Feasibility Study of a 3D CFD Solution for FSI Investigations on NREL 5MW Wind Turbine Blade

Master of Science Thesis

KTH School of Industrial Engineering and Management Energy Technology EGI-2015-006MSC EKV1077

Division of Heat and Power Technology SE-100 44 STOCKHOLM

Feasibility Study of a 3D CFD

Solution for FSI Investigations on

NREL 5MW Wind Turbine Blade

Giacomo Bernardi

ii

Feasibility Study of a 3D CFD Solution for FSI Investigations on NREL 5MW Wind

Turbine Blade

Giacomo Bernardi

Approved Examiner

Björn Laumert

Supervisor

Nenad Glodic

Commissioner Contact person

Abstract

With the increase in length of wind turbine blades flutter is becoming a potential design

constrain, hence the interest in computational tools for fluid-structure interaction studies. The

general approach to this problem makes use of simplified aerodynamic computational tools.

Scope of this work is to investigate the outcomes of a 3D CFD simulation of a complete wind

turbine blade, both in terms of numerical results and computational cost. The model studied is a

5MW theoretical wind turbine from NREL. The simulation was performed with ANSYS-CFX,

with different volume mesh and turbulence model, in steady-state and transient mode. The

convergence history and computational time was analyzed, and the pressure distribution was

compared to a high fidelity numerical result of the same blade. All the model studied were about

two orders of magnitude lighter than the reference in computation time, while showing

comparable results in most of the cases. The results were affected more by turbulence model

than mesh density, and some turbulence models did not converge to a solution. In general seems

possible to obtain good results from a complete 3D CFD simulation while keeping the

computational cost reasonably low. Attention should be paid to mesh quality.

iii

Acknowledgments

First and foremost I wish to thank my supervisor, Nenad Glodic, for his support throughout this

project with his knowledge and comments and for his help managing its many changes.

This achievement would not have been possible without the support of my parents, and would

have been much more difficult without all my friends at KTH. Thank you all!

Contents v

List of Figures vii

List of Tables ix

Abbreviations xi

1 Introduction 1

1.1 Renewable energy and power generation . . . 1

1.2 Wind energy . . . 4

1.2.1 Wind energy history . . . 4

1.2.2 The wind turbine . . . 7

1.2.3 Offshore wind turbines . . . 10

1.2.4 Floating wind turbines . . . 13

1.3 The NREL offshore 5 MW baseline wind turbine . . . 14

2 Background 17 2.1 Aeroelasticity . . . 17

2.2 Aeroelastic modeling for wind turbine rotor . . . 20

2.2.1 Aerodynamic models . . . 20

2.2.1.1 BEM model . . . 20

2.2.1.2 3D CFD model . . . 20

2.2.2 Structural models . . . 23

2.2.2.1 FEM model . . . 23

2.2.2.2 Multy-body and modal shape approach . . . 23

2.3 Turbulence . . . 24

2.3.1 Statistical Turbulence models (RANS) . . . 25

2.3.1.1 Eddy-viscosity models . . . 27

Zero-equation models . . . 28

One-equation models . . . 28

Two-equation models . . . 28

The k-" model . . . 29

The k-ω model . . . 30

The Shear Stress Transport model . . . 30

Production limiter and curvature correction . . . 32

2.3.1.2 Explicit Algebraic Reynolds Stress Models . . . 33

v

3 Objectives 35

4 Method of Attack 37

5 Numerical techniques 39

5.1 Workflow . . . 40

5.2 ICEM . . . 40

5.2.1 Inner and outer mesh . . . 42

5.3 CFD simulation . . . 44

5.3.1 Steady-state simulation . . . 45

5.3.1.1 Effect of turbulence models . . . 46

5.3.1.2 Effect of meshing techniques . . . 46

5.3.1.3 Effect of turbulence correction factors . . . 47

5.3.2 Transient simulation . . . 47

6 Results and Discussion 49 6.1 Steady-state results . . . 49

6.1.1 Turbulence models comparison . . . 49

6.1.2 Mesh sets comparison . . . 51

6.1.2.1 Inner domain mesh . . . 51

6.1.2.2 Outer domain mesh . . . 53

6.1.3 Correction factor comparison . . . 57

6.1.4 Discussion . . . 59

6.2 Transient results . . . 60

6.2.1 C_P results . . . 60

6.2.1.1 Comparison with steady-state results . . . 60

6.2.1.2 Inner domain mesh comparison . . . 61

6.2.1.3 Outer domain mesh comparison . . . 63

6.2.2 Discussion . . . 66

7 Final discussion and Conclusions 67 7.1 Future work . . . 69

Bibliography 71

1.1 Estimated renewable energy share of global electricity production, 2011 . . . 1

1.2 Renewable power capacities, EU-27, BRICS, 2011 . . . 2

1.3 Renewable energy share of installed capacity of electricity production, 2011 . . 2

1.4 Electrical generating capacity of renewable energy plant . . . 3

1.5 Persian windmill . . . 4

1.6 C. Brush’s wind turbine . . . 5

1.7 Smith-Putnam wind turbine . . . 6

1.8 Wind turbine components . . . 7

1.9 Aerodynamic force near the blade tip . . . 8

1.10 Loads on a wind turbine blade . . . 9

1.11 Cumulative offshore wind capacity and capacity share in Europe . . . 11

1.12 Tower/foundation/anchor cost, including installation . . . 12

1.13 Floating wind turbine structures . . . 14

2.1 Collar diagram . . . 17

2.2 Increase of wing incidence due to wing twist . . . 19

2.3 Coupling of bending and torsional oscillations . . . 19

4.1 Blade geometry . . . 38

5.1 Inner and outer domain geometry . . . 41

5.2 Domain boundary conditions . . . 41

5.3 Inner domain mesh, blade section . . . 43

5.4 Outer domain mesh . . . 44

5.5 Span positions for results analysis . . . 45

6.1 Pressure distribution: steady-state, turbulence models . . . 50

6.2 Pressure distribution: steady-state, inner domain mesh . . . 52

6.3 Residuals time history: steady-state, inner domain mesh . . . 53

6.4 Pressure distribution: steady-state, outer domain mesh . . . 54

6.5 Residuals time history: steady-state, outer domain mesh . . . 55

6.6 Pressure distribution: steady-state, outer domain mesh (2) . . . 56

6.7 Residuals time history: steady-state, outer domain mesh (2) . . . 57

6.8 Pressure distribution: steady-state, turbulence correction factors . . . 58

6.9 Pressure distribution: steady-state and transient on Case 1 . . . 61

6.10 Pressure distribution: transient, inner domain mesh . . . 62

6.11 Pressure distribution: transient, outer domain mesh . . . 64

6.12 Pressure distribution: transient, outer domain mesh (2) . . . 65

vii

1.1 Load factor for renewable and conventional energy . . . 3

1.2 Physical and operational characteristics of the NREL 5 MW wind turbine . . . . 15

5.1 Inner domain mesh sets . . . 42

5.2 Outer domain mesh sets . . . 43

5.3 Physical values . . . 45

5.4 Inner mesh studies . . . 46

5.5 Outer mesh studies . . . 47

5.6 Turbulence correction factors studies . . . 47

6.1 Computational time: steady-state, turbulence models . . . 51

6.2 Computational time: steady-state, inner domain mesh . . . 53

6.3 Computational time: steady-state, outer domain mesh . . . 56

6.4 Computational time: steady-state, turbulence correction factors . . . 58

6.5 Computational time: reference and transient . . . 60

6.6 Computational time: transient, inner domain mesh . . . 63

6.7 Computational time: transient, outer domain mesh . . . 65

ix

AC Aerodynamic Center

BEM Blade Element Momentum method CFD Computational Fluid Dynamics CG Center of Gravity

DES Detached Eddy Simulation DNS Direct Numerical Solution

DRSM Differential Reynolds Stress Model EARSM Explicit Algebraic Reynolds Stress Model EVTE Eddy-Viscosity Transport Equation FEM Finite Element Method

FSI Fluid Structure Interaction HAWT Horizontal Axis Wind Turbine LES Large Eddy Simulation

PS Blade Pressure Side

RANS Reynolds Averaged Navier-Stokes equations SS Blade Suction Side

SST Shear Stress Transport

xi

Introduction

1.1 Renewable energy and power generation

According the International Energy Agency renewable energy is:“. . . derived from natural pro- cesses that are replenished constantly. In its various forms, it derives directly from the sun, or from heat generated deep within the earth. Included in the definition is electricity and heat generated from solar, wind, ocean, hydropower, biomass, geothermal resources, and biofuels and hydrogen de- rived from renewable resources”[1]. In 2010 renewable energy supplied approximately 16.7% of the global final energy consumption (power generation, heating and cooling, transport fuels, and rural/off-grids energy services)[2], and the share of renewable energy in power production is even bigger.

At the end of 2011 renewable energy accounted for more than 25% of the global generating ca- pacity and provided an estimated 20.3% of the total consumption (fig. 1.1). This difference be- tween generating capacity and power produc- tion is caused by the variability of most of the renewable energy sources, such as wind and so- lar energy[2].

Hydropower is the main source of renewable power worldwide, followed by wind power,

FIGURE 1.1: Estimated renewable en- ergy share of global electricity produc-

tion, 2011[2]

1

biomass power and solar photovoltaic. For the EU-27 countries the photovoltaic capacity is higher than biomass power (fig. 1.2).

FIGURE1.2: Renewable power capacities, EU-27, BRICS, 2011[2]

The success of renewable energy for power

FIGURE 1.3: Renewable energy share of installed capacity of electricity production, 2011[2]

generation can be seen in its fast growth.

During 2011 renewable energy has been al- most half of the new electric production ca- pacity globally installed, estimated in 208 GW.

Wind power, solar photovoltaic, and hydropower are in order the most installed renewable sources during 2011, see figure 1.3.

Some useful data about renewable energy and load factor can be found in the Digest of United Kingdom energy statistics (DUKES) for 2012[3], issued by the Department of Energy and Cli- mate Change (DECC), a British government department. Table 1.1 shows the plant load factor for different renewable and conventional electricity generation, as comparison. Plant load fac- tor is defined as the plant actual yearly energy output over the potential output if the plant operates at full capacity for one year.

Hydropower is the most reliable among the renewable sources but is not expected to grow much more, as can be seen in figure 1.4. Photovoltaic is growing at a fast pace, but it is still the renewable energy with the lowest plant load factor. Certainly photovoltaic is the youngest renewable technology, but so far it cannot withstand the comparison with wind energy. Wind energy shows a fast, longer, growth in installed capacity, a more mature technology and a more

Renewables

Wind 27.1%

- onshore 26.2%

- offshore 29.8%

Photovoltaic 8.3%

Hydro 35.4%

- small scale 37.1%

- large scale 35.1%

Conventional

Combined cycle gas turbine 61.9%

Nuclear 60.1%

Coal fired 42.2%

TABLE1.1: Plant load factor for renewable and conventional electricity generation (average 2007-2011)[3]

favorable plant load factor. Can also be observed the steady growth in offshore wind energy, which as well features a higher load factor. This important characteristic, liable to be quickly improved, and the limitation in new onshore wind farms, makes offshore wind energy the most interesting renewable energy source for the next future[3].

FIGURE1.4: Electrical generating capacity of renewable energy plant[3]

1.2 Wind energy

1.2.1 Wind energy history

Wind energy has accompanied mankind for most of its history. At about 3100 B.C. Egyptians were using sails of linen or papyrus to propel small boats southward against the stream of river Nile, and sailboats were replaced only many centuries later by steamboats[4]. From this ini- tial use as boat propulsion, wind energy later became also source of mechanical power. The first windmills (fig. 1.5) appeared in the Persian region of Sistan, between seventh and ninth century[5]. From there windmills spread to central Asia and later in China and India [6].

Windmills started to be used in northwestern Europe during the twelfth century[7].

FIGURE1.5: Persian windmill[8]

The first windmill in Europe was built in England in 1137 by William of Almoner, of Le- icester. The original design, called post-mill, where the whole tower rotates to face the wind, is thought to be developed independently from the carousel design used in Persia and most likely inspired by the design of waterwheels. From England, post-mills spread in the rest of Europe in a west-to-east direction; in the 1300s windmills were used in Spain, France, Belgium, the Netherlands, Denmark, the German principalities and the Italian states. As they became more common, windmills became also bigger and their design had to be improved for that;

hence the tower-mill design was developed, where only the upper part of the tower with the sails, the wind shaft and the brake wheel rotate to the wind[4].

With migration from Europe to the New World, windmills were introduced in the colonies, but they were not as successful as in Europe. Beside being big and expensive to build, a tower- mill requires constant human attention, a problem in the rural areas in the colonies with their

lack of labor force. Furthermore, the abundance of rivers made watermills more suitable for the development of the colonies. Windmills were more common on islands and along the shore for grain grinding. As the migration moved to the arid and windy West, wind power became more important[4]. Here the American windmill design was developed during the 19th cen- tury by D. Halladay (1854) and by Reverend L. H. Wheeler (1867)[9]. Light, small, movable, self-regulating, cheap and easy to maintain, these windmills were used for grain grinding and water pumping across the West plains. Railroad companies were the first users of Halladay’s windmills, used to supply water to steam locomotives[4]. About two decades later, the first experiments of wind energy production were performed in Europe and US. In July 1887, in the town of Marykirk in Aberdeenshire, Scotland, Professor James Blyth of Anderson’s College installed the first windmill for electricity

production, a 33-feet tall cloth-sailed tur- bine, charging accumulators developed by Camille Alphonse Faure to power the lights of his cottage. Later he built a wind turbine to supply emergency power to the Montrose Lu- natic Asylum[10].

Few months later the Blyth’s experiment, in winter 1887-1888 in Ohio, US Professor Charles F. Brush built a massive 12 kW wind turbine, with a rotor diameter of 50 meters and 144 rotor blades (fig. 1.6). The machine served for twenty years, until Brush decided to take down the sails in 1908[4].

FIGURE1.6: C. Brush’s wind turbine[4]

In 1890s, danish scientist Poul la Cour began test on wind turbines, he was the first to discover that a fast rotating wind turbine with fewer blades is more efficient in electricity production.

In 1903 he founded the Society of Wind Electrician, with the first course held the following year[11]. Commercial use of wind power had to wait until 1927, when Joe and Marcellus Jacobs opened in Minneapolis the Jacobs Wind factory, selling wind turbines used on farms to charge batteries and power lighting.

FIGURE1.7: Smith-Putnam wind turbine [12]

Year 1941 saw the first megawatt-size wind tur- bine, the Smith-Putnam wind turbine, installed at Grandpa’s Knob, Castelton, Vermont. It was a two-bladed, variable pitch, downwind wind turbine, with a 53-meters diameter rotor and 1.25 MW power output (fig. 1.7). It was af- fected by a succession of problems and never run unattended. It was dismantled in 1945 af- ter a blade failure[13]. In 1956 la Cour’s stu- dent Johannes Juul built in Gedser, Denmark a three-bladed, upwind wind turbine capable of 200 kW inspiring the design of later wind turbines. He is also the inventor of the emer- gency aerodynamic tip brake. The turbine run for eleven years and was refurbished in the mid 1970s at the request of NASA.

The Oil Crisis in 1970s drove the United States government to begin, in cooperation with NASA, research into large commercial wind turbines, resulting in the first windfarm, con- sisting of 20 turbines, built in 1980 in New Hampshire. Eleven years later, the first offshore windfarm was built in Vindeby, in the southern part of Denmark, with eleven turbines of 450 kW each[11].

At these days, the Enercon E–126 is the world’s biggest wind turbine, capable of 7.5 MW of electricity output, featuring a 135-meters high concrete tower and three segmented steel- composite hybrid blades for its 127-meters wide rotor[14].

1.2.2 The wind turbine

Wind turbines transform the kinetic energy of the wind in electrical energy, rotating the shaft of a generator. The maximum theoretical power available for the turbine is:

P_{M AX} = 1/2 ˙mV₀²= 1/2ρAV₀³

whereV₀is the wind speed,ρ the air density and A the rotor area; showing the importance of wind speed for the power output. To extract all the power available, the wind turbine should bring down to zero the speed of the air flowing through the turbine disc, which is physically impossible. Therefore a theoretical maximum power coefficient is imposed by the mass and energy conservation, called Betz limit, equal to 0.593. Modern wind turbines are capable of really high power coefficient, up to 0.5[15]. The main components of a HAWT can be seen in fig.1.8.

1. Foundation

2. Connection to the electric grid 3. Tower

4. Access ladder 5. Yaw control 6. Nacelle 7. Generator 8. Anemometer 9. Brake

10. Gearbox 11. Rotor blade 12. Blade pitch control 13. Rotor hub

FIGURE1.8: Wind turbine components[16]

The rotor, generally made of three blades, generates the torque necessary to rotate the shaft by means of the tangential component of the total aerodynamic force acting on the blades. Since the rotational speed of the rotor is much lower than the speed of the generator, a gearbox has to be used to connect the slow shaft of the former with the fast shaft of the latter. On the fast shaft,

between the gearbox and the generator, sits a brake, either electric or mechanical, to prevent the rotor from overspeeding in case of failure in the generator or in the grid connection. Usually the generator employed in wind turbines is of the asynchronous type, so the shaft rotational speed has to be bigger and really close to the synchronous speed, requiring the turbine to work at constant speed. To maximize the power output, a control unit reads the wind direction and actuate a yaw motor in order to keep the turbine always against wind. Another purpose of the control unit is to prevent the generator from operating above its rated nominal power, causing a possible failure. To do so, when the nominal power is reached and the wind speed is high enough to exceed the maximum torque, the control unit increases the blade pitch (they rotate against the wind), reducing the angle of attach of the blade and hence the torque generated by the rotor. In case the wind speed is too high for a safe operation (cut-off speed), the controller brakes the rotor and put the blades in neutral position[17].

v A

v B u

w

Plane of rotation F

F

FIGURE1.9: Aerodynamic force near the blade tip, redrawn from[17]

When the wind turbine is stationary, the wind generates on the blades a small aerodynamic force, almost perpendicular to the rotor plane; the small tangential component generates the torque necessary to start the wind turbine, which slowly accelerates(fig.1.9 A). As the rotor accelerates, the total air speedw seen by the blade, sum of the wind speed v and the tangential speedu, rotates towards the rotor plane and increases in magnitude; the effect is a reduction of the high angle of attach the blade experiences at start up, hence the blade becomes more efficient and produce a high lift forceL and a relatively small drag D (fig.1.9 B). The projection of these two forces on the rotor plane create the tangential force P_T, and consequently the

torque that drives the generator; the orthogonal componentP_N tends to bend the tower and the blades and does not produce work (fig.1.10)[17].

Rotor plane φ

P_T

D P_N R L

w

FIGURE1.10: Loads on a wind turbine blade, redrawn form[15]

The strategies to control the rotational speed of the rotor and prevent it form overspeeding are:

• stall regulation;

• pitch regulation;

• yaw control.

The first two are the most common.

Stall regulation is the most simple mechanically, since the blade does not need to be pitched and can hence be rigidly fixed to the hub. This type of turbine generally operates at almost constant speed, therefore as the wind speed increases the angle of attack also increases, eventu- ally leading to stall. In this way the tangential load can be reduced to keep the rotor at nominal speed. This control strategy lacks in finely controlling the power output; at start up, when the generator is connected instantaneously to the grid, the turbine causes an overshoot in power output. To prevent the turbine from overspeeding in case of a generator failure, an aerody- namic brake is needed. In case of stall regulated wind turbines, the blade tip can be rotated towards the wind, acting as a brake. This system is activated by centrifugal force.

Pitch regulation uses some sort of actuators to change the pitch of the blade, resulting in a smoother power output at start up. On the other hand, the pitch regulation system is not fast enough to filter all the turbulence from the airflow, hence the power output has a bigger fluc- tuation about the nominal value, compared to a stall regulated wind turbine. Since the blade

pitch can be changed, the blade can act as an aerodynamic brake, so tip brakes are no longer necessary.

Yaw control is used in stall and pitch regulated wind turbines to rotate the nacelle against wind and hence extracting the maximum power from the wind. For this reason the yaw con- trol can also be used to reduce the power extraction by turning the rotor away from the wind and therefore reducing the airflow trough the rotor. This control method is not commonly used for large machines[15].

1.2.3 Offshore wind turbines

Wind turbines need flat areas with strong, constant wind and proper separation to avoid recip- rocal interference; sea is the natural answer to these requirements. The flat, smooth water sur- face creates much less turbulence than any land surface, allowing the wind to flow undisturbed and faster. While onshore wind turbines rely on winds from the irregular cyclonic-anticyclonic structures in the atmosphere, offshore wind turbines harvest the energy of the regular breezes from land and sea and vice-versa caused by the uneven heating of land and water masses. Fur- thermore, offshore wind turbines are not constrained in the maximum dimension by road transport of their components, unlike onshore wind turbines.

On the other hand, offshore wind turbines are more expensive to manufacture, place and main- tain. The basic onshore wind turbine has to be modified in order to withstand the harsher en- vironment: corrosion protection of the tower and the blades; sealing of the nacelle to protect the sensitive internal components from seawater spray and, in case, heating units for low tem- perature operation; more reliable mechanical components for longer maintenance intervals.

To connect the turbine tower to the foundation in the seabed, a support structure has to be built and transported to the final site; particular, and often expensive vessels, are used during the installation of the complete structure. Clearly, sending the personnel and the equipment necessary for the maintenance of an offshore wind turbine is much more expensive than for an onshore wind turbine. The connection to the power grid with underwater cables, further increases the complexity, and hence the cost, of an offshore wind farm. For these reasons big wind turbines are used for offshore energy production, in order to minimize the number of supporting structures and grid connection points required for total power output; this results also in more complex turbines and structures.

In 1991 Denmark inaugurated the first offshore wind farm, capable of 4.95 MW with eleven turbines. Twenty years later the total installed capacity was about 3 GW, almost all of it in Europe. United Kingdom is the leading country in offshore wind energy production, with more than half of the total European capacity (fig.1.11)[18].

FIGURE1.11: Cumulative offshore wind capacity 1991-2010 and share of installed capacity in Europe in 2010[18]

The support structures used for offshore wind turbines have been previously well tested in the oil and gas industry, but so far only grounded structures are used in wind energy [19].

Different designs are available, depending on the depth and soil characteristic, the main are:

gravity-based: durable and simple but heavy and expensive for depth above ten meters[20];

monopile: simple and well documented, requires scour protection and is affected by large hydrodynamic loads[21];

suction bucket: cheap and suitable for high depth, difficult to transport and install, not suit- able for rocky soil[22];

tripile: simple and more stiff than monopile, it is heavy and requires a large amount of steel [23];

tripod: good stiffness and resistance against overturning, complex structure with risk of fa- tigue[24];

jacket: light and good overturning resistance, complex to manufacture and transport[25].

The main problem with grounded offshore wind turbines is that the cost of the support struc- ture grows with the water depth, because of the bigger amount of material used but also because

of the larger overturning moment at their foundations (fig.1.12). Some countries, e.g., USA, Japan, Korea, have small amount of shallow waters for wind energy production and a big po- tential of strong, constant winds lies in deep water areas; this is the reason of the recent big interest in floating structures for wind turbines[19].

FIGURE1.12: Tower/foundation/anchor cost, including installation [26]

1.2.4 Floating wind turbines

In a floating structure the support is provided by the water by means of buoyancy, while some mooring lines are only used to keep the platform in place. As grounded structures, floating structures are widely used in the oil and gas industry for deep water operation, and these de- signs, plus some hybrids, are under study for wind turbine support. So far only prototypes have been built. This section is based on[27].

The spar buoy structure consists of a long cylinder with air in its top and ballast in its bottom (fig.1.13 A–left). The big volume of air provides the buoyancy required to support the whole structure, while the ballast keeps the center of gravity below the center of buoyancy, making the structure stable. The longer is the cylinder and the heavier is the ballast, the more stable is the structure. This kind of structure, with its small cross section at sea surface, is poorly affected by wave motion. The Hywind prototype, by Statoil, is an example of this structure.

The tension leg platform (TLP) is an underwater structure anchored to the sea bed by some tension legs (fig.1.13 A–center). The buoyancy force is bigger than the weight of the structure, hence the mooring cables are under tension, stabilizing the structure. The whole platform is kept underwater, with only the connection structure to the wind turbine tower affected by the wave loads. Blue H is testing a 3/4 scale prototype of this design.

The barge floater is a semi-submersible floating structure, widely used in oil and gas industry (fig.1.13 A–right). The structure can be sailed to the site already completely installed, and can be sailed back to a harbor for maintenance purpose, making it cheaper to install and operate.

On the other hand, its big floating structure at sea surface makes it sensitive to wave motion.

A possible improvement of this design, is to add to the barge a wave energy device. These com- ponents are used to extract the energy of the waves and convert it in electricity. The structure becomes clearly more complex and expensive, but generates more energy and the wave energy device greatly reduces the movements of the platform, improving the efficiency of the wind turbine. Due to the particular design, the structure rotates autonomously against the waves, which are usually propagating in the same direction of the wind, so the yaw motor will not be necessary. A prototype is under study, the Poseidon 37[28].

A possible improvement of the spar buoy is to strengthen it with tension wires, as in the Sway prototype (fig.1.13 B). In this way the structure can be made stiff enough while saving steel and weight. If the rotor is placed downwind of the tower, the tension wires can continue up to the top of the structure, making it more stiff; with the downwind rotor, the structure will turn against wind, not requiring any yaw control.

The tri-floater uses three shorter semi-submerged spars (fig.1.13 C). To compensate the loads from the waves, a dynamic ballast is used to stabilize the structure. A full scale prototype, WindFloat, is under test with a 2 MW wind turbine.

FIGURE1.13: Floating wind turbine structures[26, 29, 30]

1.3 The NREL offshore 5 MW baseline wind turbine

The United States, among others, are interested in the big potential of offshore wind energy.

The Mineral Management Service published in 2006 the results of a study about offshore wind energy[31]. According data from the U.S. Department of Energy (DOE), a potential available energy of 900 GW, almost the total capacity installed in the U.S., is available in waters beyond five nautical miles from shore; the so called Outer Continental Shelf (OCS), defined as “all submerged lands, its subsoil, and seabed that belong to the United States and are lying seaward and outside of the states’ jurisdiction”[32].

This report[31] showed also that only about 10% of this energy potential is in shallow waters area (depth< 30 m). In order to study and evaluate available and new technologies in offshore wind energy, the DOE’s National Renewable Energy Laboratory (NREL) defined a realistic standardized design of a wind turbine for offshore operation[33]. NREL based its design on publicly available data from real wind turbines and theoretical designs used in offshore wind energy simulations. In particular NREL used data from the largest wind turbine prototypes available at that time¹, the Repower 5M and the Multibrid M5000, both rated at a power out- put of 5 MW. Many theoretical studies (DOWEC, RECOFF, WindPACT among others) are based on conceptual design of 5-6 MW. The resulting design is a 5 MW, upwind, three-bladed

1February 2009

wind turbine, operating at variable speed and pitch controlled[33]. The main physical and operational characteristics are listed in table 1.2.

Blade

Length (w.r.t. Root Along Preconed Axis) 61.5 m

Mass 17,740.0 kg

First Mass Moment of Inertia (w.r.t. Root) 363,231.0 kg m Second Mass Moment of Inertia (w.r.t. Root) 11,776,047.0 kg m²

Rated tip speed 80.0 m s⁻¹

Rotor

Diameter 126.0 m

Mass 110,000.0 kg

Shaft tilt 5.0 deg

Precone 2.5 deg

Hub

Diameter 3.0 m

Mass 56,780.0 kg

Hub Inertia about Low-Speed Shaft 115,926.0 kg m²

Height above ground 90.0 m

Nacelle

Mass 240,000.0 kg

Nacelle Inertia about Yaw Axis 2,607,890.0 kg m²

Tower

Mass 347,460.0 kg

Height above ground 87.6 m

Operation

Cut-in (@ 6.9 rpm) 3.0 m s⁻¹

Rated (@ 12.1 rpm) 11.4 m s⁻¹

Cut-out 25.0 m s⁻¹

TABLE1.2: Physical and operational characteristics of the NREL offshore 5 MW baseline wind turbine[33]

Background

2.1 Aeroelasticity

Aeroelasticity can be considered as the result of the mutual interaction of three main disci- plines: dynamics, solid mechanics and (unsteady) aerodynamics.[34] This can be easily visual- ized with theCollar diagram (fig.2.1).

Elastic forces Inertial forces

Aerodynamic forces

(Dynamics)

(Solid mechanics) (Fluid)

FIGURE2.1: Collar diagram[34]

Pairing the vertices of the triangle, three important scientific fields can be found:

• Stability and control: inertial and aerodynamics forces,

• Structural vibrations: inertial and elastic forces,

• Static aeroelasticity: aerodynamic and elastic forces.

17

The Collar diagram can be further expanded to include, for example, the stress induced by high temperature (aerothermoelasticity) or the dynamics of the control system (aeroservoelas- ticity)[34].

Aeroelastic problems can be divided indynamic and static aeroelasticity, respectively if inertial and unsteady aerodynamic loads are or not involved. In case of a long, slender wing in subsonic flow they are[35, 36]:

• static aeroelasticity:

– divergence, – control reversal;

• dynamic aeroelasticity:

– flutter

classical flutter, stall flutter, – buffeting.

The aeroelastic results and methods used in wing design can be used also in wind turbine blades, since they share similar geometric and aerodynamic characteristics. While control reversal is of no interest in case of wind turbine blades, buffeting, caused by the interaction of a lifting surface with a strongly turbulent flow, is generally expected only in downwind wind turbines.

Furthermore, classical flutter mainly affects pitch–regulated wind turbines while stall flutter mainly stall–regulated wind turbines. Here static aeroelastic torsional divergence and dynamic aeroelastic bending–torsional flutter will be studied[37].

Divergence is the case of a statically unstable fluid-structure interaction. In the case of a wing section with its aerodynamic center ahead of its center of torsion (fig.2.2) the lift force will twist the section to a higher angle of attach and hence further increase the lift and so on. The torsional stiffness counteracts this torsion, if the air speed is smaller than a limit speed called divergence speed the internal elastic force and the external aerodynamic force converge to an equilibrium point[36]; if this speed is exceeded the section will indefinitely twist beyond the elastic limit and eventually causing the structure to fail [34]. Even if aeroelastic divergence is avoided, in straight wing, and therefore wind turbine blades, often this torsion causes an increase of the aerodynamic load on the outer section of the blade and hence an higher bending moment at the blade root, possibly causing a collision between the blade and the supporting tower in an upwind HAWT.

Flutter occurs in case of unfavorable coupling between flexural and torsional modes. While an elastic system with one degree of freedom cannot be unstable without negative spring or

Lift

Center of twist Wing twist

Aerodynamic center

FIGURE2.2: Increase of wing incidence due to wing twist, redrawn from[36]

damping force, in case of two degrees of freedom the force associated with each degree of free- dom can interact with each other causing a diverging oscillation. While pure bending and pure torsional oscillation are quickly damped by aerodynamic forces, the inertial and aerodynamic forces in case of a combined torsional–flexural oscillation can excite the structure beyond its structural damping.

For example if the torsional and bending oscillation are 90° out of phase, i.e. the torsion is at its maximum at zero bending and vice versa (see fig. 2.3) the twisting causes a positive angle of attack and therefore a force in the direction of motion. The situation is reversed when the wing moves downwards. In practical cases the difference in phase angle would not be as large as 90°, but the same effect applies[36].

Motion

Flexural axis

Positive geometric Negative geometric

incidence producing positive lift incidence producing negative lift

of wing

FIGURE2.3: Coupling of bending and torsional oscillations and destabilizing effect of geo- metric incidence, redrawn from[36]

2.2 Aeroelastic modeling for wind turbine rotor

In order to perform a computational simulation of the dynamic performance of a wind tur- bine, an aerodynamic model has to be used in order to determine the loads necessary for a structural model to determine the dynamic response of the rotor.

Some common aerodynamic and structural models are available and used in different aeroelas- tic codes developed by universities, research centers and private companies. Both time-domain and frequency-domain codes are used.

2.2.1 Aerodynamic models

Typically two types of aerodynamic computational models are used for wind turbine aeroelas- tic codes: BEM model and CFD model.

2.2.1.1 BEM model

The Blade Element Momentum (BEM) method is widely used because of its low computational cost. In its classic formulation, the BEM method takes in account the angular momentum of the flow trough a wind turbine and the deflection in the incoming near flow caused by the presence of the blade; it is then an improvement of the simpler 1D-momentum disk actuator model. Two important assumptions, and hence limits, of the BEM method is that it assumes the disk to be of an infinite number of blades and no radial dependency, neglecting tip and 3D effects. Furthermore it can not model transient effects, like dynamic stall. With the Prandtl’s tip loss factor, the effect of a finite number of blades can be simulated, and usually this modified BEM model is used to compute the torque and drag acting on the wind turbine. For aeroelastic simulations the unsteady aerodynamics have to be modeled by means of different models as dynamic wake model and dynamic stall model.

2.2.1.2 3D CFD model

With a 3D CFD model, the three governing equations of fluid dynamics - the continuity, mo- mentum and energy equations - are numerically solved. The equations for a viscous flow, also known as Navier-Stokes equations are[38]:

Continuity equation

∂ ρ

∂ t + ∇ · (ρV) = 0 (2.1)

Momentum equations

x component: ∂ (ρu)

∂ t + ∇ · (ρuV) = −∂ p

∂ x +∂ τ_{x x}

∂ x +∂ τ_{y x}

∂ y +∂ τ_{z x}

∂ z + ρf_x (2.2a) y component: ∂ (ρv)

∂ t + ∇ · (ρvV) = −∂ p

∂ y +∂ τ_xy

∂ x +∂ τ_yy

∂ y +∂ τ_zy

∂ z + ρf_y (2.2b) z component: ∂ (ρw)

∂ t + ∇ · (ρwV) = −∂ p

∂ z +∂ τ_{x z}

∂ x +∂ τ_{y z}

∂ y +∂ τ_{z z}

∂ z + ρf_z (2.2c)

Energy equation

∂

∂ t

 ρ

 e+V²

2



+ ∇ ·

 ρ

 e+V²

2

 V



= ρ˙q + ∇ · (k∇T ) − ∇ · (Vp) +∂ (uτ_{x x})

∂ x +∂ € uτ_{y x}Š

∂ y +∂ (uτ_{z x})

∂ z (2.3)

+∂ € vτ_xyŠ

∂ x +∂ € vτ_yyŠ

∂ y +∂ € vτ_zyŠ

∂ z +∂ (wτ_{x z})

∂ x +∂ € wτ_{y z}Š

∂ y +∂ (wτ_{z z})

∂ z + ρf · V

The system includes seven unknown variables,ρ, p, u, v, w, T and e and five equations. The relation between the thermodynamic variablesρ, p, T and e can be found first assuming the gas to be a perfect gas, which equation of state is:

p= ρRT

where R is the specific gas constant. To close the model an equation for internal energy is required, like for example

e= e(T, p) and in case of a calorically perfect gas it will be

e= c_vT

wherec_v is the specific heat at constant volume[38].

For aerodynamic problems is often possible to consider the fluid newtonian, in which the

shear stress is proportional to the time rate of strain, i.e., velocity gradients. For such fluid the viscous stresses are

τ_{x x}= λ(∇ · V) + 2µ∂ u

∂ x τ_yy= λ(∇ · V) + 2µ∂ v

∂ y τ_{z z}= λ(∇ · V) + 2µ∂ w

∂ z (2.4) τ_xy= τ_{y x}= µ

∂ v

∂ x+∂ u

∂ y



τ_{x z}= τ_{z x}= µ∂ u

∂ z +∂ w

∂ x

‹

τ_{y z}= τ_zy= µ

∂ w

∂ y +∂ v

∂ z



whereµ is the dynamic viscosity relating stresses to deformation and λ is the second viscosity coefficient relating stresses to volumetric deformation. The second viscosity coefficient is dif- ficult to determine, with Stokes estimated it to beλ = −²₃µ, and is therefore often neglected [39]. In this case the momentum equations become

x component:

∂ (ρu)

∂ t + ∇ · (ρuV) = −∂ p

∂ x + ∂

∂ x

 2µ∂ u

∂ x

‹ + ∂

∂ y

 µ∂ v

∂ x+∂ u

∂ y

 + ∂

∂ z

• µ∂ u

∂ z +∂ w

∂ x

‹˜

+ρf_x (2.5a)

y component:

∂ (ρv)

∂ t + ∇ · (ρvV) = −∂ p

∂ y + ∂

∂ x

 µ

∂ v

∂ x+∂ u

∂ y



+ ∂

∂ y

 2µ∂ v

∂ y

 + ∂

∂ z

 µ

∂ w

∂ y +∂ v

∂ z



+ρf_y (2.5b)

z component:

∂ (ρw)

∂ t + ∇ · (ρwV) = −∂ p

∂ z + ∂

∂ x

• µ∂ u

∂ z +∂ w

∂ x

‹˜

+ ∂

∂ y

 µ

∂ w

∂ y +∂ v

∂ z



+ ∂

∂ z

 2µ∂ w

∂ z

‹

+ρf_z (2.5c)

And the energy equation becomes

∂

∂ t

 ρ

 e+V²

2



+ ∇ ·

 ρ

 e+V²

2

 V



= ρ˙q + ∇ · (k∇T ) − ∇ · (Vp) + µ

– 2

∂ u

∂ x

‹2

+ 2

∂ v

∂ y

2

+ 2∂ w

∂ z

‹2

(2.6) +

∂ u

∂ y +∂ v

∂ x

2

+∂ u

∂ z +∂ w

∂ x

‹2

+

∂ v

∂ z +∂ w

∂ y

2™

+ ρf · V

The result is a system of nonlinear partial differential equations, very difficult to solve analyt- ically, since a general close-form solution to these equations has not been found yet[38]. For this reason they are solved numerically integrating them over finite control volumes, which is

what a CFD code does.

With this method, the mass, momentum and energy conservation equations, for a viscous, compressible flow are numerically solved, requiring high computational resources. The most challenging part is to model the turbulence of the model, due to the wide range of vortex length scales. A common approach is to split the total pressure and velocity fields in a (time)averaged and a fluctuating part, the so calledReynolds decomposition, the famous RANS equations. Dif- ferent turbulence models can be used with this method, modeling the whole range of turbu- lence structures. In case of separation, more advanced and more computationally expensive turbulence models are required. The Large Eddy Simulation (LES) model solves the exact equations for the large scale turbulent structures, while the smaller scales are modeled in a sim- plified manner. In order to model the strongly separated flow areas without affecting to much the computational cost, an hybrid approach between RANS and LES, called DES (Detached Eddy Simulation) can be used; in this case the LES model is used only in separated flow areas, while the RANS model is used in the rest of the flow field.

2.2.2 Structural models

The blade structure can be modeled with a classic 3D FEM model or with simpler multi-body and modal shape approach

2.2.2.1 FEM model

In rare cases this approach has been used to model the complete 3D structure of the blade, leading to complex models of thousands of shell elements. Due to its high computational cost a simple 1D model is preferred to the full 3D model. It consist of an elastic beam clamped at the blade root and free at the tip. The model can also be extended to take in account non- uniformity and anisotropic beam.

2.2.2.2 Multy-body and modal shape approach

A simpler approach is to consider the blade as a series of rigid elements hinged together and linked with springs and dampers to model the structure stiffness and damping (multi-body approach). Solving the equation of motion:

M¨x+ C˙x + Kx = F_g (2.7)

where M is the mass matrix, C the damping matrix, K the stiffness matrix and F_g the gener- alized force vector related to the external loads, the displacement vector x can be found. The elements of the vector x are the degrees of freedom of the system.

To reduce the number of DOFs and hence the computational time of the problem, the modal shape function can be used. A deflection shape in this method is defined as a linear combi- nation of a few but physically realistic basic functions, which are often the deflection shapes corresponding to the eigenmodes with the lowest eigenfrequencies; in this way only three or four eigenmodes are used, two flapwise and one or two edgewise.

2.3 Turbulence

Turbulence is a chaotic fluctuation in time and space in the flow field. It is characterized by high Reynolds number, meaning the inertia forces in the fluid are significant compared to viscous forces[40].

Turbulent flows can be described with the Navier-Stokes equations, but their direct numerical solution (DNS) is not computationally feasible at realistic Reynolds numbers, because of the wide spectrum of sizes of the swirling flow structures (eddies). For example, in case of a tur- bulent flow not undergoing any rapid change in the mean flow, the turbulence can be assumed in quasi-equilibrium, meaning that the dissipation at small scales is in balance with the energy transfer from the largest scale to the smaller. In this case the ratio between the largest scalesΛ (i.e. the thickness of the boundary layer of a wall-bounded shear flow) and the smallest scaleη (called the Kolmogorov¹length scale) can be estimated as[41]:

Λ

η ∼ Re^3/4

and the smallest length scale would be much smaller than the smallest finite volume mesh that can be practically used in numerical analysis.

1The smallest length scale is assumed to be independent of the outer geometrical restriction and depends only on the viscosityν and viscous dissipation " as: η = ν³/"
^1/4.

To predict the effect of turbulence without the use of direct simulation, different turbulence models have been developed:

• RANS:

– Eddy-viscosity models, – Reynolds stress models;

• LES;

• Hybrid LES-RANS.

2.3.1 Statistical Turbulence models (RANS)

When the timescale of the problem is much larger than the turbulent timescale, the turbulent flow can be seen as some averaged characteristics with some time-dependent fluctuating com- ponents. The so calledReynolds decomposition splits the total velocity and pressure fields into a mean and a fluctuating part

u_i(x, t) = U_i(x) + u_i⁰(x, t)

and

U_i= 1

∆t

t+∆t

Z

t

u_id t

where∆t is a time scale that is large compared to the turbulence time scale but small compared to the time scale to which the equations are solved.

For incompressible flows the continuity condition is

∂ u_i

∂ x_i = 0

and the same condition holds for both the mean velocity and the fluctuating part

∂ U_i

∂ x_i = 0 ∂ u⁰_i

∂ x_i = 0

The mean flow equation, usually referred to as Reynolds equation, is derived from the incom- pressible version of the Navier-Stokes equation for the total instantaneous velocity:

∂ u_i

∂ t + u_j∂ u_i

∂ x_j = −1 ρ

∂ p

∂ x_i + ν ∂²u_i

∂ x_j∂ x_j (2.8)

and it is

∂ U_i

∂ t + U_j∂ U_i

∂ x_j = −1 ρ

∂ P

∂ x_j + ∂

∂ x_j

‚ ν∂ U_i

∂ x_j − u_i⁰u⁰_j

Œ

(2.9) where can be seen the turbulence interaction term in the mean flow equation in a role similar to the viscous stress tensor. The turbulent stress, or Reynolds stress, tensor can be defined as

−ρu_i⁰u⁰_j

This tensor is symmetric and thereby has six independent components. Therefore, after aver- aging there are ten unknowns: the threeU_i components, pressureP and the six components of the Reynolds stress tensor; but only four equations: the three components of the Reynolds equation and the continuity equation. This is the so called turbulence closure problem, mean- ing a model is needed for the Reynolds stress tensor.

The stress tensor for a Newtonian incompressible flow can be written

− pδi j+ µ

‚∂ u_i

∂ x_j +∂ u_j

∂ x_i

Œ

(2.10)

where the pressure gives the isotropic part of the stress tensor.

The Reynolds stress tensor can also be rewritten to isolate the isotropic part. Introducing the definition of kinetic energy (per unit mass) of the turbulent fluctuations

k≡1 2u_k⁰u_k⁰ the Reynolds stress tensor becomes

−ρu_i⁰u⁰_j= −2

3ρkδ_{i j}− ρ



u_i⁰u⁰_j−2 3kδ_{i j}‹

where the first term is the isotropic component, in analogy with the contribution form the pressure, and accounts for two thirds of the turbulence stress, while the anisotropic component can be described with an eddy-viscosity concept[41].

2.3.1.1 Eddy-viscosity based models of the turbulent stress tensor

The turbulence shear stress can be described in terms of a turbulent viscosity. In analogy to the Newtonian fluid stress description(2.10), the turbulence stress can be written

− ρu_i⁰u⁰_j= −2

3ρkδ_{i j}+ ρν_T

‚∂ U_i

∂ x_j +∂ U_j

∂ x_i

Œ

(2.11)

Whileν is a property of the fluid, ν_T is a property of the flow. The anisotropic part can be rewritten as

ρν_T

‚∂ U_i

∂ x_j +∂ U_j

∂ x_i

Œ

= 2ρν_TS_{i j} (2.12)

whereS_{i j}= ¹₂

∂ Ui

∂ xj +^{∂ U}_{∂ x}^j

i



is the mean strain rate tensor.

Since the stress tensor is symmetric, it means that it can only depend on the symmetric, strain part of the mean velocity gradient tensor. It thereby assumes the turbulent stress not to depend directly on the antisymmetric mean rotation rate tensorΩ_{i j}= ¹₂

∂ Ui

∂ xj −^{∂ U}_{∂ x}^j

i

[41].

To obtain the kinetic energy of the turbulent velocity fluctuation for the isotropic part of the turbulence tensor, equation (2.9) is subtracted from (2.8) to obtain the fluctuating part of the velocity, then multiplied byu_i⁰and averaged

‚ ∂

∂ t + U_j ∂

∂ x_j

Œ

k= − ∂

∂ x_j

‚ 1

2u_i⁰u_i⁰u⁰_j+ 1

ρu⁰_jp⁰− ν ∂ k

∂ x_j

Œ

− ν∂ u_i⁰

∂ x_j

∂ u_i⁰

∂ x_j − u_i⁰u⁰_j∂ U_i

∂ x_j (2.13)

The first group of terms in the right hand side represents spatial redistribution (or transport);

the first is the net effect of turbulent diffusion of u_i⁰u_i⁰/2 by the velocity fluctuation u⁰_j, the second can be seen as turbulent diffusion caused by the pressure fluctuation and the third as the viscous diffusion ofk. The second term in the right hand side of equation (2.13) is the viscous dissipation,", of turbulence kinetic energy (the transfer of kinetic energy into heat) and the last term is the transfer of energy from the mean flow to the turbulent fluctuation, therefore named the production term.

‚ ∂

∂ t + U_j ∂

∂ x_j

Œ

k= Dk

Dt = P − " + D

With the production, dissipation and diffusion terms defined as

P = −u_i⁰u⁰_j∂ U_i

∂ x_j

" = ν∂ u_i⁰

∂ x_j

∂ u_i⁰

∂ x_j D = ∂

∂ x_j

‚ ν∂ k

∂ x_j −1

2u_i⁰u_i⁰u⁰_j− 1 ρu⁰_jp⁰

Œ

Zero-equation models With this simple model, there is no transport of any component of Reynolds stress tensor. The isotropic part of the turbulent stress tensor is incorporated in a modified pressure

p^∗= p +2 3ρδ_{i j}k

and the turbulent viscosity of the anisotropic part is constant in the flow field and proportional to a turbulent velocity scale and a geometric length scale (i.e. wall distance, wake thickness, etc.)

ν_T ∼ V · L

One-equation models In this kind of models, only one quantity is transported, k or ν_T. For example in the Prandtl’s one equation model the kinetic energy transport equation is

∂ k

∂ t + U_j ∂ k

∂ x_j = P − " + D = τ_{i j}∂ U_i

∂ x_j − C_Dk^3/2 l + ∂

∂ x_j

–

ν + ν_T σ_K

 ∂ k

∂ x_j

™

where

τ_{i j}= 2ν_TS_{i j}−2

3kδ_{i j}, C_D= 0.08 , ν_T = k^1/2l , σ_K= 1.0 l is the turbulent length scale[42].

Two-equation models With two equation turbulence models, both the velocity and the length scale are solved using separate transport equation. In these models the Reynolds stresses are related to the mean velocity gradients and the turbulent viscosity (2.11). The turbulent (eddy) viscosity is modeled as the product of turbulent velocity and turbulent length scale, k^1/2andk^3/2/" respectively

ν_T = C_µk²

" or ν_T = k ω

In two equation models the turbulent velocity scale is related to the turbulent kinetic energy, which is provided from the solution of its transport equation. The turbulent length scale is computed from the turbulent kinetic energy and its dissipation rate. The dissipation rate of the turbulent kinetic energy is provided from the solution of its transport equation[40].

The k-" model The transport equations for the turbulent kinetic energy and its dissipation rate are

∂ (ρk)

∂ t +∂ €ρUjkŠ

∂ x_j = P − ρ" + ∂

∂ x_j

–

µ +µ_T σ_K

 ∂ k

∂ x_j

™

(2.14)

∂ (ρ")

∂ t +∂ €ρUj"Š

∂ x_j = (C_"1P − C_"2")"

k + ∂

∂ x_j

–

µ +µ_T σ_"

 ∂ "

∂ x_j

™

P = τ_{i j}∂ U_i

∂ x_j τ_{i j}= µ_T



2S_{i j}−2 3

∂ U_k

∂ x_kδ_{i j}



−2 3ρkδ_{i j}

(2.15) S_{i j}= 1

2

‚∂ U_i

∂ x_j +∂ U_j

∂ x_i

Œ

µ_T = C_µρk²

"

The model coefficients are

C_µ= 0.09, C_"1= 1.44 , C_"2= 1.92 , σ_K = 1.0 , σ_"= 1.3

The k-" model is singular at the wall; hence it requires near-wall correction with a wall damping function[40].

The k-ω model (Wilcox 1988) In this model, proposed by Wilcox, ω is interpreted as the inverse timescale of the large eddies, and the transport equations are[43]

∂ (ρk)

∂ t +∂ €ρU_jkŠ

∂ x_j = P − β^∗ρωk + ∂

∂ x_j

–

µ + σ_Kρk ω

 ∂ k

∂ x_j

™

(2.16)

∂ (ρω)

∂ t +∂ €ρU_jωŠ

∂ x_j = γω

k P − βρω²+ ∂

∂ x_j

–

µ + σ_ωρk ω

∂ ω

∂ x_j

™

The model coefficients are

β^∗= 0.09 , γ = 5/9 ≈ 0.56 , β = 3/40 , σ^K= 0.5 , σ_ω= 0.5

The production term is defined in eq.(2.15), according the Boussinesq assumption. The k- ω model is not singular on the wall, so can be integrated to the wall without wall damping functions, but is unphysically sensitive to the free stream conditions[40].

The Shear Stress Transport (SST) model (SST-2003) Menter’s SST model is an evolution of the baseline (BSL) model from the same author. In the BSL model the k-ω model is used near the wall, switching to the k-" model for the outer region, in order to avoid the drawbacks of these two models. The (modified) transport equations for k-" are added to the transport equations for k-ω, with a blending function F₁, going from one near the surface to zero outside the boundary layer.

∂ (ρk)

∂ t +∂ €ρU_jkŠ

∂ x_j = P − β^∗ρωk + ∂

∂ x_j

–

(µ + σ_Kµ_T) ∂ k

∂ x_j

™

(2.17)

∂ (ρω)

∂ t +∂ €ρU_jωŠ

∂ x_j = γ

ν_TP − βρω²+ ∂

∂ x_j

–

(µ + σ_ωµ_T)∂ ω

∂ x_j

™

+ 2(1 − F₁)ρσ_ω2 ω

∂ k

∂ x_j

∂ ω

∂ x_j The coefficients of the new model are a linear combination of the corresponding coefficients of the underlying models, (1) for the k-ω model and (2) for the k-" model:

φ = F₁φ₁+ (1 − F₁)φ₂