Navier-Stokes modelling of offshore wind turbines using the SPH method
Jean-Marie Le Goff
Master of Science Thesis KTH School of Industrial Engineering and Management Energy Technology EGI_2017-0071-MSC EKV1197 Division of Heat & Power SE-100 44 STOCKHOLM Master of Science Thesis EGI_2017-0071-MSC EKV1197
Navier-Stokes modelling of offshore wind turbines using the SPH method
Jean-Marie Le Goff
Approved Examiner Supervisor 2017-08-15 Miroslav Petrov - KTH/ITM/EGI Miroslav Petrov Commissioner Contact person EDF R&D, France Agnès Leroy Christophe Peyrard
ABSTRACT
This Master Thesis has been realized as a result of an internship supervised by Agnès Leroy and Christophe Peyrard.at LNHE (Laboratoire National d'Hydraulique et Environnement), a part of EDF R&D in Chatou, France. The aim was to use the Smoothed Particle Hydrodynamics (SPH) method in order to study the forces generated by ocean waves on two different offshore wind turbine structures.
The SPH method is a Lagrangian computational method, first used in astrophysics but soon extended to the study of free surface flows. Developed by EDF R&D and in cooperation with other institutes and universities, GPUSPH is an open source software based on the SPH method to simulate complex free surface flows.
The first part of the internship was dedicated to the study of the forces applied on a monopile structure by breaking waves, while the second part was dedicated to the study of the forces induced on a gravity-based foundation by regular waves. Both studies were done using GPUSPH, and the numerical results are compared to experimental results obtained in 2003 and 2015 within EDF R&D facilities. SAMMANFATTNING Detta examensarbete har resulterat efter en praktikperiod hos LNHE (Laboratoire National d'Hydraulique et Environnement), en del av EDF R&D i Chatou, Frankrike, under handledning av Agnès Leroy och Christophe Peyrard. Syftet var att använda metoden för smoothed particle hydrodynamics (SPH) för att studera krafterna som genereras av vågor på två typer av fundament till havsbaserade vindkraftverk. SPH-metoden är en Lagrange beräkningsmetod, som först användes i astrofysik men snart utvidgades också till studier av fria ytflöden. Vidare har GPUSPH utvecklats av EDF R&D i samarbete med andra institut och universitet, som är en öppen källkodsprogram baserad på SPH- metoden för att simulera komplexa fria ytflöden. Den första delen av arbetet ägnades åt att studera de krafter som brytande vågor tillämpar på en monopile-struktur, medan den andra delen var avsedd för studien av de krafter som inducerades på en tyngdbaserad grundfundament som utsätts för vanliga vågor. Båda studierna gjordes med hjälp av GPUSPH, och de numeriska resultaten jämfördes med tidigare experimentella resultat som uppnåddes år 2003 och 2015 inom EDF:s forskningsanläggningar. Contents
Introduction 10
1 Background on offshore wind turbines 11 1.1 Historical reminder about offshore wind power ...... 13 1.1.1 Offshore wind power in the world and in Europe ...... 13 1.1.2 Offshore wind power in France ...... 19 1.2 Questions about fluid dynamics while studying offshore turbine ...... 20 1.2.1 Simplified mathematical model ...... 20 1.2.2 Numerical model based on potential flows ...... 21 1.2.3 Computational fluid dynamics (CFD) ...... 22 1.2.4 New numerical methods ...... 22
2 The SPH method 23 2.1 Navier-Stokes equations for weakly compressible flows ...... 23 2.2 Mathematical principles of the SPH method ...... 24 2.2.1 Continuous interpolation ...... 25 2.2.2 Discrete interpolation ...... 26 2.2.3 Choice of discrete operators ...... 28 2.2.4 Boundary conditions ...... 28 2.2.5 Moving objects ...... 30
3 First studied case : waves breaking on a monopile structure 31 3.1 Description of the experiments ...... 31 3.2 Description of the shoaling phenomemon ...... 32 3.3 Features of the GPUSPH simulations ...... 34 3.3.1 Configurations of the several cases ...... 34 3.3.2 Numerical options ...... 35 3.3.3 Construction of the geometry files ...... 35 3.4 Results and analysis ...... 37 3.4.1 Computational times ...... 37 3.4.2 Mean free surface elevation results ...... 37 3.4.3 Mean horizontal force results ...... 39
4 Second studied case : waves impacting on a gravity-based foundation (GBF) 43 4.1 Description of the experiments ...... 43 4.2 Features of the GPUSPH simulations ...... 43 4.2.1 Configuration of the several cases ...... 43 4.2.2 Numerical options ...... 45 4.3 Results and analysis ...... 45 4.3.1 Computational times ...... 45 4.3.2 Mean free surface elevation results ...... 46 4.3.3 Mean horizontal force results ...... 54 4.3.4 Mean overturning moment results ...... 59 4.4 Possible improvements ...... 63 4.4.1 Computational times ...... 63
1 4.4.2 Mean free surface elevation results ...... 63 4.4.3 Mean horizontal force results ...... 64 4.4.4 Mean overturning moment results ...... 65
Appendix A Consistency of the continuous SPH interpolation 67
Appendix B Plan of the gravity-based foundation 69
Bibliography 71
2 List of Figures
1.1 The Smith-Putnam turbine prototype in Vermont, USA (1940)...... 13 1.2 The Gedser turbine prototype in Gedser, Denmark (1957)...... 14 1.3 Evolution of the size and the power capacity of wind turbine from the 1980’s up to now.. 14 1.4 Share of main countries in European offshore wind power capacity [9]...... 15 1.5 Picture of the different existing offshore wind turbine structures [2]...... 16 1.6 Share of foundations in the European offshore wind farms [2]...... 16 1.7 Hywind, the first tested floating structure from the Norvegian company Statoil (2001)... 17 1.8 Picture of the different existing floating offshore wind turbine structures [26]...... 17 1.9 Windfloat, the structure from the American company Principle Power (installed in 2011). 18 1.10 Floatgen, the floating structure from Ideol, installed by the end 2017...... 18 1.11 The floating structure proposed by the Dutch company SBM Offshore...... 19 1.12 Map presenting the different locations of the French awarded site, and the companies in charge of each project...... 20 1.13 Numerical model of the GBF studied by EDF R&D and Innosea [23]...... 21
2.1 Illustration of a 2D-kernel function...... 25 2.2 Illustration of a 2D-kernel function for discrete particles. The particles b ∈ F are in blue. 27 2.3 Illustration of a 2D-kernel near a boundary ∂Ω...... 29 2.4 Illustration of a 2D-kernel near a boundary ∂Ω with discrete particles and boundary segments...... 30
3.1 Sketch of the experimental facilites [13]...... 31 3.2 Scheme of the propagating regular waves arriving in a non constant depth area...... 33 3.3 Representation of the shoaling phenomemon...... 34 3.4 Visualization of the 3D modelling of EOL95 case in ParaView...... 35 3.5 Visualization of the 3D particle filling from Crixus for EOL95 case in ParaView. Red particles are fluid particles, blue and gray particles are boundary particles...... 36 3.6 Visualization of the 3D particle filling around the monopile in ParaView...... 36 3.7 Visualization of the pressure field in the flow induced by the wavemaker in EOL95 case, t = 60 s...... 37 3.8 Mean free surface elevation near the pile in EOL95 case...... 38 3.9 Mean free surface elevation near the pile in EOL96 case...... 38 3.10 Mean force Fx applied on the pile in EOL95 case...... 39 3.11 Mean force Fx applied on the pile in EOL96 case...... 39 3.12 Mean force Fx applied on the pile in EOL54 case...... 40 3.13 Mean force Fx applied on the pile in EOL79 case...... 40
4.1 Scheme of the experimental facilites [23]...... 43 4.2 Visualization of the 3D modelling of GBF02R case in ParaView...... 44 4.3 Visualization of the 3D modelling of the GBF in ParaView...... 45 4.4 Free surface elevation near the GBF in GBF02R case...... 46 4.5 Mean free surface elevation near the GBF in GBF02R case...... 47 4.6 Mean wave height evolution before the GBF in GBF02R case...... 47 4.7 Fourier analysis of the free surface elevation in GBF02R case...... 48 4.8 Free surface elevation near the GBF in GBF04R case...... 48
3 4.9 Mean free surface elevation near the GBF in GBF04R case...... 49 4.10 Mean wave height evolution before the GBF in GBF04R case...... 49 4.11 Fourier analysis of the free surface elevation in GBF04R case...... 50 4.12 Free surface elevation near the GBF in GBF05R case...... 50 4.13 Mean free surface elevation near the GBF in GBF05R case...... 51 4.14 Mean wave height evolution before the GBF in GBF05R case...... 51 4.15 Fourier analysis of the free surface elevation in GBF05R case...... 52 4.16 Free surface elevation near the GBF in GBFU1R case...... 52 4.17 Mean free surface elevation near the GBF in GBFU1R case...... 53 4.18 Mean wave height evolution before the GBF in GBFU1R case...... 53 4.19 Fourier analysis of the free surface elevation in GBFU1R case...... 54 4.20 Forces Fx applied on the GBF in GBF02R case...... 54 4.21 Mean force Fx applied on the GBF in GBF02R case...... 55 4.22 Forces Fx applied on the GBF in GBF04R case...... 56 4.23 Mean force Fx applied on the GBF in GBF04R case...... 56 4.24 Forces Fx applied on the GBF in GBF05R case...... 57 4.25 Mean force Fx applied on the GBF in GBF05R case...... 57 4.26 Over-turning moments My applied on the GBF in GBFU1R case...... 58 4.27 Forces Fx applied on the GBF in GBFU1R case...... 58 4.28 Mean force Fx applied on the GBF in GBFU1R case...... 59 4.29 Over-turning moments My applied on the GBF in GBF02R case...... 59 4.30 Mean over-turning moment My applied on the GBF in GBF02R case...... 60 4.31 Over-turning moments My applied on the GBF in GBF04R case...... 60 4.32 Mean over-turning moment My applied on the GBF in GBF04R case...... 61 4.33 Over-turning moments My applied on the GBF in GBF05R case...... 61 4.34 Mean over-turning moment My applied on the GBF in GBF05R case...... 62 4.35 Over-turning moments My applied on the GBF in GBFU1R case...... 62 4.36 Mean over-turning moment My applied on the GBF in GBFU1R case...... 63 4.37 Free surface elevation near the GBF in GBF02R case...... 64 4.38 Mean free surface elevation near the GBF in GBF02R case...... 64 4.39 Forces Fx applied on the GBF in GBF02R case...... 65 4.40 Mean force Fx applied on the GBF in GBF02R case...... 65 4.41 Over-turning moments My applied on the GBF in GBF02R case...... 66 4.42 Mean over-turning moment My applied on the GBF in GBF02R case...... 66 4.43 3D-model of the DeepCwind floatting wind turbine...... 67
4 List of Tables
1.1 Overview of the European offshore wind power capacity [9]...... 15
3.1 Characteristics of the simulations run for the EOL95 case [12]...... 34 3.2 Computational characteristics of the simulations...... 37
4.1 Characteristics of the simulations run for the EOL95 case [12]...... 44 4.2 Computational characteristics of the simulations...... 45 4.3 Computational characteristics of the simulations...... 63
5 6 Nomenclature
A corresponds to an arbitrary field, Aa = A(~ra), As = A(~rs) and Aab = Aa − Ab.
αW Normalizing constant of the kernel function ...... ∆t Timestep ...... (s)
∆x Spatial discretization ...... (m)
δ Dirac distribution ...... (m−1)
γa Renormalization factor ...... µ Dynamic viscosity ...... (kg.m−1.s−1)
ν Kinematic viscosity ...... (m2.s−1)
Ωr Compact support ......
Fx Mean force in the direction of propagation of waves...... (N)
∂Ωr Boundary of the compact support ...... ρ Density...... (kg.m−3)
−3 ρ0 Reference density ...... (kg.m ) S Strain rate tensor ...... (s−1)
∇~ Nabla operator used in vector analysis ......
~g Gravity ...... (m.s−2)
~ + Ga Antisymmetric classical SPH gradient operator ...... ~ − Ga Symmetric classical SPH gradient operator ...... ~ γ,+ Ga Antisymmetric USAW SPH gradient operator ...... ~ γ,− Ga Symmetric USAW SPH gradient operator ...... ~r Position ...... (m)
~v Velocity...... (m.s−1)
ξ Constant of the Tait state equation ......
C Courant-Friedrichs-Lewy number ......
c Speed of sound ...... (m.s−1)
−1 c0 Numerical speed of sound ...... (m.s )
Cd Drag coefficient ......
7 Cm Inertia coefficient......
− Da Symmetric classical SPH divergence operator ...... γ,− Da Symmetric USAW SPH divergence operator ...... h Water depth ...... (m) L Characteristic length of the flow ...... (m)
La Classical SPH Laplacian operator ......
γ La USAW SPH gradient operator ...... Ma Mach number ...... p Pressure ...... (P a) p1, p2 Constants of the Lennard-Jones boundaries model ...... r0 Initial particle spacing ...... (m) T Period ...... (s)
3 Va Volume of the particle a ...... (m )
−1 wh Kernel function ...... (m )
8 Acknowledgements I want to thank Agnès Leroy and Christophe Peyrard for being my supervisors in EDF during this internship, for helping me and answering all of my questions. Thanks a lot also for helping me find my first job as an engineer, I am very indebted to you. Thank you to Damien Violeau for the support during the internship. Thank you to Antoine Joly, Alex Ghaïtanellis, Rémi Carmigniani, Thomas Fonty, Constantin Kuznetzov, Adrien Bourgoin who helped me, in one way or another, during my work in EDF. Finally, thank you to Miroslav Petrov for being my supervisor in KTH and supporting me for red tape.
9 Introduction In naval and ocean engineering, the hydrodynamic sizing of offshore structures and ships is the focal point of many projects. Engineers need to do lots of calculations, they use semi-empirical models or simplified model, and these models need to be calibrated or validated. Until recently, numerical hydrodynamics was not really convenient for industry, because it took too much time to get results for studies. So, the use of experimental model was an important step during the design and the study of new system, regardless of whether it is a ship, an offshore platform or a system of energy conversion. However, the experimental means, such as towing tanks or wave tanks, are often expensive and the experimental trials require a lot of time to set up, to use and to postprocess the results. Thus, with the recent emergence of numerical tools, engineers have now been able to use numerical model to simulate flows, systems behaviour or energy conversion, for instance. Numerical software often have a calculation time that is higher than the real time simulated, but they have many advantages, as easily modifiable inputs of simulations and multiple runs at the same time. But as numerical models are based on assumptions or approximations, they cannot totally substitute the experimental models. So, it is important to often compare the performances of a software with the same results obtain with experimental means, in order to see if such a numerical model is reliable to study a system. This internship is dedicated to the use of a rather new numerical method called the Smoothed Particle Hydrodynamics (SPH) method, to study the hydrodynamic loads applied on offshore wind turbines structures. The SPH method is an innovative computational method. It was first proposed in 1977, almost simultaneously by Leon Lucy [14] and Robert Gingold and Joe J. Monaghan [8]. It was first dedicated to study complex astrophysical phenomena, such as the creation of stars. Then it was extended to the simulation of free surface flows. This computational method is quite innovative, as it is based on a meshless interpolation to determine the properties of the fluid domain at multiple points.This makes it interesting to simulate flows involving complex interfaces since no tracking of the interfaces is required. In EDF R&D, a software called GPUSPH is developped jointly with several other institutes. This open source software can run on multiple graphics cards and can simulate 3D flows and complex phenomena such as breaking waves. The first part of the internship was dedicated to simulations of breaking waves on a cylinder, representing a monopile structure of an offshore wind turbine, and to compare the results with experiments made at EDF R&D. The second part was about simulations of wave induced loads on a gravitational base foundation of an offshore wind turbine, and to compare the results with experiments made at EDF R&D.
10 Chapter 1
Background on offshore wind turbines
1.1 Historical reminder about offshore wind power 1.1.1 Offshore wind power in the world and in Europe The concept of wind turbine is ancient. First, it was used as windmill in order to mill grain (to produce flour for instance) or to pump water. Then, at the end of the 19th century, a Danish scientist called Poul La Cour created the first producing electricity wind turbine, by connecting a windmill to an electricity generator. From these first works, several wind turbines prototypes were created and tested during the 20th century, like for instance the Smith-Putnam turbine prototype, as shown in Fig. 1.1 below.
Figure 1.1 – The Smith-Putnam turbine prototype in Vermont, USA (1940).
11 Figure 1.2 – The Gedser turbine prototype in Gedser, Denmark (1957).
Fig. 1.2 is a picture of the wind turbine prototype in Gedser, designed in 1957 by Johannes Juul, who was one of the student of Poul La Cour in Askov University (Denmark). That prototype defined the first 3-bladed upwind rotor wind turbine, with stall control and induction generator, and it made possible the commerciAl breakthrough of wind power generation. Finally, in the 1980’s, the robust and reliable "Danish-concept" (3-bladed horizontal axis wind turbine) initiated by the Gedser wind turbine was selected as the state-of-art commercial wind turbine technology.
Figure 1.3 – Evolution of the size and the power capacity of wind turbine from the 1980’s up to now.
Then, from the 1980’s up to now, the size and power rate of wind turbines have never stopped increasing (see 1.3). In less than 30 years, the standard size of wind turbines has been multiplied by more than 10, and their standard power rate has increased from about 50 kW to 8 MW. But the potential of offshore wind power is way more important than the onshore’s one. Thus, in the 1990’s, the first prototypes of offshore wind turbines were tested off Denmark, by the companies Vestas, Alstom, Siemens and Gamesa. The first operational offshore wind farm was Vindeby’s, commissioned in 1991, with 11 turbines, for a total capacity of 4.95 MW. Then, still in 1991, the first UK wind farms opened at Delabole, in Cornwall, with 10 wind turbines, for a total capacity of 4 MW. As for onshore wind turbines, the power capacity of new installed turbines has never stop increasing. So,in 2002, when the Danish offshore wind farm of Horns Rev was commissioned, it was the biggest offshore wind farm never built, with an installed capacity of 160 MW. Nowadays, the world’s largest offshore wind farm is located off Cumbria, in England, with an installed power capacity of 367 MW. Table 1.1 and Fig.1.4 show the total installed wind power capacity in Europe, and the share of the main countries in that capacity.
12 Table 1.1 – Overview of the European offshore wind power capacity [9].
Country Nb of offshore farms Nb of wind turbines Capacity (MW) United Kingdom 28 1,472 5,156 Germany 18 947 4,108 Denmark 13 517 1,271 Netherlands 6 365 1,118 Belgium 6 182 712 Sweden 5 86 202 Finland 2 11 32 Ireland 1 7 25 Spain 1 1 5 Norway 1 1 2 TOTAL 81 3589 12631
Figure 1.4 – Share of main countries in European offshore wind power capacity [9].
Offshore wind turbines involve complex technical questions. They are submitted to wind loads, as onshore wind turbines, but the foundation of the turbine is also submitted to loads, from the sea, that imposes to find robust, reliable but not too expensive structure to carry the wind turbine. As for onshore wind turbines, several type of foundations were created and tested :
• the monopile structure, a cylindrical structure whose base is anchored onto the sea bed
• the gravity-based foundation (GBF) or gravity-based structure (GBS), a structure similar to the monopile structure, but with an expanded and weighted base that is just laid on the sea bed • the tripod structure, a monopile structure supported by 3 smaller piles, anchored onto the sea bed • the jacket structure (inspired from offshore oil platforms), a four-leg structure with steel frames between the legs, anchored onto the sea bed
• the tri-pile structure, 3 cylindrical structure similare to the monopile structure, anchored onto the sea bed and supporting the wind turbine
Fig. 1.5 and 1.6 show the different existing structure, and their share in the European offshore wind farms. It can easily be seen that monopile structure is the foundation type that is the most used in Europe, probably because it is easier to install and to maintain.
13 Figure 1.5 – Picture of the different existing offshore wind turbine structures [2].
Figure 1.6 – Share of foundations in the European offshore wind farms [2].
Finally, for few years, a new type of offshore wind turbine has been tested : floating offshore wind turbine. The potential of floating wind turbines is greater than the one of fixed-bottom wind turbines, as they enable to reach locations far from the coasts, where traditional offshore wind turbines cannot be installed because of too important depth. It is estimated that in Europe, 80% of the offshore wind resource is located in areas where the offshore depth is 60 m or more, so too deep for traditional fixed- bottom turbines. For the moment, there are not any operative farms of floating wind turbines, but only prototypes test still being tested and studied. The very first floating wind turbine that was tested was the Hywind prototype, proposed by the Norvegian company Statoil in 2009 (see Fig.1.7). It is expected that the Hywind-Scotland farm will be commissioned in 2018. It will a farm composed of 5 wind turbines, for a total capacity of 30 MW.
14 Figure 1.7 – Hywind, the first tested floating structure from the Norvegian company Statoil (2001).
On Fig.1.8 are represented the different floating structure that are proposed for the moment. They are:
• the barge foundation, prototype proposed by the French company Ideol, to be commissioned this year • the semi-submersible foundation, prototype tested in Japan, off Fukushima, and in Europe with the Windfloat project
• the spar foundation, prototype tested in Europe with the Hywind project and in Japan • the tension-leg platform (TLP)n not installed yet
Figure 1.8 – Picture of the different existing floating offshore wind turbine structures [26].
Fig.1.9, Fig.1.10 and Fig.1.11 represent three prototypes of floating wind turbine, each using different floating technologies.
15 Figure 1.9 – Windfloat, the structure from the American company Principle Power (installed in 2011).
Figure 1.10 – Floatgen, the floating structure from Ideol, installed by the end 2017.
16 Figure 1.11 – The floating structure proposed by the Dutch company SBM Offshore.
1.1.2 Offshore wind power in France As compared to many other European countries, France is late in offshore wind power development. In 2004, a first call for tenders, named “Centrale Eoliennes en mer”, was awarded by Enertrag France and Areva, and a 105 MW offshore farm was supposed to be built off Veulettes-sur-Mer (Seine-Maritime). But after some NIMBY issues (“Not In My BackYard“), the project was cancelled. Then, in 2011, a new call for tenders was launched, originally for 6 sites but awarded by 4 sites : Fécamp (Seine-Maritime), Courseulles (Calvados), Saint-Brieuc (Côtes d’Armor) and Saint-Nazaire (Loire-Atlantique). Those 4 farms should be built between 2019 and 2021, for a total capacity of 1.9 GW. Finally, a second call for tenders was launched in 2013, awarded by the 2 last sites of the 2011 initial project : Le Tréport (Seine-Maritime) and Îles d’Yeu & de Noirmoutier (Vendée). Those 2 farms will be built between 2021 and 2023, for a total capacity of 1 GW. Fig.1.12 shows the location of those future French wind farms.
17 Figure 1.12 – Map presenting the different locations of the French awarded site, and the companies in charge of each project.
A third call for tenders was supposed has been launched in end 2017, about the project of offshore wind farms of Dunkerque (Nord) and Île d’Oléron (Charente-Maritime). The discussions with politics, sea users and inhabitnats are expected to take several months before awarding.
1.2 Questions about fluid dynamics while studying offshore tur- bine
Before testing full scale prototypes, engineers studied offshore structure using models. Historically, the very first model used by engineers are the ones used in test tanks, using wave, current or wind generating systems in order to sudy different characteristics of the offshore strucure. But there are more and more numerical tools used by engineers, in combination with physical models, in order to study and design offshore structure, especially when engineers focus on hydrodynamic loads.
1.2.1 Simplified mathematical model The hydrodynamic loads applied on offshore structures are due to two main phenomena: waves and current. Thus, different dimensionless parameters have been created in order to have a quick overview of the phenomena that are leading the flow, and so on the leading loads applied on the offshore structure. First let us introduce the Reynolds number :
ρvL Re = (1.1) µ with ρ the fluid density, v the velocity of the fluid, L the characteristic length of the flow (for instance, when the fluid meets an object, L corresponds to the characteristic length of the object) and µ the molecular dynamic viscosity of the fluid. The Reynolds number indicates the importance of the inertia forces compared with the viscous forces. For high Reynolds number, that means that the flow is turbulent and that the inertia loads are way more important than the viscous ones. On the contrary, if the Reynolds number is low, that the flow is laminar and that the forces due to viscous effects are more important
18 than the inertia ones. Another dimensionless parameter that is important for engineers is the Keulegan- Carpenter number. Several definitions of this number exist, but the simplest one to use is : vT 2πA KC = = (1.2) L L with T and A the period and the amplitude of the supposed sinusoidal motion of the fluid. The Keulegan- Carpenter number describes the relative importance of drag forces over inertia ones. For high values of the Keulegan-Carpenter number that means that the drag loads dominate, whereas if KC < 20, it means that inertia forces dominate. Finally, a dimensionless parameter is also used by engineers. It is the diffraction parameter: L N = (1.3) D λ When the Keulegan-Carpenter number is high, it means that the flow separation around the body is important, thus that the viscous loads dominate. Thus, if ND < 0.2 (small bodies hypothesis), engineers often use a simplified and reliable mathematical model called Morison equation, in order to obtain an approximate value of the forces applied on the structure, in the direction of the flow. The Morison equation reads: ∂v 1 F (t) = C ρA + ρC Dv(t)|v(t)| (1.4) m ∂t 2 d
where Cm is the inertia coefficient, Cd the drag coefficient (both obtained using tables or graphs depending on the Reynolds number, the roughness, ...), A is the cross-sectional are of the body, D the cross-sectional length of the body, ρ the density of the fluid and u(t) the velocity of the fluid. For instance, if the studied body is a cylinder with a diameter D, the Morison equation reads:
D2 ∂v 1 F (t) = C ρπ + ρC Dv(t)|v(t)| (1.5) m 4 ∂t 2 d This model of force is often used by engineers and provides a quick and quite good estimation of the efforts applied of a small structure (in comparison with the wavelength of the fluid motion). However, in some cases, the Morison equation shows limitations and cannot be used because it gives bad approximations or underestimation of the efforts. For instance, in the framework of the study of a GBF structure with a conical base [23], EDF R&D and Innosea noted the Morison equation combined with a discretization of the structure as a stack of cylinders provides results that are far from those obtained during experiments in a flume.
Figure 1.13 – Numerical model of the GBF studied by EDF R&D and Innosea [23].
1.2.2 Numerical model based on potential flows If the Reynolds number is low, the flow can be considered as inviscid and irrotational. Moreover, if KC < 20, the flow is attached to the body and the viscous effects only occur in the boundary layer, that means that the inertia forces dominate the flow around the body. Then, the Navier-Stokes equations used in fluid dynamics can be simplified and the velocity (i.e. a vectorial field) and the pressure (i.e. a scalar) field can be fully described by a potential field (then those 4 variables are gathered into a single one).
19 Then, the solution of the Navier-Stokes equations only depends on the boundaries of the fluid domain, so the softwares based on that numerical model are called Boundary Elements Method (BEM) codes (for instance, the open-source software NEMOH developped by the laboratory LHEEA, from École Centrale de Nantes). Only the meshes of the boundaries of the fluid domain are necessary while using those softwares. Those kind of software can provides good results, while studying the efforts on a structure generated by the diffraction and the radiation of the flow due to the presence of the studied structure, for instance. In comparison with other numerical tools in fluid dynamics, potential-based codes also have quicker calculation times. However, as they do not take into account the viscosity of the fluid, some phenomemon cannot be studied by such codes, like turbulence effects, viscous efforts or damping, for instance.
1.2.3 Computational fluid dynamics (CFD) CFD often refers to the softwares that numerically solve the Navier-Stokes equations without gathering the thermodynamic parameter of the flow into one single variable, as it is done in potential-based software. Moreover, such softwares use a 3D mesh of the fluid domain, so the Navier-Stokes equations are based on an Eulerian specification of the flow field. It exists lots of CFD softwares, that realise a compromise between the chosen hypothesis for the flow and the necessary calculation time. For the same spatial and temporal discretizations, a CFD software generally has a higher calculation time than potential based softwares. However, such codes are able to simulate more phenomena. For instance, in the study of wind turbine structure, CFD softwares are useful to determine the drag efforts or the effects of turbulence on the structure.
1.2.4 New numerical methods Some phenomena are quite hard to simulate with these previous softwares. For instance, the study of violent wave on structures (involving violent deformation of the free surface’s shape) is quite hard with them. So, there are now new numerical methods that exist to study such complex phenomena. Among them, the Smoothed-Particle Hydrodynamics (SPH) method is well adapted for this kind of studyes. Contrary to classical CFD software, SPH method-based softwares use a Lagrangian specification of the flow field, that means that the Navier-Stokes equations are nol solved on a mesh of the fluid domain but on a set of particle of fluid that can move. Thus, such codes makes it possible to get a more precise shape of the free surface (and then better simulating breaking waves, for instance). However, for similare spatial and temporal discretization, the SPH method has longer calculation time than the previsouly mentioned numerical methods. So, through this internship, the objective is to study offshore wind turbine structure using the SPH method and see if it is a good numerical tool in order to predict the loads applied on offshore structure. But before presenting the simulations that were done and the obtained results, the mathematical model and theory of the SPH method will be introduced, in order to understand the main differences of that method with classical meshed method.
20 Chapter 2
The SPH method
2.1 Navier-Stokes equations for weakly compressible flows
A weakly compressible flow of a Newtonian fluid is described by the Navier-Stokes equations, which describe its motion through a system of equations that links the position the velocity and the pressure in the flow. These governing equations are mathematical formulations of the following conservation laws of physics [24]:
– the conservation of the mass of the fluid
– the conservation of momentum (from Newton’s laws)
As the SPH method is a Lagrangian computational method, the following equations will be given in their Lagrangian form. Considering that there are not processes that create or destroy mass in the flow (such as nuclear processes for instance), the mass conservation equation (also called the continuity equation) reads: dρ + ρ∇~ .~v = 0 (2.1) dt with the density of the fluid ρ and the velocity ~v. On the otger hand, the momentum equation for a fluid particle reads: d~v 1 1 = − ∇~ p + ∇~ · (µS) + ~g (2.2) dt ρ ρ with the pressure p, the gravity ~g, the dynamic viscosity µ (constant for Newtonian fluid) and the strain rate tensor S. The rate of strain tensor is given by the formula:
1 S = [∇~v + (∇~v)T] (2.3) 2 If µ is constant, then the following equation is obtained:
∇~ .(µS) = µ∇~ .(∇~v) (2.4) = µ∇~ 2~v
The kinematic viscosity reads: µ ν = (2.5) ρ Thus, (2.4) now reads: ∇~ · (µS) = ν∇~ 2~v (2.6)
The fluid velocity is simply defined by: d~r = ~v (2.7) dt
21 with the position ~r. It can noticed that those 7 equations (a vectorial equation is a set of 3 scalar equations) are linked by 8 unknowns, so there is a need of an additional equation to be able to solve this system. Thus, a state equation describing the compressibility of the fluid is added [24]: " # ρ c2 ρ ξ p = 0 − 1 (2.8) ξ ρ0
with ξ a constant parameter such as ξ = 7, c the speed of sound in the considered fluid and ρ0 the reference value of the density of the fluid. Nevertheless, in numerical simulations in hydrodynamics, the spatial and temporal discretizations are linked through a parameter called the Courant number C (also known as the Courant–Friedrichs–Lewy condition, to ensure the stability of the scheme with regards to advection): v∆t C = ≤ C (2.9) ∆x max with v the magnitude of the velocity, ∆t the temporal discretization and ∆x the spatial discretization (here, it is limited to the 1D case, but the definition in 3D is very similar). Typically, it is condidered that Cmax = 1. Thus, if the fluid is water, the value of the speed of sound is c ∼ 1480 m.s−1. If the velocity magnitude ∆t 1 in the CFL condition is chosen as the speed of sound, the value of ∆x should not exceed 1480 . In hydrodynamics, the SPH method is often used because it enables to get a good representation of the shape of the free surface if the chosen spatial discretization is short enough. So, if a relatively small spatial discretization is chosen, the CFL condition forces the temporal discretization to be very short, and it is not affordable for both computational time and memory use during a simulation. Moreover, choosing a too high value for the speed of sound can lead to instability of the results. Therefore, the equation (2.8) is turned into the following equation: " # ρ c2 ρ ξ p = 0 0 − 1 (2.10) ξ ρ0
with the numerical speed of sound c0 defined by the following formula, after numerical experiments [11]: p c0 = 10 max(vmax, ghmax) (2.11) with vmax the maximum estimated velocity in the flow, g the value of the gravity and hmax the maximum water depth. Even if the speed of sound is lowered to use affordable temporal and spatial discretizations, the flow can still be considered weakly compressible if the following condition is verified: v Ma = max ≤ 0.1 (2.12) c0 with Ma the Mach number of the studied flow. To sum up, the final system of equations that is consid- ered here to study a weakly compressible flow is: dρ ~ + ρ∇ · ~v = 0 dt d~v = − 1 ∇~ p + ν∇~ 2~v + ~g dt ρ (2.13) d~r = ~v dt ξ ρ c2 p = 0 0 ρ − 1 ξ ρ0
2.2 Mathematical principles of the SPH method
The SPH method was firstly used in the 1970s in astrophysics. It was developed by Gingold and Monaghan [8] and Lucy [14] at the same time. The principle of the method is to divide the fluid into
22 particles that contain physical quantities (velocity, pressure, density, volume, ...) and that interact with each other. Contrary to classical numerical schemes used in fluid dynamics, the SPH method is a mesh-free Lagrangian method : the previously described system of equations is solved for each fluid particle. However, the way to compute physical quantities through interpolations is different from classical methods using meshes, because it cannot be based on a grid. The SPH method is based on two successive steps ([18]) : – a continuous interpolation – a discrete interpolation
2.2.1 Continuous interpolation The first step of the SPH method is a continuous interpolation. It is performed with a kernel function that smooths the studied fields. If a scalar field A and a continuous domain Ω are considered, the value of A at the position ~r is: Z A(~r) = A(r~0)δ(~r − r~0)dr~0 (2.14) Ω with δ the Dirac distribution. This formula gives the exact value of the field A at the position ~r but it is 0 only theoretical because it is impossible to numerically compute it. Hence, a kernel function wh(~r − r~ ), of scale h (the smoothing length), is used to computed an approached value of the field A at the position ~r. The kernel function wh has non-zero values on a compact support Ωr, and is null outside of this compact support. Then, the value of the field A at the position ~r approached by: Z 0 0 0 [A]c(~r) = A(r~ )wh(~r − r~ )dr~ Ωr Z (2.15) 0 0 0 = A(r~ )wh(~r − r~ )dr~ Ω
The kernel function wh has several properties. First, it must tend to the Dirac distribution when the smoothing lenght h tends to zero. Then, in the SPH literature, there are different used kernel functions (for instance Schoenberg Mn splines, see [18]). All of the kernel functions existing are spheric, that 0 means that the value of wh only depends on |~r − r~ |. Finally, an other important property of the kernel functions is that they are normalized using the condition: Z Z 0 0 0 0 wh(~r − r~ )dr~ = wh(~r − r~ )dr~ = 1 (2.16) Ω Ωr This condition makes it possible to interpolate exactly the value of the field A if it is a constant field.
Figure 2.1 – Illustration of a 2D-kernel function.
The kernel functions used in SPH are usually bell-shaped, like shown in Fig.2.1. In GPUSPH, the chosen kernel function is the C2 Wendland kernel, which is defined by: w (|~r − r~0|) = αW f (q) h hd W ~0 (2.17) q = |~r−r | h
where αW is a normalizing constant. In 3D, it is equal to: 21 α = (2.18) W 16π
23 The definition of the function fW is: ( q 4 1 − (1 + 2q) 0 ≤ q ≤ qmax fW (q) = 2 (2.19) 0 qmax < q
where qmax = 2 is the dimensionless size of the kernel support. The definition of the gradient of this kernel is: ∂w (~r − r~0) α ~r − r~0 ~ ~0 h W 0 ∇wh(~r − r ) = = fW (q) (2.20) ∂r hd+1 |~r − r~0| 0 where fW is defined by: ( q 3 0 −5q 1 − 0 ≤ q ≤ qmax fW (q) = 2 (2.21) 0 qmax < q Using those formulas, the approximation of the field ∇~ A reads:
Z ~0 ∂A(r ) 0 0 [∇~ A]c(~r) = wh(~r − r~ )dr~ (2.22) ~0 Ωr ∂r Thanks to an integration by part, this also reads:
Z Z ~0 0 0 0 0 0 ∂wh(~r − r ) 0 [∇~ A]c(~r) = − A(r~ )wh(~r − r~ )~n(r~ )dΓ − A(r~ ) dr~ (2.23) ∂Ωr Ωr ∂~r
0 with ∂Ωr the boundary of Ωr and ~n(r~ ) the inward unit normal to the boundary of Ωr. On the boundary of the compact support Ωr, the kernel function wh is equal to zero. So the first term in (2.23) is equal to zero. Then, it has been previously said that the kernel function is symmetric. Thus, the kernel gradient is antisymmetric: ∂w (~r − r~0) ∂w (~r − r~0) ∇~ w (~r − r~0) = h = − h (2.24) h ∂~r ∂~r Using those properties, the approximation of the field ∇~ A now reads: Z 0 0 0 [∇~ A]c(~r) = A(r~ )∇~ wh(~r − r~ )dr~ (2.25) Ωr This formula gives an approximate value of the gradient of A using the value of the field A. The gradient of A can also be written as: A A ∇~ A = ∇~ ρ + ρ∇~ (2.26) ρ ρ Combining (2.26) with the previous formulas leads to the final expression of the gradient of the field A:
Z " ~0 ~0 # ρ(r ) ρ(r ) 0 0 0 [∇~ A]c(~r) = A(~r) + A(r~ ) ∇~ wh(~r − r~ )dr~ (2.27) Ωr ρ(~r) ρ(~r) InA, one can find the calculations from [20] that show the consistency of the continuous interpolation method described above.
2.2.2 Discrete interpolation The second step of the SPH method is a discrete interpolation. In this part, several discrete operators will be introduced as in [24] and [25]. Assuming that a is a particle, ~ra is the position of the particle a, the discrete interpolation obtained from (2.15) reads: X [A]d( ~ra) = VbAbwab (2.28) b∈F where F stands for all fluid particle, Ab = A(~rb) is the value of the field A at the position ~rb of the mb particle b, Vb = is the volume of the fluid particle b and wab = wh(| ~ra − ~rb|). An illustration of the ρb discretization is shown in Fig.2.4.
24 Figure 2.2 – Illustration of a 2D-kernel function for discrete particles. The particles b ∈ F are in blue.
Using the same principle, the SPH gradient of A obtained from (2.27) reads: X Aa Ab G~ +{A } = ρ m + ∇~ w ≈ (∇~ A) (2.29) a b a b ρ2 ρ2 ab a b∈F a b It can be easily seen that if the field A is a constant field, this formula does not lead to a null gradient. Thus, using the following formula: 1 ∇~ A = [∇~ (ρA) − A∇~ ρ] (2.30) ρ an other expression of the SPH gradient of A can be obtained. This version is written G~ − in order to make the difference with the first version G~ +:
~ − 1 X ~ ~ Ga {Ab} = − mbAab∇wab ≈ (∇A)a (2.31) ρa b∈F
− with Aab = Aa − Ab. The gradient G~ has the advantage to be null when the studied field is constant. However, it does not conserve the linear momentum. Hence, both versions of the gradient will be used, but their use will depend on the property that should be satisfied for the field. In the same way, we can define an "SPH divergence", which is, for a field A~:
− ~ 1 X ~ ~ ~ ~ Da {Ab} = − mbAab · ∇wab ≈ (∇ · A)a (2.32) ρa b∈F And for a field A and its diffusion coefficient B, a SPH Laplacian can also be defined as :
X Aab L {B ,A } = V (B + B ) ~r · ∇~ w ≈ [∇~ · (B∇~ A)] (2.33) a b b b a b r2 ab ab a b∈F ab
and just as in (2.33), the SPH Laplacian for a vector field A can be defined as :
~ X Aab L~ {B , A~ } = V (B + B ) ~r · ∇~ w (2.34) a b b b a b r2 ab ab b∈F ab
Moreover, as the SPH method is based on a collection of particles, false pressure modes can appear in the fluid domain. Several methods exist to avoid those kind of problems, as the Ferrari correction used in GPUSPH. This correction is based on a modified continuity equation with an additional term representing density diffusion:
dρa X ~rab = −ρaDa~v + ηF Vbca,b ρab · ∇~ wab (2.35) dt rab b∈P
with ca,b defined as: ca,b = max(ca, cb) (2.36) and c defined as: a s ξ−1 ρa ca = c0 (2.37) ρ0
25 In (2.35), ηF is a damping coefficient called the Ferrari coefficient. As a conclusion, the discrete equation solved in GPUSPH are : dρa − P ~rab ~ + ρaDa {~vb} − ηF Vbca,b ρab · ∇wab = 0 rab dt b∈P d ~va 1 ~ 1 ~ = − Ga{pb} + La{µb, ~vb} + ~g dt ρa ρa (2.38) d ~ra = ~va dt 2 ξ ρ0c0 ρa pa = − 1 ξ ρ0
However, in GPUSPH, there are also equations to model turbulence (using the k − modeling equations, see [11]), but the equations are not introduced in that report as they were not used in the framework of that work.
2.2.3 Choice of discrete operators
The choice of the form of discrete SPH operators previously introduced is based on the derivation of the Lagrangian variational principle [24]. It is possible to find other forms of the SPH divergence, gradient or Laplacian, as long as they respect this variational principle (see [3]).
2.2.4 Boundary conditions
The treatment of boundary conditions is complex in SPH due to the Lagrangian nature of the method. Without treatment, this leads to wrong near-wall behavior, particle leakage and numerical instability. Several methods have been developed to overcome this issue. The available formulations in GPUSPH are listed below.
Lennard-Jones boundaries
The first type of boundary formulation available in GPUSPH is based on the Lennard-Jones potential (see [17] and [19]). In this model, the boundary particles, separated from a considered fluid particle by a distance r, apply on this particle a force f~(r) per unit mass, whose expression is :
r p1 r p2 ~r f~(r) = D 0 − 0 (2.39) r r r2
where p1 and p2 are two constants that must verify p1 > p2 to ensure that this force is repulsive. Generally, the constants are taken as p1 = 4 and p2 = 2. r0 corresponds to the initial particle spacing and the coefficient D is chosen asn D = 5gh with g the gravity and h the water depth. This model is easy to implement, computationally cheap but it represents fictitious forces and i thus inaccurate.
Dynamic boundaries
The second model uses several layers of fixed particles to represent the boundary. It is called the dynamic boundaries model [4]. The layers Serve to fill the kernel support, and there is no need to apply an artificial force onto the fluid particles. They verify the same equations of continuity and state as the fluid particles, but have a fixed velocity equal to zero (or the wall’s velocity). This method provides good results but is difficult to implement with complex shapes of the boundaries, especially those with wide angles. With this formulation, as well as with Lennard-Jones boundaries, the discrete set of equations solved by GPUSPH is the system (2.38).
26 Figure 2.3 – Illustration of a 2D-kernel near a boundary ∂Ω.
Semi-analytical wall boundaries The last model used in GPUS PH is the semi-analytical boundaries model [7]. When a particle is close to the boundary, the interpolation kernel is truncated (see Fig.2.4). Let Ωa be the truncated kernel that is associated to a fluid particle a,close to a boundary, the renormalization factor γa can be introduced as : Z 0 0 γa = wh( ~ra − r~ )dr~ (2.40) Ωa to ensure that the condition (2.16) is always fulfilled. For a continuous field A, the renormalized interpolated value can be written as : 1 Z γ ~0 ~0 ~0 [A]c ( ~ra) = A(r )wh( ~ra − r )dr (2.41) γa Ωa That leads to the following discrete interpolation :
γ 1 X [A]d ( ~ra) = VbAbwab (2.42) γa b∈F
Without neglecting the boundary in (2.23), the gradient of a field A now reads : Z Z 1 0 0 0 0 0 0 ∇~ A( ~ra) = A(r~ )∇~ wh( ~ra − r~ )dr~ − A(r~ )wh( ~ra − r~ )~ndr~ (2.43) γa Ω∩Ωa ∂Ω∩Ωa that leads to the following discrete interpolation : ! 1 X X ∇~ A( ~ra) = VbAb∇~ wab − As∇~ γas (2.44) γa b∈P s∈S
with : Z ∇~ γas = wh( ~ra − ~rs) ~nsdS (2.45) ∂Ωs∩Ωa where ∂Ωs is the portion of boundary spanned by the segment s. The boundary is discretised using these segments s ∈ S (for 3D cases, they are not segments but triangles, but the principle is the same. In (2.42), segments do not appear in the sum over fluid particles. Moreover, there are fluid particles on the boundaries, called vertex particles v ∈ V . They are situated at the tips of the triangles. They have truncated volumes and are taken into account in the sum over b in (2.44), with P = F ∪ V (see 2.4).
27 Figure 2.4 – Illustration of a 2D-kernel near a boundary ∂Ω with discrete particles and boundary segments.
With this change of SPH interpolator, the previously introduced discrete operators now read [7]: ~ γ,+ ρa P Aa Ab ~ ρa P Aa As ~ G {Ab} = mb 2 + 2 ∇wab − ρs 2 + 2 ∇γas a γa ρ ρ γa ρ ρ b∈P a b s∈S a s γ,− 1 P 1 P G~ {Ab} = − mbAab∇~ wab + ρsAas∇~ γas a ρaγa ρaγa b∈P s∈S (2.46) γ,− ~ 1 P ~ ~ 1 P ~ ~ Da {Ab} = − mbAab · ∇wab + ρsAas · ∇γas ρaγa ρaγa b∈P s∈S γ 1 P Aab ~ 1 P ~ ~ ~ L {Bb,Ab} = Vb(Ba + Bb) 2 ~rab · ∇wab − [Ba(∇A)a + Bs(∇A)s] · ∇γas a γa r γa b∈P ab s∈S
Thus, with the semi-analytical boundaries, the discrete equations solved by GPUSPH are given by: dρa γ,− ηF P ~rab ~ + ρaDa {~vb} − Vbca,b ρab · ∇wab = 0 = 0 γa rab dt b∈P d ~va 1 ~ γ 1 ~ γ = − Ga{pb} + La{µb, ~vb} + ~g dt ρa ρa (2.47) d ~ra = ~va dt 2 ξ ρ0c0 ρa pa = − 1 ξ ρ0
With the semi-analytical boundaries, it is also possible to represent open boundaries, using the proposed technique in [6]. However, the equations of such a model are not introduced in that report as it was not used during this work. Moreover, there are also equations for the k − model of turbulence with the boundaries [7], which are not used in the framework of this study.
2.2.5 Moving objects The motion of solid bodies in GPUSPH is handled by an external library dedicated to this kind of physics: the Chrono library [21]. The forces applied by the flow on the body are computed and then used by the Chrono library to determine the position of the center of gravity of the object at the next timestep, but also its new spatial orientation.
28 Chapter 3
First studied case : waves breaking on a monopile structure
3.1 Description of the experiments
The first studied case is the impact of breaking waves on a monopile strcuture. In 2002, in the laboratory LNHE, from EDF R&D, several experiments were realised to study the effects of breaking waves on a monopile structure, in shallow water depth (see [12] and [13]). The experiments were realised in a flume which is 72 m long, 1.5 m wide, 1.5 m depth. A cylinder made of PVC and plaster was representing a monopile structure. It was 1.1 m height, 0.2 m diameter and has a approximate weight of 21 kg. It was placed 0.4 m high from the flat bottom of the flume, on a slope. Two different values of the slope were tested, 2.5 % and 5%. The effect of the slope will be discussed later in this report. Fig.3.1 is a scheme from [13] representing the experimental set-up.
Figure 3.1 – Sketch of the experimental facilites [13].
The waves were generated by a computer controlled piston, equipped with an active wave absorption system. Moreover, at the extremity of the flume opposed the wave generator, rocks were placed in order to limit wave reflection from that part of the flume. The generated waves are regular waves (i.e. mathematically represented by a sinusoid with a period T and an amplitude a = H/2, with H the wave height). Let us introduce the frequency ω: 2π ω = (3.1) T Then, from the linear theory and the Airy wave theory, one can obtain a relation called the dispersion relation, that reads [16]:
ω2 = gk tanh(kh) (3.2) with g the gravity, h > 0 the offshore depth and k the wave number, that is also given by the following formula: 2π k = (3.3) λ
29 with λ the wavelength. So, for a given period, wave height and offshore depth, k can be first computed iteratively using (3.2), and then using the wavemaker theories from [5] and [10], the stroke S of the piston can be calculated through the following formula: H S = (3.4) kh Finally, the positionx of the piston can be obtained through: S x = cos(ωt) (3.5) 2 3.2 Description of the shoaling phenomemon
Several parameters exist to describe waves. Among them, the steepness is a way to characterize nonlinearity of the waves. For a wave with an amplitude A and a wave number k, is defined as: 2πA = kA = (3.6) k The higher is, the more important the effects of nonlinearity are. When is too high, it can lead to breaking waves. Let us assume a regular wave, coming from “infinity” (i.e. depth is constant), propagating in one direction ~ex with an amplitude A∞ > 0, a frequency ω∞ and a wavenumber k∞. Thus, the free surface elevation η due to that wave reads:
η = A∞ cos(k∞x − ω∞t) (3.7) One can show that the average energy in a wavetrain beneath a unit square of the free surface, for a regular wave of amplitude A, is [15]: 1 E = ρgA2 (3.8) 2
Thus, E∞ can be defined as: 1 E = ρgA2 (3.9) ∞ 2 ∞
Let us introduce the group velocity Cg as [15]: dω C = (3.10) g dk
Cg corresponds to the travel speed of energy in a wave. Using 3.2 for such a regular wave, the group velocity Cg reads: ω 2kh C = 1 + (3.11) g 2k sinh 2kh
As for E∞, Cg,∞ reads: ω∞ 2k∞h Cg,∞ = 1 + (3.12) 2k∞ sinh 2k∞h Now, let us assume that this wave coming from infinity arrives on the coast and the depth is not constant anymore. To simplify the problem, let us consider a slight slope that leads to a constant decreasing of the offshore depth, only depending on x in the direction of propagation of waves, as shown on Fig.3.1. Thus, A, w and k are not constant anymore and are now functions of x (the waves are considered steady so that no function depends on t).
30 Figure 3.2 – Scheme of the propagating regular waves arriving in a non constant depth area.
In [15], under the assumptions of steady waves and non dissipative propagation of waves, one can obtain that ω is constant. So, using 3.2 and the properties of the function tanh, as h is a stricly decreasing 2π function of x, k is a stricly increasing function of x. As k = λ , λ is then a strictly decreasing function x. Then, one can show that:
dEC g = 0 (3.13) dx So the following relation can be obtained:
∀x, E(x)Cg(x) = E∞Cg,∞ (3.14) using 3.8, the amplitude A reads, as a function of x: s Cg,∞ A(x) = A∞ (3.15) Cg(x)
Thus, one can show that is A a strictly increasing function of x [15]. Finally, the steepness of the waves (x) is an increasing function of x. So, the closer from the coast the wave, the steeper it is, and then breaking waves can appear. That is why, in the study of the monopile structure, the cylinder is posi- tioned on a slope, to force waves breaking on the structure and then study the impact of breaking waves on the supported efforts by the cylinder.
In order to illustrate these formulae, let us consider a wave with an offshore amplitude A∞ = 0.1 m, a frequency ω = 0.5 rad.s−1. The offshore depth is 1 m and the wave is arriving on a 2.5 %-slope. Ac- cording to the previous formula, at a certain time t, the shape of the free surface elevation is represented on Fig.3.3. Indeed, this simplified model does not exactly fit the physical reality of shoaling, but it is a good way to explain and represent that complex phenomemon. On Fig.3.3, one can easily see that while arriving on the constant slope, the wavelength is reduced and the amplitude of the wave is increased. 2πA Thus, the steepness = k increases, which inevitably leads to breaking waves.
31 Figure 3.3 – Representation of the shoaling phenomemon.
3.3 Features of the GPUSPH simulations
3.3.1 Configurations of the several cases
This work focuses on 4 different cases : EOL95, EOL96, EOL54 and EOL79 (see [12]). The differences between these 4 experiments were the value of the slope where the monopile is located, the offshore depth of the tank, the wave period and the offshore wave height. During a previous work realised by Louise Fratter during an internship in EDF R&D, a first case called EOL 96 [12]) was simulated in GPUSPH, with a given spatial discretisation (i.e. size of particles) of 5 cm. So, this case was studied again, with a smaller size of particle, and other cases were also simulated. The geometry files that were created for these cases do not completely fit the real geometry of the problem used during experiments, for several reasons. First, damping usually occurs while simulating wave propagation on long distances using the SPH method. Moreoever, reducing the geometry makes it possible to reduce the number of fluid particles used during the calculation, and thus reducing both memory use and computation time. So, instead of creating a 72 m length flume as the one used during the experiments, the length of the numerical flume was shortened to 54.5 m, that mean that the distance between the wave generator and the beginning of the slope is reduced. The width of the numerical tank is 1.5 m, like during the experiments, and its height is 2 m. Then, considering the water in the tank, it was chosen to set the density to 1000 kg.m−3 and its kinematic viscosity to 1.0 · 10−6 m2.s−1, as these values are essentially the same as those during the experiements. Tables 3.1 sums up the characteristics that differ from one case to another:
Table 3.1 – Characteristics of the simulations run for the EOL95 case [12].
Cases EOL95 EOL96 EOL54 EOL79 Slope 2.5 % 2.5 % 5 % 5 % Length of the tank before the slope 4.5 m 4.5 m 12.5 m 12.5 m Wave period 2.4 s 2.4 s 1.8 s 2.1 s Offshore wave height 0.325 m 0.377 m 0.260 m 0.230 m Offshore depth 1 m 1 m 0.8 m 0.7 m
32 3.3.2 Numerical options As the cylinder diameter is small compared to the basin width, it has been assumed that the cylinder has negligible effects on the flow in the direction perpendicular to the flow direction. Thus, the walls of the numerical flume were removed and replaced by periodic boundaries (i.e. the fields on the left boundary is identical to the field on the right boundary). This choice also makes it possible to reduce the number of particles by deleting the ones that represent the walls, and thus to reduce the calculation time. As they are better for the computation of efforts, the semi-analytical boundary conditions were chosen during these simulations for all the other boundaries of the domain. For the fours cases, the size particle was set to 2.5 cm, the numerical speed of sound was set to 40 m.s−1 and the value of the Ferrari coefficient was set to 0.1.
3.3.3 Construction of the geometry files The geometry files were done using SALOME, an open source scientific tool dedicated to preprocessing and postprocessing of numerical simulations [22]. Before running the simulation, the fluid domain must be initialised (i.e. filled with fluid particles). This task is done using Crixus, a software delivered with GPUSPH [1]. Fig. 3.4, 3.5 and 3.6 represent the 3D modelling of the cases on SALOME and the particle filling realised with Crixus. Crixus fills the fluid domain using a cartesian grid. Actually, and it can be easily seen on Fig. 3.6, fluid particles are highly ordered. So, at the beginning of a simulation using GPUSPH, fluid particles are rearranged, what creates fake and ephemeral pressure or velocity fields in the fluid domain. So, before really starting the simulation (by moving the wave generator for instance), it is better to wait for some iterations in order to make those spurious fields vanish.
Figure 3.4 – Visualization of the 3D modelling of EOL95 case in ParaView.
33 Figure 3.5 – Visualization of the 3D particle filling from Crixus for EOL95 case in ParaView. Red particles are fluid particles, blue and gray particles are boundary particles.
Figure 3.6 – Visualization of the 3D particle filling around the monopile in ParaView.
34 3.4 Results and analysis 3.4.1 Computational times All the simulations were run on the computer cluser Porthos, owned by EDF R&D. The graphics cards used are Nvidia GK110GL Quadro (K5200).Tables 3.2 summarizes the simulated physical times, number of particles and computational times for the four simulated cases.
Table 3.2 – Computational characteristics of the simulations.
Cases EOL95 EOL96 EOL54 EOL79 Physical simulated time 120 s 120 s 120 s 120 s Simulation time ∼ 14 days ∼ 14 days ∼ 7 days ∼ 7 days Number of GPUs 6 12 6 6 Number of particles 4 571 582 4 571 582 1 907 151 1 651 031 Number of iterations ∼ 538000 ∼ 538000 ∼ 500000 ∼ 500000
The results obtained from the simulations correspond to the mean horizontal force Fx applied on the monopile in the direction of propagation of waves over one period and the mean free surface elevation near the monopile over one period. These results are gathered in the following subsections and compared with the results available in the reports [12] and [13]. For the case EOL96, the results obtained with GPUSPH were also compared to the results obtained during the previous internship, during which the particle size was 5 cm, in order to see the impacts of the particle refinement on the numerical results. The 3D-visualization of the results makes possible to check whether there are errors are not and if breaking occurs at the expected location. A screenshot of one of the simulation is provided in Fig. 3.7.
Figure 3.7 – Visualization of the pressure field in the flow induced by the wavemaker in EOL95 case, t = 60 s.
3.4.2 Mean free surface elevation results As there were no experimental results available for the mean free surface elevation near the monopile for EOL54 and EOL79 cases, the results are only introduced for EOL95 and EOL96 cases.
35 Figure 3.8 – Mean free surface elevation near the pile in EOL95 case.
Figure 3.9 – Mean free surface elevation near the pile in EOL96 case.
Fig. 3.8 and 3.9 show the mean free surface elevation in front of the cylinder for the EOL95 and EOL96 cases. Fig. 3.8 shows that the mean free surface elevation around the pile is underestimated compared to the experiments. In that case, it was experimentally observed that there were breaking waves on the pile, but that was not observed numerically. For the EOL96 case, one can see that refining (i.e. decreasing the size of particles) has an impact on the numerical results. Fig. 3.9 shows that the shape of the free surface elevation around the monopile is correct but is also underestimated, whereas it was a bit overestimated with the coarser size of particles. Experimentally, it was observed that there were breaking waves before the pile. While watching the results on ParaView, one can also see breaking waves before the monopile but closer than in the experiments, which explains the obtained free surface elevation.
36 3.4.3 Mean horizontal force results
Figure 3.10 – Mean force Fx applied on the pile in EOL95 case.
Figure 3.11 – Mean force Fx applied on the pile in EOL96 case.
37 Figure 3.12 – Mean force Fx applied on the pile in EOL54 case.
Figure 3.13 – Mean force Fx applied on the pile in EOL79 case.
Fig. 3.10 to 3.13 show the mean horizontal force applied on the pile for the four studied cases. For the EOL95 case, one can observe on Fig. 3.10 lower mean efforts applied on the monopile than during the experiments. This is because there are no breaking waves on the monopile, that is why there is no peak of effort. For EOL96, one can see on Fig. 3.11 that there is a peak of the mean horizontal force applied on the pile in the numerical results, whereas there was not during the experiments. This is explained by the fact that in the simulation, waves broke closer to the pile than during the experiments, that is why there is a peak of effort. This peak was not seen during the simulation with the coarser size of particle because that size was not small enough to involve wave breaking. For the cases EOL54, Fig. 3.12 shows that the shape of the mean effort seems accurate, but the peak of efforts is underestimated by GPUSPH. In EOL79 case, Fig. 3.13 the shape of the computed efforts by GPUSPH seems quite accurate and even the peak of effort is well estimated and close to the experimental result. However, for both cases, there were available data about the free surface elevation, so it is hard to comment or
38 interpret the numerical results obtained with GPUSPH.
The results obtained for these simulations show that it is complex to well simulate shoaling phenomemon with GPUSPH. First, because of particles rearrangement at the beginnning of the simulation, coupled with compressibility effects due to the solved system of equations, the still water level in the simulations is underestimated compared to the experiments. Moreover the chosen wave maker movement does not correspond exactly to the experimental wave maker motion. Actually, during the experiments, an activa wave absorption system was used in order to be as close as possible to the desired waves. In GPUSPH, that model has not been implemented yet. Thus, the wave generation is not the same than during the experiments, which can also explain why waves do not break exactly at the same location in the numerical simulations and in the experiments.
39 40 Chapter 4
Second studied case : waves impacting on a gravity-based foundation (GBF)
4.1 Description of the experiments
The second studied case is the impact of regular waves on a gravity-based foundation. In 2015, in the laboratory LNHE, from EDF R&D, several experiments were performed to study the effects of waves and current on a GBF. The experiments were realised in exactly the same flume than for the previous case. The GBF was made of inox elements, surrounded by a concrete ring. The model of the studied GBF is in Appendix B. Fig.4.1 is a scheme from [23] representing the experimental set-up. Contrary to the study of the monopile, the study of the GBF focused on the wave induced loads on the structure, but not on breaking waves induced loads. That is why the structure is not positionned on a slope here.
Figure 4.1 – Scheme of the experimental facilites [23].
On Fig.4.1, WG stands for “wave gauges” and ADV stands for “Acoustic Doppler Velocimeter”.
4.2 Features of the GPUSPH simulations 4.2.1 Configuration of the several cases This work focuses on 4 different cases : GBF02R, GBF04R, GBF05R and GBFU1R (see [23]). As for EOL cases in the previous chapter, the dimension of the GBF were kept identical to the dimensions of the model used during the experiments (see Appendix B), but the dimensions of the numerical tank in GPUSPH were modified from the dimensions of the trial tank. However, in order to give waves enough
41 space to develop, and to prevent from effects of wave reflection on the GBF, in the 4 studied cases, the length of the tank between the piston and the GBF was set to 2 wavelengths of the studied waves. In order to avoid wave reflection from the back side of the flume and to reach affordable computation time, the length of the tank after the GBF was set to 3 wavelengths. After those 3 wavelengths, a 6.25 %-slope was created in order to act as an absorbing beach in experimental tanks, so that the waves can lose energy and are not reflected toward the GBF. Like in the first study case, the width of the numerical tank is 1.5 m, its height is 2 m, the density of the water is 1000 kg.m−3 and its kinematic viscosity is 1.0 · 10−6 m2.s−1. The characteristics of each case are displayed in Tables 4.1.
Table 4.1 – Characteristics of the simulations run for the EOL95 case [12].
Cases GBF02R GBF04R GBF05R GBFU1R Length of the tank 37.75 m 41.75 m 40.6 m 43.75 m Length of the tank before the GBF 5.47 m 6.68 m 7.2 m 18 m Wave period 1.282 s 1.461 s 1.535 s 1.625 s Offshore wave height 0.124 m 0.207 m 0.289 m 0.304 m Offshore depth 0.67 m 0.67 m 0.67 m 0.76 m
Fig. 4.2 and 4.3 represent screenshots of the geometry file realized on SALOME
Figure 4.2 – Visualization of the 3D modelling of GBF02R case in ParaView.
42 Figure 4.3 – Visualization of the 3D modelling of the GBF in ParaView.
4.2.2 Numerical options
The numerical options for the simulations were the same than during the first study:
• the walls on the sides of the tank are implemented as periodic boundaries
• the other boundaries of the domain were treated with semi-analytical boundary conditions
• the numerical speed of sound was set to 40 m.s−1
• the value of the Ferrari coefficient was set to 0.1
The only difference concerns the size of particle: it was set to 2.4186 cm for the GBF02R, whereas for GBF04R, GBF05R and GBFU1R cases it was set to 2.22355 cm (this is due to meshing with SALOME).
4.3 Results and analysis
4.3.1 Computational times
As for the first study, all the simulation were run on the computer cluser Porthos, owned by EDF R&D, on graphics cards Nvidia GK110GL Quadro (K5200). Tables 4.2 summarizes the simulated physical times, number of particles and computational times for the four simulated cases.
Table 4.2 – Computational characteristics of the simulations.
Cases GBF02R GBF04R GBF05R GBFU1R Physical simulated time 120 s 120 s 120 s 120 s Simulation time ∼ 5 days ∼ 5.5 days ∼ 5 days ∼ 7 days Number of GPUs 6 6 6 6 Number of particles 1 731 077 2 344 799 2 324 444 2 709 756 Number of iterations ∼ 508000 ∼ 550000 ∼ 550000 ∼ 550000
43 4.3.2 Mean free surface elevation results Fig. 4.4 to 4.7 show the free surface results for the case GBF02R: free surface elevation near the GBF in a given temporal window, mean free surface elevation near the GBF during a period, evolution of the mean wave height along the tank and spectrum from the Fourier analysis of the free surface elevation along the tank. On can see on Fig. 4.4 and 4.5 that, like in the EOL cases, the free surface elevation is a bit underestimated by GPUSPH. However, the gap between the numerical results and the experimental ones is not as important as it was in the EOL cases. Fig. 4.6 shows that along the tank, one can see a downward trend of the mean wave height. This can come from the numerical dissipation due to interpolation, but Fig. 4.7 shows that the effects of the second mode are increased as the wave travels through the tank. Thus, non-linear effects can occur and can also explain why the free surface elevation near the GBF is a bit underestimated by GPUSPH. In the next section, two possible solutions to improve these results will be introduced and the results will be compared. It is also important to keep in mind that the motion of the wave maker in the simulations is not perfectly identical to the one of the wave maker used in the experiments, as the one numerically modeled does not have an active wave absorption system.
Figure 4.4 – Free surface elevation near the GBF in GBF02R case.
44 Figure 4.5 – Mean free surface elevation near the GBF in GBF02R case.
Figure 4.6 – Mean wave height evolution before the GBF in GBF02R case.
45 Figure 4.7 – Fourier analysis of the free surface elevation in GBF02R case.
Fig. 4.8 to 4.11 show the free surface results for the case GBF04R. As in the GBF02R case, one can see that the free surface elevation is underestimated, and this is explained by numerical dissipation and non-linear effects that appear along the tank.
Figure 4.8 – Free surface elevation near the GBF in GBF04R case.
46 Figure 4.9 – Mean free surface elevation near the GBF in GBF04R case.
Figure 4.10 – Mean wave height evolution before the GBF in GBF04R case.
47 Figure 4.11 – Fourier analysis of the free surface elevation in GBF04R case.
Fig. 4.12 to 4.15 show the free surface results for the case GBF05R. In that case, Fig. 4.12 and 4.13 show that the peak of the free surface elevation near the GBF are underestimated compared to the experimental results, and that the troughs are overestimated. On Fig. 4.14 it is complicated to detect a downward trend of the mean wave height as in the GBF02R case, however Fig. 4.15 shows that the effect of the first mode decrease along the tank, while the effect of the second ones are increased, which can explain why the free surface elevation in the numerical results is different from the one in the experimental results.
Figure 4.12 – Free surface elevation near the GBF in GBF05R case.
48 Figure 4.13 – Mean free surface elevation near the GBF in GBF05R case.
Figure 4.14 – Mean wave height evolution before the GBF in GBF05R case.
49 Figure 4.15 – Fourier analysis of the free surface elevation in GBF05R case.
Fig. 4.16 to 4.19 show the free surface results for the case GBFU1R. The numerical results in this case are similar to the ones in the GBF05R case. Peaks and troughs of the free surface elevation respectively are underestimated and overestimated, and this can also be due to non-linear effects.
Figure 4.16 – Free surface elevation near the GBF in GBFU1R case.
50 Figure 4.17 – Mean free surface elevation near the GBF in GBFU1R case.
Figure 4.18 – Mean wave height evolution before the GBF in GBFU1R case.
51 Figure 4.19 – Fourier analysis of the free surface elevation in GBFU1R case.
4.3.3 Mean horizontal force results Fig. 4.20 and 4.21 show the horizontal force applied on the GBF results for the GBF02R case: horizontal force in a given temporal window and mean horizontal force during a period. One can see that the horizontal force obtained with GPUSPH has a good shape compared to the experimental results. However, like in EOL cases, as the free surface elevation is underestimated, the horizontal force numerically obtained is also underestimated compared to the experimental results.
Figure 4.20 – Forces Fx applied on the GBF in GBF02R case.
Fig. 4.22 and 4.23 show the horizontal force applied on the GBF results for the GBF04R case. One can see that the horizontal force obtained with GPUSPH has a good shape compared to the experimental resutls. Like in EOL cases, as the free surface elevation is underestimated, the horizontal force numerically obtained is also underestimated compared to the experimental results. However, the difference between the numerical free surface elevation and the experimental one in that case is smaller than in the GBF02R
52 Figure 4.21 – Mean force Fx applied on the GBF in GBF02R case. cas, which explains why the differences about the horizontal force are smaller.
53 Figure 4.22 – Forces Fx applied on the GBF in GBF04R case.
Figure 4.23 – Mean force Fx applied on the GBF in GBF04R case.
Fig. 4.24 and 4.25 show the horizontal force applied on the GBF results for the GBF05R case. As the free surface elevation for that case was not well estimated, the horizontal force applied on the GBF obtained with GPUSPH is different from the one experimentally obtained. Moreover, one can see that there is a phase delay between the numerical results and the experimental results. This may be due to non-linear effects that appear along the tank.
54 Figure 4.24 – Forces Fx applied on the GBF in GBF05R case.
Figure 4.25 – Mean force Fx applied on the GBF in GBF05R case.
However, while observing the results, one can notice that at the precise location of the connexion of two GPU cards in the simulation, there were sometimes a spurious overpressure created in the field (certainly due to a problem while two GPU cards exchange information). This overpressure that is numerically created can also explain why there is a delay betwwen the experimental results and the numerical ones. Fig. 4.26 shows that problem for the GBF04R case.
55 Figure 4.26 – Over-turning moments My applied on the GBF in GBFU1R case.
Fig. 4.27 and 4.28 show the horizontal force applied on the GBF results for the GBFU1R case. As the free surface elevation for that case was not well estimated, the horizontal force applied on the GBF obtained with GPUSPH is different from the one experimentally obtained. As in the GBF05R case, one can see that there is a phase delay between the numerical results and the experimental results. This may be due to non-linear effects that appear along the tank.
Figure 4.27 – Forces Fx applied on the GBF in GBFU1R case.
56 Figure 4.28 – Mean force Fx applied on the GBF in GBFU1R case.
4.3.4 Mean overturning moment results Fig. 4.29 and 4.30 show the overturning moment applied on the GBF results for the GBF02R case: overturning moment in a given temporal window and mean overturning moment during a period. One can see that the overturning moment obtained with GPUSPH is underestimated compared to the exper- imental resutls. Again, this is explained by the fact that the free surface elevation numerically obtained is different from the one experimentally obtained.
Figure 4.29 – Over-turning moments My applied on the GBF in GBF02R case.
57 Figure 4.30 – Mean over-turning moment My applied on the GBF in GBF02R case.
Fig. 4.31 and 4.32 show the overturning moment applied on the GBF results for the GBF04R case. For that case, like for the horizontal force, the overturning moment applied on the GBF is a bit underestimated but its shape is close to the one observed during the experiments.
Figure 4.31 – Over-turning moments My applied on the GBF in GBF04R case.
58 Figure 4.32 – Mean over-turning moment My applied on the GBF in GBF04R case.
Fig. 4.33 and 4.34 show the overturning moment applied on the GBF results for the GBF05R case. In that case, he shape of the numerical results look correct but the overturning moment is a bit undersetimated compared to the experimental results.
Figure 4.33 – Over-turning moments My applied on the GBF in GBF05R case.
59 Figure 4.34 – Mean over-turning moment My applied on the GBF in GBF05R case.
Fig. 4.35 and 4.36 show the overturning moment applied on the GBF results for the GBFU1R case. In that case, the shape of the overturning moment does not fit the experimental results. There is also a phase delay between the numerical results and the experimental ones. Moreover, the overturning moment numerically obtained is underestimated compared to the one experimentally obtained.
Figure 4.35 – Over-turning moments My applied on the GBF in GBFU1R case.
60 Figure 4.36 – Mean over-turning moment My applied on the GBF in GBFU1R case.
4.4 Possible improvements
In order to obtain more accurate numerical results, two solutions were implemented. The first one corresponds to the same simulation, but run with a smaller size of particle. The size of particle was chosen so that the ratio between the size of particle and the length of the tank before the GBF was the same than in GBF04R, because in that case, the shapes and the amplitudes of the obtained results were satisfying.The second solution that was implemented was the same simulation, but that time run using walls on the sides of the tank. Actually, the bottom part of the GBF is larger than the top part, so it can influence the flow around it and using walls instead of periodic condition on the side could show different results.
4.4.1 Computational times Table 4.3 summarizes the simulated physical times, number of particles and computational times for the three simulated cases.
Table 4.3 – Computational characteristics of the simulations..
Cases GBF02R GBF02R refined GBF02R with walls Physical simulated time 120 s 120 s 120 s Simulation time ∼ 5 days ∼ 5.5 days ∼ 5 days Number of GPUs 6 6 6 Number of particles 1 731 077 2 344 799 2 324 444 Number of iterations ∼ 508000 ∼ 550000 ∼ 550000
4.4.2 Mean free surface elevation results
61 Figure 4.37 – Free surface elevation near the GBF in GBF02R case.
Figure 4.38 – Mean free surface elevation near the GBF in GBF02R case.
Fig. 4.37 and 4.38 show the free surface elevation near the GBF in a given temporal window and the mean free surface elevation near the GBF during a period for all the tested cases. One can see that replacing the periodic boundaries on the sides of the tank by walls leads to an overestimation of the troughs of the free surface elevation, whereas the refinement has nearly no effects on that. Moreover, one can see that these two solutions do not improve the results concerning the peaks of the free surface elevation.
4.4.3 Mean horizontal force results
62 Figure 4.39 – Forces Fx applied on the GBF in GBF02R case.
Figure 4.40 – Mean force Fx applied on the GBF in GBF02R case.
Fig. 4.39 and 4.40 show the horizontal force applied on the GBF in a given temporal window and the mean horizontal force applied on the GBF during a period for all the tested cases. One can see that, as the improvement of the free surface elevation estimation were not significant for these two tested solutions, there are no significant improvements concerning the estimation of the horizontal force applied on the GBF.
4.4.4 Mean overturning moment results
63 Figure 4.41 – Over-turning moments My applied on the GBF in GBF02R case.
Figure 4.42 – Mean over-turning moment My applied on the GBF in GBF02R case.
Fig. 4.29 and 4.30 show the overturning moment applied on the GBF in a given temporal window and the mean overturning moment applied on the GBF during a period for all the tested cases. One can see that, as the improvement of the free surface elevation estimation were not significant for these two tested solutions, there are no significant improvements concerning the estimation of the overturning moment applied on the GBF.
One can see that these two tested solutions does not really improve the results, as the results have the same order of magnitude than the first ones, and present quite similar shapes. Moreover, it took about 2 days more than in the initial to fully run the simulation. This is normal normal because in both cases the number of particles is increased. So, the initial simulation looks better from the point of view on an engineer.
64 Conclusion In this work, the GPUSPH software was used to simulate the impact of waves on two types of offshore wind energy structures: a monopile and a gravity-based foundation. Regarding the obtained results through these two study cases, several remarks can be made. First, as observed in both cases, as long as the correct free surface shape is correctly simulated, the efforts and moments are correctly predicted. This was the case on the GBF simulations, however on the monopile cases a good prediction of the water elevation was not achieved. Thus, the predicted efforts were not matching the experimental measurements. This may be due to a difference between the motion of the wave maker in the experiments and the simulations. The mean water level ended up always uderestimated, which affected wave breaking. Implementing an active wavemaker system in GPUSPH could help improving the results. The SPH method requires great computation time while using acceptable size of particle in order to obtain accurate results. Thus, it is important to well master the parameters of the simulation (the size of the partciles, the boundary conditions, ...) in order to optimize the computational time and find a good compromise between the accuracy of the results and the computational time. The very end of the internship was dedicated to the study of the motion of a floatting wind turbine, proposed by the American consortium DeepCwind. Even if no results were obtained for that case, the geometry files to study that case with GPUSPH are fully available to keep on this work. Fig. 4.39 represents the 3D-model made during the internship for this floatting wind turbine.
Figure 4.43 – 3D-model of the DeepCwind floatting wind turbine.
65 66 Appendix A
Consistency of the continuous SPH interpolation
In this part, to simplify the calculations, only the two dimensions problem is considered, but the expla- nations would be the same with 3 dimensions [20]. Two fluid particles, that are closed to each other (i.e. that are located in the kernel of the other one during an interpolation), are considered. One represented by the vector ~r = (x, y) and the other one represented by the vector r~0 = (x0, y0). The Taylor series of A at r~0 using ~r can be written:
∂A(~r) ∂A(~r) 1 ∂2A(~r) A(r~0) = A(~r) + (x0 − x) + (y0 − y) + (x0 − x)2 ∂x ∂y 2 ∂x2 (A.1) 1 ∂2A(~r) ∂2A(~r) +(y0 − y)2 + (x0 − x)(y0 − y) + O(h3) 2 ∂y2 ∂x∂y
If the continuous interpolation, using the kernel function wh, is applied to this formula, it reads: Z Z Z 0 0 0 0 0 ∂A(~r) 0 0 0 A(r~ )wh(~r − r~ )dr~ = A(~r) wh(~r − r~ )dr~ + (x − x)wh(~r − r~ )dr~ Ωr Ωr ∂x Ωr Z ∂A(~r) 0 0 0 + (y − y)wh(~r − r~ )dr~ ∂y Ωr 1 ∂2A(~r) Z 0 2 ~0 ~0 + 2 (x − x) wh(~r − r )dr (A.2) 2 ∂ x Ωr 1 ∂2A(~r) Z 0 2 ~0 ~0 + 2 (y − y) wh(~r − r )dr 2 ∂ y Ωr 2 Z ∂ A(~r) 0 0 0 0 3 + (x − x)(y − y)wh(~r − r~ )dr~ + O(h ) ∂x∂y Ωr Using the property of symmetry of the kernel function, the following relations can be obtained: Z Z 0 0 0 0 0 0 (x − x)wh(~r − r~ )dr~ = (y − y)wh(~r − r~ )dr~ = 0 Ωr Ωr Z (A.3) 0 0 0 0 (x − x)(y − y )wh(~r − r~ )dr~ = 0 Ωr Moreover, the calculations lead to the following result: 1 ∂2A(~r) Z 1 ∂2A(~r) Z 0 2 ~0 ~0 0 2 ~0 ~0 2 2 (x − x) wh(~r − r )dr + 2 (y − y) wh(~r − r )dr = O(h ) (A.4) 2 ∂ x Ωr 2 ∂ y Ωr Using the fact that the kernel function is normalized, the following results is obtained: Z 0 0 0 2 A(r~ )wh(~r − r~ )dr~ = A(~r) + O(h ) (A.5) Ωr
67 Thus, in the continuous case, the interpolation scheme proposed in the SPH method to interpolate a field has a h-quadratic convergence rate. So, the continuous interpolation used in the SPH theory is a consistent mathematical method to interpolate the value of a field A.
In the same way, the integration using the gradient of the kernel function can be calculated. This leads to: Z Z Z 0 0 0 0 0 ∂A(~r) 0 0 0 A(r~ )∇~ wh(~r − r~ )dr~ = A(~r) ∇~ wh(~r − r~ )dr~ + (x − x)∇~ wh(~r − r~ )dr~ Ωr Ωr ∂x Ωr Z ∂A(~r) 0 0 0 + (y − y)∇~ wh(~r − r~ )dr~ ∂y Ωr 1 ∂2A(~r) Z 0 2 ~ ~0 ~0 + 2 (x − x) ∇wh(~r − r )dr (A.6) 2 ∂ x Ωr 1 ∂2A(~r) Z 0 2 ~ ~0 ~0 + 2 (y − y) ∇wh(~r − r )dr 2 ∂ y Ωr 2 Z ∂ A(~r) 0 0 0 0 2 + (x − x)(y − y)∇~ wh(~r − r~ )dr~ + O~ (h ) ∂x∂y Ωr Using the property of symmetry of the kernel function, the following relations can be obtained: Z 0 0 ∇~ wh(~r − r~ )dr~ = (0, 0) Ωr Z 0 0 0 (x − x)∇~ wh(~r − r~ )dr~ = (1, 0) Ωr Z 0 0 0 (y − y)∇~ wh(~r − r~ )dr~ = (0, 1) Ωr Z Z Z 0 2 0 0 0 2 0 0 0 0 0 0 (x − x) ∇~ wh(~r − r~ )dr~ = (y − y) ∇~ wh(~r − r~ )dr~ = (x − x)(y − y)∇~ wh(~r − r~ )dr~ = (0, 0) Ωr Ωr Ωr (A.7) Thus, the interpolated gradient of a function reads: Z 0 0 0 2 A(r~ )∇~ wh(~r − r~ )dr~ = ∇~ A(~r) + O~ (h ) (A.8) Ωr Therefore, in the continuous case, as for the interpolation of a field A, the interpolation scheme proposed in the SPH method to interpolate a gradient has a h-quadratic convergence rate. So, the continuous interpolation used in the SPH theory is a consistent mathematical method to interpolate the value of a field A.
68 Appendix B
Plan of the gravity-based foundation
Unit : mm
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