Wind Turbine Simulations using
Actuator Line Model
Author: Andrea MATIZ CHICACAUSA
Supervisor: PhD. Omar L ´OPEZ Evaluator: PhD. Andr´es GONZALES
A thesis submitted in fulfilment of the requirements for the degree of Master in Mechanical Engineering
in the
Departamento de Ingenier´ıa Mec´anica Facultad de Ingenier´ıas
Universidad de Los Andes
July 2015 Bogot´a, Colombia
I, AndreaMatiz Chicacausa, declare that this thesis titled, ’Wind Turbine Simulations
using Actuator Line Model’ and the work presented in it are my own. I confirm that:
This work was done wholly or mainly while in candidature for a research degree
at this University.
Where any part of this thesis has previously been submitted for a degree or any
other qualification at this University or any other institution, this has been clearly stated.
Where I have consulted the published work of others, this is always clearly
at-tributed.
Where I have quoted from the work of others, the source is always given. With
the exception of such quotations, this thesis is entirely my own work.
I have acknowledged all main sources of help.
Where the thesis is based on work done by myself jointly with others, I have made
clear exactly what was done by others and what I have contributed myself.
Signed:
Date:
Abstract
Facultad de Ingenier´ıas
Departamento de Ingenier´ıa Mec´anica
Master in Mechanical Engineering
Wind Turbine Simulations using Actuator Line Model
by AndreaMatiz Chicacausa
Computational simulations of fluid dynamics around wind turbines have become and important tool in the wind energy research field due to the opportunity to gain good insight of the physical phenomena with less expensive cost compared with experimenta-tion. However, the computational cost of such simulations is still high and efforts have been made in order to simplify simulations keeping the accuracy in the results. Actua-tor line is a technique proposed by Sørensen and Shen [22] to simplify such simulations replacing the actual geometry of a blade by a line over which punctual body forces will be computed and projected on the flow. The use of this technique makes simulations with fewer grid points able to resolve wake structures behind the turbine. This thesis proposes the use of this technique to simulate a full wind turbine (rotor and tower) in order to achieve conclusions regarding the accuracy of the model to predict forces and to finally prove the capability of the model to simulate the tower. Results from the use of the AL technique and the aerodynamic response to the tower presence are shown and compared to experimental data from UEA NASA Ames Phase VI Experiment.
As always there are many people to whom I am deeply thankful. First of all, I would like to express my deepest gratitude to my thesis supervisor Omar Lopez who was always present during this work, his constant support and patience encourage me to continue during the most difficult parts of this work. This work could not have been possible without the patient advice and time invested by Ivan Herraez who spent precious time teaching me and discussing with me the main topic of this thesis.
This thesis would not have been possible without funding from the Turbine Simulation, Software Development and Aerodynamics Group of Fraunhofer IWES Institute and his head Dr. Bernhard Stoevensandt who helped me even in difficult moments for the group and himself.
Many thanks to the Turbulence, Stochastic and Wind Energy Group of ForWind Insti-tute of Oldenburg University, specially the CFD team for providing all resources needed, the productive discussions and support from colleagues made every day of work of this thesis worth it. I am deeply indebted to Bastian Dose and Hamid Rahimi who helped me more than needed not just for this work but also in my daily life in a foreign country and made of my days in Germany something more than work.
Thanks to the computer time provided by the Facility for Large-scale Computations in Wind Energy Research (FLOW) at University of Oldenburg. Many thanks to Dr. Scott Schreck from the National Renewable Energy Laboratory who provided the experimental data.
Last but not least to my family, without their support and sometimes sacrifices nothing of this could not have been possible. Finally, many thanks to Carlos Benavides for his support in critical moments.
Declaration of Authorship i
Abstract ii
Acknowledgements iii
Contents iv
List of Figures vi
List of Tables viii
1 Introduction 1
1.1 Motivation . . . 2
1.2 Previous Work . . . 4
1.3 Unsteady Aerodynamics Experiment Phase VI . . . 6
1.3.1 Turbine Description . . . 7
1.3.2 Test Description . . . 9
1.4 Present Work . . . 10
1.4.1 Objectives. . . 10
1.4.2 Thesis Outline . . . 10
2 Theoretical Background 11 2.1 Governing equations . . . 11
2.1.1 Turbulence . . . 11
2.2 Numerical Methods. . . 13
2.2.1 Finite Volume Method . . . 13
2.2.2 SIMPLE, PISO and PIMPLE . . . 16
2.3 Wind Turbines Aerodynamic and BEM . . . 19
2.4 Actuator Line Model (ALM) . . . 21
3 Numerical Simulations and Results 23 3.1 Computational Tools and Setup . . . 23
3.1.1 Navier-Stokes Solver . . . 23
3.1.2 Mesh Generator . . . 24
3.1.3 Actuator Line Solver . . . 24
3.2 CFD Simulations . . . 24
3.2.1 Rotor-Actuator Line Simulations . . . 25
3.2.1.1 Grid dependence . . . 27
3.2.1.2 Epsilon . . . 27
3.2.2 Tower-Actuator Line Simulations . . . 29
3.2.2.1 Implementation . . . 29
3.2.2.2 Tower-AL . . . 31
3.2.3 Rotor and Tower-Actuator Line Simulations. . . 34
3.2.4 Three-dimensional rotor and tower-AL . . . 36
4 Conclusions 41 4.1 Conclusions . . . 41
4.2 Further Work . . . 42
A Blade chord and Twist Distribution 43
B k−ω SST Model: Mathematical Expressions [12]. 45
1.1 Wind Turbines hisctoric increment of size [15]. . . 1
1.2 NREL Phase VI Turbine. . . 3
1.3 Design Models of Wind Turbines through complexity levels. BEM [3], Free Vortex Wake and CFD [17] . . . 4
1.4 NREL Phase VI Blades [5]. . . 7
1.5 Reynolds Number local to the blade distribution for different wind speeds. 8 1.6 Mach Number local to the blade distribution for different wind speed. . . 9
2.1 Structured and unstructured mesh for the finite volume method [27]. . . . 14
2.2 Schematic representation of a mesh for finite volume method [27].. . . 15
2.3 SIMPLE algorithm [27]. . . 17
2.4 PIMPLE algorithm. . . 18
2.5 Acutator Disc and Stream Tube. . . 19
2.6 Blade Element [3]. . . 20
2.7 Blade Element velocities and forces [3].. . . 21
3.1 Computational Domain. . . 25
3.2 Schematic top view of the domain and the refinement boxes. . . 26
3.3 Output power vs. number of elements in the domain. . . 28
3.4 Normal force coefficient along the blade. . . 28
3.5 Tangential force coefficient along the blade. . . 29
3.6 Computational Domain (Cylinder Simulations). . . 30
3.7 Pressure distribution on the cylinder surface at Re= 2.6×105. . . 31
3.8 Lift and drag forces from the cylinder simulation. . . 32
3.9 Forces oscillating with the actuator line model after modifying the lift and drag coefficients on the code. . . 32
3.10 Deficit of velocity comparisson between results from simulation of a cylin-der, actuator line model and actuator line modified. . . 33
3.11 Vorticity produced by the Actuator Line tower. . . 33
3.12 Output Power spectrum forU∞= 7m/s . . . 34
3.13 Normal force coefficient . . . 35
3.14 Tangential force coefficient. . . 35
3.15 Normal force coefficient for 5 sections on the blade 30%, 47%, 63%, 80% and 95% atU∞= 7m/s . . . 36
3.16 Three dimensional mesh, detail of the root of the blade. . . 37
3.17 Instantaneousy+ distribution over the blade. . . 37
3.18 Pressure coefficient distribution for 5 sections on the blade 30%, 47%, 63%, 80% and 95% at U∞= 7m/s . . . 39
1.1 NREL Phase VI Turbine Dimensions . . . 7
1.2 NREL Phase VI Blade chord and twist distributions . . . 9
3.1 Mesh characteristics . . . 26
3.2 Boundary Conditions for rotor and tower simulations . . . 27
3.3 Boundary Conditions for 3D rotor and tower-actuator line . . . 38
Introduction
The use of renewable energies to supply an increasing demand of energy over the world seems to be the trend in the near future. So far, this kind of alternative energies are the key to a sustainable way of progress since the use of wind or sunlight, for instance, is an unlimited, clean and free source of energy.
As the use and demand of these sources of energy increase the size of the wind turbines has grown up as well (see Figure1.1). According to the International Energy Outlook of 2013 by the Energy Information Administration EIA, in 2010 the wind energy production was 2.5% of the total electricity consumed and it was estimated that for the year 2020 the wind energy generation will reach 700 GW. However, nowadays the wind installations grew by 44% worldwide and it is expected a generation of nearly 2000 GW by 2030 [1].
Figure 1.1: Wind Turbines hisctoric increment of size [15].
The constant increment of the wind turbines size and the complexities present in the evolution of the wind industry are the reason of the growing need for highly reliable devices to transform the energy from the source to everyday electricity.
Moreover, the significant size that the wind turbines have achieved nowadays is an im-portant aspect that demands better/reliable design models in order to avoid expensive costs of maintenance, that in some seasons of the year is impossible to carry out. There-fore the wind energy industry needs are clear: reliable, accurate and affordable design
models that improve the loads prediction during the operation of the wind turbine on the field. And this is just possible by understanding the physical effects behind the operation and states of these devices.
This section aims to present the importance of researching in wind energy field specially in the topic regarding to this thesis: the accurate prediction of aerodynamic loads by means of Computational Fluid Dynamics (CFD) models validating the Actuator Line (AL) model and the study of the aerodynamic effects.
First, it is presented the motivation for this research work. Second, it will be shown some previous works done on this topic. And finally, the objectives proposed and the outline of this document.
1.1
Motivation
In Wind Energy and especially regarding to wind turbines technologies, aerodynamics is one of the most important aspects. We must not forget that a wind turbine is, after all, a machine that transforms kinetic energy contained in the wind into mechanical power through an aerodynamic process.
The importance of aerodynamics lies on the large amount of parameters that play a big role in the process of energy transformation. The current design models of the rotor are still subject of large uncertainties due to several phenomena that can be classified in two groups: periodic and aperiodic [10]. These uncertainties sources can be produced by three-dimensional effects, unsteady effects, de-attached flow effects (stall), tower effects, among others [18].
These phenomena are highly difficult to measure, asses or model and the unknown responses of the turbine under those effects, different work conditions and configuration produce an unexpected and erratic behaviour, placing higher loads on the rotor than that they were designed for, and finally making a not-so-reliable device increasing the cost of energy in some cases.
The upwind configuration of wind turbines, that is when the air flow comes form the front of the rotor, has been the most popular the last decades. The main reason for this is the low frequency noise found in downwind turbines when the blade passes through the tower wake. However, downwind configuration present some advantages like the flexibility on blade design and yaw angle since the interference between rotor and tower is avoided.
In 2001 the National Renewable Energy Laboratory (NREL) invited several experts from different institutes to participate in a blind prediction of loads and performance of an instrumented wind turbine that was tested under controlled conditions in the NASA Ames wind tunnel [21]. The results from the comparison between the data measured showed an important lack of accuracy among the predictions; moreover, the comparison between the participants shown wide variations between various design codes.
The evident and significant problems within the wind turbines designing process have yield in making efforts to understand the complexities of the several phenomena taking place in the wind turbines operation. A database of experimental data was obtained by the NREL-NasaAmes Unsteady Aerodynamic Experiment (UAE) in 2001 [5] (see
Figure 1.2: NREL Phase VI Turbine.
Figure 1.2); later and with the objective to reduce the uncertainties present in design models, performance and loads calculations the Energy Research Center of The Nether-lands (ECN) led the Model Experiments in Controlled Conditions (MEXICO) project, which provided a experimental measurements database but with some different measured methods as for example Particle Image Velocimetry (PIV) [19].
Those databases sought to validate the existent engineering models and specially in the case of MEXICO to provide a validation tool for Navier-Stokes based calculation techniques. In conclusion, the aerodynamic modelling of wind turbines ranged from the basic Blade Element and Momentum theory (BEM) to engineering models based on BEM and vortex wake methods to CFD methods to solve the Navier-Stokes equations (see Figure1.3).
Although engineering models based on BEM are currently the most used methods and provide high predictive confidence levels in the designing of wind turbines the CFD method is a more useful tool from which it is possible to achieve a deep insight and physical realistic simulation of the turbine flow field, its behaviour and surrounding.
In the last decade CFD simulations have been a commonly used tool that gives good insight about the phenomena happening during a wind turbine operation. However, the simulation of such big machines require to handle complex geometries and in most cases simulate transient phenomena making the method a expensive and complex technique.
Figure 1.3: Design Models of Wind Turbines through complexity levels. BEM [3],
Free Vortex Wake and CFD [17]
The large amount of computational resources (memory and simulation time) needed and the numerical issues associated plus the complexity of the phenomena to simulate have led to developed simplified models based on actuators that capture the essence of the wind turbine behaviour reducing the computational resources needs. Although these models do not describe completely the underlying physics behind the operation they are accurate when computing specific quantities of interest.
The actuator disk concept, for example, is based on the representation of the physical rotor as equivalent forces distributed on a permeable disc of zero thickness in a flow domain. With a similar concept in mind Sørensen and Shen [22] proposed an actuator line (AL) model which was later slightly modified by Michelsen [13].
The AL model combines a three-dimensional Navier-Stokes equations solver with a tech-nique where body forces are distributed along a line that represents a blade. The ad-vantage of this technique against the actuator disc model is its discrete nature that it is able to resolve tip and root vortices. Moreover, with the AL technique there is not need of resolving blade boundary layer instead the computational resources are employed to simulate the dynamics of the flow structure [7]; therefore, information about the wake structure and azimuthal distribution of velocity induction factors can be obtained with the use of simple meshes.
1.2
Previous Work
Numerical simulation for wind turbines application started with a low acceptance since at that time they were too demanding in terms of computational resources and time of processing. However, as the computational technologies progress they became more accepted and currently a powerful tool for accurate aerodynamic predictions with a lower cost than required for full scale experiments in wind tunnel. These simulations are commonly carried out by solving the three-dimensional Navier-Stokes equations both in steady and unsteady approaches [6, 9]. Besides the computation of the governing equations the turbulence can be modelled by three main methods Reynolds Averaged Equations (RANS), LES and DES. However, those last ones are more demanding in
terms of mesh quality and computational time hence RANS modelling of turbulence is the most commonly used. For wind turbine applications several tools have been used, from commercial codes developed by ANSYS likeFluent to open source codes like
OpenFoam to in-house developed codes like ellipSys3D [14].
The challenges that simulate a full wind turbine present are time and computational consuming. First, mesh generation of complex geometries as it is the case for blades is one of the most time demanding tasks. Second, handling the relative movement between the rotor and tower. And finally, to capture the unsteady phenomena happening on the flow over the rotor and surroundings in a reduced-cost way.
Different techniques have been adopted to face those challenges. Concerning the relative movement between rotor and tower can be found for both upwind and downwind: Sliding interface mesh [8], a General Grid Interface (GGI) method [9] and overset grid or so-called chimera grids [31] are the most used techniques.
In order to reduce the computation time, techniques based on actuators have been developed. In 2002, Sørensen and Shen [22] proposed a model to study three-dimensional flow field over wind turbine rotors. This model combined a three-dimensional Navier-Stokes Solver with an actuator line that represent the blade geometry over which the loads are distributed. The main goal of this model was to analyse and validate some engineering models.
In 2003, Michelsen [13] reformulated the AL model in terms of the primitive variables: pressure and velocity (p−V) and use the in-house Navier-Stokes equations solver ellip-Sys3Dto study the aerodynamic behaviour of coned rotors, rotors exposed to yaw inflow and tunnel blockage. Besides, the AL model was used to analyse some of the assump-tions of BEM method such as tip loss correction models. Michelsen [13] also proposed a three dimensional actuator line changing the form of the regularization kernel since the original model was physically inaccurate when distributing loads near the tip.
Since its formulation AL model has been subject of several studies especially directed to analyse the effect in the results when changing the parameters associated. The use of a suitable and its influence is one of the most studied [11,20,26]; however, there is not a definitely conclusion about the bestvalue.
Studies using the AL model have been often directed to analyse wake structures and tip vortices. Ivanell et al. [7] studied the behaviour of wakes behind the wind turbines with special interest in the structure and position of the tip vortex and the circulation on the blades. Regarding to it was concluded that the wake expansion does not depend on it; however, it influences the phase between the root and tip vortices; therefore, they suggested reduce it as much as possible keeping a good agreement between numerical stability and accuracy.
Troldborg et al. [26] studied the wake behind the wind turbine operating in a uniform inflow combining LES simulations and the AL technique and, varying inflow conditions combining AL technique and URANS computations [25]. There, it was investigated the sensitivity of the computed solutions to the regularization parameter . Simulations with four different values were performed 1.5∆r, 2.0∆r, 2.5∆r and 3.0∆r where ∆r is the cell side length in the equidistant region. As conclusion from his work it is seen that increasing causes a more smooth variation of the velocity near the root and tip. On the other hand if is chosen too large it will be difficult to distinct the pattern from
the tip and root vortices. In this work = 2∆r showed a good compromise between reducing oscillations without smoothing them out too much.
A comparison between actuator disk and actuator line was performed by Martinez et al. [11] in terms of predicting wind turbine power production and wake velocity deficits and conclusions about good practices were attempted. Besides, the parameters that affect the performance of these models were analysed; for example, grid resolution, use of tip and root loss correction and the way the forces are projected onto the flow field. To do these, they implemented a CFD solver in OpenFOAM and compare the results with a NREL-developed code that uses BEM theory.
From the performance of the AL model it was concluded that the power output increases with mesh refinement and asbecomes larger the predicted power increases as well. The largerthe smaller the rate of change of predicted power as a function of grid resolution. And as closest this parameter to the characteristic length of the blade section gives better results. Regarding to the actuator points were the forces will be smeared is said that they should be enough to predict the force field along the blades and to have a smooth distribution through the blade, this criterion should be the basis for establishing the blade resolution. A very important conclusion is get from this work: the time step size have a big impact on simulation and it is restricted by the tip speed which should not pass through more than one cell each time-step. This condition is more restrictive than the typical Courant-Friedrichs-Lewy condition (CFL= 1).
It is evident the importance of studying the dependence between AL performance and grid refinement level. Shives and Crawford [20] performed a numerical tests to determine mesh density and to provide force distribution guidelines. This was carried out by using an infinite span wing and finite span wing with constant circulation distributions. They found that the parameter should be somehow to be related to the local airfoil chord lengthc. Although, they did not conclude to an specific value, they suggested the ideal of defining based on the airfoil pressure distribution, which depends of the angle of attack (AoA). This would means to change this parameter in each iteration.
Lately, Nilsson et al. [14] validated the capability of the AL model to capture vortex structures in the near weak behind the MEXICO rotor comparing the simulation results with PIV measurement data finding good agreement between the wake expansion and tip vortex circulation.
Overall, the AL model has been tested for several cases and has showed a good perfor-mance. Although, definitely guidance with respect to the use and the influence of the parameters associated has not been completely concluded some basic assumptions are been made and it is clear that it has to be tested and analysed for other cases.
1.3
Unsteady Aerodynamics Experiment Phase VI
The National Renewable Energy Laboratory (NREL) of the United States conducted the Unsteady Aerodynamics Experiment (UAE) in order to provide accurate data represent-ing a wind turbine in field operation; in this way the characterization of the forces actrepresent-ing in a turbine to get a better understanding of the loads was possible; hence, optimize blade and rotor designs and validate the existent models and BEM assumptions.
The experiment was performed in the NASA-Ames wind tunnel with a two-blade wind turbine of 10 meter diameter. Several measurement campaigns were carried out in order to test the wind turbine under different operational conditions and to obtain data to analyse specific effects.
The information in this section about experiment, test turbine and test description was found in [5]. For detailed information refer to the reference or to other technical reports.
1.3.1 Turbine Description
The test turbine was a 10 meter diameter, stall-regulated with full span pitch control with a power rating of 20 kW, shown in Figure1.4.
It was two, twisted and tapered blades and it was tested in both upwind and downwind configurations.
Figure 1.4: NREL Phase VI Blades [5].
The blade’s cross section is S809 airfoil designed by NREL. The S809 is a 21% thick-ness airfoil, designed for wind energy applications, specifically for horizontal axis wind turbines (HAWT). It was optimised to improve wind energy power production and is less sensitive to leading edge roughness; for this airfoil there is available several data that includes pressure distribution, separation boundary locations and drag coefficient. The two-dimensional blade profile data and, chord and twist distribution are included in AppendixA. Some other turbine parameters are shown in Table 1.1.
Rotor diameter 10.058 m.
Hub height 12.192 m.
Tower Diameter 0.4 m.
Tower clearance 1.401 m.
Airfoil Data
As mentioned the blades are conformed of the S809 airfoil and its polars were took from [5] for Reynolds number ranging between 300000 and 1000000. They were obtained at the Colorado State University for Reynolds number until 650000 and at Ohio State University for higher values.
The local Reynolds number of the blade varies in function of the chord as shown in Equation 1.1 and its distribution spam wise is shown in in Figure 1.5 for wind speed from 5 to 9m/s; then, accordingly the airfoil data was setted up in the directory for AL simulations.
Re= Ubladec
ν , (1.1)
where Ublade is the local blade velocity as express in (1.2), c is the chord and ν is the
kinematic viscosity.
Ublade=
q
U2
ax+Urot2 , (1.2)
where, Uax is the velocity in the axial direction (stream direction); and Urot is the
rotational speed (Urot=ωr, where r is the the radius of the blade).
Figure 1.5: Reynolds Number local to the blade distribution for different wind speeds.
The local Mach number
M a= Ublade
a
whereais the velocity of sound in air (343m/s) was computed as well in order to determine the compressibility conditions of the rotor. The distribution along the spam, shown in Figure1.6makes evident the incompressibility of the flow as the Mach number is always less than 0.3.
Figure 1.6: Mach Number local to the blade distribution for different wind speed.
1.3.2 Test Description
Several measure campaigns were performed, some to emulate field operation and some other to collect data to explore specific phenomena. Tests were run in upwind and downwind configuration. The angle of attack and dynamic pressure were measured using five-hole probes installed at five different spam sections over the blade. The span position, chord and twist distribution are shown in Table 1.2.
Radius [m] r/R [-] Chord [m] Twist [◦]
1.510 0.30 0.711 14.292
2.343 0.466 0.627 4.715
3.173 0.631 0.543 1.150
4.023 0.80 0.457 -0.381
4.780 0.95 0.381 -1.469
Table 1.2: NREL Phase VI Blade chord and twist distributions
For the purpose of this thesis the results were compared with two specific campaigns: test sequence B and H, the downwind and upwind baseline tests. Both were performed with a locked yaw angle, 72 RPM and wind speed ranging from 5 m/s to 25 m/s; however the numerical results were compared with the campaign at 7 m/s. The blade tip pitch was 3◦ constant. Data was recorded during 36 blade rotations that is 30 seconds; measurements were sampled at 520.83 Hz.
The test sequence B was in downwind configuration with a teetered turbine and a cone angle of 3.4◦. For low wind speed yaw angles of±180◦ were achieved and for high wind speed angles of −20◦ to 10◦. The test sequence H was in upwind configuration in a
rigid turbine with a 0◦ cone angle. For low wind speed yaw angles of −30◦ to 180◦ were achieved and for high wind speed angles of±10◦.
1.4
Present Work
The AL model has several input variables that depend strongly from each other and are significant for the results. Currently there are no clear conclusions about the method to determine these parameters. This work aims to analyse how these parameters influence the results and determine if it is possible to use the same simplification to simulate with the AL model a tower hence to simulate a full wind turbine actuator line and achieve conclusions about the effect of the tower in the aerodynamic loads over the rotor.
To fulfil this intention the next objectives were proposed:
1.4.1 Objectives
The main objective of this thesis is to simulate the NREL- UAE Phase VI turbine in downwind configuration using the AL model.
The next specific objectives are proposed:
Specific Objectives
• To generate the mesh and simulate the rotor using the AL model.
• To analyse the influence of the level of mesh refinement on the results of the simulation.
• To analyse the influence of the parameters from the actuator line model (regular-ization kernel and number of actuator points) on the results of the simulation.
• To analyse the influence of the time step (∆t) on the results of the simulation.
• To implement the actuator line model to simulate the tower.
• To simulate the full wind turbine using the actuator line model.
• To compare the results from simulations with experimental data.
1.4.2 Thesis Outline
This document is divided in five chapters. The present, Chapter1is devoted to explain the motivation of this thesis, the state of art of the use of actuator line and finally to mention the objectives of this work. Next section, Chapter 2 will provide a theoretical frame, briefly explaining the governing equations, the actuator line model and some other concepts used in the development of this work. Chapter 3 will show the simulations performed, the pre, post-processing and results. Finally, Chapter 4 will mention the conclusions, discussion and further possible work.
Theoretical Background
This section introduce some basic information about Numerical Simulations, mentioning the governing equations of fluid dynamics as a keystone upon the computational fluid dynamics; the numerical models to solve the governing equations and the methods to model turbulence. The Actuator Line model is explained as it was proposed.
2.1
Governing equations
The governing equations for a viscous, unsteady, three-dimensional, incompressible flow are the Navier-Stokes Equations: continuity and momentum conservation equation, (2.1) and (2.2) respectively.
∇ ·V= 0 (2.1)
∂V
∂t + [∇V]V=−
1
ρ∇p+ µ ρ∇
2V+f (2.2)
whereV is the velocity vector, p is the pressureρ is the density and µis the kinematic viscosity.
The Navier-Stokes equations were formulated after Navier and Stokes added the New-tonian viscous term to the momentum equations. Therefore in principle these equations are in fact the momentum equations for viscous flows; however, nowadays the com-plete set of equations (continuity and momentum) is called Navier-Stokes or Governing Equations for viscous flows.
2.1.1 Turbulence
The viscosity plays an important role in flows, it has a destabilizing effect on the fluids yielding to random phenomena calledturbulence [29].
The Reynolds number (Re) is one of the most important parameters to characterize flows. Shown in (2.3), it is a dimensionless number that correlates the inertial and the viscous forces.
Re= ρV L
µ = V L
ν (2.3)
whereV andLare characteristic velocity and length scales of the flow,µis the viscosity and ν is the kinematic viscosity.
At high Reynolds Numbers flow is considered turbulent, which means that it is subject of high frequency, random fluctuations of various flow properties, and generation of large eddies and vortex takes place in this regime.
Turbulence Modelling
In numerical simulations, turbulence modelling is one of the most important aspects, together with mesh generation and the implementation of numerical methods have a very important effect in the accuracy of the simulation [30]. Although not easy due to the complex nature of turbulence’an ideal model should introduce the minimum amount of complexity while capturing the essence of the relevant physics. (Wilcox [30])’ From Reynolds, Bussinesq, von K´arm´an and Prandtl, many authors have studied turbulence and attempted to predict the turbulent properties on a flow.
Prandtl attempted to develop a realistic mathematical expression to model the turbulent stresses for which he proposed a model that relates the eddy viscosity and the kinetic energy of the turbulent fluctuations k. This model was called one equation model of turbulence. However the model was incomplete since the length scale (eddies size) was not included.
In 1942, Kolmogorov proposed his turbulence model introducing a new parameter ω
that represents the turbulence frequency Wilcox [30]. This model was called the k−ω
model and since it satisfy another differential transport equation was also refereed as two equation model of turbulence.
RANS and k−ω SST Model
Reynolds Averaged Navier Stokes focus on the mean value of the flow velocity field and the sum of the mean value of the fluctuations of velocity, pressure, shear stress, etc. This lead to rewrite the Navier-Stokes equations in terms of mean turbulent variables.
Leading to expressions for the velocity components (u, v, w) and pressure field like (2.4).
u= ¯u+u0 v= ¯v+v0 w= ¯w+w0 p= ¯p+p0, (2.4) where the time mean ¯u of a turbulent function u(x, y, z, t) is (2.5) and the fluctuation
u0 is the deviation of ufrom its mean value ¯u: u0 =u−u¯
¯
u= 1
T
Z T
0
udt, (2.5)
Although the fluctuation has a mean value equal zero its mean square value (¯u0)2 it is not, and measure the turbulence intensity.
(¯u0)2 = 1
T
Z T
0
Substituting (2.4) in (2.2) gives as result the so-called turbulent stresses which are convective acceleration terms. These terms are unknown, therefore there are need three additional equations that involve unknown variables. This is known as the closure problem since it is need to posse an equal number of equation for all the unknown quantities.
The turbulence models based on the equation for the turbulence kinetic energykare the most used in CFD simulations. These models can be classified in: one equation models or incomplete models since they do not take into account the turbulence length; and two equation models or complete models that provides an equation for the turbulence length scale. The k−ω,k−and k−ω SST models are some of the most known and use.
The k−ω Shear Stress Tensor (SST) used in the simulations of this thesis is a two equation model that use the formulation of Kolmogorov’sk−ω model in the inner parts of the boundary layer and is able to switch to the k− model when describing the behaviour of free-stream.
This model solve two transport equations one for the turbulent kinetic energykand one for the specific dissipation rate or turbulent frequency ω and from them to obtain the behaviour of the kinematic eddy viscosityν. These equations ((B.1), (B.2) and (B.3) ) can be found in AppendixB.
2.2
Numerical Methods
So far it has not been found a general solution for the Navier-Stokes equations and, as they are a coupled system of non-linear partial differential equations it is just possible to solve analytically very simple cases. For this reason the governing equations are solved by means of numerical methods that seek to produce an approximate solution by iterative means.
The first step to obtain an approximate solution is the numerical discretization of the governing equations. This process consist of formulate each equation that is proposed over a continuous domain in discrete points or volumes in the domain. In other words the partial differential equations are replaced by algebraic expressions on each node in terms of the flow-field variables; for instance, velocity, pressure and temperature, as functions of the position on the domain (coordinates x, y and z). Therefore the flow field is described over the complete domain where it is search a solution.
Some of the discretization techniques are: finite elements, finite differences and finite volumes. All methods in CFD use those techniques to solve the governing equations. OpenFoam is based on finite volume discretization.
2.2.1 Finite Volume Method
This method subdivides the computational domain into control volumes where the con-servation laws have to be fulfilled locally in each one of them. For time dependent simula-tions, the time is divided in time intervals so-called time step, for each iteration/time-step the governing equations will be solved in the entire domain.
Figure 2.1: Structured and unstructured mesh for the finite volume method [27].
This method discretizes the governing equations presented in the conservative form (2.7). As this method works with control volumes, it can handle different kind of grids; structured or unstructured, making this method more flexible and useful for the imple-mentation on complex geometries as it is shown in Figure 2.1.
Z
S
ρφV·ndS
| {z }
convection
= Z
S
Γ∇φ·ndS
| {z }
diffusion
+ Z
Ω
qφdΩ
| {z }
source
(2.7)
where, ρ is the density, φ is the intensive property, Γ is a diffusive coefficient like the kinematic viscosityµ.
When the finite volume method is applied the domain is considered as a finite number of volumes surrounding nodal points,P,W andE, as it is shown in Figure2.2. First, the equation (2.8) is discretized in time, then at the time tk the spatial domain is divided
into finite volumes that have the reference point P at the center.
∂u ∂t =
uk+1−uk
∆t , (2.8)
Those control volumes have their interior boundaries placed at the points w between
W and P; and, e between P and E. At the nodal point P the discretization gives Equation 2.9 that in words states that the subtraction between the diffusive flux of φ
leaving the east face and the diffusive flux of φ entering the west face represents the generation of φ.
Z ∆V d dx Γdφ dx dV + Z ∆V SdV =
ΓAdφ dx
e
−
ΓAdφ
dx
w
Figure 2.2: Schematic representation of a mesh for finite volume method [27].
whereA is the cross-sectional area of the control volume face, ∆V is the volume and ¯S
is the average value of sourceS over the control volume [28].
In Equation2.9 the values of the diffusion coefficient Γ and the gradient ∂φ∂x at east (e) and west (w) are required. To calculate them it is used the central differencing approach that in a uniform grid linearly interpolated values for Γw and Γe are given by2.10 and
2.11.
Γw =
ΓW + ΓP
2 Γe=
ΓP + ΓE
2 (2.10)
ΓAdφ
dx
w
= ΓwAw
φP −φW
δxW P
ΓAdφ dx
e
= ΓeAe
φE−φP
δxP E
(2.11)
Substituting 2.10 and 2.11 into Equation2.9 gives Equation2.12 that can be arranged as Equation 2.13. The finite volumes method generate algebraic expressions at the reference point based on the the values at the neighbouring points. The resulting system of equations has to be solved simultaneously.
ΓeAe
φE −φP
δxP E
−ΓwAw
φP −φW
δxW P
+S= 0 (2.12)
Γe
δxP E
Ae+
Γw
δxW P
Aw−Sp
| {z }
aP
φP =
Γw
δxW P
Aw
| {z }
aW
φW +
Γe
δxP E
Ae
| {z }
aE
The system of equations resulting from the discretization process is solved by numerical methods whose complexity depends on the dimensionality and geometry of the physical problem; and whether the equations are linear or non-linear. These numerical methods are classified in direct and iterative methods [27]. The most well known direct method is theGauss elimination, which derives from the basis of systematic reduction of large systems of equations to small ones; however, for large systems of equations, this method becomes computationally expensive. The iterative methods are based on the repetition of an algorithm leading to a convergence of the solution. This kind of methods start with an initial guess and use the equation to improve the solution until convergences is achieved; therefore, it is less expensive compared with direct methods. Jacobi and
Gauss Seidel are some of the most common iterative methods use in CFD.
2.2.2 SIMPLE, PISO and PIMPLE
For an incompressible flow the governing equations are dependent of pressure since in each momentum equation the fluid flow is driven by pressure gradients. Therefore, the system of equations consist of four equations: one continuity equation and three equations for momentum conservation; all of them function of velocity V and pressure
p. However, there is not an independent transport equation for pressure; hence, the governing equations have to be solved for pressure and velocity fields coupled.
The most common method to solve the pressure-velocity coupling issue is an iterative method called SIMPLE, that stands from Semi-Implicit Method for Pressure-Linkage Equations developed by Patankar and Spalding [16]. In this scheme, an initial pressure field is guessed and used to solve the momentum equations. Then, from the continuity equation, a corrected pressure is deduced, which is used to update velocity and pres-sure fields. Through an iteration process the prespres-sure and velocity fields are improved until convergence is achieved for both fields. The algorithm of this scheme is shown in Figure 2.3.
Another scheme commonly use to solve the pressure-velocity coupling is the Pressure Implicit with Splitting Operators (PISO) algorithm which involves an additional pressure correction equation to improve convergence. The non-linear effects of the velocity are avoided by setting a small time step, leading to a Courant number1 below one. This scheme is an extension of the SIMPLE algorithm that has been also adapted for the computation of unsteady problems.
Depending on the problem keeping a Courant number below one means to set a time step so small that the complete simulation becomes slow and computational expensive. For this reason, in transient situations, the momentum equation is solved in an outer loop while the continuity equation can be recalculated as many times as the outer loop iterations. This procedure is called PIMPLE since it is a merged algorithm from SIMPLE and PISO and it is shown in Figure2.4.
1
Courant number is a condition for convergence and stability while solving partial equations by means of numerical methods. Co=u∆t
∆x in simple words it states that a particle of a fluid just pass trough one
Begin
Solve the momen-tum equations
Solve the pressure correction equation
Correct pressure and velocity fields
Solve all other transport equations (turbulence)
Check residuals
is solution converged?
End Replace
p∗ = p; V∗ = V,
Initial guess: p∗,V∗
V∗
p0
p,V
k,ω
yes no
Begin
Solve the momen-tum equations
Solve the pressure correction equation
Correct pressure and velocity fields
Solve the second pres-sure correction equation
Correct pressure and velocity fields
Solve all other transport equations (turbulence)
Check residuals Replace
p∗ = p; V∗ = V
is solution converged?
is solution converged
End
Initial guess: p∗,V∗
V∗
p0
p,V
k,ω
yes no
no
yes
2.3
Wind Turbines Aerodynamic and BEM
The most basic and commonly used method to design rotors and compute loads is the Momentum and Blade Element theory, in short BEM. The Momentum Theory is based on the assumptions of: i) infinite number of blades; ii) uniform thrust over the rotor; iii) non-rotating wake; and iv) infinite blade spam (two dimensional assumption).
With an infinite number of blades it is assumed that the rotor is in fact a continuous disk that produce a drop of pressure in the stream tube, as the pressure drops it produces a change of momentum equal to the change of velocity as it is express in (2.14) and shown in Figure 2.5.
p+d −p−d
Ad= (U∞−Uw)ρAdU∞(1−a)
| {z }
Change of momentum
, (2.14)
where, U∞ is the wind free stream velocity, the subscriptd indicate the conditions on the actuator disc, Uw the far wake wind velocity anda is the induction factor which is
product of a velocity variation induced by the disc on the free stream velocity. That
Figure 2.5: Acutator Disc and Stream Tube.
Where, V0 and P0 represent the wind free stream velocity (U∞) and pressure; and u1
the far wake velocity (Uw).
drop of pressure produces a force F on the actuator disc, known as thrust, that is transformed in power. Both thrust and power can be non-dimensionalized to give the respective coefficients (2.16) (2.17) which depend strongly of the induction factora.
F Ud
| {z }
Power
CP =
F Ud 1
2ρU∞3 Ad
= 4a(1−a)2 (2.16)
CT =
F Ud 1
2ρU∞2 Ad
= 4a(1−a) (2.17)
In reality the air passing through the blades (no infinite number of blades) exerts a torque on the rotor, the reaction to this effect is an equal and opposite torque exerted on the air. This reaction makes the particles of air rotates in an opposite direction to that of the rotor with a tangential and axial velocity components, hence the wake behind rotates.
On the other hand, the Blade Element Method (BEM) assume a infinite blade span (two dimensional assumption) therefore the aerodynamic forces can be calculated by two dimensional airfoil parameters for a determined angle of attack. This is done by assuming a single element of the blade as it is shown in Figure 2.6 that has not effect on the other elements.
Figure 2.6: Blade Element [3].
The angle of attack defined as the angle between the chord of the airfoil and the wind speed can be computed from the velocity components local to a blade section, the wind velocity and the rotational speed of the rotor. Knowing the angle of attack and the coefficients of lift and drag the forces on the blade can be calculated an they are shown in Figure 2.7.
Those forces are in charge of the change of momentum on an specific element and they are radial independent, which means that they do not affect other elements of the blade.
From Figure2.6and Figure2.7the obtained relative velocity to the blade (Equation2.18) acts at an inflow angle φto the rotational plane.
Vrel=
p
U2
∞(1−a)2+ω2r2(1 +a0)2 (2.18)
The inflow angle φ can be defined as (2.19) and the angle of attack α according to Figure 2.7is defined as (2.20).
Figure 2.7: Blade Element velocities and forces [3].
sinφ= U∞(1−a)
Vrel
cosφ= ωr(1−a0)
Vrel
, (2.19)
α=φ−β, (2.20)
where, ω is the rotational speed,aanda0are the axial and tangential induction factors respectively, andβ is the pitch angle.
From (2.18) it is possible to compute the lift and drag forces on a blade element of δr
length, as it is express in Equation2.21 and Equation2.22
δL= 1 2ρV
2
relcClδr (2.21)
δD= 1 2ρV
2
relcCdδr (2.22)
Nowadays, BEM is the most commonly used method to design blades and to compute loads. However, real wind turbines are subject of three-dimensional effects associated due to the discrete number and the finite span of blades that produces changes on the aerodynamic loads. For this reason, correction models for three-dimensional, rotational effects as well as for other parameters such as the presence of the tower and the so-called tower shadow have been developed, though they are not accurate enough.
2.4
Actuator Line Model (ALM)
The actuator line model was proposed as a simplified way to simulate rotors of wind turbines and to solve accurately the wake and the structures on it.
This method adds to the Navier-Stokes Equations a source term related to body forcesf.
These body forces are the aerodynamic loads computed with the basic BEM approach (2.23) that use two dimensional airfoil lift and drag coefficients corrected for three-dimensional effects to compute the tangential and axial force to the blade airfoil.
f2D = dF dr = 1 2ρU 2
relc(CLeL+CDeD), (2.23)
whereCLandCD are the lift and drag coefficients respectively that depend on the angle
of attack (AoA) and Reynolds number (Re);cis the chord, and eL andeD are the unit
vectors in direction of lift and drag.
These forces have to be smear over the line to avoid singular behaviour; therefore, by taking the convolution of the computed normal force and a regularization kernelη that
takes the form of a three-dimensional Gaussian distribution function (2.25) is produced the distributed loads f (2.24).
f =f∗η, (2.24)
where
η(r) =
1
3π3/2exp
−(r/)2
, (2.25)
where, adjust the distribution of the force and r is the distance between the measured point and the initial force points on the blade. In this way the loads are distributed smoothly on more than one grid point.
The three dimensional distribution presents physics inconsistency when distributing loads further than the physical tip of the blade. For this reason Michelsen [13] pro-posed a two dimensional regularization function following the form of a two dimensional Gaussian distribution shown in Equation 2.26. Besides, he reformulated the model in the primitive variables, pressure and velocity in order to couple it with the Navier Stokes Equations SolverellipSys3D
η2D = 1
2πexp
−(r/)2
. (2.26)
Among the advantages of the Actuator Line Method are: the need of simpler meshes with less grid points to capture the aerodynamics of the blade compared with three dimensional blade meshes; since it is a simplified model, it does not simulate the actual geometry of the blade therefore, it is not able to resolve blade boundary layer nonetheless the computational resources can be employed in solving other effects. Compared with other simplified methods (i.e. actuator disc) actuator line provides information about the kinematics of the flow and allows for a detailed study of wake structures, tip and root vortex for instance.
Numerical Simulations and
Results
This chapter is devoted to describe the simulations performed and, the pre and post processing stages. Giving details of the domain, mesh, boundary conditions and the results obtained.
First, a simulation of the rotor with the actuator line model was performed in order to asses the grid dependence and regularization factor effect on the results. Second, the actuator line model was tested and implemented to model the tower, to do this, simulations around a cylinder were performed in order to produce a velocity field to compare with the actuator line model results. Third, rotor and tower were modelled with the actuator line aiming to determine if this simplification shows the predicted effect of the tower on the rotor aerodynamics. Finally, the actual geometry of the rotor with the implemented actuator line tower was simulated to observe if the tower shadow phenomena can be studied with this model.
3.1
Computational Tools and Setup
3.1.1 Navier-Stokes Solver
OpenFOAMR(Open Field Operation and Manipulation) was used for all computations
and simulation. It is a free, open-source CFD software package that provides a large amount of features to solve several cases involving partial differential equations com-putations, however it is mostly employed in solving an extensive range of fluid flows of different levels of complexity. It includes tools for meshing, pre and post-processing. Al-most all those features run in parallel using the Message Passing Interface (MPI) library making of this package a very flexible tool. Moreover by being open source, OpenFOAM allows the user to customise and extend its functionality as long as it is desire since it is C++ based.
OpenFOAM is based in Finite Volume Method and has several numerical schemes im-plemented to carried out the time and spatial discretization. It counts as well with a wide variety of solvers to simulate different flows and different conditions. Among others, icoFoam a transient solver for incompressible, laminar flow of Newtonian flows;
simpleFoam a steady state solver for incompressible, turbulent flow; pisoFoam a tran-sient solver for incompressible flow; and,pimpleFoama large time-step solver for incom-pressible flow using a SIMPLE-PISO merged algorithm.
3.1.2 Mesh Generator
Meshes were generated with the mesh generator tools provided by OpenFOAM,blockMesh
which generates meshes of blocks of hexahedral cells according to a directory that spec-ifies the points coordinates, the block orientation, and faces comformation. The tool for unstructured meshes,snappyHexMeshwas used and for three dimensional blades the in-house1 automatic Blade Block Mesher (BBM)was employed.
3.1.3 Actuator Line Solver
NREL developed a set of classes based on C++ to couple with CFD solvers based on OpenFOAM that allow the users to investigate wind turbines and wind farms per-formance under different atmospheric conditions and terrain. This application, called SOWFA (Simulator fOr Wind Farm Applications) includes a high fidelity analysis of wind plant and wind turbine fluid physics and structural response [4].
SOWFA includes among other tools an Actuator Line turbine model class that does not include the hub nor the tower effects. However, it takes into account the geometry of the turbine (tower height and hub size) to locate the rotor on the computational domain.
The solverpimpleFoamwas modified and compiled with the AL model class included in SOWFA in order to run transient simulations.
3.2
CFD Simulations
First, simulations of the NREL-Nasa AMES UAE Phase VI rotor were performed for different wind speed (5, 6, 7 8 and 9m/s) varying the mesh refinement and the regular-ization parameter . This was done to analyse the influence of the parameters and the mesh refinement.
In order to implement the acuator line model for tower modelling, simulations of flow around a cylinder were performed in order to stablish a reference point of the velocity field. Then, simulations of a full wind turbine (rotor and tower) using the AL model were performed in order to achieve conclusions about: i) validation of AL model for tower applications; ii) accuracy of the results after simplifying the full turbine to an AL model; and finally, iii) effect of the presence of the tower in the rotor aerodynamics.
Lastly, a simulation of the actual geometry of the rotor and the tower AL was performed aiming to analyse the effects of the tower AL on the rotor aerodynamics and validate its implementation.
This section pretend to present the set up for the three cases performed, the character-istics of the mesh, the imposed boundary conditions and the parameters used.
1
3.2.1 Rotor-Actuator Line Simulations
As it was mentioned simulation for wind speed between 5 and 9 m/s were performed. First to analyse the influence of the mesh in the results and to establish the character-istics for the next simulations. Second, simulations changing the regularization factor
were performed to analyse its influence on the results and to obtain a definitely and general guidance about the use of this parameter for the next simulations. For this, a base case was created to start the first round of simulations and from it the parameters subject of this work were changed.
Mesh Generation
One of the main advantages of using AL model is that there is no need of meshing complex geometries as it is the case when generating the mesh for full three-dimensional blades. Just a much more simple mesh and some constraints to obtain an accurate and stable simulation are needed.
The domain, shown in Figure3.1and generated withblockMesh, has a rectangular shape big enough to avoid the effect that outer boundaries can have on the vortex shedding frequency and the velocity field. It consist of a mesh of 10Dlength in the axial direction and 5D in the cross-flow direction; the length was setted 1/3 in front and 2/3 behind the rotor.
Figure 3.1: Computational Domain.
It was designed in a way that fulfil the requirements established by SOWFA developers listed below.
• +xmust be east, +y must be north and +z must be up2.
• Use at least 20 grid cells across the rotor diameter2.
• Use at least 50 grid cells across the rotor diameter if it is desired to resolve tip/root vortices2.
The base mesh follows the recommendations mentioned above and three more meshes were generated with different refinement boxes in order to analyse the influence of the grid on the results and to stablish the required mesh for next simulations. The schematic view of the mesh and the refinement is shown in Figure 3.1 and Figure 3.2 and the characteristics of each mesh is shown in Table 3.1with the cell size in the refined part.
Figure 3.2: Schematic top view of the domain and the refinement boxes.
Refinement level Number of cells [M.] Cell size [m]
Base Mesh 6.1 0.2
First box 7.4 0.15
Second box 11.7 0.09
Third box 20.1 0.05
Table 3.1: Mesh characteristics
Boundary Conditions
The imposed boundary conditions are shown in Table 3.2, where I is the turbulence intensity, k turbulence kinetic energy, ω turbulent frequency, l turbulence length (l = 0.07D) and Cµ is a empirical non-dimensional constant 0.09.
Boundary U p k Omega
Inlet U∞ dpdy = 0 32(U∞I)2 k
3 2
kCµl
Outlet ddyU = 0 p∞ dkdy = 0 dωdy = 0 Ground 0 dpdy = 0 dkdy = 0 dωdy = 0 Top and sides ddyU = 0 dpdy = 0 dkdy = 0 dωdy = 0
Table 3.2: Boundary Conditions for rotor and tower simulations
Base Case
This case was setted with the mesh that follows the recommendations described before with approximately 4 million cells.
Based on the mesh, was setted equal to 0.4 from ∆x = 2 and the number of actuator points was equal to 40 following ∆b∆x = 0.753, where ∆bis the width of the discrete blade section and ∆x is the grid resolution.
The time step is restricted by the tip speed. Since the tip of the blade should not pass through more than one cell each time step. This condition is more important than the typical Courant-Friedrichs-Lewi number (CF L= 1) [11] leading to a fixed time step equal to 0.01989 s.
3.2.1.1 Grid dependence
The first parameter to analyse was the grid dependence of results. Three different cases were performed for three meshes with different level of refinement as shown in previous section.
Figure 3.3 shows the comparison of the rotor power for meshes with different levels of refinement. It is clear that the baseline guidance for mesh was not enough and the results start to convergence to the experimental data (continuous line) from meshes with more than 7 million cells which means one level of refinement and a cell size of 0.15 m in the more refined areas as it was shown in Figure 3.2 and it is 2.5 times smaller than the length of the chord on the tip of the blade.
3.2.1.2 Epsilon
It has been proposed thatshould be related to a representative length of the blade by Shives and Crawford [20] and [24] among others; because, this parameter regulates the distance over the forces are projected. In this work it is propose a value ofequal to the chord of the blade; therefore, simulations were performed with values equal to the chord of the blade in the positions were the experimental measurements were taken. That is
= 0.38, 0.46, 0.54, 0.63 and 0.71
Comparing the experimental normal and tangential force and the results from the nu-merical simulation good agreement was found specially for the normal force coefficient
Figure 3.3: Output power vs. number of elements in the domain.
(see Figure 3.4) and better agreement was obtained with lower values, or values close to the chord of the tip of the blade. On the other hand, for tangential force coefficient, shown in Figure 3.5, bigger discrepancies were found specially close to outermost part of the blade; while it is true that as higher results seems to converge, for those values it is shown that they are far from the experimental data as better agreement is shown by lower values of; from a physical point of view as higherthe forces will be smeared far away from the physical blade geometry.
Figure 3.4: Normal force coefficient along the blade.
The discrepancies found close to the tip area can be explained by the two-dimensional nature of the actuator line model and its lack of geometrical-three-dimensional correc-tions.
Figure 3.5: Tangential force coefficient along the blade.
3.2.2 Tower-Actuator Line Simulations
The actuator line model was used and modified in order to model not just the rotor both the tower as well hence to asses the capability of the model to influence the rotor aerodynamics.
3.2.2.1 Implementation
To simulate the tower with the AL model, simulations of the flow around a cylinder were performed in order to obtain a reference point of the tower conditions (velocity field) and afterwards compare it with the tower-AL results.
Cylinder Simulations
First, simulations of flow around a cylinder of 0.4m diameter were performed at wind speeds between 5 and 9m/s. The Reynolds number for this conditions varies between 1.3×106 and 2.4×106. This range fit in the critical regime where the drag coefficient curve is very stepped therefore that value changes greatly from one wind speed to other.
Mesh generation
A circular domain, shown in Figure3.6, ten times bigger than the inner bluff body was generated with blockMesh. Higher refinement was done close to the cylinder surface in order to capture the stagnation and separation point, and the vortex shedding accurately.
Figure 3.6: Computational Domain (Cylinder Simulations).
Boundary Conditions and Turbulence Properties
The boundary conditions for all cases were as follows:
• Inlet: Velocity uniform flow was specified
• Outlet: Pressure was set to a fixed value zero.
• Periodic: This boundary type simulates an infinitely long cylinder.
• Cylinder Surface: Wall conditions; pressure is set to zero gradient and velocity is set to zero.
Thek−ωSST turbulence model was used for both RANS and URANS simulations due to its wide documented well behaviour.
The values of the turbulence kinetic energy (k), the turbulence frequency ω and, the turbulence length (l) are given in (3.1), (3.2) and (3.3) according to Stringer et al. [23]; at the wallkwall is set to zero andωwall is computed with a standard logarithmic
wall function provided by OpenFOAM. Cµ is the empirical non-dimensional constant
(Cµ= 0.09) andν is the kinematic viscosity.
k= 3
2(U∞I)
2
(3.1)
ω= k
3 2
kCµl
(3.2)
l= 0.07D (3.3)
Validation
To validate the cylinder simulations, results were compared with experimental results from Achenbach [2] as it is shown in Figure3.7. Grid dependence and turbulence model comparison were performed showing independence of results for both aspects.
Figure 3.7: Pressure distribution on the cylinder surface atRe= 2.6×105.
For Reynolds number from 1×105 to 1×106 the flow is in the so-called critique regime where the drag coefficient drops hastily. The difficulty to validate the simulations of the flow around the cylinder during this regime relies in the several differences found in literature about the critical Reynolds number at the drag drop starts. These differences are product mainly of experimental set up, turbulence intensity and they are specially sensitive to the surface smoothness.
The lift and drag forces oscillated trough the simulation as it is presented in Figure3.8. The averaged drag coefficient obtained in this simulations was 0.6 that agrees with the results reported on [2]. The averaged value for lift coefficient is zero and the Strouhal number Stwas 0.2
3.2.2.2 Tower-AL
From actuator line simulations the fluctuations reported previously could not be seeing. The reason for this is that the model takes the lift and drag coefficient from a tabulated data depending of the angle of attack, those data remain constant during the simula-tion thus the fluctuasimula-tion observed before is not reproduced; moreover such fluctuasimula-tion produce the vortex shedding behind the cylinder then the actuator line model is not reproducing that condition.
To reproduce the mentioned oscillation the code of the actuator line was modified re-placing the constant lift and drag coefficients by functions of time (3.4) and (3.5).
Figure 3.8: Lift and drag forces from the cylinder simulation.
CL=CL0cos(ωt) (3.4)
CD =CD0cos(2ωt) (3.5)
where CL0 and CD0 are the averaged values found in the previous section and ω = 22
rad/scalculated from the Strouhal number asSt= f LV and as it is well knownω= 2πf.
The results from this modification produce the same oscillation of those coefficients shown in Figure3.9improving the results when comparing the deficit of velocity behind the tower as it can be observed in Figure 3.10 the drop of velocity with the modified actuator line is closer to the deficit obtained from the simulations of the cylinder.
Figure 3.9: Forces oscillating with the actuator line model after modifying the lift and drag coefficients on the code.
Figure 3.10: Deficit of velocity comparisson between results from simulation of a cylinder, actuator line model and actuator line modified.
However, fluctuations of the vortex structures from the tower were not evident because the source of shedding is too small to be observed. Although the vortex shedding from the tower affects the rotor aerodynamics, Figure3.11shows the lack of fluctuations that does not permit to properly observe the unsteady nature of the interaction between the vortex shedding from the tower and the blade.
3.2.3 Rotor and Tower-Actuator Line Simulations
Finally, a simulation of a rotor and tower using the AL model was performed in order to analyse the capability of the model to simulate a tower thus to influence the rotor aerodynamics.
The mesh and boundary conditions were the same employed in the previous simulations. From the first part, the rotor-actuator line simulations it was obtained the mesh inde-pendence study then a domain of 7 million cells was used in this simulation. Regarding the regularization parameter, the chord of the tip of the blade was used (= 0.38). To model the tower it was used the modified version of the model with the lift and drag coefficients as function of time.
To accurately quantify the frequency of the effect of the tower on the rotor aerodynamics, power spectra were computed with the time records of AL and experimental data using a Fast-Fourier Transform (FFT) algorithm. Experimental measurements were sampled at 520.83Hzover 30 seconds for which the power spectra showed important content up to 30 Hz. Moreover, spectral peaks of significant magnitude were found in the range 0
Hz to 10 Hz.
The power spectrum in Figure 3.12 shows the comparison between the actuator line model and the experimental data at wind speed of 7m/s and it is evidence of the effect of the presence of the AL-tower as the highest spectral peaks are found for 2P, 4P and 6P meaning that the power drops as the blades encounter the tower shadow once per revolution (P = 1.2Hz); for the experimental data, although, the highest peak is at 2P there are spectral peaks every 1P due to unbalance of the turbine during the experiment.
Figure 3.12: Output Power spectrum forU∞= 7m/s
Normal and tangential force coefficients were compared with experimental data, Fig-ure 3.13 and Figure 3.14 show the averaged coefficients for 5 section along the blade corresponding to the pressure tap locations in the experiment. Good agreement is found proving that the global aerodynamic parameters are well represented by the AL model.
At the root and the tip of the blade bigger discrepancies were found due to three-dimensional effects present in those locations and that they are not properly captured
Figure 3.13: Normal force coefficient
Figure 3.14: Tangential force coefficient
by the model. Although the Glauert correctionF = π2 cos−1exp(−f), wheref is given by f = B2rRsin(φ)−r , at root and tip is applied it is not enough to accurately predict the forces. This is shown as well in Figure3.15as increasing span wise from the root better agreement is found until 80% of the blade. For the outermost part of the blade, section 95%, the model does not represent properly the effect.
Figure 3.15: Normal force coefficient for 5 sections on the blade 30%, 47%, 63%, 80%
and 95% atU∞= 7m/s
3.2.4 Three-dimensional rotor and tower-AL
Rotor tower interaction was also studied with simulations performed by using a three dimensional model of the actual rotor geometry. To handle the relative movement be-tween rotor and tower while keeping a low computational cost a dynamic mesh approach was used.
Mesh of the blades in Figure 3.16 was generated with the in-house4 automatic Blade Block Mesher BBM, tool based on openFoamblockMeshandsnappyHexMeshthe result-ing domain consistresult-ing of 11.6 million cells, of cylindrical shape, 10D large and a radius
of 3D. The rotor was placed at 1/3 of the domain length from the inlet. The y+ was kept at a maximum value of 100, as it is presented in Figure3.17since good results were obtained with this mesh resolution.
Figure 3.16: Three dimensional mesh, detail of the root of the blade.
Figure 3.17: Instantaneousy+ distribution over the blade.
The Unsteady Reynolds Averaged -URANS computations were carried out using the
PIMPLE algorithm to solve the coupled velocity-pressure equations and the turbulence was modelled with the k-ω Shear Stress Tensor model. The boundary conditions in Table3.3were similar to the previous simulations adding the wall condition to the blade surfaces.
