by enrico giuseppe agostino antonini · 2018. 11. 17. · enrico giuseppe agostino antonini doctor...

CFD-based Methodology for Wind Farm Layout Optimization

by

Enrico Giuseppe Agostino Antonini

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Department of Mechanical and Industrial EngineeringUniversity of Toronto

c© Copyright 2018 by Enrico Giuseppe Agostino Antonini

Abstract

CFD-based Methodology for Wind Farm Layout Optimization

Enrico Giuseppe Agostino Antonini

Doctor of Philosophy

Department of Mechanical and Industrial Engineering

University of Toronto

2018

Driven by concerns on climate change and global warming, increasing oil prices, government support

and public receptiveness, wind energy harvesting is emerging as one of the fastest growing renewable

energy technologies. Most wind energy is nowadays produced by wind farms, which consist of hundreds

of turbines to take advantage of economies of scale. Wind farm performance is however affected by

the wakes generated by the turbines which can significantly diminish their annual energy production.

Accurate wake effect predictions and reliable wind farm layout design become therefore critical aspects

to the economic success of a wind farm project. The present research project aims therefore to define an

innovative design framework that integrates accurate wake effect predictions for the development of the

next generation wind farms as part of the strategy for promoting the transition to a renewable energy

generation.

Computational fluid dynamics (CFD) provides a unique tool to simulate wind turbine wakes because

of its capability to provide a complete solution for the flow field in complex configurations. Nevertheless,

CFD simulations are strongly influenced by the choice of the turbulence model used to close the Reynolds-

averaged Navier-Stokes (RANS) equations. We therefore conducted an analysis of different turbulence

models and their influence on the results of CFD wind turbine simulations to suggest the most suitable

for such applications. Even though proper turbulence modeling is adopted, several studies showed

however that the effectiveness of RANS models in wind farm simulations has not always been consistent.

We therefore hypothesized this limitation to arise from uncertainties generated by the wind direction

variability and proposed a modeling framework that, by accounting for such uncertainties, consistently

improved the agreement of the CFD predictions with the experimental observations. To integrate the

CFD models in a design methodology, we developed an innovative continuous adjoint formulation for

gradient calculations within the framework of a gradient-based wind farm layout optimization. By testing

this optimization methodology under different wind farm configurations, wind resource distributions and

terrain topography, we showed that this unique CFD-based design framework effectively improved the

annual energy production of a proposed wind farm by optimally siting its turbines.

ii

Acknowledgements

With this short note, I would like to express my sincere gratitude to all people and institutions that

supported me during this 4-year-long research at the University of Toronto.

First, I would like to thank my advisor, Professor Cristina Amon, for the opportunity that she gave

me to be part of her research group and for her guidance and inspiration during my studies. Under her

supervision, I could fully dedicate to my own path and have the freedom and independence of pursuing my

research interests. I would also like to thank the other members of my supervisory committee, Professor

Pierre Sullivan and Professor Jim Wallace, for their comments and recommendations to develop and

improve the quality of my research project. I am also grateful to Professor Amy Bilton and Professor

Peter Hamlington for their willingness to be my thesis examiners and for their feedback on my thesis.

At the University of Toronto, I have had the pleasure of being part of a wonderful research group

at the ATOMS lab, without which my research experience would not have been so enjoyable. Very

special thanks go to Dr. David Romero, who was a supportive mentor and provided me with insightful

comments and advice during the course of this research. I thank all the lab members with whom I

have had the chance to develop a profound professional and personal relationship: Aditya, Alba, Armin,

Carlos, Daniela, Danyal, David G., Elyar, Hank, Harmit, Helen, Jim, Jonathan, Juan, Khash, Leslie,

Matthew, Omri, Patrick, Ruslan, Ryan, Sami, Sean, Summer, Tij.

This research would not have been possible without the support from several institutions. The

Department of Mechanical and Industrial Engineering at the University of Toronto provided the funding,

the material and the courses through which I gained a precious knowledge for this project, and a

stimulating environment where I could explore the work of other brilliant researchers. This research

was also funded in part by the Hatch Graduate Scholarship for Sustainable Energy Research and the

Metcalfe Family Graduate Fellowship for Sustainable Energy Research.

The most important acknowledgement goes to my wife, Miriam Dal Bosco, for her immense love and

unconditional support throughout this entire life experience. She always believed in me and was a great

source of encouragement. I would also like to thank my family and, particularly, my parents for their

support from back home.

Enrico G. A. Antonini

Toronto, Canada

Summer of 2018

iii

Contents

Acknowledgements iii

Table of Contents iv

List of Tables vii

List of Figures ix

1 Introduction 1

1.1 State of the art in wind farm wake modeling and layout optimization . . . . . . . . . . . . 4

1.1.1 Wake modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Layout optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.3 Research gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Thesis objective and proposed investigations . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Analysis of Turbulence Models for Wind Turbine Wake Simulations 16

2.1 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Sexbierum wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.2 Nibe wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 CFD methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Turbulence modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Actuator disk modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.3 Surface boundary layer modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Turbulence model constants for SBL and wind turbine simulations . . . . . . . . . . . . . 24

2.4 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

iv

2.4.2 Wall functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3 Mesh sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Wind Farm Simulations incorporating Wind Direction Uncertainty 40

3.1 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Sexbierum wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.2 Nibe wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.3 Horns Rev wind farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 CFD modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.1 Turbulence modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Wind turbine and surface boundary layer modeling . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 Wind turbine modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


3.4 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49


3.4.2 Wall functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Modeling uncertainty with simulation ensembles (MUSE) . . . . . . . . . . . . . . . . . . 51

3.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6.2 Results with MUSE Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Continuous Adjoint Formulation for WFLO: A 2D Implementation 61

4.1 The adjoint method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Continuous adjoint formulation for the wind farm layout optimization problem . . . . . . 63

4.2.1 Adjoint equations for the laminar and frozen-turbulence cases . . . . . . . . . . . . 69

4.2.2 Adjoint equations for the turbulent case . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


4.3.2 Verification results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4 Optimization methodology for the WFLO problem . . . . . . . . . . . . . . . . . . . . . . 79

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

v

5 Toward WFLO in Complex Terrain 87

5.1 Wind turbine and surface boundary layer modeling . . . . . . . . . . . . . . . . . . . . . . 87

5.1.1 Wind turbine modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88


5.2 Verification of the 3D continuous adjoint formulation for WFLO . . . . . . . . . . . . . . 89

5.3 Optimization methodology for the WFLO problem in complex terrains . . . . . . . . . . . 92

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Conclusions and Future Directions 97

6.1 Outcomes and their impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Bibliography 101

vi

List of Tables

2.1 Wind turbine characteristics and wind conditions of Sexbierum and Nibe wind farms . . . 17

2.2 Baseline turbulence model constants derived for a turbulence decay exponent of 1.2 . . . . 27

2.3 Modified turbulence model constants derived for a turbulence decay exponent of 0.9 . . . 27

2.4 Modified turbulence model constants derived for a turbulence decay exponent of 0.6 . . . 27

2.5 Mesh sensitivity analysis for the Sexbierum case. The monitored values reported in the

table are the rotor normal average velocity and the streamwise velocity at 2.5D down-

stream. For each of these two quantities, the value obtained with each of the turbulence

models is reported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Mesh sensitivity analysis for the Nibe case. The monitored values reported in the table

are the rotor normal average velocity and the streamwise velocity at 2.5D downstream.

For each of these two quantities, the value obtained with each of the turbulence models

is reported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7 Root-mean-square errors (RMSE) between the experimental data and the simulations

results when considering the wind direction range between -30 and 30. The RMSEs are

classified by case (Sexbierum and Nibe), by quantity of interest (normalized wind speed

(NWS), normalized turbulence kinetic energy (NTKE), and turbulence intensity (TI)),

by downstream distance, and by turbulence model. . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Wind turbine characteristics and wind conditions of Sexbierum and Nibe wind farms . . . 42

3.2 Turbulence model constants for SBL and wind turbine simulations [6, 7] . . . . . . . . . . 46

3.3 Root-mean-square errors (RMSE) between the experimental data and the simulations

results when considering the wind direction range ±30 for the Sexbierum and Nibe cases,

and ±15 for the Horns Rev case. The RMSEs are classified by case (Sexbierum, Nibe,

and Horns Rev), by quantity of interest (normalized wind speed (NWS) and normalized

power (NP)), by downstream distance, and by turbulence model. . . . . . . . . . . . . . . 54

vii

3.4 Coefficients of the fitting function of Eq. 3.18 for each of the normalized power distribu-

tions plotted in Fig. 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 Percentage difference in the absolute value and angular difference in the direction of the

gradients computed by the central difference approach and the adjoint method in the

laminar case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77



frozen-turbulence case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78



turbulent case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4 Normalized AEP and improvement with respect to the initial configuration obtained at

the end of the optimization for each of the initial wind farm layouts and for each of the

wind roses (WR) used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1 Relative magnitude of the gradient computed with the adjoint method, ‖∇Jrel‖, and

errors in magnitude, err∇J , and direction, errθ, between the gradient calculated with the

adjoint method and the central difference approach on a grid of locations downstream of

the turbine for the frozen-turbulence case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Relative magnitude of the gradient computed with the adjoint method, ‖∇Jrel‖, and

errors in magnitude, err∇J , and direction, errθ, between the gradient calculated with the

adjoint method and the central difference approach on a grid of locations downstream of

the turbine for the turbulent case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3 Normalized AEP and improvement with respect to the initial configuration obtained at

the end of the optimization for each of the initial wind farm layouts and for each of the

cases tested. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

viii

List of Figures

1.1 Energy balance in 2015 [83]. The figure on the left shows the world primary energy supply

by fuel, which is then transformed/refined into the final energy consumption, shown on

the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 World electricity generation by fuel in 2015 [83]. . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Global distribution of annual average onshore wind power potential [106]. . . . . . . . . . 4

1.4 Wake generated by a single turbine. V0 = 10 m/s, CT = 0.7. Distances are non-

dimensional, normalized by the diameter of the turbine rotor. [131] . . . . . . . . . . . . . 7

2.1 Performance curves of the Sexbierum and Nibe wind turbines . . . . . . . . . . . . . . . . 18

2.2 Schematic layouts of the domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Schematic of the boundary conditions used in the simulations. . . . . . . . . . . . . . . . . 29

2.4 Wind speed and turbulence kinetic energy downstream the Sexbierum wind turbine as

a function of wind direction for the k − ε model with the baseline and modified sets of

coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Wind speed and turbulence kinetic energy downstream the Sexbierum wind turbine as

a function of wind direction for the k − ω model with the baseline and modified sets of

coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6 Wind speed and turbulence kinetic energy downstream the Sexbierum wind turbine as a

function of wind direction for the SST k − ω model with the baseline and modified sets

of coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7 Wind speed and turbulence kinetic energy downstream the Sexbierum wind turbine as a

function of wind direction for the RSM with the baseline and modified sets of coefficients 35

2.8 Wind speed and turbulence intensity downstream the Nibe wind turbine as a function of

wind direction for the k − ε model with the baseline and modified sets of coefficients . . . 37

ix


wind direction for the k − ω model with the baseline and modified sets of coefficients . . . 37


wind direction for the SST k − ω model with the baseline and modified sets of coefficients 38


wind direction for the RSM with the baseline and modified sets of coefficients . . . . . . . 38

3.1 Performance curves of the Sexbierum, Nibe, and Horns Rev wind turbines . . . . . . . . . 43

3.2 Schematic layouts of the domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Example of weighted averaging. A moving weighted averaging process based on a Gaussian

distribution is applied to all wind directions . A Gaussian distribution centered at θ = 0

with σθ = 5 is plotted for clarification purposes. . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Normalized wind speed downstream the Sexbierum wind turbine as a function of wind

direction for the different turbulence models and for different downstream distance. . . . . 56

3.5 Normalized wind speed downstream the Nibe wind turbine as a function of wind direction

for the different turbulence models and for different downstream distance. . . . . . . . . . 56

3.6 Normalized power production of turbine 17 operating in the wake of turbine 07 at the

Horns Rev wind farm as a function of wind direction for different turbulence models. The

continuous lines correspond to Eq. 3.18 fitted to the normalized power data for each of

the turbulence models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.7 Normalized averaged wind speed downstream the Sexbierum wind turbine as a function

of wind direction for the different turbulence models and for different downstream distance. 59

3.8 Normalized averaged wind speed downstream the Nibe wind turbine as a function of wind

direction for the different turbulence models and for different downstream distance. . . . . 59

3.9 Normalized averaged power production of turbine 17 operating in the wake of turbine 07

at the Horns Rev wind farm as a function of wind direction for different turbulence models. 59

4.1 Schematic illustrating the calculation of material partial derivatives. . . . . . . . . . . . . 67

4.2 Schematic of the layout for the verification case. The two grey areas represent the volumes

where the wind turbine momentum sources are applied. The black dots indicate the

different positions where the second wind turbine is placed when the gradient is calculated. 74

4.3 Gradient computation for the laminar case. The figures show the results obtained by the

adjoint method compared to a central difference discretization approach for the different

lateral positions of the second wind turbine at 10D and 15D downstream. . . . . . . . . . 76

x

4.4 Gradient computation for the frozen-turbulence case. The figures show the results ob-

tained by the adjoint method compared to a central difference discretization approach for

the different lateral positions of the second wind turbine at 10D and 15D downstream. . . 77



the different lateral positions of the second wind turbine at 10D and 15D downstream. . . 77

4.6 Flow chart reproducing the optimization methodology used to solve the wind farm layout

optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.7 Schamatic of the 2D domain used to test the optimzation methodology. . . . . . . . . . . 80

4.8 Initial layouts used as inputs for the optimization process. . . . . . . . . . . . . . . . . . . 81

4.9 Optimization results obtained from a regular initial layout. For each wind rose on the

left, the final optimal layout is shown in the center along with the value of the normalized

annual energy production (AEP) through the iterations on the right. . . . . . . . . . . . . 82

4.10 Optimization results obtained from a random initial layout. For each wind rose on the

left, the final optimal layout is shown in the center along with the value of the normalized

annual energy production (AEP) through the iterations on the right. . . . . . . . . . . . . 83

5.1 Schematic of the layout for the verification case. The two grey areas represent the volumes

where the wind turbine momentum sources are applied. The black dots indicate the

different positions where the second wind turbine is placed when the gradient is calculated. 89



the different lateral positions of the second wind turbine at 4D and 6D downstream. . . . 90

5.3 Gradient computation for the turbulent case. The figures show the results obtained by the

adjoint method compared to a central difference discretization approach for the different

lateral positions of the second wind turbine at 4D and 6D downstream. . . . . . . . . . . 91

5.4 Flow chart reproducing the optimization methodology used to solve the wind farm layout

optimization problem in complex terrians . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5 Illustration of the domains used for the application of the 2D and 3D methodologies. . . . 93

5.6 Wind rose and initial layouts used as inputs for the optimization process. . . . . . . . . . 94

5.7 Optimal layouts obtained at the end of the optimization process starting from a regular

initial layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

xi

5.8 Optimal layouts obtained at the end of the optimization process starting from a random

initial layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

xii

Chapter 1

Introduction

In recent years, there has been a growing interest on renewable energy resources, such as solar, geother-

mal, wave, and wind. These resources provide energy derived from natural processes that are replenished

constantly over human timescales. Since they are not direct sources of pollution and greenhouse gases,

they can be considered the ideal candidates to tackle one of the most concerning problems of our time:

global warming. This is defined as the observed century-scale increase in planet’s average surface tem-

perature. The Intergovernmental Panel on Climate Change (IPCC) Fifth Assessment Report concluded

in 2013 that “it is extremely likely that human influence has been the dominant cause of the observed

warming since the mid-20th century” [148]. The largest human influence has been identified in the car-

bon dioxide and other human-made emissions into the atmosphere. To take action against this problem,

most of world governments (195 as of June 2018) have signed the Paris Agreement [156] by which the

signing countries aim to limit the increase of the global average temperature to below 1.5 C above

pre-industrial levels.

The current energy scenario, which can be visualized in Fig. 1.1 and 1.2, indicates that many efforts,

technological advancements, and regulations are still pending for a transition to a sustainable energy

generation future. Based on the International Energy Agency’s 2017 report [83], renewables contributed

to 18.8% of the global energy consumption and 23% of the global electricity generation in 2015. On the

other hand, fossil fuels, which are responsible for most of the greenhouse gas emissions, contributed to

79.3% and 66.4% of the global energy consumption and electricity generation, respectively.

Concerns on climate change and global warming, coupled with growing oil prices and government

support, are however driving increasing renewable energy legislation, incentives and commercialization

[14, 22, 135]. According to the International Renewable Energy Agency [84], renewable energies are

1

Chapter 1. Introduction 2

Figure 1.1: Energy balance in 2015 [83]. The figure on the left shows the world primary energy supplyby fuel, which is then transformed/refined into the final energy consumption, shown on the right.

Figure 1.2: World electricity generation by fuel in 2015 [83].


projected to reach a 36% share in the global energy consumption by 2030, doubling the renewable

energy use with respect to 2010. This scenario envisions wind energy as one of the fastest growing

renewable technologies with an grow rate of about 8% per year for the foreseeable future [62]. This

growing trend can already be observed in markets such as North America, Asia, Africa and Europe [82].

For example, Europe is set to have 20% of its energy demand met by renewables by 2020, where wind

is expected to supply about 12%-13% of total demand [51]. This has created a huge market for wind-

energy-related products and services, and has led to significant developments in wind energy science,

technology and policy [26].

Beside the positive consequences that wind energy generation would have on the environment, other

effects are commonly seen as positive on the social and political scenes. By observing the global map of

wind energy resource, shown in Fig. 1.3, it is possible to notice that wind energy is a resource widely

distributed on the planet. Higher wind power potentials are seen near coastal regions and over large flat

lands. As opposed to fossil-fuel reserves that are concentrated in few territories, the wide distribution

and accessibility that characterize wind energy as well as other renewable energies is certainly going to

promote a transition towards a decentralization of energy provision and systems. A growing consensus

now views this transition as a key strategy to address energy security issues in terms of availability,

affordability, and resilience [21, 22, 157]. In fact, energy independence and security could have a strong

impact on geopolitical tensions and armed conflicts that have tragically occurred in modern times.

Figure 1.3 also shows that wind power potential is quite variable from region to region. This potential

is the most important factor in determining the future success of a wind energy project [120]. Therefore,

the first step in the design of a wind energy project is the definition of the site thanks to scientific

forecasting methods and by considering the proximity to population centers. To consider is also that, as

more wind capacity is developed on-shore, both land use and environmental impact become increasingly

important constraints during the design process [136]. Given current trends, it is expected that new

installations would be increasingly likely to be close to human dwellings, thus stressing the importance

of environmental and public policy aspects of the design of wind energy systems [136, 163, 169, 171].

Most wind energy is nowadays produced by wind farms, which consist of hundreds of turbines and

take advantage of economies of scale to reduce procurement, permit issuance and project development

costs. Wind farm performance is however affected by the wakes generated by the turbines [159] which

can diminish their annual energy production by 10%-20% [72]. The wakes are regions of low wind speed

that are the result of the kinetic energy extracted by the turbines. The wakes lower the speed of the wind

entering the turbines placed downstream and, consequently, reduce their power production. Accurate

wake effect predictions and reliable wind farm layout design become therefore critical aspects to the


Figure 1.3: Global distribution of annual average onshore wind power potential [106].

economic success of a wind farm project.

The present research project aims therefore to define an innovative design framework that integrates

accurate wake effect predictions for the development of the next generation wind farms as part of

the strategy for promoting the transition to a renewable energy generation and reducing greenhouse

emissions. A review of the state of the art in wind farm wake modeling and layout optimization is

conducted and current research gaps are identified. Based on the literature review, a project plan is

defined which has the potential to advance our knowledge of wind farm modeling and simulation and to

create new and reliable design frameworks for the wind turbine industry.

1.1 State of the art in wind farm wake modeling and layout

optimization

This section aims to give an overview of the currently used models to simulate the wake effects in wind

farms, the algorithms to solve the wind farm layout optimization problem, and how they are combined.

1.1.1 Wake modeling

One important phenomenon that has to be investigated with regards to wind farms is the wake generated

by wind turbines, which lowers the wind speed experience by the turbines placed downstream and, as

a consequence, reduces their power production [11, 13, 15, 147]. Accurate modeling of wake effects is

therefore crucial for a correct estimation of the annual energy production and for an optimal design of

the wind turbines placement. Different approaches exist to model wind turbine wakes, namely analytical


and numerical models [159]. Whereas analytical wake models have the advantage of being simple and

computationally efficient, numerical models, which rely on Computational Fluid Dynamics (CFD), offer

higher accuracy and flexibility to handle different ambient conditions.

The wakes generated by wind turbines can be generally divided in two distinct regions: the near

wake, where the influence of the rotor blades on the flow field can be clearly distinguished, and the far

wake, where the effect of rotor blades is diminished by the mixing of undisturbed flow and the flow inside

the wake. The near-wake effects are usually restricted to a region up to 2-5 diameters downstream the

wind turbine, even though the transition from one region to the other is gradual and not well defined.

The near-wake models are used for predicting performance and loads of wind turbines. Typical

models describing the near wake are the Blade Element-Momentum (BEM) models [16, 74], vortex wake

models [5, 152] and CFD models [63, 75]. While some of these methods have been applied to study

the flow through wind turbines, they are computationally expensive or, in some cases, inadequate for

wind farm calculations. Far-wake models can be divided in analytical models, which are based on self-

similar velocity deficit profiles obtained from experimental and theoretical work, and CFD models, which

calculate the flow magnitudes at every point of the flow field [137]. From the viewpoint of wind farm

simulations, the far-wake models are the most suitable for this kind of simulations where the local effects

of rotor blades are no longer important due to the distance between wind turbines, and these will be

analyzed in the following sections.

Analytical models

Analytical wake models are based on self-similar velocity deficit profiles obtained from experimental and

theoretical work and have the advantage of being simple and computationally efficient. Among the most

used analytical wake models are the ones developed by Jensen [87], Larsen [103], and Frandsen et al. [56].

The constitutive equations use empirical constants that have to be tuned on experimental data [37, 118].

Ad hoc models for wakes overlapping from multiple turbines need also to be introduced for wind farm

power calculations [93, 99]: they usually assume simple superimposition of turbine wakes and neglect the

complex turbulent mixing occurring in wind farms. Because of these simplifications, the analytical wake

models are not capable of accurately dealing with flow structures introduced by atmospheric conditions,

changes in terrain roughness, speed-up effects around turbines or terrain features, and complex flow

phenomena such as wake meandering [9, 12, 18, 38, 102, 139, 150].

Jensen model In the development of his model, Jensen assumed a linearly expanding wake with

a velocity deficit that is only dependent of the distance behind the rotor [87]. The following expressions


are derived to describe the wake radius and center-line velocity:

R = R0 + αx;V

V0= 1− 1−

√1− CT

(1 + αx/R0)2

where R and R0 are the wake and rotor radius, respectively, V and V0 are the center-line wake velocity

and undisturbed wind speed, respectively, and x is the distance downstream the rotor. The value of α

is generally taken to be 0.075, which is adequate for land cases, whereas for offshore applications a value

of 0.04 is recommended.

Larsen model The Larsen model is based upon the assumption that Prandtl’s turbulent boundary

layer equations can be used to describe the wake behind a turbine, considering the flow incompressible,

stationary and axisymmetric [103]. The equation used to determine the velocity deficit at the position

x downstream (obtained with a first-order approximation) and the radius of the wake are:

V

V0= 1− 1

9

(CTA (x+ x0)

−2)1/3

R3/2

(3c21CTA (x+ x0)

)−1/2 −(

35

2π

)3/10 (3c21)−1/5

2

R =

(35

2π

)1/5 (3c21)1/5

(CTA (x+ x0))1/3

where A is the rotor area, c1 is the non-dimensional mixing length and x0 is another constant that

denotes the turbine’s position with respect to the applied coordinate system. The equations used to

estimate these two constants were given in Ref. [121] and were determined empirically.

Frandsen model A more recent analytical model has been proposed by Frandsen et al. [56] to

predict the wind speed deficit in wind farms with a regular layout, i.e. with straight rows of wind

turbines and equidistant spacing between units in each row and equidistant spacing between rows. The

model includes three different wake regimes. In the first regime single or multiple wake flow is present

without interaction between wakes, the second regime takes into account the mixing of different wakes,

and the third regime models the interaction between wake flow and the planetary boundary layer. The

wake radius, the wake expansion parameter β, and the wake velocity were given for the case without

wake interaction by:

R = R0

(βk/2 + α

x

2R0

)1/k

; β =1 +√

1− CT2√

1− CT=

(ReffR

)2

;V

V0= 1− 1

2

(1−

√1− 2

A0

ACT

)

where the values of k and α are 2 and 0.7, respectively.


(a) Jensen model (b) Larsen model (c) Frandsen model

Figure 1.4: Wake generated by a single turbine. V0 = 10 m/s, CT = 0.7. Distances are non-dimensional,normalized by the diameter of the turbine rotor. [131]

CFD models

The use of CFD wake models in wind farm investigations has been undergoing a rapid growth thanks

to improvements of computational technologies and resources. The first CFD study dates back to 1985,

when Crespo et al. [43] developed a CFD code to analyze the wake of wind turbines in the atmospheric

surface layer. Since then, especially in the last 15 years, many works have been proposed in literature

that covered numerical, modeling, and accuracy issues of CFD wind turbine simulations [137].

The CFD models calculate the flow magnitudes at every point of the flow field: they are based on

the solution of the Navier-Stokes (NS) equations. Two different approaches are commonly used to solve

the NS equations. The first one uses a time-averaging procedure for the solution, producing the so-called

Reynolds-averaged Navier-Stokes (RANS) equations, which require additional turbulence modeling to

close the system of equations. The second approach aims to reduce the range of time- and length-scales

of turbulence via a high-pass filtering, producing the so-called large eddy simulations (LES), which

also require additional modeling for small turbulence scales. CFD models require a substantially larger

computer capacity than analytical models, although their requirements are well within the capabilities

of modern computers, not only in the case of single wakes but also for multiple wakes occurring in a wind

farm. On the other hand, the solution of the entire flow field allows these models to be more accurate

and to take into account also terrain topography and atmospheric conditions.

RANS models RANS models used in wind farm simulation can be divided in two groups based

on the assumptions they use to consider the pressure gradient in the turbine region, namely parabolic

and elliptic models [23]. The parabolic models are obtained by neglecting both the diffusion term and

the pressure gradient in the stream-wise direction, which allows to shift the solution technique from

time marching to space marching, reducing the computational time required to converge to the final

solution. The solution of the flow field starts downstream the wind turbine where the pressure gradient


no longer dominates the flow by prescribing a velocity profile. Due to this assumption, the parabolic

models cannot solve the near-wake region where pressure gradient dominates the flow, and they can be

used only when there is a predominant flow direction in the simulation. One of the first parabolic models

was developed by Ainslie [2], who assumed axial symmetry for the flow and an eddy viscosity method

for turbulence. A different model that does not assume axial symmetry is UPMWAKE, developed by

Crespo and Hernandez [41]. The UPMWAKE model was the foundation for the WAKEFARM model

of the Energy Research Centre of the Netherlands (ECN) [138]. The UPMWAKE model was extended

to UPMPARK to calculate parks with many turbines, using a sequence of single-wake computations,

where the outflow of one wake forms the inflow condition on the next one [40]. The elliptic models are

based on the full solution of RANS equations, which allows to solve the flow field also in the near-wake

region. One of the first elliptic codes was developed by Crespo and Hernandez [42], who extended the

original parabolic code to an elliptic version. Other implementations of elliptic RANS models can be

found in Ref. [18, 27, 28, 29, 73, 90, 107, 111, 113, 143, 144] where these models are shown to be

capable of simulating different wind farm configurations, atmospheric conditions, terrain roughness and

topography.

Many RANS studies in literature focus on the turbulence modeling and how it affects the wake

velocity and energy predictions. One of the most used turbulence models is the k−ε model, which found

implementation in many works, e.g. [24, 41, 123, 133]. The results of the simulations using the k−ε model

showed quite good agreement with experimental measurements when the CFD codes used the parabolic

RANS equations (i.e., the pressure gradient is neglected and the velocity profile is prescribed behind the

wind turbine), whereas the agreement was poor when the full RANS equations was employed (elliptic

equations). This limitation was first observed by Rethore [132], who suggested that the cause may lie

in the limited validity of the eddy viscosity assumption (Boussinesq approximation) in the near-wake

region. Another turbulence model widely used is the k−ω model, whose most notable implementations

were conducted by Prospathopoulos et al. [127, 128]. Similarly to the k − ε model, the results of the

simulations using the k − ω model showed poor agreement with experimental observations. A different

approach that does not make use of the Boussinesq hypothesis and computes directly the Reynolds

stresses is the Reynolds stress model (RSM), which was tested by Cabezon et al. [24] in comparison

with the standard k − ε model.

Due to the aforementioned limitations, several authors have proposed modifications of the original

models to improve agreement with experimental data. El Kasmi and Masson [48] modified the k − ε

model adding a source term to the transport equation for the turbulent energy dissipation in a region in

close proximity to the rotor. Other modified versions use a realizable eddy viscosity formulation model


[24] or flow-dependent turbulent coefficients [158]. Prospathopoulos et al. [128] proposed a modification

of the k − ω model adjusting the turbulence model coefficients according to a lower turbulence decay.

Furthermore, the realizability constraint was applied to the k − ω model and its results were compared

to the standard model in Ref. [128].

LES models LES methods are based on filtered NS equations: they resolve the large energy-

containing eddies whereas they introduce mathematical models for smaller eddies that are strongly

affected by molecular viscosity and dissipation. Although LES models currently represent the simulation

tool with the highest fidelity, they are computationally too expensive for a design perspective. In fact,

they are mainly used to provide valuable and detailed information about the dynamics of wind turbine

wakes, to develop, test, and improve less expensive models, or to parametrize wind farm effects in weather

and climate models [20, 108].

Most LES studies on wind farm wakes have been conducted over the last 15 years, when sufficient

computational resources became available to the research community. LESs for wind farms were first

applied by Jimenez et al. [88] who showed the potential of using such simulations to capture the details

of wake flows. Subsequent studies have focused on the characterization of the wake flow dynamics for

different ambient and operating conditions [1, 33, 34, 35, 39, 44, 55, 81, 89, 161, 167]. More recent

works have instead focused on the hypothetical case of infinite wind farms to study the effect that these

would have on the atmospheric boundary layer and to understand the mechanisms of energy extraction

[25, 170, 168].

1.1.2 Layout optimization

The wind farm layout optimization (WFLO) has become a significant topic as a consequence of the

installation of an increasing number of wind farms. The main objective of a wind farm is to maximize

power production while minimizing costs, subject to constraints such as inter-turbine spacing constraints,

environmental setbacks, noise emissions, among others [94]. In this optimization process the decision

variables are the turbine locations and the objective is commonly the maximization of the annual energy

production of the farm or the minimization of its cost (i.e. initial investment and net present value

during the entire wind-farm life span) [77]. As regards to the domain representation, there have been

two approaches: continuous (the turbines can be placed anywhere in the domain) and discrete (the

domain is partitioned into a set of cells, at the center of which the turbines can be placed) [140]. The

last approach allows to lower considerably the possible locations for the wind turbines and therefore all

the interactions between turbines can be pre-calculated.


Common approaches found in the WFLO literature have focused on minimizing turbine wake in-

teractions based on analytical wake models to estimate the losses generated by the wakes. This is

because optimization methodologies that are integrated with CFD models face practical limits in terms

of computer requirements: optimization processes require a large number of evaluations, and each CFD

evaluation will usually have a significant computational cost. The number of required evaluations strictly

depends on the particular problem being solved as well as the optimization algorithm being used, but

it scales with the number of optimization variables (i.e., the number of turbines), and generally ranges

from hundreds to hundreds of thousands. A clear consequence is that a CFD-based optimization is only

possible when the number of required evaluations is relatively low and, moreover, when the duration of

a single CFD computation does not exceed a few hours at most [154]. These constraints have prevented

the use of CFD models in the WFLO problem.

The first studies on the coupling of CFD models with optimization to tackle the WFLO problem have

been recently conducted. Kuo et al. [100] proposed an algorithm that couples CFD with mixed-integer

programming (MIP) to optimize layouts on complex terrains. Thanks to the proposed methodology,

the study achieved a convenient trade-off between computational cost and solution quality. King et

al. [95, 96] developed a gradient-based approach to solve the WFLO problem which used an adjoint

method in its discrete formulation. A similar study was conducted by Funke et al. [57] on tidal turbine

array optimisation using a discrete adjoint approach. The adjoint method is a means to compute the

gradient required by gradient-based optimization methods when the objective function depends on a set

of state variables (for this case, the NS equations) [61]. Adjoint methods can generally be divided into

discrete and continuous methods [154]. Their main characteristic is that the total cost for the gradient

computation does not depend on the number of design variables but is approximately equal to that of a

single CFD evaluation. Thanks to this feature, the adjoint methods have been widely and successfully

used in aerodynamic shape optimization and geophysical tomography (e.g., [4, 85, 86, 114, 116, 122]).

Due to the complexity of the turbine wake interactions, heuristic methods, such as genetic algorithm

(GA), particle swarm optimization (PSO), and simulated annealing (SA), are widely used for layout

optimization [142]. Nonetheless, mathematical programming approaches have also been applied to the

layout problem. These two main approaches to solve the WFLO problem will be analyzed briefly.

Heuristic methods Genetic algorithm is a common and effective optimization algorithm, which

emulates the natural process of evolution as a means of progressing toward an optimum. The foundation

of GA is based on the theory of natural selection, where individuals having certain positive characteristics

(parents) have a better chance to survive and reproduce, and hence transfer their characteristics to their


offspring. Random modification of solutions (mutation) is also applied to introduce new features into

the population which might have been left out at the beginning of the optimization process. Among all

optimization methods reported in the literature, GA has been the most utilized algorithm to solve the

wind farm layout design problem. Applications can be found in Ref. [31, 49, 68, 101, 112, 141, 169, 171].

The idea of PSO originated from the concept of social model simulations that came from a swarm

of birds flocking. This simple idea of particles swarming towards a set of possible good solutions is also

suitable for wind farm layout design. The application of PSO to the WFLO problem is relatively new

and recent publications [32, 78, 79, 80, 124, 130, 162] have shown that PSO has comparable performance

to GA.

Simulated annealing is a well-known optimization algorithm that has been successfully applied to

a number of complex optimization problems. This algorithm draws from the idea of heat treatments

in metallurgy that allow a material to reach its equilibrium state. As opposed to GA and PSO, which

maintain a population of solutions in each iteration, SA maintains a single solution which is perturbed

throughout the execution of the algorithm. Successful applications of this algorithm to the WFLO

problem can be found in Ref. [17, 76].

Mathematical programming Many authors have proposed mathematical programming approaches

to tackle the WFLO problem. The most commonly used approach is to formulate mixed integer pro-

gramming (MIP) problem, which consists of an objective function and a mix of integer and continuous

variables and constraints. Examples of this applications have been proposed by Zhang et al. [172] and

Turner et al. [155]. Other mathematical programming approaches applied to the WFLO are pattern

search [45], random search [52], and interior-point method [119]. Recently, non-linear mathematical

programming that uses exact gradient information has showed great potential in tackling this prob-

lem. In particular, gradient-based methodologies using the exact derivatives of the objective function

and constraints were demonstrated to outperform genetic algorithms in terms of solution quality and

computational cost [70, 71].

1.1.3 Research gaps

The literature analyzed in the previous sections shows that research gaps are currently present for

problems that have not been answered appropriately or at all in the context of both wake modeling and

layout optimization. These can be identified as following:

• A consistent comparison of the influence of the different turbulence models on the wind farm

simulations is still missing. In fact, the k − ε and k − ω formulations followed independent paths


with regards to both model tuning and experimental validation. For example, the turbulence

constants of the k − ε and k − ω models for atmospheric surface layer and wake simulations were

determined respectively by Crespo et al. [43] and by Prospathopoulos et al. [127] with no formal

consistency between each other. The same can also be said for the RSM, where no such study

was conducted. This lack of consistency prevented an effective comparison. Also, one of the

most promising turbulence models, the shear-stress transport (SST) k − ω model, widely used in

aeronautical applications, is still missing from the literature about wind turbine simulations.

• Although RANS turbulence modeling in the context of wind turbine wake predictions has been

widely studied in literature, its effectiveness has not been always consistent. A general level

of agreement exists in identifying the k − ε and k − ω models as the least accurate for wake

predictions. For this reason, many authors have proposed modifications of the original models

to improve agreement with experimental data. However, some studies showed that CFD wake

models employing the original k − ε and k − ω models were able to provide good agreement with

experimental observations. For instance, in Ref. [12, 15] the k − ε model was used with the full

RANS equations resulting in accurate predictions of the power production of turbines operating

in wake conditions. In addition, the k − ε and k − ω models were shown to be as accurate as

the SST k − ω and Reynolds stress models in Ref. [7, 128]. The reason for this inconsistency

of wake model predictions found in some studies is very likely related to unsteady and large-

scale phenomena that, while affecting the experimental measurements, are not taken into account

in the simulations [59]. RANS wake models are usually set up as steady simulations whereas

experimental measurements are affected by unsteady phenomena that naturally occur in ambient

conditions [97]. These unsteady phenomena can be quantified by the uncertainty that is associated

with the statistics of wind speed and direction. Because wind speed variability is accounted for in

RANS simulations by the turbulence models and, in particular, by the turbulence kinetic energy,

the most significant contribution to the aforementioned limitations is therefore expected to be given

by the wind direction variability. This variability has, in fact, been shown to have a strong impact

on turbine-wake characteristics, such as velocity deficit [126]. Direct methods currently available

to account for these unsteady flow phenomena are unsteady RANS simulations (URANS) or LES,

which are computationally more expensive by orders of magnitude.

• Many different approaches have been proposed to tackle the WFLO problem, and have shown

promising results in terms of both effectiveness, optimal design and computational cost. The

common feature of the described algorithms is that they rely on analytical wake models (mostly


Jensen model) to compute the wake effect in wind farm. As pointed out in the previous section,

these models are effectively suitable for an optimization process when the wind farm is on a flat

terrain. However, their accuracy is proved to be low compared to other more complex and accurate

models, such as the CFD models that can deal with terrain features and different atmospheric

conditions. WFLO using CFD models is currently in its early stages, mainly because of practical

limits of CFD optimization in terms of computer requirements and optimization algorithms.

1.2 Thesis objective and proposed investigations

The goal of the present research project is to develop an accurate, efficient, and fast CFD-based opti-

mization methodology for wind farm layouts. It will take advantage of the most advanced optimization

algorithms for CFD applications and novel formulations of existing algorithms will be developed for the

WFLO problem.

The project is divided in three main tasks. The first task will be focused on the development of a

state-of-the-art CFD model able to accurately simulate the effects of wake losses in a wind farm. The

proposed CFD model will be validated against publicly available experimental data, in terms of both

flow field and turbulence quantities. The CFD package that will be used for this task is OpenFOAM,

an open-source CFD software package which provides out-of-the-box support for parallel computing in

clusters. Moreover, OpenFOAM offers users complete freedom to customize and extend its existing

functionality. We will leverage this flexibility to develop and implement wind turbine models based on

the actuator disk technique and elliptic RANS equations. Current research gaps in RANS models for

wind farms will be investigated and solutions proposed.

The second task will focus on the development of a fast optimization algorithm that can be fully

integrated with OpenFOAM to optimize wind farm layouts. In order to reduce the associated compu-

tational cost, the application of the most advanced optimization algorithms for CFD applications to

the WFLO problem will be investigated. A novel formulation of one of the most promising optimiza-

tion algorithms for CFD applications, the adjoint method, will be developed for the WFLO problem

and implemented in OpenFOAM. This process will also take advantage of existing and ongoing work

in our research group in both multi-objective evolutionary strategies and mathematical programming

approaches which maximize the expected power output.

In the third and final task, the proposed CFD-based optimization approach will be tested on wind

farm layouts under several scenarios: different wind farm configurations, wind resource distributions and

terrain topography. This objective will allow to have an innovative, fast, and accurate tool available to


the wind energy community, enabling a paradigm shift to simulation-based wind farm optimization, as

opposed to current work that relies only on low-accuracy, analytical models. This novel approach will

allow wind farm designers to further reduce in the cost of wind energy, allowing the electricity produced

by wind farms to be competitive with fossil fuels, thus ensuring the continued growth of wind energy.

The proposed investigations of the present project in wake modeling and layout optimization are

summarized as follows:

• A study is conducted to compare in a consistent way the principal turbulence models for RANS

equations present in literature, namely the k−ε, k−ω, and Reynolds stress model, to introduce the

SSTk−ω model as a innovative turbulence model for wind turbine simulations, and to investigate

and assess the influence of different turbulence models on the results of the CFD simulations.

Consistent turbulence model constants for atmospheric surface layer and wake flows are derived

according to appropriate experimental observations.

• An investigation is conducted on the limitations and inconsistency of the RANS models in the

predictions of wake effects in wind farms. The discrepancies and the inconsistency of the turbulence

models are hypothesized to arise from wind direction uncertainty caused by large-scale unsteady

phenomena, which though present in the experimental measurements are not accounted for in

the simulations. An approach is therefore proposed to overcome these limitations by Modeling

Uncertainty using Simulation Ensembles (MUSE), i.e., a set of CFD results for different wind

directions to generate a single CFD prediction. The predictions of CFD model are post-processed

with this innovative method for CFD simulations that accounts for the wind direction uncertainty

associated with the specific wind farm data set.

• An optimization methodology is presented for the WFLO problem that integrates the high accuracy

and flexibility offered by the CFD models and that overcomes the computationally high costs of a

CFD-based optimization. To this end, an adjoint method is developed and used in its continuous

formulation for the gradient computation. The adjoint formulation is derived for three different flow

scenarios, namely, laminar, frozen-turbulence, and turbulent flows. The derived adjoint equations

are implemented in OpenFOAM by taking advantage of the top-level syntax of the code and of

the similarity between the Navier-Stokes and adjoint equations. The gradient calculation using the

developed adjoint method is implemented in a gradient-based optimization methodology to solve

the WFLO problem.


1.3 Thesis outline

Chapter 2 presents an analysis of the turbulence models for wind turbine wake simulations in atmospheric

boundary layers. These models are compared in a consistent way by defining appropriate turbulence

model constants. The results are validated against publicly available experimental data. Modifications

are also proposed to improve agreement with experimental observations. Chapter 3 illustrates an inves-

tigation of the limitations and inconsistency of the previously analyzed turbulence models. As a result

of this investigation, an approach to overcome these limitations is proposed. In Chapter 4, an innovative

continuous adjoint formulation for the wind farm layout optimization is formulated and implemented for

a 2D case. This formulation is first verified against traditional approaches for gradient computation and

then applied to optimize the layout of a 16-turbine wind farm. Chapter 6 presents instead preliminary

results of the developed adjoint formulation in 3D cases. These results fully demonstrate that this CFD

methodology can effectively optimize wind farm layouts with different wind farm configurations, wind

resource distributions and terrain topography. In Chapter 6, conclusions are drawn and future directions

are suggested.

Chapter 2

Analysis of Turbulence Models for

Wind Turbine Wake Simulations

The work presented in this chapter aims to compare in a consistent way the principal turbulence models

present in literature, namely the k − ε, k − ω, and Reynolds stress model, to introduce the SST k − ω

model as a innovative turbulence model for wind turbine simulations, and to investigate and assess the

influence of the different turbulence models on the results of the CFD simulations. The comparison is

made consistent by a proper adjustment of the turbulence model constants according to appropriate

experimental observations of atmospheric surface layer and wake flows. The assessment of the tur-

bulence models is conducted by comparing the CFD results with the publicly available experimental

measurements of the velocity field and turbulence quantities from two stand-alone wind turbines in the

Sexbierum and Nibe wind farms, respectively. Modifications of the derived turbulence model constants

are also investigated in order to improve agreement with experimental data.

The outline of the chapter is the following. In Sec. 2.1, we present the case studies that will be used

to test and validate the CFD model with the different turbulence models. The CFD methodology is

described in Sec. 2.2 along with an overview of turbulence models used to close the RANS equations.

Section 2.3 describes how the turbulence model constants have been reformulated for wind farm appli-

cations. In Sec. 2.4, we provide details about the numerical implementation and boundary conditions

of the CFD model. Section 2.5 presents the results obtained by RANS simulations with the different

turbulence models. Conclusions are summarized in Sec. 2.6.

16

Chapter 2. Analysis of Turbulence Models for Wind Turbine Wake Simulations 17

Table 2.1: Wind turbine characteristics and wind conditions of Sexbierum and Nibe wind farms

Wind turbine D [m] H [m] Uinf [m/s] TIx [%] CT [-]Sexbierum 30.1 35 10 10 0.75Nibe 40 45 8.5 10 0.82

2.1 Case studies

The validation of the CFD model with the different turbulence closures and SBL equations is con-

ducted using the experimental data sets from the Sexbierum [36] and Nibe [151] wind farms. Table 2.1

summarizes the wind turbine characteristics and wind conditions.

2.1.1 Sexbierum wind farm

The Dutch Experimental Wind Farm at Sexbierum is located in the Northern part of The Netherlands at

approximately 4 km distance from the seashore. The wind farm is located in flat homogeneous terrain,

mainly grassland used by farmers. The wind farm has a total of 5.4 MW installed capacity consisting of

18 turbines of 300 kW rated power each. The wind turbines in the wind farm are HOLEC machines with

three WPS 30/3 blades, a rotor diameter of 30.1 m, and a hub height of 35 m. Performance curves are

reported in Fig. 2.1a. The campaign concerned measurement of the wind speed, turbulence and shear

stress behind a single wind turbine at distances of 2.5, 5.5 and 8 rotor diameters, respectively. The free

stream wind conditions at hub height were Uinf = 10 m/s and TIx = 10%. For these conditions, the

thrust coefficient was CT = 0.75.

2.1.2 Nibe wind farm

The Nibe wind farm is located on a coastal site near Aalborg in the norther Jutland, Denmark. It is

constituted by two machines (A and B) located 200 m apart from each other along an approximately

North-South axis, which runs parallel to the coast line. To the west there is a fetch of at least 6 km

over open, shallow water. On the landward site, the ground surrounding the site is flat, grass-covered,

and free of significant obstacles. The two wind turbines are almost identical, both with a rated power of

630 kW. The rotor diameter is 40 m, the hub height is 45 m. Performance curves are reported in Fig.

2.1b. The data examined here correspond to the turbine B operating alone, and measurements of wind

speed and turbulence are available behind the turbine at distances of 2.5, 4 and 7.5 rotor diameters,

respectively. The free stream wind conditions at hub height were Uinf = 8.5 m/s and TIx = 10%. For

these conditions, the thrust coefficient was estimated to be CT = 0.82.


(a) Sexbierum wind turbine (b) Nibe wind turbine

Figure 2.1: Performance curves of the Sexbierum and Nibe wind turbines

2.2 CFD methodology

The simulation model is based on the Reynolds-averaged Navier-Stokes (RANS) equations for incom-

pressible, steady flows, which require additional turbulence modeling to solve the nonlinear Reynolds

stress term and to close the system of equations. The set of equations is then composed of the continuity

equation,

∂Ui∂xi

= 0, (2.1)

and the three momentum equations,

Uj∂Ui∂xj

= −1

ρ

∂p

∂xi+

∂

∂xj

[ν

(∂Ui∂xj

+∂Uj∂xi

)− uiuj

]+f

ρ, (2.2)

where Ui,j is the mean velocity component, p is the mean pressure, ρ and ν are the fluid density and

kinematic viscosity, respectively, f is the source term, and i, j are indexes over the coordinate directions.

The Reynolds stress term uiuj is computed with the transport equations for turbulence, whose number

depends on the particular choice of the turbulence model. OpenFOAM [153] is employed to solve this

set of equations, using a control-volume-based technique to transform the governing flow equations into

algebraic expressions that can be solved numerically. The discretization of the governing equations

is based on the second-order upwind scheme, which is applied for the interpolation of velocities and

turbulent quantities. The semi-implicit method for pressure-linked equations (SIMPLE) algorithms is

used to solve simultaneously the set of equations by an iterative scheme.


2.2.1 Turbulence modeling

This section provides an overview of the turbulence models employed to close the RANS equations,

whose effect on the simulations will be discussed in the results section.

Standard k − ε model

The standard k−ε turbulence model was first proposed by Jones and Launder [92] and was subsequently

revised by Launder and Sharma [105] with the introduction of the currently used empirical constants.

It was the first two-equation model used in applied computational fluid dynamics and is still the most

widely used in many fields [125]. The model is based on the turbulent-viscosity hypothesis (Boussinesq

approximation) that relates the Reynolds stresses to the mean flow according to the following equation:

−uiuj = νt

(∂Ui∂xj

+∂Uj∂xi

)− 2

3kδij = 2νtSij −

2

3kδij , (2.3)

where k is the turbulence kinetic energy, Sij the mean strain-rate tensor, and νt is the eddy viscosity

computed as follows:

νt = Cµk2

ε. (2.4)

The turbulence kinetic energy, k, and turbulent dissipation rate, ε, are obtained from two transport

equations. The standard values of the model constants present in the k and ε equations were chosen in

order to impose certain experimental constraints and are the following: Cµ = 0.09, C1ε = 1.44, C2ε =

1.92, σk = 1, σε = 1.3. In spite of its broad rage of applicability and accurate results for simple flows,

the k−ε model has shown some limitations: it can be quite inaccurate for complex flows, in particular in

the presence of large adverse pressure gradients [166]. Also, particular near-wall treatments are usually

included since the model showed to not perform well for near-wall regions.

Standard k − ω model

Different formulations of the k−ω turbulence model were proposed in the past, but the standard model

adopted today is the one formulated by Wilcox [165], which has been more extensively tested than any

other. The model is based on the Boussinesq approximation, and the main difference with respect to

the k − ε model is the use of the specific dissipation rate (also called turbulence frequency), ω, in place

of the turbulent dissipation rate, ε. Two transport equations are use to calculate the value of k and ω,


whereas the eddy viscosity has the following definition:

νt =k

ω. (2.5)

The standard constants in the k and ω transport equations take the following values: β∗ = 0.09, β =

0.075, α = 0.556, σ∗ = 0.5, σ = 0.5.

A close similarity can be observed between the k − ε and the k − ω models when a transformation

is applied to the ω equation by adopting the definition of ω as ε/ (Cµk) and by using the following

transformed constants [46]:

T = 1/ω,

Cµ = β∗, C1ε = 1 + α,

C2ε = 1 + β/β∗, σε = 1/σ.

(2.6)

It can be seen that the values of the transformed constants are similar but not exactly equal to the

original k − ε values, mainly because of the calibration of the constants with different (but consistent)

experimental data. This transformation reproduces the standard ε model with an additional term Sω

which is defined as:

Sω =2

T(ν + σνt)

(|∇k|2

k− ∇k · ∇ε

ε

). (2.7)

This source term in the dissipation equation distinguishes the k − ε and the k − ω models and acts

mainly in the inner region of boundary layers (near walls). This characteristic helps to explain why the

k − ω model performs better than the k − ε model for boundary-layer flows, both in its treatment of

the viscous near-wall region and in its accounting for the effects of streamwise pressure gradient [125].

Two important limitations have however to be highlighted: the first is that the model showed problems

when dealing with non-turbulent free-stream boundaries so that particular (non-physical) boundary

conditions are usually required; the second is that it overpredicts the level of shear stress in adverse

pressure-gradient boundary layers [166].

SST k − ω model

The shear-stress transport (SST) k− ω turbulence model was formulated by Menter [109] and has been

found to be quite effective in predicting many aeronautical flows [46]. The reason for this is that it was

designed to yield the best behavior of the k− ε and the k−ω models: it retains the robust and accurate

formulation of the Wilcox k − ω model in the near wall region, and takes advantage of the freestream

independence of the k− ε model in the outer part of the boundary layer. A blending function takes care


of the switch between the two models according to the distance from a wall. The blending function F1

is designed to be one in the near wall region (leading to a k − ω model) and zero away from the surface

(leading to a k − ε model). The constants of the model are also calculated by interpolation of the two

original models as follows:

φ = F1φ1 + (1− F1)φ2. (2.8)

The constants of set 1 are from the k−ω model (except σk1 which is slightly different): β∗ = 0.09, β1 =

0.075, γ1 = 0.556, σk1 = 0.85, σω1 = 0.5. The constants of set 2 are from the k − ε model, derived

through Eq. 2.6: β∗ = 0.09, β2 = 0.0828, γ2 = 0.44, σk2 = 1, σω2 = 0.856.

The other important improvement introduced by Menter in the SST k − ω model with respect to

the parent models is in the shear-stress predictions in adverse pressure-gradient boundary layers. The

tendency to overestimate the shear stress is fixed by imposing a bound on the stress-intensity ratio,

|uiuj |/k. This ratio is often denoted a1 and in many flows is approximately equal to 0.3, with lower

values in adverse pressure gradients. The bound is introduced with a new definition of the eddy viscosity:

νt =a1k

max (a1ω; 2|Ωij |F2), (2.9)

where Ωij is the mean flow rotation tensor and F2 is a function that is one for boundary-layer flows and

zero for free-shear layers.

Reynolds stress model

In the Reynolds stress models, the individual Reynolds stresses are directly computed and consequently

the turbulent-viscosity hypothesis is not needed. Six transport equations take care of each Reynolds

stress. There exist different approaches to model the terms in the transport equations that have brought

about different RSMs: among the most used are the Launder-Reece-Rodi (LRR) model by Launder

et al. [104] and the Speziale-Sarkar-Gatski (SSG) model by Speziale et al. [146]. In this work it has

been chosen to use the Gibson-Launder (GB) model [60] which was developed and calibrated with the

purpose to accurately simulate atmospheric boundary layers. This model, as the other RSMs, has six

equations to compute each of the six Reynolds stresses and an equation for the turbulent dissipation

rate. The standard coefficients of this model are the following: Cµ = 0.09, C1ε = 1.44, C2ε = 1.92, σR =

0.8197, σε = 1.3, C1 = 1.8, C2 = 0.6, C ′1 = 0.5, C ′2 = 0.3. The first five coefficients are alike the ones

in the k − ε model, whereas the others are used to model different terms in the transport equations of

the Reynolds stresses.


The RSM is potentially the most general and physically the most complete, since it calculates each

of the Reynolds stresses. For this reason, it has also the potential to accurately predict anisotropic

turbulent flows, which is an important advantage compared to the eddy viscosity models limited by the

Boussinesq approximation and the assumption of isotropic flows. On the other hand, the RSM requires

significantly more computational time and CPU memory compared to the simpler two-equation models.

2.2.2 Actuator disk modeling

The wind turbine has been modeled as an actuator disk whose main feature is to apply a distributed

force, defined as axial momentum source, F , over a cylindrical volume, defined by the rotor swept area.

The actuator disk model, even though it does not provide a detailed description of the wind turbine

geometry, is able to capture adequately the wake effect generated by the wind turbine and to compute its

power output, as required for the employment in wind turbine and wind farm simulations [3, 133, 145].

From the definition of thrust coefficient, it can be derived that the axial force is a function of the reference

wind speed:

F =1

2ρπD2

4CTU

2inf , (2.10)

where ρ is the air density, D is the rotor diameter, Uinf is the upstream wind speed, and CT is the

thrust coefficient, obtained from the thrust coefficient curve of the wind turbine at the specified Uinf .

The power generated can be computed as the product of the axial force and the average velocity over

the actuator disk volume V :

P = FUx = F1

V

∫V

UxdV. (2.11)

2.2.3 Surface boundary layer modeling

The simulations of wind turbines have necessary to take into account the wind conditions and charac-

teristics usually encountered in real flows, which are referred to as atmospheric boundary layers (ABL).

The starting point is the characterization of the mean wind shear profile. For an homogeneous and

stationary flow, the shear profile can be described, according to Panofsky and Dutton [115], as

∂Ux∂z

=u∗κl, (2.12)

where U is the mean wind speed, z is the height above ground, u∗ is the local friction velocity, l is the

local length scale, and κ is the von Karman constant (≈ 0.4). Within the ABL, the friction velocity

is expected to decrease with z, vanishing at the edge of the ABL. The expression to account for this


variation is the following:

u∗ = u∗0

(1− z

zmax

)α, (2.13)

where zmax is the height of the ABL and α depends on the state of the boundary layer, ranging from

2/3 to 3/2 [69]. The height of an ABL can extend up to some kilometers, depending on the atmospheric

stability [115]. The first 10% of the ABL, which is usually called the surface boundary layer (SBL),

can be approximated by a constant friction velocity equal to u∗0. Also, in the SBL, the length scale is

assumed equal to the height (lSL = z).

The length scale, l, is influenced by the atmospheric stability, which describes the combined effects

of mechanical turbulence and heat convection, and the height of the ABL. Three classes of atmospheric

stability can be defined: unstable, neutral, and stable conditions. The case studies analyzed in this

work will take into account only the surface boundary layer in neutral conditions, which is a reasonable

approximation up to a height of at least 100m [115]. Under these hypothesis, a logarithmic velocity

profile can be derived from Eq. 2.12 by integration:

Ux =u∗0κln

(z

z0

)(2.14)

where z0 is the surface roughness length. This parameter is solely used for describing the wind speed

profile, in fact, it is not a physical length, but rather a length scale representing the roughness of the

ground (reference values for different terrain types can be found in Ref. [115]). The friction velocity can

be calculated once a reference velocity is known at a specific height:

u∗0 =κUx,ref

ln(zrefz0

) . (2.15)

Introducing the equation for the wind profile into the turbulence models, it can be derived that the

turbulence kinetic energy, turbulent dissipation rate, and specific dissipation rate have the following

expressions, respectively [134, 127]:

k =u2∗0√Cµ

, ε =u3∗0κz

, ω =u∗0√β∗κz

. (2.16)

Average values for the Reynolds stresses were extrapolated by Panofsky and Dutton [115] from

different experimental data sets. The values of the Reynolds stresses reflect the anysotropic nature

of the atmospheric boundary layer and these are given as a function of the friction velocity: uxux =

(2.39u∗0)2, uyuy = (1.92u∗0)

2, uzuz = (1.25u∗0)

2, uxuz = −u2

∗0, uxuy = uyuz = 0. From the value


of the xx-Reynolds stress, Prospathopoulos et al. [127] derived a useful relation between the surface

roughness length and the streamwise turbulence intensity, TIx, which is a common parameter used to

characterize the flow turbulent conditions. Following the definition of turbulence intensity, it is possible

to write:

TIx =ux

Ux,ref= 2.39

u∗0Ux,ref

. (2.17)

Introducing Eq. 2.15 on the right hand side of Eq. 2.17, it is possible to rearrange the equation in order

to find the value of the surface roughness length as a function of the turbulence intensity:

z0 = zrefexp

(−0.980

TIx

). (2.18)

Starting from the definition of turbulence kinetic energy, it is also straightforward to derive a relation

between the turbulence kinetic energy and the streamwise turbulence intensity:

k =1

2(uxux + uyuy + uzuz) = 5.48u2

∗0 = 0.959TI2xU

2x,ref . (2.19)

2.3 Turbulence model constants for SBL and wind turbine sim-

ulations

The standard coefficients of the turbulence models previously described have been calibrated on several

and various experimental data sets, and therefore represent a compromise to give the best performance

for a range of flows [125]. The conditions that a wind turbine simulation has to deal with represent

a particular subset of the entire range of the turbulence model applicability. In particular, two main

phenomena occurring in this application can be identified: the SBL and the wake generated by the

wind turbine that propagates in a SBL. Taking this into account, it is possible to reduce the range

of applicability of the turbulence models to the particular flow situations previously mentioned and

recalibrate the turbulence model constants with more convenient measurements from SBL and wake

flows. Crespo et al. [43] and Prospathopoulos et al. [127] were the first to propose a modification

of the constants for the k − ε and the k − ω models, respectively, for wind turbine simulation is SBL

flows. A consistent adjustment of the coefficients for the aforementioned turbulence models has not been

formulated yet and it is proposed in this study in accordance with previous works and convenient data

sets for wind turbine simulations. Furthermore, modifications of the derived turbulence model constants

are also investigated in order to improve agreement with experimental data considering different values of


turbulence decay. In total, three sets of coefficients (one baseline and two modifications) are determined

and tested on wind turbine simulations.

The first coefficient analyzed is Cµ, equivalent to β∗, which appears in the definition of the turbu-

lent viscosity (Eq. 2.4). The standard value was determined according to measurements from simple

turbulent shear flows and the logarithmic region of boundary layers: in these particular situations it is

possible to demonstrate that [125]

Cµ =

(|uiuj |k

)2

. (2.20)

The stress-intensity ratio was measured to be approximately 0.3 in those flows and Cµ was calculated

accordingly, giving the standard value of 0.09. When dealing with SBL, the stress-intensity ratio assumes,

as shown in Sec. 2.2.3, the following value:

|uiuj |k

=u2∗0k

= 0.182. (2.21)

Therefore, the value of Cµ is changed in this work to 0.0333, as also reported in Ref. [43, 127]. On

the other hand, there are no specific measurements from wind turbine wake flows that can support

the validity of this coefficient also in wind turbine wake simulations. Nevertheless, the value of the

stress-intensity ratio is supposed to be valid also in the wake of wind turbines operating in SBL flows,

based on the consideration that turbulence “remembers” the upstream conditions much longer than the

average wind speed. In particular, the anisotropy present in the SBL is expected to be retained at a

some level also in wind turbine wakes, implying that the proposed value can be a good approximation

for both cases. Beside this consideration, Durbin and Petterson [46], in a more general discussion on

the applicability of the turbulence models, suggested that the value of Cµ should be adjusted based on

more recent experimental data, proposing a value close to the one introduced for SBL and wake flows.

The second coefficient analyzed is C2ε (equivalent to β = (C2ε − 1)β∗), which controls the decaying

of turbulence. It is possible to show that in the particular case of homogeneous, isotropic turbulence, the

decaying of turbulence is controlled by a power-law solution [125]. The decay exponent, n, characterizing

the solution is correlated to the C2ε coefficient according to the following relation:

C2ε =n+ 1

n. (2.22)

Measurements of grid turbulence in wind tunnels give a value for n in the range of 1.3 ± 0.2 [46]. The

standard values of the k− ε and k−ω models used a value of 1.09 and 1.2, respectively. The turbulence

decay is expected to behave in a similar manner also in the SBL and wake flows, and therefore the


value of the decay exponent can be considered to fall in the same range. However, Prospathopoulos et

al. [128] suggested that the exponent could be lower for the anisotropic SBL flow. They tested that

assumption on a standard k − ω model, providing some improvements to the simulation predictions.

Following their approach, three values were chosen in this work: 1.2 (the baseline one), 0.9, and 0.6 (the

two modifications). The turbulence model constants were computed accordingly in order to guarantee

consistency.

The coefficient C1ε (equivalent to α = C1ε−1), for given values of Cµ and C2ε, controls the spreading

rate of free-shear flows [46]. The standard value was chosen so that the basic model would give a

reasonable value for the spreading rate in mixing layers. What actually determines the spreading rate

of free-shear flows in numerical simulations is the difference C2ε −C1ε: a difference of about 0.45− 0.50

gives a good estimation of this quantity and this is how the standard coefficients were determined [46].

Mixing layers are an occurring phenomenon in the wakes of wind turbine and the constraint previously

mentioned has to be taken into account. This was not the case when the modifications of the standard

coefficients were first proposed in Ref. [43, 127] for SBL, and a different modification is proposed here

on the basis of the aforementioned constraint. Given the values of of Cµ and C2ε previously determined,

values of C1ε are determined according to the difference C2ε − C1ε.

The value of the coefficient σε (equivalent to 1/σ) can be established by examining the log region in

boundary layers. Under this condition the following equation must hold [125]:

κ2 = σε√Cµ (C2ε − C1ε) (2.23)

from which it is possible to calculate the value for σε, given the other coefficients already determined

and the value of the von Karman constant.

The last coefficient that has to be discussed is σk (equivalent to 1/σ∗). Differently from the other

coefficients, there are no particular cases with whom its value can be determined. In fact, for the standard

k − ε model its value is set to 1, whereas for the k − ω model its value is kept equal to σ, as a tradeoff

among a broad range of experimental observations [166]. This second approach has been chosen in this

work.

The final sets of coefficients for each turbulence model are summarized in Tab. 2.2, 2.3, and 2.4 for

the baseline and modifications, respectively.


Table 2.2: Baseline turbulence model constants derived for a turbulence decay exponent of 1.2

Turbulence model Turbulence constantsk − ε Cµ = 0.0333 C1ε = 1.42 C2ε = 1.83 σk = 2.25 σε = 2.25k − ω β∗ = 0.0333 α = 0.42 β = 0.0277 σ∗ = 0.45 σ = 0.45

SST k − ω β∗ = 0.0333γ1 = 0.42 β1 = 0.0277 σk1 = 0.45 σω1 = 0.45γ2 = 0.42 β2 = 0.0277 σk2 = 0.45 σω2 = 0.45

RSM Cµ = 0.0333C1ε = 1.42 C2ε = 1.83 σR = 0.8197 σε = 2.25C1 = 1.8 C2 = 0.6 C ′1 = 0.5 C ′2 = 0.3

Table 2.3: Modified turbulence model constants derived for a turbulence decay exponent of 0.9




Table 2.4: Modified turbulence model constants derived for a turbulence decay exponent of 0.6





(a) Top view

(b) Lateral view (c) Front view

Figure 2.2: Schematic layouts of the domain

2.4 Numerical setup

The computational domain and mesh of the two cases were generated with blockMesh and snappy-

HexMesh, two mesh utilities of OpenFOAM for mesh generation and refinement, respectively. The

Cartesian coordinate system is defined with x, y, and z being respectively the streamwise, lateral and

vertical directions. Figure 2.2 illustrates schematic layouts of the domain. The dimensions of the domain

are a function of the rotor diameter. The domain includes the actuator disk region and a refined region

surrounding the disk with a double mesh resolution in order to capture the most significant gradients in

the flow field.

The dimensions of the domain were carefully determined in order not to influence the flow-field

solution and to avoid useless domain regions. In particular, larger dimensions were tested and were

subsequently decreased according to the following rule: a smaller domain is accepted only if the flow

solution does not vary by more that 1% with respect to the largest domain tested (with dimensions

as double as the ones presented here), ideally considered as the solution of an infinite domain. The

dimensions that need a detailed discussion are the distance between the inlet and the wind turbine, and

the height of the domain. In the first case, there has to be enough distance before the wind turbine to


Figure 2.3: Schematic of the boundary conditions used in the simulations.

allow the flow field perturbed by the wind turbine to propagate upstream without being influences by the

inlet boundary conditions. A distance of 3D was found to correctly satisfy this condition. In the second

case, a too short domain height would cause flow blockage and would promote a faster, non-physical

wake recovery. A height of 5D was determined according to these considerations and to practical wind

engineering reference guideline, which suggests a value of 5H, being H the height of any obstacle (in this

case the wind turbine rotor). The other dimensions were basically chosen in order to have the flow-field

solution as far as the experimental measurements are available for comparison.

For the solution of the RANS equations, the convergence criterion was set so that the residuals of

all the equations were below 10−5. A stricter convergence criterion was found to provide a negligible

difference on the solution.

2.4.1 Boundary conditions

The inlet boundary condition was defined with the equations relative to the SBL. Given the flow char-

acteristics, i.e., Uinf , TIx, and H, the values for z0 and u∗0 were derived with Eqs. 2.15 and 2.18. The

velocity, turbulence kinetic energy (or Reynolds stresses), and turbulence dissipation rate (or specific

dissipation rate) were then prescribed according to Eqs. 2.14 and 2.16, depending on the turbulence

model used. The outlet boundary condition was defined as a pressure outlet, with zero gradient for the

velocity and turbulence quantities. The top boundary condition was defined by prescribing constant

values of velocity, turbulence kinetic energy (or Reynolds stresses), and turbulence dissipation rate (or

specific dissipation rate) at the domain height, whereas zero gradient was set for the pressure. The side

boundary condition was defined as zero gradient for all the variables. The ground was defined as a rough

wall, with wall functions that took care of the turbulence quantities. A schematic of these boundary

condition is illustrated in Fig. 2.3.


2.4.2 Wall functions

A proper treatment of the ground surface is essential to correctly simulate SBL flows. A general require-

ment of CFD simulations consists in having a very fine mesh in proximity of any surface in order to

capture the large velocity gradients and to compute a correct wall shear stress. In SBL simulations, this

is impossible because the surface roughness prevents a full solution of the boundary layers. In fact, the

first wall-adjacent cell should be at least the double of the surface roughness, which is in conflict with

the requirement of a high mesh resolution. In these cases, wall functions based on log-law boundary

layers for rough walls are used to calculate the turbulent viscosity and wall shear stress. Blocken et al.

[19] discussed the problem of the wall treatment for these particular flows, suggesting remedies when the

simulations are run with Ansys Fluent or CFX (which adopt wall functions based on an equivalent sand-

grain roughness, kS , equivalent to approximately 30z0). OpenFOAM, differently from the previously

mentioned CFD packages, has a wall function which is based on the actual surface roughness length, z0,

and which is derived from Eq. 2.14. This was used in the present work and allowed to have a higher

resolution close to the wall than the one reached with Ansys Fluent and CFX. A value of approximately

0.01D for the first cell at the wall was found to guarantee a correct simulation of SBL flows, achieving

horizontally homogeneity (i.e., zero streamwise gradients) of the SBL in an empty domain. This value

is also consistent with other works present in literature [128, 24].

2.4.3 Mesh sensitivity analysis

A mesh sensitivity analysis was conducted in order to reduce spatial discretization errors in the CFD

simulations and to guarantee a mesh-independent solution. Different grid resolutions were tested for

each turbulence model and the relative error of the designated flow variables was measured. The global

grid spacing was decreased progressively by a factor of 1.5, starting from the coarsest case where the

global spacing was 0.225D. The resolution in the refined region surrounding the wind turbine was as

double as the global resolution. In the region close to the wall, the resolution was also higher: the first

cell at the wall was fixed to a height of 0.01D and this value was progressively increased moving away

from the wall, up to the size given by the global resolution. The height of the region where this mesh

refinement took place was 0.5D.

Given the aforementioned considerations, four different global grid spacings were tested, namely

0.225D, 0.150D, 0.100D, and 0.067D (see Tab. 2.5 and 2.6). The values that were monitoread are the

rotor normal average velocity and the streamwise velocity and turbulence quantities at two locations

downstream the wind turbine, namely 2.5D and 8D for the Sexbierum case, and 2.5D and 7.5D for the


Table 2.5: Mesh sensitivity analysis for the Sexbierum case. The monitored values reported in the tableare the rotor normal average velocity and the streamwise velocity at 2.5D downstream. For each of thesetwo quantities, the value obtained with each of the turbulence models is reported.

Rotor normal average velocity [m/s] Streamwise velocity at 2.5D [m/s]Grid resolution [D] k − ε k − ω SST k − ω RSM k − ε k − ω SST k − ω RSM0.225 7.82 7.86 7.26 7.35 8.13 8.24 6.05 5.560.150 7.89 7.93 7.38 7.46 8.11 8.22 5.86 5.580.100 7.92 7.97 7.44 7.48 8.10 8.22 5.73 5.530.067 7.99 8.02 7.51 7.53 8.08 8.20 5.69 5.58

Table 2.6: Mesh sensitivity analysis for the Nibe case. The monitored values reported in the table arethe rotor normal average velocity and the streamwise velocity at 2.5D downstream. For each of thesetwo quantities, the value obtained with each of the turbulence models is reported.

Rotor normal average velocity [m/s] Streamwise velocity at 2.5D [m/s]Grid resolution [D] k − ε k − ω SST k − ω RSM k − ε k − ω SST k − ω RSM0.225 5.84 5.92 3.95 4.09 6.69 6.89 4.15 3.810.150 5.92 6.00 4.37 4.65 6.68 6.88 3.81 3.720.100 6.39 6.44 5.75 5.82 6.66 6.84 3.78 3.830.067 6.42 6.49 5.80 5.87 6.66 6.83 3.77 3.85

Nibe case. The number of cells obtained for the different resolutions was approximately 40·, 130·, 400·,

and 1300 · 103, respectively. A global grid spacing of 0.1D was found to guarantee a mesh independent

solution: the percentage difference of the calculated velocities and turbulence quantities for all the

turbulence models with respect to a lower grid spacing (0.067D) was found to be less than 1%. This

result is consistent with other computational studies on wind turbine wake simulations [24, 128, 158].

2.5 Results and discussion

This section includes the results obtained from the developed CFD wake model when applied to the

stand-alone Sexbierum and Nibe wind turbine cases. The simulated wind speed and turbulent quantities

were compared with the real wind turbine measurements in order to assess the implemented turbulence

models and the CFD model as a whole. Root-mean-square errors (RMSE) were calculated between the

experimental data and the simulations results and are reported in Tab. 2.7. The simulations were solved

with simpleFoam, the OpenFOAM steady-state solver for incompressible, turbulent flows, that run on

a Inter(R) Core(TM) i7-4790 computer with 3.60 Ghz clock time using 6 processors. The number of

iterations required to reach the convergence of the solution was about 400 for the k−ε and k−ω models,

300 for the SST k−ω model, and 800 for the RSM. The computational time required for the simulations

to converge ranged from approximately 20 minutes for the SST k−ω model to 40 minutes for th RSM.

Figures 2.4, 2.5, 2.6, and 2.7 show the normalized wind speed and the turbulence kinetic energy


downstream the Sexbierum wind turbine as a function of the wind direction. The comparison with ex-

perimental data was conducted at three downstream locations, namely 2.5D, 5.5D, and 8D downstream

the wind turbine, in order to assess the numerical results obtained with the four turbulence models. The

wind direction refers to relative direction of the incoming flow where 0 indicates the direction behind

the center of the rotor at which the maximum wind speed deficit is expected.

With regard to the results for the baseline coefficients, the wind speed was captured well by the

SST k − ω and Reynolds stress models for the three locations, but it was highly overestimated by the

k − ε and k − ω models, especially at 2.5D downstream where the RMSEs were the highest (0.1859

and 0.1936, respectively). This overestimation is consistent with previous works that highlighted the

limitations of these two models [132, 128]. The reason of the failure is very likely caused by the incorrect

prediction of the eddy viscosity in situations of adverse pressure gradients, such as the one experienced in

the near wake of the wind turbine. Indeed, the results with the SST k−ω model, whose eddy viscosity

is bounded to prevent the aforementioned behavior, were accurate and very similar to the predictions of

the RSM, which does not rely on the the turbulent-viscosity hypothesis.

Similar results can be observed for the turbulence kinetic energy in the Sexbierum case. The k − ε

and k − ω models were unable to predict the peaks of turbulence kinetic energy generated by the tip

vortexes which are present in the near wake of a wind turbine. Further downstream, the turbulence

kinetic energy profile was more homogeneous and the predictions of the two models were improved. The

SST k − ω and Reynolds stress models provided a turbulence kinetic energy profile which was similar

to the experimental data but the predicted value was slightly underestimated. Also in this case, their

predictions improved in the far wake providing a good agreement with the experimental data.

Figures 2.8, 2.9, 2.10, and 2.11 show the normalized wind speed and the turbulence intensity down-

stream the Nibe wind turbine as a function of the wind direction. The streamwise turbulence intensity

could not be computed directly from the eddy-viscosity CFD simulations (k− ε, k−ω and SST k−ω),

but was instead obtained with the inverse of Eq. 2.19 assuming that the anisotropy present in the SBL is

retained also in the wind turbine wake. Similarly to the previous case, the comparison with experimental

data was conducted at three downstream locations, namely 2.5D, 4D, and 7.5D downstream the wind

turbine.

The prediction of wind speed in the Nibe case by the baseline turbulence models presented similar

characteristics to the previous case. At the location 2.5D downstream the wind turbine the results

provided by the SST k−ω and Reynolds stress models matched very well the experimental observations

(RMSEs of 0.0614 and 0.0510, respectively), whereas the agreement was not as good for the locations

further downstream where an underestimation of the wind speed was observed. This underestimation


Table 2.7: Root-mean-square errors (RMSE) between the experimental data and the simulations resultswhen considering the wind direction range between -30 and 30. The RMSEs are classified by case(Sexbierum and Nibe), by quantity of interest (normalized wind speed (NWS), normalized turbulencekinetic energy (NTKE), and turbulence intensity (TI)), by downstream distance, and by turbulencemodel.

k − ε k − ωWind Turbine Quantity Distance baseline 1 2 baseline 1 2

Sexbierum

NWS2.5D 0.1859 0.1691 0.1328 0.1936 0.1796 0.14805.5D 0.0730 0.0656 0.0502 0.0804 0.0747 0.06108D 0.0545 0.0510 0.0449 0.0603 0.0577 0.0518

NTKE2.5D 0.0114 0.0116 0.0121 0.0114 0.0115 0.01185.5D 0.0044 0.0046 0.0050 0.0044 0.0045 0.00488D 0.0042 0.0042 0.0038 0.0042 0.0041 0.0040

Nibe

NWS2.5D 0.1848 0.1625 0.1119 0.1936 0.1757 0.13244D 0.0683 0.0560 0.0386 0.0738 0.0641 0.04287.5D 0.0453 0.0425 0.0424 0.0468 0.0439 0.0394

TI2.5D 0.0300 0.0282 0.0190 0.0301 0.0290 0.02234D 0.0239 0.0233 0.0182 0.0236 0.0235 0.02067.5D 0.0217 0.0210 0.0180 0.0215 0.0212 0.0195

spaceSST k − ω RSM

Wind Turbine Quantity Distance baseline 1 2 baseline 1 2

Sexbierum

NWS2.5D 0.0947 0.0916 0.0878 0.0797 0.0794 0.07895.5D 0.0468 0.0512 0.0601 0.0462 0.0436 0.04278D 0.0480 0.0514 0.0584 0.0431 0.0419 0.0425

NTKE2.5D 0.0148 0.0155 0.0169 0.0149 0.0147 0.01455.5D 0.0057 0.0063 0.0073 0.0045 0.0047 0.00538D 0.0038 0.0037 0.0040 0.0036 0.0035 0.0034

Nibe

NWS2.5D 0.0614 0.0568 0.0531 0.0510 0.0496 0.04754D 0.0619 0.0752 0.0912 0.0776 0.0730 0.07057.5D 0.0505 0.0595 0.0723 0.0435 0.0417 0.0440

TI2.5D 0.0202 0.0241 0.0313 0.0147 0.0141 0.01374D 0.0149 0.0157 0.0207 0.0182 0.0176 0.01297.5D 0.0163 0.0147 0.0146 0.0208 0.0178 0.0137


(a) 2.5D downstream (b) 5.5D downstream (c) 8D downstream

(d) 2.5D downstream (e) 5.5D downstream (f) 8D downstream

Figure 2.4: Wind speed and turbulence kinetic energy downstream the Sexbierum wind turbine as afunction of wind direction for the k − ε model with the baseline and modified sets of coefficients



Figure 2.5: Wind speed and turbulence kinetic energy downstream the Sexbierum wind turbine as afunction of wind direction for the k − ω model with the baseline and modified sets of coefficients




Figure 2.6: Wind speed and turbulence kinetic energy downstream the Sexbierum wind turbine as afunction of wind direction for the SST k − ω model with the baseline and modified sets of coefficients



Figure 2.7: Wind speed and turbulence kinetic energy downstream the Sexbierum wind turbine as afunction of wind direction for the RSM with the baseline and modified sets of coefficients


of the far-wake velocity is likely caused by unsteady phenomena (meandering of the wake) that were

not taken into account in the simulations. The k − ε and k − ω models, instead, provided a wind speed

significantly higher than the observed (the highest RMSEs were 0.1848 and 0.1936, respectively).

Focusing on the turbulence intensity provided by the baseline turbulence models, it is possible to see

that the k − ε and k − ω models exhibited the same incorrect behavior as previously discussed. The

predicted turbulence intensity profile did not show the characteristic peaks in the near wake of the wind

turbine, and the predicted values were also generally higher that the experimental data for all the three

locations. A much better agreement was observed using the SST k−ω and Reynolds stress models, both

in terms of intensity and profile. The similar values of turbulence intensity obtained by the SST k − ω

and Reynolds stress models supports the assumption made to calculate it. A remark has to be made

on the low level of turbulence intensity observed experimentally for wind directions higher than 20

(right side of the figures): winds coming from those directions experience open, shallow water and are

characterized by lower turbulence intensity which was not taken into account in the simulations.

The modified sets of turbulence constants, determined following the proposition of Prospathopoulos

et al. [128], had the effect of decreasing the wind speed and the wind speed recovery. This is particularly

evident for the k−ε and k−ω models, whose predictions are improved with respect to the baseline set of

coefficients in the two cases analyzed. The highest RMSEs decreased respectively to 0.1328 and 0.1480

for the Sexbierum case and to 0.1119 and 0.1324 for the Nibe case. The improvement was even more

evident for the other locations. This trend suggests that decreasing the decay exponent is beneficial

for the k − ε and k − ω models. On the other hand, the effect of decreasing the decay exponent was

deleterious on the predictions of the SST k − ω model, especially for the locations at 5.5D and 8D for

the Sexbierum case, and 4D and 7.5D for the Nibe case. No significant effect was instead observed for

the prediction of the RSM, where the results changed negligibly.

These modified sets of turbulence constants influenced also the turbulence quantities. In particular,

it is possible to notice that the predictions of the k − ε and k − ω models were improved with respect

to the models using the baseline coefficients: the turbulence kinetic energy in the Sexbierum case and

the turbulence intensity in the Nibe case showed the characteristic peaks in the near wake of the wind

turbines. Instead, this influence was not beneficial for the SST k − ω model, whose results were more

inaccurate. With regard to the RSM, no significant effect was observed as for the wind speed.


(a) 2.5D downstream (b) 4D downstream (c) 7.5D downstream

(d) 2.5D downstream (e) 4D downstream (f) 7.5D downstream

Figure 2.8: Wind speed and turbulence intensity downstream the Nibe wind turbine as a function ofwind direction for the k − ε model with the baseline and modified sets of coefficients



Figure 2.9: Wind speed and turbulence intensity downstream the Nibe wind turbine as a function ofwind direction for the k − ω model with the baseline and modified sets of coefficients




Figure 2.10: Wind speed and turbulence intensity downstream the Nibe wind turbine as a function ofwind direction for the SST k − ω model with the baseline and modified sets of coefficients



Figure 2.11: Wind speed and turbulence intensity downstream the Nibe wind turbine as a function ofwind direction for the RSM with the baseline and modified sets of coefficients


2.6 Conclusions

The present study was conducted in order to compare in a consistent way the principal turbulence models

present in literature, namely the k−ε, k−ω, and Reynolds stress model, to introduce the SST k−ω model

as a innovative turbulence model for wind turbine simulations, and to investigate and assess the influence

of the different turbulence models on the results of the CFD simulations. The turbulence models were

implemented in simulations of two stand-alone wind turbines modeled with the constant-distribution

actuator disk approach. The wind turbines operated in atmospheric environment which was modeled

with the atmospheric surface layer theory. Consistent turbulence model constants for atmospheric surface

layer and wake flows were derived according to appropriate experimental observations.

The results considered in this study included quantities such as wind speed, turbulence kinetic energy

and turbulence intensity in the wake region of the two stand-alone turbines. The results showed that

the SST k − ω model performed as good as the the RSM, which is recognized as the most complete

model with general applicability. The results obtained with these two models and with the baseline set

of coefficients matched quite accurately the experimental observation, both in terms of wind speed and

turbulence quantities. On the other hand, the simulations using the k − ε and k − ω models provided

poor predictions of wake flows, as already documented in literature.

Modified sets of coefficients were also investigated in order to improve agreement with experimental

data. These sets of coefficients improved the predictions of the k−ε and k−ω models, which were however

not as good as the predictions from the baseline SST k−ω and RSM. The effect of the modified sets of

coefficients on these latter models was not effective, and was even deleterious for the SST k − ω.

From the results of this study it is possible to conclude that the SST k−ω can be used as an effective

turbulence model for wind turbine simulations without any particular modification of its coefficients. Its

results were showed to be similar to those of the RSM model but obtained at a much faster computation

time.

Chapter 3

Wind Farm Simulations

incorporating Wind Direction

Uncertainty

In the present chapter, we aim to investigate the limitations and inconsistency of the RANS wake models

and to propose an innovative approach to overcome them. A CFD model was initially developed using

the actuator disk technique to simulate the wind turbines and the surface boundary layer approximation

to simulate the ambient conditions. The developed CFD model was implemented to simulate three

different wind farms, namely, Sexbierum, Nibe, and Horns Rev, with publicly available experimental

measurements. The main turbulence models present in literature and available in common CFD software

packages, namely, the k− ε, k−ω, SST k−ω and Reynolds stress models, were used to close the RANS

equations and their results compared. Following the same approach of other studies (e.g., [24, 41,

133, 127, 128, 143]), the validation of the developed CFD model with different turbulence closures was

conducted by comparing the CFD predictions with both observed wind speeds and power production of

the selected wind farms.

To account for the wind direction variability, we subsequently introduced a method to model the wind

direction uncertainty using simulation ensembles, i.e., a set of CFD results for different wind directions

is post-processed to generate a single CFD prediction. Our results showed that RANS simulations using

the SST k − ω and Reynolds stress models were consistently more accurate when considering wind

speeds and power production in the wake region of the considered wind farms. These are therefore

40

Chapter 3. Wind Farm Simulations incorporating Wind Direction Uncertainty 41

to be preferred over the k − ε and k − ω models. The result also showed that the proposed approach

for considering wind direction uncertainty was able to overcome the limitations and inconsistency of

previous works. Overall, this method improved the agreement between the experimental data and CFD

models with the suggested turbulence closures by accounting for the uncertainty in the wind direction

reported in the data sets.

The outline of the chapter is the following. In Sec. 3.1, we present the case studies that will be used to

test and validate the CFD model and the innovative post-processing technique. The CFD methodology

is described in Sec. 3.2 along with an overview of turbulence models used to close the RANS equations.

Section 3.3 describes the modeling assumptions for the wind turbines and the surface boundary layer that

are integrated in the CFD model. In Sec. 3.4, we provide details about the numerical implementation

and boundary conditions of the CFD model. The innovative post-post-processing technique, called in

this paper MUSE (Modeling Uncertainty with Simulation Ensembles), is presented in Sec 3.5. Section

3.6 compares the results obtained directly from the RANS simulations and the results obtained after the

averaging process with the MUSE method. Conclusions are summarized in Sec. 3.7.

3.1 Case studies

The validation of the proposed approach for CFD models, which will be described in the following

sections, was conducted using the experimental data sets from the Sexbierum [36], Nibe [151], and

Horns Rev [72] wind farms. Table 3.1 summarizes the wind turbine characteristics and wind conditions.

3.1.1 Sexbierum wind farm

The Dutch Experimental Wind Farm at Sexbierum is located in the Northern part of The Netherlands at

approximately 4 km distance from the seashore. The wind farm is located in flat homogeneous terrain,

mainly grassland used by farmers. The wind farm has a total of 5.4 MW installed capacity consisting of

18 turbines of 300 kW rated power each. The wind turbines in the wind farm are HOLEC machines with

three WPS 30/3 blades, a rotor diameter of 30.1 m, and a hub height of 35 m. Performance curves are

reported in Fig. 3.1a. The campaign concerned measurement of the wind speed, turbulence and shear

stress behind a single wind turbine at distances of 2.5, 5.5 and 8 rotor diameters, respectively. The free

stream wind conditions at hub height were Uinf = 10 m/s and TIx = 10%. For these conditions, the

thrust coefficient was CT = 0.75. The wind direction bin width was 2.5.


Table 3.1: Wind turbine characteristics and wind conditions of Sexbierum and Nibe wind farms

Wind farm D [m] H [m] Uinf [m/s] TIx [%] CT [-]Sexbierum 30.1 35 10 10 0.75Nibe 40 45 8.5 10 0.82Horns Rev 80 70 8 8 0.80

3.1.2 Nibe wind farm

The Nibe wind farm is located on a coastal site near Aalborg in the norther Jutland, Denmark. It is

constituted by two machines (A and B) located 200 m apart from each other along an approximately

North-South axis, which runs parallel to the coast line. To the west there is a fetch of at least 6 km

over open, shallow water. On the landward site, the ground surrounding the site is flat, grass-covered,

and free of significant obstacles. The two wind turbines are almost identical, both with a rated power of

630 kW. The rotor diameter is 40 m, the hub height is 45 m. Performance curves are reported in Fig.

3.1b. The data examined here correspond to the turbine B operating alone, and measurements of wind

speed and turbulence are available behind the turbine at distances of 2.5, 4 and 7.5 rotor diameters,

respectively. The free stream wind conditions at hub height were Uinf = 8.5 m/s and TIx = 10%. For

these conditions, the thrust coefficient was estimated to be CT = 0.82. The wind direction bin width

was 2.5.

3.1.3 Horns Rev wind farm

The Horns Rev wind farm is located 14 km from the west coast of Denmark. It has a rated capacity

of 160 MW comprising 80 wind turbines, which are arranged in a regular array of 8 by 10 turbines.

The wind turbines are installed with an internal spacing along the main directions (West-East) of 7D,

whereas the diagonal wind turbine spacing is either 9.4D or 10.4D. The wind farm comprises Vestas

V80 turbines, which are 2 MW pitch-controlled, variable speed wind turbines with an 80 m diameter

and a 70 m hub height. The performance of these turbines is reported in Fig. 3.1c. The data from

the measurement campaign include the power output from the different rows of wind turbines and for

different wind directions and sectors. In this work, two turbines were considered, namely, the turbines

07 and 17, which are 7D apart along the West-East direction with the turbine 07 facing undisturbed

winds coming from West. The power measurements as a function of wind direction are taken from Ref.

[117] where the free stream wind conditions were Uinf = 8 m/s and TIx = 8%. For these conditions,

the thrust coefficient was estimated to be CT = 0.80. The wind direction bin width was 5.


(a) Sexbierum wind turbine (b) Nibe wind turbine

(c) Horns Rev wind turbine

Figure 3.1: Performance curves of the Sexbierum, Nibe, and Horns Rev wind turbines


3.2 CFD modeling

Reynolds-averaged Navier-Stokes (RANS) equations for incompressible, steady flows are chosen as the

basis of the simulation model in this study. They require additional turbulence modeling to solve the

nonlinear Reynolds stress term and to close the system of equations. The set of equations is composed

of the continuity equation,

∂Ui∂xi

= 0, (3.1)

and three momentum equations,

Uj∂Ui∂xj

= −1

ρ

∂p

∂xi+

∂

∂xj

[ν

(∂Ui∂xj

+∂Uj∂xi

)− uiuj

]+f

ρ, (3.2)

where Ui,j is the mean velocity component, p is the mean pressure, ρ and ν are the fluid density and

kinematic viscosity, respectively, f is the source term, and i, j are indexes over the coordinate directions.

Transport equations for turbulence quantities are employed to compute the Reynolds stress terms, uiuj ,

and their number depends on the particular choice of the turbulence model. OpenFOAM [153] is em-

ployed to solve this set of equations, using a control-volume-based technique to transform the governing

flow equations into algebraic expressions that can be solved numerically. The discretization of the gov-

erning equations is based on the second-order upwind scheme, which is applied for the interpolation of

velocities and turbulent quantities. The semi-implicit method for pressure-linked equations (SIMPLE)

algorithms is used to solve simultaneously the set of equations by an iterative scheme.

3.2.1 Turbulence modeling

Four different turbulence models are used to close the RANS equations and compared to each other:

k − ε, k − ω, SST k − ω, and Reynolds stress models. A more detailed description of the constitutive

equations of the turbulence models used in this study can be found in Ref. [7].

The standard k − ε turbulence model, first developed by Jones and Launder [92] and subsequently

revised by Launder and Sharma [105], is currently the most widely used model in many fields [125].

The model is based on the turbulent-viscosity hypothesis (Boussinesq approximation) that relates the

Reynolds stresses to the mean flow via the eddy viscosity. In spite of its broad rage of applicability and

accurate results for simple flows, the k− ε model has shown some limitations: it can be quite inaccurate

for complex flows, in particular in the presence of large adverse pressure gradients [166]. Also, special

near-wall treatments are usually required since the model has been shown to underperform in near-wall

regions.


In this study, we adopt the standard k−ω model formulated by Wilcox [165] which, similarly to the

k− ε model, relies on the Boussinesq approximation to calculate the Reynold stresses. The k−ω model

has been shown to perform better than the k − ε model for boundary-layer flows, both in its treatment

of the viscous near-wall region and in its accounting for the effects of streamwise pressure gradient [125].

However, the model showed problems when dealing with non-turbulent free-stream boundaries so that

special boundary conditions are usually required. Also, it was shown that the k − ω model overpredicts

the level of shear stress in adverse pressure-gradient boundary layers [166].

The shear-stress transport (SST) k − ω turbulence model was formulated by Menter [109] and has

been found to be quite effective in predicting many aeronautical flows [46]. The reason for this is that

it was designed to yield the best behavior of the k − ε and the k − ω models: it retains the robust and

accurate formulation of the Wilcox k − ω model in the near wall region, and takes advantage of the

freestream independence of the k− ε model in the outer part of the boundary layer. A blending function

takes care of the switch between the two models according to the distance from a wall. A new definition

for the turbulence viscosity was introduced by Menter to prevent the tendency to overestimate the shear

stress intensity in adverse pressure-gradient boundary layers.

In the Reynolds stress models, the individual Reynolds stresses are directly computed and con-

sequently the turbulent-viscosity hypothesis is not needed. Different formulations have been used to

model the terms in the transport equations of the Reynolds stresses [104, 146]. However, in this work

the Gibson-Launder (GB) model [60] was chosen because it was developed and calibrated specifically to

simulate atmospheric boundary layers. The RSM has six equations to compute each of the six Reynolds

stresses and an equation for the turbulent dissipation rate. Thanks to the calculation of all six Reynolds

stresses, the model can accurately predict anisotropic turbulent flows, which is an important advantage

compared to the eddy viscosity models limited by the Boussinesq approximation and the assumption

of isotropic turbulence. On the other hand, the RSM requires significantly more computational time

compared to the simpler two-equation models.

Application-specific turbulence model constants

The commonly used values for the coefficients of the turbulence models previously described have been

calibrated on several and various experimental data sets, and therefore represent a compromise to give

the best performance for a wide range of flow conditions [125]. The conditions that a wind turbine

simulation has to deal with represent a particular subset of the entire range of the turbulence model

applicability. In particular, two main phenomena occurring in this application can be identified: the

surface boundary layer (SBL) and the wake generated by the wind turbine that propagates in a SBL.


Table 3.2: Turbulence model constants for SBL and wind turbine simulations [6, 7]




Taking this into account, it is possible to reduce the range of applicability of the turbulence models

to the particular flow characteristics of wind farms in the surface boundary layer by recalibrating the

turbulence model constants based on experimental measurements. For example, Antonini et al. [6, 7]

derived and tested consistent turbulence model constants for each of the aforementioned turbulence

models for SBL and wind turbine simulations (see Tab. 3.2). In this work, we used these constants

for all our predictions, so that our results reflect the most-accurate versions of each turbulence model,

recalibrated for our specific application.

3.3 Wind turbine and surface boundary layer modeling

This section provides a description of the approach used to model in our computational domain the wind

turbines and the surface boundary layer where the turbines operate.

3.3.1 Wind turbine modeling

The wind turbine was modeled as an actuator disk which is characterized by a cylindrical volume, defined

by the rotor swept area, where a distributed force, defined as axial momentum source, F , is applied.

The main limitation of this model is that it does not provide a detailed description of the wind turbine

geometry and therefore cannot capture the flow dynamics occurring on the rotor blades. However, it

is able to capture adequately the wake effect generated by the wind turbine and to compute its power

output, as required for the employment in wind turbine and wind farm simulations [3, 132, 145]. From

the definition of thrust coefficient, it can be derived that the axial force is a function of the reference

wind speed:

F =1

2ρπD2

4CTU

2ref , (3.3)

where ρ is the air density, D is the rotor diameter, Uref is the upstream wind speed, and CT is the

thrust coefficient, obtained from the thrust coefficient curve of the wind turbine at the specified Uref .




P = FUx = F1

V

∫V

UxdV. (3.4)

In the case of wind turbines that operate in the wake of others, the value for the upstream wind speed

is not readily available. Ideally, the reference wind speed for a turbine operating in wake condition is

the speed that would be present at the turbine location without the turbine itself. Evaluating this speed

would require therefore an additional simulation for each given wind boundary condition. Because of

the high computational cost associated, two different approaches were used in literature to estimate the

reference wind speed with simpler procedures. A common choice is to use the wind speed upstream of

the rotor, at a distance of one (1D) or two (2D) rotor diameters as an estimate of the reference speed.

However, this method cannot always guarantee an accurate estimation, so a validation step is typically

recommended on a case-by-case basis. A different approach was introduced by Prospathopoulos et al.

[128], which is based on an iterative calculation that uses the definition of the axial induction factor, a,

and thrust coefficient curve of the specific wind turbine. In the present work, however, the authors chose

to calculate the reference wind speed using the ideal method: an additional simulation for each given

wind boundary condition was run to calculate the speed that would be present at the turbine location

without the turbine itself. This procedure was conducted only for the case of multiple wind turbines,

and it allowed to have the most accurate solution for the reference wind speed.


The atmospheric boundary layers (ABL) is used to model the wind conditions and characteristics usually

encountered in real wind turbine and wind farm flows. For an homogeneous and stationary flow, the

wind shear profile can be described, according to Panofsky and Dutton [115], as:

∂Ux∂z

=u∗κl, (3.5)

where Ux is the mean streamwise wind speed, z is the height above ground, u∗ is the local friction

velocity, l is the local length scale, and κ is the von Karman constant (≈ 0.4). Within the ABL, the

friction velocity decreases with z, vanishing at the edge of the ABL according the following relation:

u∗ = u∗0

(1− z

zmax

)α, (3.6)

where zmax is the height of the ABL and α depends on the state of the boundary layer, ranging from

2/3 to 3/2 [69]. The height of an ABL can extend up to some kilometers, depending on the atmospheric


stability [115]. The first 10% of the ABL, which is usually called the surface boundary layer (SBL),

can be approximated by a constant friction velocity equal to u∗0. Also, in the SBL, the length scale is

assumed equal to the height (lSL = z).

The length scale, l, is influenced by the atmospheric stability, which describes the combined effects of

mechanical turbulence and heat convection, and the height of the ABL [115]. Three classes of atmospheric

stability can be defined: unstable, neutral, and stable conditions. The experimental data set for two

of the case studies analyzed in this work, Sexbierum and Nibe, was reported to experience neutral or

near-neutral conditions, whereas, for the Horns Rev case, the experimental measurements were recorded

under unstable conditions. In spite of the substantial difference in the phenomena that drive the different

stability conditions, the measurements obtained under neutral and unstable conditions are usually very

close to each other in terms of both wind speed/power production and turbulence intensity. This can

be seen with the measurement campaign conducted at the Horns Rev wind farm [72], where neutral

and unstable conditions are even grouped together because of their similarity. For this reason, the

case studies analyzed in this work will take into account only the surface boundary layer under neutral

conditions, which is a reasonable approximation for the lowest part of the atmospheric boundary layer

where wind turbines operate [115].

Under the previously discussed hypothesis, a logarithmic velocity profile can be derived from Eq. 3.5

by integration:

Ux =u∗0κln

(z

z0

), (3.7)

where z0 is the surface roughness length. This parameter is solely used for describing the wind speed

profile, in fact, it is not a physical length, but rather a length scale representing the roughness of the

ground (reference values for different terrain types can be found in Ref. [115]). The friction velocity can

be calculated once a reference velocity is known at a specific height:

u∗0 =κUx,ref

ln(zrefz0

) . (3.8)

Introducing the equation for the wind profile into the turbulence models, it can be derived that the

turbulent kinetic energy, turbulent dissipation rate, and specific dissipation rate have the following

expressions, respectively [134, 127]:

k =u2∗0√Cµ

, ε =u3∗0κz

, ω =u∗0√β∗κz

. (3.9)


Panofsky and Dutton [115] extrapolated average values for the Reynolds stresses from different ex-

perimental data sets and these are given as a function of the friction velocity: uxux = (2.39u∗0)2,

uyuy = (1.92u∗0)2, uzuz = (1.25u∗0)

2, uxuz = −u2

∗0, uxuy = uyuz = 0. From the value of the xx-

Reynolds stress, Prospathopoulos et al. [127] derived a useful relation between the surface roughness

length and the streamwise turbulence intensity, TIx, which is a common parameter used to characterize

the flow turbulent conditions. Following the definition of turbulence intensity, it is possible to write:

TIx =ux

Ux,ref= 2.39

u∗0Ux,ref

. (3.10)

Introducing Eq. 3.8 on the right hand side of Eq. 3.10, it is possible to rearrange the equation in order

to find the value of the surface roughness length as a function of the turbulence intensity:

z0 = zrefexp

(−0.980

TIx

). (3.11)

Starting from the definition of turbulent kinetic energy, it is also straightforward to derive a relation

between the turbulent kinetic energy and the streamwise turbulence intensity:

k =1

2(uxux + uyuy + uzuz) = 5.48u2

∗0 = 0.959TI2xU

2x,ref . (3.12)

3.4 Numerical setup

The computational domain and mesh of the three cases were generated with blockMesh and snappy-

HexMesh, two mesh utilities of OpenFOAM for mesh generation and refinement, respectively. The

Cartesian coordinate system is defined with x, y, and z being respectively the streamwise, lateral and

vertical directions. Figure 3.2 illustrates schematic layouts of the domain used for the Sexbierum and

Nibe cases. The dimensions of the domain are a function of the rotor diameter (D). The domain includes

the actuator disk region and a refined region surrounding the disk with a double mesh resolution in order

to capture the most significant gradients in the flow field. For the Horns Rev case, two turbines were

instead included in the domain.

The dimensions of the domain were carefully determined in order to not influence the flow-field

solution and to avoid useless domain regions. A detailed sensitivity analysis can be found in Ref. [6, 7]

and the same procedure was followed in this work. The upstream distance from the wind turbine and

the domain height were set to 3D and 5D, respectively. For the Horns Rev case, the second wind turbine

was place at a distance of 7D downstream. The other dimensions were basically chosen in order to have


(a) Top view

(b) Lateral view (c) Front view

Figure 3.2: Schematic layouts of the domain

the flow-field solution as far as the experimental measurements are available for comparison. The global

grid spacing was set to 0.1D, whereas the resolution in the refined region surrounding the wind turbine

was as double as the global resolution in order to capture the main gradients in the flow field. In the

region close to the wall, the resolution was also higher: the first cell at the wall was fixed to a height

of 0.01D and this value was progressively increased moving away from the wall, up to the size given by

the global resolution. The height of the region where this mesh refinement took place was 0.5D. For

the solution of the RANS equations, the convergence criterion was set so that the residuals of all the

equations were below 10−5.


The inlet boundary condition was defined with the equations relative to the SBL. Given the flow char-

acteristics, i.e., Uinf , TIx, and H, the values for z0 and u∗0 were derived with Eqs. 3.8 and 3.11. The


dissipation rate) were then prescribed according to Eqs. 3.7 and 3.9, depending on the turbulence model

used. The outlet boundary condition was defined as a pressure outlet, with zero gradient for the velocity

and turbulence quantities. The top boundary condition was defined by prescribing constant values of



dissipation rate) at the domain height, whereas zero gradient was set for the pressure. The side boundary

condition was defined as zero gradient for all the variables. The ground was defined as a rough wall,

with wall functions that took care of the turbulence quantities.

3.4.2 Wall functions

A proper treatment of the ground surface is essential to correctly simulate SBL flows. A general require-

ment of CFD simulations consists in having a very fine mesh in proximity of any surface in order to

capture the large velocity gradients and to compute a correct wall shear stress. In SBL simulations, this

is impossible because the surface roughness prevents a full solution of the boundary layers. In fact, the

first wall-adjacent cell should be at least the double of the surface roughness, which is in conflict with

the requirement of a high mesh resolution. In these cases, wall functions based on log-law boundary

layers for rough walls are used to calculate the turbulent viscosity and wall shear stress. Blocken et al.

[19] discussed the problem of the wall treatment for these particular flows, suggesting remedies when

the simulations are run with ANSYS Fluent or CFX, which adopt wall functions based on an equiva-

lent sand-grain roughness, kS , equivalent to approximately 30z0. In contrast, OpenFOAM uses a wall

function based on the actual surface roughness length, z0, that is derived from Eq. 3.7. This was used

in the present work and allowed us to have a higher resolution close to the wall than the one possible

with ANSYS Fluent and CFX. A value of approximately 0.01D for the height of the first cell at the wall

was found to guarantee a correct simulation of SBL flows, achieving horizontally homogeneity (i.e., zero

streamwise gradients) of the SBL in an empty domain. This value is also consistent with other works

present in literature [24, 128].

3.5 Modeling uncertainty with simulation ensembles (MUSE)

The results from simulations run with steady conditions are not directly comparable with results from

field measurements. The reason is that field measurements are given as an average of recordings in a

certain period of time, classified by a specific range of wind direction, wind speed, turbulence intensity

and atmospheric stability. Within each period, wind intensity and direction change in time and produce

an uncertainty associated with the average value of a measuring period. The wind direction variability is

expected to have the most significant impact on turbine-wake characteristics, such as velocity deficit [126].

Causes of uncertainty on the wind direction can be identified in spatial and temporal de-correlation of the

wind direction between the measurement and the turbine locations, large-scale turbulence of the incoming


boundary layer flow that drives the meandering of the wakes [50], and sensor inaccuracy and uncertainty,

among others [72]. Gaumond et al. [58] suggested that the discrepancies of numerical simulations for

narrow wind direction sectors found in different studies are not caused by wake modeling inaccuracies

but rather by the large wind direction uncertainty included in the data sets. Subsequently, Gaumond

et al. [59] proposed a method to take into account the wind direction uncertainty post-processing the

results from analytical wake models, improving the agreement of the results. This technique is applied

in this study in order to compare the results from the CFD wake model with field measurements. With

this approach, we propose to Model Uncertainty with Simulation Ensembles (MUSE): we show that a

weighted average of several CFD RANS results covering a wide range of wind directions can effectively

take into account the large-scale turbulence of the incoming boundary layer flow causing wind direction

variability. As such, this method can be considered a computationally faster alternative to URANS or

LES models, which are usually needed to simulate large-scale flow phenomena with transient changes in

the flow field.

The wind direction uncertainty is assumed to have a Gaussian distribution around an average value.

The probability density of the Gaussian distribution relative to a wind direction, θ, is thus:

fg (θ) =1

σθ√

2πexp

[−(θ − θ

)22σ2

θ

], (3.13)

where θ is the average wind direction and σθ is the standard deviation associated with the wind di-

rection. The standard deviation of this distribution is usually provided by the experimental data set.

For the Nibe wind farm, the standard deviation was σθ = 5. For the Sexbierum wind farm, the wind

direction uncertainty was not reported in the data set. Nevertheless, Pena et al. [118] derived from

numerical simulations of the same case and experimental measurements of a different site with similar

wind conditions a value in the range of 2−3.5. A value of 3 is therefore adopted in this study. Lastly,

for the Horns Rev wind farm, Hansen et al. [72] stated that the wind direction uncertainty associated

with the measurements could reach values of more than 7 because of the large distance between the

wind measuring station and the operating wind turbines. However, Gaumond et al. [59] in their post-

processing calculations with the wind direction uncertainty found that values in the range of 5 − 7

provided better wake deficit predictions. Therefore, in this study, a value of 6 was chosen as a trade-off

between the previous considerations.

Once the flow field is obtained from the steady-state simulations, the velocity downstream the wind

turbine can be expressed as a function of the wind direction, θ, and the downstream distance, d, given


Figure 3.3: Example of weighted averaging. A moving weighted averaging process based on a Gaussiandistribution is applied to all wind directions . A Gaussian distribution centered at θ = 0 with σθ = 5

is plotted for clarification purposes.

the aforementioned input conditions:

U = f (θ, d) . (3.14)

A method to model wind direction uncertainty with simulation ensembles is then used: a weighted

average is applied to a set of steady-state CFD results for different wind directions to generate a single

CFD prediction based on the given standard deviation associated with the data set. The resulting

velocity for a specific direction, θ, and distance, d, is obtained as a weighted averaged of the simulated

velocities in the range of θ ± 3σθ at the same distance, where the weights are given by the Gaussian

distribution:

U(θ, d)

=

∫ θ+3σθ

θ−3σθ

U (θ, d) fg (θ) dθ. (3.15)

The process is repeated for all the directions at the same downstream distance. The same method can

also be applied to the power generation of a turbine operating in wake conditions where the resulting

power can be calculated as follows:

P(θ, d)

=

∫ θ+3σθ

θ−3σθ

P (θ, d) fg (θ) dθ. (3.16)

Figure 3.3 shows an example of the averaged results obtained from a CFD simulation when the MUSE

method is applied. It can be seen that the averaging process has the effect of decreasing the center-line

wind speed deficit and broadening the wake width.


Table 3.3: Root-mean-square errors (RMSE) between the experimental data and the simulations resultswhen considering the wind direction range ±30 for the Sexbierum and Nibe cases, and ±15 for theHorns Rev case. The RMSEs are classified by case (Sexbierum, Nibe, and Horns Rev), by quantity ofinterest (normalized wind speed (NWS) and normalized power (NP)), by downstream distance, and byturbulence model.

k − ε k − ωWind Turbine Quantity Distance Original MUSE Original MUSE

Sexbierum NWS2.5D 0.1859 0.1890 0.1936 0.19575.5D 0.0730 0.0757 0.0804 0.08168D 0.0545 0.0556 0.0603 0.0601

Nibe NWS2.5D 0.1848 0.1910 0.1936 0.19834D 0.0683 0.0787 0.0738 0.08147.5D 0.0453 0.0536 0.0468 0.0528

Horns Rev NP 7D 0.0247 0.0503 0.0442 0.0798

SST k − ω RSMWind Turbine Quantity Distance Original MUSE Original MUSE

Sexbierum NWS2.5D 0.0947 0.0933 0.0797 0.06175.5D 0.0468 0.0401 0.0462 0.03388D 0.0480 0.0435 0.0431 0.0425

Nibe NWS2.5D 0.0614 0.0694 0.0510 0.04654D 0.0619 0.0462 0.0776 0.05827.5D 0.0505 0.0401 0.0435 0.0423

Horns Rev NP 7D 0.1349 0.0461 0.0866 0.0392

3.6 Results and discussion

This section includes the results obtained directly from the CFD RANS simulations and the results

obtained after the averaging process with the MUSE method. This comparison aims to highlight the

importance of taking into account the wind direction uncertainty when comparing simulation and ex-

perimental results. A quantitative comparison, which will be used in the analysis of the results, is made

in Tab. 3.3 where the root-mean-square errors (RMSE) were calculated between the experimental data

and the CFD results. The RMSE is defined according to:

RMSE =

√∑Ni=1 (yi,exp − yi,CFD)

2

N, (3.17)

where yi,exp and yi,CFD are the experimental and simulated quantities of interest, respectively, and N

is the number of experimental observations.

3.6.1 Simulation results

Figures 3.4 and 3.5 show the normalized wind speed downstream the Sexbierum and Nibe wind turbines,

respectively, for different downstream distances and for different turbulence models. The wind direction


Table 3.4: Coefficients of the fitting function of Eq. 3.18 for each of the normalized power distributionsplotted in Fig. 3.6

a0 a1 a2 a3 a4

k − ε 0.0175 0.3694 0.0004 0.0077 0.0320k − ω 0.0183 0.2675 0.0004 0.0033 0.0236SST k − ω 0.0126 0.6402 0.0050 0.0178 0.0496RSM 0.0137 0.5251 0.0000 0.0101 0.0347

in the figures refers to the relative direction of the incoming flow where 0 indicates the direction for which

the maximum wind speed deficit is expected. In the Sexbierum case, the wind speed was captured well

by the SST k−ω and Reynolds stress models for the three locations, whereas was highly overestimated

by the k−ε and k−ω models, especially at 2.5D downstream where the RMSEs were the highest (0.1859

and 0.1936, respectively). The results for the Nibe case showed that the wind speed was captured well

by the SST k − ω and Reynolds stress models for the location at 2.5D downstream (RMSEs of 0.0614

and 0.0510, respectively), whereas it was underestimated for the other two locations. The k−ε and k−ω

models, instead, failed to capture the wind speed at 2.5 and 4D downstream, whereas more accurate

results were obtained in the far wake location.

Figure 3.6 show instead the normalized power production of turbine 17 operating in the wake of

turbine 07 at the Horns Rev wind farm as a function of wind direction for different turbulence models.

In this case, the wind direction is indicated with respect to North, therefore the two turbines are aligned

with the incoming flow along a West-East axis at 270. The calculations of the power production

were repeated every 2.5 starting from the direction of 270 where the two turbines were aligned with

the incoming wind speed. Because of the scattering in these simulation results, the normalized power

distribution was fitted with the following expression:

Pn (θ) = 1−[a0 +

(a1 + a2θ + a3θ

2)

exp(−a4θ

2)], (3.18)

where the variables a0, a1, a2, a3, and a4 are determined by fitting Eq. 3.18 to the power values, as

function of the normalized wind direction θ (see Fig. 3.6 for the resulting fitting curves). This fitting

function was introduced by Hansen et al. [72] to characterize the power deficit distributions when the

results are scattered. The coefficients of the fitting function obtained for each of the normalized power

distributions are reported in Tab. 3.4. The fitting function is conveniently used in place of the individual

power generation data as the input for the MUSE method in the following section. To note in the results

for the normalized power production is that the simulations using the k − ε and k − ω models were

very accurate (RMSEs of 0.0247 and 0.0442, respectively) as opposed to the results obtained with the



Figure 3.4: Normalized wind speed downstream the Sexbierum wind turbine as a function of winddirection for the different turbulence models and for different downstream distance.


Figure 3.5: Normalized wind speed downstream the Nibe wind turbine as a function of wind directionfor the different turbulence models and for different downstream distance.

(a) 7.0D downstream

Figure 3.6: Normalized power production of turbine 17 operating in the wake of turbine 07 at the HornsRev wind farm as a function of wind direction for different turbulence models. The continuous linescorrespond to Eq. 3.18 fitted to the normalized power data for each of the turbulence models.


SST k − ω and Reynolds stress models (RMSEs of 0.1349 and 0.0866, respectively).

The results obtained from the simulations showed the particular characteristics of the turbulence

models as well as the inconsistency in wake effect predictions that was found also in literature. The

k − ε and k − ω models provided very similar results between each other. The only difference between

the two models lies in a source term in the ω equation that, in these particular simulations, did not

produce a significant contribution. Their inaccuracy, which is usually agreed upon in literature, can be

seen particularly in the Sixberium case and in the near wake of the Nibe wind turbine as opposed to the

accurate wind speed predictions provided by the SST k − ω and Reynolds stress models. However, the

results using the k− ε and k− ω models were more accurate that those of the SST k− ω and Reynolds

stress models for the far wake of the Nibe wind turbine and for the Horns Rev wind farm.

Better predictions are usually expected by the SST k − ω model and Reynolds stress models. The

advantage of the former relies on the bound that is introduced in the eddy viscosity of the model and

that overcomes the limitation of its parent models. The RSM has instead the advantage of solving all

the Reynolds stresses and therefore it is not affected by the eddy-viscosity approximation. Nevertheless,

the results clearly showed that the predictions had high discrepancies for the Horns Rev case and at the

locations of 4 and 7.5D for the Nibe case.

The discrepancies and the inconsistency of the turbulence models are believed to rely in the wind

direction uncertainty, which was not taken into account in the simulations. The result of using the

MUSE method is showed in the next section where these limitations are overcome.

3.6.2 Results with MUSE Method

The results from the CFD simulations were post-processed with the MUSE method using a Gaussian

distribution for the wind direction uncertainty. An averaging process was used to take into account the

wind direction variability characterizing the specific site.

Figures 3.8 and 3.7 show the post-processed wind speed in the wake of the Nibe and Sexbierum wind

turbines, respectively. The averaged wind speed from the Sexbierum case did not vary much with respect

to the original predictions. This is due to the fact that the wind direction uncertainty was low for this

case and the effect on the results was not significant. On the other hand, the averaged results from the

Nibe case showed a clear improvement with respect to the original predictions when the SST k−ω and

Reynolds stress models were used: at 5.5D, the RMSEs decreased from 0.0619 and 0.0776 to 0.0462

and 0.0582, respectively, whereas at 7.5D, the RMSEs decreased from 0.0505 and 0.0435 to 0.0401 and

0.0392, respectively. In this case, the wind direction uncertainty was higher and, therefore, a wider


range of wind speeds was included in the averaging process, resulting in a more significant effect on the

results. As expected, no improvements were registered for the k − ε and k − ω models, which were seen

to overestimate the wind speed in the original results. The inclusion of the wind speed uncertainty for

these two models was even more deleterious for the predictions.

Figure 3.9 shows instead the normalized averaged power production of turbine 17 operating in the

wake of turbine 07 at the Horns Rev wind farm as a function of wind direction for different turbulence

models. The MUSE method was applied to the fitted power distribution curve in order to have more

data to process. The results showed a significant difference with respect to the original simulation

results. This is due to the high wind direction uncertainty that was associated with the data set and

that was used to post-process the simulation results. It is possible to notice that the predictions of power

production given by using the SST k − ω and Reynolds stress models were much more accurate than

in the previous case: the RMSEs decreased from 0.1349 and 0.0866 to 0.0461 and 0.0392, respectively.

On the other hand, when using the k− ε and k−ω models, the post-processed results provided a higher

power production and the discrepancy became consistent with the other wind farm simulations (e.g.

[24, 143, 7]) where the wake wind speed was overestimated.

The inclusion of the wind direction uncertainty with the proposed MUSE method showed that the

results directly obtained from the CFD simulation are not always comparable with the experimental

observations. This is particularly noticeable in the far wake regions when the wind direction uncer-

tainty was relatively high. By using the MUSE method, the RANS simulations using the SST k − ω

and Reynolds stress models were shown to be consistently more accurate for wake predictions and are

therefore to be preferred over the k− ε and k− ω models. This result clarifies and gives an explanation

to the sometime inconsistent behavior of the turbulence models highlighted in our results and in the

literature. Overall, this method improved the predictions of the CFD RANS model when either of the

suggested turbulence models is used. The improvements are more significant where large-scale unsteady

phenomena resulted in uncertainty in the wind direction as reported in the experimental data sets.

3.7 Conclusions

In the present work, we conducted an investigation of the limitations and inconsistency of the RANS

wake models in the predictions of wake effects in wind farms. A CFD model was developed which used

the actuator disk technique to simulate the wind turbines and the surface boundary layer approximation

to simulate the ambient conditions. The developed CFD model was implemented for three different wind

farms, namely, Sexbierum, Nibe, and Horns Rev, with publicly available experimental measurements.



Figure 3.7: Normalized averaged wind speed downstream the Sexbierum wind turbine as a function ofwind direction for the different turbulence models and for different downstream distance.


Figure 3.8: Normalized averaged wind speed downstream the Nibe wind turbine as a function of winddirection for the different turbulence models and for different downstream distance.

(a) 7.0D downstream

Figure 3.9: Normalized averaged power production of turbine 17 operating in the wake of turbine 07 atthe Horns Rev wind farm as a function of wind direction for different turbulence models.


The main turbulence models present in literature, namely, the k − ε, k − ω, SST k − ω and Reynolds

stress models were used to close the RANS equations and their results compared.

The results obtained from the simulations were compared to the experimental measurements of wind

speeds and power production in the wake region of the selected wind farms. The CFD results showed

the inconsistency in wake effect predictions that was also found in the literature. The simulation using

the k− ε and k−ω models provided inaccurate predictions for the Sixberium case and in the near wake

of the Nibe wind turbine. Despite their known limitations, their results were more accurate for the far

wake of the Nibe wind turbine and for the Horns Rev wind farm. On the other hand, the SST k−ω and

Reynolds stress models provided opposite results: accurate wind speed predictions for the Sexberium

case and in the near wake of the Nibe wind turbine and high discrepancies for the Horns Rev wind farm.

The discrepancies and the inconsistency of the turbulence models were hypothesized to arise from

wind direction uncertainty caused by large-scale unsteady phenomena, which though present in the ex-

perimental measurements were not accounted for in the simulations. We therefore proposed an approach

to overcome these limitations by Modeling Uncertainty using Simulation Ensembles (MUSE), i.e., a set

of CFD results for different wind directions to generate a single CFD prediction. The predictions of

CFD model were post-processed with this innovative method for CFD simulations that accounts for the

wind direction uncertainty associated with the specific wind farm data set. The proposed MUSE method

can be considered a computationally faster alternative to URANS or LES models, when the goal is to

account for the effect of large-scale, transient flow phenomena causing wind direction variability.

The results in terms of wind speed and power output showed that this technique corrects the pre-

dictions of the CFD model which would be otherwise inaccurate. Specifically, RANS simulations using

the SST k − ω and Reynolds stress models were shown to be consistently more accurate for wake pre-

dictions and are therefore to be preferred over the k− ε and k−ω models. The results showed also that

the discrepancy found between CFD models using the SST k − ω or Reynolds stress models and field

measurements are not related to the inaccuracy of the CFD models but to the uncertainty in the wind

direction.

Chapter 4

Continuous Adjoint Formulation for

WFLO: A 2D Implementation

In the present chapter, we describe an optimization methodology that integrates the high accuracy and

flexibility offered by the CFD models and that overcomes the computationally high costs of a CFD-

based optimization. To this end, we present an adjoint method in its continuous formulation for the

gradient computation. To the authors’ best knowledge, this is the first continuous formulation of the

adjoint method applied to the gradient computation in the WFLO problem. The continuous adjoint

formulation allows for a derivation of the general adjoint equations, before any discretization is being

applied, and therefore allows also for a more flexible implementation in CFD software packages. Here we

present a formulation for different conditions in the flow equations, namely, laminar, frozen-turbulence

and turbulent flows. To verify the developed formulation, gradients calculated under these different

flow conditions are compared with gradients computed with traditional central-difference schemes. The

gradient calculation using the developed adjoint method is then incorporated into a gradient-based

optimization methodology and applied to a set of 2D case studies with a wide variety of wind resource

profiles.

The outline of the chapter is the following. In Sec. 4.1, we present general framework of the adjoint

method and its computational advantage. The continuous adjoint formulation is described in Sec. 4.2

for the three cases of the flow equations. Section 4.3 illustrates the verification of the developed adjoint

method by comparing its results with a central difference discretization approach. In Sec. 4.4, we present

the wind farm layout optimization methodology and its application on a 2D wind farm. Conclusions are

summarized in Sec. 4.5.

61

Chapter 4. Continuous Adjoint Formulation for WFLO: A 2D Implementation 62

4.1 The adjoint method

This section illustrates the general framework underlying the adjoint method, as first conceptualized by

Jameson [85], and its computational advantages. Suppose that the governing equations of a system can

be expressed as G (φ,α) = 0, where G is the set of differential equations expressed in vector form (e.g.,

the RANS equations), φ is the vector of state variables (e.g., the flow field variables), and α is a vector

of the design variables (e.g., the wind turbine coordinates). The system of the governing equations

implicitly states that the state variables, φ, are function of the design variables, α. A scalar objective

function that measures a quantity of interest (e.g., the total power/energy production of the wind farm)

can be expressed as the integral of a user-defined function, J [φ (α) ,α], over a certain volume, ΩO.

Optimization problems are formulated such that the optimal design variables need to be found within

certain constraints to maximize the objective function, namely:

maxα

∫ΩO

J [φ (α) ,α] dΩ,

subject to G [φ (α) ,α] = 0 in Ω,

k (α) = 0,

h (α) ≤ 0,

(4.1)

where h and k are additional equality and inequality constraints on the design variables α, such as upper

and lower bounds on a control input (e.g., wind farm site boundaries, wind turbines interspacing), and

Ω is the entire domain over which the constraints are applied.

Gradient-based optimization algorithms require the gradient of the objective function with respect

to all of the control parameters, i.e., d(∫

ΩOJdΩ

)/dα. The derivation of this gradient with the adjoint

method can be shown starting from the definition of the Lagrangian function:

L =

∫ΩO

JdΩ +

∫Ω

φTGdΩ, (4.2)

where φ is the vector of the Lagrange multipliers, also called adjoint variables. As G is everywhere zero

by construction, the Lagrangian and its variation are always equal in value to the objective function and

its variation (i.e., L = J and δL = δ(∫

ΩOJdΩ

)) while φ can be arbitrarily chosen. By applying the


chain rule, the variation of the Lagrangian can be shown to be:

δL =∂

∂α

(∫ΩO

JdΩ

)δα+

∂

∂α

(∫Ω

φTGdΩ

)δα+

+

∫ΩO

(∂J

∂φ+ φT

∂G

∂φ

)dφ

dαδαdΩ +

∫Ω\ΩO

(φT

∂G

∂φ

)dφ

dαδαdΩ.

(4.3)

Since the objective function is typically a simple user-defined function, its partial derivatives with respect

to the design variables, ∂(∫

ΩOJdΩ

)/∂α, and state variables, ∂J/∂φ, are straightforward to calculate

numerically and analytically, respectively. The partial derivative of the constraints with respect to the

design variables, ∂(∫

ΩφTGdΩ

)/∂α, is equal to zero whenever the state equations, G, are and remain

satisfied. However, this is not the case when volume source terms in the state equations are themselves

a function of the design variables. In such cases, in the partial derivative, a term representing the

source position shift needs to be defined and this creates an imbalance in the state equations that can

be computed numerically. The partial derivative of the constraints with respect to the state variables,

∂G/∂φ, can be determined with classic derivation rules. Lastly, dφ/dα is the most expensive term

to compute for high-dimensional design and state spaces. In the adjoint approach, this last term is

eliminated by choosing the adjoint variables such that:

∂J

∂φ+ φT

∂G

∂φ= 0 in ΩO,

φT∂G

∂φ= 0 in Ω \ ΩO,

(4.4)

These are called the adjoint equations and their solution usually requires a computational time that is

comparable to the solution of the flow equations. With the values of the adjoint variables, it is therefore

easy to calculate the total gradient needed for the optimization algorithm:

d

dα

(∫ΩO

JdΩ

)=dL

dα=

∂

∂α

(∫ΩO

JdΩ

)+

∂

∂α

(∫Ω

φTGdΩ

). (4.5)

4.2 Continuous adjoint formulation for the wind farm layout

optimization problem

The formulation of the adjoint method needs to be derived for every problem to which it is applied.

Specifically, this requires the derivation and the calculation of the partial derivative terms highlighted

in the previous section. The formulation of the adjoint method for the wind farm layout optimization


problem starts with the definition of the design variables, α, which are the coordinates of the wind

turbines within the wind farm:

α = [(x1, x2)k] k = 1, ...,K, (4.6)

where K is the total number of wind turbines.

Each wind turbines is modeled as an actuator disk, which is characterized by a cylindrical volume,

defined by the rotor swept area, where a distributed force, defined as axial momentum source, F , is

applied. The actuator disk model offers a convenient trade-off between computational cost and accuracy:

even if it does not provide a detailed description of the wind turbine geometry it is able to adequately

capture its wake effect for the intended application in wind farm simulations and optimization [145, 3].

The axial force applied on the flow field as function of the reference wind speed is:

F =1

2ρπD2

4CTU

2ref , (4.7)





P = Fvn = F1

V

∫V

vndΩ = f

∫V

vndΩ, (4.8)

where vn is the average normal velocity over the wind turbine rotor volume, V . Because of the 2-

dimensional problem formulation, the actual wind turbine rotor volume is a rectangular prism with unit

height and rectangular base whose width is equal to the rotor diameter.

The objective function for the WFLO is the annual energy production (AEP) of the wind farm, which

is a function of both the wind farm layout and the wind resource. Without loss of generality, herein we

present a simplified version of the problem in which the objective function is the total power produced

by the wind turbines under a single wind state, i.e.:

P =

∫ΩO

JdΩ =

∫ΩO

fkvndΩ, (4.9)


where

ΩO =

K∑k=1

Vk, (4.10)

fk ∈ Vk. (4.11)

For WFLO, given the statistical distribution of wind speeds and directions, the formulation above (Eq.

4.9) is the elemental building block required to calculate the expected AEP, as follows:

AEP = 8760

S∑s=1

T∑t=1

ps,tPs,t, (4.12)

where S and T are the number of wind speed and direction bins, respectively, ps,t is the probability of

occurrence for each wind speed and direction bin, and Ps,t is the total power produced by the turbines,

Eq. 4.9, for a given wind speed and direction.

The governing equations for the present WFLO are the Navier-Stokes equations (i.e., the continuity

(C) and momentum (M) equations) for steady, incompressible flow, which in the context of the adjoint

formulation are the constraints of the problem, G:

C :∂vi∂xi

= 0 in Ω, (4.13)

Mi :

vj∂vi∂xj

+∂p∗

∂xi− ∂

∂xj(2νeffSij) = 0 in Ω− ΩO,

vj∂vi∂xj

+∂p∗

∂xi− ∂

∂xj(2νeffSij) + fk,i = 0 in Vk k = 1, ...,K,

(4.14)

where vi,j is the mean velocity component; p∗ is the mean kinematic pressure (i.e., p/ρ); fk is the

constant source term generated by the k-th turbine (modeled as an actuator disk); i, j are indexes over

the coordinate directions; Sij is the mean rate-of-strain tensor, defined as:

Sij =1

2

(∂vi∂xj

+∂vj∂xi

); (4.15)

and νeff is the effective viscosity, which for a laminar case is simply equal to the fluid kinematic viscosity,

νeff = ν = µ/ρ, and for turbulent flows is the sum of the fluid and turbulent viscosity, νeff = ν + νturb.

The turbulent viscosity is the result of the Boussinesq’s hypothesis to model the Reynolds stresses that

arise after the Reynolds averaging operation. Turbulence modeling is required to calculate this viscosity

and close the Navier-Stokes equations, and, consequently, more equations need to be considered as

constraints of the problem. In the present formulation we adopted the k − ω turbulence model to close


the RANS equation, but the current approach can similarly be extended to other one- and two-equation

turbulence models. For the k−ω turbulence model, the turbulent viscosity is given by the ratio between

the turbulence kinetic energy, k, and the specific dissipation rate, ω. The equations to calculate these

two variables are the following:

T1 : vj∂k

∂xj− 2

k

ωS2 + β∗kω − ∂

∂xj

[(ν + σ∗

k

ω

)∂k

∂xj

]= 0 in Ω, (4.16)

T2 : vj∂ω

∂xj− 2αS2 + βω2 − ∂

∂xj

[(ν + σ

k

ω

)∂ω

∂xj

]= 0 in Ω, (4.17)

where α, β, β∗, σ, σ∗ are empirical constants of the model, and S is the modulus of the mean rate-of-

strain tensor, defined as:

S =√SijSij . (4.18)

The state variables, also called field variables, φ, for this problem are therefore:

φ =

(p∗, vi) Laminar case,

(p∗, vi, k, ω) Turbulent case.

(4.19)

The derivation of the adjoint formulation depends on the state equations that govern the original

system and on the problem assumptions. Three cases can be identified for the adjoint formulation:

laminar, turbulent, and frozen-turbulence cases. The first two cases are the results of the original

laminar and turbulent equations respectively, whereas the latter is the results of turbulent equations

when the frozen-turbulence hypothesis is used. Under this assumption, the variation of the turbulence

field is neglected (i.e., the turbulent viscosity, although non-uniform in space, is assumed not to depend

on the mean velocities), and only changes of the mean flow are taken into account, described through

the system for continuity and momentum. Although the gradient obtained by freezing the turbulence

field is incomplete, this assumption is a convenient simplification that is considered standard for adjoint

methods [47], and as such it is also considered in the present work. Our results, shown later in this

chapter, will show that this assumption introduces inaccuracies that we deemed inacceptable for the

purpose of optimization.

The Lagrangian function that results for the laminar and frozen-turbulence cases is:

L =

∫ΩO

fkvndΩ +

∫Ω

(p, vi) · (C,Mi) dΩ, (4.20)


(a) Volume shift perpendicular to axial direction. (b) Volume shift longitudinal to axial direction.

Figure 4.1: Schematic illustrating the calculation of material partial derivatives.

whereas for the turbulent case, the Lagrangian function is given by:

L =

∫ΩO

fkvndΩ +

∫Ω

(p, vi, k, ω

)· (C,Mi, T1, T2) dΩ, (4.21)

where the adjoint variables, φ, for each of the cases are the following:

φ =

(p, vi) Laminar and frozen-turbulence cases,(p, vi, k, ω

)Turbulent case.

(4.22)

The derivation of the terms given in Eq. 4.3 and used to calculate the total gradient follows classical

techniques from calculus of variations. The first terms analyzed here are the partial derivatives with

respect to the design variables. As opposed to the traditional shape optimization formulations, the

WFLO problem does not require surface sensitivities which would in turn require surface displacements

and mesh deformation. In the current problem, the calculation of the wind turbine position sensitivities

entails instead the displacements of the volumes where the source terms are applied without the need

of mesh deformation. In fact, the source terms will be shifted over different regions depending of the

chosen displacements to calculate the partial derivatives with respect to the design variables (see Fig.

4.1 for clarification purposes).

When considering the partial derivative of the objective function, the resulting expression is the

following:

∂

∂α

(∫ΩO

JdΩ

)=

∂

∂α

(∫ΩO

fkvndΩ

). (4.23)

The calculation of this term requires the normal average velocity over the shifted volumes illustrated in

Fig. 4.1 for each of the design variables. This can be computed numerically with a central difference


discretization at no additional cost given that the field variables are already known:

∂

∂α

(∫ΩO

JdΩ

)=

∂

∂ [(x1, x2)k]

(∫Vk

fkvndΩ

)k = 1, ...,K. (4.24)

When considering instead the partial derivative of the augmented constraints, it can be seen that

the displacements of the source terms affect only the momentum equations, whereas the other state

equations remain equal to zero. The resulting expression is therefore the following:

∂

∂α

(∫Ω

φTGdΩ

)=

∂

∂α

(∫Ω

vi ·MidΩ

). (4.25)

If the state equations remained satisfied everywhere in Ω when calculating the partial derivative in Eq.

4.25, this term would result equal to zero. However, because the turbine (source term) positions are the

design variables of the problem and their displacements need to be defined and applied to calculate the

derivative, an imbalance in the momentum equations is created. This imbalance is equal to the source

term and is present with a positive value in the regions where the shifted volumes do not overlap with

the original turbine volume or with a negative value where the original volume does not overlap with

shifted volumes. In formulas, this can be expressed as:

∂

∂α

(∫Ω

φTGdΩ

)=

∂

∂ [(x1, x2)k]

(∫Vk,imb

fkvndΩ

)k = 1, ...,K, (4.26)

where Vk,imb refers to the region where the momentum equations result imbalanced when a central

difference approximation is used to discretize the derivative and a volume shift is applied. The final

result can be computed numerically once the adjoint variables are known.

The two partial derivatives calculated in Eq. 4.25 and 4.26 are the fundamental components used to

obtain the final value of the objective function gradient. However, other terms are needed to calculate

the required adjoint variables. For this purpose, the last two terms given in Eq. 4.3, namely the partial

derivatives with respect to the field variables, need to be calculated. First, the partial derivative of the

objective function can be easily calculated as:

∫ΩO

∂J

∂φδφdΩ =

∫ΩO

∂

∂φ(fkvn) δvndΩ =

∫ΩO

fk,iδvidΩ. (4.27)

The second term is the partial derivative of the augmented constraints and it requires instead a few

algebraic manipulations to obtain a convenient relation that can be subsequently used for the adjoint


equations. To derive it, the following rule for integration by parts for multivariable calculus will be used:

∫Ω

∂g

∂xihdΩ =

∫Γ

ghnidΓ−∫

Ω

∂h

∂xigdΩ. (4.28)

Also, depending on the cases previously discussed (i.e., laminar, frozen-turbulence, and turbulent),

different equation will be derived. The following sections will illustrate the two different derivations.

4.2.1 Adjoint equations for the laminar and frozen-turbulence cases

The adjoint equations and boundary conditions for the laminar and frozen-turbulence cases are derived

by developing the last term required:

∫Ω

φT∂G

∂φδφdΩ =

∫Ω

p

−∂δvi∂xi

dΩ+

∫Ω

vi

δvj

∂vi∂xj

+ vj∂δvi∂xj

+∂δp∗

∂xi− ∂

∂xj(2νeffδSij)

dΩ, (4.29)

where

δSij =1

2

(∂δvi∂xj

+∂δvj∂xi

). (4.30)

The integration by parts is used (twice for the diffusion term) to obtain the final expression:

∫Ω

φT∂G

∂φδφdΩ =−

∫Ω

− ∂p

∂xiδvi − vi

∂vi∂xj

δvj + vj∂vi∂xj

δvi +∂vi∂xi

δp∗ +∂

∂xj

(2νeff Sij

)δvi

dΩ+

+

∫Γ

− pniδvi + vivjnjδvi + viniδp

∗ − vi2νeffδSijnj + 2νeff Sijnjδvi

dΓ,

(4.31)

where

Sij =1

2

(∂vi∂xj

+∂vj∂xi

). (4.32)

After collecting terms with the variations of the field variables, the resulting expression is the following:

∫Ω

φT∂G

∂φδφdΩ =−

∫Ω

δp∗

[∂vi∂xi

]+ δvi

[vj∂vi∂xj− ∂p

∂xi− vj

∂vj∂xi

+∂

∂xj

(2νeff Sij

)]dΩ+

+

∫Γ

δp∗[vini

]+ δvi

[vivjnj − pni + 2νeff Sijnj

]− vi2νeffδSijnj

dΓ.

(4.33)

The adjoint equations can be eventually determined to eliminate the variations of the field variables.

By summing the partial derivatives of the objective function in Eq. 4.27 and of the augmented constraints


in Eq. 4.33, the adjoint equations for the laminar and frozen-turbulence cases are:

∂vi∂xi

= 0 in Ω, (4.34)−vj

∂vi∂xj

+∂p

∂xi− ∂

∂xj

(2νeff Sij

)+ vj

∂vj∂xi

= 0 in Ω− ΩO,

−vj∂vi∂xj

+∂p

∂xi− ∂

∂xj

(2νeff Sij

)+ vj

∂vj∂xi

+ fk,i = 0 in Vk k = 1, ...,K,

(4.35)

δp∗[vini

]= 0 in Γ, (4.36)

δvi

[vivjnj − pni + 2νeff Sijnj

]= 0 in Γ, (4.37)

vi2νeffδSijnj = 0 in Γ. (4.38)

4.2.2 Adjoint equations for the turbulent case

The adjoint equations and boundary conditions for the turbulent case are derived, as in the previous

case, by developing the last term required:

∫Ω

φT∂G

∂φδφdΩ =

∫Ω

p

−∂δvi∂xi

dΩ +

∫Ω

vi

δvj

∂vi∂xj

+ vj∂δvi∂xj

+∂δp∗

∂xi+

− ∂

∂xj

[2

(δk

ω− k

ω2δω

)Sij + 2

(ν +

k

ω

)δSij

]dΩ+

+

∫Ω

k

δvj

∂k

∂xj+ vj

∂δk

∂xj+

− 2δk

ωS2 + 2

k

ω2δωS2 − 2

k

ωδ(S2)

+ β∗δkω + β∗kδω+

− ∂

∂xj

[σ∗(δk

ω− k

ω2δω

)∂k

∂xj+

(ν + σ∗

k

ω

)∂δk

∂xj

]dΩ+

+

∫Ω

ω

δvj

∂ω

∂xj+ vj

∂δω

∂xj− 2αδ

(S2)

+ 2βωδω+

− ∂

∂xj

[σ

(δk

ω− k

ω2δω

)∂ω

∂xj+

(ν + σ

k

ω

)∂δω

∂xj

]dΩ,

(4.39)

where

δ(S2)

= 2Sij∂δvi∂xj

. (4.40)


The integration by parts is used (twice for the diffusion term) to obtain the final expression:

∫Ω

φT∂G

∂φdΩ = −

∫Ω

− ∂p

∂xiδvi − vi

∂vi∂xj

δvj + vj∂vi∂xj

δvi +∂vi∂xi

δp∗ +

+∂

∂xj

[2

(ν +

k

ω

)Sij

]δvi −

∂vi∂xj

[2

(δk

ω− k

ω2δω

)Sij

]+

− k ∂k∂xj

δvj + vj∂k

∂xjδk − ∂

∂xj

(4kk

ωSij

)δvi+

− k(−2

S2

ωδk + 2

k

ω2S2δω + β∗ωδk + β∗kδω

)+

+∂

∂xj

[(ν + σ∗

k

ω

)∂k

∂xj

]δk − ∂k

∂xj

[σ∗(δk

ω− k

ω2δω

)∂k

∂xj

]+

− ω ∂ω∂xj

δvj + vj∂ω

∂xjδω − ∂

∂xj(4ωαSij) δvi − 2βωωδω+

+∂

∂xj

[(ν + σ

k

ω

)∂ω

∂xj

]δω − ∂ω

∂xj

[σ

(δk

ω− k

ω2δω

)∂ω

∂xj

]dΩ+

+

∫Γ

−pniδvi + vivjnjδvi + viniδp∗+

− vi[2

(δk

ω− k

ω2δω

)Sij + 2

(ν +

k

ω

)δSij

]nj + 2

(ν +

k

ω

)Sijnjδvi+

+ kvjnjδk − 4kk

ωSijnjδvi+

− k[σ∗(δk

ω− k

ω2δω

)∂k

∂xj+

(ν + σ∗

k

ω

)∂δk

∂xj

]nj+

+

(ν + σ∗

k

ω

)∂k

∂xjnjδk+

+ ωvjnjδω − 4ωαSijnjδvi+

− ω[σ

(δk

ω− k

ω2δω

)∂ω

∂xj+

(ν + σ∗

k

ω

)∂δω

∂xj

]nj+

+

(ν + σ

k

ω

)∂ω

∂xjnjδω

dΓ.

(4.41)


After collecting terms with the variations of the field variables, the resulting expression is the following:

∫Ω

φT∂G

∂φδφdΩ = −

∫Ω

δp∗

∂vi∂xi

+

+ δvi

vj∂vi∂xj− ∂p

∂xi− vj

∂vj∂xi− k ∂k

∂xi− ω ∂ω

∂xi+

+∂

∂xj

[2

(ν +

k

ω

)Sij

]− ∂

∂xj

(4kk

ωSij

)− ∂

∂xj(4ωαSij)

+

+ δk

vj∂k

∂xj+

∂

∂xj

[(ν + σ∗

k

ω

)∂k

∂xj

]− 1

ω2Sij

∂vi∂xj

+

−σ∗

ω

∂k

∂xj

∂k

∂xj− σ

ω

∂ω

∂xj

∂ω

∂xj+ 2

k

ωS2 − β∗kω

+

+ δω

vj∂ω

∂xj+

∂

∂xj

[(ν + σ

k

ω

)∂ω

∂xj

]+

k

ω22Sij

∂vi∂xj

+

+σ∗k

ω2

∂k

∂xj

∂k

∂xj+σk

ω2

∂ω

∂xj

∂ω

∂xj− 2

kk

ω2S2 − β∗kk − 2βωω

dΩ+

+

∫Γ

δp∗ [vini] +

+ δvi

[vivjnj − pni + 2

(ν +

k

ω

)Sijnj +

−(

4kk

ω+ 4ωα

)Sijnj

]− vi2

(ν +

k

ω

)δSijnj+

+ δk

[kvjnj +

(ν + σ∗

k

ω

)∂k

∂xjnj −

viωSijnj +

− σ∗k

ω

∂k

∂xj− σω

ω

∂ω

∂xj

]− k

(ν + σ∗

k

ω

)∂δk

∂xjnj+

+ δω

[ωvjnj +

(ν + σ

k

ω

)∂ω

∂xjnj +

vik

ω2Sijnj +

+σ∗kk

ω2

∂k

∂xj+σωk

ω2

∂ω

∂xj

]− ω

(ν + σ

k

ω

)∂δω

∂xjnj

dΓ.

(4.42)

By eliminating the variations of the field variables and by summing the partial derivatives of the objective

function in Eq. 4.27 and of the augmented constraints in Eq. 4.42, the adjoint equations for the turbulent


case are:

∂vi∂xi

= 0 in Ω, (4.43)

−vj∂vi∂xj

+∂p

∂xi− ∂

∂xj

[2

(ν +

k

ω

)Sij

]+ vj

∂vj∂xi

+

+k∂k

∂xi+ ω

∂ω

∂xi+

∂

∂xj

[4

(kk

ω+ αω

)Sij

]= 0 in Ω− ΩO,

−vj∂vi∂xj

+∂p

∂xi− ∂

∂xj

[2

(ν +

k

ω

)Sij

]+ vj

∂vj∂xi

+

+k∂k

∂xi+ ω

∂ω

∂xi+

∂

∂xj

[4

(kk

ω+ αω

)Sij

]+ fk,i = 0 in Vk k = 1, ...,K,

(4.44)

− vj∂k

∂xj− ∂

∂xj

[(ν + σ∗

k

ω

)∂k

∂xj

]+

1

ω2Sij

∂vi∂xj

+

+σ∗

ω

∂k

∂xj

∂k

∂xj+σ

ω

∂ω

∂xj

∂ω

∂xj− 2

k

ωS2 + β∗kω = 0 in Ω,

(4.45)

− vj∂ω

∂xj− ∂

∂xj

[(ν + σ

k

ω

)∂ω

∂xj

]− k

ω22Sij

∂vi∂xj

+

− σ∗k

ω2

∂k

∂xj

∂k

∂xj− σk

ω2

∂ω

∂xj

∂ω

∂xj+ 2

kk

ω2S2 + β∗kk + 2βωω = 0 in Ω,

(4.46)

δp∗[vini

]= 0 in Γ, (4.47)

δvi

[vivjnj − pni + 2

(ν +

k

ω

)Sijnj − 4

(kk

ω+ ωα

)Sijnj

]= 0 in Γ, (4.48)

vi2

(ν +

k

ω

)δSijnj = 0 in Γ, (4.49)

δk

[kvjnj +

(ν + σ∗

k

ω

)∂k

∂xjnj −

viωSijnj −

σ∗k

ω

∂k

∂xj− σω

ω

∂ω

∂xj

]= 0 in Γ, (4.50)

k

(ν + σ∗

k

ω

)∂δk

∂xjnj = 0 in Γ, (4.51)

δω

[ωvjnj +

(ν + σ

k

ω

)∂ω

∂xjnj +

vik

ω2Sijnj +

σ∗kk

ω2

∂k

∂xj+σωk

ω2

∂ω

∂xj

]= 0 in Γ, (4.52)

ω

(ν + σ

k

ω

)∂δω

∂xjnj = 0 in Γ. (4.53)

4.3 Verification

In this section, the verification of the proposed continuous adjoint method is carried out by comparing

its results in terms of gradient computation with a finite difference discretization, defined according to:

d

dα

(∫ΩO

JdΩ

)=

∫ΩO

J (αn + δαn) dΩ−∫

ΩO

J (αn − δαn) dΩ

2δαnn = 1, ..., N, (4.54)


where δαn is a small variation of the n-th design variable. As opposed to the adjoint method, the

gradient calculated with the central difference discratization requires two function evaluations for each

of the design variables. This form of verification has been extensively used in literature to assess the

accuracy of the derivatives (e.g., [4, 30, 47, 116]). It is important to note that, although the continuous

adjoint formulation calculates the exact gradient, its implementation requires the discretization of the

equations over the same computational mesh of the primal simulation. Hence, both the central difference

and adjoint method results are approximations to the underlying gradient, for which there are no closed-

form expressions available.

The continuous adjoint method was therefore applied to the verification case shown in Fig. 4.2, which

consists in finding the gradient of the power production of a turbine placed in the wake of another with

respect to its turbine coordinates. For this case, a 2D domain was used and the results were obtained

for the three formulations of the adjoint equations. To have a robust verification and to capture both

stream-wise and cross-wise variations, the gradient components were calculated for the two directions of

the domain and on a regular grid of downstream and cross-stream positions. The downstream positions

were set to 10D and 15D, with a cross-stream spacing of 0.5D, where D is the turbine rotor diameter

equal to 80 m.

Figure 4.2: Schematic of the layout for the verification case. The two grey areas represent the volumeswhere the wind turbine momentum sources are applied. The black dots indicate the different positionswhere the second wind turbine is placed when the gradient is calculated.

The mathematical formulation of this adjoint problem was implemented in OpenFOAM [153]. The

implementation took advantage of the top-level syntax of the code, which is very close to the conventional

mathematical notation for tensors and partial differential equations [164]. Thanks to the high degree of


similarity between the state and the adjoint equations, the latter ones were elegantly coded in a similar

way to the NS equations. Second-order discretization was subsequently applied to the adjoint equations

for the interpolation of the adjoint variables. Similarly to the NS equations, the semi-implicit method

for pressure-linked equations (SIMPLE) algorithms was used to solve simultaneously the set of adjoint

equations by an iterative scheme.


The boundary conditions for the wind turbines simulation were set with realistic values for wind speed

and turbulence quantities. Even though the simulated system is 2-dimensional, an atmospheric boundary

layer (ABL) was assumed to be present at the inlet. The relations that govern an ABL can be found in

Ref. [6]. Given the ABL characterizing parameters, i.e., undisturbed speed at reference height, Uref=10

m/s, reference height, H=60 m, and surface roughness, z0=0.0018 m, the turbulence kinetic energy and

specific dissipation rate were calculated and then prescribed at the inlet along with the wind speed.

The outlet boundary condition was defined as pressure outlet, with zero gradient for the velocity and

turbulence quantities. The side boundary condition was defined as zero gradient for all the variables.

With regards to the adjoint simulation, the definition of the boundary conditions requires a specific

discussion. In traditional cases of shape design optimization, the boundaries of the domain are subject to

variation depending on the design variables. Therefore the variation of the boundaries and the definition

of the adjoint boundary conditions are crucial for an accurate gradient calculation (see Ref. [149] for

a detailed analysis). In the present WFLO problem, the boundaries are not subject to any variation

and they do not carry any objective function. Because of these circumstances, their influence on the

gradient calculation was seen to be negligible. The boundary conditions on Γ for the present problem

can therefore be fulfilled by the following solution:

vi = 0 in Γ,

∂p

∂xi= 0 in Γ,

k = 0 in Γ,

ω = 0 in Γ.

(4.55)

This solution enabled a straightforward implementation, a stable convergence of the iterative scheme,

and accurate results when considering the gradient calculation.


(a) 10D downstream (b) 15D downstream

Figure 4.3: Gradient computation for the laminar case. The figures show the results obtained by theadjoint method compared to a central difference discretization approach for the different lateral positionsof the second wind turbine at 10D and 15D downstream.

4.3.2 Verification results

The results of the developed adjoint formulation were compared to those of a central difference (CD)

approach, which can be considered as the best achievable solution for numerical gradient computations.

Figures 4.3, 4.4 and 4.5 show the values of the gradient obtained by the three different adjoint formu-

lations and the CD approach. Tables 4.1, 4.2 and 4.3 report evaluation metrics when comparing the

adjoint formulations and the CD approach. These are: the relative magnitude of the gradient computed

with the adjoint method:

‖∇Jrel‖ =‖∇JAM‖‖∇JAM,max‖

· 100; (4.56)

the percentage difference in the absolute value of the gradients computed by the adjoint method with

respect to the central difference approach:

err∇J =‖∇JAM‖ − ‖∇JCD‖‖∇JCD,max‖

· 100; (4.57)

the angular difference in the direction between the gradient calculated with the adjoint method and the

central difference approach, according to:

errθ = ‖θAM − θCD‖ . (4.58)

For the laminar case, the adjoint method could capture almost exactly the gradient given by the central

difference approach for all the downstream positions. A maximum difference of 1.2% in the absolute value



Figure 4.4: Gradient computation for the frozen-turbulence case. The figures show the results obtainedby the adjoint method compared to a central difference discretization approach for the different lateralpositions of the second wind turbine at 10D and 15D downstream.



10D downstream 15D downstreamCross-stream [D] ‖∇Jrel‖ [%] err∇J [%] errθ [] ‖∇Jrel‖ [%] err∇J [%] errθ []

0.0 1.2 1.2 0.0 1.0 0.8 0.00.5 100.0 0.1 0.3 99.7 0.5 0.31.0 47.8 0.7 0.1 46.8 0.9 0.01.5 0.4 0.4 1.1 0.3 0.1 0.92.0 0.4 0.3 1.3 0.3 0.1 1.12.5 0.4 0.3 1.5 0.3 0.1 1.4

Table 4.1: Percentage difference in the absolute value and angular difference in the direction of thegradients computed by the central difference approach and the adjoint method in the laminar case.



0.0 18.4 12.0 0.0 11.8 10.5 0.00.5 65.1 3.0 9.2 39.4 0.6 14.31.0 97.6 1.1 4.5 63.2 3.5 7.21.5 100.0 0.5 3.2 73.4 3.7 5.42.0 37.8 1.1 12.0 52.7 3.6 7.52.5 10.1 3.7 72.3 14.5 0.1 40.3

Table 4.2: Percentage difference in the absolute value and angular difference in the direction of thegradients computed by the central difference approach and the adjoint method in the frozen-turbulencecase.


0.0 9.9 3.6 0.0 5.1 3.9 0.00.5 62.0 0.1 2.8 37.6 2.4 5.31.0 95.1 1.2 1.3 62.1 4.5 2.51.5 100.0 0.5 1.3 72.6 4.6 2.02.0 34.6 1.4 6.0 53.0 3.4 3.62.5 6.6 0.2 47.8 12.7 1.7 22.2

Table 4.3: Percentage difference in the absolute value and angular difference in the direction of thegradients computed by the central difference approach and the adjoint method in the turbulent case.

and a maximum difference of 1.5 in the direction were observed. When the frozen-turbulence assumption

was used for the turbulence equations, higher discrepancies were seen between the results of the two

methods, with a maximum difference of 12.0% in absolute value and a maximum difference of 72.3 in the

direction. These higher discrepancies are explainable by the fact that the frozen-turbulence assumption

is a simplification of the adjoint derivation and, consequently, the gradient obtained is incomplete.

The fully turbulent formulation of the adjoint method had results that overall were comparable to the

laminar case in terms of accuracy both in the absolute value and direction. However, for both the frozen-

turbulence and turbulent results, high discrepancies were observed in the direction values for the cases

where the second turbine was in the outermost lateral positions (2.0 and 2.5D cross-stream positions).

This is related to the fact that the two components of the gradient approached a value of zero and the

small discrepancies observed in streamwise components generated a large direction variation. In these

positions, the magnitude of the gradient is however significantly lower and almost negligible with respect

to the wake region and therefore this behavior is not expected to affect the optimization process.

In terms of computational cost, the central difference approach required 4 simulations to compute

the gradient. However, CD requires in general 2N system solutions, where N is the number of design

variables (in the WFLO, there are two design variables per wind turbine). Instead, the adjoint method


required the solution of the adjoint system and 4 numerical integrations (with negligible computational

cost) to compute the gradient, regardless of the number of design variables. For the present verification

case where the gradient of only one turbine was calculated, the computational cost of the adjoint method

was 75% lower than the central difference approach. Higher reductions are however expected when more

wind turbines are considered, such as the application case in the following section.

Overall, the results presented for this verification case showed that the adjoint method can effectively

replace a traditional central difference approach for gradient computation and significantly reduce the

amount of time required for the process. The adjoint method will be therefore integrated, in place

of a central difference discretization, in the gradient-based optimization methodology illustrated in the

following section.

4.4 Optimization methodology for the WFLO problem

The gradient calculation performed with the adjoint method is an essential component of the optimization

methodology, which aims to find the optimal placement of a given number of wind turbines within a

wind farm domain to maximize the AEP. The optimization methodology, illustrated in Fig. 4.6, starts

with the wind rose, i.e., the site-specific statistical distribution of the wind resource, and an initial wind

farm layout. Initially, CFD simulations are used to calculate the AEP of that particular configuration.

If convergence/termination criteria in the iterative loop are not met, adjoint CFD simulations are then

used to calculate the gradient of the objective function, i.e., the gradient of the AEP with respect

to the turbine positions. Using the calculated gradients, turbine positions are updated to create a

new turbine layout, which is then evaluated with a CFD simulation. These steps are repeated until

convergence/termination criteria are met.

This methodology was entirely developed and implemented to run autonomously. Two separate

routines were coded to take care of the calculation of the AEP and of the gradient. In these routines,

the original and adjoint CFD simulations are automatically set up and run using OpenFOAM, and the

results of interest are generated and passed to the rest of the loop. The overall optimization process,

which consists of running the two aforementioned routines, updating the turbine position, and monitoring

the convergence, is handled by a sequential quadratic programming (SQP) algorithm as implemented in

the open-source library NLopt [91, 98]. A stopping criterion is applied so that the optimization run is

stopped when changes in the objective function from one iteration to the next are less than 0.01%.

The developed optimization methodology was applied to optimize a 2D wind farm layout consisting

of 16 wind turbines using the fully turbulent formulation. The domain used for the CFD and adjoint


Figure 4.6: Flow chart reproducing the optimization methodology used to solve the wind farm layoutoptimization problem

Figure 4.7: Schamatic of the 2D domain used to test the optimzation methodology.

simulations had a dimension of 40Dx40D, whereas the wind turbines were constrained to be within a

17D-radius circumference, as shown in Fig 4.7. This circular sub-domain was introduced as a compu-

tational convenience, to allow us to efficiently rotate the wind farm layout depending on the specified

wind direction, rather than rotating the boundary conditions. A minimum distance of 1D between

wind turbines was set as additional constraint to avoid any overlap of wind turbines generated by the

optimization algorithm.

The convergence behavior of gradient-based optimization methods is highly dependent on the starting

design configuration [70, 71]. Due to the local character of gradient information, gradient-based opti-

mization methods are only guaranteed to perform local optimization, i.e., they converge to the nearest

locally optimal solution, but they cannot guarantee that the globally optimal solution is found. To over-


(a) Regular initial layout (b) Random initial layout

Figure 4.8: Initial layouts used as inputs for the optimization process.

Regular initial layout Random initial layoutNorm. AEP Impr. [%] Norm. AEP Impr. [%]

1-dir. uniform WR 1.073 37.1 1.080 15.02-dir. uniform WR 0.973 27.1 1.070 13.93-dir. uniform WR 1.045 12.9 1.069 12.14-dir. uniform WR 0.958 25.6 1.069 16.46-dir. uniform WR 1.023 10.8 1.053 10.812-dir. uniform WR 0.995 8.0 1.034 9.612-dir. non-uniform WR 1.013 7.1 1.019 7.6

Table 4.4: Normalized AEP and improvement with respect to the initial configuration obtained at theend of the optimization for each of the initial wind farm layouts and for each of the wind roses (WR)used.

come this limitation, the initial solution should be located in the basin of attraction of a global optimum

to get a global optimum solution. Because of that, and to analyze the dependence of the gradient-based

methodology to the initial configuration, two initial layouts were tested, namely a regular layout (com-

monly used in wind farms) and a random layout (see Fig. 4.8). With these initial layouts, seven different

wind roses were chosen to conduct a comprehensive analysis of the proposed methodology. Six of these

wind roses were composed of evenly weighted wind directions, whereas the last one was representative

of a more realistic scenario with a predominant wind direction (see Fig. 4.9 and 4.10). For all wind

resource distributions, the wind speed was considered constant at 10 m/s.

The results obtained from the optimization are reported in Fig. 4.9 and 4.10 for a regular and a

random initial layout, respectively. For each wind rose reported on the left, the final optimal layout is

shown in the center along with the value of the normalized AEP through the iterations on the right. The

normalized AEP is defined as the ratio between the actual AEP and the AEP that would be generated

by the same turbines operating in isolation. Convergence was reached in approximately 30-60 iterations

of the optimization loop for all the cased analyzed. The results of the optimization process in terms of

normalized AEP are reported in Tab. 4.4.


Figure 4.9: Optimization results obtained from a regular initial layout. For each wind rose on the left,the final optimal layout is shown in the center along with the value of the normalized annual energyproduction (AEP) through the iterations on the right.


Figure 4.10: Optimization results obtained from a random initial layout. For each wind rose on the left,the final optimal layout is shown in the center along with the value of the normalized annual energyproduction (AEP) through the iterations on the right.


For the cases with a regular initial layout, it is possible to notice that for evenly weighted wind

roses the generated optimal layouts retained a certain degree of regularity. For example, the layouts

for 4 and 12 directions were symmetric about a 90 rotation; the layouts for 2 and 6 directions were

instead symmetric about a horizontal or vertical line passing by the center of the domain. These

characteristics of symmetry of the layouts for evenly weighted wind directions and for uniform wind

speed are usually considered an indication of an effective optimization algorithm. Obviously, these

characteristics of regularity could not be seen in the optimal layouts for other case of a random initial

layout.

Looking at the results in terms of normalized AEP for the optimal layouts, it is interesting to notice

that the actual AEP was greater, up to a maximum of 8%, than the one that would be generated by the

same turbines operating in isolation. The reason for this interesting result can be explained by looking at

two different aspects. First, the CFD simulations allow to accurately resolve the flow field and to capture

the real behavior of wakes. In fact, besides creating a wind speed reduction in the wake of wind turbines,

wakes also generate a local speed-up effect just outside of the wake region to compensate the wind speed

deficit within the wake. The optimization algorithm took advantage of this flow characteristic and,

depending on the wind rose, tried to move the wind turbines toward these favorable positions where the

speedups were higher. This speedup effect is more pronounced in 2D simulations like the ones conducted

in the present study, whereas its effect should be lower in 3D simulations [95]. Second, the wind farm

had a low turbine density because it consisted only of 16 wind turbines that could be spread over a wide

area. This fact allowed the optimization algorithms to find favorable locations with speed-up effects for

almost all the turbines. If the wind farm have had a higher wind turbine density, it is unlikely that the

normalized AEP would have had a value greater than 1 because of the higher wake losses. It is also

possible to notice that as the wind roses became less sparse, the maximum normalized AEP dropped

from 1.07-1.08 for both the unidirectional wind roses to 1.00-1.01 for the 12-direction wind roses. This

indicates that as the wind rose has more wind directions, it is more difficult to find favorable positions

with speed-up effects for all the wind turbines and some of them end up in locations of partial wake

shading.

With regards to the convergence behavior of the optimization algorithm, the results show that,

as expected, the optimal layouts found with the proposed methodology were highly dependent on the

initial layout. The optimal layouts obtained starting from a regular layout were very different from those

obtained when the optimization started from an random layout. In terms of AEP, it was observed that

the optimal layouts obtained when a random layout was used as a starting point resulted in higher energy

than those obtained starting from a regular layout, on average by 4.5%. In both cases, the algorithm can


be said to have converged to a local maxima, though the maximum of the objective function was higher

in the random layout case. Another interesting observation that can be made from the results is that, for

a regular initial layout and when the wind rose had directions aligned with the wind turbine arrangement

(2- and 4-direction wind roses), the optimization algorithm seemed to be trapped in a local optimum in

the early stages of the optimization loop. This issue was not observed when the optimizations started

from an random initial layout. This is evidence of the non-linear nature of the optimization problem,

with strong interactions between the wind resource profile and the wind turbine layout.

Overall, the developed optimization methodology based on the adjoint method for the gradient

computation could effectively improve the AEP of a given wind farm layout by changing its turbine

positions. The improvements ranged from about 7% for the case of a 12-direction non-uniform wind rose

and regular initial layout, to 37% for the case of a unidirectional wind rose and regular initial layout. On

average, the improvements were of 18% and 12% for the regular and random initial layouts, respectively.

4.5 Conclusions

In the present chapter, we developed an optimization methodology for the WFLO problem that integrates

the high accuracy and flexibility offered by the CFD models and that overcomes the computationally

high costs of a CFD-based optimization. To this end, we developed and used an adjoint method in its

continuous formulation for the gradient computation. The adjoint formulation was derived for three

different flow scenarios, namely, laminar, frozen-turbulence, and turbulent flows. The derived adjoint

equations were implemented in OpenFOAM by taking advantage of the top-level syntax of the code and

of the similarity between the Navier-Stokes and adjoint equations. The gradient calculation using the

developed adjoint method was implemented in a gradient-based optimization methodology to solve the

2D WFLO problem.

The developed adjoint formulation was first verified on a simplified 2-turbine wind farm where only

the gradient of the wind turbine operating in wake conditions was considered. The results obtained

with adjoint method in terms of gradient computations where generally accurate when compared to the

results obtained with a central difference discretization. The calculated gradients showed higher accuracy

for laminar and turbulent flow regimes, while lower accuracy was observed for the frozen-turbulence

case. Overall, the results showed that the adjoint method could effectively replace a traditional central

difference approach for gradient computation and significantly reduce the amount of time required for

the process.

The gradient calculation performed with the adjoint method was implemented in an optimization


methodology to solve a 2D WFLO problem. A hypothetical 16-turbine wind farm was defined with

two different initial layouts, with turbine positions either set on a regular grid or random. Seven wind

roses were also defined as inputs for the optimization for a comprehensive analysis. Six of them were

formed by evenly weighted wind directions and one with a realistic wind rose with a predominant wind

direction. The optimization methodology could effectively improve the AEP of the given wind farm

layouts by changing its turbine positions on average by 18% and 12% for the regular and random initial

layouts, respectively.

The proposed approach exhibited the typical local convergence behavior of gradient-based optimiza-

tion approaches. In this work, we used two different initial layouts (symmetric/regular, and random)

for the optimization, since we were interested in observing the behavior of the algorithm in these cases.

However, in practical applications where more assurance is needed of having obtained a globally optimal

solution, different optimization approaches could be used. At the very least, multiple optimization runs

with different initial layouts must be performed, and the best solution among them should be chosen.

Overall, the developed continuous adjoint formulation and its results showed that significant im-

provements can be achieved in terms of computational time with respect to traditional approaches for

gradient computation, such as the central difference approach, without diminishing the accuracy. The

developed gradient-based optimization methodology using the adjoint method showed instead that it is

possible to effectively improve the wind farm AEP. The optimization methodology showed also to benefit

from the use of CFD models which offer a more detailed representation of the flow field in wind farms

by, for example, capturing speed-up effects just outside of wake regions that cannot be observed with

analytical wake models. This 2D continuous adjoint formulation for the WFLO is a first step toward a

more general 3D formulation that could enable an optimization of wind farm layouts in complex terrain.

Chapter 5

Toward WFLO in Complex Terrain

In the present chapter, we show the application of a gradient-based WFLO methodology to 3D wind

farms in both flat and complex terrain. The calculations of the gradients are performed by a contin-

uous adjoint method for turbulent flows. To verify the adjoint formulation, gradients calculated with

this method are compared with gradients computed with traditional central-difference schemes in a 3D

domain. The gradient calculation using the verified adjoint method is then incorporated into a gradient-

based optimization methodology and applied to a set of 3D case studies with flat and complex terrains.

Additionally, we compare the results of the 2D and 3D methodologies when these are applied to the

same initial configurations to highlight the advantages that the latter has over the former.

The outline of the chapter is the following. Section 5.1 describes the modeling assumptions for the

wind turbines and the surface boundary layer that are integrated in the CFD and adjoint simulations.

In Sec. 5.2, we verify the continuous adjoint formulation by comparing the results obtained under the

frozen-turbulence and fully turbulent flow conditions with a central difference discretization method.

In Sec. 5.3, we present the wind farm layout optimization methodology that incorporates the terrain

topography and its application to a wind farm both in flat and complex terrains. We also compare this

methodology with the previously developed 2D implementation. Conclusions are summarized in Sec.

5.4.

5.1 Wind turbine and surface boundary layer modeling

This section provides a description of the approach used to model in our computational domain the wind

turbines and the surface boundary layer where the turbines operate.

87

Chapter 5. Toward WFLO in Complex Terrain 88

5.1.1 Wind turbine modeling

The wind turbines were modeled as actuator disks which are characterized by a cylindrical volume,

defined by the rotor swept area, where a distributed force, defined as axial momentum source, F , is

applied. From the definition of thrust coefficient, it can be derived that the axial force is a function of

the reference wind speed:

F =1

2ρπD2

4CTU

2ref , (5.1)





P = FUx = F1

V

∫V

UxdV. (5.2)

In the present study, a reference turbine with 80 m rotor diameter and 70 m hub height is chosen and

modeled in the wind farm simulations. It is also assumed to operate at 10 m/s and to have a thrust

coefficient of 0.75.


The atmospheric boundary layers (ABL) is used to model the wind conditions and characteristics usually

encountered in real wind turbine and wind farm flows. For an homogeneous and stationary flow, the

wind shear profile can be described, according to Panofsky and Dutton [115], as:

∂Ux∂z

=u∗κl, (5.3)

where Ux is the mean streamwise wind speed, z is the height above ground, u∗ is the local friction

velocity, l is the local length scale, and κ is the von Karman constant (≈ 0.4). In the present study, only

the lower part of the atmospheric boundary layer (ABL) under neutral conditions is considered. The

lowest part of the ABL, which is usually called the surface boundary layer (SBL), can be approximated

by a constant friction velocity equal to u∗0. Also, in the SBL, the length scale is assumed equal to the

height (lSL = z). Under these assumptions, a logarithmic velocity profile can be derived from Eq. 5.3

by integration:

Ux =u∗0κln

(z

z0

), (5.4)


Figure 5.1: Schematic of the layout for the verification case. The two grey areas represent the volumeswhere the wind turbine momentum sources are applied. The black dots indicate the different positionswhere the second wind turbine is placed when the gradient is calculated.

where z0 is the surface roughness length. This equation will be used to model the atmospheric conditions

in the wind farm simulations, assuming that the turbines are places over grass terrain with a surface

roughness of 0.0018 m.

5.2 Verification of the 3D continuous adjoint formulation for

WFLO

In this section, the verification of the continuous adjoint method is conducted for 3D simulations. Its

results in terms of gradient computation were compared with a finite difference discretization. This

form of verification has been extensively used in literature to assess the accuracy of the derivatives (e.g.,

[4, 8, 30, 47, 116]). The continuous adjoint method was therefore applied to the verification case shown

in Fig. 5.1, which consists in finding the gradient of the power production of a turbine placed in the

wake of another with respect to its turbine coordinates. For this case, a 3D domain was used and

the results were obtained for the frozen-turbulence and turbulent formulations of the adjoint equations.

The gradient components were calculated for the two directions of the domain and for a regular grid

of downstream and cross-stream positions. The downstream positions were set to 4D and 6D, with a

cross-stream spacing of 0.5D, where D is the turbine rotor diameter.

The results of the adjoint formulation were compared to those of a central difference (CD) approach,

which can be considered as the best achievable solution for numerical gradient computations. Figures

5.2 and 5.3 show the values of the gradient obtained by the two different adjoint formulations and the




CD approach. Tables 5.1 and 5.2 report evaluation metrics when comparing the adjoint formulations

and the CD approach. These are: the relative magnitude of the gradient computed with the adjoint

method:

‖∇Jrel‖ =‖∇JAM‖‖∇JAM,max‖

· 100; (5.5)

the percentage difference in the absolute value of the gradients computed by the adjoint method with

respect to the central difference approach:

err∇J =‖∇JAM‖ − ‖∇JCD‖‖∇JCD,max‖

· 100; (5.6)

the angular difference in the direction between the gradient calculated with the adjoint method and the

central difference approach, according to:

errθ = ‖θAM − θCD‖ . (5.7)

When the frozen-turbulence assumption was used for the adjoint equations, high discrepancies were

seen between the results of the two methods, with a maximum difference of 24.1% in absolute value and

a maximum difference of 121.5 in the direction. On the other hand, the fully turbulent formulation of

the adjoint method had results that overall were accurate for most part of wake region with a maximum

discrepancy of 5.7% in the absolute value. High discrepancies in the gradient direction can instead

be observed for the cases where the second turbine was in the outermost lateral positions. This is



Figure 5.3: Gradient computation for the turbulent case. The figures show the results obtained by theadjoint method compared to a central difference discretization approach for the different lateral positionsof the second wind turbine at 4D and 6D downstream.


0.0 39.0 24.1 0.0 19.4 20.4 0.00.5 100.0 9.0 10.6 63.2 5.1 14.01.0 93.9 9.0 9.0 72.3 4.1 11.81.5 26.2 7.8 48.7 30.5 6.6 38.72.0 20.8 20.4 103.2 20.9 18.7 92.02.5 20.6 21.4 121.5 20.7 21.6 126.8

Table 5.1: Relative magnitude of the gradient computed with the adjoint method, ‖∇Jrel‖, and errors inmagnitude, err∇J , and direction, errθ, between the gradient calculated with the adjoint method and thecentral difference approach on a grid of locations downstream of the turbine for the frozen-turbulencecase.


0.0 19.4 1.2 0.0 10.9 4.1 0.00.5 100.0 1.2 0.7 63.6 1.3 1.31.0 99.3 7.1 2.2 75.7 1.9 2.41.5 21.5 1.0 19.5 27.4 1.1 14.42.0 6.9 4.7 91.4 7.6 3.6 71.62.5 6.5 5.4 118.7 6.6 5.7 123.6

Table 5.2: Relative magnitude of the gradient computed with the adjoint method, ‖∇Jrel‖, and errorsin magnitude, err∇J , and direction, errθ, between the gradient calculated with the adjoint method andthe central difference approach on a grid of locations downstream of the turbine for the turbulent case.


explainable by the fact that the two components of the gradient approached a value of zero and even

a small discrepancy in one of the two can generate a large direction variation. In these positions,

the magnitude of the gradient is however significantly lower and almost negligible with respect to the

wake region and therefore this behavior is not expected to affect the optimization process. Overall, the

continuous adjoint method for 3D simulations had results that were very similar to those obtained for

2D simulations. In fact, the verification showed again the accuracy of the turbulent formulation and the

significant saving in computational cost. The successful verification of the 3D formulation of the adjoint

method enables its application to any kind of 3D system and therefore it will be implemented in the

optimization methodology for complex terrains in the following section.

5.3 Optimization methodology for the WFLO problem in com-

plex terrains

The gradient calculation performed with the adjoint method is used in a gradient-based optimization

methodology to find the optimal placement of a given number of wind turbines within a wind farm

domain to maximize the AEP. The optimization methodology, illustrated in Fig. 5.4, starts with the

wind rose, i.e., the site-specific statistical distribution of the wind resource, and an initial wind farm

layout. The wind farm is positioned on a terrain for which the geometry is known. Initially, CFD

simulations are used to calculate the AEP of that particular configuration. If convergence/termination

criteria in the iterative loop are not met, adjoint CFD simulations are then used to calculate the gradient

of the objective function, i.e., the gradient of the AEP with respect to the turbine positions. Using the

calculated gradients, turbine positions are updated to create a new turbine layout. These steps are

repeated until convergence/termination criteria are met.

The developed optimization methodology was applied to optimize a wind farm layout consisting

of 5 wind turbines using the fully turbulent formulation. The domain used for the CFD and adjoint

simulations had a dimension of 12Dx12Dx6D, whereas the wind turbines were constrained to be within

a 3D-radius circumference. A minimum distance of 1D between wind turbines was set as additional

constraint to avoid any overlap of wind turbines generated by the optimization algorithm. The terrains

used to test the optimization were of two kinds, flat and complex. For the complex terrain, we defined a

Gaussian-shaped hill in the middle of the domain, whose geometry is given by the following expression:

z (x, y) = H exp

−

[(x− x0)

2

2σ2+

(y − y0)2

2σ2

], (5.8)


Figure 5.4: Flow chart reproducing the optimization methodology used to solve the wind farm layoutoptimization problem in complex terrians

(a) 2D case, flat terrain (b) 3D case, flat terrain (c) 3D case, complex terrain

Figure 5.5: Illustration of the domains used for the application of the 2D and 3D methodologies.

where H is the maximum height of the hill, σ is the standard deviation of the Gaussian function, and x0

and y0 are the coordinates of the maximum height of the hill. For the present case, the height and the

standard deviation were set to 0.5D and 1D, respectively. To highlight the advantages of the present

methodology for 3D systems, this was also compared to the previously developed 2D methodology for the

same initial configurations and boundary conditions. As such, we had in total three different domains,

as shown in Fig. 5.5. To analyze the dependence of the gradient-based methodology to the initial

configuration, two initial layouts were tested, namely a regular layout (commonly used in wind farms)

and a random layout (see Fig. 5.6). The wind rose used to characterize the wind distribution was instead

assumed to be composed of six evenly weighted wind directions with a constant wind speed of 10 m/s.

The results obtained from the optimization are reported in Fig. 5.7 and 5.8 for a regular and a

random initial layout, respectively. The normalized AEP is defined as the ratio between the actual AEP

and the AEP that would be generated by the same turbines operating in isolation on a flat terrain.

Convergence was reached within 10 iterations of the optimization loop for all the cased analyzed. The


(a) Wind rose (b) Regular initial layout (c) Random initial layout

Figure 5.6: Wind rose and initial layouts used as inputs for the optimization process.

Regular initial layout Random initial layoutNorm. AEP Impr. [%] Norm. AEP Impr. [%]

2D case, flat terrain 1.032 13.9 0.982 6.03D case, flat terrain 0.972 6.1 0.964 2.83D case, complex terrain 1.013 3.0 1.019 3.2

Table 5.3: Normalized AEP and improvement with respect to the initial configuration obtained at theend of the optimization for each of the initial wind farm layouts and for each of the cases tested.

results of the optimization process in terms of normalized AEP are reported in Tab. 5.3.

For the cases with a regular initial layout, it is possible to notice that the generated optimal layouts

retained a certain degree of regularity, which can be explained by the symmetry of both the wind rose

and the initial layout. The layouts were, in fact, symmetric about a horizontal or vertical line passing

by the center of the domain. These characteristics of symmetry of the layouts for evenly weighted wind

directions and for uniform wind speed are usually considered an indication of an effective optimization

algorithm.


Figure 5.7: Optimal layouts obtained at the end of the optimization process starting from a regularinitial layout.



Figure 5.8: Optimal layouts obtained at the end of the optimization process starting from a randominitial layout.

Differences can be highlighted between 2D and 3D results for a flat terrain scenario. It is possible

to notice that even if the layouts look quite similar, the turbines in the 2D cases are further apart.

This can be explained by the different modeling implementation of the wind turbines. Whereas the 3D

case used actual cylinders to model the wind turbines, in the 2D case the actual turbine volumes were

modeled with a rectangular prism with unit height and rectangular base whose width is equal to the

rotor diameter. Because of this, the 2D modeling inherently generates wake effects that are higher and

therefore induces the optimization to move the turbines further apart.

Looking at the results in terms of normalized AEP for the optimal layouts, it is interesting to notice

that the actual AEP was greater in some cases than the one that would be generated by the same

turbines operating in isolation and on a flat terrain. For instance, this occurred in the 2D case with a

regular initial layout. The reason for this result can be explained by the fact that the CFD simulations

allow to accurately resolve the flow field and to capture the local speed-up effect just outside of the

wake regions. The optimization algorithm took advantage of this feature and moved the turbines toward

these favorable positions where the speedups were higher. Although the optimal layout of the 3D cases

in flat terrain were very similar to the 2D cases, the AEP results of the former ones did not clearly

show to benefit from this flow characteristic. As also suggested in previous studies [95], this indicates

that speedup effects generated by the turbine presence are more pronounced in 2D simulations. A value

of normalized AEP greater than one was observed also for the 3D cases in complex terrain. This can

be explained by the fact that the Gaussian-shaped hill, similarly to the turbines, induced a speed-up

effect above it that created an increase in the energy captured by the turbines. A consequence of this

effect can be seen in the optimal layout for the 3D case in complex terrain with a random initial layout

(see Fig. 5.8c). In fact, differently from the other two cases with the same initial layout, it is possible

to notice that one of the turbines remains in the middle of the domain on top of the hill where the


speedup and therefore the energy increase are expected to be maximum. These results ultimately show

the advantages that CFD models have over the simplified wake models currently used for the wind farm

layout optimization, which cannot capture the complex flow dynamics occurring in wind farms.

Overall, the optimization methodology applied to 3D systems and based on the adjoint method for

the gradient computation could effectively improve the AEP of a given wind farm layout by changing

its turbine positions. The improvements ranged from about 3% for the 3D case in flat terrain with a

random initial layout, to 6% for the 3D case in flat terrain with a regular initial layout.

5.4 Conclusions

In the present work, we applied a gradient-based WFLO methodology to 3D wind farms in both flat

and complex terrain. This methodology integrates the high accuracy and flexibility offered by the CFD

models and uses the adjoint method to calculate the required gradients. The adjoint method was first

verified on a simplified 3D 2-turbine wind farm where only the gradient of the wind turbine operating

in wake conditions was considered. The results obtained with adjoint method using the fully turbulent

formulation were generally accurate in terms of gradient computations when compared to the results

obtained with a central difference discretization. The gradient calculation performed with the adjoint

method was therefore used in the optimization methodology to solve the 3D WFLO problem.

A hypothetical 5-turbine wind farm was defined with two different initial layouts, with turbine

positions either set on a regular grid or random. A wind rose with six evenly weighted wind directions

and constant wind speed was defined as input for the optimization. Two different kinds of terrain were

considered, flat and complex. For the flat terrain scenario, the present methodology for 3D systems

was also compared to the previously developed 2D methodology for the same initial configurations and

boundary conditions. For the complex terrain, we defined instead a Gaussian-shaped hill in the middle

of the domain, which could only be handled by the present 3D methodology.

The optimization methodology for 3D wind farms could effectively improve the AEP of the given

wind farm layouts from 3% to 6% depending of the initial layout and terrain topography. It was, in

fact, the first CFD-based methodology being applied to the wind farm layout optimization in complex

terrain. The methodology showed also the benefits of using CFD models which offer a more detailed

representation of the flow field in wind farms by, for example, capturing speed-up effects just outside of

wake regions and around terrain features.

Chapter 6

Conclusions and Future Directions

In this chapter, we present a summary of the conclusions of the works that contributed to the final CFD-

based methodology for wind farm layout optimization and we also present directions and suggestions for

future research.

6.1 Outcomes and their impact

The main contribution of this thesis is a new methodology for the wind farm layout optimization that

integrates CFD models and takes advantage of an innovative formulation of the adjoint method for

gradient calculations. Within this overarching contribution, important results can be highlighted for

each of the works conducted in this thesis:

• We compared in a consistent way the principal turbulence models for RANS equations present in

literature, namely the k− ε, k− ω, and Reynolds stress model, and we introduced the SST k− ω

model as a innovative turbulence model for wind turbine simulations. We showed that the SST k−

ω model performed as good as the the RSM when calculating turbine-wake characteristics, such as

wind speed, turbulence kinetic energy and turbulence intensity, whereas the simulations using the

k − ε and k − ω models provided poor predictions of wake flows. As a result of this comparison,

we showed that the SST k − ω can be used as an effective turbulence model for wind turbine

simulations.

• We investigated the limitations and inconsistency of the RANS models in the predictions of wake

effects in wind farms. We hypothesized that the discrepancies found in some cases between experi-

mental and simulation results arise from wind direction uncertainty caused by large-scale unsteady

97

Chapter 6. Conclusions and Future Directions 98

phenomena, which though present in the experimental measurements were not accounted for in the

simulations. We therefore proposed an approach to account for the wind direction uncertainty by

modeling it using simulation ensembles. Our results showed that the proposed method significantly

improves the agreement of the CFD predictions with the available experimental observations when

wind speed and power production in wake regions are considered. These results suggested that

the discrepancies between CFD predictions and experimental data reported in previous works,

attributed to inaccuracy of the CFD models, can be explained instead by the uncertainty in the

wind direction reported in the data sets.

• We presented an innovative continuous adjoint formulation for gradient calculations within the

framework of a gradient-based wind farm layout optimization. The optimization methodology

integrates CFD models and, thanks to the adjoint method, overcomes the computationally high

costs of a CFD-based optimization. We derived adjoint formulations for different conditions in the

flow equations, namely, laminar, frozen-turbulence and turbulent flows. The proposed formulation

was implemented in a 2D domain and successfully verified by comparing the calculated gradients

with finite-difference approximations. Gradient calculations using the developed adjoint method

were implemented in a gradient-based optimization methodology to solve a 2D wind farm layout

optimization problem under a wide array of wind resource scenarios. Our results showed that the

annual energy production of a given wind farm layout can be effectively improved within 30 to 60

iterations, depending on the initial layout and wind resource distribution. Improvements in AEP

were found to be in the range of 7-37%, with an average of 15%.

• We improved the developed optimization methodology so to handle 3D systems and therefore

to account for the terrain topography. The adjoint method for 3D simulations was verified by

comparing the calculated gradients with a finite-difference discretization method. We applied the

gradient-based optimization methodology using the verified adjoint method to a wind farm both

in flat and complex terrain. We showed that the methodology could effectively improve the AEP

of the given wind farm layouts by changing the turbine positions on average by 3% and 6% for the

regular and random initial layouts, respectively. It was, in fact, the first CFD-based methodology

being applied to the wind farm layout optimization in complex terrain. The methodology showed

the benefits of using CFD models which offer a more detailed representation of the flow field in

wind farms by, for example, capturing speed-up effects just outside of wake regions and around

terrain features that cannot be observed with analytical wake models.

Overall, we showed that CFD RANS models can effectively be used in wind farm simulations when proper


turbulence modeling is chosen and when uncertainties in wind direction are properly considered. We

enabled the use of these CFD models in the wind farm layout optimization by developing an innovative

adjoint formulation for gradient computations. Our contributions advanced our knowledge of wind farm

modeling and simulation and created a new and reliable CFD-based design framework for the wind

turbine industry.

6.2 Future directions

In this section, we suggest some future research directions that arise thanks to the successful development

of the proposed CFD-based methodology or from aspects that have not been considered in this study.

Application of the proposed methodology to real wind farms

The developed CFD-based methodology for WFLO has not been applied to real wind farms. We consider

this as the first step that should be conducted after this work. Simulation and optimization of real

wind farms demand however publicly available experimental measurements for the model validation and

extensive computational resources for the optimization process. Whereas measurements from real wind

farms are becoming more accessible thanks to a growing wind energy community [10, 28, 139], access to

sufficient computational resources is currently seen as the bottleneck for such applications.

Uncertainty quantification of RANS models for wind farm simulations

CFD, especially RANS, models represent a significant part of the simulation tools used for wind farm

modeling and performance estimation and are expected to be next widely-used simulation tool for wind

farm design and optimization. In the present study, we investigated the effect of the wind direction

uncertainty and we proposed a method to effectively account for it. However, significant uncertainties

are believed to be generated by the modeling assumptions for the Reynolds stresses in the momentum

equations [66, 67, 160]. The quantification of such uncertainty is of fundamental importance to enhance

the predictive capabilities of RANS simulations in the context of wind farm simulations.

CFD-based wind farm layout optimization under uncertainty

In the present study when developing the optimization methodology, we implicitly assumed that the

inputs and boundary conditions for the optimization did not have any uncertainty associated. This is

of course not the case: the wind resource can have variations from year to year, the terrain roughness

is usually not constant over the entire wind farm land, and the terrain topography is generally not


accurate. Uncertainties for these and other variables need necessary to be accounted for in a robust

optimization methodology. Some studies have been conducted to account for the uncertainty of some of

these factors in wind farm simulation and optimization [54, 110, 129], but a comprehensive methodology

is still missing, especially when CFD models are used.

Optimal control of wind farm energy production

After a wind farm has been designed and installed in place, optimal control of energy production becomes

the next target for the wind farm operators. Wind farm control has been extensively studied in literature.

Simultaneously controlling the performance of each wind turbine has been shown to be an effective

strategy for optimal energy extraction [53]. Techniques for performance control can be identified as

pitch and yaw control [65]. Adjoint methods are suitable for this kind of applications [64] and we believe

that the developed adjoint formulation could be used as the basis for an innovative methodology for

optimal control of wind farm energy production.

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