized, compared, and contrasted. 

Wind turbines harness the energy in the wind by extracting useful work. The power in the wind 

is extracted by a rotor, which consists of blades appended to a hub. The rotor extracts 

mechanical energy in the form of torque and rotational speed. Typically, this mechanical energy 

is converted to electrical energy by having the rotor run an electric generator, although the 

mechanical energy can also be used directly for applications such as pumping water. 

All wind turbines can be characterized as either horizontal axis wind turbines (HAWTs) or 

vertical axis wind turbines (VAWTs). A HAWT is labeled as such since its rotor spins about an 

axis horizontal to the Earth’s surface. The rotor of a VAWT spins about an axis perpendicular 

(vertical) to the Earth’s surface. This report is devoted entirely to the analysis of HAWTs. 

Rotors of HAWTs are placed on towers to position them where the wind speed is fastest and 

exhibits the most power. A nacelle typically resides atop the tower and contains the support 

structure for the rotor, the rotor shaft, a gearbox, and the electric generator. The gearbox is used 

to transform the low-speed, high-torque power of the rotor to high-speed, low-torque power that 

can run the electric generator. 

Rotors can be positioned upwind or downwind of the tower. Downwind rotor configurations can 

track the wind automatically as wind directions change, similar to a weather vane (a passive yaw 

control system). However, the wind must flow around the tower to reach the rotor of a 

downwind turbine. This results in complex flow patterns and periodic fluctuations in 

aerodynamic loads, which have important dynamical effects on the turbine structure. Flow 

passing through the rotor plane is unobstructed by the tower for upwind rotor configurations. 
 4

However, these turbines must possess an active yaw control system to keep the rotor aligned 

with the wind. 

The hub structure connects the blades to the drive shaft. Hubs are generally characterized as 

either rigid or teetering. As the name implies, in rigid hub designs, the hub is rigidly attached to 

the drive shaft. In contrast, teetered hubs are connected to the drive shaft by means of a teeter 

pin, a bearing that permits the rotor to rock into and out of the plane of rotation. Teetered hubs 

have the benefit that bending moments brought about by thrust forces acting on the blades are 

not transferred to the nacelle and tower structure. Consequently, the nacelle and tower structures 

of turbines with rigid hubs must be designed more robustly than those with teetered hubs. 

Rotor power is typically controlled by either pitch or stall regulation. Stall-regulated wind 

turbines take advantage of the fact that aerodynamic lift, and inevitably power, are reduced when 

the blades stall in exceptionally high winds. In pitch-controlled machines, power is regulated by 

varying the pitch of the blades (that is, the orientation of the blades about the blade axis) through 

the use of complex control mechanisms. In stall-regulated wind turbines, the blades of the rotor 

are rigidly connected to the hub. Though the hub design is simpler for stall-regulated machines, 

the drive train (rotor shaft, gearbox, and generator) must be more robustly designed than in pitchregulated machines so they can withstand extreme winds. These forces can be shed if a pitching 

mechanism is employed. 
 5

2. Wind Turbine Aerodynamics Theory 

Accurate models of the aerodynamic aspects of wind turbines are essential to successfully design 

and analyze wind energy systems. Wind turbine aerodynamic models are used to relate windinflow conditions to loads applied on the turbine. One must enter the field of wind turbine 

aerodynamics analysis with an appreciation for the complexity of the subject, however. After 

years of investigation and testing, researchers throughout Europe and the United States have 

concluded that the typical atmospheric conditions under which a wind turbine operates are 

incredibly complex. The underlying flow physics of wind turbine operation are also much more 

complex than originally thought. 

The true fluid flow passing around and through a wind turbine is governed by the firstprinciples-based Navier-Stokes equations. Unfortunately, these equations are so complex that 

analytical solutions have been found for only a few simple cases. Although numerical solutions 

may be found via a computer, most first-principles-based solutions are computationally intense 

and the cost of such detailed simulations prohibits their use in the design and analysis 

environment. As an alternative, most wind turbine designers and analysts have opted to develop 

models that incorporate two-dimensional airfoil analysis techniques and quasi-steady flow fields. 

The subsequent analysis develops the most common aerodynamics theory employed in the wind 

turbine design and analysis environment: blade element momentum theory (BEM), the 

fundamental aerodynamic model used by the AeroDyn aerodynamics subroutines of the 

FAST_AD design code. The final form of the BEM equations developed here, however, is not 

as common [see Eq. (2.73)], but enhances the visibility of the model’s weaknesses. This 

attribute will become more apparent in the results and discussion of Chapter 6. To help the 

reader appreciate the complexities of real fluid flow, we discuss natural wind characteristics 

qualitatively in section 2.1. Following a procedure similar to that used by McGowan and 

Manwell (2000), the BEM equations are developed in sections 2.2–2.5, with increasing 

complexity added in each section. A brief qualitative discussion of advanced aerodynamics 

topics is presented in section 2.6. This chapter and the aerodynamics theories developed are 

devoted entirely to HAWTs. Aerodynamic theories for VAWTs and other wind turbine 

configurations are beyond the scope of this report. 

2.1 Natural Wind Characteristics 

For practical reasons, wind turbines are positioned near ground level. As such, boundary layer 

effects with the Earth dominate airflow passing through wind turbines. Although the easiest way 

to regard the airflow is as steady, homogenous, uniform, and fixed in direction (see Fig. 2.1a), 

boundary layer flow is anything but. As a direct result of viscosity and the no-slip condition with 

the Earth’s surface, boundary layer flow exhibits a velocity gradient with elevation. The 

simplest description of this tendency is the uniform wind shear model depicted in Fig. 2.1b. The 

boundary layer flow passing through a wind turbine is inherently inhomogeneous, unsteady, and 

turbulent, consisting of rapid velocity and pressure fluctuations. Turbulence occurs since inertial 

effects increasingly overwhelm viscous stresses inherent in the flow stream as the flow speed 

increases (even at very small wind speeds for air), resulting in intrinsically unstable flow that can 

become turbulent with even the slightest flow perturbation. Irregular terrain, such as mountains, 
 6

valleys, houses, and trees increases the likelihood of turbulence at even the lowest wind speeds. 

Wind turbines with downwind rotors experience additional turbulence from the wake of the air 

passing around the tower and nacelle. Turbulent flow is depicted graphically in Fig. 2.1c. 

Figure 2.1: Natural wind characteristics 

The situation is further complicated when wind directions change or the turbine becomes yawed. 

A rotor not aligned with the wind experiences cross-flow components whose magnitude and 

direction relative to the rotor change as the blades rotate. In such cases, flow separation and 

reattachment, spanwise flow, and three-dimensional effects become increasingly important. 

Moreover, the instability of the flow over the rotor blades is provoked by surface imperfections, 

squashed bugs, surface-water droplets, or ice buildup. Furthermore, blade vibration and torsional 

oscillation have important consequences to the form of the fluid flow. Vortices formed via flow 

interactions with the blade, hub, and tip also affect the character of the flow field in an integral 

way. Clearly, wind turbine aerodynamics is not an easy concept to grasp. 

2.2 Rankine-Froude Actuator Disc Theory and Betz Limit 

To begin to understand the complex physics involved in wind turbine aerodynamics, we should 

analyze a simple one-dimensional model. In the one-dimensional Rankine-Froude actuator disc 

model, the rotor is represented by an “actuator disc,” through which the static pressure has a 

jump discontinuity. Consider a control volume fixed in space whose external boundaries are the 

surface of a stream tube whose fluid passes through the rotor disc, a cross-section of the stream 

tube upwind of the rotor, and a cross-section of the stream tube downwind of the rotor. A simple 

schematic of this control volume is given in Fig. 2.2. 

The following assumptions are made in the actuator disc model: 

(1) Wind is steady, homogenous, and fixed in direction. 

(2) Air is incompressible, inviscid, and irrotational. 

(3) Both the flow and the thrust are uniform across the disc. The flow is uniform at the upwind 

(station 0) and downwind (station 3) boundaries of the control volume. 

a) Uniform 

Wind b) Uniform 

Wind Shear

c) Turbulent 

Wind 
 7

Figure 2.2: Control volume for actuator disc model 

(4) The upwind and downwind boundaries are far enough removed from the rotor that the static 

pressure at these points is equal to the unobstructed ambient static pressure. The static 

pressure on the stream tube portion of the boundary is also equal to the unobstructed ambient 

static pressure. 

For a wind turbine rotor to act as an actuator disc, the rotor would have to be composed of an 

infinite number of very thin, dragless blades that rotate with a tip speed much higher than that of 

the incoming wind. Also, station 1 is designated to be slightly upwind and station 2 slightly 

downwind of the rotor. 

Since air does not pass through the stream tube portion of the control volume boundary by 

definition of a stream tube, applying the conservation of mass1 to the control volume yields: 

V0 A0 =V1A1 =V2 A2 =V3A3 (2.1) 

where Vi is the wind speed at station i and Ai is the cross section area of station i. Since V1 is 

equal to V2 according to assumption (3), and since A1 equals A2 by necessity, the components of 

the wind speed and cross sectional area at the plane of the disc are designated without subscripts 

for the remainder of this analysis (i.e., V = V1 = V2 and A = A1 = A2). 

The thrust at the rotor disc, T, can be found by applying the conservation of linear momentum2

 to 

the control volume in the axial direction. This process results in: 

__________________ 

1 The conservation of mass, which asserts that the instantaneous rate of change of mass within a control volume 

must equal the net flux of mass out of the control volume, can be stated as: 

∫∫∫ + ∫∫ ⋅ = ∂

∂

S

dS 0

V

dV

t

ρu n

ρ

2

 The conservation of linear momentum, which asserts that the instantaneous rate of change of linear momentum in 

the control volume plus the net flux of linear momentum out of the control volume must equal the sum of the net 

pressure forces and the net body force exerted by the surroundings on the control volume, can be stated as: 

() ( ) ∫∫∫ ∫∫ ∫∫ ∫∫∫ + ⋅ = − +

∂

∂

V S S V

dV dS p dS dV

t

ρu ρu u n n ρf

 0 1 2 3
 8

2

3 3

2 T = ρA0V0 − ρA V (2.2) 

or equivalently, using Eq. (2.1): 

( ) T = ρAV V0 −V3 (2.3) 

where ρ is the density of the air. The thrust at the rotor disc, T, is also the differential pressure 

between stations 1 and 2 multiplied by the disc area: 

T = (p1 − p2 )A (2.4) 

where pi is the static pressure at station i. Since no work is done on either side of the turbine 

rotor, Bernoulli’s equation3 can be applied to obtain the pressures incorporated into Eq. (2.4): 

2

1

2

0 0 V

2

1 V p 2

1

p + ρ = + ρ (2.5) 

and 

2

2

2

3 3 V

2

1 V p 2

1

p + ρ = + ρ (2.6) 

where pi is again the static pressure at station i. Pressures p0 and p3 are identical by assumption 

(4), so pressures p1 and p2 can be eliminated from the thrust Eq. (2.4) with the help of Eqs. (2.5) 

and Eq. (2.6) to obtain: 

( ) 2

3

2 A V0 V

2

1 T = ρ − (2.7) 

Eliminating the thrust at the rotor disc from Eqs. (2.3) and (2.7), the velocity of the flow through 

the rotor disc is the average of the upwind (free stream) and downwind velocities: 

2

V V V 0 + 3 = (2.8) 

__________________ 

3

 Converting the conservation of linear momentum to differential form and considering steady, incompressible, and 

irrotational flows subject to conservative body forces only, one can derive Bernoulli’s equation to be: 

constan t

2

1

p + ρu ⋅ u + ρυ =

where the quantity υ is the body-force potential. For example, if gravity is considered to be the only conservative 

body force acting on the system, the body-force potential, υ, would equal the product of the gravitational 

acceleration and the elevation relative to some datum. 
 9

An axial induction (or interference) factor, a, is customarily defined as the fractional decrease in 

wind velocity between the free stream and the rotor plane: 

0

0

V

V V

a − = (2.9) 

or equivalently: 

V V (1 a) = 0 − (2.10) 

and 

V V (1 2a) 3 = 0 − (2.11) 

The velocity lost at the rotor plane, V0 – V, in Eq. (2.9) is known as the induced velocity. 

As a increases from zero, the downwind flow speed steadily decreases until, at a = 1⁄2, it has 

completely stopped and the simple theory is no longer applicable. 

Substituting for V3 from Eq. (2.11), Eq. (2.7) can be rewritten in a more useful manner: 

AV 4a( ) 1 a

2

1 T 2 = ρ 0 − (2.12) 

The power extracted from the wind by the rotor, P, is the product of the thrust, T, from Eq. (2.12) 

and the wind velocity at the rotor plane, V, from Eq. (2.10): 

( ) 3 2

0 AV 4a 1 a

2

1 P = ρ − (2.13) 

The commonly used nondimensional power coefficient, CP, representing the fraction of available 

power in the wind that is extracted by the turbine, is defined as: 

3

0

P

AV

2

1

P C

ρ

= (2.14) 

Substituting the extracted power from Eq. (2.13) into Eq. (2.14) yields: 

( )

2

P C = 4a 1− a (2.15) 

The theoretical maximum power coefficient from an idealized rotor, CPmax, known as Betz limit, 

can be found by setting the derivative of Eq. (2.15) with respect to a equal to zero, and solving 

for a: 
 10

( ) 3

1 4 1 3a 0 ... yields ...a

a

CP 2 = − = = ∂∂ (2.16) 

Substituting this result into Eq. (2.15) yields: 

0.59259

27

16 CP max = ≈ (2.17) 

The maximum possible efficiency for an idealized wind turbine is roughly 59.3%. In practice, 

three effects prohibit a real wind turbine from achieving this efficiency: 

(1) Rotation of the wake caused by the spinning rotor. 

(2) Finite numbers of blades. 

(3) Viscid flow causes nonzero aerodynamic drag. 

All these effects will be considered in the sections that follow. 

This one-dimensional model is simple and does not describe the true nature of the physical flow 

around wind turbines; however, it does bring to light several concepts that are key to 

understanding wind turbine operation. If the rotor is to extract any power from the wind, the 

wind must slow down as it passes through the rotor. An ideal wind turbine would have to slow 

the wind velocity at the rotor plane to two-thirds of the free stream value if it is to extract power 

at maximum efficiency. Thus, from continuity [Eq. (2.1)] the effective upstream area is less than 

the swept area of the rotor and the area of the wake downstream is greater than the swept area of 

the rotor. For an ideal wind turbine operating at maximum efficiency, the effective upstream 

area is two-thirds the swept area and the area of the wake downstream is twice the area swept by 

the rotor. Finally, extracting all the power available in the wind is theoretically impossible. In 

practice, a real wind turbine generally does not achieve a power coefficient of more than 45%. 

2.3 Effects of Wake Rotation on Betz Limit 

In the previous model, assumption (1) stated that the rotor imparted no angular momentum to the

wake. However, conserving angular momentum necessitates rotation of the wake if the rotor is 

to extract useful torque. Moreover, the flow behind the rotor will rotate in the opposite direction 

as the rotor in reaction to the flow imparting torque on the rotor as shown in Fig. 2.3: 

Figure 2.3: Wake rotation 
 11

Since the wake is now rotating, it exhibits rotational kinetic energy that reduces the amount of 

available energy that can be extracted as useful work. As explained in the subsequent analysis, a 

rotor generating high torque at low speed will generate less power than a rotor generating low 

torque at high speed. The typical multibladed farm windmill is a notable example of a wind 

turbine whose efficiency is severely limited by wake rotation. 

The Rankine-Froude actuator disc model can be easily extended to account for this wake 

rotation. To do this, assumptions (1) and (3) from the previous analysis can be relaxed and three 

supplementary assumptions added: 

(1) The control volume used in the previous one-dimensional model can be split into many 

noninteracting annular stream tube control volumes. 

(2) The flow entering the control volume(s) far upstream remains purely axial and uniform. 

(3) The angular speed of the wake far downstream of the rotor is low so the static pressure far 

downstream can still be assumed to be identical to the unobstructed ambient static pressure. 

Assumption (1) disregards assumption (3) from the previous analysis and allows for the local 

pressures, axial velocities (and induction factor), angular velocities, thrust, and power to all be 

functions of the annular radius. This point of view is illustrated in Fig. 2.4: 

Figure 2.4: Annular stream tube control volumes

Assumption (3) in the analysis of the Rankine-Froude actuator disc model could have been 

relaxed in the analysis if the variables Ai, T, and P were replaced with the differentials, dAi, dT, 

and dP, respectively. In this case, the differential area of an annular ring at station i, dAi, is 

defined as: 

i i i dA = 2πr dr (2.18) 

where ri and dri are the radius and differential radius, respectively, of an annular ring at station i. 

To be precise, the radius and differential radius of the annular ring of Fig. 2.4 are shown as being 

the radius and differential radius, respectively, of an annular ring at station 3, or r3 and dr3. 

Although wake rotation is now included in the analysis, the assumption that the flow is 

irrotational [assumption (2) from the previous analysis] has not been lifted. The definition of 

 0 1 2 3

r 

dr 
 12

rotational flow is any flow with nonzero vorticity. A fluid particle that moves in a circular trend 

but always retains its original orientation has zero vorticity and is considered to be irrotational. 

A classic example of this type of flow is a potential vortex. 

Since applying the conservation of mass is not affected by wake rotation, the conservation of 

mass for an annular stream tube control volume can be found by substituting dAi from Eq. (2.18) 

in for Ai in Eq. (2.1) to yield: 

0 0 0 3 3 3 V 2πr dr =V 2πrdr =V 2πr dr (2.19) 

where the components of velocity and annular area at the plane of the disc are designated without 

subscripts (V = V1 = V2, r = r1 = r2, and dr = dr1 = dr2). This designation is made throughout the 

remainder of the analysis. 

The conservation of angular momentum4

 about an axis consistent with the control volume’s axis 

of symmetry can be applied to obtain the differential torque at the rotor disc, dQ, resulting in: 

3 3 3

2

3 3 dQ = ρω r V 2πr dr (2.20) 

where ω3 is the angular velocity of the wake at station 3 defined so the differential rotor torque is 

positive. By assumption (1), the angular velocity of the flow stream at station i, ωi, is also a 

function of the annular radius. 

Since the flow entering the control volume(s) far upstream is purely axial according to 

assumption (2), applying the conservation of angular momentum between stations 0 and 1 

necessitates that there is no mechanism to impart rotational momentum to the stream ahead of 

the rotor. Thus, 

0 ω0 =ω1 = (2.21) 

Similarly, angular momentum is conserved between stations 2 and 3: 

2

3 3

2

2 ω r =ω r (2.22) 

The angular velocity of the flow stream is discontinuous across the rotor plane because torque is 

exerted onto the rotor. Also, since the downwind radius of the control volume is larger than the 

radius of the control volume at the plane of the rotor, the wake rotation slows with distance from 

the rotor. This latter outcome correlates agreeably with assumption (3). 

__________________ 

4

 The conservation of angular momentum, which asserts that the instantaneous change rate of angular momentum in 

the control volume plus the net flux of angular momentum out of the control volume must equal the sum of the net 

pressure moments and the net body moments exerted by the surroundings on the control volume, can be stated as: 

∫∫∫ ( ) ( )( ) × + ∫∫ × ⋅ = −∫∫ × +∫∫∫ × ∂

∂

V

dV

S

p dS

S

dS

V

dV

t

ρr u ρr u u n r n ρr f
 13

By combining Eqs. (2.20), (2.19), and (2.22), the differential rotor torque can be rewritten as: 

dQ r V 2 rdr 2 = ρω2 π (2.23) 

In this form, as the torque of the rotor is increased, the angular speed of the wake increases 

accordingly. 

Since applying the conservation of linear momentum in the axial direction is unaffected by wake 

rotation, the differential thrust at the rotor disc, dT, can be found by substituting dAi from Eq. 

(2.18) at station 1 (equivalently station 2) in for A in Eq. (2.12) to yield: 

2 rdrV 4a( ) 1 a

2

1 dT 2 = ρ π 0 − (2.24) 

where the induction factor, a, is now a function of the annular radius. 

The differential thrust at the rotor disc, dT, is also governed by Eq. (2.4) if dAi from Eq. (2.18) at 

station 1 (equivalently station 2) is again substituted for A. This substitution results in: 

dT ( p p )2 rdr = 1 − 2 π (2.25) 

where the local pressures at stations 1 and 2, p1 and p2 respectively, are now both functions of the 

annular radius. 

To arrive at a description of the wake rotation, it is beneficial to find the difference in pressures 

p1 and p2 not by Bernoulli’s equation, but5: 

( ) V 2 rdr

dQ V r

2

1 V p 21

p

2 2

2

2

2

2

1

π

Ω

+ ρ = + ρ +ω + (2.26) 

The product of the differential rotor torque, dQ, and the angular speed of the rotor, Ω, in Eq. 

(2.26) represents the differential power extracted by the turbine, dP. 

Substituting Eq. (2.23) for dQ in Eq. (2.26), solving for the difference in pressure, p1 – p2, and 

substituting the result into Eq. (2.25) results in an alternative relation for the differential rotor 

thrust, dT: 

__________________ 

5

 The approximate form of the energy equation is a replacement for Bernoulli’s equation when viscous, heat transfer, 

work insertion, and work extraction effects are important. The approximate form of the energy equation can be 

written as: 

L gh

m

Wt 2

2u 2 2

1

2p

m

2 Wp

1u1 2

1

1p ρ

ρ

ρα

ρ

+ ρα + = + + + 






 14

r 2 rdr

2

1 dT 2 ρ Ω ω2 ω2 π 

 



 = + (2.27) 

An angular induction factor, a’, is customarily defined as: 

Ω

ω

2

a' 2 = (2.28) 

or equivalently: 

2 a' ω2 = Ω (2.29) 

and 

2

3

3

r

r 2 a' 















ω = Ω (2.30) 

by virtue of Eq. (2.22). The 1⁄2 is used in the definition of the angular induction factor a’ since 

the angular velocity of the flow stream at the plane of the rotor (at the jump discontinuity), ω, is 

the average value of the angular velocity just upwind and downwind of the rotor: 

2

ω1 ω 2 ω+ = (2.31) 

or: 

2

ω2 ω = (2.32) 

Ω

ω

a' = (2.33) 

and 

ω = Ωa' (2.34) 

by virtue of Eqs. (2.21) and (2.28). 

The differential rotor thrust, dT, and differential rotor torque, dQ, can now be written in terms of 

the two induction factors, a and a’: 

dT r 4a'(1 a') rdr 2 2 = ρΩ + π (2.35) 
 15

and 

dQ V r 4a'(1 a) rdr 2 = ρ 0Ω − π (2.36) 

Equating the differential rotor thrust from Eq. (2.35) with Eq. (2.24), the two induction factors 

are related: 

( )

( )

2

V0

r

a' 1 a'

a 1 a















 = +

− Ω (2.37) 

The local speed ratio, λr, is defined as the square root of the right hand side of this equation: 

0

r V

Ωr λ = (2.38) 

This variable is an expanded definition of the more commonly used tip speed ratio, defined as 

the dimensionless number corresponding to the blade tip speed, ΩR, divided by the velocity of 

the free stream wind speed, V0: 

V0

ΩR λ = (2.39) 

where R is the radius of the turbine blades. 

Substituting the local speed ratio, λr, from Eq. (2.38) into Eq. (2.37), and solving for a’, the 

angular induction factor is written in terms of the axial induction factor and the local speed ratio: 

( ) 













 = + a 1− a − 1 4 1

2

1

a' 2 λr

 (2.40) 

As a increases from zero, the magnitude of a’ (and thus the magnitude of the angular speed of 

the wake accordingly) also increases. The magnitude of this increase is limited by the local 

speed ratio. The faster the blade spins relative to a given free stream wind speed, the smaller the 

angular induction factor and angular speed of the wake. 

Following a procedure similar to that used when finding Betz limit in section 2.2, the equations 

developed thus far can be used to modify the theoretical maximum power coefficient for an 

idealized rotor, CPmax, accounting for wake rotation. The results of this analysis are presented 

graphically in Fig. 2.5: 
 16

Figure 2.5: Theoretical maximum power coefficient

2.4 Blade Element Theory 

In the previous two sections, forces exerted on a wind turbine rotor by the flow stream (thrust 

and torque) were found by applying principles of mass and momentum conservation on annular 

control volumes surrounding the flow. This type of analysis is commonly referred to as 

momentum theory. However, unless some flow state is assumed, there are not enough equations 

to solve for the differential rotor thrust, dT, and differential rotor torque, dQ, at a given span 

location on the rotor (a specified r) even if the air density, ρ, free stream wind speed, V0, and 

angular speed of the rotor, Ω, are known. That is, even though the angular induction factor, a’, is 

related to the axial induction factor, a, by Eq. (2.40), the value of a must be assumed at the span 

location in question, if the differential rotor thrust, dT, and differential rotor torque, dQ, are to be 

discerned. Thus, additional equations concerning the state of flow must be developed. 

This state of flow is governed by the characteristics of the rotor blades, such as airfoil shape and 

twist distribution. The type of analysis that uses these blade geometry properties to determine 

the forces exerted on a wind turbine by the flow stream is referred to as blade element theory. 

As the word element in the theory title suggests, blade element theory again uses several annular 

stream tube control volumes. At the rotor plane, the boundaries of these control volumes 

effectively split the blade into a number of distinct elements, each of length dr. At each element, 

blade geometry and flow stream properties can be related to a differential rotor thrust, dT, and a 

differential rotor torque, dQ, if a number of assumptions are made. In blade element theory, the 

following assumptions are customarily made: 

(1) Just as the annular stream tube control volumes used in the wake rotation analysis were 

assumed to be noninteracting [assumption (1)], it is assumed that there is no interaction 

between the analysis of each blade element. 

(2) The forces exerted on the blade elements by the flow stream are determined solely by the 

two-dimensional lift and drag characteristics of the blade element airfoil shape and 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5 6 7 8 9 10

Tip Speed Ratio

Power Coefficient

No Wake Rotation (Betz Limit)

Including Wake Rotation
 17

orientation relative to the incoming flow. The term two-dimensional implies that no threedimensional flow effects are assumed to occur around each blade element. 

As discussed in Chapter 6, these assumptions are flawed and will bring about disagreements 

between modeled parameters and true physical parameters, including force coefficients, pressure 

distributions, and angle of attack distributions. 

The forces exerted on an object by fluid as it flows over the object are due to pressure and 

viscous stresses. On the upper surface of an airfoil, a two-dimensional cross-section of a blade, 

the pressure is less than that of the incoming flow stream and effectively “sucks” the airfoil 

upward, normal to the incoming flow stream. By contrast, the pressure on the lower surface of 

the airfoil is greater than that of the incoming flow stream and effectively “pushes” the airfoil 

upward, normal to the incoming flow stream. The components of the pressure distribution 

parallel to the incoming flow stream tend to slow the velocity of the incoming flow relative to 

the airfoil, as do the viscous stresses. A depiction of the pressure and viscous stress distribution 

around a typical airfoil is given in Fig. 2.6: 

Figure 2.6: Pressure and viscous stresses exerted on an airfoil

On an airfoil, the resultants of these forces are usually resolved into two forces and one moment. 

The component of the net force acting normal to the incoming flow stream is known as the lift 

force and the component of the net force acting parallel to the incoming flow stream is known as 

the drag force. The effective moment, known as the pitching moment, is usually defined about 

an axis normal to the airfoil cross-section, located a quarter of the distance from the leading edge 

to the trailing edge of the airfoil. The pressure distribution on both sides of the airfoil contributes 

to the lift (see Fig. 2.6). The part of the drag force related to the pressure distribution around the 

airfoil is known as the pressure drag. The part of the drag force related to the viscous stresses is 

known as the skin-friction drag. Their sum, the total drag force, is commonly referred to as form 

drag. The viscous stresses generally make a negligible contribution to the lift force. 

As an alternative to the lift and drag forces, analyses are commonly made with normal and 

tangential forces. The normal force is the component of the net force acting normal to the chord 

line and the tangential force is the component of the net force acting parallel to the chord line 

(the straight line that connects the leading and trailing edges of the airfoil). For a given flow 

condition, the relationships between the lift and drag forces and the normal and tangential forces 

are purely vectorial as determined by the angle of attack of the incoming flow stream. The angle 

Suction Side

Pressure Side

Surface Viscous 

Shear Stresses
 18

of attack, α, is defined as the angle between the incoming flow stream and the chord line of the 

airfoil. These characteristics are depicted in Fig. 2.7. 

Figure 2.7: Resultant forces exerted on an airfoil

The relationships between the resultant forces are: 

Normal Force = Lift Force× cos(α)+ Drag Force× sin(α) (2.41) 

and 

Tangential Force = Lift Force× sin(α)− Drag Force×cos(α) (2.42) 

or equivalently: 

Lift Force = Normal Force× cos(α)+Tangential Force× sin(α) (2.43) 

and 

Drag Force = Normal Force× sin(α)−Tangential Force×cos(α) (2.44) 

The resultant forces are normally characterized by dimensionless coefficients. The lift and drag 

coefficients, CL and CD respectively, are defined as: 

V c

2

1

Lift Force / Unit Span C

2

rel

L

ρ

= (2.45) 

and 

Drag Force

Lift Force

Normal Force

Tangential

Force Pitching Moment

 Incoming 

Flow Stream

α

α

Chord Line 
 19

V c

2

1

Drag Force / Unit Span C

2

rel

D

ρ

= (2.46) 

where Vrel is the velocity of the incoming flow stream relative to the airfoil and c is the chord 

length, the distance between the leading edge and trailing edge measured along the chord line. In 

blade element theory, the lift and drag forces per unit span employed in these equations can be 

thought of as the differential lift and drag forces acting on the blade element in question divided 

by the differential length of the element, dr. 

Similarly, the dimensionless normal and tangential force coefficients, CN and CT respectively, are 

defined as: 

V c

2

1

Normal Force / Unit Span C

2

rel

N

ρ

= (2.47) 

and 

V c

2

1

Tangential Force / Unit Span C

2

rel

T

ρ

= (2.48) 

Finally, the dimensionless pitching moment coefficient, CM, is defined as: 

2 2

rel

M

V c

2

1

Pitching Moment / Unit Span C

ρ

= (2.49) 

In the denominator of each coefficient is the common product 1⁄2ρVrel

2

, commonly referred to as 

the dynamic pressure, QNORM: 

2 Vrel 2

1 QNORM = ρ (2.50) 

These dimensionless coefficients quantitatively describe how flow past an airfoil of a given 

profile will exert forces on the airfoil. The dimensionless coefficients are heavily dependent on 

the airfoil profile and angle of attack and to a lesser, but generally substantial extent, on the size 

of the airfoil and the relative speed of the incoming flow. The dependence of the dimensionless 

coefficients to the size of airfoil and the relative speed are generally quantified using the 

nondimensional Reynolds number, Re, defined as the ratio of inertia forces to viscous forces: 
 20

μ

ρV c Re rel = (2.51) 

where μ is the viscosity of the fluid. 

For any given airfoil, the dimensionless coefficient airfoil data are typically presented in a 

graphical format as plots of the coefficients versus angle of attack. A different set of curves is 

needed for each Reynolds number considered. An example set of dimensionless coefficient 

airfoil data is given in Tables 5.2–5.7 and Figs. 5.1 and 5.2 for the S809 airfoil used on the UAE 

wind turbine, which is analyzed later. These airfoil data are obtained by testing scaled-down 

airfoil sections in a wind tunnel where flow conditions can be carefully controlled and twodimensional flow can be closely approximated. 

At low angles of attack, the dimensionless lift coefficient increases linearly with angle of attack 

and drag is reasonably small. Flow is attached to the airfoil throughout this regime. At an angle 

of attack of roughly 6° or 7°, the flow on the upper surface of the airfoil begins to separate and a 

condition known as stall begins to develop. The dimensionless lift coefficient peaks and the 

dimensionless drag coefficient increases as stall increases. Dimensionless coefficient airfoil data 

are generally unavailable for angles of attack much beyond stall. This unavailability is 

unfortunate for a wind turbine analyst since wind turbine airfoils do operate in this regime, 

especially under heavy winds and yawed conditions. In the post-stall regime, the flow becomes 

decreasingly dependent on the airfoil profile and behaves similarly to high angle of attack flow 

around flat plates. Using this notion, models have been developed that are commonly used to 

extrapolate the available airfoil data across the entire 360° range of potential angles of attack. 

The most popular Viterna-Corrigan post-stall model is developed in Viterna and Corrigan (1981) 

and summarized in Eggleston and Stoddard (1987) and Hansen and Laino (1998). A detailed 

description of the dimensionless pitching moment coefficient is beyond the scope of this work. 

The differential rotor thrust, dT, and a differential rotor torque, dQ, acting on each blade element 

as described by blade element theory can be found by analyzing the geometry of Fig. 2.8. In this 

figure, the blade is specified as propagating to the left as a result of blade rotation. The pitching 

moment is absent because it contributes nothing to the rotor thrust and torque. 

In this figure, θP is the blade collective pitch angle measured relative to the point of zero twist, θT

is the local blade twist angle, and θPT is their sum. The local blade twist angle is a function of 

the annular radius and does not change with time (assuming the blade is structurally rigid and 

does not feather). The blade collective pitch angle can change with time to adjust for different 

operating conditions (for a pitch-regulated wind turbines only), but is constant across the entire 

length of the blade (the rigid blade is connected at the root by a pitching mechanism). The angle 

θ is the angle of the relative incoming flow stream with respect to the plane of rotation and is 

equal to the sum of θPT and α. 
 21

Figure 2.8: Blade element theory geometry

The velocity of the incoming flow stream relative to the blade element, Vrel, is the vector sum of 

the axial inflow velocity at the rotor plane, V, the inflow velocity caused by the rotation of the 

blade, Ωr, and the inflow velocity caused by wake rotation at the rotor plane, ωr (r is the radius 

of the blade element whose cross section is shown in the figure). With the help of Eqs. (2.10) 

and (2.34), Vrel is found to be: 

[ ] ( ) [ ] ( ) 2 2

rel 0 V = V 1− a + Ωr 1+ a' (2.52) 

Also, using geometry, a relationship can be developed relating a, a’, and θ : 

( ) ( )

r( ) 1 a'

V 1 a

tan 0

+

− = Ω

θ (2.53) 

dQ Drag Force

Lift Force 

Normal

Force

Tangential Force

 Incoming 

θ Flow Stream

α

Chord Line 

dT

Plane of 

Rotation

Vrel

V=V0(1-a)

(Ω+ω)r=Ωr(1+a’)

θPT

θP

θT
 22

Similarly, a relationship can be developed relating Vrel, a, and θ: 

( )

sin( ) θ

V 1 a V 0

rel

− = (2.54) 

The relationships between the resultant forces are: 

dT = Lift Force× cos(θ )+ Drag Force× sin(θ ) (2.55) 

and 

dQ = r[ ] Lift Force× sin(θ )− Drag Force×cos(θ ) (2.56) 

or equivalently: 

( ) ( ) PT PT dT = Normal Force×cos θ −Tangential Force× sin θ (2.57) 

and 

[ ] ( ) ( ) PT PT dQ = r Normal Force× sin θ +Tangential Force×cos θ (2.58) 

To be precise, the lift, drag, normal, and tangential forces shown in Fig. 2.8 and used in the 

equations represent the differential components of the force imparted by the flow stream on the 

blade element whose cross section is shown. Also, all the differential forces shown in Fig. 2.8 

and used in the equations represent the differential forces acting on a single blade. If the wind 

turbine rotor consists of B identical blades, the differential rotor thrust, dT, and differential rotor 

torque, dQ, equal the following when substituting the dimensionless coefficients for the forces: 

V [ ] C cos() () C sin cdr

2

1 dT B L D

2 = ρ rel θ + θ (2.59) 

and 

V [ ] C sin() () C cos crdr

2

1 dQ B L D

2 = ρ rel θ − θ (2.60) 

or equivalently: 

V [ ] C cos() () C sin cdr

2

1 dT B N PT T PT

2 = ρ rel θ − θ (2.61) 

and 
 23

V [ ] C sin() () C cos crdr

2

1 dQ B N PT T PT

2 = ρ rel θ + θ (2.62) 

dT and dQ now represent the total differential thrust and differential torque, respectively, acting 

on an annular ring of radius r and thickness dr, just as they were represented in sections 2.2 and 

2.3. Moreover, just as the induction factors and velocities are functions of the annular radius as 

in section 2.3, the dimensionless coefficients and the chord length used in the previous equations 

are all permitted to be functions of the annular radius. 

2.5 Blade Element Momentum Theory and Performance Prediction 

Combining the results of the previous analyses enables one to model the performance of a rotor 

whose airfoil properties, size, and twist distribution are known. This type of analysis is BEM 

theory, which is based on the fact that the differential rotor thrust, dT, and differential rotor 

torque, dQ, derived from momentum theory and blade element theory must be equivalent. 

Before these equivalency relationships are applied, the momentum theory equations are 

customarily adjusted to account for the finite number of blades that are inherent in real wind 

turbine rotors (the number of blades, B, cannot be infinite). The most common adjustment is the 

straightforward tip loss (or correction) factor, F, developed by Prandtl as summarized by Wilson 

et al. (1976). Although originally developed to account for the finite number of blades, as the 

name suggests, the tip loss factor additionally accounts for the finiteness of the blades (the radius 

of the blades, R, cannot be infinite). Because a blade has a suction surface and a pressure 

surface, air tends to flow over the blade tip from the lower (pressure) surface to the upper 

(suction) surface, effectively reducing the resulting forces in the vicinity of the tip. The tip loss 

factor is defined as: 

( )

( )















 



 



 =

− − − θ

π

2r sin

B R r

1

cos e

2 F (2.63) 

where the resulting inverse cosine term is assumed to be expressed in radians. F is very near 

unity except when B is small, θ is large, or r approaches R, in which case F tends toward zero. 

Prandtl’s tip loss factor is incorporated into the momentum theory equations [Eqs. (2.24) and 

(2.36)]: 

dT F V 4a(1 a) rdr 2 = ρ 0 − π (2.64) 

and 

dQ F V r 4a'(1 a) rdr 2 = ρ 0Ω − π (2.65) 

The tip loss factor does not affect the blade element theory equations. 
 24

Although less common, a hub loss factor may also be incorporated into the momentum theory 

equations. Incorporating Prandtl’s hub loss factor is similar to incorporating the tip loss factor, 

but is beyond the scope of this work. Eggleston and Stoddard (1987) describe this model. 

Equating the differential rotor thrust from moment theory [Eq. (2.64)] and blade element theory 

[Eq. (2.59)] and incorporating the value of Vrel from Eq. (2.54) results in: 

( )

( )

( )

( )



 





 + 

 









 



  =



 









 





− θ

θ

θ π

θ

C cos

C sin 1

4F

C

2 r

Bc

cos

sin

1 a

a

L

L D

2

 (2.66) 

Equating the differential rotor torque from moment theory [Eq. (2.65)] and blade element theory 

[Eq. (2.60)] and incorporating the value of Vrel from Eq. (2.54) results in: 

( )

( ) ( ) ( )



 





 − 

 









 



  =



 





− 



 





















θ π θ θ

Ω θ

C sin

C

cos

1

4F

C

2 r

Bc

1 a

a'

cos

sin

V

r

L

L D

0

 (2.67) 

The first term on the right-hand side of each of these equations is known as the local solidity, σ: 

2 r

Bc ' π

σ = (2.68) 

This variable is an expanded definition of the more commonly used rotor solidity, defined as the 

dimensionless number corresponding to the ratio of the total blade area to the total area swept by 

the rotor (rotor plane area): 

Bc

R

σ

π = (2.69) 

Eq. (2.69) assumes that the average chord length of each blade is c . This assumption is not

employed in Eq. (2.68). 

Eliminating a’ from Eq. (2.67) using Eq. (2.53), then eliminating a from Eq. (2.66) and the 

revised Eq. (2.67) results in: 

( ) () ()

() () [ ] () () 



























+ + −

− 



 



 =

θ λ θ λ θ θ

θ λ θ θ

σ sin cos

C

C sin cos

cos sin sin '

4F C

r

L

D

r

r

L (2.70) 

The local solidity (σ’) from Eq. (2.68) and the local speed ratio (λr) from Eq. (2.38) have been 

employed when developing Eq. (2.70). 
 25

Accepted practice is to set the dimensionless drag coefficient, CD, equal to zero when analyzing 

the performance of a given wind turbine. For wind turbines that consist of aerodynamically 

effective blades operating in a condition where drag is small, this simplification introduces 

negligible errors. With CD equal to zero, Eqs. (2.66), (2.67), and (2.70) simplify to: 

( )

( ) 4F

'C

cos

sin

1 a

a L

2 σ

θ

θ  =



 









 





− (2.71) 

( )

( ) ( )



 









 



  =



 





− 



 





θ

σ

θ

θ λ

cos

1

4F

'C

1 a

a'

cos

sin ' L (2.72) 

and 

( ) ( ) ( )

() ()



 





+

− 



 



 = θ λ θ

θ λ θ θ

σ sin cos

cos sin sin '

4F C

r

r

L (2.73) 

Note that σ’ and λr have again been employed when developing Eqs. (2.71) and (2.72). 

Equation (2.73) explicitly expresses the lift coefficient (CL) as a function of the angle of attack 

(α) at a specific annular radius (r) for a wind turbine operating at a constant wind speed (V0) 

whose number of blades (B), blade radius (R), pitch (θP), twist distribution (θT), chord 

distribution (c), and rotational speed (Ω) are fixed. Equation (2.73) is BEM theory’s prediction 

of the feasible CL versus α operating points for a fixed wind turbine operating at a static 

operating condition. 

Equation (2.73) is not a function of the airfoil profile of the blades, which would be manifested 

in the dimensionless coefficient airfoil data (CL and CD or equivalently CN and CT). To see this, 

the reader should recall that α is incorporated into θ and F; r is incorporated into F, σ’, and λr; 

V0 is incorporated into λr; B is incorporated into F and σ’; R is incorporated into F; θP and θT are 

incorporated into θ and F; c is incorporated into σ’; Ω is incorporated into λr; and that no other 

variables are inherent in Eq. (2.73). Thus, the curve presented by Eq. (2.73) is valid for any wind 

turbine operating at a condition where drag is small and the parameters are fixed. 

For a wind turbine whose blade airfoil profiles are fixed along with the various parameters 

specified previously, BEM theory predicts that the actual aerodynamic operating condition (lift 

coefficient and angle of attack pairs) at each blade element will be the point where the BEM CL

versus α curve [Eq. (2.73)] intersects the CL versus α curve characterizing the aerodynamic 

properties of the given airfoil. This notion is a manifestation of assumption (2) in section 2.4. 

The intersection point can be found either graphically (easiest for hand –computations) or 

numerically (iteratively), which is easiest for computer-based solutions. In practice, the iterative 

method is usually employed with Eqs. (2.71), (2.72), and (2.53) instead of Eq. (2.73). An 

example of the graphical solution is presented in Chapter 6. The graphical solution enhances the 

visibility of the weaknesses in the model. 
 26

Once the lift coefficient and angle of attack are found for each blade element, the total 

aerodynamic forces exerted on the blades by the flow stream can be found by simple integration 

(or by a simple summation approximating the integral for the case when only a finite number of 

blade elements are used). This enables one to determine the overall power output of the wind 

turbine as well as the aerodynamic loads experienced by its various members. A derivation of 

the overall rotor power coefficient is presented next. 

The power extracted from the wind by the rotor, P, employed in the definition of the 

nondimensional power coefficient, CP, of Eq. (2.14), may be found: 

∫ P = ΩdQ (2.74) 

Exactly as stated in the paragraph following Eq. (2.26), the product of the differential rotor 

torque, dQ, and the angular speed of the rotor, Ω, in Eq. (2.74) represents the differential power 

extracted by the turbine, dP. 

Using the differential rotor torque from blade element theory [Eq. (2.60)] and incorporating the 

value of Vrel from Eq. (2.54), Eq. (2.74) can be rewritten: 

( )

( ) [ ] C sin() () C cos crdr

sin

V 1 a

2

1 P B L D

2

0 R

H

θ θ

θ

Ω ρ − 



 



 − = ∫ (2.75) 

where H is the radius of the rotor hub. Although Eq. (2.73), describing BEM theory’s prediction 

of the feasible CL versus α operating points for a fixed wind turbine operating at a static 

operating condition, is found assuming drag is small and thus negligible. This expression for the 

total power extracted by the rotor includes drag. This too is common practice. 

Substituting this expression for the total power extracted by the rotor into the equation for 

dimensionless power coefficient [Eq. (2.14)] and simplifying with help from Eq. (2.68) for the 

local solidity, Eq. (2.38) for the local speed ratio, and Eq. (2.39) for the tip speed ratio results in: 

( )

( ) ( ) ( ) ∫ 



 



 − − = R

H

L

D

2

L

2 2

r

P dr

C

C cos

sin

sin

' 1 a C

R

2 C θ θ

θ

σ λ

λ (2.76) 

Eliminating a’ from Eq. (2.67) using Eq. (2.53), then eliminating a from Eq. (2.76) and the 

revised Eq. (2.67) results in: 

( )

( )

( ) ( ) ( ) ( ) ( ) ∫ 



 



 −

























− +

= R

H

L

D

2

2

L

2

2 L

2 2

r

2

L

2

r

P dr

C

C cos sin

16F

' C

2F

'C cos

cos

sin

sin

' C

R

2 C θ θ

σ θ σ θ

λ θ

θ

σ λ

λ (2.77) 
 27

Equation (2.77) can be simplified by substituting the value of CL from Eq. (2.73) into all but the 

rightmost CL in Eq. (2.77). This action results in: 

() () () [ ][ ] () () ( )

( ) ∫ 



 



 = − + − R

H

L

D

r r

2 2

P r dr

C sin

C cos

sin cos sin sin cos 1

R

2 C

θ

θ λ θ θ λ θ θ λ θ

λ (2.78) 

2.6 Advanced Aerodynamics Topics 

BEM theory assumes that the flow field around the wind turbine rotor is in equilibrium at every 

time step considered (quasi-equilibrium). During the operation of a real wind turbine, however, 

the flow field is always changing, even during normal operation. Consequently, BEM theory 

computations may not reflect reality when examining unsteady events. The AeroDyn 

aerodynamics subroutines employed in the FAST_AD design code use the BEM theory with 

modifications to improve the results for unsteady events. This section briefly highlights 

advanced aerodynamics topics that address these conditions. 

Simple uniform wind shear is modeled using a power-law profile, in which the incoming wind 

speed, V0, becomes dependent on the elevation above the ground. Horizontal variation in 

incoming wind speed is usually modeled with a linear velocity profile. Wind-inflow data are 

usually stored in files that permit modelers to specify wind parameters that change with time 

(time-series inputs). Alternatively, several models have been developed to simulate full field 

turbulent flow. In these models, inflow data can have velocity components in three dimensions 

that vary across the entire rotor plane. These velocity components have an effect on the local 

inflow angles. 

For downwind rotor configurations, aerodynamic loads on the blades are affected by the tower 

wake, commonly called the tower shadow. The tower shadow or wake is modeled as a reduction 

in wind velocity according to a cosine law characterized by the width of the flow stream at the 

rotor plane exhibiting a velocity deficit and the velocity deficit at the center of the wake. 

When wind directions change, or when the rotor exhibits a yaw error, the axial induction is 

affected by both a cyclic variation in axial flow velocities relative to the blades and a deflected 

rotor wake. The cyclic variation in axial flow is accounted for by including the axial component 

of the blade velocity in the equations for the relative velocity of the incoming flow stream. The 

deflected wake is accounted for by multiplying the axial induction factor, a, by a so-called 

skewed-wake correction factor as described by Wilson et al. (1999). 

Pressure gradients develop across the blade span in the three-dimensional flow environment of 

rotary wing aerodynamics. Ensuing spanwise flow velocity helps maintain attached flow on the 

inboard parts of the blades, delaying stall and increasing lift. These so-called three-dimensional 

stall delay effects are accounted for by modifying the aerodynamic force coefficient-angle of 

attack relationships in the stall region for the inboard blade elements. Du and Selig (1998) 

outline the most popular stall delay model. 

For blades that are downwind of the tower, turbines that are operating in yawed conditions, or 

even turbines operating in a turbulent flow, the aerodynamic environment is rapidly changing. 

During such dynamic conditions, stall may be delayed or may take place earlier than expected 
 28

(known as dynamic stall). The most common dynamic stall model is the Beddoes-Leishman 

model, summarized by Leishman and Beddoes (1989). 

In turbulent conditions, the extended flow field around the rotor cannot react promptly enough to 

establish steady-state conditions instantaneously. Thus, the aerodynamic conditions at the rotor 

are not necessarily the conditions seen just ahead of the rotor. The response of the extended flow 

field to turbulent events is referred to as dynamic inflow. An overview of the popular Pitt and 

Peters dynamic inflow model is given in Suzuki and Hansen (1998). 
 29

3. Structural Modeling Theory 

Once wind-inflow conditions are correlated to applied loads on the wind turbine, structural 

models are needed to accurately predict and understand the complex interactions between their 

dynamically active members. Accurate structural models are thus essential to successfully 

design and analyze wind energy systems. 

The subsequent analysis develops the fundamental structural models employed in the FAST_AD 

design code for two-bladed HAWTs. The modifications needed to extend the model to threebladed HAWTS are beyond the scope of this work. Wind turbine geometry, coordinate systems, 

and degrees of freedom (DOFs) are first discussed in section 3.1. Since FAST_AD models the 

blades and tower as flexible bodies, their deflections are presented next in section 3.2. 

Expressions relating to the kinematics and kinetics of wind turbine motion are developed in 

sections 3.3 and 3.4, respectively. Finally, Kane’s equations of motion, which describe the 

force-acceleration relationships of the entire wind turbine system, are presented in section 3.5. 

Discrepancies between the models developed herein and those given by Wilson et al. (1999) and 

implemented in the FAST_AD design code and the Modes preprocessor code are noted using 

footnotes where appropriate. This chapter is devoted entirely to the structural models employed 

in the FAST_AD design code. Structural models, such as the commonly used equivalent-springhinge models of flexible blades, though simpler, are beyond the scope of this work. 

3.1 Geometry, Coordinate Systems, and Degrees of Freedom 

The FAST_AD design code models a wind turbine structurally as a combination of six rigid and 

four flexible members. The six rigid bodies are the Earth, nacelle, tower-top base plate, 

armature, hub, and gears. The four flexible bodies are the two blades, tower, and drive shaft. 

The model connects these bodies through 15 DOFs. Blade deflections account for six DOFs: 

two arise from the first flapwise, two from the second flapwise, and two from the first edgewise 

natural vibration mode of each blade. Tower deflections account for four DOFs: the first two 

natural vibration modes in each longitudinal and lateral direction. Rotor teeter, rotor speed 

variation, drive train flexibility, and nacelle yaw and tilt account for the remaining five DOFs. 

Each blade can be regarded as having a structural pretwist, but no torsional freedom is modeled. 

FAST_AD employs several reference frames for ease in conceptualizing geometry and 

developing kinematics and kinetics expressions. Most reference frames6

, corresponding to a 

coordinate system formed by a dextral set of orthogonal unit vectors (denoted by a bold 

lowercase script letter) are fixed in one of the rigid bodies (denoted by an uppercase character). 

__________________ 

6

 Wilson et al. (1999) claim that every reference frame is fixed in one of these six rigid bodies. In particular, 

reference frame d is claimed to be fixed in the rigid nacelle (N). However, this claim can only be true if the nacelle 

tilt, represented by DOF variable q5 is held constant. FAST_AD code developers should consider revising this 

statement. 
 30

These coordinate systems are listed in Table 3.17

. A number of points on the wind turbine are 

also labeled for convenience. These are listed in Table 3.2. 

Table 3.1: Coordinate System Descriptions7 

a E (Earth) Inertial coordinates

b B (Base plate) Tower-top coordinates

d Yaw coordinates

c N (Nacelle) Tilt coordinates

e L (Low speed shaft) Azimuth coordinates

f H (Hub / rotor) Teeter coordinates

g H (Hub / rotor) Delta-3 coordinates

i H (Hub / rotor) Coning coordinates

Unit Vector 

Triad Description Body Fixed In

N/A

Table 3.2: System Points 

Point Description

O B (Base plate) Tower-top

D N (Nacelle) Nacelle center of mass

P L (Low speed shaft) Teeter pin

Q H (Hub / rotor) Intersection of blade inertia axes

C H (Hub / rotor) Hub center of mass

S Center of blade element in deflected blade

S' H (Hub / rotor) Center of blade element in undeflected blade

T Center of tower element in deflected tower

Body Fixed In

Blade element

Tower element

The variable names of each DOF used by FAST_AD are listed in Table 3.38. Other commonly 

used coordinates are listed in Table 3.4 (for angles9

) and Table 3.5 (for distances). 

__________________ 

7

 This table differs from the corresponding table given in Wilson et al. (1999), not just from the discrepancy of 

reference frame d indicated previously. The bodies containing the reference frames e, f, g, and i are also different. 

The corresponding table given in Wilson et al. (1999) is in error since reference frame e cannot be fixed in the hub 

as listed if any teeter motion is taking place. Teetered rotors consist of blades rigidly attached to a hub that teeters 

about a pin located on the low speed shaft. If teeter motion is taking place, reference frame e can be considered 

fixed on the rotor side of the low speed shaft and reference frames f, g, and i can be considered fixed in the teetering 

hub (rotor). FAST_AD code developers should consider revising this table. 

8

 This table differs from the corresponding table given in Wilson et al. (1999) by the description of the DOF variable 

q15 (azimuth angle, generator side). Wilson et al. (1999 ) claim that q15 represents the angle of the high-speed shaft 

(i.e., the generator side of the gearbox) when in fact FAST_AD models q15 as the angle of the low-speed shaft near 

the entrance to the gearbox. That is, the only difference between the parameter q15 and q4 in FAST_AD is the twist 

of the low-speed shaft because of its torsional flexibility. The angle of the high-speed shaft should be roughly n

times the angle of the low-speed shaft, where n is the gearbox ratio. FAST_AD code developers should consider 

revising this table. 

9

 This table differs from the corresponding table given in Wilson et al. (1999) by the definition of the distributed 

structural pretwist angle, θS(r). The corresponding table given in Wilson et al. (1999) is in error since the distributed 
 31

Table 3.3: Degree of Freedom Variables8

q 1 Blade 1 flapwise tip displacement for natural mode 1

q 2 Blade 2 flapwise tip displacement for natural mode 1

q 3 e 2 = f 2 Teeter angle

q 4 c 1 = e 1 Azimuth angle, rotor side of drive train

q 5 d 3 = c 3 Nacelle tilt angle

q 6 b 2 = d 2 Nacelle yaw angle

q 7 a 1 Longitudinal tower top displacement for natural mode 1

q 8 a 3 Latitudinal tower top displacement for natural mode 1

q 9 a 1 Longitudinal tower top displacement for natural mode 2

q 10 a 3 Latitudinal tower top displacement for natural mode 2

q 11 Blade 1 flapwise tip displacement for natural mode 2

q 12 Blade 2 flapwise tip displacement for natural mode 2

q 13 Blade 1 edgewise tip displacement for natural mode 1

q 14 Blade 2 edgewise tip displacement for natural mode 1

q 15 c 1 = e 1

Azimuth angle, generator side of drive train, low speed shaft 

side of gearbox

Variable

Direction / 

Rotation Axis Description

localized blade 

coordinates

localized blade 

coordinates

localized blade 

coordinates

Table 3.4: Other Angle Variables9 

θ 7 a 3 Longitudinal tower-top rotation

θ 8 a 1 Latitudinal tower-top rotation

z(4 ) c 1 = e 1 Blade 1 zero azimuth offset

δ 3 f 1 = g 1 Delta-3 angle (teeter hinge orientation)

β 1 g 2 = i 2 Blade 1 coning angle

β 2 g 2 = i 2 Blade 2 coning angle

θ S (r) -i 3 Distibuted structural pretwist of each blade

θ P -i 3 Blade collective pitch

Angle Description

Rotation 

Axis

__________________ 

structural pretwist angle cannot be a positive rotation about the i3-axis if it is to rotate the local principal axes to their 

proper orientation. A positive rotation about the i3-axis would twist the blade element toward stall, not feather. In 

actuality, the inboard blade elements should be twisted toward feather for maximum aerodynamic efficiency. In 

essence, the sign convention FAST_AD uses to define the blade structural pretwist angle causes FAST_AD to 

model the blades with a mirror image of the stiffness distribution physically relevant to the wind turbine being 

modeled. When running FAST_AD, a simple fix would be to reverse the sign of all of the distributed structural 

pretwist angles in the input file (i.e., if the structural twist of an inboard element should be 15°, then the input should 

be written as -15° in the input file). Unfortunately, that is not how Wilson et al. (1999) explain it to develop the 

input file. Consequently, users of FAST_AD will undoubtedly enter in the structural pretwist parameters 

incorrectly. FAST_AD code developers should consider revising the code to correct for this significant error! 
 32

Table 3.5: Other Distance Variables 

Variable Direction Description

H S a 2 Height of the rigid base of the tower

H a 2 Length of the flexible portion of the tower

H H a 2

Elevation of the hub (hub height) relative to the Earth’s surface assuming 

negligible nacelle tilt and tower deflection

TWRHTOFFSET c 2 Distance between the hub and the tower-top base plate

u 7 a 1 Total tower-top displacement in the longitudinal direction 

u 8 a 3 Total tower-top displacement in the lateral direction 

D N c 1 = e 1 Distance between the tower-top base plate and the teeter pin

D NM1 c 1 = e 1 Distance between the tower-top base plate and the nacelle center of mass

D NM2 c 2 Distance between the origin of c and the nacelle center of mass

R U -f 1 = -g 1 Distance from the teeter pin to the blade axes intersection (undersling)

R UM -f 1 = -g 1 Distance from the teeter pin to the hub center of mass

R H i 3 Hub radius; radius of the rigid portion of the rotor

R i 3 Total radius of the rotor

Relationships between the various coordinate systems, points, DOFs, and other angles and 

distances are illustrated graphically in Fig. 3.1. FAST_AD employs the convention that 

downwind displacements are positive displacements. 

In FAST_AD, the bottom part of the tower can be modeled as rigid to a height HS; thus, the 

length of the flexible part of the tower, H, is defined as: 

H = HH −TWRHTOFFSET − HS (3.1) 

where HH is the elevation of the hub (hub height) relative to the Earth’s surface and 

TWRHTOFFSET is the vertical distance between the hub and the tower-top base plate, both 

specified while assuming that the tower deflection and nacelle tilt are negligible. 

The sums of the tip deflections for both natural modes in the longitudinal and lateral deflections 

form the total longitudinal and lateral displacements of the tower-top base plate, u7 and u8,

respectively: 

7 7 9 u = q + q (3.2) 

and 

 8 8 10 u = q + q (3.3)
 33

Figure 3.1: FAST_AD coordinate system illustrations

Tower Coordinates 

a2

θ8

Deflected 

Tower 

a1 a3

θ7

u7; Longitudinal 

 Direction

u8; Lateral Direction 

HS

b1

b2

b3

H

O 

E 

B 

Wind 

Direction

O

TWRHTOFFSET

q5

DN

P 

D

DNM1

DNM2

N

d1

c1=e1

c2

d3=c3

c3=d3

d2=b2

Wind 

Direction

Nacelle Coordinates

b1

b2=d2

b3

d1

d3=c3

O

q6

B

Base Plate Coordinates

N

 Coning Axis= 

Projection of g2=i2-Axis 

 Onto c2-c3=e2-e3-Plane 

q4+z(4)

δ3

c3=d3

c2

e3

e2=f2

Teeter Axis

Projection of g3-Axis 

Onto c2-c3=e2-e3-

Blade 1 

Blade 2 

Projection of f3-Axis 

Onto c2-c3=e2-e3-

L 

Azimuth Coordinates: 

Looking Upwind From Downwind

Teeter Coordinates 

P

Q

q3 q3

δ3

δ3

β1

β2

e3

e1=c1

e2=f2

f3

g3

g2=i2

f1=g1

RU

L 

H

Blade 1

Blade 2 

Wind 

Direction

θs(r)+θP

i1

i2=g2

S’ 

Out-Of-Plane Direction

In-Plane Direction 

Local Flapwise Direction

Local Edgewise Direction

Approx. 

Wind 

Direction

Local Blade Element Coordinates
 34

Since the generalized coordinates associated with the tip deflections are functions of time, so too 

are the total longitudinal and lateral displacements of the tower-top base plate. 

In section 3.2, dealing with deflections of the tower and blades, the assumption is employed that 

the deflections are small. With this assumption, the tower-top rotations in both the longitudinal 

and lateral directions, θ7 and θ8 respectively, can be approximated: 

( ) ( ) 













 = − +

= =

9

h H

2T

7

h H

1T

7 q dh

d h

q dh

dφ h φ θ (3.4) 

and 

( ) ( )

10

h H

2T

8

h H

1T

8 q dh

d h

q dh

d h

= =

= + φ φ θ (3.5) 

where φ1T(h) and φ2T(h) are the first and second natural mode shapes of the tower, respectively. 

In these expressions, the elevation along the flexible part of the tower, h, ranges from zero to H. 

Note that h equals zero at an elevation of HS relative to the Earth’s surface. Also, the derivatives 

of the mode shapes are evaluated at an elevation of h = H as indicated. The derivation of these 

natural mode shapes of the tower is presented in section 3.2 where the tower is assumed to 

deflect in the longitudinal and lateral directions independently; yet, the natural mode shapes in 

each direction are assumed to be identical in each direction. The negative sign is present in Eq. 

(3.4) since positive longitudinal displacements of the tower-top base plate tend to rotate the base 

plate about the negative a3-axis. The generalized coordinates associated with the tip deflections 

are functions of time; so are the longitudinal and lateral tower-top rotations of the plate. 

Attached to the tower-top base plate is a yaw bearing (O). The yaw bearing allows everything 

atop the tower to rotate (q6) as winds change direction. The yaw bearing also has the flexibility 

to allow everything atop the tower to tilt (q5) when responding to wind loads. The nacelle houses 

the generator and gearbox and supports the rotor. The center of mass of the nacelle (D) is related 

to the tower-top base plate by the position vector r

OD10: 

__________________ 

10 This is in contrast to the position vector connecting the nacelle mass center to the tower-top base plate and the 

position vector connecting the teeter pin to the tower-top base plate employed by the FAST_AD design code: 

1 2 r c c NM 1 NM 2

OD = D + D

and 

1 r cN

OP = D

as indicated by Wilson et al. (1999). These equations and the equations presented in the main body of the text are 

clearly in disagreement by the absence of the TWRHTOFFSET parameter in the former. The TWRHTOFFSET

parameter is a recent addition to FAST_AD (version 3.00) and was added to improve the calculations of the 

resultant forces and moments about the tower-top base plate. Before the addition of this parameter, the tower-top 

base plate was assumed to be at the same elevation as the hub, which physically is not the case. Unfortunately, the 

TWRHTOFFSET parameter was only incorporated into the subroutines associated with the calculations of the output 

forces and moments about the tower-top base plate. The TWRHTOFFSET parameter is not used in any of the 
 35

( ) 1 2 r D c D TWRHTOFFSET c NM 1 NM 2

OD = + + (3.6) 

Blade 1 is at an azimuth angle of q4. The zero-azimuth reference position can be located by the 

azimuth offset parameter z(4). Blade 2 is naturally 180° out of phase with blade 1. Drive train 

flexibility allows the induction generator to see an angular velocity that is different than n times 

the angular velocity of the rotor, where n is the gearbox ratio. The twist of the low-speed shaft 

is, because of its torsional flexibility, modeled with the q15 parameter. The azimuth angle q15, is 

essentially the sum of the azimuth angle, q4, and the twist of the low-speed shaft. 

If applicable to the wind turbine under consideration, teeter motion of the rotor is about a pin (P) 

fixed on the low-speed shaft. The position vector connecting the teeter pin to the tower-top base 

plate, r

OP, is10: 

1 2 r D c TWRHTOFFSETc N

OP = + (3.7) 

For an upwind turbine configuration, the distance between the tower-top base plate and the teeter 

pin in the c1 = e1 direction, DN, must be less than zero. 

The delta-3 angle, δ3, orients the axis of the unconed rotor blades so it is no longer perpendicular 

to the axis of the teeter pin. In this case, the teeter motion has both a flapping and a pitching 

component. If the delta-3 angle is zero, teeter motion is purely flapping motion. 

Each blade can be coned a different amount (β1 for blade 1 and β2 for blade 2), though the 

coning angles are constant, not changing with time. Coning begins in the very center of the hub 

at point Q, which is offset of the teeter pin (P) by a distance RU along the central axis of the hub 

(undersling length). The position vector connecting the blade axes intersection point and the 

teeter pin, r

PQ, is: 

1 r g U

PQ = −R (3.8) 

Located between point P and point Q, along the central axis of the hub, is the hub center of mass 

(C) (not shown in Fig. 3.1). The position vector connecting the hub center of mass and the teeter 

pin, r

PC, is: 

1 r g UM

PC = −R (3.9) 

Similar to the tower, the root of each rotor blade can be considered rigid to some radius RH

representing the robustness of the hub (hub radius). The length of the flexible part of each blade 

is thus R – RH, where R is the total radius of the rotor (also not shown in Fig. 3.1). The flexible 

part of each blade is assumed to deflect in the local flapwise (out-of-plane of rotor if pitch and 

__________________ 

calculations associated with the kinematics and kinetics of the equations of motion. FAST_AD code developers 

should consider revising the code to thoroughly implement the addition of this new parameter. 
 36

twist distribution equal zero) and local edgewise (in-plane of rotor if pitch and twist distribution 

equal zero) directions independently. Local means that the flapwise and edgewise directions are 

unique to each blade element as defined by the sum of the distributed structural pretwist angle, 

θS(r)

9

, and the blade collective pitch angle. Unlike the tower, the natural mode shapes in each 

direction are permitted to be different. The natural mode shapes for each blade are assumed to 

be identical. 

Because each blade can have some distributed structural pretwist, defining the deflection in two 

directions is complicated. The most viable method is to define the total blade curvature as the 

combination of the local curvature in each local blade element direction (flapwise or edgewise), 

resolved into in-plane and out-of-plane components by orienting them with the structural 

pretwist and blade collective pitch angles. This curvature can then be integrated twice to get the 

total deflection shape. Assuming that the blade deflections are small, the local curvatures in the 

flapwise and edgewise directions at a span of r, and time t, κF(r,t) and κE(r,t) respectively, for 

blade 1, are11: 

( ) ( ) ( )

2

2BF

2

2 11

1BF

2

F 1 dr

d r

q dr

d r

r,t q φ φ κ = + (3.10) 

and 

( ) ( )

2

1BE

2

E 13 dr

d r

r,t q φ κ = (3.11) 

where φ1BF(r) and φ2BF(r) are the first and second natural mode shapes, respectively, of the blades 

in the flapwise direction and φ1BE(r) is first natural mode shape of the blades in the edgewise 

direction. The dependency on these curvatures with time is inherent in the generalized 

coordinates q1, q11, and q13. In these expressions, the radius along the flexible part of the blade, 

__________________ 

11 The curvature of a curve y(x), κ(x), is: 

( )

( )

( ) 3 / 2 2

2

2

dx

dy x 1

dx

d y x

x





















 



 +

κ =

If the curve is composed only of small deflections, then: 

( ) 1

dx

dy x

<<

and the curvature simplifies to the following form: 

( ) ( )

2

2

dx

d y x

κ x =
 37

r, ranges from zero to R – RH. Note that r equals zero at a span of RH relative to the axis of the 

hub. The derivation of these natural mode shapes of the blades is presented in section 3.2. 

The curvatures in the out-of-plane and in-plane directions at a span of r, and time t, κO(r,t) and 

κI(r,t) respectively, for blade 1, are12: 

( ) ( ) ( ) [ ] ( )

( ) [ ] ( ) 2 S P

1BE

2

13

2 S P

2BF

2

2 11

1BF

2

O 1

sin r

dr

d r

q

cos r

dr

d r

q dr

d r

r,t q

θ θ φ

θ θ φ φ κ

 +



 





 + +



 



 = +

 (3.12) 

and 

( ) ( ) ( ) [ ] ( )

( ) [ ] ( ) 2 S P

1BE

2

13

2 S P

2BF

2

2 11

1BF

2

I 1

cos r

dr

d r

q

sin r

dr

d r

q dr

d r

r,t q

θ θ φ

θ θ φ φ κ

 +



 





 + +



 



 = − +

 (3.13) 

or equivalently: 

( ) ( ) ( ) ( )

2

3

2

2 13

2

2

2 11

1

2

O 1 dr

d r

q dr

d r

q dr

d r

r,t q φ φ φ κ = + + (3.14) 

and 

( ) ( ) ( ) ( )

2

3

2

2 13

2

2

2 11

1

2

I 1 dr

d r

q dr

d r

q dr

d r

r,t q ψ ψ ψ κ = + + (3.15) 

where the twisted shape functions [the φi(r)’s and ψi(r)’s for i = 1, 2, and 3] are defined as: 

( ) ( ) [ ] ( ) 2 S P

1BF

2

2

1

2

cos r

dr

d r

dr

d r θ θ φ φ  +



 



 = (3.16) 

__________________ 

12 Since the distributed structural pretwist angle, θS(r), is defined in the opposite sense as Wilson et al. (1999), the 

signs of these equations are different. This, however, is not the only significant difference between the in-plane and 

out-of-plane curvature equations presented here and the equations presented in Wilson et al. and implemented in the 

FAST_AD design code. The in-plane and out-of-plane curvatures and corresponding twisted shape functions 

developed in FAST_AD as indicated by Wilson et al. leave out the blade collective pitch angle from these 

equations! Thus, when modeling a wind turbine, FAST_AD in essence regards the stiffness orientation of the blades 

to be fixed when the blades pitch, which physically is not the case. FAST_AD code developers should consider 

revising the code to correct for this significant error! 
 38

( ) ( ) [ ] ( ) 2 S P

2BF

2

2

2

2

cos r

dr

d r

dr

d r θ θ φ φ  +



 



 = (3.17) 

( ) ( ) [ ] ( ) 2 S P

1BE

2

2

3

2

sin r

dr

d r

dr

d r θ θ φ φ  +



 



 = (3.18) 

( ) ( ) [ ] ( ) 2 S P

1BF

2

2

1

2

sin r

dr

d r

dr

d r θ θ ψ φ  +



 



 = − (3.19) 

( ) ( ) [ ] ( ) 2 S P

2BF

2

2

2

2

sin r

dr

d r

dr

d r θ θ ψ φ  +



 



 = − (3.20) 

and 

( ) ( ) [ ] ( ) 2 S P

1BE

2

2

3

2

cos r

dr

d r

dr

d r θ θ ψ φ  +



 



 = (3.21) 

The curvatures can be integrated over r to obtain the deflections of blade 1 in the out-of-plane 

(i1) and in-plane (i2) directions at a span of r, and time t, u(r,t) and v(r,t) respectively. Since the 

deflections at the root of each blade (a span RH relative to the hub axis or r = 0) must be zero, the 

deflections of blade 1 in the out-of-plane and in-plane directions become: 

() ( ) ∫ ∫ −



  = R RH

0

r

0 O u r,t κ r' ,t dr' dr (3.22) 

and 

() ( ) ∫ ∫ −



  = R RH

0

r

0 I v r,t κ r' ,t dr' dr (3.23) 

or equivalently: 

u( ) r,t q (r) q (r) q (r) = 1φ1 + 11φ2 + 13φ3 (3.24) 

and 

v( ) r,t q (r) q (r) q (r) = 1ψ 1 + 11ψ 2 + 13ψ 3 (3.25) 

where r’ is a dummy variable representing the span along the flexible part of the blade. 

As discussed in section 3.2, an axial (radial or span-wise) deflection of the blades will directly 

result from any lateral deflection if the flexible blades are assumed to remain fixed in length. By 
 39

an extension of the derivations of section 3.2, this axial deflection for blade 1 at a span of r, and 

time t, w(r,t), is: 

( ) ( ) ( ) dr' r'

v r' ,t

r'

u r' ,t

2

1

w r,t r

0

2 2

∫ 



















 





∂

∂

+ 



 





∂

∂ = (3.26) 

A positive axial deflection is directed along the negative i3-axis. 

Substituting Eqs. (3.24) and (3.25) into Eq. (3.26), results in: 

( ) ( ) 33 1 11 12 11 13 23 1 13 13

2

22 13

2

11 11

2

1 q S q S q S 2q q S 2q q S 2q q S

2

1

w r,t = + + + + + (3.27) 

where the symmetric, three by three, matrix [S] is: 

( ) ( ) ( ) ( ) dr' dr'

d r'

dr'

d r'

dr'

d r'

dr'

d r' S r

0

j i j i

ij ∫ 















 









 



  +



 









 



 = φ φ ψ ψ (3.28) 

The position vector connecting any point S on the deflected blade 1 to the blade axes intersection 

point (O), r

QS, is: 

( ) ( ) [ ( )] 1 2 3 r u r,t i v r,t i r R w r,t i H

QS = + + + − (3.29) 

The position vector connecting any point S’ on the undeflected blade 1 to the blade axis 

intersection point (O), r

QS’, is: 

( ) 3 r i H

QS' = r + R (3.30) 

For blade 2, only q2 needs to be substituted in for q1, q12 in for q11, and q14 in for q13. The twisted 

shape functions [the φi(r)’s and ψi(r)’s for i = 1, 2, and 3] are not dependent on the blade 

considered. 

Fixed quantities (not dependent on time) can be translated to any coordinate system by simple 

coordinate transformations: 

From tower top to inertial: 

( ) ( )

() () () () ()

() () () () () 







































−

−

=





















3

2

1

3

2

1

b

b

b

a

a

a

7 8 7 8 8

7 8 7 8 8

7 7

sin sin cos sin cos

sin cos cos cos sin

cos sin 0

θ θ θ θ θ

θ θ θ θ θ

θ θ

 (3.31) 

From yaw to tower top: 
 40

( ) ( )

() () 







































−

=





















3

2

1

3

2

1

d

d

d

b

b

b

6 6

6 6

sin q 0 cos q

0 1 0

cos q 0 sin q

 (3.32) 

From tilt to yaw: 

( ) ( )

() () 





































 −

=





















3

2

1

3

2

1

c

c

c

d

d

d

0 0 1

sin q cos q 0

cos q sin q 0

5 5

5 5

 (3.33) 

From azimuth to tilt: 

[ ] ( ) [ ] ( )

[ ] ( ) [ ] ( ) 







































+ +

= + − +





















3

2

1

3

2

1

e

e

e

c

c

c

0 sin q z 4 cos q z 4

0 cos q z 4 sin q z 4

1 0 0

4 4

4 4 (3.34) 

From teeter to azimuth: 

( ) ( )

() () 







































−

=





















3

2

1

3

2

1

f

f

f

e

e

e

3 3

3 3

sin q 0 cos q

0 1 0

cos q 0 sin q

 (3.35) 

From delta-3 to teeter: 

() ()

() () 







































= −





















3

2

1

3

2

1

g

g

g

f

f

f

3 3

3 3

0 sin cos

0 cos sin

1 0 0

δ δ

δ δ (3.36) 

From coning to delta-3: 

( ) ( )

() () 







































−

=





















3

2

1

3

2

1

i

i

i

g

g

g

β β

β β

sin 0 cos

0 1 0

cos 0 sin

 (3.37) 

Equation (3.37) is applicable to either blade. When representing blade 1, β equals β1 and the 

reference frame i is specific to blade 1. When representing blade 2, β equals β2 and the reference 

frame i is specific to blade 2. 
 41

3.2 Blade and Tower Deflections 

The structural model of FAST_AD considers the blades and tower to be flexible cantilevered 

beams with continuously distributed mass and stiffness. In theory, such bodies possess an 

infinite number of DOFs, since an infinite number of coordinates are needed to specify the 

position of every point on the body. In practice, such bodies are modeled as a linear sum of 

known shapes of the dominant normal vibration modes. This technique is known as the normal 

mode summation method and reduces the number of DOFs from infinity to N, the number of 

normal modes considered to be dominant. With this method, the lateral deflection 

(perpendicular to the undeformed beam) anywhere on the flexible beam at any time, u(z,t), is 

given as the summation of the products of each normal mode shape, φa(z), and their associated 

generalized coordinate, qa(t): 

( ) ( ) () ∑=

=

N

a 1

a a u z,t φ z q t (3.38) 

The normal mode shape for mode a, φa(z), is purely a function of the distance z along the beam 

(z = 0 at the fixed end and z = Z at the free end) and the generalized coordinate associated with 

normal mode a, qa(t), is purely a function of time t. Each normal mode has an associated natural 

frequency, ωa, and phase, ψa. The generalized coordinate associated with a normal mode is 

customarily allowed to be the deflection of the free end of the cantilever beam; thus, each normal 

mode shape is dimensionless and normalized so it is equal to unity at the free end. 

When each normal mode shape is known, N parameters are required to specify the deflection of 

the flexible body at any time. Thus, alternatively, the lateral deflection of the flexible body 

could be expressed using N other functions, φb(z), not unique to each normal mode: 

( ) ( ) () ∑

+ −

=

=

N p 1

b p

b b u z,t φ z c t (3.39) 

where cb(t) is the generalized coordinate associated with the function φb(z). The φb(z)’s are 

known as shape functions and the parameter p is chosen for convenience. 

Since the shape functions are not unique to each normal mode, meaning that each normal mode 

is related to all shape functions, there is a relationship such that the normal mode shapes form a 

linear combination of the shape functions: 

() () ( ) ∑

+ −

=

= =

N p 1

b p

φ a z Ca,bφb z a 1,2,...,N (3.40) 

where Ca,b is the constant proportionality coefficient associated with the bth shape function and 

the a

th normal mode. This is known as the Rayleigh-Ritz method. 

In FAST_AD, each normal mode shape is assumed to be expressible as a polynomial; thus, the 

bth shape function is defined as: 
 42

( )

b

b Z

z

z 



 





φ = (3.41) 

Since the slope of a cantilevered beam must be zero at the fixed end, p must be no smaller than 

two if the shape functions are to accurately represent the normal mode shapes. Thus, the 

FAST_AD design code requires that p equal two. FAST_AD allows N to be as high as five. In 

FAST_AD, the constant proportionality coefficients associated with each shape function and 

normal mode are parameters requested in the input file. A preprocessor code entitled Modes 

enables users of FAST_AD to obtain these parameters. The theory employed by Modes is also 

developed here. 

Using Lagrange’s equations for a conservative, scleronomic system13, the equations of motion 

for an N-DOF system are equivalent to: 

∑ ∑ () () ( ) + −

=

+ −

=

+ = = + + −

N p 1

j p

N p 1

j p

ij j ij j m c t k c t 0 i p, p 1,...,N p 1 (3.42) 

where the generalized mass and stiffness, mij and kij respectively, are defined in terms of the 

kinetic energy, T, and potential energy, V: 

∑ ∑ () () + −

=

+ −

=

=

N p 1

i p

N p 1

j p

ij i j m c t c t

2

1 T   (3.43) 

and 

∑ ∑ () () + −

=

+ −

=

=

N p 1

i p

N p 1

j p

ij i j k c t c t

2

1 V (3.44) 

Now, when the flexible beam is vibrating at a specific natural mode, say a = m, the following 

conditions result: 

__________________ 

13 Lagrange’s equations are the scalar equivalents of Newton’s law of motion: 

F = ma

Lagrange’s equations for a conservative, scleronomic system can be stated in terms of the standard form of 

Lagrange’s equations: 

0 ( ) i 1,2, ,N

q

L

q

L

dt

d

i i

...  = = ∂

∂ −















∂

∂

where the Lagrangian function, L, is defined as: 

L = T −V
 43

( ) ( )





 + = = 0 otherwise

Q sin t for a m

q t a a a

a

ω ψ (3.45) 

and 

c ( )t C q (t) (b p, p 1, ,N p 1) b = m,b m = + ... + − (3.46) 

where Qa is the amplitude of the deflection of the tip of the flexible beam associated with natural 

mode a. 

Substituting Eq. (3.45) into Eq. (3.46), then substituting the resulting equation into Eq. (3.42), 

results in (the subscript has been dropped from the specific natural mode): 

( ) m k C 0 ( ) i p, p 1, ,N p 1

N p 1

j p

ij ij j

2

∑ − + = = + + −

+ −

=

ω ... (3.47) 

Equation (3.47) can be written in matrix form: 

( [M ] [K]){C} {0} 2 −ω + = (3.48) 

where the generalized mass matrix, [M], and generalized stiffness matrix, [K], are both N × N

matrices and the coefficient vector, {C}, is an N × 1 vector. The determinant of the matrix 

premultiplying the coefficient vector results in an Nth-degree algebraic equation in ω2

, which is 

called the characteristic equation. The N roots, ωa

2

, are the eigenvalues, each being the square of 

the natural frequency associated with normal mode a. The eigenvector associated with each 

eigenvalue, {C}a, defines the constant proportionality coefficients associated with normal mode 

a (the Ca,b’s). 

In FAST_AD and Modes, the tower is modeled as an inverted cantilever beam with a point mass 

affixed to its free end. The point mass, MTop, represents the combined mass of the base plate, 

nacelle, hub, and blades. The tower is assumed to deflect in the longitudinal and lateral 

directions independently. The stiffness distributions in each direction are assumed to be 

identical; consequently, the associated natural mode shapes and frequencies are assumed to be 

identical in each direction. 

The kinetic energy of the tower has a component associated with the distributed mass of the 

beam and a component associated with the point mass: 

T = TBeam +TTop (3.49) 

The kinetic energy of a beam as developed in Thomson and Dahleh (1998) is: 
 44

∑ ∑ ( ) ( ) ( ) () () ∫ + −

=

+ −

=





 



 =

N p 1

i p

N p 1

j p

i j

H

0 Beam T i j h h h dh c t c t

2

1 T μ φ φ   (3.50) 

where μT(h) is the distributed lineal density of the beam (tower). 

The kinetic energy of the affixed point mass is: 

∑ ∑( ) ()() () () + −

=

+ −

=

=

N p 1

i p

N p 1

j p

Top Top i j i j M H H c t c t

2

1 T φ φ   (3.51) 

Since each shape function equals unity at the free end [see Eq. (3.41)], the kinetic energy of the 

affixed point mass can be simplified: 

∑ ∑( ) () () + −

=

+ −

=

=

N p 1

i p

N p 1

j p

Top Top i j M c t c t

2

1 T   (3.52) 

Equations (3.43), (3.49), (3.50), and (3.52) show that the generalized mass of the tower is: 

() () () ∫ = +

H

0

mij MTop μT h φ i h φ j h dh (3.53) 

The potential energy of the tower has a component associated with the distributed stiffness of the 

beam and a component associated with gravity: 

V =VBeam +VGravity (3.54) 

The potential energy of a beam as developed in Thomson and Dahleh (1998) is: 

( ) ( ) ( ) ∑ ∑ () () ∫ + −

=

+ −

= 













 =

N p 1

i p

N p 1

j p

i j

H

0 2

j

2

2

i

2

Beam T dh c t c t

dh

d h

dh

d h EI h

2

1 V φ φ (3.55) 

where EI T(h) is the distributed stiffness of the beam (tower). 

Gravity tends to reduce the stiffness of the tower. The potential energy caused by gravity of an 

inverted beam with a point mass affixed to its free end is the product of the gravitational force 

and the distance upon which this force acts (the negative sign promotes the notion that gravity 

reduces the stiffness): 

( ) ( )( ) 

  = − + ∫

H

0 Gravity Top T V g M v H ,t μ h v h,t dh (3.56) 

where v(h,t) is the axial deflection of the flexible cantilever beam at time t and an original 

elevation h. The axial deflection is not due to compression of the tower. Instead, the axial 
 45

deflection is a combined result of assuming the flexible beam remains fixed in length (measured 

along the beam’s central axis) and the fact that the free end of a cantilevered beam must move 

closer to the fixed end when the beam deflects laterally. That is, the axial deflection is directly 

and interdependently related to the lateral deflection. This relationship can be obtained by 

examining the geometry of the deflection(s) as depicted graphically in Fig. 3.2. 

Figure 3.2: Tower deflection geometry

Expanding the lateral and axial deflections about any elevation h using a first order Taylor series 

approximation results in: 

( ) () ( )dh

h

u h,t

u h dh,t u h,t

∂

∂

+ = + (3.57) 

and 

( ) () ( )dh

h

v h,t

v h dh,t v h,t

∂

∂

+ = + (3.58) 

The Pythagorean theorem can be applied to the geometry of the deflection of a tower element 

(see the dashed circle magnification of Fig. 3.2) to easily obtain: 

( ) ( ) ( )

h

v h,t 2

h

v h,t

h

u h,t

2 2

∂

∂ = 



 





∂

∂

+ 



 





∂

∂ (3.59) 

Equation (3.59) can be solved for the gradient of the axial deflection of the tower using the 

quadratic formula. This action results in: 

( ) ( ) 2

h

u h,t 1 1

h

v h,t





 





∂

∂ = ± − ∂

∂ (3.60) 

The lateral deflection of the tower is assumed to be small so that: 

h 

HS,, h=0

h=H u(h,t)

v(h,t)

Deflected 

Tower 

g u(h,t)

v(h,t)

u(h+dh,t)

v(h+dh,t) dh

dh
 46

( ) 1

h

u h,t

<<

∂

∂ (3.61) 

Applying Eq. (3.61) to Eq. (3.60) results in two simplified solutions to the gradient of the axial 

deflection of the tower: 

( ) ( ) ( ) 2 2

h

u h,t

2

1 and

h

u h,t

2

1 2

h

v h,t





 





∂

∂





 





∂

∂ = − ∂

∂ (3.62) 

However, if the lateral deflection of the tower is small so must the axial deflection: 

( ) 1

h

v h,t

<<

∂

∂ (3.63) 

Consequently, the only valid solution is the latter: 

( ) ( ) 2

h

u h,t

2

1

h

v h,t





 





∂

∂ = ∂

∂ (3.64) 

This expression can be integrated over h to obtain the axial deflection of the tower as a function 

of the tower’s lateral deflection. Since the slope of a tower must be zero at the fixed end (an 

elevation HS above the surface of the Earth or h = 0), the axial deflection of the tower becomes: 

( ) ( ) dh' h'

u h' ,t

2

1

v h,t

h

0

2

∫ 



 





∂

∂ = (3.65) 

where h’ is a dummy variable representing the elevation along the flexible part of the tower. 

Employing the shape functions through the use of Eq. (3.39), Eq. (3.65) can be rewritten: 

( ) ( ) ( )

dh' c () () t c t

dh'

d h'

dh'

d h'

2

1

v h,t i j

N p 1

i p

N p 1

j p

h

0

j i

∑ ∑ ∫ + −

=

+ −

= 













 = φ φ (3.66) 

Substituting Eq. (3.66) into Eq. (3.56) results in the following equation for the potential energy 

of the tower caused by gravity: 

( ) ( )

( ) ( ) ( )dh' dh c () () t c t

dh'

d h'

dh'

d h' h

dh

dh

d h

dh

d h M

2

1 V g

i j

H

0

h

0

j i

T

N p 1

i p

N p 1

j p

H

0

j i

Gravity Top



























 = − +

∫ ∫

∑ ∑ ∫ + −

=

+ −

=

φ φ

μ

φ φ

 (3.67) 

Equations (3.44), (3.54), (3.55), and (3.67) show that the generalized stiffness of the tower is: 
 47

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) 



 





















+

= −

∫ ∫ ∫

∫

H

0

j h

0

i

T

j H

0

i

Top

2

j

2

H

0 2

i

2

ij T

dh' dh

dh'

d h'

dh'

d h' dh h

dh

d h

dh

d h

g M

dh

dh

d h

dh

d h k EI h

φ φ

μ φ φ

φ φ

(3.68) 

After integrating by parts and simplifying, Eq. (3.68) is equivalent to14: 

( ) ( ) ( )

( ) ( ) ( ) dh

dh

d h

dh

d h

g M h' dh'

dh

dh

d h

dh

d h k EI h

j H

0

i H

h Top T

2

j

2

H

0 2

i

2

ij T

φ φ

μ

φ φ

∫ ∫

∫



  +

= −

 (3.69) 

In FAST_AD and Modes, each blade is modeled as a rotating cantilever beam with a point mass 

affixed to its free end. This time labeled MTip, the point mass represents the mass of the tip brake 

if applicable. In the calculation of the generalized mass and generalized stiffness, the cantilever 

beam is assumed to rotate with an angular speed, Ω, about an axis perpendicular to the axis of 

the beam, which is equivalent to assuming that there is no coning angle, β, associated with each 

blade15 and no teeter motion. The flexible part of each blade is assumed to deflect in the 

flapwise (out-of-plane of rotor if pitch and twist distribution equal zero) and edgewise (in-plane 

of rotor if pitch and twist distribution equal zero) directions independently. Unlike the tower, the 

stiffness distributions in each direction are permitted to be different. 

The kinetic energy of the blades in the reference frame rotating with the rotor is identical in form 

to the kinetic energy of the tower [reference Eq. (3.49), Eq. (3.50), and Eq. (3.52)] since the 

rotation of the blades contributes nothing to the kinetic energy in this reference frame. Thus, the 

generalized mass of each blade can be written in a form similar to Eq. (3.53): 

__________________ 

14 This is in contrast to the generalized stiffness of the tower used by FAST_AD and program Modes specified as: 

( ) ( ) ( ) [ ] ( ) () ( ) ( ) dh

dh

d h

dh

d h dh g M H h h

dh

d h

dh

d h k EI h j H

0

i

2 Top T

j

2

H

0 2

i

2

ij T

φ φ

μ φ φ

∫ ∫ = − + −

as indicated by Wilson et al. (1999). The equation for the generalized stiffness of the tower as implemented in 

Modes preprocessor code and FAST_AD is valid only if the distributed lineal density of the tower is constant with 

elevation (i.e., not dependent on the elevation h along the flexible portion of the tower). For single section tubular 

towers, this assumption is valid. However, if the distributed lineal density of the tower is not constant with 

elevation, than the mode shapes outputted by Modes and the generalized stiffness employed by FAST_AD will be in 

error. This error is more substantial when the distributed lineal density has greater variability. Code developers of 

FAST_AD and Modes should consider revising the code to correct for this error! 

15 For turbines with small coning angles (β << 1 radians), this assumption begets negligible errors. However, for 

turbines with relatively large coning angles, the errors brought about by this assumption may be significant. Code 

developers of FAST_AD and Modes should consider enhancing the applicability of the codes by incorporating a 

coning angle into the computation of the generalized mass and stiffness of the blades. 
 48

() () () ∫

−

= +

R RH

0 ij Tip B i j m M μ r φ r φ r dr (3.70) 

where μ B(r) is the distributed lineal density of the beam (blades). Equation (3.70) is valid for 

both the flapwise and edgewise directions. 

Neglecting gravity, the potential energy of each blade has a component associated with the 

distributed stiffness of the beam and a component associated with centrifugal stiffening as a 

result of rotor rotation: 

V =VBeam +VRotation (3.71) 

The potential energy component of each blade associated with the distributed stiffness of the 

beam is identical in form to that of Eq. (3.55). The potential energy caused by rotor rotation is: 

( ) ( )( ) ( ) 

  = − + + ∫

R−RH

0 Tip H B H

2

Rotation V Ω M Rv R R ,t μ r R r v r,t dr (3.72) 

The centrifugal forces tend to increase the stiffness of each blade. 

The potential energy caused by rotor rotation is essentially the product of the centrifugal forces 

and the distances upon which they act. For example, the centrifugal force acting on a blade’s tip 

brake as used in Eq. (3.72) is Ω2

MTipR, and the distance upon which this force acts is v(R-RH,t). 

This expression assumes centrifugal forces acting on a blade do not change when the blade 

deflects. To be technically correct, the centrifugal forces should be regarded as functions of the 

blade deflection since, for example, the centrifugal force acting on a blade’s tip brake would be 

slightly less than Ω2MTipR if the tip was deflected substantially since the distance to the tip from 

the center of rotation would be shorter than R by a distance v(R-RH,t). If this correlation between 

centrifugal force and deflection were accounted for, however, terms involving v(r,t)

2

 would 

appear in Eq. (3.72). This action would prevent the centrifugal forces from being derivable from 

a conservative force field and would beget considerable difficulties in the ensuing analysis. 

Therefore, FAST_AD and Modes assume that the centrifugal forces are not functions of the 

blade deflection. If the deflections of the blades are assumed to be small (the blades are 

relatively stiff), this action introduces negligible errors. 

As for the tower, the axial (radial or span-wise) deflection of the flexible cantilever beam at time 

t and an original span r, v(r,t), is a direct result of the assumption that the flexible beam remains 

fixed in length (measured along the beam’s central axis) when deflecting. Thus, assuming small 

deflections, the axial deflection of each blade is directly related to its lateral deflection [see Eqs. 

(3.65) and (3.66) for towers]: 

( ) ( ) dr' r'

u r' ,t

2

1

v r,t r

0

2

∫ 



 





∂

∂ = (3.73) 

or 
 49

( ) ( ) ( )

dr' c () () t c t

dr'

d r'

dr'

d r'

2

1

v r,t i j

N p 1

i p

N p 1

j p

r

0

j i

∑ ∑ ∫ + −

=

+ −

= 













 = φ φ (3.74) 

where r’ is a dummy variable representing the span along the flexible part of each blade. 

Substituting Eq. (3.74) into Eq. (3.72) results in Eq. 3.75 for the potential energy of each blade 

caused by rotor rotation: 

( ) ( )

( )( ) ( ) ( )dr' dr c () () t c t

dr'

d r'

dr'

d r' r R r

dr

dr

d r

dr

d r M R

2

1 V

i j

R R

0

r

0

j i

B H

N p 1

i p

N p 1

j p

R R

0

j i

Tip

2

Rotation

H

H























+





 = +

∫ ∫

∑ ∑ ∫

−

+ −

=

+ −

=

−

φ φ

μ

φ φ Ω

 (3.75) 

Consequently, the generalized stiffness of each blade is: 

( ) ( ) ( )

( ) ( )

( )( ) ( ) ( ) 





















+







+

= +

∫ ∫

∫

∫

−

−

−

H

H

H

R R

0

j r

0

i

B H

j R R

0

i

Tip

2

2

j

2

R R

0 2

i

2

ij B

dr' dr

dr'

d r'

dr'

d r' r R r

dr

dr

d r

dr

d r M R

dr

dr

d r

dr

d r k EI r

φ φ

μ

φ φ Ω

φ φ

 (3.76) 

where EI B(r) is the distributed stiffness of the beam (blades). 

After integrating by parts and simplifying, it can be shown that Eq. (3.76) is equivalent to16: 

__________________ 

16 This is in contrast to the generalized stiffness of each blade as specified by Wilson et al. (1999): 

( ) ( ) ( )

( )( ) ( ) ( ) Ω ( ) θ φ φ Ω μ

φ φ

dr m sin

dr

d r

dr

d r M R r' R r' dr'

dr

dr

d r

dr

d r k EI r

ij

j 2 R R

0

i R R

r Tip B H

2

2

j

2

R R

0 2

i

2

ij B

H H

H

+ 

  + +

= +

∫ ∫

∫ − −

−

where θ equals π/2 for edgewise motion and zero for flapwise motion. The only difference between this equation 

and the corresponding equation in the main body of the thesis is the addition of the term containing sin(θ). The term 

associated with the sin(θ) however, is not implemented in the FAST_AD nor the Modes preprocessor code, thus 

FAST_AD and Modes do compute the generalized stiffness in the flapwise and edgewise directions correctly. Code 

developers should consider revising this equation in the FAST_AD user’s manual. 
 50

( ) ( ) ( )

( )( ) ( ) ( ) dr

dr

d r

dr

d r M R r' R r' dr'

dr

dr

d r

dr

d r k EI r

j R R

0

i R R

r Tip B H

2

2

j

2

R R

0 2

i

2

ij B

H H

H

φ φ Ω μ

φ φ

∫ ∫

∫ − −

−



  + +

= +

 (3.77) 

The distributed stiffness of the flexible part of each blade is generally greater edgewise than 

flapwise. Otherwise there is essentially no difference in the computation of the generalized 

stiffness of the blades in the flapwise and edgewise directions, since centrifugal forces are 

assumed to be unaffected by blade deflection. 

3.3 Kinematics 

The geometry, coordinate systems, and DOFs of a two-bladed HAWT as modeled by FAST_AD 

and discussed in the previous two sections can be used to develop the kinematics expressions for 

the entire structure. 

Applying the addition theorem for angular velocities17 yields the following form of the angular 

velocity of the hub in the inertial reference frame, EωH: 

E H E B B N N L L H ω = ω + ω + ω + ω (3.78) 

where EωB is the angular velocity of the tower-top base plate in the inertial reference frame, BωN

is the angular velocity of the nacelle relative to the tower-top base plate, NωL

 is the angular 

velocity of the low-speed shaft relative to the nacelle, and LωH is the angular velocity of the hub 

(rotor) relative to the low-speed shaft. The angular velocity of the tower-top base plate in the 

inertial reference frame, EωB, is related to the deflection of the tower18: 

1 3

E B ω a a θ 8 θ7   = + (3.79) 

The angular velocity of the nacelle relative to the tower-top base plate, BωN, has a component 

associated with the rate of yaw and the tilt rate: 

__________________ 

17 The addition theorem for angular velocities, which asserts that the angular velocity of rigid body B in reference 

frame A can be expressed using n auxiliary reference frame A1, A2,..., An, can be stated as: 

A B A A A A A A A B ω = ω 1 + 1ω 2 +"+ n-1

ω n + nω

18 Wilson et al. (1999) define the angular velocity of the hub in the inertial reference frame: 

1 3

E B ω a a θ 8 θ7   = −

This definition is equivalent to the definition in the main body of the report, however, since Wilson et al. (1999) 

define the time derivative of the longitudinal tower-top rotation without a negative sign [see Eq. (3.4)]. To be 

consistent with sign conventions and coordinate systems, it is more proper to define the angular velocity of the hub 

in the inertial reference frame as is done in the main body of the report. 
 51

2 3

B N ω d d 6 5 = q + q (3.80) 

The time derivative of the azimuth angle relates the angular velocity of the low-speed shaft to 

that of the nacelle: 

1

N L ω e4 = q (3.81) 

Finally, the teeter rate relates the angular velocity of the hub (rotor) to that of the low-speed 

shaft: 

2

L H ω g3 = q (3.82) 

The previous five equations can be combined to give the following form of the angular velocity 

of the hub in the inertial reference frame: 

1 3 2 3 1 2

E H ω a a d d e g 8 7 6 5 4 3 q q q q   =θ +θ + + + + (3.83) 

If the velocity of the axial deflection of the tower is assumed to be negligible, the velocity of the 

tower-top base plate (O) in the inertial reference frame, Ev

O, is19: 

1 3

E O v a a 7 8 = u + u (3.84) 

The velocity of any other point T on the flexible tower in the inertial frame, Ev

T

, is: 

[ ] ( ) ( ) [ ( ) ( )] 1 3

E T v q h q h a q h q h a 7φ1T 9φ 2T 8φ1T 10φ 2T =  +  +  +  (3.85) 

where h is the elevation of point T along the flexible part of the tower and ranges from zero to H. 

Again h equals zero at an elevation of HS relative to the Earth’s surface. 

The velocity of the nacelle center of mass (D) in the inertial reference frame, Ev

D, is: 

E D E O E N OD v = v + ω × r (3.86) 

where EωN is the angular velocity of the nacelle in the inertial reference frame (EωN = 

E

ωB + 

B

ωN). Similarly, the velocity of the teeter pin (P) in the inertial reference frame, Ev

P

, is: 

E P E O E N OP v = v + ω × r (3.87) 

__________________ 

19 This is exactly the same expression for the velocity of the tower-top base plate in the inertial reference frame 

given by Wilson et al. (1999) and coded into FAST_AD. Since FAST_AD assumes that the towers are relatively 

rigid, and consequently, that deflection is small, this approximation introduces negligible errors. For relatively soft 

towers, however, this approximation may not fare so well. 
 52

The velocities of the blade axes intersection point (Q) and the hub center of mass (C) in the 

inertial reference frame, E

v

Q and E

v

C

 respectively, are: 

E Q E P E H PQ v = v + ω × r (3.88) 

and 

E C E P E H PC v = v + ω × r (3.89) 

Finally, the velocity of any point S on blade 1 in the inertial reference frame, E

v

S

, is: 

E S E Q H S E H QS v = v + v + ω × r (3.90) 

where Hv

S

 is the velocity of the point S on blade 1 relative to the rotating reference frame fixed in 

the hub (rotor) which is simply the time derivative of equation (3.29) performed while holding 

the i’s constant: 

( ) ( ) ( ) 1 2 3

H S v = u r,t i + v r,t i − w r,t i (3.91) 

Many of the previously developed equations can be combined to expand the velocity of any 

point S on blade 1 in the inertial reference frame20: 

( )( ) ( )

( )

( ) QS H S

1 3 2 3 1 2

PQ

1 3 2 3 1 2

OP

1 3 1 3 2 3

E S

a a d d d g r v

a a d d d g r

v a a a a d d r

+ + + + + × +

+ + + + + × +

= + + + + + + + × +

8 7 6 5 4 3

8 7 6 5 4 3

7 9 8 10 8 7 6 5

q q q q

q q q q

q q q q q q

     

     

       

θ θ

θ θ

θ θ

 (3.92) 

When expanded as in Eq. (3.92), all the velocities of the various points on the wind turbine 

system in the inertial reference frame can be expressed as: 

X

t

15

r 1

r

X

r v v q v E X E E +



 



 = ∑=

 (3.93) 

__________________ 

20 Wilson et al. (1999) expand the velocity of any point S on blade 1 in the inertial reference frame: 

( )( ) ( )

( )

( )

( ) ( ) 1 3 2 3 1 2 3

QS H S

1 3 2 3 1 2

PQ

1 3 2 3 1 2

OP

1 3 1 3 2 3

E S

a a d d d g i

a a d d d g r v

a a d d d g r

v a a a a d d r

q q q q r

q q q q

q q q q

q q q q q q

8 7 6 5 4 3

8 7 6 5 4 3

8 7 6 5 4 3

7 9 8 10 8 7 6 5

+ + + + + ×

+ + + + + × + +

+ + + + + × +

= + + + + + + + × +

     

     

     

       

θ θ

θ θ

θ θ

θ θ

This equation is clearly in error caused by the additional line of terms (bottom row). However, according to code 

developers of FAST_AD, Eq. (3.92) is actually implemented in the FAST_AD design code. These code developers 

should consider revising this equation in the FAST_AD user’s manual. 
 53

where E

v

X

r is the r

th partial velocity21 associated with point X, which can be a function of time 

and the generalized coordinates but not of their time derivatives, and Ev

X

t is the sum of all the 

terms not of this form. Similarly, the angular velocity of any reference frame X in the inertial 

reference frame, E

ω

X

, can be expressed as: 

X

t

X E

r

E X E ω ω + ω



 



 = ∑=

15

r 1

r q (3.94) 

where Eω

X

r is the r

th partial angular velocity associated with reference frame X, which can be a 

function of time and the generalized coordinates but not of their time derivatives, and E

ω

X

t is the 

sum of all the terms not of this form. 

As an example, the partial velocities associated with a point T in the tower are: 

( )

( )

( )

( )

( ) r 1,2, ,15

0 otherwise

h for r 10

h for r 8

h for r 9

h for r 7

2T

1T

2T

1T

= ...



















=

=

=

=

=

3

3

1

1

T

r

E

a

a

a

a

v

φ

φ

φ

φ

 (3.95) 

It is also evident that: 

= 0 T

t

E v (3.96) 

__________________ 

21 To be technically correct, the definition of partial velocities (the E

v

X

r‘s) arise when the velocity of a point X in 

inertial frame E is expressed as: 

X

t

N

r 1

r

X

r v v u v E X E E +



 



 = ∑=

where N is the number of DOFs required to specify the position of the entire system under consideration and ur is 

the r

th generalized speed. A generalized speed is essentially a composite of the velocities of the generalized 

coordinates: 

u Y q Zr ( ) r 1,2, ,N

N

s 1

r rs s  + = " 

 



 = ∑=

where the coefficients Yrs can be a function of time and the generalized coordinates but not of their time derivatives 

and the terms Zr are the sum of all of the terms not of this form. Thus, the labeling of partial velocities as in Eq. 

(3.93) is equally valid if Yrs and Zr are chosen: 

( ) r,s 1,2, ,N

0 otherwise

1 fors r

Yrs = "





 = =

and 

Z 0 (r 1,2, ,N) r = = "
 54

Accelerations can be found by taking time derivatives of the velocities. When differentiating Eq. 

(3.93) with respect to time, one arrives at the acceleration of any point X in the inertial frame, 

E

a

X

: 

( ) ( ) X

t

15

r 1

r

X

r

15

r 1

r

X

r dt

d

q dt

d

a v q v v E X E E E +



 



  +



 



 = ∑ ∑ = =

  (3.97) 

or alternatively: 

X

t

15

r 1

r

X

r a v q a E X E E +



 



 = ∑=

 (3.98) 

where, 

( ) ( ) X

t

15

r 1

r

X

r dt

d

q dt

d

a v v X E E

t

E +



 



 = ∑=

 (3.99) 

3.4 Kinetics 

The kinematics expressions for the entire wind turbine structure found in section 3.3 can be used 

to form kinetics expressions. Kane’s equations of motion (see section 3.5) use two sets of scalar 

quantities called generalized inertia forces, Fr*’s, and generalized active forces, Fr’s: 

F * ( ) m ( ) r 1,2, ,15

v

i 1

r = ∑ ⋅ − Xi = ... =

Xi E Xi

r

E v a (3.100) 

and 

F ( ) ( ) r 1,2, ,15

v

i 1

r = ∑ ⋅ = ... =

Xi Xi

r

E v F (3.101) 

where v is the number of particles with mass in the system, E

v

Xir is the rth partial velocity 

associated with particle Xi, mXi is the mass of particle Xi, E

aXi is the acceleration of particle Xi in 

the inertial frame, and FXi is the resultant of all applied forces acting on particle Xi. If the system 

is composed of a number of rigid bodies instead of particles, the generalized inertia forces can be 

simplified: 

F * ( ) () m ( ) r 1,2, ,15

w

i 1

r = ∑ ⋅ − Xi + ⋅ −  = ... =

i i Xi E Xi

r

X E X E

r

E v a ω H (3.102) 

where w is the number of rigid bodies with mass in the system, E

v

Xir is the rth partial velocity 

associated with the center of mass of rigid body Xi, mXi is the mass of rigid body Xi, EaXi is the 

acceleration of the center of mass of rigid body Xi in the inertial frame, EωXir is the r

th partial 
 55

angular velocity associated with rigid body Xi, and E

ĤXi is the time derivative of the angular 

momentum of rigid body Xi about its center of mass, in the inertial frame. For ease in 

computation, E

ĤXi can be written in terms of a body-fixed coordinate system22: 

( ) i i E Xi E Xi ' E X X H = H + ω × H (3.103) 

where (ĤXi)’ is the time derivative of the angular momentum of rigid body Xi about its center of 

mass, relative to the body-fixed coordinate system [(ĤXi)’ contains time derivatives of the 

angular velocity of the body but the moments and products of inertia are all constant], E

ωXi is the 

angular velocity associated with rigid body Xi in the inertial frame (equal to the angular velocity 

of the body-fixed coordinate system in the inertial frame), and EHXi is the angular momentum of 

rigid body Xi about its center of mass, in the inertial frame. 

For the wind turbine modeled in FAST_AD, the mass of the tower, nacelle, hub, and blades 

contribute to the total generalized inertia forces: 

F * F * F * F * F * (r 1,2, ,15) r = r T+ r N+ r H+ r B = ... (3.104) 

The generalized inertia forces associated with the tower, Fr*|T, result from the tower’s distributed 

lineal density, μT(h): 

F * () ( ) h dh r 1,2, ,15

H

0 r T = −∫ μT

E vrT ⋅E aT = ... (3.105) 

where E

v

T

r is the r

th partial velocity associated with point T in the tower and E

a

T

 is the 

acceleration of the same point in the inertial frame. 

Tower deflections, yaw rates, and tilt rates contribute to the generalized inertia forces associated 

with the nacelle, Fr*|D: 

F * ( m ) ( ) (r 1,2, ,15) N

D

r N =E vr ⋅ − E a D + Eωr

N ⋅ −EH D = ... (3.106) 

where E

v

D

r is the r

th partial velocity associated with the center of mass (point D) of the nacelle, 

mN is the mass of the nacelle, Ea

D is the acceleration of the center of mass of the nacelle in the 

inertial frame, E

ωN

r is the r

th partial angular velocity associated with the nacelle, and E

ĤD is the 

time derivative of angular momentum of the nacelle about its center of mass, in the inertial 

frame. 

__________________ 

22 FAST_AD employs dyadic notation, in which case E

ĤXi is written: 

E Xi Xi E Xi E Xi Xi E ω

Xi H = I ⋅ ω + ω × I ⋅ 

where I

Xi is the inertia dyadic of rigid body Xi about its center of mass, relative to the body-fixed coordinate system 

(i.e., I

Xi, which contains the moments and products of inertia, is constant), E

ώXi is the angular acceleration associated 

with rigid body Xi in the inertial frame, and E

ωXi is the angular velocity associated with rigid body Xi in the inertial 

frame (equal to the angular velocity of the body-fixed coordinate system in the inertial frame). 
 56

Similarly, the generalized inertia forces associated with the hub, Fr*|H, are: 

F * ( m ) ( ) (r 1,2, ,15) H

C

r H =E vr ⋅ − E aC + Eωr

H ⋅ −EH C = ... (3.107) 

where Ev

C

r is the r

th partial velocity associated with the center of mass (point C) of the hub, mH is 

the hub mass, EaC is the acceleration of the center of mass of the hub in the inertial frame, Eω

H

r is 

the r

th partial angular velocity associated with the hub, and E

ĤC is the time derivative of angular 

momentum of the hub about its center of mass, in the inertial frame. 

Finally, the distributed lineal density of the blades, μB(r), contribute to the generalized inertia 

forces associated with the blades, Fr*|B, in the same way the distributed lineal density of the 

tower contributed to its corresponding generalized inertia forces: 

( )

() ( ) r dr r 1,2, ,15

F * r dr

2

R R

0 B 2

1

R R

0 r B B 1

H

H

⋅ = ...

= − ⋅ −

∫

∫

−

−

2 2

1 1

S E S

r

E

S E S

r

E

v a

v a

μ

μ

 (3.108) 

where Ev

Si

r is the r

th partial velocity associated with point S in blade i (i = 1 and 2) and E

aSi is the 

acceleration of the same point in the inertial frame. 

The resultant of all applied forces acting on elements of the wind turbine contributes to the total 

generalized active forces that govern the equations of motion. These forces include aerodynamic 

forces (see Chapter 2); elastic forces from the tower, blade, and drive train flexibility; elastic 

forces from the springs inherent in the yaw drive (yaw and tilt motion) and teetering device; 

gravitational forces; generator forces; and damping forces: 

F F F F F F (r 1,2, ,15) Damp r = r Aero+ r Elastic + r Grav+ r Generator+ r = ... (3.109) 

An exhaustive development of all these local generalized active forces is beyond the scope of 

this work. Brief descriptions of a few follow. 

If FAero

Si is the resultant of all aerodynamic forces acting on point S in blade i (i = 1 and 2), then 

the generalized active aerodynamic forces, Fr|Aero, are: 

F dr dr ( ) r 1,2, ,15 2

R R

0 1 Aero

R R

0 r Aero Aero

H H

= ∫ ⋅ + ∫ ⋅ = ... − − S2 2 S1v 1 F v F S

r

S E

r

E (3.110) 

FAST_AD assumes aerodynamic forces act only on the rotor blades. FAST_AD neglects any 

aerodynamic forces that are imparted on the tower or the rest of the wind turbine structure, 

though FAST_AD does assume that the tower forms a wake (tower shadow) that affects the 

aerodynamics of downwind turbines (see section 2.6). 

Instead of defining the generalized active elastic restoring forces of the tower and blades with 

Eq. (3.101), FAST_AD uses the equivalent form: 
 57

( ) r 1,2, ,15

q

V F

r

r = ... ∂

∂ = (3.111) 

where the potential energy, V: 

∑∑ () () = =

=

N

i 1

N

j 1

ij i j k q t q t

2

1 V (3.112) 

where N is the number of DOFs, kij is the generalized stiffness, and the qi(t)’s are the generalized 

coordinates pertaining to the flexible body. There is a subtle difference between Eqs. (3.112) 

and (3.44). Since Eq. (3.112) contains qi(t)’s and Eq. (3.44) contains ci(t)’s, the generalized 

stiffness inherent in Eq. (3.112) is related to the integrals of the partial derivatives of the normal 

mode shapes [the φi(z)’s], whereas the generalized stiffness inherent in Eq. (3.44) is related to the 

integrals of the partial derivatives of the shape functions [the φi(z)’s]. 

Combining the elastic restoring forces and the gravitational forces of the tower into a single set 

of forces, the generalized active forces attributed to the tower deflection are found by replacing 

the φi(h)’s with φi(h)’s in the generalized stiffness of the tower [Eq. (3.69)], substituting the 

result into Eq. (3.112), and then substituting the combined result into Eq. (3.111). Since the 

normal mode shapes are orthogonal to each other, the cross-coupled terms will drop out of this 

process (the generalized stiffness terms, kij, involving i ≠ j will be zero). 

A similar process is performed when computing the generalized active elastic restoring forces of 

the blades. However, since the centrifugal stiffening effects are accounted for in the generalized 

inertia force of the blades [see Eq. (3.108)], the centrifugal stiffening terms must be dropped 

from the generalized stiffness of the blades given in Eq. (3.77). The potential energy of the 

flexing beam is the only potential energy term substituted into Eq. (3.111) when finding the 

generalized active elastic restoring forces of the blades. Since the centrifugal force terms are 

included when calculating the natural mode shapes but not in the immediate discussion of the 

generalized blade stiffness, the cross-coupled generalized stiffness terms will not drop out. 

Structural damping inherent in the tower and blades is modeled in FAST_AD using the Rayleigh 

damping technique. To characterize the magnitude of this damping, modal damping ratios are 

defined in the FAST_AD input file. 

The flexibility of the drive shaft is modeled with a linear torsional spring and a linear viscous 

damper. The flexibility of the yaw drive, which allows everything atop the tower-top base plate 

to yaw and tilt, is modeled with a linear torsional spring and a linear torsional viscous damper in 

both the yaw and tilt directions. The teetering device is modeled with a constant coulomb 

damping moment (that is, constant in magnitude, but always opposing the teeter motion), a linear 

torsional viscous damper, and a nonlinear, cubic torsional spring. A teeter stop, which prevents 

excessive teeter motion, is modeled with a linear spring. 
 58

3.5 Kane’s Equations of Motion 

FAST_AD uses Kane’s method to set up equations of motion, which are then solved by 

numerical integration (see Chapter 4). By a direct result of Newton’s laws of motion, Kane’s 

equations of motion for a simple holonomic system can be stated as (Kane and Levinson 1985): 

F F * 0 (r 1,2, ,15) r + r = = ... (3.113) 

Substituting the equations presented in the previous sections of this chapter into Eq. (3.113) 

results in a set of 15 coupled, dynamic equations that prescribe the motion of the entire wind 

turbine structure as a function of time. 
 59

4. FAST_AD Design Code Overview and Limitations 

The FAST_AD design code is a medium-complexity code used to (1) model a wind turbine 

structurally given the turbine layout and aerodynamic and mechanical properties of its members 

and (2) simulate the wind turbine’s aerodynamic and structural response by imposing complex 

virtual wind-inflow conditions. Outputs of these simulations include time-series data on the 

loads and responses of the structural members of the wind turbine. Post-processing codes can 

than be used to analyze these data, enabling researchers and designers to efficiently and safely 

design, analyze, and improve wind energy systems and lower the cost of wind-generated 

electricity. Wind energy researchers and designers rely heavily on computer modeling tools such 

as FAST_AD to accurately predict, quantify, and understand the complex physical interactions 

that characterize component loads and overall wind turbine performance. 

FAST_AD is a streamlined code developed through a partnership between NREL and Oregon 

State University. The aerodynamics theories are modeled using the AeroDyn subroutine 

package (indicated by the AD postscript), Developed by engineering faculty and staff at the 

University of Utah. The FAST_AD design code can be used to model both two- and threebladed HAWTs; though as in Chapter 3, this chapter focuses solely on the two-bladed HAWT 

configuration. The modifications needed to extend the code to three-bladed HAWTS are beyond 

the scope of this work. An overview of the code is presented in section 4.1 and some of its 

limitations are given in section 4.2. 

4.1 FAST_AD Design Code Overview 

To simulate a wind turbine’s aerodynamic and structural response to various wind-inflow 

conditions, FAST_AD uses BEM theory aerodynamic models and Kane’s structural dynamics 

models as developed in Chapters 2 and 3. Before a wind turbine simulation can be performed in 

FAST_AD, the parameters of the models must first be defined in an input file. The main input 

file of FAST_AD contains 200+ parameters that must be specified. Details regarding the layout 

of this input file are beyond the scope of this work and are well documented in Wilson et al. 

(1999). An example input file is given in Appendix I for test case S0700000 of the Blind 

Comparison study of the UAE wind turbine. A description of the UAE wind turbine and the 

Blind Comparison exercise is presented in Chapter 5, and the steps needed to develop this input 

file are given in Chapter 6. 

Many of the input file parameters pertain to the geometrical layout of the turbine and mechanical 

properties of its members, most of which are described thoroughly in Chapter 3. Another group 

of input file parameters pertains to the initial conditions of the system (the configuration of the 

wind turbine at time, t = 0). All 15 of the model’s DOFs can be switched on or off to 

systematically reduce or increase the complexity of the model. The switch for the DOF variable 

q4 is a bit broader, enabling the analyst to toggle between one of several generator models: 

constant speed, induction generator, start up, shut down, variable speed, and idling (no generator 

torque). Several pitch control methodologies—fixed pitch, power control, and speed control—

can also be modeled. 
 60

Aerodynamic properties of the rotor blades form the basis of another significant fraction of input 

file parameters. In FAST_AD, each blade element can be modeled with its own unique airfoil 

table, which describes the dimensionless coefficient airfoil data (CL and CD) as a function of 

angle of attack (α). Since wind turbine airfoils operate in both the attached and separated flow 

regimes, especially under heavy winds and yawed conditions, these airfoil tables must 

encompass the entire 360° range of potential angles of attack. The input file also enables the 

analyst to choose whether to include axial induction, wake rotation, tower shadow, dynamic stall, 

and dynamic inflow effects when simulating. The wind-inflows can be modeled as uniform, 

unvarying wind shear, or fully unsteady, inhomogeneous, and turbulent. Theories and 

descriptions of each of these simple and advanced models are presented in Chapter 2. 

During a simulation, FAST_AD numerically iterates23 to find the actual operating condition of 

each blade element at each time segment (to find the intersection point of the BEM CL versus α

curve and the CL versus α curve characterizing the aerodynamic properties of the element’s 

airfoil; or equivalently, to find the axial and tangential induction factors). The iterative method 

is employed with forms of Eqs. (2.71), (2.72), and (2.53) where the dimensionless drag 

coefficients, the CD’s, are assumed negligible and are set equal to zero. Throughout this process, 

the velocities of the incoming flow stream relative to each blade element are adjusted from the 

form given in Eq. (2.52) to account for motions associated with deflections of the tower, nacelle, 

and blades. 

Since the only other externally applied load is gravity, once the aerodynamic forces are 

determined, Kane’s equations of motion [see Eq. (3.113)] can be used to solve for the ensuing 

velocities of every constituent of the wind turbine modeled. To do this, Eq. (3.113) is written as: 

C q f () ( ) q ,q 0 r 1,2, ,15

15

s 1

∑ r s s + r r r = = ... =

 (4.1) 

and is then expanded into matrix form: 

( )

( )

( )

( )







































−

−

−

−

=

















































































15 15 15

r r r

2 2 2

1 1 1

15

s

2

1

151 15 2 1515

r s

21 2 2 215

11 12 115

f q ,q

f q ,q

f q ,q

f q ,q

q

q

q

q

C C C

C

C C C

C C C



#



#







#



#





... ... ...

# # % #

# ... ... ... #

# % # #

#

... ... ...

 (4.2) 

where the Crs’s are known coefficients of the accelerations of each DOF variable and the fr’s 

contain lower order terms. At each time step, the first step of the numerical solution to Eq. (4.2) 

is to use a fourth-order Adams-Beshforth predictor method to determine the lower order terms 

that make up the right-hand side of these equations. A Gauss elimination technique is then 

__________________ 

23 However, no iteration is necessary when employing the dynamic inflow model. 
 61

performed to solve for the accelerations of the DOF variables. Next, these accelerations are used 

to improve the estimate made by the predictor. After several iterations, a fourth-order AdamsMounton corrector is used to make a final estimate and a final determination of the accelerations, 

giving the final solution for this time step. Since this predictor-corrector method is not selfstarting, a fourth-order Runge-Kutta method is used to determine the solution for the first four 

time steps. The solution algorithm is very efficient, so the code is indeed fast. The time step or 

time increment, dt, and time duration of the simulation, tMax, are additional parameters designated 

in the input file. 

Once the applied loads and motion of the entire system are characterized for a given time step, 

loads (forces and moments) at various critical points on the wind turbine are found by 

performing simple composites (summations) of these loads. A list of the selection of output 

parameters available to the analyst is given in Wilson et al. (1999). Derivations of the equations 

used to calculate all these parameters are beyond the scope of this work. A list of desired output 

parameters, referred to as channels, is one of the final sets of parameters defined in the 

FAST_AD input file. 

4.2 FAST_AD Design Code Limitations 

The FAST_AD design code is a noteworthy tool for researchers and designers of wind energy 

systems. It is significantly more developed than codes such as YawDyn (Hansen and Laino 

1998) and WT_Perf (Buhl 2000), enabling analysts to study the complex interactions between 

structural members of a wind turbine during normal and extreme operating conditions. WT_Perf 

is essentially a wind turbine performance prediction code only. YawDyn is more advanced than 

WT_Perf, permitting an analyst to model several structural degrees of freedom, but exhibits few 

of the capabilities available in FAST_AD. FAST_AD is also significantly less sophisticated, 

easier to use, and faster to operate than ADAMS®, which is an advanced, commercial, 

multibody-dynamics code developed by Mechanical Dynamics, Inc. capable of modeling 

virtually any type of mechanical device. 

Regardless of its attributes, omitting a discussion of FAST_AD’s limitations would be careless 

when performing simulations in FAST_AD and when discussing the significance of the results 

(see Chapters 6 and 7). As presented in the wind turbine aerodynamics discussion and in 

Chapter 2, one limitation is that FAST_AD, or more specifically AeroDyn, regards the drag 

coefficient as negligible when iterating to find an aerodynamic operating point. For wind 

turbines consisting of aerodynamically effective blades operating in the attached flow regime, 

this notion is justified since drag is very small in this operating state. However, for conditions 

around stall and in the post-stall regime, essentially all airfoils begin to exhibit significant drag. 

At this point, one should be reluctant to accept all simulated results as accurate, since 

fundamental assumptions regarding the models from which those results were obtained were 

violated in the process. 

Another limitation is that blade pitching moments (or equivalently feather moments) are 

neglected; that is, FAST_AD assumes that none of the effective aerodynamic forces acting on 

the blades cause the blades to twist. This prohibits FAST_AD from modeling important 

dynamical conditions such as flutter. Flutter is a destructive concatenation of torsional and 

flapping oscillations that has been known to catastrophically twist improperly designed blades 
 62

off the pitching mechanism. For wind turbines with blades that have inherently high torsional 

stiffness or coincident inertial and aerodynamic pitching axes, flutter is generally negligible. 

With the lack of either condition, however, FAST_AD has limited ability to model this important 

dynamical event. Moreover, neglecting pitching moment calculations causes the predicted stress 

levels at the blade root to be underpredicted regardless of blade design. 

FAST_AD also neglects the computation of any radial aerodynamic loads, which may contribute 

to blade root stresses on real wind turbines. If the radial loads or blade deflections are 

substantial, the absence of radial load considerations may also bring about underpredicted root 

bending moments. 

One of FAST_AD’s minor limitations is that it assumes the natural mode shapes of the tower in 

the longitudinal and lateral directions are identical. This limitation is deemed minor since most 

wind turbines use axisymmetric towers. 

With similar merit, the absence of a coning angle in the generalized mass and generalized 

stiffness equations is another minor limitation. This should be a concern only if the blade coning 

angles or teeter motion are significant, which is not the case for most utility-scale wind turbines. 

Another limitation is that FAST_AD regards most deflections to be small. For example, the 

assumption that tower deflections are small is employed in the development of Eqs. (3.4), (3.5), 

(3.61), and (3.63). For relatively stiff towers, this is entirely valid. For relatively soft towers, 

however, this assumption begets errors that may be substantial. Similarly, the assumption that 

rotor blade deflections are small is employed in the development of the blade deflection 

equations [see Eqs. (3.10) and (3.11)]. An equivalent limitation is inherent in the blade coning 

angles, since the determination of the blade mode shapes and centrifugal stiffening effects are 

heavily dependent on the assumption that the blade coning angles are small (see footnote 15). 

Most utility-scale wind turbines are relatively rigid; thus, these limitations are reasonably minor. 

As turbines become larger and inherently more flexible, however, these limitations become 

increasingly important and the models will have to be expanded to incorporate the effects of 

increasing flexibility. 

Errors in the structural models employed by FAST_AD impose other limitations. These errors 

are covered in detail in Chapter 3 (see footnotes 9, 10, 12, and 14) and are briefly discussed here. 

The most significant error is the incorrect sign convention of the distributed structural pretwist 

angle, which defines the blades as having the mirror image of the stiffness distribution physically 

relevant to the modeled wind turbine (see footnote 9). For blades with substantial stiffness 

orientation variability, the consequences of this error are difficult to characterize. 

With similar merit, the absence of the blade collective pitch angle in the equations defining the 

stiffness orientation of the blades gives rise to important dynamical consequences that are 

difficult to characterize (see footnote 12). Fixed pitch (stall-controlled) turbines and pitchcontrolled turbines that maintain small pitching angles will most likely have little sensitivity to 

this error. However, in cases where the pitching angles are large or change rapidly to adjust for 

wind gusts, etc., the dynamical outcomes are expected to be more significant. 
 63

A less substantial error is the absence of the TWRHTOFFSET parameter in the kinematics and 

kinetics terms related to the equations of motion of the wind turbine system (see footnote 10). 

This error is less substantial since the tilting motion of typical nacelles is small; consequently, 

the dynamic effects coupled to this parameter when tilting (increased blade motions relative to 

the tower-top base plate) are also small. 

The error in the equation for the generalized stiffness of the tower (see footnote 14) employed in 

FAST_AD and the Modes preprocessor code is also less substantial. The assumption that the 

lineal density of the tower is constant with elevation is unsubstantial, since the resulting mode 

shapes have a negligible effect on the dynamics of the rest of the wind turbine system and only a 

minor effect on the longitudinal and lateral moments in the tower base. 

FAST_AD code developers should consider revising the code to correct for all these errors, 

since errors limit the validity of the solutions in many situations! An exhaustive study should 

also be performed on the other models and equations employed by FAST_AD and all its 

preprocessor and postprocessor codes (both structural and aerodynamic) not described in this 

report. Time constraints prevented us from studying and developing every model and equation 

employed by FAST_AD and all its preprocessor and postprocessor codes, though there may well 

be other errors. For example, both the Modes and FAST_AD programs output natural 

frequencies of the blades and tower, which are estimated from the distributed structural 

parameters input into the model. After running a few simulations, we noticed that the output 

frequencies from FAST_AD do not match those output by Modes when the same structural 

parameters are input. We concluded that there must be an error in the computation of these 

natural frequencies. A case for other errors, especially in determining a few output channels, is 

given in Chapter 7. 

These errors are separate from any bugs in the coding. We did not perform an exhaustive 

overview of the coding. 

Above all, the simulated results obtained from FAST_AD are only as good as the models from 

which they were derived. This has more to do with our limited understanding of the physics 

involved in wind turbine dynamics than our limited attempt to model all physical realizable 

scenarios. As discussed in Chapters 6 and 7, our limited understanding of the physics of threedimensional flow in a rotating environment overwhelms most of the limitations in FAST_AD. 
 64

5. UAE and Blind Comparison Overview 

In recent years, the wind turbine research community has made significant advances in 

simulating wind turbine operation and predicting performance and component loads. Design 

codes such as FAST_AD have been successfully used to improve wind energy systems and 

lower the cost of wind-generated electricity on a noteworthy scale. Though detailed, the models 

employed in these codes, such as those developed in the previous chapters, are surprisingly 

deficient. As discussed in Chapter 2, these deficiencies include the inability to accurately model 

the effects of inhomogeneous, unsteady, and turbulent inflows and operating states such as 

severe yaw error, dynamic stall, and blade/tower shadow interaction. Perhaps more 

disconcerting is that the models also tend to miss the mark when predicting loads and responses, 

even under relatively homogeneous and steady wind-inflow conditions. This is unfortunate, 

since the entire spectrum of physically realizable operational states must be considered when 

estimating extreme loads needed in the successful design of long-lasting turbines. Moreover, if a 

design is to be truly optimized, load predictions from all operating states must be accurate. 

Currently, wind turbine designers rely heavily on time-tested safety factors to account for these 

deficiencies. Such a design methodology greatly limits the cost effectiveness of new wind 

turbine designs. 

To address these issues, NREL planned and conducted the comprehensive test program known as 

the UAE. Its objectives are daunting: “to provide accurate and reliable experimental 

measurements, having highly spatial and temporal resolution, for a realistic rotating blade 

geometry, under closely matched conditions of dynamic similarity, and in the presence of strictly 

controlled inflow conditions” (Simms et al. 2001). These objectives were met by testing a 

heavily instrumented, 10-m diameter, 20-kW wind turbine, known as the UAE research wind 

turbine, in the world’s largest wind tunnel located at the NASA-Ames Research Center. These 

tests were completed in May 2000, following three weeks of exhaustive testing. Information 

obtained from these tests includes data regarding the full-scale, three-dimensional steady and 

unsteady behavior of HAWT aerodynamics and structural dynamics including inflow conditions, 

wake form, airfoil aerodynamic pressure distributions, and machine responses and loads; 

information that is invaluable for improving and validating enhanced engineering design codes 

such as FAST_AD for HAWTs. 

To test the validity and range of applicability of models used in design codes, NREL solicited 

wind turbine modeling experts from around the world to predict loads and performance of the 

UAE wind turbine at selected operating conditions. These predictions were done “blind” in that 

the only information given to the participants regarded basic machine parameters of the UAE 

wind turbine and essential wind-inflow data, not measured wind tunnel experimental data. 

During a later comparison phase, the predictions from all the Blind Comparison modelers were 

presented and compared with the experimentally measured data. 

A summary of the UAE wind turbine configuration and technical specifications is presented in 

section 5.1. An overview of the wind tunnel tests and the quality and reliability of the resulting 

data are given in section 5.2. A description of the Blind Comparison exercise is presented in 

section 5.3. 
 65

5.1 UAE Wind Turbine Specifications 

Since the UAE wind turbine was designed specifically for research and testing, extensive 

documentation on the machine’s configuration and its technical specifications is readily available 

(see Simms et al. 1999; Hand 2001; and Hand et al. 2001). An exhaustive accounting of the 

machine’s parameters will not be presented here. This section will summarize some of the wind 

turbine’s key characteristics that are needed to develop the input files, specifically for FAST_AD 

and its configuration during the Blind Comparison operating conditions. The approximations 

and assumptions made and the steps taken to transform these data into specific input file entries 

are given in Chapter 6. A discussion of the tools and techniques used to obtain the data is 

beyond the scope of this work. 

The basic machine characteristics include: 

 Number of blades: 2 

 Rotor configuration: upwind or downwind (to transition from a downwind to upwind 

configuration, the nacelle is rotated 180o and both blades are detached from the hub, rotated 

180o, and reattached to keep the suction sides of the blades downwind) 

 Power regulation: stall controlled (fixed pitch) 

 Rotor type: teetered or rigid (to transition from a teetered rotor to a rigid hub configuration, 

the teeter dampers are removed and replaced with rigid links); the test cases considered for 

the Blind Comparison exercise employed only the rigid rotor type 

 Rotor rotational direction: counterclockwise when looking downwind (for both rotor 

configurations) 

 Rated turbine power: 19.8 kW 

 Cut-in wind speed 6 m/s 

 Hub height above ground level: 12.192 m 

 Blade tip radius measured from hub centerline: 5.029 m 

 Distance to blade root mounted strain gages measured from hub centerline: 0.432 m 

 Distance from yaw axis to blade axis (yaw moment arm): 1.401 m 

 Distance from yaw axis to teeter pin (load path): 1.469 m 

 Synchronous rotational speed of low-speed shaft: 71.63 rpm 

 Nacelle tilt angle: 0o 

 Blade precone angles: locked at one of three positions: 0o, 3.4o, or 18o 

 Blade collective pitch angles: locked at one of five positions: 0o, 2o, 3o, 4o, or 6o measured 

relative to the blade tip; the test cases considered for the Blind Comparison exercise all 

employed the 3o blade collective tip pitch position 

Each rotor blade of the UAE wind turbine is tapered and twisted (see Table 5.1). All local twist 

angles are relative to the point of zero twist at the 3.772 m station (75% span) and convention is 

positive toward feather (same as pitch convention). Between the 0.508 m and 0.883 m stations, 

the blade root is cylindrical. This transitions to an airfoil shape between the 0.883 m and 1.257 

m stations. 
 66

Table 5.1: Blade Chord and Twist Distributions 

Distance from Spanwise Local Chord Local Local

Rotor Center Fraction Length (c ) Twist (θ T ) Thickness

m (/ 5.029 m) m deg m

0.000 0.000 hub hub hub

0.508 0.101 0.218 0.000 0.218

0.660 0.131 0.218 0.000 0.218

0.883 0.176 0.183 0.000 0.183

1.008 0.200 0.349 6.700 0.163

1.067 0.212 0.441 9.900 0.154

1.133 0.225 0.544 13.400 0.154

1.257 0.250 0.737 20.040 0.154

1.343 0.267 0.728 18.074 0.153

1.510 0.300 0.711 14.292 0.149

1.648 0.328 0.697 11.909 0.146

1.952 0.388 0.666 7.979 0.140

2.257 0.449 0.636 5.308 0.133

2.343 0.466 0.627 4.715 0.131

2.562 0.509 0.605 3.425 0.127

2.867 0.570 0.574 2.083 0.120

3.172 0.631 0.543 1.150 0.114

3.185 0.633 0.542 1.115 0.114

3.476 0.691 0.512 0.494 0.107

3.781 0.752 0.482 -0.015 0.101

4.023 0.800 0.457 -0.381 0.096

4.086 0.812 0.451 -0.475 0.094

4.391 0.873 0.420 -0.920 0.088

4.696 0.934 0.389 -1.352 0.081

4.780 0.950 0.381 -1.469 0.080

5.000 0.994 0.358 -1.775 0.075

5.029 1.000 0.355 -1.815 0.074

Except at the root region, the entire blade exhibits an NREL S809 airfoil shape (blade profile). 

Airfoils of this shape have been extensively tested in wind tunnels to obtain two-dimensional 

aerodynamic coefficients. The dimensionless coefficient airfoil data resulting from tests taken at 

Colorado State University (CSU), Ohio State University (OSU), and the Delft University of 

Technology (DUT) are given in Tables 5.2–5.7 and are illustrated in Figs. 5.1 and 5.2. 
 67

Table 5.2: NREL S809 Airfoil Data from CSU; Re = 300,000 

Angle of Dim.less Dim.less

Attack (α ) Lift Coeff. Drag Coeff.

deg (CL ) (C D )

0.00 0.105 0.0117

1.99 0.307 0.0116

4.08 0.545 0.0139

6.11 0.748 0.0135

8.14 0.880 0.0198

10.20 0.878 0.0360

11.20 0.870 0.0446

12.20 0.854 0.0496

13.10 0.877 0.0619

14.10 0.894 0.0731

15.20 0.891 0.0865

16.30 0.745 0.2200

17.20 0.591 0.2480

18.10 0.592 0.2650

19.20 0.580 0.2790

20.20 0.604 0.2980

22.10 0.588 0.3230

26.20 0.669 0.4120

30.20 0.946 0.6330

35.20 1.020 0.7990

40.30 1.080 0.9830

45.20 1.230 1.3100

45.10 1.130 1.2100

50.00 1.210 1.5100

60.00 1.050 1.8600

69.90 0.805 2.2700

80.00 0.456 2.1600

90.00 0.128 2.2400
 68

Table 5.3: NREL S809 Airfoil Data from CSU; Re = 500,000 

Angle of Dim.less Dim.less

Attack (α ) Lift Coeff. Drag Coeff.

deg (CL ) (C D )

-2.230 -0.060 0.006

-0.161 0.156 0.004

1.840 0.369 0.006

3.880 0.571 0.008

5.890 0.755 0.009

7.890 0.860 0.017

8.950 0.887 0.024

9.910 0.869 0.035

10.900 0.868 0.039

12.000 0.894 0.048

12.900 0.938 0.061

14.000 0.929 0.074

14.900 0.908 0.080

16.000 0.912 0.106

17.000 0.655 0.271

18.000 0.588 0.265

19.000 0.587 0.281

20.000 0.597 0.299

22.000 0.603 0.326

24.000 0.647 0.375

26.000 0.683 0.419

28.100 0.745 0.482

30.000 0.824 0.560

35.000 1.050 0.817

40.000 1.140 1.030

45.000 1.200 1.260

50.000 1.120 1.380

55.000 1.170 1.700

60.000 1.080 1.870

65.000 0.940 1.980

70.000 0.857 2.190

74.900 0.666 2.170

79.900 0.472 2.210

84.800 0.356 2.320

89.900 0.142 2.090
 69

Table 5.4: NREL S809 Airfoil Data from CSU; Re = 650,000 

Angle of Dim.less Dim.less

Attack (α ) Lift Coeff. Drag Coeff.

deg (CL ) (C D )

-0.25 0.151 0.002

1.75 0.354 0.001

3.81 0.561 0.002

5.92 0.765 0.006

7.94 0.860 0.015

9.98 0.848 0.031

11.00 0.892 0.043

12.00 0.888 0.049

13.00 0.927 0.043

14.00 0.910 0.075

15.00 0.910 0.075

16.00 0.928 0.107

17.00 0.686 0.278

18.00 0.639 0.276

19.00 0.576 0.273

20.00 0.552 0.275

22.00 0.596 0.323

23.90 0.649 0.370

26.00 0.680 0.417

30.00 0.851 0.576

35.00 1.010 0.789

40.00 1.120 1.030

45.00 1.120 1.190

50.00 1.100 1.360

55.30 1.080 1.580

60.20 0.931 1.620

65.20 0.968 2.000

70.20 0.776 2.040

75.20 0.630 2.130

80.20 0.485 2.320

85.10 0.289 2.140

90.20 0.109 2.270
 70

Table 5.5: NREL S809 Airfoil Data from OSU; Re = 750,000 

Angle of Dim.less Dim.less Reynolds

Attack (α ) Lift Coeff. Drag Coeff. Number (Re )

deg (CL ) (C D ) × 10 6

-20.1 -0.56 0.3027 0.76

-18.1 -0.67 0.3069 0.75

-16.1 -0.79 0.1928 0.75

-14.2 -0.84 0.0898 0.75

-12.2 -0.70 0.0553 0.75

-10.1 -0.63 0.0390 0.73

-8.2 -0.56 0.0233 0.74

-6.1 -0.64 0.0112 0.74

-4.1 -0.42 -0.0004 0.76

-2.1 -0.21 -0.0003 0.75

0.1 0.05 0.0029 0.75

2.0 0.30 0.0056 0.74

4.1 0.54 0.0067 0.75

6.2 0.79 0.0085 0.74

8.1 0.90 0.0127 0.75

10.2 0.93 0.0274 0.75

11.3 0.92 0.0303 0.74

12.1 0.95 0.0369 0.74

13.2 0.99 0.0509 0.74

14.2 1.01 0.0648 0.74

15.3 1.02 0.0776 0.74

16.3 1.00 0.0917 0.74

17.1 0.94 0.0994 0.73

18.1 0.85 0.2306 0.77

19.1 0.70 0.3142 0.76

20.1 0.66 0.3186 0.76

22.0 0.70 0.3694 0.76

24.1 0.79 0.4457 0.77

26.2 0.88 0.5260 0.76
 71

Table 5.6: NREL S809 Airfoil Data from OSU; Re = 1,000,000 

Angle of Dim.less Dim.less Reynolds

Attack (α ) Lift Coeff. Drag Coeff. Number (Re )

deg (CL ) (C D ) × 10 6

-20.1 -0.55 0.2983 1.01

-18.2 -0.65 0.2955 1.02

-16.2 -0.80 0.1826 1.01

-14.1 -0.79 0.0793 0.99

-12.1 -0.70 0.0547 1.01

-10.2 -0.63 0.0401 1.00

-8.2 -0.58 0.0266 1.00

-6.2 -0.61 0.0183 1.00

-4.1 -0.40 0.0004 0.99

-2.1 -0.16 0.0009 1.00

0.0 0.07 0.0022 1.01

2.1 0.30 0.0037 1.00

4.1 0.55 0.0050 1.00

6.1 0.79 0.0063 1.00

8.2 0.90 0.0096 1.00

10.1 0.94 0.0231 1.00

11.2 0.93 0.0236 1.00

12.2 0.97 0.0368 1.00

13.3 1.00 0.0551 0.99

14.2 1.02 0.0618 0.99

15.2 1.03 0.0705 0.99

16.2 1.01 0.0880 1.00

17.2 0.95 0.1043 0.99

18.1 0.90 0.1325 1.00

19.2 0.78 0.3474 1.02

20.0 0.67 0.3211 1.02

22.1 0.70 0.3699 1.02

24.0 0.77 0.4348 1.03

26.1 0.91 0.5356 1.04
 72

Table 5.7: NREL S809 Airfoil Data from DUT; Re = 1,000,000 

Angle of Dim.less Dim.less

Attack (α ) Lift Coeff. Drag Coeff.

deg (CL ) (C D )

-1.04 0.019 0.0095

-0.01 0.139 0.0094

1.02 0.258 0.0096

2.05 0.378 0.0099

3.07 0.497 0.0100

4.10 0.617 0.0100

5.13 0.736 0.0097

6.16 0.851 0.0095

7.18 0.913 0.0127

8.20 0.952 0.0169

9.21 0.973 0.0247

10.20 0.952 0.0375

11.21 0.947 0.0725

12.23 1.007 0.0636

13.22 1.031 0.0703

14.23 1.055 0.0828

15.23 1.062 0.1081

16.22 1.043 0.1425

17.21 0.969 0.1853

18.19 0.938 0.1853

19.18 0.929 0.1853

20.16 0.923 0.1853

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

-25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Angle of Attack [deg]

Lift Force Coefficient

CSU, Re=300,000

CSU, Re=500,000

CSU, Re=650,000

OSU, Re=750,000

OSU, Re=1,000,000

DUT, Re=1,000,000

Figure 5.1: NREL S809 dimensionless lift force coefficient airfoil data
 73

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

-25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Angle of Attack [deg]

Drag Force Coefficient

CSU, Re=300,000

CSU, Re=500,000

CSU, Re=650,000

OSU, Re=750,000

OSU, Re=1,000,000

DUT, Re=1,000,000

Figure 5.2: NREL S809 dimensionless drag force coefficient airfoil data

Structurally speaking, the rotor blades are manufactured with a fiberglass composite skin and a 

load-carrying unidirectional carbon fiber and ±45o S-glass fiber hybrid composite D-spar. 

Unfortunately, structural properties of the blades used on the UAE wind turbine are unavailable. 

The blade manufacturer, Composite Engineering, estimated mass and stiffness distributions for a 

similar blade: a constant 0.457 m chord, highly twisted blade, used during UAE Phase V24

testing. Data from Composite Engineering are listed in Table 5.8. These estimates were made 

using a computer model and include the effects of pressure measurement instrumentation and 

counterweights, as well as a root-mounted camera attachment. 

__________________ 

24 The UAE tests performed at the NASA-Ames Research Center in Moffett Field, California, encompass essentially 

Phase VI of a wind turbine analysis venture that has been going on since the early 1990s. Throughout the many 

phases of the project, several turbine configurations have been employed. 
 74

Table 5.8: Structural Properties of the UAE Phase V Blades 

Distance Blade Local Local Local

from Rotor Spanwise Lineal Flapwise Edgewise Aerodynamic

Center Fraction Density Stiffness Stiffness Twist (θ T )

m (/ 5.029 m) (μ B ) kg/m (EI ) Nm 2 (EI ) Nm 2 deg

0.402 0.080 473,517 473,517 0.000

0.508 0.101 30.14 2,320,700 2,322,100 0.000

0.749 0.149 45.18 1,302,400 1,556,800 43.824

1.006 0.200 46.09 710,230 1,332,800 35.840

1.257 0.250 31.53 471,680 1,165,553 30.054

1.509 0.300 16.92 232,180 997,640 24.246

2.012 0.400 13.49 149,420 737,010 16.639

2.515 0.500 12.95 123,480 650,900 11.431

3.018 0.600 12.06 105,560 583,420 7.702

3.520 0.700 11.19 84,468 512,420 4.922

4.023 0.800 10.22 65,974 436,440 2.775

4.526 0.900 9.25 46,953 387,600 1.070

5.029 1.000 9.32 46,953 365,070 0.000

The differences between the UAE Phase V blades and the tapered, twisted blades used in the 

UAE Phase VI (i.e., NASA-Ames Research Center) tests are illustrated in Table 5.9. 

Table 5.9: Differences between the UAE Phase V and Phase VI Blades 

Property UAE Phase V Blade UAE Phase VI Blade Details

Mass [kg] 69.605 60.2 Average value per blade

Root mounted camera 

attached? Yes No The Phase V camera mass is 13 kg; the mass is equally distributed between 0.508 and 1.006 m radius

Distance to c.g. from 

center of rotation [m] 1.999 2.266 Average value per blade

Chord distribution Constant, 0.457 m beginning at 

0.723 m from center of rotation Tapered This will affect the mass and stiffness distribution

Twist distribution Ideal twist distribution (highly 

twisted) Slightly less twisted This will affect the out-of-plane and in-plane natural 

frequency calculations

1st in-plane natural 

frequency [Hz]

8.065 8.981 Average value per blade relative to the condition when 

the tip chord is parallel to the plane of rotation

1st out-of-plane natural 

frequency [Hz]

4.865 7.2815 "

2nd out-of-plane natural 

frequency [Hz]

20.7 29.75 "

1st in-plane damping 

ratio 0.7650% Data not available Average value per blade

1st out-of-plane 

damping ratio

0.9250% Data not available "

2nd out-of-plane 

damping ratio

Data not available Data not available

Characteristics of the power train are: 

 Power train inertia: 144–170 kgm2

 Power train stiffness: 194,000–204,000 Nm/rad considering the low-speed shaft, gearbox, 

and high-speed shaft as one unit 

 Power train natural frequency: 5.71–5.85 Hz 
 75

 Power train damping ratio: 0.06–0.08 

 Gearbox ratio: 25.13 to 1 

 Rated generator power: 20 kW 

 Synchronous speed of generator: 1,800 rpm on the high-speed shaft side 

 Generator slip: 1.69% at rated power 

 Power train efficiency: about 78% including the combined effect of mechanical losses in the 

gearbox and mechanical and electrical losses in the generator 

 Low-speed torque during startup: 2,210–2,261 Nm 

 Maximum braking torque: 115 Nm 

The tower is essentially a steel weldment assembly. A short and broad tubular steel lower 

section is connected to a long and narrow tubular steel upper section with a small conical 

segment. The base of the conical section is 3.4 m above the base of the tower and the top of the 

conical section is 3.9 m above the base of the tower. The bottom of the tower is connected to a 

circular base plate with 12 flanges. Other structural characteristics of the tower and nacelle 

include: 

 Tower length: 11.5 m 

 Tower material: ASTM A106 schedule 40 and schedule 80 Type B pipe 

 Tower Modulus of Elasticity: 200 GPa 

 Tower shear Modulus: 77 GPa 

 Tower density: 7,860 kg/m3

 

 Tower base plate diameter/thickness: 1.829 m/0.01905 m (3⁄4 inch) 

 Base plate flange thickness: 0.01905 m (3⁄4 inch) 

 Base plate flange shape: 45 ̊-45 ̊-90 ̊ triangle whose equal-length sides are 0.6097 m in length 

 Lower tower outside diameter/wall thickness: 0.6096 m/0.0175 m 

 Upper tower outside diameter/wall thickness: 0.4064 m/0.0214 m 

 Tower mass: 3,317 kg including the effects of paint, welds, tower base plate, and 12 base 

plate flanges 

 1st longitudinal/lateral natural frequency of tower alone: 3.91–4.03 Hz25

 2nd longitudinal/lateral natural frequency of tower alone: 19.50–19.91 Hz25

 Damping ratio associated with 1st natural frequency of tower: 0.5–0.625

 Damping ratio associated with 2nd natural frequency of tower: 0.7–0.925

 Nacelle mass: 1,712 kg including the mass of the hub, boom, and pitch shafts 

 Nacelle inertia about yaw axis: 3,789 kgm2

 including the effects of the hub, boom, and pitch 

shafts 

 Mass atop tower: 1,832 kg including the mass of the hub, boom, pitch shafts, and blades 

 Inertia atop tower about yaw axis: approximately 4,640 kgm2

 including the effects of the hub, 

boom, pitch shafts, and blades 

Because the UAE wind turbine was designed exclusively with research in mind, the UAE wind 

turbine is extensively instrumented. Bending moments are measured with strain gages installed 

on the low-speed shaft and at the root of each blade. In the wind tunnel tests, the rotor blade 

labeled “Blade 3” (out of two blades) was extensively instrumented and the structural properties 

of the other blade, “Blade 1,” were balanced to match those of Blade 3. On Blade 3, the strain 

__________________ 

25 These values were obtained by examining modal test videos of the tower alone (see Hand 2001). 
 76

gages measuring out-of-plane and in-plane moments at the root are applied to a steel pitch shaft 

adjacent to the blade attachment location and are aligned with the blade’s tip chord line. The 

rotor torque is measured with additional strain gages installed on the low-speed shaft. A 

tension/compression load cell is used to measure moments about the yaw bearing. The yaw 

angle, azimuth angle of the instrumented blade (Blade 3), blade pitch angles, and blade flap 

angles were all measured with digital position encoders. 

NREL has spent considerable time, energy, and money developing and perfecting 

instrumentation systems that can directly measure and characterize the aerodynamic forces 

acting on a rotating wind turbine blade. Aerodynamic coefficients for a particular radial station 

are found by integrating measured pressure values at that span location. To measure pressures, a 

grouping of 22 pressure taps, each connected to pressure transducer, is mounted chordwise 

around Blade 3. Each pressure tap opening is mounted flush to the airfoil surface to prevent the 

taps from disturbing the flow. The pressure transducers are mounted close to the pressure taps to 

mitigate potential dynamic effects within the tubes. These pressure tap groupings are installed at 

five primary spanwise locations: 30%, 46.67%, 63.33%, 80%, and 95% span. 

Table 5.10: Aerodynamic Coefficient Pressure Tap Grouping Properties 

Distance from Spanwise Number Of Local Local Chord Local

Rotor Center Fraction Pressure Twist (q T ) Length (c ) Thickness

m (/ 5.029 m) Taps deg m m

1.509 0.300 22 14.321 0.711 0.149

2.347 0.467 22 4.692 0.627 0.131

3.185 0.633 22 1.115 0.542 0.114

4.023 0.800 22 -0.381 0.457 0.096

4.778 0.950 22 -1.466 0.381 0.080

Five-hole probes are used to measure local flow angles on the rotating blade. Like the pressure 

tap groupings, the probes are mounted at five primary spanwise locations, each located 4% span 

outboard of its corresponding pressure tap grouping, except for the 95% span station, where the 

probe is mounted 4% span inboard. Angle of attack values at the pressure tap grouping span 

stations are derived from the local flow angle data by applying an upwash correction algorithm. 

Unfortunately, neither the probes nor the upwash correction algorithm are precise (±1 ̊ each). 

Simms et al. (1999) comprehensively document information regarding the instrumentation 

systems used by the UAE research wind turbine. 

5.2 NASA-Ames Wind Tunnel Testing 

The UAE test program objectives summarized in the introduction of this chapter were met by 

testing the UAE research wind turbine in the world’s largest wind tunnel located at the NASAAmes Research Center in Moffett Field (Silicon Valley), California. The so-called National 

Full-Scale Aerodynamics Complex wind tunnel has an enormous 24.4 m (80 ft) by 36.6 m (120 

ft) test section, permitting the 10 m (33 ft) in diameter UAE wind turbine to be tested with 

minimal contribution of blockage effects. That is, the test section boundaries have a negligible 

influence on the induction of the stream tube passing through the plane of the rotor. 
 77

Although the test section is enormous, flow through the wind tunnel is highly controlled and 

flow quality is consistently maintained. Flow is virtually steady, homogenous, uniform, and 

fixed in direction corresponding with assumptions (1) and (3) from section 2.2 and assumption 

(2) from section 2.3. Flow speed deviates by no more than 0.25% across an empty test section, 

and the flow velocity vector diverges by no more than 0.5o from the test section axis. Thus, 

testing in a highly controlled wind tunnel environment eliminates many of the complex 

characteristics of the natural, outdoor wind turbine environment—factors that complicate a 

detailed understanding of wind turbine aerodynamics. 

To garner the most benefits associated with the wind tunnel tests, NREL formed a panel of 

advisers from around the world to guide in the selection of particular configurations and 

conditions under which the wind turbine should be operated in the wind tunnel. In the end, more 

than 1,700 tests, including variations in turbine configuration, pitch angles, yaw positions, and 

wind velocities, were conducted, resulting in almost any conceivable operation regime, including 

steady state, post stall, and dynamic stall. Throughout each test, the turbine was extensively 

instrumented to measure quantities such as inflow conditions, wake form, airfoil aerodynamic 

pressure distributions, and machine responses and loads. Certain measurements, such as angle of 

attack, were taken only during specific tests to thwart the obstruction of other measurements 

made on the wind turbine. Each test consisted of 30 s of data-recording time, corresponding to 

roughly 36 rotor revolutions. 

Data obtained via the UAE wind tunnel tests are accurate and repeatable. The accuracies of the 

instrumentation systems are well documented by Simms et al. (1999). Exhaustive calibration 

procedures were performed between each wind tunnel test. During the wind tunnel tests, 

multiple independent data systems were also conducted to cross check incoming values, 

continuously ensuring high levels of data accuracy. Comparing the resulting data from similar 

tests demonstrates how astonishingly repeatable the data are, as data sets taken weeks apart are 

virtually identical. The acquired data represent the finest, most accurate set of wind turbine 

aerodynamic and induced flow field data available today. Simms et al. (2001) quantify 

additional information on data quality, accuracy, and repeatability. 

5.3 Blind Comparison Overview 

The key reasons for the Blind Comparison exercise were to (1) isolate deficiencies in the models 

employed in current design codes, (2) assess the range of applicability and efficiency of current 

design codes, and (3) obtain results that will guide the direction of future wind turbine modeling 

research. Thirty experts from 18 organizations around the world, representing an international 

conglomeration of industry, academic, and government wind turbine modeling specialists, 

participated and employed 19 wind turbine modeling tools. All participants were asked to 

provide documentation on their solution methodology, including information regarding model 

input parameters, other assumptions and approximations, theory employed in the models and the 

solution algorithm involved, computer performance, and post-processing tools. 

The Blind Comparison exercise was executed to expose deficiencies linked with the modeling 

tools and the modelers’ tactics. Shortcomings in the modeling tools originate from simulating 

operation states outside the scope of the models with which they were developed, flawed models, 

and code errors. Though all modelers were provided with the same turbine specifications (see 
 78

section 5.1), the development of model input files is a discriminatory process that relies on 

modeler experience and intuition. For example, with the abundance of aerodynamic data 

provided (see Tables 5.2– 5.7 and Figs. 5.1 and 5.2), modelers must discern which data are most 

appropriate in each situation. The choice and method of extrapolating two-dimensional airfoil 

data to account for the effects of stall delay across the inboard stations of the blade (see the 

discussion of advanced aerodynamics topics in section 2.6) and deep stall is another example of 

modeler discernment. As demonstrated in Chapter 6, the selection of these input file parameters 

drastically affects the resulting predictions; thus, the Blind Comparison exercise also exposes 

deficiencies linked to the modelers’ course of action. 

Only 20 test cases were chosen for the Blind Comparison exercise, since a considerable amount 

of participant effort is required for each. Because the aerodynamic forces determine component 

loads and turbine performance, all the cases were chosen to target specific aerodynamic 

phenomena. The major differences between individual test cases were nominal wind speed and 

turbine yaw error. Nonzero yaw cases correspond to operating states in which the angle of attack 

is driven dynamically back and forth through static stall. Key parameters of each test scenario 

are listed in Table 5.11. 

Table 5.11: Blind Comparison Test Scenarios 

Blade Average

Average Average Blade Collective Rotor

Wind Yaw Air Precone Tip Pitch Rotational

Speed Error Density Angle Angle Speed

Test Name m/s deg kg/m 3 deg deg rpm

S0700000 7.0 0.0 1.246 0.0 3.0 71.9

S1000000 10.0 0.0 1.246 0.0 3.0 72.1

S1300000 13.1 0.0 1.227 0.0 3.0 72.1

S1500000 15.1 0.0 1.224 0.0 3.0 72.1

S2000000 20.1 0.1 1.221 0.0 3.0 72.0

S2500000 25.1 0.1 1.220 0.0 3.0 72.1

S1000100 10.1 10.0 1.246 0.0 3.0 72.1

S1300100 13.1 10.1 1.227 0.0 3.0 72.1

S1500100 15.1 10.0 1.224 0.0 3.0 72.1

S1000300 10.1 30.2 1.246 0.0 3.0 72.0

S1300300 13.0 30.0 1.227 0.0 3.0 72.2

S1500300 15.1 29.9 1.225 0.0 3.0 72.2

S1000600 10.1 60.0 1.246 0.0 3.0 71.7

S1500600 15.1 60.0 1.225 0.0 3.0 71.9

E070000A 7.0 0.1 1.234 3.4 3.0 71.9

E170000A 17.1 0.1 1.234 3.4 3.0 72.0

E070020A 7.1 20.0 1.234 3.4 3.0 71.9

E170020A 17.2 20.2 1.234 3.4 3.0 72.2

E07M020A 7.0 -19.9 1.235 3.4 3.0 71.9

E17M020A 17.2 -20.0 1.234 3.4 3.0 72.1

Measurement conventions were established to ensure continuity between modelers’ predictions 

for the comparison phase, which was staged at the end of the Blind Comparison exercise. The 

Blind Comparison participants were asked to perform an equivalent simulation of each test case 

and to azimuth average the resulting time series data into 1o azimuth angle bins (centered on the 
 79

integer). The requested output channels and their associated measurement conventions are listed 

in Table 5.12. The requested data represent less than 1% of all data obtained during the UAE 

wind tunnel tests. 
 80

Table 5.12: Blind Comparison Output Channels and Measurement Conventions 

Channel Channel Channel

Name Channel Description Units Measurement Convention

YAW Yaw error deg Positive clockwise looking down

VTUN Wind speed m/s N/A

B3AZI Blade 3 azimuth angle deg

Counterclockwise when looking 

downwind (for both upwind and 

downwind configurations)

B3FLAP Blade 3 coning (flap) angle deg Positive downwind

B3PITCH Blade 3 collective tip pitch angle deg Positive towards feather

RPM Rotor rotational speed rpm N/A

B3RFB Blade 3 root flap bending moment Nm Moment vector parallel to tip chord; 

positive for downwind forces

B3REB Blade 3 root edge bending moment Nm Moment vector perpendicular to tip 

chord; positive in direction of rotation

LSSTQ Low speed shaft torque Nm Positive in direction of rotation

NAYM Nacelle yaw moment Nm Positive to realign turbine in wind from 

negative yaw error

QNORM30 Local dynamic pressure at 30.0% span Pa N/A

QNORM47 Local dynamic pressure at 46.7% span Pa "

QNORM63 Local dynamic pressure at 63.3% span Pa "

QNORM80 Local dynamic pressure at 80.0% span Pa "

QNORM95 Local dynamic pressure at 95.0% span Pa "

CN30 30.0% span normal force coefficient - Standard definition

CT30 30.0% span tangential force coefficient - "

CM30 30.0% span pitching moment coefficient - "

CN47 46.7% span normal force coefficient - "

CT47 46.7% span tangential force coefficient - "

CM47 46.7% span pitching moment coefficient - "

CN63 63.3% span normal force coefficient - "

CT63 63.3% span tangential force coefficient - "

CM63 63.3% span pitching moment coefficient - "

CN80 80.0% span normal force coefficient - "

CT80 80.0% span tangential force coefficient - "

CM80 80.0% span pitching moment coefficient - "

CN95 95.0% span normal force coefficient - "

CT95 95.0% span tangential force coefficient - "

CM95 95.0% span pitching moment coefficient - "

N30 30.0% span normal force coefficient N/unit span Standard definition

T30 30.0% span tangential force coefficient N/unit span "

M30 30.0% span pitching moment coefficient Nm/unit span "

N47 46.7% span normal force coefficient N/unit span "

T47 46.7% span tangential force coefficient N/unit span "

M47 46.7% span pitching moment coefficient Nm/unit span "

N63 63.3% span normal force coefficient N/unit span "

T63 63.3% span tangential force coefficient N/unit span "

M63 63.3% span pitching moment coefficient Nm/unit span "

N80 80.0% span normal force coefficient N/unit span "

T80 80.0% span tangential force coefficient N/unit span "

M80 80.0% span pitching moment coefficient Nm/unit span "

N95 95.0% span normal force coefficient N/unit span "

T95 95.0% span tangential force coefficient N/unit span "

M95 95.0% span pitching moment coefficient Nm/unit span "
 81

6. Blind Comparison Input Parameters and Results 

FAST_AD has yet to be validated. Validation of wind turbine design codes such as FAST_AD 

involves comparing simulated load and performance predictions with actual experimental test 

data and assessing the degree to which they correspond. To facilitate validation of FAST_AD, 

the 20 test cases chosen for the Blind Comparison exercise are simulated. This permits a 

comparison between load and performance predictions made by FAST_AD and those obtained 

experimentally (permitting one to test how accurate the code predicts), and a comparison to load 

and performance predictions made by other design codes used worldwide (permitting one to test 

how FAST_AD’s predictions compare to those of the wider wind turbine design and analysis 

community). The steps taken to perform simulations of the UAE research wind turbine under 

these scenarios and some of the results obtained are presented in this chapter. The steps taken to 

transform the UAE wind turbine specifications (summarized in Chapter 5) into specific input file 

entries are documented in section 6.1. A comparison and discussion of some of the load 

predictions made by FAST_AD and those obtained by other Blind Comparison participants and 

those obtained experimentally in the NASA-Ames wind tunnel are presented in section 6.2. 

The goal of this chapter is not to determine whether the aerodynamic and structural models 

employed in FAST_AD are better than entirely dissimilar models employed by other design 

codes (which employ CFD, prescribed-wake, performance-only, or other solution algorithms). 

As a consequence, the results and discussion presented in section 6.2 are devoted entirely to the 

comparison of load and performance values obtained by means of the NASA-Ames wind tunnel 

tests, predictions made by FAST_AD, and predictions made only by Blind Comparison 

participants who employed the AeroDyn aerodynamics subroutines when determining 

aerodynamic forces acting on the rotor blades. All five Blind Comparison predictions that use 

the AeroDyn aerodynamics subroutines are presented. The reader is encouraged to examine 

Simms et al. (2001) for the results and conclusions of the entire Blind Comparison study. For 

anonymity, the load and performance predictions made by other Blind Comparison participants 

are labeled without names in section 6.2. 

6.1 FAST_AD Input Files Development for the Blind Comparison Study 

To transform the specifications of the UAE wind turbine presented in Chapter 5 into viable 

parameters needed to define the FAST_AD input file(s), several assumptions and approximations 

must be made. These assumptions and approximations, as well as other steps taken to develop 

the input file(s) for FAST_AD for the Blind Comparison test cases simulated, are documented in 

this section. An example input file is given in Appendix I for test case S0700000 of the Blind 

Comparison study of the UAE wind turbine. Input files for the other test cases are similar. 

Dissimilarities in the input files among the test cases are also discussed. 

Following the practice of several Blind Comparison participants, an azimuth angle increment or 

azimuth step, dψ, of about 1o is used throughout all the simulations. In wind turbine analysis, 

reasoning in terms of an azimuth step, instead of a time step, is often convenient, though the 

main input file of FAST_AD requires the designation of a time step in seconds instead of 

degrees. The following simple formula can be applied to convert an azimuth step in degrees to a 

time step, dt, in seconds: 
 82

































=

1rev

360

1min

60 sec d

dt D

Ω

ψ

 (6.1) 

where the average angular speed of the rotor, Ω, is assumed to be expressed in revolutions-perminute (rpm). Using an azimuth step of 1o and an average rotor speed of 72 rpm, the time 

increment used for all simulations is 0.002 s, rounded to the nearest thousandth of a second. 

Each simulation is run for 30 s, corresponding to roughly 36 revolutions of the rotor. To avoid 

recording data associated with the simulation startup transients26, only data from the last 10 s of 

simulation time are recorded for data analysis purposes. Several test cases were run to verify 

that all startup transients die out before 20 s of simulation time pass. 

All DOFs except yaw, tilt, and teeter are included (switched on) in all simulations. The yaw 

DOF is excluded to ensure that the yaw position of the nacelle and yaw error remain fixed 

throughout any individual simulation. The tilt DOF is excluded since information regarding the 

stiffness of the yaw bearing in the tilt direction is unavailable and since the yaw bearing is 

assumed to be stiff enough to prevent any substantial tilting motion of the nacelle. Since the 

UAE wind turbine exhibits a rigid rotor type for each Blind Comparison test case, the teeter DOF 

switch is also turned off. 

FAST_AD considers the wind turbine modeled to spin clockwise when looking downwind and 

has the convention that a positive yaw angle is counterclockwise when looking downward for 

upwind and downwind rotor configurations. Since the sign conventions of both parameters are 

opposite to those used by the UAE wind turbine considered in the Blind Comparison test cases, 

equivalency is maintained and interpretation of proper sign conventions (of, for example, root 

bending moments) is not difficult. This is because the orientation of a blade advancing into a 

region of increased relative wind speed is identical in either perspective. The validity of this 

point is demonstrated in Fig. 6.1. 

Since most blade load outputs available with FAST_AD are for Blade 1 only, Blade 1 in 

FAST_AD is used to model Blade 3 (the instrumented blade) of the UAE wind turbine. To 

ensure a proper azimuth angle convention, the azimuth offset angle, z(4) is set fixed at 270o. The 

steady-state angular velocity of the rotor is defined to be the average rotor rotational speed listed 

in Table 5.11 for each Blind Comparison test scenario simulated. Similarly, the fixed yaw angle 

is set to the yaw error listed in Table 5.11 (these parameters are specified differently for each test 

case). All other initial conditions (initial wind turbine component displacements) are set equal to 

zero for each simulation run. 

__________________ 

26 The model starts out with initial displacements but zero initial inertias Once several seconds of the simulation 

have progressed, the inertias of most components of the turbine will no longer be zero and remain that way. The 

start-up transients are the events that must take place to make these inertias follow some quasi-steady-state pattern. 
 83

Figure 6.1: Azimuth and yaw angle sign conventions and model equivalency

The pitch mechanism is modeled as fixed (stall regulation). Since FAST_AD assumes that the 

blade collective pitch angle is measured relative to the point of zero aerodynamic twist and since 

the UAE wind turbine’s tip twist is –1.815o relative to the point of zero aerodynamic twist at the 

75% span location, the blade collective pitch angle is set at 4.815o to ensure the angle to the tip 

chord line from the rotor plane is 3.0o for all Blind Comparison test cases. The fixed blade 

precone angles, β1 and β2, are set to zero for the upwind rotor configurations and 3.4o for the 

downwind rotor configurations as designated in Table 5.11. 

Since the rotor is nonteetering, both the undersling length, RU, and the delta-3 angle, δ3, are fixed 

at zero. The locations of the hub and nacelle center of masses are unknown; thus, the parameters 

RUM, DNM1, and DNM2 are also set to zero. The designation of RUM = 0 is valid since blade coning 

angles are small or zero for each Blind Comparison test case; thus, the hub center of mass is 

most likely close to the blade axis intersection point. The designations of DNM1 = 0 and DNM2 = 0 

are valid since the yaw and tilt DOFs are switched off and since the nacelle yaw moment is one 

of the noncrucial output channels, according to the directors of the Blind Comparison exercise. 

The distance to the teeter pin (or equivalently, to the blade axis intersection point since RUM is set 

to zero), DN, is set to –1.401 m for the upwind rotor configurations and +1.401 m for the 

downwind rotor configurations. 

Since inertial properties of the UAE wind turbine hub separate from the nacelle are not available 

and since the rotor is nonteetering, the hub is modeled as having no mass and its inertial effects 

are included with the inertial attributes of the nacelle. Consequently, the lumped mass of the 

Upwind Config. Dnwind. Config. Upwind Config. Dnwind. Config.

UAE / Blind Comparison Conventions FAST_AD Conventions 

Top View:

Looking 

Downward 

Downwind View:

Looking Upwind 

Upwind View:

Looking 

Downwind 

+YAW 

+Ω

Wind 

Dir. 

+YAW

+Ω

Wind

Dir.

+q6

+Ω

Wind 

Dir.

+q6

+Ω

Wind

Dir.

+AZI 

Blade 3

+AZI

Blade 3

+AZI 

Blade 3 

+AZI

Blade 3

z(4)+q4

Blade 1

z(4)+q4

Blade 1

z(4)+q4

Blade 1

z(4)+q4

Blade 1 
 84

nacelle is set at 1,712.0 kg, the inertia of the nacelle about the yaw axis is set at 3,789.0 kgm2

, 

and the hub mass and inertia are set equal to zero. 

The electric generator is modeled as an induction generator with a linear generator constant. The 

generator inertia is set at 161.5 kgm2, the average value from the range provided (144–170 

kgm2). FAST_AD permits the designation of independent gearbox and generator efficiencies. 

Since only an effective power train efficiency of the UAE wind turbine is provided (78%), the 

gearbox efficiency is set at 100% and the generator efficiency is set at 78%. The synchronous 

speed of the generator (zero-slip condition) is 1,800 rpm on the high-speed shaft side. This 

speed must be designated in the FAST_AD input file relative to the low-speed shaft (Ω0). Since 

the gearbox ratio is 25.13:1, Ω0 is set at 71.63 rpm. Likewise, the rated speed of the induction 

generator must be specified relative to the low-speed shaft (ΩR). The rated speed, synchronous 

speed, and slip of a linearly modeled induction generator are related: 

0

R 0 generatorslip atrated power Ω

Ω − Ω = (6.2) 

With a generator slip at rated power of 1.69% for the UAE wind turbine, the rated speed of the 

induction generator relative to the low-speed shaft, ΩR, is set at 72.82 rpm. The linear generator 

constant, CINGEN, is equal to the low-speed shaft torque at rated power, LSSTQR, divided by the 

product of the gearbox ratio squared, YN2

, and the difference in rated and synchronous speeds 

relative to the low-speed shaft measured in radians-per-second: 

( ) 













 −

















=

1rev

2 rad YN

1min

60 sec LSSTQ

CINGEN

R 0

2

R

π Ω Ω

 (6.3) 

The low-speed shaft torque at rated power, LSSTQR, is the ratio of the induction generator’s rated 

power , PR, and the rated speed of the low-speed shaft measured in radians-per-second: 

































=

1rev

2 rad

1min

60 sec P

LSSTQ

R

R

R

π Ω

 (6.4) 

With a rated generator power of 20 kW (20,000 W), LSSTQR is 2,622 Nm and CINGEN is set at 

32.7539 Nm/(rad/s). Though not needed in any simulations, the high-speed shaft startup torque 

is set at 88.9574 Nm in the FAST_AD input file. This is the average value of the startup lowspeed shaft torque divided by gearbox ratio from the range of startup low-speed shaft torques 

provided (2,210–2,261 Nm). 
 85

Flexibility of the drive shaft is modeled with a linear torsional spring and a linear viscous 

damper, both specified as constants in the FAST_AD input file. The equivalent drive train 

torsional spring constant, ZKDRV, is set at 199,000 Nm/rad, the average value from the range 

provided (194,000–204,000 Nm/rad). The equivalent drive train torsional damper constant, 

CDRV, is equal to twice the product of ZKDRV and the power train damping ratio, ζDRV, divided 

by the power train natural frequency in radians-per-second, ωnDRV: 

nDRV

ZKDRV DRV CDRV 2

ω

⋅ζ = (6.5) 

The average power train damping ratio from the range provided (0.06–0.08) is 0.07. The 

average power train natural frequency from the range provided (5.71–5.85 Hz) is 5.78 Hz, or 

36.3168 rad/s. Thus, CDRV is approximated as 767.138 Nm/(rad/s). 

The entire length of the tower (11.5 m) is modeled as flexible. The rigid length of the tower, HS, 

is thus set to zero. The distance from the tower top base plate (yaw bearing) to the rotor axis, 

TWRHTOFFSET, is set at 0.692 m, the difference between the hub height above ground level 

(12.192 m) and the length of the tower (11.5 m). 

Several approximations are made to estimate the structural properties of the tower and its 

corresponding natural mode shapes. To estimate the distributed lineal density of the tower, 

μT(h), estimates of masses of various parts of the tower are made and scaled so their sum, the 

total approximated tower mass, is equal to the actual measured tower mass (3,317 kg). First, the 

mass of several sections of the tower (upper segment, lower segment, conical segment, base 

plate, and flanges) are estimated, summed, and compared to the total tower mass provided for the 

UAE wind turbine. The details of each calculation follow without explanation, but should be 

easily inferred from the data given in Chapter 5: 

( )

m

kg 203.445

0.0214 m

2

0.4064 m

2

0.4064 m

m

kg uppersegment 7,860

2 2

T 3

=































 



 



  − −  



 



 





μ = π

 (6.6) 

( ) 11.5m 3.9 m 1,546.182kg

m

kg mass of uppertower segment 203.445  − = 

 



 = (6.7) 

( )

m

kg 255.862

0.0175m

2

0.6096 m

2

0.6096 m

m

kg lower segment 7,860

2 2

T 3

=































 



 



  − −  



 



 





μ = π

 (6.8) 
 86

( ) 3.4m 869.931kg

m

kg mass of lowertower segment 255.862  = 

 



 = (6.9) 

( )

114.827 kg

3.9m 3.4m

m

kg 255.862

m

kg 203.445

2

1

mass of conical tower segment

=

 − 

 



 = +

 (6.10) 

( ) 0.01905m 393.400 kg 2

1.829 m

m

kg mass of base plate 7,860

2

3 =















 



 



 



 



 = π (6.11) 

( )( ) m m kg

m

kg mass of gle flange 0.6097 0.01905 27.830

2

1 sin 7,860 2

3 =



 









 



 = (6.12) 

and 

( )( ) 12 27.830 kg 3,258.305 kg

total tower mass 1,546.182kg 869.931kg 114.827 kg 393.400 kg

+ =

= + + +

 (6.13) 

The estimated total tower mass given in Eq. (6.13) is 1.80% less than the total tower mass 

measured. The difference is most likely due to the addition of paint and welds in the latter. 

Consequently, the lineal densities of the upper and lower tower segments calculated above are 

scaled up by a factor of 1.018–207.107 kg/m and 260.467 kg/m respectively. These calibrated 

distributed lineal density parameters are used during calculations of the tower’s mode shapes in 

program Modes and specification of these tower properties in the input file of FAST_AD. 

Since the mass distribution is already calibrated and since the stiffness distribution, EIT(h), is the 

only other set of tower parameters that is not well characterized, estimates of the stiffness 

distribution are calibrated to ensure that the first natural frequency predicted by Modes of the 

tower alone is equal to 3.97 Hz, the average value from the range provided (3.91–4.03 Hz). A 

preliminary estimate of the distributed stiffness of the tower is made by estimating the stiffness 

of the upper and lower tower segments (again, the details of each calculation follow without 

explanation but should be easily inferred from the data given in Chapter 5): 

( )( )

2

4 4

T

96,211,189.2 Nm

0.0214m

2

0.4064m

2

0.4064m

4

EI uppersegment 200GPa

=































 



 



  − −  



 = π

(6.14) 

and 
 87

( )( )

2

4 4

T

285,556,680.0 Nm

0.0175m

2

0.6096 m

2

0.6096 m

4

EI lower segment 200GPa

=































 



 



  − −  



 = π

(6.15) 

To calibrate this stiffness distribution, the first two natural frequencies of the tower alone are 

predicted by program Modes and compared to the natural frequencies of the tower specified in 

Chapter 5. This is done by specifying the parameters into a Modes input file (reference 

Appendix II for layout) and running the program with the end mass, MTop, equal to zero. The 

resulting frequency predictions are higher than those specified in Chapter 5. Both distributed 

stiffness values must be scaled by a factor of 0.826 to achieve equivalency of the first natural 

frequency. The resulting calibrated stiffness is 79,470,442.3 Nm2

 for the upper tower section 

and 235,869,817.7 Nm2

 for the lower tower section. The resulting predicted second natural 

frequency is 19.898 Hz, which is within the range of the values provided (19.50–19.91 Hz). 

To obtain the proper mode shapes needed for the FAST_AD input file (tower mode shapes that 

take into account the mass atop the tower), the tower end mass is specified to be the combined 

mass of the nacelle and rotor (1,832 kg) in the Modes input file and Modes is run again. A 

summary of the calibrated distributed tower parameters and the resulting mode shapes is given in 

Tables 6.1 and 6.2. 

Table 6.1: Distributed Properties of the UAE Tower 

Ht. Above Fractional Lineal Tower

Ground (h ) Distance Density Stiffness

m (h / 11.5 m) (μ T ) kg/m (EI T ) Nm 2

0.000 0.00000 260.467 235,869,817.7

3.400 0.29565 260.467 235,869,817.7

3.401 0.29574 260.467 235,869,817.7

3.899 0.33904 207.107 79,470,442.3

3.900 0.33913 207.107 79,470,442.3

11.500 1.00000 207.107 79,470,442.3

Table 6.2: Mode Shapes of the UAE Tower (Including Mass atop Tower) 

Shape Shape 1st Mode Const. 2nd Mode Const.

Function Function Proportionality Proportionality

Number (b ) (φ b 's) Coef.s (C1,b 's) Coef.s (C2,b 's)

2 (h /H )

2 1.187 -29.592

3 (h /H )

3 -2.497 -1.335

4 (h /H )

4 6.824 85.063

5 (h /H )

5 -6.654 -62.876

6 (h /H )

6 2.140 9.740

The modal damping ratio associated with the first natural frequency of the tower is set at 0.55, 

the average value from the range provided (0.5–0.6). The modal damping ratio associated with 
 88

the second natural frequency of the tower is set at 0.8, again the average value from the range 

provided (0.7–0.9). The maximum number of constant length increments along the tower 

allowable for integration of elastic forces is chosen (20). 

Though never used because the yaw DOF is switched off, parameters associated with flexibility 

of the yaw drive in the yaw direction are specified in the FAST_AD input file as accurately as 

possible. Since the yaw bearing is very stiff, the stiffness of the yaw drive in the yaw direction is 

equal to the torsional stiffness of the tower. To determine this stiffness, the torsional stiffness of 

the upper and lower tower segments are first estimated as follows27 (again, the details of each 

calculation follow without explanation but should be easily inferred from the data given in 

Chapter 5): 

( )

( )

rad

Nm 9,747,713

0.0214 m

2

0.4064 m

2

0.4064 m

11.5m 3.9m 2

77GPa

torsional stiffness of uppersegment

4 4

=































 



 



  − −  





−

=

π (6.16) 

and 

( )

( ) rad

Nm 0.0175m 64,670,189

2

0.6096 m

2

0.6096 m

3.4m 2

77GPa

torsional stiffness of lowersegment

4 4

=































 



 



  − −  





=

π (6.17) 

The stiffness of the upper and lower tower segments combine in series to determine the 

equivalent torsional yaw spring constant, ZKYAW28: 

rad

Nm 8,471,000

rad

Nm 64,670,189

rad

Nm 9,747,713

rad

Nm 64,670,189

rad

Nm 9,747,713

ZKYAW =

+





 









 





= (6.18) 

__________________ 

27 The torsional stiffness of an linearly elastic, isotropic, and homogenous rod is defined as: 

L

GJ k =

28 The effective stiffness of set of two linearly elastic springs in series is: 

1 2

1 2

eff k k

k k k

+ =
 89

Similarly, the torsional damping constant of the yaw drive in the yaw direction, ZCYAW, is equal 

to the torsional damping of the tower. Assuming a damping ratio of 2%, ZCYAW is estimated29: 

( ) ( ) rad / sec

Nm 4,640 kgm 7,930

rad

Nm ZCYAW 2 0.02 8,471,000 2  = 

 



 = (6.19) 

The hub radius, or equivalently, the rigid blade length, RH, is set at 0.432 m, the distance from 

the hub centerline to the root-mounted strain gages. 0.432 m is chosen instead of 0.508 m, the 

actual hub radius, to ensure that the blade root stresses are characterized correctly. The length of 

the flexible part of the blade is thus designated as 4.597 m (5.029 m rotor radius minus 0.432 m 

hub radius). 

Like the tower, several approximations are made to estimate the structural properties of the 

blades and their corresponding natural mode shapes. Estimations of the structural properties of 

the blades are more difficult, however, since detailed information on the blades used during the 

NASA-Ames wind tunnel tests is unavailable. The flapwise and edgewise stiffness distributions 

listed in Table 5.8 are for the blades of the wind turbine used during Phase V testing and the 

differences between these blades and those used during the NASA-Ames wind tunnel tests (UAE 

Phase VI testing) are listed in Table 5.9. Also, the natural frequencies listed in Table 5.9 are outof-plane and in-plane frequencies relative to the condition in which the tip chord is parallel to the 

rotor plane. Program Modes allows its user to input a structural pretwist distribution so the 

natural frequencies can be found in the out-of-plane and in-plane directions, allowing the user to 

match physically measured frequencies. Once such frequencies are matched, the distributed 

structural pretwist must be zeroed out to obtain the proper mode shapes needed for the 

FAST_AD input file. 

With this in mind, the first step to obtain structural properties of the UAE Phase VI blades and 

their corresponding mode shapes is to adjust the mass and stiffness distributions of the Phase V 

turbine so that program Modes accurately predicts the experimentally measured out-of-plane and 

in-plane frequencies given in Table 5.9. After adequate information on the mass and stiffness 

distributions for the Phase V blades is in-hand, these parameters are modified to account for the 

differences between the UAE Phase V and Phase VI blades listed in Table 5.9. Finally, the 

stiffness distributions are calibrated by ensuring that the natural frequencies of the UAE Phase 

VI blades predicted by Modes are equivalent to those corresponding frequencies also listed in 

Table 5.9. 

To calibrate the mass and stiffness distributions of the Phase V blades, similar steps are taken to 

those used to calibrate the mass and stiffness distributions of the tower. The mass distribution of 

the Phase V turbine is first tuned so a computation of the blade mass is equal to the actual 

measured blade mass (69.605 kg). The blade mass is estimated with the data provided in Table 

5.8 by dividing the blade into a number of elements, calculating their masses, and summing. 

__________________ 

29 The torsional damping constant for a linearly elastic, isotropic, and homogenous rod is defined as: 

c = 2ζ kJ
 90

This is done by using elements that are centered at the radial distances listed in Table 5.8 (the 

nodes between the elements are located halfway between these radial distances and the elements 

at the endpoints are only half-elements) as illustrated in Table 6.3. Each element’s contribution 

to the center of gravity (c.g.) of the blade is found by multiplying the element’s mass by the 

radial distance to its center and dividing by the total blade mass. 

Table 6.3: Computation of the UAE Phase V Blade Mass 

Distance Blade

from Rotor Lineal Element Element Element

Center Density Length Mass Contribution

m (μ B ) kg/m m kg to c.g. m

0.402

0.508 30.14 0.121 3.632 0.024

0.749 45.18 0.249 11.250 0.108

1.006 46.09 0.254 11.707 0.151

1.257 31.53 0.252 7.931 0.128

1.509 16.92 0.378 6.387 0.124

2.012 13.49 0.503 6.785 0.175

2.515 12.95 0.503 6.514 0.210

3.018 12.06 0.503 6.060 0.234

3.520 11.19 0.503 5.623 0.254

4.023 10.22 0.503 5.141 0.265

4.526 9.25 0.503 4.653 0.270

5.029 9.32 0.252 2.344 0.151

TOTALS 4.521 78.026 2.093

Properties of an Element Centered at r

The estimated blade mass given in Table 6.3 is considerably larger than that measured and listed 

in Table 5.9. A scaling factor of 0.892 applied to the distributed lineal density of the blades, 

μB(r), is required to calibrate the two masses. Since the predicted blade c.g. of 2.093 m from the 

center of rotation is close to the value of 1.999 m listed in Table 5.9 without any modification to 

the mass distribution, this is purely a magnitude adjustment. 

Now that a calibrated distributed lineal density of the Phase V blades is acquired, the magnitudes 

of the flapwise and edgewise stiffness distributions of the Phase V turbine blades (listed in Table 

5.8) are scaled (calibrated) so the out-of-plane and in-plane natural frequencies predicted by 

program Modes match those given in Table 5.9 (approximately 4.865 and 8.065 Hz for the first 

natural modes, respectively). Magnitude scaling factors of 0.72 and 0.83 applied to the flapwise 

and edgewise stiffness distributions are necessary to achieve equivalency of these first natural 

frequencies. Since the actual structural pretwist distribution is unknown, it is assumed to 

coincide with the aerodynamic twist distribution during this process. Also, the end mass, MTip, is 

set equal to zero in this process because there are no tip brakes. Centrifugal stiffening effects are 

not accounted for in this process. A summary of the calibrated distributed structural properties 

of the UAE Phase V blades is given in Table 6.4. 
 91

Table 6.4: Calibrated Properties of the UAE Phase V Blades 

Distance Distance Blade Local Local Local

from Rotor Along Blade Radial Lineal Flapwise Edgewise Structural

Center From Root (r) Fraction Density Stiffness Stiffness Twist (θ S )

m m (r / 4.597 m) (μ B ) kg/m (EI ) Nm 2 (EI ) Nm 2 deg

0.432 0.000 0.00000 0.000 340,932.2 393,019.1 0.000

0.508 0.076 0.01653 26.885 1,670,904.0 1,927,343.0 0.000

0.749 0.317 0.06896 40.301 937,728.0 1,292,144.0 43.824

1.006 0.574 0.12486 41.112 511,365.6 1,106,224.0 35.840

1.257 0.825 0.17946 28.128 339,609.7 967,409.1 30.054

1.509 1.077 0.23428 15.093 167,169.6 828,041.2 24.246

2.012 1.580 0.34370 12.033 107,582.4 611,718.3 16.639

2.515 2.083 0.45312 11.551 88,905.6 540,247.0 11.431

3.018 2.586 0.56254 10.758 76,003.2 484,238.6 7.702

3.520 3.088 0.67174 9.981 60,817.0 425,308.6 4.922

4.023 3.591 0.78116 9.116 47,501.3 362,245.2 2.775

4.526 4.094 0.89058 8.251 33,806.2 321,708.0 1.070

5.029 4.597 1.00000 8.313 33,806.2 303,008.1 0.000

These parameters can now be modified to account for the differences between the UAE Phase V 

and Phase VI blades listed in Table 5.9. To do this, values of the inboard-most lineal density 

parameters listed in Table 6.4 are first lowered to account for the absence of the root-mounted 

camera in the UAE Phase VI blades (the Phase V camera mass is 13 kg, equally distributed 

between the 0.508 and 1.006 m stations). Next, the mass distribution of the entire blade is 

adjusted to account for blade taper (the UAE Phase VI blades are larger near the root and smaller 

near the tip than the Phase V blades). This is done by multiplying the mass distribution by a 

linear function equal to unity at a span of 4.023 m, the location where the chord length of the 

Phase VI blades equals the chord length of the Phase V blades (a position on the two blades 

where the lineal densities are most likely identical). The slope of this linear function is chosen to 

be –0.021/m such that the c.g. of the blade is calculated to be 2.266 m, the distance to the c.g. of 

the Phase VI blade from the center of rotation (the slope should be less than zero to ensure that 

the adjustment factor is smaller at the outboard stations and larger at the inboard stations). The 

magnitude of the resulting mass distribution is then scaled by a slight amount (0.966), so the 

calculated blade mass is equal to the actual measured blade mass (60.2 kg). The results of these 

steps are illustrated in Table 6.5. 
 92

Table 6.5: Distributed Mass Properties of the UAE Phase VI Blades 

Distance Blade

from Rotor Lineal Element Element Element

Center Density Length Mass Contribution

m (μ B ) kg/m m kg to c.g. m

0.402

0.508 8.298 0.121 1.000 0.008

0.749 22.109 0.249 5.505 0.068

1.006 32.416 0.254 8.234 0.137

1.257 28.750 0.252 7.231 0.151

1.509 15.349 0.378 5.794 0.145

2.012 12.115 0.503 6.094 0.203

2.515 11.512 0.503 5.791 0.242

3.018 10.611 0.503 5.332 0.267

3.520 9.744 0.503 4.896 0.286

4.023 8.806 0.503 4.430 0.296

4.526 7.886 0.503 3.967 0.298

5.029 7.861 0.252 1.977 0.165

TOTALS 4.521 60.250 2.267

Properties of an Element Centered at r

Now that a calibrated distributed lineal density of the Phase VI blades is acquired, the flapwise 

and edgewise stiffness distributions listed in Table 6.4 are adjusted to account for taper and tuned 

so the out-of-plane and in-plane natural frequencies predicted by program Modes match those 

given in Table 5.9 (approximately 7.282 and 8.981 Hz for the first natural modes, respectively). 

This is done by a procedure similar to that described in the previous paragraph. For the flapwise 

stiffness distribution, the slope of the linear function is chosen to be –0.451/m and the magnitude 

scaling factor is chosen to be 1.50. For the edgewise stiffness distribution, the slope of the linear 

function is chosen to be –0.051/m and the magnitude scaling factor is set to unity. Using these 

adjustment factors, the resulting natural frequencies are 7.28 Hz (first out-of-plane), 30.79 Hz 

(second out-of-plane), and 8.985 Hz (first in-plane), which are all very close to the values 

provided for the UAE Phase VI blade in Table 5.9. As was the case for the stiffness calibrations 

of the Phase V blades, the structural pretwist distribution is assumed to coincide with the 

aerodynamic twist distribution during this process since the actual distribution is unknown. 

Also, the end mass, MTip, is set equal to zero in this process because there are no tip brakes. 

Again, centrifugal stiffening effects are not accounted for in this process. A summary of the 

calibrated distributed structural properties of the UAE Phase VI blades is given in Table 6.6. 
 93

Table 6.6: Calibrated Properties of the UAE Phase VI Blades 

Distance Distance Blade Local Local Local

from Rotor Along Blade Radial Lineal Flapwise Edgewise Structural

Center From Root (r) Fraction Density Stiffness Stiffness Twist (θ S )

m m (r / 4.597 m) (μ B ) kg/m (EI ) Nm 2 (EI ) Nm 2 deg

0.432 0.000 0.00000 0.000 511,398.4 393,019.1 0.000

0.508 0.076 0.01653 8.298 2,506,356.0 1,927,343.0 0.000

0.749 0.317 0.06896 22.109 3,483,529.2 1,507,898.5 0.000

1.006 0.574 0.12486 32.416 1,810,745.8 1,276,435.4 6.593

1.257 0.825 0.17946 28.750 1,144,892.0 1,103,877.7 20.040

1.509 1.077 0.23428 15.349 535,063.2 934,207.7 14.315

2.012 1.580 0.34370 12.115 307,733.2 674,456.7 7.454

2.515 2.083 0.45312 11.512 224,056.5 581,796.3 3.702

3.018 2.586 0.56254 10.611 165,678.0 509,058.2 1.621

3.520 3.088 0.67174 9.744 111,920.2 436,219.0 0.421

4.023 3.591 0.78116 8.806 71,251.9 362,245.2 -0.381

4.526 4.094 0.89058 7.886 39,205.7 313,455.2 -1.111

5.029 4.597 1.00000 7.861 27,702.2 287,462.0 -1.815

To obtain the proper mode shapes needed for the FAST_AD input file (local flapwise and 

edgewise blade mode shapes instead of out-of-plane and in-plane mode shapes) the distributed 

structural pretwist is zeroed out in the Modes input file and Modes is rerun. Centrifugal 

stiffening effects are finally accounted for in this process by specifying a typical rotor rotational 

speed of 72 rpm. A summary of the resulting mode shapes is given in Table 6.7. 

Table 6.7: Mode Shapes of the UAE Phase VI Blades 

1st Edgewise

Shape Shape 1st Mode Const. 2nd Mode Const. Mode Const.

Function Function Proportionality Proportionality Proportionality

Number (b ) (φ b 's) Coef.s (C1,b 's) Coef.s (C2,b 's) Coef.s (C1,b 's)

2 [r/(R-R H )]2 0.448 -2.067 1.023

3 [r/(R-R H )]3 -1.694 5.5 -0.056

4 [r/(R-R H )]4 8.139 -27.516 1.351

5 [r/(R-R H )]5 -8.976 46.7 -2.187

6 [r/(R-R H )]6 3.083 -21.616 0.869

Flapwise Direction

The modal damping ratio associated with the first natural frequency of the blades in both the 

flapwise and edgewise directions is assumed to be the same as the in-plane and out-of-plane 

damping ratios of the UAE Phase V blade, 0.9250% and 0.7650% respectively. The modal 

damping ratio associated with the second flapwise natural frequency of the blades is assumed to 

equal 1.345%, typical for utility-scale turbines. Though not used for any outputs or postprocessed data presented in this report, the locations of the stations along the blade for local 

moment outputs are chosen to coincide with the pressure tap stations (see Table 5.10). The 

maximum number of constant length increments along the blades allowable for integration of 
 94

aerodynamic and elastic forces is chosen (20). Consequently, each blade element is 0.22985 m 

in length [(5.029 m – 0.432 m)/20]. 

Axial and tangential induction factors are included in all aerodynamics computations by the 

choice of “SWIRL” for the induction factor model parameter of the main FAST_AD input file. 

The default tolerance for induction factor convergence (0.005) is specified. Both the dynamic 

inflow and dynamic stall models are employed for all Blind Comparison simulations run. 

Following the practice of a well-respected Blind Comparison participant, the tower shadow halfwidth at the plane of the rotor is assumed to be 0.4 m and the velocity deficit at this same point is 

assumed to be 0.35. Consequently, the dimensionless tower shadow width at the rotor plane (the 

tower shadow width at the rotor plane divided by the rotor radius) is specified as 0.15908. The 

tower shadow models are switched off for the upwind rotor configurations. 

The air density is defined to be the average air density listed in Table 5.11 for each Blind 

Comparison test scenario. Since the flow in the wind tunnel is virtually steady, homogenous, 

uniform, and fixed in direction (see section 5.2), constant hub height wind data are used instead 

of full-field wind data. The constant wind speed value is specified in a wind data file differently 

for each Blind Comparison test case simulated, by using the average wind speed values listed in 

Table 5.11. An example wind data input file is given in Appendix III for test case S0700000 of 

the Blind Comparison study of the UAE wind turbine. Wind data files for other Blind 

Comparison test scenarios are very similar. 

In FAST_AD, each blade element can be modeled with its own unique airfoil table. Eight airfoil 

tables are used in each simulation (see Table 6.8 and Appendix IV). All airfoil data tables are 

the same ones used as, and are obtained from, a well-respected Blind Comparison participant 

who employed the AeroDyn aerodynamics subroutines during simulations. All the airfoil data 

table files whose names begin with “S809_CLN” represent dimensionless coefficient airfoil data 

for the S809 airfoil taken from the OSU wind tunnel at a Reynolds number of 750,000 and 

extrapolated to account for deep stall effects. The deep stall characteristics are found using a 

utility program titled FoilCheck, which employs the Viterna-Corrigan equations of post stall. An 

aspect ratio of 11 is used during the application of these equations. All but the last “S809_CLN” 

airfoil data table files are additionally modified to account for stall delay effects. The stall delay 

characteristics are found using a procedure similar to that outlined in Du and Selig (1998). 
 95

Table 6.8: Airfoil Data Table File Descriptions 

Table

Identifier Filename File Description

1 cylinder.dat Dimensionless coefficient airfoil data for a 

smooth circular cylinder

2 S809_CLN_129.dat Dimensionless coefficient airfoil data for the 

S809 airfoil at a span of 12.9%

3 S809_CLN_189.dat Dimensionless coefficient airfoil data for the 

S809 airfoil at a span of 18.9%

4 S809_CLN_242.dat Dimensionless coefficient airfoil data for the 

S809 airfoil at a span of 24.2%

5 S809_CLN_298.dat Dimensionless coefficient airfoil data for the 

S809 airfoil at a span of 29.8%

6 S809_CLN_354.dat Dimensionless coefficient airfoil data for the 

S809 airfoil at a span of 35.4%

7 S809_CLN_410.dat Dimensionless coefficient airfoil data for the 

S809 airfoil at a span of 41.0%

8 S809_CLN_Outboard.dat Dimensionless coefficient airfoil data for the 

S809 airfoil at all other outboard span stations

Airfoil data tables are correlated to each blade element by prescribing an airfoil table identifier at 

the node between each element. Since 20 elements are chosen for integrating aerodynamic and 

elastic forces, there are 21 such nodes. The first is located at a distance along the blade axis from 

the center of rotation equal to that of the hub radius (taken here to be 0.432 m) and the last node 

is located at the blade tip. Since FAST_AD requires each blade element to have the same length, 

he nodal locations cannot be aligned with the exact span station for which each airfoil data table 

is derived. Therefore, the airfoil data tables are matched to the most well-suited node. 

Aerodynamic properties associated with each node are listed in Table 6.9. 
 96

Table 6.9: Nodal Aerodynamic Properties 

Prescribed

Distance Distance Airfoil Local

from Rotor Spanwise Along Blade Radial Data Chord Local Local

Node Center Fraction From Root (r) Fraction Table Length (c ) Twist (θ T ) Thickness

Number m (/ 5.029 m) m (r / 4.597 m) Identifier m deg m

1 0.432 0.086 0.000 0.000 1 0.218 0.00 1.000

2 0.662 0.132 0.230 0.050 2 0.209 0.00 1.000

3 0.892 0.177 0.460 0.100 3 0.283 3.66 0.707

4 1.122 0.223 0.690 0.150 4 0.526 12.78 0.351

5 1.351 0.269 0.919 0.200 4 0.727 17.89 0.209

6 1.581 0.314 1.149 0.250 5 0.704 13.33 0.209

7 1.811 0.360 1.379 0.300 6 0.680 10.19 0.209

8 2.041 0.406 1.609 0.350 7 0.657 7.24 0.209

9 2.271 0.452 1.839 0.400 8 0.634 5.52 0.209

10 2.501 0.497 2.069 0.450 8 0.611 3.81 0.209

11 2.731 0.543 2.299 0.500 8 0.588 2.81 0.209

12 2.960 0.589 2.528 0.550 8 0.565 1.86 0.209

13 3.190 0.634 2.758 0.600 8 0.542 1.21 0.209

14 3.420 0.680 2.988 0.650 8 0.518 0.66 0.209

15 3.650 0.726 3.218 0.700 8 0.495 0.21 0.209

16 3.880 0.771 3.448 0.750 8 0.472 -0.15 0.209

17 4.110 0.817 3.678 0.800 8 0.448 -0.51 0.209

18 4.339 0.863 3.907 0.850 8 0.425 -0.84 0.209

19 4.569 0.909 4.137 0.900 8 0.402 -1.17 0.209

20 4.799 0.954 4.367 0.950 8 0.378 -1.49 0.209

21 5.029 1.000 4.597 1.000 8 0.355 -1.82 0.209

Node Location

One of the final sets of parameters defined in the FAST_AD input file is a list of desired output 

channels. Of the numerous output channels available, those requested include instantaneous 

simulation clock time, nacelle yaw angle, wind direction, wind speed, Blade 1 azimuth angle 

position, Blade 1 out-of-plane tip deflection, Blade 1 pitch angle, rotor rotational speed, Blade 1 

out-of-plane and in-plane bending moments, low-speed shaft torque, and nacelle yaw moment. 

Additionally, detailed aerodynamic data, including instantaneous angle of attack, lift and drag 

coefficients, axial and tangential induction factors, differential thrust, and differential torque 

force30, are requested at several nodes between the blade elements. This is achieved by setting 

the “PRINT” flag at each node. Since the nodal locations cannot be aligned with the exact span 

stations where the pressure tap groupings are positioned (see discussion in the previous 

paragraph), the detailed instantaneous aerodynamic data are requested at the two nodes 

surrounding each pressure tap grouping span station unless the station is significantly closer to 

one of the two adjoining nodes, in which case, only the closest node is chosen. Comparing the 

pressure tap grouping span stations given in Table 5.10 with the node locations given in Table 

__________________ 

30 The differential thrust and differential torque force output channels are labeled as ForcN and ForcT respectively 

and are identified as differential normal and tangential forces acting on the blade element in the YawDyn User’s 

Guide by Hansen and Laino (1998). This identification is misleading since these forces are not perpendicular and 

parallel to the local chord line, which is the UAE’s measurement convention. Instead, these forces are defined to be 

normal and tangent to the plane of rotation, respectively, for blades with no coning, which are more commonly 

identified as thrust and torque forces. 
 97

6.9, the “PRINT” flag is seen to be most properly designated at nodes 5, 6, 9, 10, 13, 16, 17, and 

20. The 30% span pressure tap grouping is located 68.431% of the way between nodes 5 and 6, 

the 46.67% span pressure tap grouping is located 33.093% of the way between nodes 9 and 10, 

and the 80% span pressure tap grouping is located 62.407% of the way between nodes 16 and 17. 

The 63.33% span pressure tap grouping is aligned very closely with node 13 and the 95% span 

pressure tap grouping is aligned very closely with node 20. 

For each simulation run, these output channels yield time-series data on the loads and responses 

of the selected members of the wind turbine modeled. That is, at each time increment between 

the simulation time of 20 and 30 s, an instantaneous value of each output channel is listed. The 

time-series data associated with the aerodynamic parameters of the blade nodes are given in a 

separate file than the time-series data for the other output channels selected. After each 

simulation is run, these two output data files are concatenated into one file using an MS Excel 

spreadsheet. All cells are formatted to retain 6 decimal places to minimize any round-off errors. 

Next, the post-processing code Crunch is employed to use these data and develop the output 

channels and associated measurement conventions requested for the Blind Comparison exercise 

and to azimuth average the resulting time series data into 1 ̊ azimuth angle bins. The resulting 

data file, which contains 360 rows of azimuth averaged time series data of the 45 output channels 

requested for the Blind Comparison exercise, is finally imported into a reporting spreadsheet for 

ease in developing plots of the results. Unfortunately, the version of Crunch employed (2.60) 

does not allow for selection of bin centers, so the azimuth averaged values are offset by 0.5 ̊

from Blind Comparison output convention. 

The equations employed in Crunch to develop the output channels and associated measurement 

conventions requested for the Blind Comparison exercise are relatively simple. They are 

described qualitatively and quantitatively here, and are given in detail in the example Crunch 

input file of Appendix V for the test case S0700000. The Crunch input files for the other test 

cases are the same as this input file except for the last line: the name of the input file from which 

to read the time series data. 

The yaw error, YAW, is found by subtracting the wind direction (which is always zero) from the 

nacelle yaw position. The Blade 3 flap angle of the UAE wind turbine, B3FLAP, is found by 

summing the precone angle and the inverse tangent of the ratio of the Blade 1 out-of-plane tip 

deflection (of the FAST_AD turbine) and the rotor radius: 





 



 − − = +

R

Blade1Out Of PlaneTip Displacement B3FLAP β 1 atan (6.20) 

To orient the moments with respect to the tip chord line properly, the Blade 3 root flap and root 

edge bending moments of the UAE wind turbine, B3RFP and B3REP respectively, are found by 

applying the following vectorial relationships between the in-plane and out-of-plane bending 

moments outputted by FAST_AD: 
 98

( ) ( )

( ) ( ) D

D

In Plane Moment sin 3.0

B3RFP Out Of Plane Moment cos 3.0

− ×

= − − × +

 (6.21) 

and 

( ) ( )

( ) ( ) D

D

In Plane Moment cos 3.0

B3REP Out Of Plane Moment sin 3.0

− ×

= − − − × +

 (6.22) 

To ensure the proper sign convention, the nacelle yaw moment of the UAE wind turbine, NAYM, 

is found by multiplying the nacelle yaw moment output by FAST_AD by –1. In addition, all the 

torque and moment outputs provided by FAST_AD are multiplied by 1,000, since FAST_AD 

outputs moments in kNm and the unit convention for the Blind Comparison exercise is Nm. 

For the cases in which the nodes do not line up with the pressure tap grouping span locations, 

linear interpolations are made between the two adjoining nodes to determine the aerodynamic 

properties at the correct span locations. This is done to find the local dimensionless normal and 

tangential force coefficients, local normal and tangential forces per unit span, and local dynamic 

pressures at each of the five pressure tap grouping span stations. Since FAST_AD cannot 

compute pitching moments or dimensionless pitching moment coefficients, all these Blind 

Comparison Output channels are zeroed out. 

Dimensionless normal and tangential force coefficients are derived from the dimensionless lift 

and drag coefficients provided at each node via forms of Eqs. (2.43) and (2.44) (the forces in 

these equations are replaced with force coefficients). Normal and tangential forces per unit span 

are derived from the differential thrust and differential torque forces (dQ divided by r) through 

the inverses of Eqs. (2.57) and (2.58) divided by the blade element length, dr: 

( ) ( )



 



 



 



 = PT + PT sin

r

dQ dT cos

dr

1 Normal Force / Unit Span θ θ (6.23) 

and 

( ) ( )



 



 



 



 = − PT + PT cos

r

dQ dT sin

dr

1 Tangential Force / Unit Span θ θ (6.24) 

The local dynamic pressure at each node is backed out of the previous results by dividing the 

force per unit span by its associated dimensionless coefficient and the local chord length. Since 

this can be done with either the normal and tangential forces and associated dimensionless force 

coefficients, the average of the two is taken. 

6.2 Blind Comparison Results and Discussion 

After the FAST_AD input files(s) are fully developed, simulations are easy to run and output 

data are obtained in minutes. Careful and meaningful examination of these data is more difficult. 

A comparison and discussion of some load predictions made by FAST_AD and other Blind 
 99

Comparison participants and data obtained experimentally in the NASA-Ames wind tunnel are 

given here. Just as the Blind Comparison exercise opted to target specific aerodynamic 

phenomena, so too, this section focuses on particular aerodynamic events. Since a prudent study 

of the more complicated downwind and nonzero yaw error cases cannot be undertaken before 

discrepancies in the simplest test cases are first understood, significantly more attention is paid to 

the upwind rotor configuration, zero yaw error cases. Moreover, concentration is given to 

comparing and discussing aerodynamic properties, including dimensionless force coefficients, 

local dynamic pressures, and the like. To facilitate validation of FAST_AD, trends in 

discrepancies between the modelers’ predictions are noted and discrepancies between the 

modelers’ predictions and those measured during the NASA-Ames tests are highlighted. 

Potential sources of the discrepancies and their effects on the design of wind turbine systems are 

noted. Probable causes of discrepancies are discussed further in the next chapter. 

Average dimensionless normal force coefficients at each pressure tap grouping span station are 

plotted against wind speed on the abscissa in Figs. 6.2–6.6. Corresponding local dynamic 

pressures are given in Figs. 6.7–6.11. All these data represent measurements and code 

predictions for the upwind rotor configuration, zero yaw error test cases at wind speeds of 7, 10, 

13, 15, 20 and 25 m/s (S**00000-series Blind Comparison test cases). The curves labeled 

“NREL” represent the data measured during the NASA-Ames wind tunnel tests. The light 

curves labeled “Others” represent data predicted by other Blind Comparison participants who 

employed the AeroDyn aerodynamics subroutines. 

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

95% Span Normal Force Coefficient

Others

FAST_AD

NREL

Figure 6.2: 95% span normal force coefficient, upwind, zero yaw
 100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

80% Span Normal Force Coefficient

Others

FAST_AD

NREL

Figure 6.3: 80% span normal force coefficient, upwind, zero yaw

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

63% Span Normal Force Coefficient

Others

FAST_AD

NREL

Figure 6.4: 63% span normal force coefficient, upwind, zero yaw
 101

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

47% Span Normal Force Coefficient

Others

FAST_AD

NREL

Figure 6.5: 47% span normal force coefficient, upwind, zero yaw

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

30% Span Normal Force Coefficient

Others

FAST_AD

NREL

Figure 6.6: 30% span normal force coefficient, upwind, zero yaw
 102

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

95% Span Local Dynamic Pressure [Pa]

Others

FAST_AD

NREL

Figure 6.7: 95% span local dynamic pressure, upwind, zero yaw

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

80% Span Local Dynamic Pressure [Pa]

Others

FAST_AD

NREL

Figure 6.8: 80% span local dynamic pressure, upwind, zero yaw
 103

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

63% Span Local Dynamic Pressure [Pa]

Others

FAST_AD

NREL

Figure 6.9: 63% span local dynamic pressure, upwind, zero yaw

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

47% Span Local Dynamic Pressure [Pa]

Others

FAST_AD

NREL

Figure 6.10: 47% span local dynamic pressure, upwind, zero yaw
 104

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

30% Span Local Dynamic Pressure [Pa]

Others

FAST_AD

NREL

Figure 6.11: 30% span local dynamic pressure, upwind, zero yaw

All modelers tend to overpredict dimensionless normal aerodynamic force coefficients at the 

outermost span location (95% span). This reveals that Prandtl’s tip loss factor (see section 2.5) 

may not properly account for the reduction in aerodynamic forces near the tip. Proper 

characterization of aerodynamic forces along the outboard portions of a blade is critical to 

successfully design wind turbines, since most rotor torque is generated from these blade sections 

when operating at or below the rated wind speed of the turbine. 

In contrast, all modelers tend to underpredict dimensionless normal force coefficients at the 

innermost span location (30% span), suggesting that more work needs to be done to improve stall 

delay models and models that account for the three-dimensional effects at the blade root. Flow 

around this part of a wind turbine blade is very complex as a result of flow disturbances around 

the hub (and around the nacelle for downwind rotor configurations). Consequently, refining 

these models may be a difficult or impossible. Proper characterization of aerodynamic forces 

along the inboard parts of a blade is especially critical when analyzing wind turbines that operate 

at high wind speeds (above rated speed) since the outboard parts of the blade are in deep stall and 

most rotor torque is generated from the innermost blade sections because of stall delay. 

Most modelers also underpredict dimensionless normal force coefficients at the midboard span 

locations. The cause of these discrepancies is most likely a combination of the discrepancy 

causes previously discussed. 

All modelers predicted local dynamic pressures accurately at all span locations and wind speeds 

for the upwind, zero yaw error test cases. As a result, the local normal aerodynamic force curves 

(not shown) follow similar trends to the curves of the dimensionless normal force coefficients. 
 105

FAST_AD predicts local dynamic pressures more accurately than the other codes in almost 

every instance. 

From a global standpoint, data scatter between load predictions tends to increase with wind 

speed. This indicates that post-stall models, such as the commonly used Viterna-Corrigan model 

(see section 2.4), are deficient. Such models will have to be refined if they are to be useful in the 

development of stall-regulated wind turbines. A significant fraction of a stall-controlled wind 

turbine blade operates in the post-stall regime throughout all but the smallest wind-inflows. 

Another probable cause of increasing data scatter with increasing wind speed is that the drag 

coefficient is neglected when iterating to find the operational state of the wind turbine in the 

AeroDyn aerodynamics subroutines. Since the UAE research wind turbine is a stall-regulated 

machine and there is significant drag in the post-stall regime, drag is not negligible, and 

AeroDyn is limited in its ability to properly model these type of operational states (see section 

4.2). AeroDyn Code developers should consider revising the aerodynamics models in future 

versions of the code by having them include the effects of drag when iterating to find the 

operational state. That is, the code should be modified to use a form of Eq. (2.70) instead of a 

form of Eq. (2.73). Aerodynamic load predictions in the post stall regime will undoubtedly be 

improved by this action. 

To examine the collective effect of the aerodynamic force discrepancies, plots of low-speed shaft 

torque and root flap bending moments for the same six cases are given in Figs. 6.12 and 6.13: 

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

2,000

5 10 15 20 25

Wind Speed [m/s]

Low Speed Shaft Torque [Nm]

Others

FAST_AD

NREL

Figure 6.12: Low-speed shaft torque, upwind, zero yaw
 106

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

5 10 15 20 25

Wind Speed [m/s]

Root Flap Bending Moment [Nm]

Others

FAST_AD

NREL

Figure 6.13: Root flap bending moment, upwind, zero yaw

Modeling tool predictions of the low-speed shaft torque are 10%–170% of their experimentally 

measured counterparts. Predictions of the root flap bending moment are 70%–160% of the 

measured values. In many instances, overpredictions of local aerodynamic forces on some parts 

of the blade and underpredictions of local aerodynamic forces on others most likely mitigated the 

discrepancies in these torque and moment predictions. However, from a structural standpoint, 

optimal blades cannot be designed properly if the assumed load distribution is erroneous. 

Like the aerodynamic force predictions, torque and moment prediction discrepancies tend to 

increase with wind speed; the causes of the discrepancies in the former predictions are most 

likely the causes for the discrepancies in the latter. Nevertheless, as demonstrated for the 

FAST_AD model in Chapter 7, even complete eliminating aerodynamic force discrepancies does 

not eliminate discrepancies in the torque and moment predictions. 

Azimuth averaged dimensionless normal force coefficients at 47% span are plotted against the 

azimuth angle on the abscissa for the downwind rotor configuration, zero yaw error, 7 and 17 

m/s test cases (E**00000A-series) in Figs. 6.14 and 6.15. These plots highlight deficiencies in 

the tower shadow model employed in the AeroDyn aerodynamics subroutines. At a wind speed 

of 7 m/s, all modelers apparently underpredicted the tower shadow width. In this case, some 

modelers apparently overpredicted and others underpredicted the velocity deficit. At a wind 

speed of 17 m/s, all modelers properly estimated the tower shadow width, but overpredicted all 

the velocity deficits. In either case, more work is needed to improve tower shadow models and 

the methods of predicting tower shadow features. This is crucial for the successful design of 

downwind turbines. 
 107

0.0

0.2

0.4

0.6

0.8

1.0

1.2

130 140 150 160 170 180 190 200 210 220 230

Blade Azimuth Angle [deg]

47% Span Normal Force Coefficient

Others

FAST_AD

NREL

Figure 6.14: 47% span normal force coefficient, 7 m/s, downwind, zero yaw

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

130 140 150 160 170 180 190 200 210 220 230

Blade Azimuth Angle [deg]

47% Span Normal Force Coefficient

Others

FAST_AD

NREL

Figure 6.15: 47% span normal force coefficient, 17 m/s, downwind, zero yaw

Azimuth averaged dimensionless normal force coefficients and local dynamic pressures for 

several span stations are plotted against the azimuth angle on the abscissa for the upwind rotor 
 108

configuration, 30 ̊ yaw error for the 15 m/s test case (S1500300) in Figs. 6.16–6.21. Though the 

causes of discrepancies in the aerodynamic forces all apply in this test case, these plots also 

highlight deficiencies in the Beddoes-Leishman dynamic stall model. In general, predictions of 

the dynamic pressure distribution fared quite well. However, many features of the dimensionless 

normal force coefficients measured in the NASA-Ames wind tunnel tests are not modeled and 

predicted by the codes. In most cases, aerodynamic loads are estimated more accurately in the 

operating states below static stall. Since all wind turbines operating in the field will frequently 

experience significant yaw errors because of wind gusts and changing wind directions, these 

comparisons underscore the difficulty in accurately predicting extreme load cases. 

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 60 120 180 240 300 360

Blade Azimuth Angle [deg]

95% Span Normal Force Coefficient

Others

FAST_AD

NREL

Above Static 

Stall

Above Static 

Stall

Below Static Stall

Figure 6.16: 95% span normal force coefficient, 15 m/s, upwind, 30 ̊ yaw
 109

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 60 120 180 240 300 360

Blade Azimuth Angle [deg]

63% Span Normal Force Coefficient

Others

FAST_AD

NREL

Above Static Stall Below Static Stall Above Static Stall

Figure 6.17: 63% span normal force coefficient, 15 m/s, upwind, 30 ̊ yaw

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 60 120 180 240 300 360

Blade Azimuth Angle [deg]

30% Span Normal Force Coefficient

Others

FAST_AD

NREL

Below Static 

Stall

Above Static Stall Above Static Stall

Figure 6.18: 30% span normal force coefficient, 15 m/s, upwind, 30 ̊ yaw
 110

0

200

400

600

800

1,000

1,200

1,400

1,600

0 60 120 180 240 300 360

Blade Azimuth Angle [deg]

95% Span Local Dynamic Pressure [Pa]

Others

FAST_AD

NREL

Above Static 

Stall

Above Static 

Stall

Below Static Stall

Figure 6.19: 95% span local dynamic pressure, 15 m/s, upwind, 30 ̊ yaw

0

100

200

300

400

500

600

700

800

0 60 120 180 240 300 360

Blade Azimuth Angle [deg]

63% Span Local Dynamic Pressure [Pa]

Others

FAST_AD

NREL

Above Static Stall Below Static Stall Above Static Stall

Figure 6.20: 63% span local dynamic pressure, 15 m/s, upwind, 30 ̊ yaw
 111

0

50

100

150

200

250

300

350

0 60 120 180 240 300 360

Blade Azimuth Angle [deg]

30% Span Local Dynamic Pressure [Pa]

Others

FAST_AD

NREL

Above Static Stall Above Static Stall Below Static 

Stall

Figure 6.21: 30% span local dynamic pressure, 15 m/s, upwind, 30 ̊ yaw

No individual code predicted experimentally measured load and performance data consistently, 

or for several of the test cases or particular span stations. Discrepancies between the code 

predictions and the experimental test data should not be faulted on inaccuracies in the test data, 

since they are precise and repeatable (see section 5.2) and since these data often represent the 

average of the modelers’ predictions. For the zero yaw error cases, load discrepancies also 

should not be faulted on the averaging that is done to obtain the load data for each wind speed. 

This is because the load spectra across azimuth angle orientation are fairly flat, especially for the 

low wind speed test cases (see Fig. 6.22). If there were tremendous jumps or spikes in the data, 

this last conclusion would be plausible. 
 112

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

0 60 120 180 240 300 360

Blade Azimuth Angle [deg]

Root Flap Bendign Moment [Nm]

FAST_AD

NREL

25 m/s

20 m/s

15 m/s

10 m/s

7 m/s

Figure 6.22: Root flap bending moment, upwind, zero yaw

In most instances, FAST_AD’s code predictions also represent the average of the other 

modelers’ predictions, revealing that the FAST_AD design code is comparable, if not somewhat 

better, at predicting loads and performances than other modeling tools that employ similar 

aerodynamics models. However, the accuracy of a modeling tool’s predictions depends as much 

on how they are used (the modelers’ tactics) as on the accuracy of the fundamental models and 

theories they employ. Most discrepancies between the load and performance predictions 

between the modelers are likely the results of choices made on how to use the turbine 

specifications outlined in section 5.1. Blind Comparison participants theorized, and the author 

agrees, that the choice of which aerodynamics data are used and the method, if any, of 

extrapolating these data to account for the effects of stall delay and deep stall explain for the 

most part large discrepancies between the aerodynamic force coefficient predictions and 

experimentally measured values. This last point will be demonstrated at the end of this chapter. 

The question of whether these large aerodynamic force coefficient discrepancies explain the 

large discrepancies in low-speed shaft torque and blade root flap bending moments is addressed 

in Chapter 7. 

Other reasons for discrepancies between load and performance predictions between the modelers 

include the possibility of modeler error or code bugs. We did not scrutinize the work done by 

other modelers to determine what all these errors may be; however, the Blind Comparison 

documentation revealed several input deck inaccuracies. For instance, one Blind Comparison 

participant misinterpreted the definition of the ForcN and ForcT output channels of the blade 

element aerodynamic predictions. This participant erroneously assumed that these aerodynamic 

forces represent the differential forces normal and tangent to the element’s chord line. In fact, 

these forces in AeroDyn represent the differential thrust and torque forces for blades that have no 

coning (see footnote 30 in section 6.1). This error clearly affects this modeler’s predictions of 

the normal and tangential forces at each pressure tap grouping span station; it also subtly 
 113

propagates into the predictions of the local dynamic pressures since the values of these 

parameters must be backed out of the available output channel data. Another Blind Comparison 

participant chose not to linearly interpolate between the two nodes surrounding the correct span 

stations where the pressure tap groupings are located and instead chose to use the closest 

possible node. The span station used for output aerodynamic data is off by more than 2% in one 

instance. Since the operational regimes and resulting aerodynamic properties depend strongly on 

the blade span location, inaccurate positioning of span stations where the aerodynamic properties 

are recorded is a source of considerable load and performance prediction discrepancy. 

Known inaccuracies with the input parameters and models that are coupled to the simulations 

performed with FAST_AD, other than those already mentioned, include: 

(1) The six inboard airfoil data tables that attempt to account for the effects of stall delay are 

not lined up to the exact span stations for which each airfoil data table is derived. 

(2) In the pre-processing of the turbine specifications, numerous linear interpolations are made 

to obtain parameters necessary for the main FAST_AD input file. Examples of parameters 

that involve linear interpolations include linear interpolations to obtain chord, twist, and 

thickness distribution parameters. No effort is made to determine whether an enhanced 

curve fitting technique is applicable. 

(3) In the post-processing of the output blade element data, numerous linear interpolations are 

made to determine aerodynamic properties at the proper pressure tap grouping span 

locations. Again, no effort is made to determine whether an enhanced curve fitting 

technique is applicable. 

(4) Several assumptions and approximations are made when estimating the structural 

properties of the blades and tower and their corresponding natural mode shapes. 

(5) Errors in the structural models employed by FAST_AD code (see Chapters 3 and 4) were 

not fixed, or even discovered, before the simulations were run. 

The consequences of these inaccuracies should be evident from the previous discussion and in 

the analysis that follows here and in Chapter 7. 

To determine whether most aerodynamic load discrepancies are the direct results of airfoil data 

table selection and extrapolation, the following analysis is performed. This analysis also 

highlights the weaknesses in the standard BEM model. 

Figure 6.23 demonstrates a graphical BEM theory hand computation of the actual aerodynamic 

operating conditions at each pressure tap grouping span location for the upwind rotor 

configuration, 7 m/s wind speed, zero yaw error (S0700000) test case. The curves labeled 

“FAST_AD Airfoil Data” illustrate the different dimensionless lift coefficient airfoil data used 

by FAST_AD at all blade elements. (The curves closest to the top of the plot represent the 

airfoil data at the innermost span stations caused by stall delay.) The curves labeled “Other 

Airfoil Data” illustrate all the other dimensionless lift coefficient airfoil data used by the other 

Blind Comparison participants who employed the AeroDyn aerodynamics subroutines in their 

Blind Comparison work. The curves labeled “BEM Equations” depict the feasible lift coefficient 

and angle of attack relations [Eq. (2.73)] at each pressure tap grouping span station as marked. 

The diamonds labeled “NREL 7 m/s” depict the actual aerodynamic operating conditions (lift 
 114

coefficient and angle of attack pairs) at the five span stations measured during the NASA-Ames 

wind tunnel tests. These angle of attack data were recorded in a different NASA-Ames test than 

the dimensionless lift coefficient data (specifically, H0700000 instead of S0700000). The angle 

of attack probes, which were used to measure the angle of attack values, were not installed on 

Blade 3 during the S-series tests so the probes would not disturb the flow around the blade and 

the resulting measurements of pressure and associated aerodynamic forces. Figure 6.24 gives a 

similar illustration for the upwind rotor configuration, 10 m/s wind speed, zero yaw error 

(S1000000) test case. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Angle of Attack [deg]

Lift Force Coefficient

Other Airfoil Data

FAST_AD Airfoil Data

BEM Equations

NREL 7 m/s

95% Span

80% Span

63% Span

47% Span

30% Span

95% Span

80% Span

63% Span

30% Span

47% Span

Figure 6.23: BEM theory hand computation, 7 m/s, upwind, zero yaw
 115

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Angle of Attack [deg]

Lift Force Coefficient

Other Airfoil Data

FAST_AD Airfoil Data

BEM Equations

NREL 10 m/s

95% Span

80% Span 63% Span 47% Span

30% Span

95% Span

80% Span

63% Span

47% Span

30% Span

Figure 6.24: BEM theory hand computation, 10 m/s, upwind, zero yaw

As stated in section 2.5, according to BEM theory, the actual aerodynamic operating point (lift 

coefficient and angle of attack pair) at each span station is the intersection point between the 

associated BEM curve and the CL versus α curve characterizing the aerodynamic properties of 

the given airfoil. These BEM curves are independent of the blades’ aerodynamic coefficients, 

and are thus equally valid for every dimensionless lift coefficient curve plotted in the figures. 

Several conclusions may be drawn from these plots. First, the correlation between airfoil data 

selection spread and predicted load discrepancies is clear. The possible range of dimensionless 

lift coefficient values that could be predicted at a particular span station and wind speed using 

BEM theory is equal to the range of dimensionless lift coefficient airfoil data values that 

intersect the BEM curve at that particular span station and wind speed. For example, 

dimensionless lift coefficient airfoil data are 0.82–1.00 for the pressure tap grouping located at 

63.33% span at a wind speed of 7 m/s (see Fig. 6.23). At this same span station and wind speed, 

the actual dimensionless normal force coefficient predictions were 0.82–1.04 (see Fig. 6.4). 

Since the normal and lift force coefficients are nearly equivalent for situations in which the angle 

of attack is small (as is the case here) [see Eq. (2.43)], the possible range of lift coefficient values 

almost entirely encapsulates the actual range predicted. 

The fact that this actual range is not entirely encapsulated by the “possible” range is explainable 

by some of the inaccuracies previously discussed. For instance, when a span station used for 

aerodynamic load data recording is positioned imprecisely in a modeling tool, the BEM curve 

characterizing the aerodynamic operational state of that station is shifted from its corresponding 

nominal position. This in essence widens the range of possible dimensionless lift coefficient 
 116

values that could be predicted at the associated nominal span station. The inaccuracy associated 

with linearly interpolating between adjoining span stations to determine the aerodynamic 

properties of a given pressure tap grouping at the proper span station is another explanation why 

the range of possible dimensionless lift coefficient values could be larger—even large enough to 

encapsulate the actual range predicted. 

A summary of “possible” and actual ranges of dimensionless lift coefficient value predictions at 

the five pressure tap grouping span stations and upwind configuration, zero yaw error, 7 and 10 

m/s wind speed test cases is given in Table 6.10. The table shows that the likelihood the 

“possible” range encapsulates the actual range increases with increased wind speed, especially at 

the middle and innermost span stations. 

Table 6.10: “Possible” and Actual Lift Coefficient Value Prediction Ranges 

Wind

Speed

m/s Min Max Min Max Min Max Min Max Min Max

Possible 0.60 - 0.70 0.76 - 0.88 0.82 - 1.00 0.86 - 1.07 0.86 - 1.06

Actual 0.10 - 0.92 0.66 - 0.94 0.82 - 1.04 0.86 - 1.08 0.86 - 1.09

Possible 0.85 - 1.08 0.88 - 1.18 0.90 - 1.36 0.62 - 1.54 0.56 - 1.60

Actual 0.40 - 1.05 0.90 - 1.08 0.85 - 1.14 0.74 - 1.01 0.77 - 1.28

63.33% Span 46.67% Span 30.00% Span

Blade Radial Location

7

10

95.00% Span 80.00% Span

The hand computations of these two test cases show why nearly everyone overpredicted normal 

force coefficients at the 95% span location. Figs. 6.23 and 6.24 show that the actual measured 

CL–α aerodynamic operating points at this span station do not even lie within the range of used 

airfoil data. This indicates that three-dimensional flow effects prohibit the use of unmodified 

two-dimensional airfoil data at the outboard span stations. Moreover, Chapter 7 shows that no 

pressure tap grouping span stations exhibited the same two-dimensional aerodynamic 

characteristics measured in the CSU, OSU, and DUT wind tunnels. This reveals the flaws in 

assumptions (1) and (2) of blade element theory (see section 2.4). That is, the pressure 

distribution across a wind turbine blade during normal operation most likely causes threedimensional flow effects, such as spanwise flow, flow circulation, and vorticity. The nature of 

this flow has strong effect on the CL–α characteristics of each blade element when contrasted 

with measurements taken in a two-dimensional flow environment (wind tunnel measurements 

made on static airfoils). If airfoil data used for modeling purposes are to be obtained from wind 

tunnel tests on static airfoils, new models that can extrapolate two-dimensional airfoil data to 

account for these three-dimensional flow effects are needed across the entire blade span. The 

only popular models that are used to extrapolate two-dimensional airfoil data to account for 

three-dimensional flow effects are stall delay models for the inboard span stations. 

A final conclusion drawn from the hand computations is that modeling tool predictions of the 

angles of attack at the various span stations are erroneous in a noteworthy number of situations. 

The misprediction of angle of attack is especially critical around the linear portion of the CL–α

curve and around stall. The slopes of the CL–α curve in these operating regimes are considerably 

steeper than in other operating regimes, and thus, a slight misprediction in angle of attack can 

easily lead to a drastic misprediction in aerodynamic properties. The effects of this detail will be 

clearly seen in the sensitivity study of the next chapter. 
 117

FAST_AD simulations use the dynamic inflow model instead of the standard BEM model. 

Nevertheless, these last few conclusions remain entirely valid, since the two theories predict 

nearly identical load and performance values for zero yaw error test cases—cases where dynamic 

stall events are minimal (Suzuki and Hansen 1998). This point is demonstrated further in the 

sensitivity study of Chapter 7. Several Blind Comparison participants who used the AeroDyn 

aerodynamics subroutines also used the dynamic inflow model during their simulations. 
 118

7. Sensitivity Study Input Parameters and Results 

Several causes of discrepancies between modeling tool load predictions are accounted for in 

Chapter 6 by taking into consideration some inconsistencies in the various input files. 

Additionally, weaknesses in the commonly used tip loss, stall delay, post stall, and other 

frequently used aerodynamic models are highlighted in Chapter 6 by examining discrepancies 

between load predictions and measured load values taken during the NASA-Ames wind tunnel 

tests. To further account for the many load prediction and measured load discrepancies observed 

in the Blind Comparison results, a sensitivity analysis is performed and discussed in this chapter. 

The sensitivity study is guided by the following questions: 

(1) What are some factors that did not contribute to the load prediction discrepancies? 

(2) What are the critical parameters that affect wind turbine load and performance prediction 

accuracy and could they have had an impact on the Blind Comparison outcome? 

(3) Do the discrepancies in predicted and measured aerodynamic force coefficient values 

account for the large discrepancies in low-speed shaft torque and blade root flap bending 

moments? If not, what are other causes and what can be done to fix them? 

(4) To have the greatest impact on wind turbine design improvement and wind-generated 

electricity cost reduction, where do ongoing wind turbine research programs need to 

concentrate? 

To address these questions, a sample of the FAST_AD model inputs is systematically perturbed 

to determine its effect on load and performance predictions. Unfortunately, time prohibits the 

study of the effects of all realizable input parameter perturbations; focus is given to aerodynamic 

parameters, since aerodynamic forces are the key factors that determine wind turbine component 

loads. Moreover, since the dynamical events associated with complicated operational states 

cannot be studied before discrepancies in the simplest test cases are first understood, attention is 

given solely to upwind rotor configuration, zero yaw error test cases. Descriptions and reasons 

for each sensitivity analysis performed and the FAST_AD input file parameter alterations needed 

to perform these analyses are given in section 7.1. (The steps taken to develop the original 

FAST_AD input file parameters are outlined in section 6.1.) The results of each sensitivity 

analysis are compared and discussed in section 7.2. Only results that are above and beyond those 

established in Chapter 6 are discussed. Questions 1–4 are addressed wherever applicable. 

7.1 Sensitivity Studies and FAST_AD Input Parameter Alterations 

Eleven sensitivity analyses are chosen and simulated in FAST_AD for the six upwind 

configuration, zero yaw error, 7, 10, 13, 15, 20, and 25 m/s (S**00000-series) test cases. The 

names of each study and the differences between their input file(s) and the original file(s) used 

for the for Blind Comparison exercise are summarized in Table 7.1. 
 119

Table 7.1: Sensitivity Study Input Parameter Alterations for FAST_AD 

Study Name Variable Description From To

IZD40 IZD(4) Azimuth angle, rotor side switch 1 - Induction generator 0 - Constant speed

DT001 DT Time increment 0.002 second 0.001 second

NR Number of blade elements 20 - maximum allowable 10

NFOIL Airfoil table ID numbers distributed across blade

THETA3 THETA Blade collective pitch angle 4.815 deg - 3.0 deg pitch at tip 3.0 deg - 1.185 deg pitch at tip

BETA(1 ) Blade 1 coning angle

BETA(2 ) Blade 2 coning angle

RHO10PL RHO Air density Exact NASA-Ames test values 10% less than these values

μ B Distributed lineal density of blade

EI Distributed flapwise stiffness across blade 

EI Distributed edgewise stiffness across blade 

PC(*,* ) Blade mode shape const. proportionality coef.s

NegTHETAS θ S Distributed structural pretwist across blade Properties of the tapered, twisted, UAE 

blade Negative of these values

NUMFOIL Number of different airfoil tables 8 6

NUMFOIL Number of different airfoil tables 8 6

θ T Distributed aerodynamic pretwist across blade 

θ S Distributed structural pretwist across blade 

Alteration(s) Made

NR10 Most well suited node given the 

number of elements

Most well suited node given the 

PRINT / NOPRINT flags - Distributed across blade number of elements

EQUIL Dynamic inflow model switch

Input Parameter(s) Modified

BETA34 0.0 deg - No preconing 3.4 deg - same as downwind 

configuration test cases

PhaseVModeShapes Properties of the tapered, twisted, UAE 

blade

Properties of the constant-chord, 

highly twisted blade used during 

Phase V testing

NASAAmesAirfoilData

NASAAmesAlpha

Properties of the tapered, twisted, UAE 

blade

Adjusted at each span location 

by the average discrepancy

OSU data at Re = 750,000 extrapolated 

for stall delay at 12.9%, 18.5%, 24.2%, 

29.8%, 35.4%, and 41.0% span

Airfoil table ID numbers distributed across blade Most well suited node given the 

number of elements

NFOIL Most well suited node given the 

number of elements

DYNIN - Dynamic inflow model EQUIL - Equilibrium, standard 

BEM model

OSU data at Re = 750,000 extrapolated 

for stall delay at 12.9%, 18.5%, 24.2%, 

29.8%, 35.4%, and 41.0% span

Effective NASA-Ames test 2D 

airfoil data at 30.0%, 46.7%, 

63.3%, 80.0%, and 95.0% span

Effective NASA-Ames test 2D 

airfoil data at 30.0%, 46.7%, 

63.3%, 80.0%, and 95.0% span

Most well suited node given the 

number of elements

Airfoil data table files

NFOIL

Airfoil table ID numbers distributed across blade Most well suited node given the 

number of elements

Airfoil data table files

The sensitivity study titled “EQUIL” is performed to demonstrate the equivalency of the 

dynamic inflow and classic BEM models when simulating zero yaw error test scenarios 

(operating scenarios in which dynamic stall events are minimal). The standard BEM model is 

selected by designating the “EQUIL” flag for the dynamic inflow model switch parameter 

instead of the “DYNIN” flag in the main FAST_AD input file. 

The modelers who used the AeroDyn aerodynamics subroutines during their Blind Comparison 

work employed several induction generator models (including constant speed or no generator 

model). To verify whether the induction generator model employed by FAST_AD has any 

impact on the resulting load predictions, the sensitivity study titled “IZD40” is carried out. In 

these simulations, the rotor is modeled with a constant speed instead of as an induction generator 

with a fixed linear generator constant. 

To determine whether the time step used in FAST_AD for the Blind Comparison work is too 

large and affected the load predictions, the test cases are rerun with a time step half the size of 

the original (0.001 instead of 0.002 s). This is done in the sensitivity study titled “DT001.” 

Of similar merit, the accuracy of the model to the number of blade elements for integrating 

aerodynamic and elastic forces is studied in the analysis titled “NR10.” This study chooses half 

as many blade elements as were chosen originally (10 instead of the maximum of 20). 

Aerodynamic characteristics associated with each of the 11 new blade nodes are listed in Table 

7.2. This study also examines the sensitivity of the load predictions to the linear interpolations 
 120

that are performed to determine the aerodynamic properties at the correct pressure tap grouping 

span locations. Detailed instantaneous aerodynamic data for determining these aerodynamic 

properties are requested at nodes 3, 4, 5, 7, 9, 10, and 11. In addition to the input file alterations 

listed in Table 7.1, the input file for the post-processing code Crunch had to be extensively 

modified to accommodate these changes. 

Table 7.2: Nodal Aerodynamic Properties for Sensitivity Study “NR10” 

Prescribed

Distance Distance Airfoil Local

from Rotor Spanwise Along Blade Radial Data Chord Local Local

Node Center Fraction From Root (r) Fraction Table Length (c ) Twist (θ T ) Thickness

Number m (/ 5.029 m) m (r / 4.597 m) Identifier m deg m

1 0.432 0.086 0.000 0.000 1 0.218 0.00 1.000

2 0.892 0.177 0.460 0.100 3 0.283 3.66 0.707

3 1.351 0.269 0.919 0.200 4 0.727 17.89 0.209

4 1.811 0.360 1.379 0.300 6 0.680 10.19 0.209

5 2.271 0.452 1.839 0.400 8 0.634 5.52 0.209

6 2.731 0.543 2.299 0.500 8 0.588 2.81 0.209

7 3.190 0.634 2.758 0.600 8 0.542 1.21 0.209

8 3.650 0.726 3.218 0.700 8 0.495 0.21 0.209

9 4.110 0.817 3.678 0.800 8 0.448 -0.51 0.209

10 4.569 0.909 4.137 0.900 8 0.402 -1.17 0.209

11 5.029 1.000 4.597 1.000 8 0.355 -1.82 0.209

Node Location

Defining the blade collective pitch angle is problematic. Many wind turbine analysts believe it 

should be defined with respect to the local aerodynamic twist at 75% blade span; others believe it 

should defined with respect to the zero twist location (which may or may not be at 75% blade 

span); still others believe it should defined with respect to the blade tip. Several Blind 

Comparison participants may have used 3 ̊ instead of 4.815 ̊ for the blade collective pitch angle 

at zero aerodynamic twist (3 ̊ is the actual angle between the tip chord line and rotor plane, 

regarded as the blade pitch angle for all NASA-Ames test cases chosen for the Blind Comparison 

exercise). The sensitivity study titled “THETA3” shows how this error will propagate through 

the simulated load predictions. To do this, the FAST_AD input parameter THETA is set fixed at 

3 ̊ instead of 4.815 ̊, corresponding to an angle between the tip chord line and rotor plane of 

1.185 ̊ (an angle of 1.815 ̊ smaller than it should be). This study additionally highlights the 

sensitivity of the model to blade torsional motion since aerodynamic responses to any significant 

torsional motion should be at least the same order of magnitude as the responses to rather small 

pitch angle perturbations. In addition to the input file alteration listed in Table 7.1, the input file 

for the post-processing code Crunch had to be modified extensively to accommodate this change. 

Similar to “THETA3,” the sensitivity study titled “BETA34” is performed to demonstrate how 

sensitive the aerodynamic models are to coning angle perturbations and how an error made in the 

blade coning angles affects the load predictions. In this study, the coning angle of each blade for 

the upwind rotor configuration is fixed at 3.4 ̊, the coning angle used for the downwind rotor 

configuration test cases. (Downwind preconing for upwind rotor configurations in FAST_AD is 

modeled using positive precone angles.) Though it is doubtful that any Blind Comparison 

participant made this error, this study additionally demonstrates the sensitivity of the model to 

blade flapping motion, since aerodynamic responses to any significant flapping motion should be 

at least the same order of magnitude as the responses to rather small precone angle perturbations. 
 121

The sensitivity analysis titled “RHO10PL” demonstrates how an underprediction in air density is 

reflected in an underprediction of the potential loads. This is done by reducing the exact NASAAmes test values by 10% for each of the six test cases considered for this study. This study has 

less to do with load prediction discrepancy explanations and more to do with wind turbine design 

considerations. For example, it is important to consider how the potential loads may change if a 

wind turbine designed reflecting conditions seen at relatively high altitudes (say Colorado) is 

installed at a lower altitude (say sea level). 

Since several assumptions and approximations were made to estimate the structural properties of 

the blades of the UAE used during Phase VI (see section 6.1) a sensitivity analysis is performed 

to determine whether any structural property inaccuracies contributed to any of the load 

prediction inaccuracies. To accomplish this, structural properties of the UAE Phase V blades are 

input into FAST_AD in the sensitivity study titled “PhaseVModeShapes.” The structural 

properties and mode shapes of the Phase V blades are considerably different than those used for 

the Phase VI testing. If differences in the load predictions are minor, inaccuracies in the 

structural properties had a negligible impact on load prediction inaccuracies. If not, inaccuracies 

in the blade structural properties probably played a role in the load prediction inaccuracies. A 

summary of the calibrated distributed structural properties of the UAE Phase V blades is given in 

Table 6.4. To obtain the proper mode shapes needed for the FAST_AD input file (local flapwise 

and edgewise blade mode shapes instead of out-of-plane and in-plane mode shapes) the 

distributed structural pretwist is zeroed out in the Modes input file and Modes is run again. 

Centrifugal stiffening effects are accounted for in this process by specifying a typical rotor 

rotational speed of 72 rpm. A summary of the mode shapes is given in Table 7.3. 

Table 7.3: Mode Shapes of the UAE Phase V Blades 

1st Edgewise

Shape Shape 1st Mode Const. 2nd Mode Const. Mode Const.

Function Function Proportionality Proportionality Proportionality

Number (b ) (φ b 's) Coef.s (C1,b 's) Coef.s (C 2,b 's) Coef.s (C1,b 's)

2 [r/(R-R H )]2 0.225 -0.795 0.931

3 [r/(R-R H )]3 0.155 -10.547 0.569

4 [r/(R-R H )]4 4.728 17.341 0.057

5 [r/(R-R H )]5 -6.612 0.487 -1.094

6 [r/(R-R H )]6 2.503 -5.485 0.536

Flapwise Direction

The sensitivity study titled “NegTHETAS” is performed to determine how a properly oriented 

structural pretwist affects the resulting load predictions. That is, this study counteracts the error 

in the structural model indicated in footnote 9 of Chapter 3 by reversing the sign of each 

distributed structural pretwist entry in the main FAST_AD input file. 

As mentioned in Chapter 6, none of the pressure tap grouping span stations exhibit the same twodimensional aerodynamic characteristics measured in the CSU, OSU, and DUT wind tunnels. 

The validity of this statement is seen in the data that follow. The aerodynamic force coefficient-
 122

angle of attack relationships at each pressure tap grouping span station (as witnessed in the threedimensional environment of the NASA-Ames wind tunnel tests) are given in Tables 7.4–7.8 and 

are graphically illustrated in Figs. 7.1–7.4. That is, these data (labeled “NREL **.*%” in the 

figures) reflect the operational environment of the wind turbine; the wind turbine was operating 

when each data point was recorded. Each data point represents the average force coefficient and 

angle of attack pairs measured at a single wind speed for the upwind configuration, zero yaw 

error test cases. (The data points associated with the smallest angles of attack correspond to data 

measured during the smallest wind speed test cases.) The angle of attack data were recorded in 

different NASA-Ames tests than the dimensionless lift coefficient data (specifically, H**00000-

series instead of S**00000-series). To prevent the angle of attack probes, which were used to 

measure the angle of attack values, from disturbing the flow around the blade and the resulting 

measurements of pressure and associated aerodynamic forces, these probes were not installed on 

Blade 3 during the S-series tests. 

Table 7.4: NREL S809 Airfoil Data from NASA-Ames Tests: 95% Span 

Wind Angle of Dim.less Dim.less Dim.less Dim.less

Speed Attack (α ) Normal Tangential Lift Coeff. Drag Coeff.

m/s deg Coeff. (C N ) Coeff. (CT ) (CL ) (C D )

5 2.6 0.3467 0.0059 0.3466 0.0098

6 4.0 0.4268 0.0221 0.4273 0.0077

7 5.3 0.5065 0.0414 0.5082 0.0056

8 6.8 0.6200 0.0624 0.6230 0.0114

9 8.4 0.6870 0.0811 0.6915 0.0201

10 9.9 0.7311 0.0993 0.7373 0.0279

11 11.3 0.7430 0.1145 0.7510 0.0333

12 12.8 0.7592 0.1325 0.7697 0.0390

13 13.9 0.7969 0.1463 0.8087 0.0494

14 15.1 0.8104 0.1492 0.8213 0.0671

15 16.1 0.7849 0.1391 0.7927 0.0840

16 17.7 0.7550 0.1012 0.7500 0.1331

17 18.9 0.7282 0.0458 0.7038 0.1925

18 20.1 0.7386 -0.0112 0.6898 0.2643

19 21.4 0.7597 -0.0233 0.6988 0.2989

20 22.5 0.7147 -0.0565 0.6387 0.3257

21 23.9 0.6499 -0.0621 0.5690 0.3201

22 24.9 0.6298 -0.0622 0.5451 0.3216

23 26.0 0.6319 -0.0627 0.5405 0.3334

24 26.9 0.6077 -0.0615 0.5141 0.3298

25 27.8 0.6061 -0.0616 0.5074 0.3372
 123

Table 7.5: NREL S809 Airfoil Data from NASA-Ames Tests: 80% Span 

Wind Angle of Dim.less Dim.less Dim.less Dim.less

Speed Attack (α ) Normal Tangential Lift Coeff. Drag Coeff.

m/s deg Coeff. (C N ) Coeff. (CT ) (CL ) (C D )

5 2.6 0.4787 0.0167 0.4790 0.0050

6 4.1 0.6108 0.0433 0.6123 0.0005

7 5.6 0.7709 0.0730 0.7743 0.0026

8 7.2 0.8534 0.1015 0.8594 0.0063

9 8.9 0.8788 0.1232 0.8873 0.0142

10 10.5 0.9474 0.1483 0.9586 0.0268

11 12.1 1.0202 0.1672 1.0326 0.0504

12 13.6 0.9857 0.1653 0.9969 0.0711

13 14.8 1.0228 0.1516 1.0276 0.1147

14 16.1 1.0591 0.0493 1.0312 0.2463

15 17.3 0.9184 -0.0015 0.8764 0.2745

16 18.8 0.9924 -0.0141 0.9349 0.3332

17 20.2 1.0902 -0.0247 1.0146 0.3996

18 21.3 1.0457 -0.0374 0.9607 0.4147

19 22.7 1.0750 -0.0463 0.9739 0.4576

20 23.9 0.9831 -0.0568 0.8758 0.4502

21 25.3 1.1523 -0.0501 1.0204 0.5377

22 26.4 1.1814 -0.0456 1.0379 0.5661

23 27.5 1.1152 -0.0540 0.9643 0.5628

24 28.7 1.1520 -0.0521 0.9855 0.5989

25 29.6 1.1393 -0.0475 0.9672 0.6040
 124

Table 7.6: NREL S809 Airfoil Data from NASA-Ames Tests: 63.33% Span 

Wind Angle of Dim.less Dim.less Dim.less Dim.less

Speed Attack (α ) Normal Tangential Lift Coeff. Drag Coeff.

m/s deg Coeff. (C N ) Coeff. (CT ) (CL ) (C D )

5 2.2 0.5479 0.0244 0.5484 -0.0033

6 4.1 0.7096 0.0573 0.7119 -0.0064

7 5.9 0.8656 0.0949 0.8708 -0.0054

8 7.8 0.9569 0.1386 0.9669 -0.0075

9 9.9 1.0804 0.1754 1.0945 0.0130

10 11.8 1.1079 0.1790 1.1211 0.0513

11 13.7 0.9257 0.0397 0.9088 0.1807

12 15.5 1.0239 0.0159 0.9909 0.2583

13 16.9 1.0926 0.0122 1.0490 0.3059

14 18.6 1.2424 0.0070 1.1797 0.3896

15 20.3 1.3169 0.0038 1.2364 0.4533

16 22.2 1.3566 -0.0034 1.2548 0.5157

17 23.7 1.3006 -0.0286 1.1794 0.5490

18 25.0 1.3348 -0.0171 1.2025 0.5796

19 26.6 1.2452 -0.0204 1.1043 0.5758

20 27.8 1.2303 -0.0356 1.0717 0.6053

21 29.5 1.1861 -0.0348 1.0152 0.6144

22 30.8 1.1859 -0.0393 0.9985 0.6410

23 32.4 1.2114 -0.0305 1.0065 0.6749

24 33.8 1.2389 -0.0317 1.0119 0.7155

25 35.2 1.2520 -0.0296 1.0060 0.7459
 125

Table 7.7: NREL S809 Airfoil Data from NASA-Ames Tests: 46.67% Span 

Wind Angle of Dim.less Dim.less Dim.less Dim.less

Speed Attack (α ) Normal Tangential Lift Coeff. Drag Coeff.

m/s deg Coeff. (C N ) Coeff. (CT ) (CL ) (C D )

5 5.0 0.5382 0.0276 0.5386 0.0194

6 7.7 0.7513 0.0694 0.7538 0.0319

7 10.4 0.9124 0.1177 0.9187 0.0489

8 12.8 1.1681 0.1855 1.1802 0.0779

9 15.5 1.2652 0.2243 1.2791 0.1220

10 18.0 1.3460 0.0069 1.2823 0.4094

11 20.4 1.4892 0.0300 1.4063 0.4910

12 22.9 1.4273 -0.0191 1.3074 0.5730

13 24.8 1.3293 -0.0412 1.1894 0.5950

14 26.8 1.2439 -0.0512 1.0872 0.6065

15 28.8 1.2576 -0.0509 1.0775 0.6505

16 30.8 1.2750 -0.0488 1.0702 0.6948

17 32.7 1.3289 -0.0454 1.0938 0.7561

18 34.1 1.3859 -0.0451 1.1223 0.8143

19 36.0 1.4434 -0.0422 1.1429 0.8825

20 37.6 1.4912 -0.0417 1.1560 0.9429

21 39.2 1.5160 -0.0383 1.1506 0.9878

22 40.5 1.5432 -0.0365 1.1498 1.0300

23 42.1 1.5567 -0.0356 1.1312 1.0701

24 43.4 1.5716 -0.0312 1.1204 1.1025

25 44.6 1.5805 -0.0296 1.1046 1.1308
 126

Table 7.8: NREL S809 Airfoil Data from NASA-Ames Tests: 30% Span 

Wind Angle of Dim.less Dim.less Dim.less Dim.less

Speed Attack (α ) Normal Tangential Lift Coeff. Drag Coeff.

m/s deg Coeff. (C N ) Coeff. (CT ) (CL ) (C D )

5 1.4 0.3651 0.0032 0.3651 0.0057

6 4.7 0.5980 0.0422 0.5994 0.0069

7 8.4 0.8048 0.0963 0.8102 0.0223

8 12.1 1.0930 0.1816 1.1068 0.0515

9 15.9 1.1980 0.2148 1.2110 0.1216

10 19.2 1.2941 0.1767 1.2802 0.2587

11 22.0 1.4932 0.1498 1.4406 0.4205

12 24.8 1.7765 0.1397 1.6713 0.6183

13 27.2 2.0447 0.1697 1.8962 0.7837

14 29.6 2.4150 0.1232 2.1607 1.0857

15 31.9 2.4481 0.0518 2.1057 1.2497

16 33.9 2.4153 0.0185 2.0150 1.3318

17 36.0 2.4058 0.0047 1.9491 1.4103

18 37.1 2.3882 -0.0018 1.9037 1.4420

19 39.1 2.3696 -0.0032 1.8369 1.4969

20 40.5 2.3898 -0.0013 1.8164 1.5530

21 42.0 2.3624 -0.0059 1.7517 1.5851

22 43.2 2.3770 -0.0023 1.7312 1.6288

23 44.6 2.3545 0.0006 1.6769 1.6528

24 45.8 2.3182 -0.0028 1.6142 1.6639

25 46.8 2.2919 -0.0002 1.5688 1.6709

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

-25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Angle of Attack [deg]

Normal Force Coefficient

Other Airfoil Data

FAST_AD Airfoil Data

NREL 95.0%

NREL 80.0%

NREL 63.3%

NREL 46.7%

NREL 30.0%

Figure 7.1: NREL S809 dimensionless normal force coefficient airfoil data
 127

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

-25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Angle of Attack [deg]

Tangential Force Coefficient

Other Airfoil Data

FAST_AD Airfoil Data

NREL 95.0%

NREL 80.0%

NREL 63.3%

NREL 46.7%

NREL 30.0%

Figure 7.2: NREL S809 dimensionless tangential force coefficient airfoil data

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

-25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Angle of Attack [deg]

Lift Force Coefficient

Other Airfoil Data

FAST_AD Airfoil Data

NREL 95.0%

NREL 80.0%

NREL 63.3%

NREL 46.7%

NREL 30.0%

Figure 7.3: NREL S809 dimensionless lift force coefficient airfoil data
 128

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

-25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Angle of Attack [deg]

Drag Force Coefficient

Other Airfoil Data

FAST_AD Airfoil Data

NREL 95.0%

NREL 80.0%

NREL 63.3%

NREL 46.7%

NREL 30.0%

Figure 7.4: NREL S809 dimensionless drag force coefficient airfoil data

In Figs. 7.1–7.4, the curves labeled “FAST_AD Airfoil Data” illustrate some of the different 

dimensionless force coefficient airfoil data used by FAST_AD at the various blade elements for 

the Blind Comparison work. The curves labeled “Other Airfoil Data” illustrate the unmodified 

airfoil data measured in the static, two-dimensional environment of the CSU, OSU, and DUT 

wind tunnel tests. The dissimilarities between the unmodified and extrapolated two-dimensional 

airfoil data and the airfoil data measured in the three-dimensional environment of the NASAAmes wind tunnel tests are clearly visible. Therefore, three-dimensional flow effects, such as 

spanwise pressure gradients, spanwise flow, flow circulation, and vorticity, prohibit the use of 

unmodified two-dimensional airfoil data at all span stations. This is manifest even in the 

supposedly attached flow, linear, pre-stall regions! The “NREL **.*%” data are surprisingly 

coherent, considering each data point was measured in a separate wind tunnel test. This further 

underscores the accuracy and repeatability of NASA-Ames wind tunnel tests (see section 5.2). 

In the sensitivity study titled “NASAAmesAirfoilData,” the five airfoil data tables that reflect the 

measurements made in the three-dimensional environment of the NASA-Ames wind tunnel tests 

(given in Tables 7.4–7.8) are used for the blade element airfoil data in FAST_AD. This study 

demonstrates how accurately the model can pinpoint the correct force coefficient-angle of attack 

pair at each span station when very accurate aerodynamic properties of the airfoil at each span 

station are given. Since the BEM curves [Eq. (2.73)] are not dependent on airfoil data, Figs. 6.23 

and 6.24 reveal that the exact force coefficient-angle of attack pairs will not be predicted 

correctly at the 7 and 10 m/s wind speeds even before the simulations are run. To be predicted 

correctly, the BEM curves in these figures would have to pass through their respective “NREL” 

data points, which they do not, in almost every instance. These figures indicate that the angle of 

attack will be mispredicted in nearly every situation. Thus, stated another way, the 

“NASAAmesAirfoilData” study demonstrates the sensitivity of the model to mispredictions in 

angle of attack. As indicated at the end of Chapter 6, this is especially critical in the attached 

flow and stall regimes. 
 129

For use in FAST_AD, the three-dimensional environment, NASA-Ames airfoil data tables must 

be extended across the entire 360 ̊ range of potential angles of attack. To obtain the absent deep 

stall force coefficient angle of attack relationships, the utility program titled “FoilCheck,” which 

employs the Viterna-Corrigan equations of post stall, is used. As in the Blind Comparison work, 

an aspect ratio of 11 is used during the application of these equations. Also, since NASA-Ames 

airfoil data tables are available only at five blade span stations, the data tables are correlated to 

the most well-suited node. As such, the airfoil data table reflecting the conditions seen at 95% 

span is coupled to nodes 19–21. Similarly, the airfoil data table reflecting the conditions seen at 

80% span is coupled to nodes 15–18, the airfoil data table for 63.33% span is coupled to nodes 

12–14, the airfoil data table for 46.67% span is coupled to nodes 8–11, and the airfoil data table 

for 30% span is coupled to nodes 2–7. The dimensionless coefficient airfoil data for a smooth 

circular cylinder are again applied at node 1, the innermost node. 

As seen in the next section, mispredictions in angle of attack for the “NASAAmesAirfoilData” 

study do have an impact on the accuracy of predicted aerodynamic loads when compared to 

experimentally measured loads (see Figs. 7.7–7.16). Predictions of low-speed shaft torque, root 

flap bending, and root edge bending moments are also inaccurate for this study (see Figs. 7.22–

7.24). This raises the question of whether these moments would be accurately predicted if the 

aerodynamic loads were predicted accurately at each span station and wind speed. In other 

words, do the discrepancies between the “NASAAmesAirfoilData” study aerodynamic load 

predictions and the loads measured during the NASA-Ames wind tunnel tests in Figs. 7.7–7.16 

account for the discrepancies in the predicted and measured moments seen in Figs. 7.22–7.24? 

This is a refinement of question (3) in the introduction to this chapter. 

This question is addressed in the sensitivity study entitled “NASAAmesAlpha.” To answer the 

question, the angle of attack misprediction must be eliminated since elimination of aerodynamic 

load discrepancies is a direct outcome of this action. The inaccuracy in angle of attack 

predictions for the “NASAAmesAirfoilData” study is characterized in Figs. 7.5 and 7.6. The 

curves given in Fig. 7.6 represent the difference between the curves given in Fig. 7.5 for each 

discrete wind speed test case. The curve labeled “Average” represents the mean of these curves. 
 130

0

5

10

15

20

25

30

35

40

45

50

20% 30% 40% 50% 60% 70% 80% 90% 100%

Spanwise Fraction

Angle of Attack [deg]

FAST_AD, NASAAmesAirfoilData

NREL 25 m/s

20 m/s

15 m/s

13 m/s

10 m/s

 7 m/s

Figure 7.5: “NASAAmesAirfoilData” study angle of attack predictions 

The curves in Fig. 7.6 clearly illustrate that the angle of attack is consistently underpredicted 

inboard, overpredicted midboard, and slightly underpredicted outboard at every wind speed. In 

the “NASAAmesAlpha” study, the aerodynamic twist distribution of both blades is modified to 

counteract these mispredictions in angle of attack by decreasing the local aerodynamic twist 

inboard, increasing it midboard, and decreasing it again outboard. Though slightly arbitrary, the 

magnitudes of these adjustments are set equal to the average angle of attack mispredictions 

shown in Fig. 7.6. The average differences are employed instead of the exact differences for 

each wind speed test case for a number of reasons: 

(1) Modifying the aerodynamic twist distribution does not completely eliminate angle of attack 

prediction inaccuracies. 

(2) The angle of attack probes are not precise and the measurements have to be upwash 

corrected, a process that too is not precise (see section 5.1). The end result is that the angle 

of attack distributions measured during the NASA-Ames wind tunnel tests are not entirely 

accurate. 

(3) Manual effort is reduced. 
 131

-5

-4

-3

-2

-1

0

1

2

3

4

20% 30% 40% 50% 60% 70% 80% 90% 100%

Spanwise Fraction

Difference in Angle of Attack [deg]

7 m/s

10 m/s

13 m/s

15 m/s

20 m/s

25 m/s

Average

Overpredicted

Underpredicted

Figure 7.6: “NASAAmesAirfoilData” study angle of attack inaccuracies

Adjusted local twist angles for the 21 blade nodes are listed in Table 7.9. The structural pretwist 

distribution is altered accordingly. The same airfoil data used in the “NASAAmesAirfoilData” 

study are used for the “NASAAmesAlpha” study. 
 132

Table 7.9: Nodal Twist Properties for Sensitivity Study “NASAAmesAlpha” 

Distance Distance

from Rotor Spanwise Along Blade Radial

Node Center Fraction From Root (r) Fraction Original Adjusted

Number m (/ 5.029 m) m (r / 4.597 m) Values Alteration Values

1 0.432 0.086 0.000 0.000 0.00 0.00 0.00

2 0.662 0.132 0.230 0.050 0.00 0.00 0.00

3 0.892 0.177 0.460 0.100 3.66 -1.30 2.36

4 1.122 0.223 0.690 0.150 12.78 -2.35 10.43

5 1.351 0.269 0.919 0.200 17.89 -2.37 15.52

6 1.581 0.314 1.149 0.250 13.33 -2.30 11.03

7 1.811 0.360 1.379 0.300 10.19 -2.02 8.17

8 2.041 0.406 1.609 0.350 7.24 -1.72 5.52

9 2.271 0.452 1.839 0.400 5.52 -1.17 4.35

10 2.501 0.497 2.069 0.450 3.81 -0.63 3.18

11 2.731 0.543 2.299 0.500 2.81 0.42 3.23

12 2.960 0.589 2.528 0.550 1.86 1.50 3.36

13 3.190 0.634 2.758 0.600 1.21 1.97 3.18

14 3.420 0.680 2.988 0.650 0.66 2.23 2.89

15 3.650 0.726 3.218 0.700 0.21 2.26 2.47

16 3.880 0.771 3.448 0.750 -0.15 2.10 1.95

17 4.110 0.817 3.678 0.800 -0.51 1.67 1.16

18 4.339 0.863 3.907 0.850 -0.84 0.76 -0.08

19 4.569 0.909 4.137 0.900 -1.17 -0.06 -1.23

20 4.799 0.954 4.367 0.950 -1.49 -0.52 -2.01

21 5.029 1.000 4.597 1.000 -1.82 -0.97 -2.79

Node Location

Local Twist (θ T )

deg

7.2 Sensitivity Study Results and Discussion 

Average dimensionless normal force coefficients at each pressure tap grouping span station 

predicted by FAST_AD for each sensitivity study are plotted against wind speed on the abscissa 

in Figs. 7.7–7.11. Corresponding tangential force coefficients are given in Figs. 7.12–7.16. 

Local dynamic pressures are given in Figs. 7.17–7.21. Plots of low-speed shaft torque, root flap 

bending moments, and root edge bending moments are given in Figs. 7.22–7.24. The curves 

labeled “FAST_AD” represent the data from the simulations performed for the Blind 

Comparison work, the results of which are described in Chapter 6. Data from the other Blind 

Comparison participants were omitted from these plots to avoid clutter and confusion. The 

curves labeled “NREL” represent the data measured during the NASA-Ames wind tunnel tests. 
 133

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

95% Span Normal Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.7: 95% span normal force coefficient, upwind, zero yaw

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

80% Span Normal Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.8: 80% span normal force coefficient, upwind, zero yaw
 134

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

63% Span Normal Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.9: 63% span normal force coefficient, upwind, zero yaw

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

47% Span Normal Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.10: 47% span normal force coefficient, upwind, zero yaw
 135

0.0

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Wind Speed [m/s]

30% Span Normal Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.11: 30% span normal force coefficient, upwind, zero yaw

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

5 10 15 20 25

Wind Speed [m/s]

95% Span Tangential Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.12: 95% span tangential force coefficient, upwind, zero yaw
 136

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

5 10 15 20 25

Wind Speed [m/s]

80% Span Tangential Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.13: 80% span tangential force coefficient, upwind, zero yaw

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

5 10 15 20 25

Wind Speed [m/s]

63% Span Tangential Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.14: 63% span tangential force coefficient, upwind, zero yaw
 137

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

5 10 15 20 25

Wind Speed [m/s]

47% Span Tangential Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.15: 47% span tangential force coefficient, upwind, zero yaw

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

5 10 15 20 25

Wind Speed [m/s]

30% Span Tangential Force Coefficient

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.16: 30% span tangential force coefficient, upwind, zero yaw
 138

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

95% Span Local Dynamic Pressure [Pa]

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.17: 95% span local dynamic pressure, upwind, zero yaw

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

80% Span Local Dynamic Pressure [Pa]

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.18: 80% span local dynamic pressure, upwind, zero yaw
 139

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

63% Span Local Dynamic Pressure [Pa]

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.19: 63% span local dynamic pressure, upwind, zero yaw

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

47% Span Local Dynamic Pressure [Pa]

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.20: 47% span local dynamic pressure, upwind, zero yaw
 140

0

200

400

600

800

1,000

1,200

1,400

5 10 15 20 25

Wind Speed [m/s]

30% Span Local Dynamic Pressure [Pa]

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.21: 30% span local dynamic pressure, upwind, zero yaw

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

2,000

5 10 15 20 25

Wind Speed [m/s]

Low Speed Shaft Torque [Nm]

EQUIL

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.22: Low-speed shaft torque, upwind, zero yaw
 141

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

5 10 15 20 25

Wind Speed [m/s]

Root Flap Bending Moment [Nm]

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.23: Root flap bending moment, upwind, zero yaw

0

100

200

300

400

500

600

700

800

900

1,000

5 10 15 20 25

Wind Speed [m/s]

Root Edge Bending Moment [Nm]

EQUIL

IZD40

DT0001

NR10

THETA3

BETA34

RHO10PL

PhaseVModeShapes

NegTHETAS

NASAAmesAirfoilData

NASAAmesAlpha

FAST_AD

NREL

Figure 7.24: Root edge bending moment, upwind, zero yaw

Though many of the data series from the sensitivity study simulations appear to be absent from 

these plots, all these “invisible” curves lie directly under the curves labeled “FAST_AD.” For 
 142

example, in the “EQUIL” study, the predicted normal and tangential force coefficients at 30% 

span are slightly offset from the original FAST_AD predictions for the Blind Comparison work 

(represented by the “FAST_AD” data series). There are also slight dissimilarities between the 

“EQUIL” and “FAST_AD” predictions of the low-speed shaft torque, root flap and edge bending 

moments, and local dynamic pressure at 95% span. However, the predictions made during the 

“EQUIL” and “FAST_AD” exercises of the remaining force coefficients and local dynamic 

pressures are nearly identical, causing the associated data series for the “EQUIL” study to 

coincide with the data series labeled “FAST_AD.” Thus, many of these “EQUIL” curves appear 

invisible. Similar explanations account for the remaining “invisible” data series. 

The similarity between the load predictions made via the classic BEM model (“EQUIL” study) 

and dynamic inflow model (“FAST_AD” study) establishes that the two theories are nearly 

equivalent for zero yaw error test cases. Addressing question (1) in the introduction to this 

chapter, we conclude that this choice of aerodynamic parameters did not contribute to any of the 

load prediction discrepancies among the various modelers and modeling tools in the Blind 

Comparison exercise for the upwind configuration, zero yaw error test cases. Furthermore, the 

choice of whether to employ the standard BEM model in place of the dynamic inflow model is 

not critical when modeling operational states where dynamic inflow effects are negligible 

[addressing question (2)]. 

For the six test cases analyzed in this sensitivity study, not once is it evident that selecting 

generator models (induction in “FAST_AD” or fixed speed in “IZD40”) influenced the load and 

performance predictions. When modeling with an induction generator, rotor speed deviates by 

no more than 0.1 rpm for the zero yaw error test cases (these data are not presented here). This 

deviation jumps up to several rpm for the nonzero yaw error test cases. Referring to question 

(1), the application of different generator models should not be a cause of load prediction 

discrepancies for the zero yaw error test cases. 

There is also no visible difference between the load predictions when a time increment of 0.001 s 

(study “DT001”) is used in place of 0.002 s. An azimuth step of 1 ̊, from which the original time 

step of 0.002 s was derived, appears to be appropriately, if not conservatively, selected. The 

time increment is thus another parameter that did not cause any performance prediction 

discrepancies seen in the Blind Comparison study, unless of course, a Blind Comparison 

participant chose a foolishly large increment. 

The load predictions made during the “NR10” sensitivity study are surprisingly similar to 

predictions made with twice as many blade elements during the original “FAST_AD” Blind 

Comparison work. Discrepancies between the load predictions made by these two studies are 

minimal whether one is considering local force coefficient values, local dynamic pressures, or 

bending moments. This implies that the number of blade elements chosen by the Blind 

Comparison participants played an insignificant role in the discrepancies among their load 

predictions if they used linear interpolations to pinpoint aerodynamic loads at the correct 

pressure tap span stations. 

Using linear interpolations to pinpoint aerodynamic loads at span stations between two adjacent 

nodes is equivalent to assuming that force coefficient-angle of attack relationships are linear 
 143

between any two points where these curves intersect the BEM curves [Eq. (2.73)]. The results of 

the “NR10” study demonstrate that linear interpolation is an appropriate technique since the load 

predictions obtained with 20 blade elements are nearly identical to those made with 10. 

Load predictions made during the “THETA3” study vary considerably from those made during 

the original “FAST_AD” Blind Comparison work. This is particularly evident for the 

aerodynamic force coefficients at the outboard span stations (80% and 95%), the low-speed shaft 

torque, and the root edge bending moment at all wind speeds. Naturally, this parameter is 

critical for accurate load and performance predictions [regarding question (2)]. Since several 

Blind Comparison participants probably used 3 ̊ instead of 4.815 ̊ for the blade collective pitch 

angle relative to zero aerodynamic twist (see section 7.1), many of the load prediction 

discrepancies evident in the Blind Comparison exercise may result directly from this error. 

Since the orientation of a blade about its central axis is critical to the resulting component loads 

(demonstrated by the “THETA3” study), the absence of any blade torsional motion modeling in 

FAST_AD may be a contributing factor in some of the load prediction inaccuracies. Conversely, 

data from the NASA-Ames wind tunnel tests (not presented here) show that tip pitch angles 

range by no more than 0.35 ̊ throughout even the most extreme test conditions (high wind speed, 

large yaw error test cases). Consequentially, the inability to model blade torsional motion likely 

produced a negligible effect on the load predictions. This will probably not be the case when 

modeling more flexible blades, however (see Chapter 4). 

There are very minor differences between aerodynamic force coefficient predictions made during 

the “BETA34” study and the “FAST_AD” simulations. The local dynamic pressure predictions 

for these two studies are nearly identical. However, the two studies have substantially different 

predictions of the torque and bending moments. The root flap bending moment predictions from 

the “BETA34” study are considerably lower than the original predictions (see Fig. 7.23). The 

shaft torque and bending moment output channels are highly sensitive to flapping angle 

perturbations; local aerodynamic loads are not. This reality is a bit puzzling. Indeed, the root 

flap bending moment should decrease when downwind preconing is prescribed, since centrifugal 

loads tend to offset thrust loads in this situation. However, the same conditions are not true for 

root edge bending moments or low-speed shaft torque. There is no mechanism for a small 

perturbation in blade coning angle to increase root edge bending moments and significantly 

affect low-speed shaft torque predictions. This leads to the suspicion that the method FAST_AD 

uses to integrate aerodynamic loads to back out bending moments may be erroneous. This 

suspicion is fortified in discussions at the end of this chapter. 

The results of the “RHO10PL” study are not surprising. As implied in Eq. (2.73) by the absence 

of the fluid density (ρ), computation of the operational aerodynamic force coefficients is 

independent of the fluid density. This is demonstrated in the results of the “RHO10PL” study; 

predictions of the dimensionless aerodynamic force coefficients obtained via the “RHO10PL” 

and “FAST_AD” simulations are identical. Also, the actual aerodynamic forces, characterized in 

predictions of the local dynamic pressures, low-speed shaft torque, and root flap and edge 

bending moments, are all scaled down proportionally with the air density. This is not surprising, 

since all these loads are linearly proportional to the air density [see, for example, Eq. (2.50)]. 

Thus, if one underpredicts air density, one will underpredict component loads by the same 
 144

fraction. Knowledge of the environment in which a turbine is to be placed is critical in the wind 

turbine design and selection process. 

The “PhaseVModeShapes” sensitivity study shows that the variation in blade structural 

properties and associated mode shapes has no effect on the predictions of local aerodynamic 

force coefficients and dynamic pressures. Also, these variations have only a small effect on the 

predictions of low-speed shaft torque and root flap and edge bending moments, key items of 

information when considering hub loads and blade life. Question (1) indicates the various 

approximations and assumptions Blind Comparison participants made with regard to the 

specification of blade structural properties had little influence on their load prediction 

discrepancies. This assertion most likely applies only since the blades of the UAE turbine are 

quite stiff. Measurements taken during the NASA-Ames wind tunnel tests show that actual flap 

angles are no larger than 0.05 ̊ (these data are not presented here). Blade structural properties 

and mode shapes may have a greater impact on the component loads of more flexible machines. 

A comparison of the “THETA3,” “BETA34,” “RHO10PL,” and “PhaseVModeShapes” 

sensitivity studies shows that the parameters most directly associated with the aerodynamics 

computations are notably more critical than structural parameters when developing models to 

consider loads and performance. Factors such as mode shapes and other distributed blade 

structural parameters have a negligible effect on the aerodynamics and only a minor effect on the 

low-speed shaft torque and root flap and edge bending moments (at least on stiff turbines). 

The results of the “NegTHETAS” study are unexpected. Since the change in blade structural 

properties and associated mode shapes has some effect on the resulting torque and moment 

predictions, reversing the sign of the structural pretwist distribution angles would likely affect 

them as well—especially since the twist distribution has a large variation (a range of 

approximately 22o). However, this is not the case. Just like the “DT001” study, there is 

absolutely no difference between the load predictions derived from the “NegTHETAS” study 

and those derived from the original Blind Comparison work (“FAST_AD”). This could mean 

that the code error uncovered in footnote 9 of Chapter 3 is inconsequential for stiff rotor blades. 

Or there may be some code feature in FAST_AD that takes the magnitudes instead of the actual 

values of the structural pretwist angles during processing. The incorrect sign of the structural 

pretwist angles likely has important consequences on the dynamical outcomes of the simulations. 

As in the “PhaseVModeShapes” study, this will probably have a greater impact on the 

component loads of more flexible machines. 

Compared to measurements taken in the NASA-Ames wind tunnel, aerodynamic force 

coefficients are more accurately predicted in the “NASAAmesAirfoilData” study than they are in 

the original “FAST_AD” work. They are more but not entirely accurate. Prediction of 

aerodynamic force coefficients is highly sensitive to mispredictions in angle of attack, especially 

in the attached flow and stall regions. For example, when using the NASA-Ames airfoil table 

data, FAST_AD overpredicts the angle of attack at 63% span at every wind speed (see Figs. 7.5 

and 7.6). At a wind speed of 7 m/s, this overprediction is about 1.6o. Figure 7.1 demonstrates 

that the flow around the 63% span station is attached at this wind speed since the symbol 
 145

associated with the 63% span, 7 m/s wind speed measurement recording lies on the linear portion 

of the force coefficient curve31. Starting at this symbol, an overprediction in angle of attack of 

about 1.6o would cause the normal force coefficient to be overpredicted by about 0.08. Indeed, 

this is exactly what is seen in Fig. 7.9. Fig. 7.1 also reveals that the 63% span radial location 

experiences stall sometime between a wind speed of 10 and 11 m/s. At a wind speed of 10 m/s, 

FAST_AD overpredicts the angle of attack by about 1.9o when using the NASA-Ames airfoil 

table data (see Fig. 7.5 or 7.6.). As shown in Fig. 7.1, an overprediction in angle of attack of this 

magnitude around stall will cause the normal force coefficient to be underpredicted by about 0.2. 

This is seen in Fig. 7.9. Similarly, the overprediction in angle of attack at this wind speed and 

span station directly results in the underprediction in the tangential force coefficient (see Fig. 

7.14), which is easily verified by examining the events around stall (see Fig. 7.2). 

This process can be used to explain the discrepancies between the predicted and measured 

aerodynamic force coefficients at every span station and wind speed. Aerodynamic load 

predictions are more accurate at high wind speeds and several other instances since the slopes of 

the force coefficient-angle of attack relationships are small in these situations, and thus a slight 

misprediction in angle of attack will not lead to any significant misprediction of the aerodynamic 

force coefficient. 

The inability to pinpoint the correct force coefficient-angle of attack pairs when using accurate 

airfoil data that take into account three-dimensional flow effect considerations, or the inability to 

accurate predict angle of attack in general, underscores a fundamental flaw in the classic BEM 

aerodynamics models. Since design tools such as FAST_AD will only significantly help to 

improve wind turbine designs if component loads are predicted accurately, refining or 

redeveloping efficient rotary wing aerodynamics models is essential. Addressing question (4), 

this work will have the greatest impact on wind turbine design improvement and wind-generated 

electricity cost reduction. When component loads are accurately predicted, safety factors may be 

relaxed and turbine designs can be optimized. The fact that the angle of attack mispredictions 

are consistent at every wind speed (see Fig. 7.5 or 7.6.) makes the feasibility of this task 

promising. Indeed, without modifications to any aerodynamics models, the aerodynamic load 

predictions made by the “NASAAmesAirfoilData” study are the most accurate that this theory 

can achieve. Without theory modification or the ability to use more accurate airfoil data that 

take three-dimensional flow into consideration, a better prediction of experimentally measured 

loads is impossible. 

Since the aerodynamic loads predicted by the “NASAAmesAirfoilData” study are more accurate 

than those predicted during the “FAST_AD” study, when using airfoil data to predict 

aerodynamic loads on wind turbines, measuring airfoil characteristics on a rotating blade may be 

wiser than measuring them on a static airfoil and applying correction algorithms to account for 

three-dimensional flow effects. Unfortunately, the cost of adequate tests is prohibitive in the 

__________________ 

31 The data points in this figure associated with the smallest angle of attacks correspond to the data measured during 

the smallest wind speed test cases. Thus, the leftmost symbol for each span station data series corresponds to the 5 

m/s wind speed test case. The next symbol to the right corresponds to the 6 m/s wind speed test case and the next 

symbol to the right of that corresponds to the 7 m/s wind speed test case. This process can repeated for all 21 wind 

speed tests, so the rightmost symbol for each span station data series corresponds to the 25 m/s wind speed test case. 
 146

design and analysis environment. The cost to perform the NASA-Ames wind tunnel tests is 

feasible only in the research environment. Furthermore, the physics of flow are not well enough 

understood to know what similitude (scaling) parameters are necessary if rotating blade data are 

to be scaled to blades of a different size. Certainly testing full-size rotating blades is impractical, 

since most utility-scale blades cannot fit in even the world’s largest wind tunnels. 

A comparison of the “DT001,” “NR10,” and “NASAAmesAirfoilData” sensitivity studies 

reveals that using accurate airfoil table data is much more important than using more blade 

elements or using smaller time steps to integrate aerodynamic and elastic forces. This is 

important because significant computer-time savings can be achieved by using fewer elements 

and larger time steps. However, when modeling blades that are considerably more flexible, the 

use of more blade elements and smaller time steps will likely become more important. 

As stated in section 7.1, the “NASAAmesAlpha” study is conducted to determine whether the 

predicted and measured aerodynamic load discrepancies account for the discrepancies between 

the measured low-speed shaft torque and root flap and edge bending moments and the predicted 

values obtained from the “NASAAmesAirfoilData” study. The “NASAAmesAlpha” study 

shows clearly that modifying the aerodynamic twist distribution eliminated the predicted and 

measured aerodynamic load discrepancies almost entirely. However, the low-speed shaft torque 

and root flap and edge bending moments are still mispredicted in this study! These inaccuracies 

are difficult to explain. 

One possible explanation is that the inaccuracies in torque and bending moments may result from 

the absence of skin friction drag effects in the aerodynamic force coefficient data obtained from 

the NASA-Ames wind tunnel tests and used in the “NASAAmesAirfoilData” and 

“NASAAmesAlpha” studies. Since the aerodynamic force coefficients are measured using 

pressure transducers, which measure only the pressure drag, viscous stress effects are neglected 

in the measurements of the aerodynamic force coefficients obtained from the NASA-Ames wind 

tunnel tests. If the viscous stress contribution to the drag force could somehow be quantified, 

measurements of the tangential force coefficients would, in essence, decrease [an increase in 

drag force is a decrease in tangential force as seen in Eq. (2.42)]. Equation (2.57) predicts that a 

decrease in tangential force will lead to an increase in thrust force and associated root flap 

bending moment. Equation (2.58) predicts that a decrease in tangential force will lead to a 

decrease in low-speed shaft torque, or equivalently root edge bending moment. Unfortunately, 

these are backward to what is needed to shift the predicted torque and moment curves to the 

curves representing the experimentally measured loads. This thought process reveals that the 

inaccuracies in predicting torque and bending moments do not result from the lack of skin 

friction drag in the aerodynamic force coefficient data. 

One possible explanation for the inaccuracy in torque and bending moment predictions obtained 

from the “NASAAmesAlpha” study is that aerodynamic load data are known to be accurately 

predicted at only five radial locations on the blade. Some of this inaccuracy may be attributable 

to potential mispredictions in aerodynamic loads at other span locations. This is unlikely to be 

the only contributing factor, however. 
 147

Instead, errors and limitations in the structural models and load routines probably form the 

crucial explanation. The belief that there are errors in the structural models and load routines is 

reinforced by the computations given in Table 7.10. This table compares actual predicted and 

measured low-speed shaft torque values (LSSTQ) with “estimates” of low-speed shaft torque 

based on the vector sum of the root flap and edge bending moments (B3RFB and B3REB

respectively). Since the root bending moments are measured fairly near to the rotor axis, the 

value of LSSTQ can be estimated from the values of B3RFB and B3REB via the following 

vectorial relationship: 

[ ( ) ( )] D D LSSTQ ≈ 2 B3REBcos 3.0 + B3RFB sin 3.0 (7.1) 

The “2” appears in Eq. (7.1) since moments from two blades give rise to the overall shaft torque. 

Application of this equation is demonstrated for the “NREL,” “FAST_AD,” and 

“NASAAmesAlpha” data in Table 7.10. 

Table 7.10: Demonstration of Torque and Bending Moment Relationship Flaws 

Wind 2*[B3REB *cos(3) Actual Average

Speed B3REB B3RFB + B3RFB *sin(3)] LSSTQ Mismatch Mismatch

Data Series m/s Nm Nm Nm Nm Error Error

7 347 1,453 845 801 5.5%

10 584 2,121 1,388 1,341 3.5%

13 619 2,497 1,498 1,309 14.4%

15 499 2,750 1,284 1,172 9.6%

20 392 3,658 1,166 1,110 5.0%

25 484 4,757 1,465 1,482 -1.1%

7 256 1,721 691 815 -15.2%

10 513 2,339 1,269 1,492 -14.9%

13 403 2,480 1,064 1,260 -15.6%

15 234 2,487 728 912 -20.2%

20 140 2,980 592 827 -28.4%

25 271 3,901 950 1,302 -27.0%

7 282 1,571 728 791 -8.0%

10 502 2,314 1,244 1,362 -8.6%

13 516 2,840 1,327 1,444 -8.1%

15 381 3,060 1,080 1,169 -7.6%

20 257 3,811 912 985 -7.4%

25 339 4,773 1,176 1,290 -8.9%

"NASAAmesAlpha" -8.1%

"NREL"

"FAST_AD"

6.2%

-20.2%

With the NASA-Ames test data (“NREL”), the “estimated” low-speed shaft torque (column 5) 

overpredicts the actual measured low-speed shaft torque (column 6) by about 6.2% on average. 

In contrast, the “estimated” low-speed shaft torque using Eq. (7.1) underpredicts the value of the 

low-speed shaft torque obtained using FAST_AD by about 10%–20% on average. Intuitively 

speaking, an “estimated” low-speed shaft torque would underpredict the actual low-speed shaft 

torque, since the bending moments at the rotor that combine to form the low-speed shaft torque 

should be smaller at the hub radius than at the rotor axis. Why “estimated” low-speed shaft 

torque overpredicts actual measured low speed shaft torque with the experimental test data is 

unclear. Nevertheless, the results demonstrate that the relationships between the low-speed shaft 

torque and root bending moments are not properly modeled in FAST_AD. 
 148

Exact identification of the errors that cause these relationship flaws is difficult without 

thoroughly examining the code. Code designers of FAST_AD suspect that there are errors in the 

loads-analysis routines. This theory is not documented here. Additional structural modeling 

errors that can contribute to the inaccuracies in torque and bending moment predictions include 

the errors uncovered in Chapter 3 and described in Chapter 4. 

Limitations in the structural models of FAST_AD that can contribute to the inaccuracies in 

torque and bending moment predictions include the inability to model pitching moments and 

radial aerodynamic forces (both are also discussed in Chapters 3 and 4), since both these 

additional blade aerodynamic loads contribute fundamentally to the stress levels at the blade 

root. Radial aerodynamic loads can contribute to blade root bending moments if the loads or 

blade deflections are substantial. Pitching moments contribute to the deflected blade shape, 

which is inherently coupled with the ensuing aerodynamic loads and resulting torque and 

bending moments (see the earlier discussion of the “THETA3” sensitivity study). 

These flaws must be pinpointed and corrected for accurate hub loading, rotor torque, and overall 

performance predictions. Nevertheless, the results of the sensitivity study demonstrate that until 

the aerodynamics models are refined or redeveloped, eradicating these flaws is less critical. Our 

limited understanding of the three-dimensional flow environment and the flow physics involved 

overwhelms most of these errors and model limitations. 
 149

8. Conclusions 

Researchers and designers use design codes such as FAST_AD to efficiently, safely, and costeffectively design and analyze wind energy systems. After developing aerodynamic and 

structural models of the UAE research wind turbine in FAST_AD, simulations are run by 

exciting the model with the same conditions seen in the NASA-Ames wind tunnel for several 

UAE test cases. To facilitate validation of FAST_AD, the load predictions are compared with 

those made by other similar modeling tools and with experimental measurements taken on the 

actual UAE wind turbine during the NASA-Ames wind tunnel tests. The experimental test data 

acquired from the NASA-Ames wind tunnel tests represent the finest, most accurate set of wind 

turbine aerodynamic and induced flow field data available. There are load magnitude 

discrepancies—inconsistencies in predicted and measured aerodynamic force coefficients, rotor 

shaft torque, and out-of-plane bending moments at the blade root—between the modeling tool 

predictions and experimental measurements over a range of operating conditions. . 

Inconsistencies in input file parameters—variations in aerodynamic force coefficient-angle of 

attack characteristics, blade node positions for data recording and analysis, and pitch angle 

convention—explain a noteworthy fraction of the load prediction discrepancies of the various 

modeling tools. Furthermore, other input file parameter inconsistencies such as variations in the 

time step, the number of blade elements, and the distributed blade structural properties 

contributed little to the load prediction discrepancies. Other reasons for load prediction 

discrepancies include modeler error and code bugs. 

Discrepancies between modeling tool load predictions and physically measured load values 

highlight weaknesses in tip loss, stall delay, post stall, and other frequently used aerodynamic 

models. A comparison of force coefficient-angle of attack relationships measured in the threedimensional environment of the NASA-Ames wind tunnel at each pressure tap grouping span 

station reveals that three-dimensional flow effects prohibit the use of unmodified twodimensional airfoil data at all span stations, not only the inboard stations as originally thought. 

These effects include spanwise pressure gradients, spanwise flow, flow circulation, and vorticity. 

Moreover, the inability to pinpoint angle of attack values, even when using accurate airfoil data 

that take into account three-dimensional flow behavior, is a fundamental fault in the classic BEM 

aerodynamic models. 

Demonstrating mispredictions in low-speed shaft torque and root flap and edge bending 

moments even when mispredictions in aerodynamic forces are eliminated at all span stations 

uncovers flaws and limitations in the structural models employed by FAST_AD and suggests 

means of model and code improvement. These flaws and limitations include an improper 

definition of the blade structural pretwist angle and the inability to model (1) radial aerodynamic 

forces acting on the blades, (2) blade pitching moments, and (3) properly understanding the 

relationship between the low-speed shaft torque and root bending moments. 

These flaws must be pinpointed and corrected for accurate hub loading, rotor torque, and overall 

performance predictions. This work should be completed before FAST_AD validation 

concludes. Nevertheless, the results demonstrate that until the aerodynamics models are refined 

or redeveloped, eradicating these flaws is less critical. Our limited understanding of the three-
 150

dimensional flow environment and the flow physics involved overwhelms most of these errors 

and model limitations. Aerodynamic models need to more accurately incorporate threedimensional flow effects and predict angle of attack distributions. Refining or redeveloping 

efficient rotor wing aerodynamics models is essential if design tools, such as FAST_AD, are to 

accurately predict component loads and significantly reduce wind-generated electricity costs. 

When component loads are accurately predicted, safety factors may be relaxed and turbine 

designs can be optimized without the fear of detrimental component failure. The consistency of 

the angle of attack mispredictions at every wind speed makes the feasibility of this task 

promising. 
 151

References 

Buhl, M.L. WT_Perf User’s Guide, National Renewable Energy Laboratory, 2000. 

Du, Z. and Selig, M.S. “A 3-D Stall-Delay Model for Horizontal Axis Wind Turbine 

Performance Prediction,” Proceedings of the 1998 ASME Wind Energy Symposium, Reno, NV: 

pp. 9–19, 1998. 

Eggleston, D.M. and Stoddard, F.S. Wind Turbine Engineering Design, Van Nostrand 

Rheinhold, New York, 1987. 

Hand, M.M. NASA-Ames Wind Tunnel Test Home Page. Golden, CO: National Renewable 

Energy Laboratory (accessed November 2000–September 2001), 

<http://wind2.nrel.gov/amestest/>. 

Hand, M.M.; Simms, D.A.; Fingersh, L.J.; Jager, D.W.; Cotrell, J.R.; Schreck, S.J.; Larwood, 

S.M. Unsteady Aerodynamics Experiment Phase VI: Wind Tunnel Test Configurations and 

Available Data Campaigns, NREL/TP-500-29955; Golden, CO: National Renewable Energy 

Laboratory, 2001. 

Hansen, A.C. and Laino, D.J. User’s Guide to the Wind Turbine Dynamics Computer Programs 

YawDyn and AeroDyn for ADAMS®, University of Utah, 1998. 

Kane, T.R. and Levinson, D.A. Dynamics: Theory and Applications, McGraw-Hill Inc., New 

York, 1985. 

Leishman, J.G. and Beddoes, T.S. “A Semi-Empirical Model for Dynamic Stall,” Journal of the 

American Helicopter Society, Volume 34(3): pp. 3–17, 1989. 

McGowan, J. and Manwell, J. Wind Energy Engineering Fundamentals Course: Course Notes 

– Part 1, University of Massachusetts, Amherst, MA: Renewable Energy Research Laboratory, 

2000. 

Simms, D.A., Hand, M.M., Fingersh, L.J., and Jager, D.W. Unsteady Aerodynamics Experiment 

Phases II-IV Test Configurations and Available Data Campaigns, NREL/TP-500-25950; 

Golden, CO: National Renewable Energy Laboratory, 1999. 

Simms, D.A., Schreck, S.J., Hand, M.M., and Fingersh, L.J. NREL Unsteady Aerodynamics 

Experiment in the NASA-Ames Wind Tunnel: A Comparison of Predictions to Measurements, 

NREL/TP-500-29494; Golden, CO: National Renewable Energy Laboratory, 2001. 

Suzuki, A. and Hansen, A.C. “Dynamic Inflow Model for YawDyn,” Proceedings of the 1998 

AWEA Windpower Conference, Bakersfield, CA: pp. 233–240, 1998. 

Thomson, W.T. and Dahleh, M.D. Theory of Vibrations with Applications, 5th Edition, PrenticeHall, Inc., New Jersey, 1998. 
 152

Viterna, L.A. and Corrigan, R.D. “Fixed Pitch Rotor Performance of Large Horizontal Axis 

Wind Turbines,” Proceedings, Workshop on Large Horizontal Axis Wind Turbines, NASA CP2230, DOE Publication CONF-810752, NASA Lewis Research Center, Cleveland, OH: pp. 69–

85, 1981. 

Wilson, R.E., Lissaman, P.B.S., and Walker, S.N. Aerodynamic Performance of Wind Turbines, 

Oregon State University, 1976. 

Wilson, R.E., Walker, S.N., and Heh, P. Technical and User’s Manual for the FAST_AD 

Advanced Dynamics Code, Oregon State University, 1999. 
 153

Appendix I. FAST_AD Input File for Test Case S0700000

S0700000.inp 

--------------------------------------------------------------------- 

---- FAST_AD CODE INPUT DATA FILE ------------------------------------- 

FAST_AD analysis of UAE S0700000. 

Compatible with FAST_AD Version 3.01b. 

-------------- FAST/AERODYN GENERAL RUN PARAMETERS ------------------ 

20.0 TSTRT Time step to begin output [sec] 

30.0 TMAX Time duration of simulation [sec]; # of revolutions = TMAX*ZD(4)/60 

0.002 DT Time increment [sec]; Azimuth Step in deg = 6*DT*ZD(4) 

0 NTSKIP Number of time steps to skip between output printings; 0 = none 

-------------- ATMOSPHERIC CONDITIONS --------------------------------- 

1.246 RHO Air density [kg/m^3] 

-------------- DEGREE OF FREEDOM SWITCHES ---------------------------- 

1 IZ(1) First flapwise blade mode switch: 1 = yes, 0 = no 

1 IZ(11) Second flapwise blade mode switch: 1 = yes, 0 = no 

1 IZ(13) First edgewise blade mode switch: 1 = yes, 0 = no 

0 IZ(3) Teeter switch: 1 = yes, 0 = no 

1 IZD(4) Azimuth angle, rotor side switch: 0 = const speed, 1 = induction generator, 2 = start up, 3 

= shut down, 4 = variable-speed,5 = idling (TGEN=0) 

1 IZD(15) Drive train flexibility switch: 1 = yes, 0 = no 

0 IZ(5) Tilt switch: 1 = yes, 0 = no 

0 IZ(6) Yaw switch: 1 = yes, 0 = no 

1 IZ(7) First tower mode switch: 1 = yes, 0 = no 

1 IZ(9) Second tower mode switch: 1 = yes, 0 = no 

0 ISHAD Tower shadow switch: 1 = include, 0 = don't include; use 0 for upwind, 1 for downwind 

rotor 

1 IDYNST Dynamic stall switch 1 = include, 0 = don't include 

-------------- INITIAL CONDITIONS ------------------------------------ 

0.0 Z(1) Blade tip initial flapwise displacement [m] 

0.0 Z(13) Blade tip initial edgewise displacement [m] 

0.0 Z(3) Initial or fixed teeter angle [deg] 

270.0 Z(4) Blade 1 zero azimuth offset [deg]; q4 = 0 deg for blade 1 left if Z(4) = 0 deg (when 

looking upwind); q4 = 0 for blade 1 up if Z(4) = 270 deg (when looking upwind); here I 

am using blade 1 in FAST_AD to represent blade 3 in the UAE, but the UAE blade 

rotates in the opposite direction as the FAST_AD model 

71.9 ZD(4) Steady state angular velocity of blades [rpm] 

0.0 Z(5) Fixed or initial tilt angle [deg] 

0.0 Z(6) Fixed or initial yaw angle [deg] 

0.0 Z(7) Initial longitudinal tower displacement [m] 

0.0 Z(8) Initial lateral tower displacement [m] 

-------------- MACHINE PARAMETERS ------------------------------------ 

5.029 RT Blade tip radius measured from hub centerline [m] 

0.432 RH Blade hub radius [m]; a.k.a distance to blade root measured from hub centerline; here I 

am using the distance to the strain gages from the hub centerline instead of 0.508 m = 

actual hub radius so that the blade root stresses will be correct 

4.815 THETA Blade collective pitch relative to the point of zero twist [deg]; ex. THETA = 4.815 deg 

yields 3.0 deg pitch at tip if the tip twist is -1.815 deg 
 154

0 PITCHMODE Pitch control mode switch: 0 = fixed pitch, 1 = power control, 2 = speed control 

0.0 STRTPTCH Time to start pitch control [sec] 

0.0 RRGAP Location of end gap between blade and partial-span aileron control device in fraction of 

tip radius [dimensionless]; use 0.0 if entire blade pitches 

0.0 RLU Undersling length = distance from teeter pin to blade axis intersection [m]; this is used in 

conjunction with coning angles BETA1 and BETA2 

0.0 RLUM Distance from teeter pin to hub center of mass[m] 

-1.401 DN Distance from yaw axis to teeter pin [m]; DN is < 0 for an upwind rotor 

0.0 DNM1 Distance to nacelle center of mass from yaw axis in c1 direction (along low speed shaft) 

[m]; this data is unavailable for the UAE 

0.0 DNM2 Distance to nacelle center of mass from yaw axis in c2 direction (vertically upwards for 

zero tilt) [m]; this data is unavailable for the UAE 

0.0 SHAFTMOMLENGTH Distance from hub to low speed shaft moment output station [m]; 

SHAFTMOMLENGHT is > 0 for upwind rotor 

12.192 HH Hub height above ground level [m] 

0.0 HS Rigid base height of tower above ground level [m] 

0.692 TWRHTOFFSET Distance from actual tower-top (yaw bearing) to rotor axis (hub height) [m] 

0.0 DELTA3 Delta3 angle [deg]; orients the teeter hinge so that it is no longer perpendicular 

to the unconed blade axis 

0.0 BETA(1) Blade 1 coning angle [deg] 

0.0 BETA(2) Blade 2 coning angle [deg] 

-------------- MASS AND INERTIA -------------------------------------- 

1712.0 XMNAC Nacelle lumped mass [kg]; here I am using the UAE Nacelle + hub + boom + 

pitch shaft mass here and am setting XMHUB = 0 

0.0 XMHUB Mass of hub [kg] 

0.0 TIPM(1) Mass of tip brake, blade 1 [kg] 

0.0 TIPM(2) Mass of tip brake, blade 2 [kg] 

0.0 HTINER Inertia of nacelle about tilt axis [kg m^2]; this data is unavailable for the UAE 

but shouldn't matter since IZ(5) = 0 (no tilt DOF) 

161.5 HSINER Inertia of generator (entire equivalent power train) [kg m^2]; here I used an 

average value from the 144 - 179 range 

3789.0 HYINER Inertia of nacelle about yaw axis [kg m^2]; here I am using the UAE Nacelle + 

hub + boom inertia here and am setting HINER = 0 

0.0 HINER Inertia of hub about teeter axis [kg m^2] 

-------------- DRIVETRAIN PARAMETERS --------------------------------- 

1.0 ETAGB Gearbox efficiency [dimensionless]; 1.0 = 100% 

0.78 ETAGEN Generator efficiency [dimensionless]; 1.0 = 100%; here I included the combined 

gearbox and generator efficiency 

72.82 OMEGR Rated speed for induction generator on low speed shaft side [rpm] 

71.63 OMEG0 Reference speed for generator = synchronous speed of generator on low speed shaft side 

[rpm] 

32.7539 CINGEN Induction generator constant [N m/(rad/sec)] 

25.13 YN Gearbox ratio [Dimensionless] 

0.0 QFL Fixed loss constant [N m] 

0.0 QVL Variable loss constant [N m] 

115.0 QBRAKE Mechanical brake torque value [N m] 

88.9574 QMOTOR Motor start-up torque (generator side) [N m] 

1.99E5 ZKDRV Equivalent drive train torsional spring [N m/rad]; includes low-speed shaft, gearbox, and 

high-speed shaft 

767.138 CDRV Equivalent drive train torsional damper [N m/(rad/sec)]; includes low-speed shaft, 

gearbox, and high-speed shaft 

9999.9 TGOFF Time to turn off generator for braking [sec] 

9999.9 TBRAKON Time to turn on brake [sec] 

9999.9 TBRAKFUL Time at which brake reaches full torque [sec] 

9999.9 TIPBRAK1 Time to apply tip brake at shut-down process [sec] 

9999.9 TIPBRAK2 Time to reach the full Cd values of tip brake [sec] 
 155

-------------- TOWER PARAMETERS -------------------------------------- 

0.55 CTWR(1) Tower 1st fore-aft mode structural damping ratio [%] 

0.55 CTWR(2) Tower 1st side-to-side mode structural damping ratio [%] 

0.8 CTWR(3) Tower 2nd fore-aft mode structural damping ratio [%] 

0.8 CTWR(4) Tower 2nd side-to-side mode structural damping ratio [%] 

0.15908 EL Tower shadow width / rotor radius [dimensionless] 

0.35 EPP Tower shadow velocity deficit [dimensionless] 

1.0 TWRMT1 Tower mass tuner, 1st mode [dimensionless] 

1.0 TWRMT2 Tower mass tuner, 2nd mode [dimensionless] 

1.0 TWRST1 Tower stiffness tuner, 1st mode [dimensionless] 

1.0 TWRST2 Tower stiffness tuner, 2nd mode [dimensionless] 

20 NXTWR Number of constant length increments along tower for integration of elastic 

forces 

6 N2 Number of input stations for distributed tower parameters; this should equal NUMINST 

in MODES 

1.0 AMSTWR Adjustment factor for mass distribution [dimensionless]; this should equal 

MASSFACT in MODES 

1.0 STFLNG Adjustment factor for longitudinal stiffness distribution [dimensionless]; this 

should equal STIFFFACT(1) in MODES 

1.0 STFLAT Adjustment factor for lateral stiffness distribution [dimensionless]; this should 

equal STIFFFACT(1) in MODES 

- RAD -- MASS -- LONG STIF ----- LAT STIF ----------------- 

 [dim.less] [kg/m] [N m^2] [N m^2] 

-------------- TOWER ----------------------------------------------- 

0.00000 260.467 235869817.7 235869817.7 

0.29565 260.467 235869817.7 235869817.7 

0.29574 260.467 235869817.7 235869817.7 

0.33904 207.107 79470442.3 79470442.3 

0.33913 207.107 79470442.3 79470442.3 

1.00000 207.107 79470442.3 79470442.3 

-------------- YAW AND TEETER PARAMETERS ----------------------------- 

8.471E6 ZKYAW Yaw spring stiffness [N m/rad] 

7930.0 ZCYAW Yaw damper [N m/(rad/sec)] 

0.0 ZKTILT Tilt spring stiffness [N m/rad] 

0.0 ZCTILT Tilt damper [N m/(rad/sec)] 

0.0 TILTSTOP1 Tilt soft stop angle [deg] 

0.0 TILTSTOP2 Tilt hard stop angle [deg] 

0.0 QKTSTOP1 Nacelle soft stop stiffness [N m/rad] 

0.0 QKTSTOP2 Nacelle hard stop stiffness [N m/rad] 

0.0 COULMB Coulomb teeter damping moment at teeter hinge, [N m] 

0 ITSPDM Teeter damper type: 0 = none, 1 = linear, 3 = user 

0.0 CTEET Teeter damper constant [N m/(rad/sec)] 

0.0 ZKTEET(1) First teeter spring coefficient for cubic curve fit [N m] 

0.0 ZKTEET(2) Second teeter spring coefficient for cubic curve fit [N m/rad] 

0.0 ZKTEET(3) Third teeter spring coefficient for cubic curve fit [N m/rad] 

0.0 QCTEET Angle where teeter damper begins [deg] 

0.0 QKTEET Angle where teeter spring begins [deg] 

0.0 TSTOP Teeter stop angle [deg] 

0.0 QKSTOP Spring stiffness constant for teeter stop [N m/rad] 

-------------- BLADE PARAMETERS -------------------------------------- 

0.925 CBLD(1) Blade first flapwise mode structural damping ratio [%] 

1.345 CBLD(2) Blade second flapwise mode structural damping ratio [%] 

0.765 CBLD(3) Blade first edgewise mode structural damping ratio [%] 

0.0 CDATIPN Drag due to tip brakes at normal operation (no braking) [m^2]; CDATIPB = 

Cd*flat plate area 

0.0 CDATIPB Drag due to tip brakes while braking [m^2]; CDATIPB = Cd*flat plate area 
 156

20 NR Number of constant length increments along blade for integration of aerodynamic forces 

13 N1 Number of input stations for distributed blade parameters; this should equal NUMINST 

in MODES 

1.0 STFFAC(1) Adjustment factor for blade 1 flapwise stiffness [dimensionless]; this should 

equal STIFFFACT(1) in MODES 

1.0 STFFAC(2) Adjustment factor for blade 2 flapwise stiffness [dimensionless]; this should 

equal STIFFFACT(1) in MODES 

1.0 STEFAC(1) Adjustment factor for blade 1 edgewise stiffness [dimensionless]; this should 

equal STIFFFACT(2) in MODES 

1.0 STEFAC(2) Adjustment factor for blade 2 edgewise stiffness [dimensionless]; this should 

equal STIFFFACT(2) in MODES 

1.0 AMSFAC(1) Adjustment factor for blade 1 mass distribution [dimensionless]; this should 

equal MASSFACT in MODES 

1.0 AMSFAC(2) Adjustment factor for blade 2 mass distribution [dimensionless]; this should 

equal MASSFACT in MODES 

1.0 CHDFAC(1) Adjustment factor for blade 1 chord distribution [dimensionless] 

1.0 CHDFAC(2) Adjustment factor for blade 2 chord distribution [dimensionless] 

1.0 TWIFAC(1) Adjustment factor for blade 1 aerodynamic twist distribution [dimensionless] 

1.0 TWIFAC(2) Adjustment factor for blade 2 aerodynamic twist distribution [dimensionless] 

1.0 TWSFAC(1) Adjustment factor for blade 1 structural twist distribution [dimensionless] 

1.0 TWSFAC(2) Adjustment factor for blade 2 structural twist distribution [dimensionless] 

5 NSPAN Number of stations along blade for local moment outputs; max = 5 

23.42,41.65,59.89,78.12,94.53 SPAN(i=1,2..NSPAN) Location of station i from blade root divided 

by blade length, RT - RH [%] 

- RAD - CHORD- THICK - AERO TWIST - MASS - FLAP STIF - STRUC TWIST - EDGE STIFF 

 [dim.less] [m] [dim.less] [deg] [kg/m] [N m^2] [deg] [N m^2] 

-------------- BLADE 1 ----------------------------------------------- 

0.00000 0.218 1.0000 0.000 0.000 511398.4 0.000 393019.1 

0.01653 0.218 1.0000 0.000 8.298 2506356.0 0.000 1927343.0 

0.06896 0.204 1.0000 0.000 22.109 3483529.2 0.000 1507898.5 

0.12486 0.346 0.4716 6.593 32.416 1810745.8 6.593 1276435.4 

0.17946 0.737 0.2090 20.040 28.750 1144892.0 20.040 1103877.7 

0.23428 0.711 0.2095 14.315 15.349 535063.2 14.315 934207.7 

0.34370 0.660 0.2095 7.454 12.115 307733.2 7.454 674456.7 

0.45312 0.610 0.2095 3.702 11.512 224056.5 3.702 581796.3 

0.56254 0.559 0.2095 1.621 10.611 165678.0 1.621 509058.2 

0.67174 0.508 0.2095 0.421 9.744 111920.2 0.421 436219.0 

0.78116 0.457 0.2095 -0.381 8.806 71251.9 -0.381 362245.2 

0.89058 0.406 0.2095 -1.111 7.886 39205.7 -1.111 313455.2 

1.00000 0.355 0.2095 -1.815 7.861 27702.2 -1.815 287462.0 

-------------- BLADE 2 ---------------------------------------------- 

0.00000 0.218 1.0000 0.000 0.000 511398.4 0.000 393019.1 

0.01653 0.218 1.0000 0.000 8.298 2506356.0 0.000 1927343.0 

0.06896 0.204 1.0000 0.000 22.109 3483529.2 0.000 1507898.5 

0.12486 0.346 0.4716 6.593 32.416 1810745.8 6.593 1276435.4 

0.17946 0.737 0.2090 20.040 28.750 1144892.0 20.040 1103877.7 

0.23428 0.711 0.2095 14.315 15.349 535063.2 14.315 934207.7 

0.34370 0.660 0.2095 7.454 12.115 307733.2 7.454 674456.7 

0.45312 0.610 0.2095 3.702 11.512 224056.5 3.702 581796.3 

0.56254 0.559 0.2095 1.621 10.611 165678.0 1.621 509058.2 

0.67174 0.508 0.2095 0.421 9.744 111920.2 0.421 436219.0 

0.78116 0.457 0.2095 -0.381 8.806 71251.9 -0.381 362245.2 

0.89058 0.406 0.2095 -1.111 7.886 39205.7 -1.111 313455.2 

1.00000 0.355 0.2095 -1.815 7.861 27702.2 -1.815 287462.0 

-------------- MODE SHAPES-------------------------------------------- 

 0.000 PC(1,1) Coeffient of x^1 for first flapwise mode shape of blades [dimensionless] 
 157

 0.448 PC(1,2) Coeffient of x^2 " 

 -1.694 PC(1,3) Coeffient of x^3 " 

 8.139 PC(1,4) Coeffient of x^4 " 

 -8.976 PC(1,5) Coeffient of x^5 " 

 3.083 PC(1,6) Coeffient of x^6 " 

 0.000 PC(2,1) Coeffient of x^1 for second flapwise mode shape of blades [dimensionless] 

 -2.067 PC(2,2) Coeffient of x^2 " 

 5.500 PC(2,3) Coeffient of x^3 " 

-27.516 PC(2,4) Coeffient of x^4 " 

 46.700 PC(2,5) Coeffient of x^5 " 

-21.616 PC(2,6) Coeffient of x^6 " 

 0.000 PC(3,1) Coeffient of x^1 for first edgewise mode shape of blades [dimensionless] 

 1.023 PC(3,2) Coeffient of x^2 " 

 -0.056 PC(3,3) Coeffient of x^3 " 

 1.351 PC(3,4) Coeffient of x^4 " 

 -2.187 PC(3,5) Coeffient of x^5 " 

 0.869 PC(3,6) Coeffient of x^6 " 

 0.000 PC(4,1) Coeffient of x^1 for first mode shape of tower [dimensionless] 

 1.187 PC(4,2) Coeffient of x^2 " 

 -2.497 PC(4,3) Coeffient of x^3 " 

 6.824 PC(4,4) Coeffient of x^4 " 

 -6.654 PC(4,5) Coeffient of x^5 " 

 2.140 PC(4,6) Coeffient of x^6 " 

 0.000 PC(5,1) Coeffient of x^1 for second mode shape of tower [dimensionless] 

-29.592 PC(5,2) Coeffient of x^2 " 

 -1.335 PC(5,3) Coeffient of x^3 " 

 85.063 PC(5,4) Coeffient of x^4 " 

-62.876 PC(5,5) Coeffient of x^5 " 

 9.740 PC(5,6) Coeffient of x^6 " 

------- OUTPUT SELECTED PARAMETERS (see list below) ------- 

10,8,7,13,1,20,15,21,22,47,50 

----------------------------------------------------------- 

---- BEGIN YAWDYN/AERODYN PARAMETER INPUT ----------------- 

----------------------------------------------------------- 

SWIRL Induction Factor Model: NONE, WAKE, SWIRL 

0.005 ATOLER Tolerance for induction factor convergence; use 0.005 

DYNIN Dynamic inflow model: DYNIN, EQUILibrium 

HH Wind data file type: HH = hub height, FF = full field 

S0700000.WND 

8 NUMFOIL Number of different airfoil tables that will be read 

..\AirfoilData\cylinder.dat 

..\AirfoilData\S809_CLN_129.DAT 

..\AirfoilData\S809_CLN_185.DAT 

..\AirfoilData\S809_CLN_242.DAT 

..\AirfoilData\S809_CLN_298.DAT 

..\AirfoilData\S809_CLN_354.DAT 

..\AirfoilData\S809_CLN_410.DAT 

..\AirfoilData\S809_CLN_Outboard.DAT 

1 NOPRINT NFOIL; PRINT or NOPRINT; NFOIL = Airfoil ID number for each element 

2 NOPRINT 

3 NOPRINT 

4 NOPRINT 

4 PRINT 

5 PRINT 

6 NOPRINT 

7 NOPRINT 
 158

8 PRINT 

8 PRINT 

8 NOPRINT 

8 NOPRINT 

8 PRINT 

8 NOPRINT 

8 NOPRINT 

8 PRINT 

8 PRINT 

8 NOPRINT 

8 NOPRINT 

8 PRINT 

8 NOPRINT 

END of ADAMS input (the word END must appear in the first three columns) 

============================================================ 

---- COMMENTS (The remainder of this file is not used) ---- 

============================================================ 

Output parameter conventions: 

YAW = Out10 - Out8 (accounting for opposite rotation of blade) 

VTUN = Out7 

B3AZI = Out13 (accounting for opposite rotation of blade) 

B3FLAP = BETA(1) + (180/Pi)*atan[Out1/(100*RT)] 

B3PITCH = Out20 - 1.815 deg (blade 1 pitch relative to the point of zero twist minus the tip-twist of 1.815 deg) 

RPM = Out15 

B3RFP = [Out21*cos(B3PITCH) + Out22*sin(B3PITCH)]*1000 

B3REP = [-Out21*sin(B3PITCH) + Out22*cos(B3PITCH)]*1000 

LSSTQ = Out47*1000 

NAYM = -Out50*1000 

--------------------------------- 

- Possible Output Quantities ---- 

--------------------------------- 

 1 = Blade 1 out-of-plane tip deflection, q1 + q11 [cm] 

 2 = Blade 1 in-plane tip deflection, q13 [cm] 

 3 = Blade 2 out-of-plane tip deflection, q2 + q12 [cm] 

 4 = Blade 2 in-plane tip deflection, q14 [cm] 

 5 = Blade 3 out-of-plane tip deflection [cm] 

 6 = Blade 3 in-plane tip deflection [cm] 

 7 = Hub-height wind speed [m/s] 

 8 = Hub-height wind direction [deg] 

 9 = Teeter angle, q3 [deg]; unavailable for nonteetering rotor 

10 = Yaw angle, q6 [deg] 

11 = Longitudinal tower top deflection, q7 + q9 [cm] 

12 = Lateral tower top deflection, q8 + q10 [cm] 

13 = Rotor azimuth angle relative to blade 1, q4 [deg] 

14 = Generator azimuth angle, q15 [deg] 

15 = Low speed shaft rotational speed [rpm] 

16 = Generator rotational speed [rpm] 

17 = Nacelle tilt angular acceleration [deg/sec^2] 

18 = Nacelle relative tilt angle, q5 [deg] 

19 = Nacelle tilt angular velocity [deg/sec] 

20 = Blade 1 pitch [deg] 

21 = Blade 1 out-of-plane bending moment at root [kNm] 

22 = Blade 1 in-plane bending moment at root [kNm] 

23 = Blade 2 out-of-plane bending moment at root [kNm] 

24 = Blade 2 in-plane bending moment at root [kNm] 
 159

25 = Blade 3 out-of-plane bending moment at root [kNm] 

26 = Blade 3 in-plane bending moment at root [kNm] 

27 = Blade 1 flapwise bending moment at span1% radius [kNm] 

28 = Blade 1 chordwise bending moment at span1% radius [kNm] 

29 = Blade 1 flapwise bending moment at span2% radius [kNm] 

30 = Blade 1 chordwise bending moment at span2% radius [kNm] 

31 = Blade 1 flapwise bending moment at span3% radius [kNm] 

32 = Blade 1 chordwise bending moment at span3% radius [kNm] 

33 = Blade 1 flapwise bending moment at span4% radius [kNm] 

34 = Blade 1 chordwise bending moment at span4% radius [kNm] 

35 = Blade 1 flapwise bending moment at span5% radius [kNm] 

36 = Blade 1 chordwise bending moment at span5% radius [kNm] 

37 = Rotor torque [kNm] 

38 = Rotor power [kW] 

39 = Rotor thrust in shaft direction [kN] 

40 = Rotor thrust in rotating frame, in direction of blade 1 (e3) [kN] 

41 = Rotor thrust in rotating frame, perpendicular to blade 1 (e2) [kN] 

42 = Tower top yaw moment [kNm] 

43 = Tower top longitudinal pitch moment [kNm] 

44 = Tower top lateral pitch moment [kNm] 

45 = Low speed shaft bending moment at hub, aligned with blade [kNm] 

46 = Low speed shaft bending moment at hub, 90 deg from blade [kNm] 

47 = Low speed shaft torque [kNm] 

48 = Generator torque [kNm] 

49 = Electrical power [kW] 

50 = Yaw moment at yaw bearing [kNm] 

51 = Pitch moment at yaw bearing [kNm] 

52 = Force on yaw bearing along nacelle direction [kN] 

53 = Force on yaw bearing in vertical direction [kN] 

54 = Force on yaw bearing in lateral direction [kN] 

55 = Force at blade root normal to blade (i1) [kN] 

56 = Force at blade root tangent to blade (i2) [kN] 

57 = Force at blade root along blade (i3) [kN] 

58 = Yaw moment at tower base (ground level) (b2) [kNm] 

59 = Longtudinal bending moment at tower base (b3) [kNm] 

60 = Lateral bending moment at tower base (b1) [kNm] 

61 = Low speed shaft bend. mom. at distance SHAFTMOMLENGTH from hub, aligned with 

blade [kNm] 

62= = Low speed shaft bend. mom. at distance SHAFTMOMLENGTH from hub, 90 deg from 

blade [kNm] 
 160

Appendix II. Modes Input Files for use in FAST_AD 

UAE_TOWER.inp 

False ISBLADE Blade/tower switch: True = Blade, False = Tower 

0.0 OMEGA Steady-state angular velocity of rotor [rpm]; ignored for towers 

0.0 PITCH Blade collective pitch relative to the point of zero structural twist [deg]; ignored for 

towers 

11.5 TOTLEN Total length of the beam [m]; RT for blades, HH - TWRHTOFFSET for towers 

0.0 RIGLEN Rigid length of the beam [m]; RH for blades, HS for towers 

1832.0 ENDMASS Lumped mass at the end of the beam [kg]; tip brake for blades, nacelle + rotor 

mass for towers 

5 N Number of modes shapes to compute = number of polynomial coefficients to use 

2 NP Order of first polymomial coefficient (i.e. x^NP) 

6 NUMINST Number of input stations for distributed parameters 

1.0 MASSFACT Adjustment factor for mass distribution [dimensionless] 

1.0 STIFFFACT(1) Adjustment factor for out-of-plane (flapwise) stiffness distribution 

[dimensionless] 

1.0 STIFFFACT(2) Adjustment factor for in-plane (edgewise) stiffness distribution [dimensionless]; 

ignored for towers (use STIFFFACT(1)) 

0.00000 260.467 235869817.7 

0.29565 260.467 235869817.7 

0.29574 260.467 235869817.7 

0.33904 207.107 79470442.3 

0.33913 207.107 79470442.3 

1.00000 207.107 79470442.3 

============================================================ 

---- COMMENTS (The remainder of this file is not used) ---- 

============================================================ 

NOTES FOR BLADES: 

If you intend to export results from Modes to FAST_AD, you must 

set both the structural twist and pitch to zero. FAST_AD needs the mode 

shapes in the local principal reference system. It will convert the 

shapes to in-plane and out-of-plane coordinates itself. 

For the lines of distributed parameters, the first column is the 

fractional distance along the flexible portion of the blade. It must go 

from 0.0 to 1.0. The second and third columns are the structural twist [deg] 

and lineal density [kg/m]. The third and fourth columns are the 

flatwise and edgewise stiffnesses [N m^2]. 

NOTE FOR TOWERS: 

For the lines of distributed parameters, the first column is the fractional 

distance along the flexible portion of the tower. It must go from 0.0 to 1.0. 

The second column is the lineal density [kg/m] and the third column is the 

stiffnesses [N m^2]. 
 161

UAE_BLADE.inp 

True ISBLADE Blade/tower switch: True = Blade, False = Tower 

72.0 OMEGA Steady-state angular velocity of rotor [rpm]; ignored for towers 

0.0 PITCH Blade collective pitch relative to the point of zero structural twist [deg]; ignored for 

towers 

5.029 TOTLEN Total length of the beam [m]; RT for blades, HH - TWRHTOFFSET for towers 

0.432 RIGLEN Rigid length of the beam [m]; RH for blades, HS for towers 

0.0 ENDMASS Lumped mass at the end of the beam [kg]; tip brake for blades, nacelle + rotor 

mass for towers 

5 N Number of modes shapes to compute = number of polynomial coefficients to use 

2 NP Order of first polymomial coefficient (i.e. x^NP) 

13 NUMINST Number of input stations for distributed parameters 

1.0 MASSFACT Adjustment factor for mass distribution [dimensionless] 

1.0 STIFFFACT(1) Adjustment factor for out-of-plane (flapwise) stiffness distribution 

[dimensionless] 

1.0 STIFFFACT(2) Adjustment factor for in-plane (edgewise) stiffness distribution [dimensionless]; 

ignored for towers (use STIFFFACT(1)) 

0.00000 0.0 0.000 511398.4 393019.1 

0.01653 0.0 8.298 2506356.0 1927343.0 

0.06896 0.0 22.109 3483529.2 1507898.5 

0.12486 0.0 32.416 1810745.8 1276435.4 

0.17946 0.0 28.750 1144892.0 1103877.7 

0.23428 0.0 15.349 535063.2 934207.7 

0.34370 0.0 12.115 307733.2 674456.7 

0.45312 0.0 11.512 224056.5 581796.3 

0.56254 0.0 10.611 165678.0 509058.2 

0.67174 0.0 9.744 111920.2 436219.0 

0.78116 0.0 8.806 71251.9 362245.2 

0.89058 0.0 7.886 39205.7 313455.2 

1.00000 0.0 7.861 27702.2 287462.0 

============================================================ 

---- COMMENTS (The remainder of this file is not used) ---- 

============================================================ 

NOTES FOR BLADES: 

If you intend to export results from Modes to FAST_AD, you must 

set both the structural twist and pitch to zero. FAST_AD needs the mode 

shapes in the local principal reference system. It will convert the 

shapes to in-plane and out-of-plane coordinates itself. 

For the lines of distributed parameters, the first column is the 

fractional distance along the flexible portion of the blade. It must go 

from 0.0 to 1.0. The second and third columns are the structural twist [deg] 

and lineal density [kg/m]. The third and fourth columns are the 

flatwise and edgewise stiffnesses [N m^2]. 

NOTE FOR TOWERS: 

For the lines of distributed parameters, the first column is the fractional 

distance along the flexible portion of the tower. It must go from 0.0 to 1.0. 

The second column is the lineal density [kg/m] and the third column is the 

stiffnesses [N m^2]. 
 162

Appendix III. Wind Data File for Test Case S0700000 

S0700000.wnd 

!UAE S0700000 

!Time Wind Wind Vert. Horiz. Vert. LinV Gust 

! Speed Dir Speed Shear Shear Shear Speed 

 0.0 7.0 0.0 0.0 0.0 0.0 0.0 0.0 

 0.1 7.0 0.0 0.0 0.0 0.0 0.0 0.0 

9999.9 7.0 0.0 0.0 0.0 0.0 0.0 0.0 
 163

Appendix IV. Airfoil Data Table Files of the S809 Airfoil 

cylinder.dat 

Title or comment line number 1 

Title or comment line number 2 

 1 Number of airfoil tables in this file 

 .00 Table ID parameter 

 150.00 Stall angle (deg) 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 .00 Zero lift angle of attack (deg) 

 .00000 Cn slope for zero lift (dimensionless) 

 .0000 Cn at stall value for positive angle of attack 

 -.8000 Cn at stall value for negative angle of attack 

 .0000 Angle of attack for minimum CD (deg) 

 .3000 Minimum CD value 

 -180.00 .000 .3000 

 .00 .000 .3000 

 180.00 .000 .3000 

S809_CLN_129.dat 

S809 Airfoil, OSU data at Re=.75 Million, Clean roughness 

Adjusted for Post Stall at r/R=.129** 

 1 Number of airfoil tables in this file 

 .00 Table ID parameter 

 30.00 Stall angle (deg)** 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 -.38 Zero lift angle of attack (deg) 

 7.12499 Cn slope for zero lift (dimensionless) 

 3.50 Cn at stall value for positive angle of attack** 

 -.8000 Cn at stall value for negative angle of attack 

 2.0000 Angle of attack for minimum CD (deg) 

 .0116 Minimum CD value 

 -180.00 .000 .1748 .0000 

 -170.00 .230 .2116 .4000 

 -160.00 .460 .3172 .1018 

 -150.00 .494 .4784 .1333 

 -140.00 .510 .6743 .1727 

 -130.00 .486 .8799 .2132 

 -120.00 .415 1.0684 .2498 

 -110.00 .302 1.2148 .2779 

 -100.00 .159 1.2989 .2933 

 -90.00 .000 1.3080 .2936 

 -80.00 -.159 1.2989 .2933 

 -70.00 -.302 1.2148 .2779 
 164

 -60.00 -.415 1.0684 .2498 

 -50.00 -.486 .8799 .2132 

 -40.00 -.510 .6743 .1727 

 -30.00 -.494 .4784 .1333 

 -20.10 -.560 .3027 .0612 

 -18.10 -.670 .3069 .0904 

 -16.10 -.790 .1928 .0293 

 -14.20 -.840 .0898 -.0090 

 -12.20 -.700 .0553 -.0045 

 -10.10 -.630 .0390 -.0044 

 -8.20 -.560 .0233 -.0051 

 -6.10 -.640 .0131 .0018 

-4.1 -0.429 0.0134 -.0216 

-2.1 -0.202 0.0119 -.0282 

0.1 0.059 0.0122 -.0346 

2 0.294 0.0116 -.0405 

4.1 0.539 0.0144 -.0455 

6.2 0.789 0.0146 -.0507 

8.1 0.970 0.0162 -.0404 

10.2 1.134 0.0274 -.0321 

11.3 1.210 0.0303 -.0281 

12.1 1.279 0.0369 -.0284 

13.2 1.375 0.0509 -.0322 

14.2 1.455 0.0648 -.0361 

15.3 1.539 0.0776 -.0363 

16.3 1.604 0.0917 -.0393 

17.1 1.639 0.0994 -.0398 

18.1 1.676 0.2306 -.0983 

19.1 1.691 0.3142 -.1242 

20.1 1.748 0.3186 -.1155 

30 2.484 0.4784 -.2459 

40 2.459 0.6743 -.2813 

50 2.133 0.8799 -.3134 

60 1.599 1.0684 -.3388 

70 0.971 1.2148 -.3557 

80 0.385 1.2989 -.3630 

90 0.000 1.3080 -.3604 

 100.00 -.159 1.2989 -.3600 

 110.00 -.302 1.2148 -.3446 

 120.00 -.415 1.0684 -.3166 

 130.00 -.486 .8799 -.2800 

 140.00 -.510 .6743 -.2394 

 150.00 -.494 .4784 -.2001 

 160.00 -.460 .3172 -.1685 

 170.00 -.230 .2116 -.5000 

 180.00 .000 .1748 .0000

S809_CLN_189.dat 

S809 Airfoil, OSU data at Re=.75 Million, Clean roughness 

Adjusted for Post Stall at r/R=.185** 

 1 Number of airfoil tables in this file 

 .00 Table ID parameter 

 30.00 Stall angle (deg)** 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 
 165

 .00 No longer used, enter zero 

 -.38 Zero lift angle of attack (deg) 

 7.12499 Cn slope for zero lift (dimensionless) 

 3.50 Cn at stall value for positive angle of attack** 

 -.8000 Cn at stall value for negative angle of attack 

 2.0000 Angle of attack for minimum CD (deg) 

 .0116 Minimum CD value 

 -180.00 .000 .1748 .0000 

 -170.00 .230 .2116 .4000 

 -160.00 .460 .3172 .1018 

 -150.00 .494 .4784 .1333 

 -140.00 .510 .6743 .1727 

 -130.00 .486 .8799 .2132 

 -120.00 .415 1.0684 .2498 

 -110.00 .302 1.2148 .2779 

 -100.00 .159 1.2989 .2933 

 -90.00 .000 1.3080 .2936 

 -80.00 -.159 1.2989 .2933 

 -70.00 -.302 1.2148 .2779 

 -60.00 -.415 1.0684 .2498 

 -50.00 -.486 .8799 .2132 

 -40.00 -.510 .6743 .1727 

 -30.00 -.494 .4784 .1333 

 -20.10 -.560 .3027 .0612 

 -18.10 -.670 .3069 .0904 

 -16.10 -.790 .1928 .0293 

 -14.20 -.840 .0898 -.0090 

 -12.20 -.700 .0553 -.0045 

 -10.10 -.630 .0390 -.0044 

 -8.20 -.560 .0233 -.0051 

 -6.10 -.640 .0131 .0018 

-4.1 -0.427 0.0134 -.0216 

-2.1 -0.204 0.0119 -.0282 

0.1 0.057 0.0122 -.0346 

2 0.295 0.0116 -.0405 

4.1 0.540 0.0144 -.0455 

6.2 0.789 0.0146 -.0507 

8.1 0.955 0.0162 -.0404 

10.2 1.090 0.0274 -.0321 

11.3 1.148 0.0303 -.0281 

12.1 1.209 0.0369 -.0284 

13.2 1.293 0.0509 -.0322 

14.2 1.360 0.0648 -.0361 

15.3 1.428 0.0776 -.0363 

16.3 1.475 0.0917 -.0393 

17.1 1.489 0.0994 -.0398 

18.1 1.500 0.2306 -.0983 

19.1 1.479 0.3142 -.1242 

20.1 1.515 0.3186 -.1155 

30 2.103 0.4784 -.2459 

40 2.089 0.6743 -.2813 

50 1.825 0.8799 -.3134 

60 1.384 1.0684 -.3388 

70 0.856 1.2148 -.3557 

80 0.352 1.2989 -.3630 

90 0.000 1.3080 -.3604 
 166

 100.00 -.159 1.2989 -.3600 

 110.00 -.302 1.2148 -.3446 

 120.00 -.415 1.0684 -.3166 

 130.00 -.486 .8799 -.2800 

 140.00 -.510 .6743 -.2394 

 150.00 -.494 .4784 -.2001 

 160.00 -.460 .3172 -.1685 

 170.00 -.230 .2116 -.5000 

 180.00 .000 .1748 .0000 

S809_CLN_242.dat 

S809 Airfoil, OSU data at Re=.75 Million, Clean roughness 

Adjusted for Post Stall at r/R=.242** 

 1 Number of airfoil tables in this file 

 .00 Table ID parameter 

 17.10 Stall angle (deg)** 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 -.38 Zero lift angle of attack (deg) 

 7.12499 Cn slope for zero lift (dimensionless) 

 2.10 Cn at stall value for positive angle of attack** 

 -.8000 Cn at stall value for negative angle of attack 

 2.0000 Angle of attack for minimum CD (deg) 

 .0116 Minimum CD value 

 -180.00 .000 .1748 .0000 

 -170.00 .230 .2116 .4000 

 -160.00 .460 .3172 .1018 

 -150.00 .494 .4784 .1333 

 -140.00 .510 .6743 .1727 

 -130.00 .486 .8799 .2132 

 -120.00 .415 1.0684 .2498 

 -110.00 .302 1.2148 .2779 

 -100.00 .159 1.2989 .2933 

 -90.00 .000 1.3080 .2936 

 -80.00 -.159 1.2989 .2933 

 -70.00 -.302 1.2148 .2779 

 -60.00 -.415 1.0684 .2498 

 -50.00 -.486 .8799 .2132 

 -40.00 -.510 .6743 .1727 

 -30.00 -.494 .4784 .1333 

 -20.10 -.560 .3027 .0612 

 -18.10 -.670 .3069 .0904 

 -16.10 -.790 .1928 .0293 

 -14.20 -.840 .0898 -.0090 

 -12.20 -.700 .0553 -.0045 

 -10.10 -.630 .0390 -.0044 

 -8.20 -.560 .0233 -.0051 

 -6.10 -.640 .0131 .0018 

-4.1 -0.425 0.0134 -.0216 

-2.1 -0.205 0.0119 -.0282 

0.1 0.055 0.0122 -.0346 

2 0.296 0.0116 -.0405 

4.1 0.540 0.0144 -.0455 

6.2 0.789 0.0146 -.0507 
 167

8.1 0.942 0.0162 -.0404 

10.2 1.054 0.0274 -.0321 

11.3 1.096 0.0303 -.0281 

12.1 1.150 0.0369 -.0284 

13.2 1.224 0.0509 -.0322 

14.2 1.281 0.0648 -.0361 

15.3 1.336 0.0776 -.0363 

16.3 1.367 0.0917 -.0393 

17.1 1.365 0.0994 -.0398 

18.1 1.353 0.2306 -.0983 

19.1 1.303 0.3142 -.1242 

20.1 1.322 0.3186 -.1155 

30 1.787 0.4784 -.2459 

40 1.781 0.6743 -.2813 

50 1.569 0.8799 -.3134 

60 1.205 1.0684 -.3388 

70 0.760 1.2148 -.3557 

80 0.323 1.2989 -.3630 

90 0.000 1.3080 -.3604 

 100.00 -.159 1.2989 -.3600 

 110.00 -.302 1.2148 -.3446 

 120.00 -.415 1.0684 -.3166 

 130.00 -.486 .8799 -.2800 

 140.00 -.510 .6743 -.2394 

 150.00 -.494 .4784 -.2001 

 160.00 -.460 .3172 -.1685 

 170.00 -.230 .2116 -.5000 

 180.00 .000 .1748 .0000 

S809_CLN_298.dat 

S809 Airfoil, OSU data at Re=.75 Million, Clean roughness 

Adjusted for Post Stall at r/R=.298** 

 1 Number of airfoil tables in this file 

 .00 Table ID parameter 

 16.30 Stall angle (deg)** 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 -.38 Zero lift angle of attack (deg) 

 7.12499 Cn slope for zero lift (dimensionless) 

 2.00 Cn at stall value for positive angle of attack** 

 -.8000 Cn at stall value for negative angle of attack 

 2.0000 Angle of attack for minimum CD (deg) 

 .0116 Minimum CD value 

 -180.00 .000 .1748 .0000 

 -170.00 .230 .2116 .4000 

 -160.00 .460 .3172 .1018 

 -150.00 .494 .4784 .1333 

 -140.00 .510 .6743 .1727 

 -130.00 .486 .8799 .2132 

 -120.00 .415 1.0684 .2498 

 -110.00 .302 1.2148 .2779 

 -100.00 .159 1.2989 .2933 

 -90.00 .000 1.3080 .2936 

 -80.00 -.159 1.2989 .2933 
 168

 -70.00 -.302 1.2148 .2779 

 -60.00 -.415 1.0684 .2498 

 -50.00 -.486 .8799 .2132 

 -40.00 -.510 .6743 .1727 

 -30.00 -.494 .4784 .1333 

 -20.10 -.560 .3027 .0612 

 -18.10 -.670 .3069 .0904 

 -16.10 -.790 .1928 .0293 

 -14.20 -.840 .0898 -.0090 

 -12.20 -.700 .0553 -.0045 

 -10.10 -.630 .0390 -.0044 

 -8.20 -.560 .0233 -.0051 

 -6.10 -.640 .0131 .0018 

-4.1 -0.424 0.0134 -.0216 

-2.1 -0.206 0.0119 -.0282 

0.1 0.054 0.0122 -.0346 

2 0.297 0.0116 -.0405 

4.1 0.540 0.0144 -.0455 

6.2 0.789 0.0146 -.0507 

8.1 0.934 0.0162 -.0404 

10.2 1.029 0.0274 -.0321 

11.3 1.061 0.0303 -.0281 

12.1 1.110 0.0369 -.0284 

13.2 1.177 0.0509 -.0322 

14.2 1.226 0.0648 -.0361 

15.3 1.272 0.0776 -.0363 

16.3 1.293 0.0917 -.0393 

17.1 1.279 0.0994 -.0398 

18.1 1.251 0.2306 -.0983 

19.1 1.181 0.3142 -.1242 

20.1 1.188 0.3186 -.1155 

30 1.569 0.4784 -.2459 

40 1.569 0.6743 -.2813 

50 1.393 0.8799 -.3134 

60 1.082 1.0684 -.3388 

70 0.694 1.2148 -.3557 

80 0.304 1.2989 -.3630 

90 0.000 1.3080 -.3604 

 100.00 -.159 1.2989 -.3600 

 110.00 -.302 1.2148 -.3446 

 120.00 -.415 1.0684 -.3166 

 130.00 -.486 .8799 -.2800 

 140.00 -.510 .6743 -.2394 

 150.00 -.494 .4784 -.2001 

 160.00 -.460 .3172 -.1685 

 170.00 -.230 .2116 -.5000 

 180.00 .000 .1748 .0000 

S809_CLN_354.dat 

S809 Airfoil, OSU data at Re=.75 Million, Clean roughness 

Adjusted for Post Stall at r/R=.354** 

 1 Number of airfoil tables in this file 

 .00 Table ID parameter 

 16.30 Stall angle (deg)** 

 .00 No longer used, enter zero 
 169

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 -.38 Zero lift angle of attack (deg) 

 7.12499 Cn slope for zero lift (dimensionless) 

 2.00 Cn at stall value for positive angle of attack** 

 -.8000 Cn at stall value for negative angle of attack 

 2.0000 Angle of attack for minimum CD (deg) 

 .0116 Minimum CD value 

 -180.00 .000 .1748 .0000 

 -170.00 .230 .2116 .4000 

 -160.00 .460 .3172 .1018 

 -150.00 .494 .4784 .1333 

 -140.00 .510 .6743 .1727 

 -130.00 .486 .8799 .2132 

 -120.00 .415 1.0684 .2498 

 -110.00 .302 1.2148 .2779 

 -100.00 .159 1.2989 .2933 

 -90.00 .000 1.3080 .2936 

 -80.00 -.159 1.2989 .2933 

 -70.00 -.302 1.2148 .2779 

 -60.00 -.415 1.0684 .2498 

 -50.00 -.486 .8799 .2132 

 -40.00 -.510 .6743 .1727 

 -30.00 -.494 .4784 .1333 

 -20.10 -.560 .3027 .0612 

 -18.10 -.670 .3069 .0904 

 -16.10 -.790 .1928 .0293 

 -14.20 -.840 .0898 -.0090 

 -12.20 -.700 .0553 -.0045 

 -10.10 -.630 .0390 -.0044 

 -8.20 -.560 .0233 -.0051 

 -6.10 -.640 .0131 .0018 

-4.1 -0.423 0.0134 -.0216 

-2.1 -0.207 0.0119 -.0282 

0.1 0.053 0.0122 -.0346 

2 0.298 0.0116 -.0405 

4.1 0.540 0.0144 -.0455 

6.2 0.790 0.0146 -.0507 

8.1 0.927 0.0162 -.0404 

10.2 1.008 0.0274 -.0321 

11.3 1.032 0.0303 -.0281 

12.1 1.077 0.0369 -.0284 

13.2 1.138 0.0509 -.0322 

14.2 1.182 0.0648 -.0361 

15.3 1.220 0.0776 -.0363 

16.3 1.233 0.0917 -.0393 

17.1 1.209 0.0994 -.0398 

18.1 1.168 0.2306 -.0983 

19.1 1.082 0.3142 -.1242 

20.1 1.079 0.3186 -.1155 

30 1.390 0.4784 -.2459 

40 1.395 0.6743 -.2813 

50 1.248 0.8799 -.3134 

60 0.981 1.0684 -.3388 

70 0.640 1.2148 -.3557 

80 0.288 1.2989 -.3630 
 170

90 0.000 1.3080 -.3604 

 100.00 -.159 1.2989 -.3600 

 110.00 -.302 1.2148 -.3446 

 120.00 -.415 1.0684 -.3166 

 130.00 -.486 .8799 -.2800 

 140.00 -.510 .6743 -.2394 

 150.00 -.494 .4784 -.2001 

 160.00 -.460 .3172 -.1685 

 170.00 -.230 .2116 -.5000 

 180.00 .000 .1748 .0000 

S809_CLN_410.dat 

S809 Airfoil, OSU data at Re=.75 Million, Clean roughness 

Adjusted for Post Stall at r/R=.41** 

 1 Number of airfoil tables in this file 

 .00 Table ID parameter 

 15.30 Stall angle (deg)** 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 -.38 Zero lift angle of attack (deg) 

 7.12499 Cn slope for zero lift (dimensionless) 

 1.90 Cn at stall value for positive angle of attack** 

 -.8000 Cn at stall value for negative angle of attack 

 2.0000 Angle of attack for minimum CD (deg) 

 .0116 Minimum CD value 

 -180.00 .000 .1748 .0000 

 -170.00 .230 .2116 .4000 

 -160.00 .460 .3172 .1018 

 -150.00 .494 .4784 .1333 

 -140.00 .510 .6743 .1727 

 -130.00 .486 .8799 .2132 

 -120.00 .415 1.0684 .2498 

 -110.00 .302 1.2148 .2779 

 -100.00 .159 1.2989 .2933 

 -90.00 .000 1.3080 .2936 

 -80.00 -.159 1.2989 .2933 

 -70.00 -.302 1.2148 .2779 

 -60.00 -.415 1.0684 .2498 

 -50.00 -.486 .8799 .2132 

 -40.00 -.510 .6743 .1727 

 -30.00 -.494 .4784 .1333 

 -20.10 -.560 .3027 .0612 

 -18.10 -.670 .3069 .0904 

 -16.10 -.790 .1928 .0293 

 -14.20 -.840 .0898 -.0090 

 -12.20 -.700 .0553 -.0045 

 -10.10 -.630 .0390 -.0044 

 -8.20 -.560 .0233 -.0051 

 -6.10 -.640 .0131 .0018 

-4.1 -0.423 0.0134 -.0216 

-2.1 -0.208 0.0119 -.0282 

0.1 0.053 0.0122 -.0346 

2 0.298 0.0116 -.0405 

4.1 0.540 0.0144 -.0455 
 171

6.2 0.790 0.0146 -.0507 

8.1 0.920 0.0162 -.0404 

10.2 0.989 0.0274 -.0321 

11.3 1.004 0.0303 -.0281 

12.1 1.046 0.0369 -.0284 

13.2 1.102 0.0509 -.0322 

14.2 1.140 0.0648 -.0361 

15.3 1.171 0.0776 -.0363 

16.3 1.176 0.0917 -.0393 

17.1 1.144 0.0994 -.0398 

18.1 1.091 0.2306 -.0983 

19.1 0.989 0.3142 -.1242 

20.1 0.977 0.3186 -.1155 

30 1.223 0.4784 -.2459 

40 1.233 0.6743 -.2813 

50 1.113 0.8799 -.3134 

60 0.886 1.0684 -.3388 

70 0.589 1.2148 -.3557 

80 0.273 1.2989 -.3630 

90 0.000 1.3080 -.3604 

 100.00 -.159 1.2989 -.3600 

 110.00 -.302 1.2148 -.3446 

 120.00 -.415 1.0684 -.3166 

 130.00 -.486 .8799 -.2800 

 140.00 -.510 .6743 -.2394 

 150.00 -.494 .4784 -.2001 

 160.00 -.460 .3172 -.1685 

 170.00 -.230 .2116 -.5000 

 180.00 .000 .1748 .0000 

S809_CLN_Outboard.dat 

S809 Airfoil, OSU data at Re=.75 Million, Clean roughness 

NREL/TP-442-7817 Appendix B, Viterna used aspect ratio=11 

 1 Number of airfoil tables in this file 

 .00 Table ID parameter 

 15.30 Stall angle (deg) 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 .00 No longer used, enter zero 

 -.38 Zero lift angle of attack (deg) 

 7.12499 Cn slope for zero lift (dimensionless) 

 1.9408 Cn at stall value for positive angle of attack 

 -.8000 Cn at stall value for negative angle of attack 

 2.0000 Angle of attack for minimum CD (deg) 

 .0116 Minimum CD value 

 -180.00 .000 .1748 .0000 

 -170.00 .230 .2116 .4000 

 -160.00 .460 .3172 .1018 

 -150.00 .494 .4784 .1333 

 -140.00 .510 .6743 .1727 

 -130.00 .486 .8799 .2132 

 -120.00 .415 1.0684 .2498 

 -110.00 .302 1.2148 .2779 

 -100.00 .159 1.2989 .2933 

 -90.00 .000 1.3080 .2936 
 172

 -80.00 -.159 1.2989 .2933 

 -70.00 -.302 1.2148 .2779 

 -60.00 -.415 1.0684 .2498 

 -50.00 -.486 .8799 .2132 

 -40.00 -.510 .6743 .1727 

 -30.00 -.494 .4784 .1333 

 -20.10 -.560 .3027 .0612 

 -18.10 -.670 .3069 .0904 

 -16.10 -.790 .1928 .0293 

 -14.20 -.840 .0898 -.0090 

 -12.20 -.700 .0553 -.0045 

 -10.10 -.630 .0390 -.0044 

 -8.20 -.560 .0233 -.0051 

 -6.10 -.640 .0131 .0018 

 -4.10 -.420 .0134 -.0216 

 -2.10 -.210 .0119 -.0282 

 .10 .050 .0122 -.0346 

 2.00 .300 .0116 -.0405 

 4.10 .540 .0144 -.0455 

 6.20 .790 .0146 -.0507 

 8.10 .900 .0162 -.0404 

 10.20 .930 .0274 -.0321 

 11.30 .920 .0303 -.0281 

 12.10 .950 .0369 -.0284 

 13.20 .990 .0509 -.0322 

 14.20 1.010 .0648 -.0361 

 15.30 1.020 .0776 -.0363 

 16.30 1.000 .0917 -.0393 

 17.10 .940 .0994 -.0398 

 18.10 .850 .2306 -.0983 

 19.10 .700 .3142 -.1242 

 20.10 .660 .3186 -.1155 

 30.00 .705 .4784 -.2459 

 40.00 .729 .6743 -.2813 

 50.00 .694 .8799 -.3134 

 60.00 .593 1.0684 -.3388 

 70.00 .432 1.2148 -.3557 

 80.00 .227 1.2989 -.3630 

 90.00 .000 1.3080 -.3604 

 100.00 -.159 1.2989 -.3600 

 110.00 -.302 1.2148 -.3446 

 120.00 -.415 1.0684 -.3166 

 130.00 -.486 .8799 -.2800 

 140.00 -.510 .6743 -.2394 

 150.00 -.494 .4784 -.2001 

 160.00 -.460 .3172 -.1685 

 170.00 -.230 .2116 -.5000 

 180.00 .000 .1748 .0000 
 173

Appendix V. Crunch Data File for Test Case S0700000 

S0700000.cru 

Input file for analysis of .all files created using FAST_AD. The .all files are a concatenation of the .out and 

felement.plt data files. Compatible with Crunch Version 2.60. 

0 Row number that contains channel titles; enter 0 if no titles are available or if titles are specified 

below 

0 Row number that contains channel units; enter 0 if no units are available or if units are specified 

below 

5 First row number of data 

0 Number of data records to be read from file; enter 0 to automatically determine which rows to read 

0.0,0.0 Start and end times of data records to be read; enter 0's to use all the data records in the file 

False Output statistics? 

False Output modified data? 

True Create tab-delimited output? 

"F11.4" Numerical-output format specifier; see manual for limitations 

False Output aggregate analysis files? enter False for separate analysis files for each input file 

"Aggregate" Root name of the aggregate files, if aggregates were specified above 

72 Number of channels in each input file 

72 Number of channels to use in analysis 

Format for column info is: Col_Title(10 char max), Units(10 char max), Orig_Col_#, Scale factor, Offset 

"Time" "(sec)" 001 1.0 0.0 

"Yaw" "(deg)" 002 1.0 0.0 

"HH_WDir" "(deg)" 003 1.0 0.0 

"HH_WSpd" "(m/s)" 004 1.0 0.0 

"RtrAzm" "(deg)" 005 1.0 0.0 

"Bld1POut" "(cm)" 006 1.0 0.0 

"Bl1Pitch" "(deg)" 007 1.0 0.0 

"RotRPM" "(RPM)"008 1.0 0.0 

"M1OuRt" "(kNm)" 009 1.0 0.0 

"M1InRt" "(kNm)" 010 1.0 0.0 

"ShafTorq" "(kNm)" 011 1.0 0.0 

"BearYawM" "(kNm)" 012 1.0 0.0 

"Time" "(sec)" 013 1.0 0.0 

"VX21" "(m/s)" 014 1.0 0.0 

"VY21" "(m/s)" 015 1.0 0.0 

"VZ21" "(m/s)" 016 1.0 0.0 

"Alpha05" "(deg)" 017 1.0 0.0 

"CLift05" "( - )" 018 1.0 0.0 

"CDrag05" "( - )" 019 1.0 0.0 

"AxInd05" "( - )" 020 1.0 0.0 

"TanInd05" "( - )" 021 1.0 0.0 

"ForcN05" "(N)" 022 1.0 0.0 

"ForcT05" "(N)" 023 1.0 0.0 

"Alpha06" "(deg)" 024 1.0 0.0 

"CLift06" "( - )" 025 1.0 0.0 
 174

"CDrag06" "( - )" 026 1.0 0.0 

"AxInd06" "( - )" 027 1.0 0.0 

"TanInd06" "( - )" 028 1.0 0.0 

"ForcN06" "(N)" 029 1.0 0.0 

"ForcT06" "(N)" 030 1.0 0.0 

"Alpha09" "(deg)" 031 1.0 0.0 

"CLift09" "( - )" 032 1.0 0.0 

"CDrag09" "( - )" 033 1.0 0.0 

"AxInd09" "( - )" 034 1.0 0.0 

"TanInd09" "( - )" 035 1.0 0.0 

"ForcN09" "(N)" 036 1.0 0.0 

"ForcT09" "(N)" 037 1.0 0.0 

"Alpha10" "(deg)" 038 1.0 0.0 

"CLift10" "( - )" 039 1.0 0.0 

"CDrag10" "( - )" 040 1.0 0.0 

"AxInd10" "( - )" 041 1.0 0.0 

"TanInd10" "( - )" 042 1.0 0.0 

"ForcN10" "(N)" 043 1.0 0.0 

"ForcT10" "(N)" 044 1.0 0.0 

"Alpha13" "(deg)" 045 1.0 0.0 

"CLift13" "( - )" 046 1.0 0.0 

"CDrag13" "( - )" 047 1.0 0.0 

"AxInd13" "( - )" 048 1.0 0.0 

"TanInd13" "( - )" 049 1.0 0.0 

"ForcN13" "(N)" 050 1.0 0.0 

"ForcT13" "(N)" 051 1.0 0.0 

"Alpha16" "(deg)" 052 1.0 0.0 

"CLift16" "( - )" 053 1.0 0.0 

"CDrag16" "( - )" 054 1.0 0.0 

"AxInd16" "( - )" 055 1.0 0.0 

"TanInd16" "( - )" 056 1.0 0.0 

"ForcN16" "(N)" 057 1.0 0.0 

"ForcT16" "(N)" 058 1.0 0.0 

"Alpha17" "(deg)" 059 1.0 0.0 

"CLift17" "( - )" 060 1.0 0.0 

"CDrag17" "( - )" 061 1.0 0.0 

"AxInd17" "( - )" 062 1.0 0.0 

"TanInd17" "( - )" 063 1.0 0.0 

"ForcN17" "(N)" 064 1.0 0.0 

"ForcT17" "(N)" 065 1.0 0.0 

"Alpha20" "(deg)" 066 1.0 0.0 

"CLift20" "( - )" 067 1.0 0.0 

"CDrag20" "( - )" 068 1.0 0.0 

"AxInd20" "( - )" 069 1.0 0.0 

"TanInd20" "( - )" 070 1.0 0.0 

"ForcN20" "(N)" 071 1.0 0.0 

"ForcT20" "(N)" 072 1.0 0.0 

0 Number of output channels to be modified by the IIR filter; next 4 lines ignored if 0 

0 

0 Filter type: 1 = LowPass, 2 = HighPass, 3 = BandPass 

0.0 Low cutoff frequency [Hz]; ignored for low-pass filters 

0.0 High cutoff frequency [Hz]; ignored for high-pass filters 

91 Number of new, calculated channels to be generated; next line ignored if 0 

Format for column info is: Col_Title(10 char max), Units(10 char max), Equation. Put each field in double quotes. 
 175

"YAW" "[deg]" "(C002-C003)" 073 

"VTUN" "[m/s]" "C004" 074 

"B3AZI" "[deg]" "C005" 075 

"B3FLAP" "[deg]" "((180.0/PI)*atan(C006/(100.0*5.029)))" 076 

"B3PITCH" "[deg]" "(C007 - 1.815)" 077 

"RPM" "[RPM]""C008" 078 

"B3RFP" "[Nm]" "((C009*cos(C077*PI/180.0) + C010*sin(C077*PI/180.0))*1000.0)" 079 

"B3REP" "[Nm]" "((-C009*sin(C077*PI/180.0) + C010*cos(C077*PI/180.0))*1000.0)" 080 

"LSSTQ" "[Nm]" "(C011*1000.0)" 081 

"NAYM" "[Nm]" "(-C012*1000.0)" 082 

"CN05" "[ - ]" "(C018*cos(C017*PI/180.0) + C019*sin(C017*PI/180.0))" 083 

"CT05" "[ - ]" "(C018*sin(C017*PI/180.0) - C019*cos(C017*PI/180.0))" 084 

"N05" "[N/m]" "((C022*cos((4.815 + 17.89)*PI/180.0) + C023*sin((4.815 + 

17.89)*PI/180.0))/0.22985)" 085 

"T05" "[N/m]" "((-C022*sin((4.815 + 17.89)*PI/180.0) + C023*cos((4.815 + 

17.89)*PI/180.0))/0.22985)" 086 

"QNORM05_N" "[Pa]" "(C085/(C083*0.727))" 087 

"QNORM05_T" "[Pa]" "(C086/(C084*0.727))" 088 

"QNORM05" "[Pa]" "((C087 + C088)/2.0)" 089 

"CN06" "[ - ]" "(C025*cos(C024*PI/180.0) + C026*sin(C024*PI/180.0))" 090 

"CT06" "[ - ]" "(C025*sin(C024*PI/180.0) - C026*cos(C024*PI/180.0))" 091 

"N06" "[N/m]" "((C029*cos((4.815 + 13.33)*PI/180.0) + C030*sin((4.815 + 

13.33)*PI/180.0))/0.22985)" 092 

"T06" "[N/m]" "((-C029*sin((4.815 + 13.33)*PI/180.0) + C030*cos((4.815 + 

13.33)*PI/180.0))/0.22985)" 093 

"QNORM06_N" "[Pa]" "(C092/(C090*0.704))" 094 

"QNORM06_T" "[Pa]" "(C093/(C091*0.704))" 095 

"QNORM06" "[Pa]" "((C094 + C095)/2.0)" 096 

"CN30" "[ - ]" "((1.0 - 0.68431)*C083 + 0.68431*C090)" 097 

"CT30" "[ - ]" "((1.0 - 0.68431)*C084 + 0.68431*C091)" 098 

"N30" "[N/m]" "((1.0 - 0.68431)*C085 + 0.68431*C092)" 099 

"T30" "[N/m]" "((1.0 - 0.68431)*C086 + 0.68431*C093)" 100 

"QNORM30" "[Pa]" "((1.0 - 0.68431)*C089 + 0.68431*C096)" 101 

"CM30" "[ - ]" "0" 102 

"M30" "[N]" "0" 103 

"CN09" "[ - ]" "(C032*cos(C031*PI/180.0) + C033*sin(C031*PI/180.0))" 104 

"CT09" "[ - ]" "(C032*sin(C031*PI/180.0) - C033*cos(C031*PI/180.0))" 105 

"N09" "[N/m]" "((C036*cos((4.815 + 5.52)*PI/180.0) + C037*sin((4.815 + 

5.52)*PI/180.0))/0.22985)" 106 

"T09" "[N/m]" "((-C036*sin((4.815 + 5.52)*PI/180.0) + C037*cos((4.815 + 

5.52)*PI/180.0))/0.22985)" 107 

"QNORM09_N" "[Pa]" "(C106/(C104*0.634))" 108 

"QNORM09_T" "[Pa]" "(C107/(C105*0.634))" 109 

"QNORM09" "[Pa]" "((C108 + C109)/2.0)" 110 

"CN10" "[ - ]" "(C039*cos(C038*PI/180.0) + C040*sin(C038*PI/180.0))" 111 

"CT10" "[ - ]" "(C039*sin(C038*PI/180.0) - C040*cos(C038*PI/180.0))" 112 

"N10" "[N/m]" "((C043*cos((4.815 + 3.81)*PI/180.0) + C044*sin((4.815 + 

3.81)*PI/180.0))/0.22985)" 113 

"T10" "[N/m]" "((-C043*sin((4.815 + 3.81)*PI/180.0) + C044*cos((4.815 + 

3.81)*PI/180.0))/0.22985)" 114 

"QNORM10_N" "[Pa]" "(C113/(C111*0.611))" 115 

"QNORM10_T" "[Pa]" "(C114/(C112*0.611))" 116 

"QNORM10" "[Pa]" "((C115 + C116)/2.0)" 117 

"CN47" "[ - ]" "((1.0 - 0.33093)*C104 + 0.33093*C111)" 118 

"CT47" "[ - ]" "((1.0 - 0.33093)*C105 + 0.33093*C112)" 119 

"N47" "[N/m]" "((1.0 - 0.33093)*C106 + 0.33093*C113)" 120 
 176

"T47" "[N/m]" "((1.0 - 0.33093)*C107 + 0.33093*C114)" 121 

"QNORM47" "[Pa]" "((1.0 - 0.33093)*C110 + 0.33093*C117)" 122 

"CM47" "[ - ]" "0" 123 

"M47" "[N]" "0" 124 

"CN63" "[ - ]" "(C046*cos(C045*PI/180.0) + C047*sin(C045*PI/180.0))" 125 

"CT63" "[ - ]" "(C046*sin(C045*PI/180.0) - C047*cos(C045*PI/180.0))" 126 

"N63" "[N/m]" "((C050*cos((4.815 + 1.21)*PI/180.0) + C051*sin((4.815 + 

1.21)*PI/180.0))/0.22985)" 127 

"T63" "[N/m]" "((-C050*sin((4.815 + 1.21)*PI/180.0) + C051*cos((4.815 + 

1.21)*PI/180.0))/0.22985)" 128 

"QNORM63_N" "[Pa]" "(C127/(C125*0.542))" 129 

"QNORM63_T" "[Pa]" "(C128/(C126*0.542))" 130 

"QNORM63" "[Pa]" "((C129 + C130)/2.0)" 131 

"CM63" "[ - ]" "0" 132 

"M63" "[N]" "0" 133 

"CN16" "[ - ]" "(C053*cos(C052*PI/180.0) + C054*sin(C052*PI/180.0))" 134 

"CT16" "[ - ]" "(C053*sin(C052*PI/180.0) - C054*cos(C052*PI/180.0))" 135 

"N16" "[N/m]" "((C057*cos((4.815 - 0.15)*PI/180.0) + C058*sin((4.815 – 

0.15)*PI/180.0))/0.22985)" 136 

"T16" "[N/m]" "((-C057*sin((4.815 - 0.15)*PI/180.0) + C058*cos((4.815 – 

0.15)*PI/180.0))/0.22985)" 137 

"QNORM16_N" "[Pa]" "(C136/(C134*0.472))" 138 

"QNORM16_T" "[Pa]" "(C137/(C135*0.472))" 139 

"QNORM16" "[Pa]" "((C138 + C139)/2.0)" 140 

"CN17" "[ - ]" "(C060*cos(C059*PI/180.0) + C061*sin(C059*PI/180.0))" 141 

"CT17" "[ - ]" "(C060*sin(C059*PI/180.0) - C061*cos(C059*PI/180.0))" 142 

"N17" "[N/m]" "((C064*cos((4.815 - 0.51)*PI/180.0) + C065*sin((4.815 – 

0.51)*PI/180.0))/0.22985)" 143 

"T17" "[N/m]" "((-C064*sin((4.815 - 0.51)*PI/180.0) + C065*cos((4.815 – 

0.51)*PI/180.0))/0.22985)" 144 

"QNORM17_N" "[Pa]" "(C143/(C141*0.448))" 145 

"QNORM17_T" "[Pa]" "(C144/(C142*0.448))" 146 

"QNORM17" "[Pa]" "((C145 + C146)/2.0)" 147 

"CN80" "[ - ]" "((1.0 - 0.62407)*C134 + 0.62407*C141)" 148 

"CT80" "[ - ]" "((1.0 - 0.62407)*C135 + 0.62407*C142)" 149 

"N80" "[N/m]" "((1.0 - 0.62407)*C136 + 0.62407*C143)" 150 

"T80" "[N/m]" "((1.0 - 0.62407)*C137 + 0.62407*C144)" 151 

"QNORM80" "[Pa]" "((1.0 - 0.62407)*C140 + 0.62407*C147)" 152 

"CM80" "[ - ]" "0" 153 

"M80" "[N]" "0" 154 

"CN95" "[ - ]" "(C067*cos(C066*PI/180.0) + C068*sin(C066*PI/180.0))" 155 

"CT95" "[ - ]" "(C067*sin(C066*PI/180.0) - C068*cos(C066*PI/180.0))" 156 

"N95" "[N/m]" "((C071*cos((4.815 - 1.49)*PI/180.0) + C072*sin((4.815 – 

1.49)*PI/180.0))/0.22985)" 157 

"T95" "[N/m]" "((-C071*sin((4.815 - 1.49)*PI/180.0) + C072*cos((4.815 – 

1.49)*PI/180.0))/0.22985)" 158 

"QNORM95_N" "[Pa]" "(C157/(C155*0.378))" 159 

"QNORM95_T" "[Pa]" "(C158/(C156*0.378))" 160 

"QNORM95" "[Pa]" "((C159 + C160)/2.0)" 161 

"CM95" "[ - ]" "0" 162 

"M95" "[N]" "0" 163 

0 Number of channels to have moving averages generated for them; next line ignored if 0 

Format for moving-average info is: Col_Title(10 char max), channel #, averaging period 

001 Channel number of time 
 177

004 Channel number of the primary wind-speed 

0 Number of channel pair(s) to have load roses generated for them; next line ignored if 0 

Format for column info is: Rose_Title(8 char max), 0 degree load, 90 degree load, # sectors 

45 Number of channels to be azimuth averaged; next four lines ignored if 0 

073,074,075,076,077,078,079,080,081,082,101,122,131,152,161,097,098,102,118,119,123,125,126,132,148,149,15

3,155,156,162,099,100,103,120,121,124,127,128,133,150,151,154,157,158,163 

360 Number of azimuth bins = number of equally spaced sectors 

005 Channel number of azimuth angle 

True Output azimuth averages to a file? 

0 Number of channel pairs that will have their crosstalk removed; next line ignored if 0 

Format for crosstalk info is: OutCol #1, OutCol #2, XT(1,1), XT(1,2), XT(2,1), XT(2,2). 

0 Number of channels to be modified by the peak finder; next line ignored if 0 

0 

0 Number of channels that will have probability density functions (PDFs) generated for them; next 2 lines 

ignored if 0 

20 Number of bins used 

Format for column info is: Column #, Minimum, Maximum. If Min=Max=0, autocalculate them. 

0 Number of channels that will have rainflow cycle counts generated for them; next 6 lines ignored if 0 

1 Rainflow counting period [sec] 

False Normalize rainflow cycle counts by bin width? 

True For bins with zero counts, output a space if we are using tab-delimited output? 

10 Number of rainflow range bins; enter 0 to output the actual cycles instead of binned cycles 

1 Number of rainflow means bins; enter 1 to output ranges only 

Format for column info is: Column #, Max Range, Min Mean, Max Mean. 

0 Number of groups of parameters that will have their extreme events recorded; next line ignored if 

0 

Format for column info is: Group_Title(100 char max), #ExtCols, ColList(#ExtCols long), #InfCols(may be 0), 

ColList(#InfCols long) 

0 Number of channels will have statistics put in separate summary files; next line ignored if 0 

0 

0 Number of channels that will have their statistics extrapolated; next line ignored if 0 

Format for statistics info is: Col_#, Hours_to_extrapolate_to, Quantile desired 

1 Number of input files to read 

S0700000.all 
REPORT DOCUMENTATION PAGE Form Approved 

 OMB NO. 0704-0188 

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson 

Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503. 

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE

December 2003 

3. REPORT TYPE AND DATES COVERED 

Technical Report 

4. TITLE AND SUBTITLE 

Modeling of the UAE Wind Turbine for Refinement of FAST_AD 

6. AUTHOR(S) 

J.M. Jonkman 

5. FUNDING NUMBERS

WER3.2010 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

National Renewable Energy Laboratory 

1617 Cole Blvd. 

Golden, CO 80401-3393 

8. PERFORMING ORGANIZATION 

REPORT NUMBER 

NREL/TP-500-34755

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES 10. SPONSORING/MONITORING 

AGENCY REPORT NUMBER

11. SUPPLEMENTARY NOTES

 

12a. DISTRIBUTION/AVAILABILITY STATEMENT

National Technical Information Service 

U.S. Department of Commerce 

5285 Port Royal Road 

 Springfield, VA 22161

12b. DISTRIBUTION CODE

 

13. ABSTRACT (Maximum 200 words)

The Unsteady Aerodynamics Experiment (UAE) research wind turbine was modeled both aerodynamically and structurally in 

the FAST_AD wind turbine design code, and its response to wind inflows was simulated for a sample of test cases. A study 

was conducted to determine why wind turbine load magnitude discrepancies—inconsistencies in aerodynamic force 

coefficients, rotor shaft torque, and out-of-plane bending moments at the blade root across a range of operating conditions—

exist between load predictions made by FAST_AD and other modeling tools and measured loads taken from the actual UAE 

wind turbine during the NASA-Ames wind tunnel tests. The acquired experimental test data represent the finest, most 

accurate set of wind turbine aerodynamic and induced flow field data available today. A sample of the FAST_AD model input 

parameters most critical to the aerodynamics computations was also systematically perturbed to determine their effect on load 

and performance predictions. Attention was focused on the simpler upwind rotor configuration, zero yaw error test cases. 

Inconsistencies in input file parameters, such as aerodynamic performance characteristics, explain a noteworthy fraction of 

the load prediction discrepancies of the various modeling tools. 

15. NUMBER OF PAGES 14. SUBJECT TERMS 

wind energy; wind turbine; FAST_AD; modeling tools; design codes; UAE; unsteady 

aerodynamics experiment 16. PRICE CODE

17. SECURITY CLASSIFICATION 

OF REPORT

Unclassified 

18. SECURITY CLASSIFICATION 

OF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATION 

OF ABSTRACT

Unclassified

20. LIMITATION OF ABSTRACT

UL 

 NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89) 

Prescribed by ANSI Std. Z39-18 

298-102