Development and validation of a tool for vertical axis tidal turbine performance evaluation in marine environments

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DEVELOPMENT AND VALIDATION OF A TOOL FOR

VERTICAL AXIS TIDAL TURBINE PERFORMANCE

EVALUATION IN MARINE ENVIRONMENTS

Relatori

Candidato

Ing. Stefania Zanforlin

Stefano Deluca

Dott. Benedetto Rocchio

_____________________________________________________________________ 29 Novembre 2018

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You’ll see it’s all a show,

Keep ‘em laughing as you go

And remember that the last laugh is on you.

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List of figures

Figure 1.1: Fuel shares in world electricity production in 2016 ... 1

Figure 1.2: Schematic representation of spring and neap tides ... 2

Figure 1.3: Schematic representation of axial-flow turbine and cross-flow turbine ... 5

Figure 1.4: Commercial HATT designs ... 7

Figure 1.5: Scheme for single streamtube model ... 8

Figure 1.6: Scheme for multiple streamtube model ... 9

Figure 1.7: Scheme for double-multiple streamtube model ... 10

Figure 2.1: DMST scheme of VATT ... 13

Figure 2.2: 𝐶𝑋 as a function of a ... 16

Figure 2.3: BET scheme of VATT. ... 16

Figure 2.4: 𝐶𝐿 and 𝐶𝐷 coefficient curves as a function of 𝛼 and 𝑅𝑒𝑐 for NACA 0015 airfoil .. 18

Figure 2.5: BET scheme for forces projection ... 18

Figure 2.6: Example of dynamic stall phenomenon during upstroke phase ... 21

Figure 2.7: Dynamic stall model phases division ... 23

Figure 2.8: Schematic representation of tip vortex... 24

Figure 2.9: Tip-loss factor for HAWT according to Glauert ... 24

Figure 2.10: Flow streamlines in the tip region, skin friction lines and z velocity component on the blade suction surface ... 25

Figure 2.11: Tip loss factor Glauert formula applied to VATT ... 26

Figure 2.12: Details of ribbons along which torque was logged during 3D CFD simulations .. 27

Figure 2.13: Examples of fTL data for two investigated rotor geometries ... 27

Figure 2.14: Influence of V1 on α for a VATT ... 28

Figure 2.15: 2D CFD streamlines plot colorized with velocity angle from -20° to 20° ... 29

Figure 2.16: Example of streamtube expansion correction ... 32

Figure 2.17: Flow diagram of MATLAB implementation of VATT performance evaluation .. 33

Figure 2.18: Flow diagram of MATLAB implementation of streamtube solver function “DMST Solve”. ... 34

Figure 3.1: Computational domain investigated through CFD ... 37

Figure 3.2: 𝑛𝑟𝑖𝑛𝑔 sensitivity analysis 𝐸𝑟 and 𝐸𝑎... 39

Figure 3.3: 𝑛𝑟𝑖𝑛𝑔 sensitivity analysis of 𝐶𝑃(𝜃) ... 40

Figure 3.4: Streamlines curvature correction factor for the investigated VATT geometry. ... 41

Figure 3.5: Streamlines curvature correction validation plots ... 42

Figure 3.6: Upstream 𝐶𝑃(𝜃) comparisons for 𝑇𝑆𝑅 = 2.1 ... 44

Figure 3.9: 𝑅𝑀𝑆𝐸𝑉𝑋_{plot as a function of 𝑇𝑆𝑅 ... 48}

Figure 3.10: Streamtubes expansion correction validation plots ... 50

Figure 3.11: Downstream 𝐶𝑃(𝜃) comparisons for 𝑇𝑆𝑅 = 2.1 ... 51

Figure 3.14: Overall turbine 2D 𝐶𝑃 against 𝑇𝑆𝑅 ... 54

Figure 3.15: Relative variation 𝛥𝐶𝑃 of 2D overall 𝐶𝑃 against 𝑇𝑆𝑅 ... 55

Figure 3.16: RMSE plot of 2D 𝐶𝑃(𝜃) against TSR ... 55

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Figure 3.18: Overall one-blade local 𝐶𝑃(𝜃) for TSR = 2.5 ... 57

Figure 3.19: Overall one-blade local 𝐶𝑃(𝜃) for TSR = 3.6 ... 57

Figure 3.20: 𝑛𝑧 sensitivity analysis Er and Ea ... 60

Figure 3.21: 𝑛𝑧 sensitivity analysis of 𝐶𝑃(𝜃) ... 60

Figure 3.22: RMSE variation with TSR for AR = 0.9 ... 63

Figure 3.25: K(μ) tip losses profiles. ... 65

Figure 3.26: Δ𝐶𝑃 variation with TSR for AR = 0.9 ... 67

Figure 3.27: Δ𝐶𝑃 variation with TSR for AR = 1.35 ... 67

Figure 3.28: 𝛥𝐶𝑃 variation with 𝑇𝑆𝑅 for AR = 1.8 ... 68

Figure 3.29: TSR-averaged |Δ𝐶𝑃| for varying AR ... 69

Figure 3.30: Overall 3D performance RMSE for AR = 0.9 ... 70

Figure 3.33: Overall turbine 𝐶𝑃 against 𝑇𝑆𝑅 for 𝐴𝑅 = 0.9 ... 72

Figure 3.36: Overall 3D 𝐶𝑃(𝜃) profile ... 75

Figure 4.1: Investigated geographic domain ... 76

Figure 4.2: Tidal energy-intensive sites in Cape Cod ... 77

Figure 4.3: Bathymetry plot of the investigated domain ... 80

Figure 4.4: Post-processed PE velocity data ... 81

Figure 4.5: DMST-predicted overall turbine 𝐶𝑃 on the investigated domain ... 83

Figure 4.6: TSR(μ) profiles for two feature cells in the domain ... 84

Figure 4.7: K(μ) profiles for two feature cells in the domain ... 85

Figure 4.8: One-blade 𝐶𝑃(𝜃) profiles for two feature cells in the domain ... 85

Figure 4.9: DMST-predicted overall turbine power P [kW] on the investigated domain ... 86

Figure 4.10: 𝑈∞(𝑧) profile for cell with highest P ... 87

Figure A.1: Upstream 𝐶𝑃(𝜃) comparisons for 𝑇𝑆𝑅 = 2.2 ... 89

Figure A.3: Upstream 𝐶𝑃(𝜃) comparisons for TSR = 2.35 ... 90

Figure A.7: Upstream 𝐶𝑃(𝜃) comparisons for 𝑇𝑆𝑅 = 3 ... 92

Figure A.9: Downstream 𝐶𝑃(𝜃) comparisons for 𝑇𝑆𝑅 = 2.2 ... 93

Figure A.15: Downstream 𝐶𝑃(𝜃) comparisons for 𝑇𝑆𝑅 = 3 ... 96

Figure A.17: Overall one-blade local 𝐶𝑃(𝜃) for TSR = 2.2 ... 97

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Figure A.23: Overall one-blade local 𝐶𝑃(𝜃) for TSR = 3 ... 100

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List of tables

Table 1.1: Characteristics of existing tidal barrage schemes ... 3

Table 1.2: Performance characteristics of selected commercial axial-flow turbines ... 5

Table 3.1: Computational mesh data ... 38

Table 3.2: Solver settings adopted ... 38

Table 3.3: Validation of streamlines curvature correction submodel. ... 42

Table 3.4: Validation of streamtube expansion submodel ... 49

Table 3.5: Validation of overall 2D DMST model ... 53

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Abstract

The interest in hydrokinetic conversion systems has significantly grown over the last decade with a special focus on cross-flow systems, generally known as Vertical Axis Tidal Turbines (VATTs). However, analyzing regions of interest for tidal energy extraction and outlining optimal rotor geometry is currently very computationally expensive via conventional 3D Computational Fluid Dynamics (CFD) methods. In this work, a VATT load prediction code based upon the Blade Element-Momentum (BEM) theory is presented and validated against high-resolution 2D and 3D CFD simulations. The standard Double-Multiple Streamtube (DMST) model is enhanced through literature-derived and original submodels that account for phenomena such as dynamic stall, tip losses, streamlines curvature and streamtubes expansion.

As a practical application, the code is employed for a site assessment analysis of the Cape Cod area to quickly highlight oceanic regions with high hydrokinetic potential, where further higher-order and more computationally expensive CFD analyses can be performed. Ocean data are obtained from data-assimilative ocean simulations predicted by the 4D regional ocean modeling system of the Multidisciplinary Simulation, Estimation, and Assimilation Systems (MSEAS) group of the Massachusetts Institute of Technology.

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Chapter 1. Introduction

The continuously increasing energy demand, the need for lowering carbon dioxide emissions together with the growing scarcity of traditional fossil fuels led to a growth of energy from alternative sources, more commonly known as renewable energy sources. The International Energy Agency estimated that renewables have grown at an average annual rate of 2% since 1990, which is slightly higher than the growth rate of world total primary energy supply [1].

Despite solar photo-voltaic (PV) and wind power being the most commonly renowned and fastest rising renewable sources, it must be pointed out that hydroelectric power also has an important role in the current world energy mix. This can be noted especially in non-OECD countries such as China, Brazil, Canada and India, where the overall average annual growth rate of hydroelectric power was 4.0% [1].

In 2016, 16.3% of world electricity was supplied by hydroelectric power, which corresponded to 68.4% of total renewable electricity [1]. Therefore, despite being one of the oldest sources of renewable energy, hydro power plants and devices are still actively researched and deployed [2].

Figure 1.1: Fuel shares in world electricity production in 2016 [1].

However, capacity limits of this source are being approached rapidly, e.g. in OECD countries hydroelectric power production grew with an average rate of 0.6% between 1990 and 2017. Therefore, new fields are under investigation.

A growingly interesting technology consists of harnessing energy from marine tides and the currents they cause, the so-called tidal energy.

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1.1. Tidal Energy

Tidal energy is a kind of renewable energy that entails converting the kinetic energy of water currents caused by tides into electricity. These arise by the interaction between Sun, Moon and Earth. Namely, the gravitational pull of the Sun and the Moon towards the Earth and centrifugal forces caused by the rotation of the Earth around the center of mass of the Earth-Moon system determine a periodical motion of the oceans, which rise and fall (tidal phenomena). The Earth’s rotation in relation to the Moon and the Sun produces 2 tidal phenomena every 24 h, 50 min and 28 s named flood current, when the flow is towards the coast, and ebb current, when away from it. The intensity of the tide may vary depending on the relative position of the 3 celestial bodies. When they are aligned, a higher spring tide is induced while a smaller neap tide happens when the Sun and the Moon are at a 90° angle [3].

Figure 1.2: Schematic representation of spring and neap tides.

As a body of water rises and falls, tidal streams are generated. The kinetic energy of these streams, whose mean velocity value theoretically approaches zero on a time period of approximately half a day and a day, can be harvested to be converted into electric energy. The main advantage of this source is that tides are easily and accurately predictable over extended periods of time, thus high load factors are achievable compared to other renewable sources [3].

The growing interest towards this new field is especially high in Europe, where a long-term strategy has been set up in order to increase the share of ocean energy (tidal and wave) in the coming years. Namely, an installed capacity of 3 GW in the UK and 3.6 GW in the EU is expected to be built by 2020 [3–5], which will be further increased to 188 GW by 2050 [4]. By then, a global installed capacity of 337 GW is estimated with an ocean energy market projected to be worth up to 53 billion € annually [5]. The effort is this direction can be also seen from the growing number of ocean energy test centers in Europe, USA, Canada and Asia [3].

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1.1.1. Tidal range technology

Traditionally, tidal energy has been harvested via tidal range power plants consisting of dams, housing turbines, constructed where the tidal range is sufficiently wide to economically generate electricity [6]. These structures can be placed either in an estuary along the coastline, creating a barrage, or off-shore, where the dam encompasses a closed surface referred to as lagoon. In both cases, the dam is operated, i.e. opened/closed, according to tidal phenomena in order to create an artificial head between the two sides. Subsequently the water is allowed through turbines for electricity production. The specific mode of operation depends on the type of turbine installed, i.e. if the turbines can use either the flood or ebb current or both.

The most renowned tidal barrage plants are reported in Table 1.1. As of 2017, most of tidal range operational plants exist in Europe, Asia and North America with a total rated power of around 522 MW, the most recent being the Sihwa Tidal Power Plant in the Republic of Korea, with rated power of 254 MW, operational since 2011 [7].

Despite this technology allows to achieve power outputs in the 1-100 MW range, important shortcomings are inherent such as the high capital costs necessary for the generally massive construction work and the environmental alterations and/or damage associated with it and plant operation [5,8,9].

Table 1.1: Characteristics of existing tidal barrage schemes [6].

Power Plant Year Capacity

(MW)

Basin area

(km2₎ Operation mode

La Rance, France 1966 240 22 Two-way with pumping

Kislaya Guba, Russia 1968 1.7 2 Two-way

Annapolis Royal Generating Station, Canada

1984 20 6 Ebb only

Jiangxia, China 1985 3.9 2 Two-way

Sihwa Tidal Power Plant 1994 254 30 Flood only

1.1.2. Tidal stream technology

In order to avoid such criticalities, it is also possible to deploy turbines in the free flow of water caused by the tide (tidal stream) to harness its kinetic energy directly. These devices, generally referred to as Marine Current Turbines (MCTs), Hydrokinetic

Turbines (HKTs), Tidal Turbines (TTs) or simply Water Turbines (WTs), are

theoretically similar to common wind turbines in that they both employ blades with an hydrofoil cross-section to generate an aerodynamic lift force (rarely drag force [10]) which puts the rotor in motion. The resulting mechanical torque is subsequently converted into electricity via a suitable generator.

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However, since tidal and wind turbines work with different fluids in two completely different environments, designs choices and materials vary. The main differences are the following:

• Fluid velocity

Common wind turbines generally operate with a wind velocity in the range of 10-20 m/s [11] while tidal streams are slower, usually with peak velocities lower than 5 m/s [5,8], and typically present no extreme conditions [10].

• Fluid density

The density of seawater amounts to 1025 kg/m3_{while air in SATP conditions reaches}

1.25 kg/m3_{. Therefore, the turbine blades are required to be thicker and use suitable}

materials in order to withstand the higher pressure caused by the interaction with water.

• Fluid viscosity

Seawater viscosity is around 0.001003 kg/m s, around 50 times higher than air which is 1.846x10-5_{kg/m s. This, together with the velocity differences, heavily influences}

the Reynolds number encountered by the two types of turbines [10]. • Fluid velocity direction

Wind direction, albeit variable, does not generally present complete inversion of direction in a short period of time which, on the contrary, is common in tidal flows. • Cavitation

Because tidal turbines employ lift forces to generate torque, it is possible that the local pressure on the suction side of the hydrofoil becomes lower than the saturated vapor pressure which entail cavitation. This must be avoided at all costs since it determines vibrations and damage to the surface of the hydrofoil and the entire turbine [10].

• Blockage effects

Unlike wind turbines, it is possible that the flow around a tidal rotor cannot be considered unconfined, i.e. the proximity to the sea floor or sea surface in shallow waters and/or to bodies of land accelerates the incoming flow. This is not necessarily a drawback but must be accounted for during experimental tests and design otherwise predicted turbine performance above theoretical limits may be achieved [5,10].

• Aggressive environment

Tidal turbines encounter a much more hostile environment than wind turbines given the high salinity of seawater together with the presence of organisms that could cause biofouling, especially in shallow waters [10]. Suitable materials such as steel or metallic alloys together with special coatings can be used to mitigate this issue [5].

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Tidal turbines designs can be categorized into axial-flow turbines (AFTs) and

cross-flow turbines (CFTs) which differ for the direction of their rotation axis, parallel and

perpendicular to the flow respectively.

Figure 1.3: Schematic representation of axial-flow turbine (left) and cross-flow turbine (right) [10]. Axial-flow turbines, also known as Horizontal Axis Tidal Turbine (HATTs) resemble common wind turbines since they present a nacelle to which the blades are attached, forming a circular frontal area, i.e. the surface swept by the blades during rotation. The nacelle houses the rotor shaft and the associated bearings together with the electrical generator, which can be coupled to the rotor shaft either directly, if a synchronous generator is employed, or by means of a gearbox, if an asynchronous generator is installed. The turbine blades may also pitch, that is to rotate around their longitudinal axis thanks to motors in order to keep an optimal hydrodynamic configuration. Moreover, the entire nacelle may also rotate (yaw) in order to keep the turbine plane perpendicular to the flow, thus maximizing the incoming flow rate.

A few commercial HATT designs are presented in Table 1.2 and shown in Figure 1.4. Table 1.2: Performance characteristics of selected commercial axial-flow turbines [10].

Developer Device Atlantis AR1000 Bourne RiverStar MCT SeaGen S Verdant Gen5 Voith 1 MW test Rated power (W) 1.00 x 106 _{5.00 x 10}4 _{2.00 x 10}6 _{1.68 x 10}3 _{1.00 x 10}6

Rated flow speed (m/s) 2.65 2.05 2.40 2.59 2.90

No. of rotors 1 1 2 3 1

Rotor diameter (m) 18 6.09 20 5 16

Rotor swept area (m2₎ ₂₅₄ ₂₉ ₃₁₄ ₂₀ ₂₀₁

Rated 𝐶𝑃 0.41 0.39 0.45 0.35 0.4

Cross-flow turbines have their rotation axis either in vertical or in horizontal position but the most common design of the two is the former, referred to as Vertical Axis Tidal

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Turbines (VATTs). In both cases, no nacelle is present, substituted by rotational

bearings to which the blades are attached. As a result, the frontal area is rectangular and the flow crosses the frontal area twice. Information about academically developed VATT models can be found in [10].

Currently, horizontal axis designs tend to be the more widespread [9]. In fact, of the 79 marine energy devices recognized and categorized by the European Marine Energy Centre (EMEC) [12], 45 are HATTs while only 14 are VATTs. This is due to HATTs designs being more technologically proven for high energy flows since quite similar to wind turbines and propellers, together with advantages over VATTs on self-starting capability and torque fluctuations during operation [3,9,10].

However, VATT technology is still under heavy research and development since it has important advantages over other designs such as:

• Omni-directionality

The turbine is able to remain operational irrespective of the direction of the incoming flow which is ideal for tidal streams. As a result, yawing and pitching devices are not necessary to reorient the rotor [10,13].

• Higher reliability

The fewer moving parts reduce the need for maintenance and lower the chances of failures, thus increasing the turbine load factor [10,13].

• Lower costs

The limited number of parts reduces the costs of design and manufacturing [10]. Moreover, for floating devices, the vertical axis of rotation allows the generator and/or gearbox to be placed outside of water, where maintenance access is easier and more economical [10].

• Higher packing factors

Given a flow of a given frontal area, the rectangular frontal area of VATTs fills it more efficiently compared to HATTs’ [10]. Moreover, it has been shown that VATTs placed in close proximity may lead to an increase in performance of the array [14,15].

• Higher theoretical efficiency

Actuator disc (AD) theory [11] can be used to evaluate a turbine power coefficient

𝐶𝑃, i.e. the rotor efficiency, defined as:

𝐶𝑃=

𝑃 1 2 𝜌𝐴𝑈∞3

(1)

Where 𝑃 is the power that the turbine has extracted from the flow, 𝜌 is the flow density, 𝐴 is the frontal area of the turbine and 𝑈∞ is the flow undisturbed velocity.

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It can be proven that, for a single actuator disc rotor such as a HATT, a maximum theoretical 𝐶𝑃 of 59.3% (16/27) [16,17] can be reached while a double actuator disc

rotor like a VATT, where the frontal area is crossed two times, has a higher limit of 64.0% (16/25) [18]. The result has also been generalized for a large number of discs for which 𝐶𝑃 approaches a theoretical limit of 66.6% (2/3) [19].

Figure 1.4: Commercial HATT designs [9]. (Top-left) MCT SeaGen S (Top-right) Voith 1 MW test (Bottom-left) Atlantis AR1000 (Bottom-right) Verdant Gen5.

Finally, it is argued that VATTs could be particularly fit for shallow, less energy intensive waters, where a smaller and more economical turbine design together with lower costs for underwater cables and installation could make the realization of an otherwise unfeasible tidal farm economically viable [10].

Therefore, VATTs appear as a promising alternative to their horizontal counterparts, especially observing that the aforementioned shortcomings can be limited by employing suitable design choices, e.g. using helical blades or hybrid designs [10].

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1.2. VATT Mathematical Models

The hydrodynamic load and performance of VATTs can be described either via

analytical models or computational fluid dynamics (CFD) methods.

Analytical models use relations that do not require to be spatially or temporally discretized in order to be solved and are mainly 3: Momentum models, Vortex models and Cascade models [20]. Given the much wider adoption of the first compared to the others [21], only momentum models will be discussed in this work.

Momentum models [11,20], also known as Blade Element-Momentum (BEM) models, are the simplest approach to VATT analysis. They are based on the physical concept that the turbine can be analyzed by using one or more streamtubes which wrap the turbine rotor. By applying Bernoulli’s equation, it can be proven that the streamwise aerodynamic force on the blades is equal to the rate of change of the momentum of the fluid acting on the blades. The same force is also equal to the average pressure times the frontal area of the turbine.

𝐹𝑇 = 1 2ρA(V∞ 2 _{− V} w2) = Δ𝑃𝑎𝑣𝑒 𝐴 = 𝑑(𝑚𝑉) 𝑑𝑡 = 𝑚̇(𝑉∞− 𝑉𝑤) = 𝜌𝐴𝑉𝑎(𝑉∞− 𝑉𝑤) (2) The first model analyzed is the single streamtube model, a schematic representation of which is shown in Figure 1.5.

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It is assumed that the flow velocity is uniform and constant throughout the swept volume of the turbine and equal to Glauert’s induced velocity for a HAWT.

𝑉𝑎 =

𝑉∞+ 𝑉𝑤

2 (3)

By knowing 𝑉𝑎 it is possible to assess the power production of the turbine and its mean

torque, considering a single blade with a chord equal to the sum of the chords of each physical blade.

Even though there are corrections to the model in order to account for static and dynamic stall, blade solidity, strut drag, turbulent wake and three-dimensional geometry, this simple model is only applicable to lightly loaded rotors with low solidity for which flow velocity azimuthal variations are not significant. Performance predictions obtained with this model should be expected to overestimate experimental results.

A more advanced approach is the multiple streamtube model [20] which makes use of a number of adjacent streamtubes completely independent from one another, as shown in Figure 1.6.

Figure 1.6: Scheme for multiple streamtube model [20].

The theoretical principles of this model are the same as the previous one, but it allows to obtain more accurate results since local conditions along the blade path could be analyzed. In fact, it is possible to account for wind shear effects, support struts, blade interference and all the aspects already dealt with in the single streamtube model.

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However, this approach is still valid for lightly loaded turbines and requires longer computational times as an iterative solution is necessary.

The most advanced momentum model is the double-multiple streamtube model (DMST) [20], presented in Figure 1.7. It is an evolution over the previous one as each parallel streamtube is divided into an upstream and downstream half linked together, for each of which an induced velocity is calculated.

Figure 1.7: Scheme for double-multiple streamtube model [20].

Each induced velocity is still considered constant throughout every half streamtube and calculated iteratively. A number of correction factors are available to account for the geometry of the rotor and of the blades and all the above-mentioned secondary effects. The predictions obtained are more accurate than the other models, given its further local approach to the problem, but still unsatisfactory in case of heavily loaded rotors for which power is overestimated.

More accurate results can be obtained by means of computational fluid dynamics methods via approximate resolution of Navier-Stokes equations over a discretized physical domain in space and time. Such approach is gaining growing consensus in the academic community [21] especially given the continually decreasing cost of computational power.

Unlike momentum models, CFD methods require the user to create a computational grid of the entire physical domain to be investigated, not only of the turbine. However, it allows to gain important insights not only on the power harvested by the rotor but also on flow features and complex phenomena which would be very hard to estimate with analytical mathematical models. Accurate velocity components near the blades, the influence of upstream blades over the downstream ones, unsteady aerodynamic

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phenomena and wake characteristics are just a few examples of the greater accuracy reachable by CFD methods [22].

Over time, approved practices have been outlined for consistent and accurate vertical axis turbines simulations [23] which are now widely adopted in academia and led to important works, the most recent and significant of which are reported below.

Wang et al. [24] analyze the performance of a VATT that is subject to displacements in a uniform flow by means of a 3D CFD with dynamic mesh. Namely, a 3-DOF motion made up of rotating, surging and yawing movement in order to better simulate a variable current environment. Results show that variation in hydrodynamic load is mostly due to the surging motion, even if yawing become more pronounced as its frequency increases. Chen et al. [25] compare the power output of a fixed-blade VATT with a variable-pitch one with a sinusoidal pattern. Different amplitudes and frequency are analyzed, showing that it is possible to greatly reduce ripple factors together with an increase in overall rotor efficiency.

Zanforlin [14] utilizes 2D CFD simulations to analyze the mutual influence of three closely placed VATTs. Such configuration leads to an acceleration of the flow due to blockage effects and a beneficial variation in the approaching flow direction, which reduces its expansion in the upstream half of the turbine and shortens the turbine wake. Moreover, the energy output is predicted in the three realistic sites in England over a 6 months period is evaluated.

Bouzaher et al. [26] show that hydrokinetic pressure on a VATT may lead to an alteration of the turbine shape that influences the overall performance of the rotor. Specifically, they employed a deforming, rotational mesh to demonstrate that efficiency increases of up to 35% are achievable.

Le et al. [27] investigate the difference in self-starting capability and torque fluctuation of a straight-bladed and helical-bladed VATTs, showing that the latter design has indeed the desired advantages over the former. Moreover, an increase in efficiency of almost 10% is also achieved.

Even if CFD allows to reach an arbitrarily high level of accuracy, it is important to notice that computational time and requirements may increase considerably [28]. For example, it is currently extremely computationally demanding to simulate and optimize the layout of vertical-axis [29] or horizontal-axis [30–32] turbine farms using 3D CFD simulations essentially because a highly refined computational grid is required close to the blades’ surfaces to properly describe the dynamics of the boundary layer. This would lead to an excessively high cells count that could only be dealt with in a timely manner by high-performance clusters. It should also be noticed that, even for a single finely-discretized turbine, it is possible to reach simulation times of almost one month despite using a powerful cluster and a highly parallelized code [22].

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Therefore, despite CFD appearing as a very powerful tool for VATT analysis, it may not be suitable for VATT design purposes or tidal power site assessment since long duration of each simulation is incompatible with iterative design practices, i.e. improving rotor design over progressive iterations, and the high number of simulations required to locate interesting tidal sites over a wide region.

For a quicker but fairly accurate results, simulations based on momentum models are universally preferred, especially for vertical axis wind turbine research and development. A literature review shows that DMST is commonly adopted to provide rotor performance predictions which match experimental and CFD results within a reasonable margin, especially for optimal working conditions [33]. Moreover, the limited computational resources required allow to conduct parametrical studies to define optimal rotor and blade geometries [34–37].

As for VATTs, the literature available on DMST applications appears limited to the works by Johnston et al. [38], who calibrated a DMST scheme over experimental data to predict the rated power of straight-bladed and helical-bladed rotors, and Lazauskas et al. [39] and Paillard et al. [40] who investigated via different BEM codes the performance of VATTs with passive and active pitching systems respectively.

Therefore, acknowledging the potential of momentum models in this still unexplored field, the objective of this work is to develop a tool based upon the DMST theory that is able to give a quick and accurate, i.e. efficient, evaluation of the power output of a straight-bladed VATT and use it for a preliminary power assessment analysis in a high-interest tidal region. Namely, the tidal power harvestable in the southern Cape Cod region, prior to environmental impact study, will be predicted thanks to dynamic ocean ﬂow data obtained from ocean circulation modeling software.

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Chapter 2. Model description

The mathematical framework of the DMST model is illustrated in the work by Paraschivoiu [41]. Despite his description is referred to vertical axis wind turbines (VAWTs), no modifications are necessary in order to apply it to a VATT.

Let us consider a horizontal slice plane of thickness Δ𝑧 of a VATT of height 𝐻 and radius 𝑅 at an arbitrary vertical position 𝑧. It is rotating at an angular velocity 𝜔 and, therefore, operates at a Tip Speed Ratio 𝑇𝑆𝑅 = 𝜔𝑅/𝑈∞. The flow, of density 𝜌, invests

the turbine with an undisturbed velocity 𝑈∞ aligned to the 𝑥-axis.

2.1. Actuator Disk theory fundamentals

Firstly, the turbine dynamics are described with the Actuator Disc (AD), i.e. momentum (M), theory. The generic scheme of a VATT analyzed with a DMST approach is illustrated in Figure 2.1.

The turbine domain is divided into a series of streamtubes, defined by their azimuthal position 𝜃, of uniform angular width Δ𝜃 = 2𝜋/𝑛𝑟𝑖𝑛𝑔 where 𝑛𝑟𝑖𝑛𝑔 is the total number

of streamtube halves.

Figure 2.1: DMST scheme of VATT (For clarity, the streamtubes do not have the same angular width).

e w

y

pstream Downstream

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Let us consider a couple of linked streamtubes at azimuthal positions 𝜃 and (2𝜋 − 𝜃), of area 𝐴1 and 𝐴2 respectively. Five different flow states can be defined, each one

represented by a different subscript. 1. Undisturbed flow 𝑼∞

In this state, the flow is not affected by the presence of the turbine. It is used as input condition for the upstream actuator disc.

2. Flow near the upstream blade 𝑼𝟏

The flow is slowed down due to the presence of the first actuator disc. 3. Equilibrium flow 𝑼𝒆

DMST theory supposes that an equilibrium condition exists between the two actuator discs, i.e. the perturbed flow behind the first actuator disc is considered as unperturbed flow for the second actuator disc.

4. Flow near the downstream blade 𝑼𝟐

The flow is further slowed down due to the presence of the downstream actuator disc.

5. Wake flow 𝑼𝒘

The flow leaves the turbine and in its perturbed state becoming the wake behind the rotor.

It is now possible to define mass, 𝑥-momentum and energy balances in their integral form over the coupled streamtubes.

𝐴𝑖 = 𝑅Δ𝜃|sin(𝜃)|1 (1) 𝑚̇ = 𝜌𝐴1𝑈1= 𝜌𝐴2𝑈2= 𝑐𝑜𝑛𝑠𝑡. (2) 𝑚̇(𝑈∞− 𝑈𝑒) = 𝐹𝑋,1 (3) 𝑚̇(𝑈𝑒− 𝑈𝑤) = 𝐹𝑋,2 (4) 1 2𝑚̇(𝑈∞ 2 _{− 𝑈} 𝑒2) = 𝐹𝑋,1𝑈1 (5) 1 2𝑚̇(𝑈𝑒 2_{− 𝑈} 𝑤2) = 𝐹𝑋,2𝑈2 (6)

Where 𝐹𝑖 is the modulus of generic thrust force along the 𝑥-axis.

It must be noted that since it is supposed 𝐴1= 𝐴2 , mass balance cannot be respected.

Nevertheless, this is generally accepted since it is not considered to be a large source of error [41]. This statement will be investigated later.

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In order to quantify the slowing down of the flow, inductions factors 𝑎𝑖 (some works

define it a 1 − 𝑎𝑖) are defined as ratio of the reduced velocity 𝑈𝑖 near the blades and the

undisturbed velocity of the considered actuator disk 𝑈𝑢𝑛𝑑.

𝑎𝑖 = 𝑈𝑖 𝑈𝑢𝑛𝑑 → { 𝑎1= 𝑈1 𝑈∞ 𝑎2 = 𝑈2 𝑈𝑒 (7)

Therefore, it can be demonstrated that:

𝑈𝑒= 𝑈∞(2𝑎1− 1) (8)

𝑈𝑤= 𝑈𝑒(2𝑎2− 1) (9)

𝑈2 = 𝑈∞(2𝑎1− 1)𝑎2 (10)

As a result, it is evident that 0.5 ≤ 𝑎𝑖 ≤ 1 since lower values would lead to an inversion

of the flow direction in the wake and higher values would cause an unphysical acceleration of the flow.

The thrust force is non-dimensionalized by employing the thrust force coefficient 𝐶𝑋.

𝐶𝑋,𝑖 = 𝐹𝑋,𝑖 1 2 𝜌𝐴𝑖𝑈𝑢𝑛𝑑2 → { 𝐶𝑋,1= 𝐹𝑋,1 1 2 𝜌𝐴1𝑈∞2 𝐶𝑋,2= 𝐹𝑋,2 1 2 𝜌𝐴2𝑈𝑒2 (11)

Finally, by combining all the previous equations, the following can be proven.

𝐶𝑋,𝑖 = 4𝑎𝑖(1 − 𝑎𝑖) (12)

However, this theoretical result shows little agreement with experimental data for 𝑎𝑖 < 0.6. Therefore, the following empirical correlation, appropriately reversed,

derived by Spera [42], has been employed. It has been designed so that it matches both momentum theory results and rotor tests.

1 − 𝑎𝑖 = 0.27𝐶𝑋,𝑖+ 0.10𝐶𝑋,𝑖3 → 𝐶𝑋,𝑖 = 𝑓𝐴𝐷(𝑎𝑖) (13)

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Figure 2.2: 𝐶𝑋 as a function of a. (Blue) Spera [42] (Red) Actuator disk theory.

Therefore, supposing 𝑎𝑖 known, it is possible to evaluate the thrust force 𝐹𝑋,𝑖 over a

generic streamtube. However, the correct value of 𝑎𝑖 for each streamtube cannot be

defined by AD theory alone.

2.2. Blade Element theory fundamentals

The Blade Element (BE) theory considers the blade divided into a series of independent elements along the spanwise direction. In the case of a VATT, blade elements of height Δ𝑧 are considered. The schematic representation of a BE framework is presented in Figure 2.3.

Figure 2.3: BET scheme of VATT.

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The nomenclature adopted for the flow velocity is analogous to the one adopted in the previous paragraph. The following is a description of a blade in the upstream path but an analogous theory can be developed for a blade in the downstream path.

It is possible to easily define the radial velocity 𝑈𝑛, the tangential velocity 𝑈𝑐 and the

relative velocity 𝑊 at an arbitrary azimuthal position 𝜃.

𝑈𝑛(𝜃) = 𝑈1𝑠𝑖𝑛(𝜃) (14)

𝑈𝑐(𝜃) = 𝜔𝑅 + 𝑈1cos(𝜃) (15)

𝑊(𝜃) = √𝑈𝑐2(𝜃) + 𝑈𝑛2(𝜃) (16)

These quantities can be used to evaluate the local angle of attack 𝛼 and the chord-based Reynolds number 𝑅𝑒𝑐 as follows.

𝛼(𝜃) = tan−1₍𝑈𝑛(𝜃) 𝑈𝑐 (𝜃) ) (17) 𝑅𝑒𝑐(𝜃) = 𝑐𝑊(𝜃) 𝜈 (18)

Where 𝜈 = 𝜇/𝜌 is the kinematic viscosity of the fluid, ratio of the fluid dynamic viscosity 𝜇 over the fluid density 𝜌.

It is now possible to evaluate the static aerodynamic lift and drag coefficients, 𝐶𝐿(𝛼, 𝑅𝑒𝑐) and 𝐶𝐷(𝛼, 𝑅𝑒𝑐), of one blade element from test data available in literature.

In this work, the database by Sheldalh et al. [43] has been employed since widely adopted in literature. The static lift and drag force curves for a NACA 0015 profile are shown in Figure 2.4. 𝐶𝐿= 𝐿 1 2 𝜌𝑐𝑊2 (19) 𝐶𝐷= 𝐷 1 2 𝜌𝑐𝑊2 (20)

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Figure 2.4: Lift and drag coefficient curves as a function of 𝛼 and 𝑅𝑒𝑐 for a NACA 0015 airfoil [43].

With reference to Figure 2.5, these non-dimensional forces can be subsequently projected along the tangential and radial directions, 𝐶𝑇 and 𝐶𝑁, and, finally, along the

direction parallel to the flow to obtain the thrust coefficient 𝐶𝑋 referred to the turbine

blade, i.e. the non-dimensional thrust force exerted by the flow on the blades.

Figure 2.5: BET scheme for forces projection.

𝐶𝑇 = 𝐶𝐿sin(𝛼) − 𝐶𝐷cos(𝛼) (21)

𝐶𝑁 = 𝐶𝐿cos(𝛼) + 𝐶𝐷sin(𝛼) (22)

𝐶𝑋= −[𝐶𝑇cos(𝜃) − 𝐶𝑁sin(𝜃)] (23)

From the definition of thrust coefficient, the thrust force 𝐹𝑋(𝜃) of one blade element at

the position 𝜃 can be evaluated. 𝐶𝑋(𝜃) = 𝐹𝑋(𝜃) 1 2 𝜌𝑐𝑊2(𝜃) → 𝐹𝑋(𝜃) = 1 2𝜌𝑐𝑊 2_(𝜃)𝐶 𝑋(𝜃) (24)

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The thrust force acting on the entire rotor is obtained by averaging 𝐹𝑋(𝜃) over the entire

revolution. 𝐹𝑋 ̅̅̅ =𝑁𝑏 2𝜋∫ 𝐹𝑋(𝜃)𝑑𝜃 2𝜋 0 (25) Once again, in order to evaluate the thrust force, an estimate of the flow velocity near the blades is needed and must be obtained through additional relations.

2.3. Blade Element-Momentum theory fundamentals

The core of the Blade Element-Momentum (BEM) theory is that by employing both the previously mentioned approaches, it is possible to reach an adequate amount of equations to close the problem.

Namely, the aerodynamic load on each streamtube can be evaluated by comparing the thrust forces computed through both theories.

The thrust force acting on a generic streamtube at position 𝜃 evaluated through BE theory is the following.

𝐹𝑋 ̅̅̅ =𝑁𝑏 2𝜋∫ 𝐹𝑋(𝜃)𝑑𝜃 𝜃+Δ𝜃₂ 𝜃−Δ𝜃₂ ~𝑁𝑏 2𝜋𝐹𝑋(𝜃)Δ𝜃 = 𝑁𝑏 𝑛𝑟𝑖𝑛𝑔 𝐹𝑋(𝜃) (26)

The first-order approximation that the thrust force 𝐹𝑋(𝜃) is constant throughout the

streamtube has been adopted. The force 𝐹𝑋(𝜃) can be evaluated through Equations

1-24 by expressing the velocity flow near the blades 𝑈𝑖 with the induction factor 𝑎𝑖.

𝑈𝑖 = 𝑎𝑖𝑈𝑢𝑛𝑑 (27)

As a result, 𝐹𝑋(𝜃) is effectively a function of 𝑎𝑖.

Finally, using Equations 11, 13, 26, the following equation can be derived. 𝑓𝐴𝐷(𝑎𝑖) 1 2𝜌𝐴𝑖𝑈𝑢𝑛𝑑 2 ₌ 𝑁𝑏 𝑛𝑟𝑖𝑛𝑔 𝐹𝑋(𝑎𝑖) (28)

It is now possible to evaluate 𝑎𝑖 for the generic streamtube, thus determining the

aerodynamic load acting on it.

However, it must be noted that given the non-linear nature of the previously described relations, an iterative solution scheme must be employed, i.e. an assumptive value of 𝑎𝑖

is adopted to evaluate both terms in Equation 28 and this procedure is repeated until they are equal. It must be also pointed out that, given the dependence of 𝑎2 from 𝑎1

visible in Equation 10, the upstream part of the rotor must be solved before proceeding to the downstream one.

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Once the correct value of 𝑎𝑖 is identified for the generic streamtube, the power extracted

from the flow by a 2D rotor 𝑃 can be assessed from the knowledge of the tangential force 𝐹𝑇. 𝐹𝑇(𝜃) = 1 2𝜌𝑐𝑊 2_(𝜃)𝐶 𝑇(𝜃) (29) 𝑃(𝜃) = 𝐹𝑇(𝜃)𝜔𝑅 (30) 𝑃 =𝑁𝑏 2𝜋∫ 𝑃(𝜃)𝑑𝜃 2𝜋 0 ~𝑁𝑏 2𝜋 ∑ 𝑃(𝜃𝑖)Δ𝜃 𝑛𝑟𝑖𝑛𝑔 𝑖=1 = 𝑁𝑏 𝑛𝑟𝑖𝑛𝑔 ∑ 𝑃(𝜃𝑖) 𝑛𝑟𝑖𝑛𝑔 𝑖=1 (31)

The power coefficient 𝐶𝑃, i.e., the efficiency of the turbine is defined as follows.

𝐶𝑃 = 𝑃 1 2 𝜌𝐴𝑈∞3 = 𝑃 𝜌𝑅𝑈∞3 (32)

Since for a 2D rotor, 𝐴 = 2𝑅.

In order to apply the previous equations to a 3D straight-bladed rotor of height 𝐻, it is sufficient to consider the vertical discretization already defined by the Blade Element theory. The number of horizontal slices in which the rotor is divided, i.e. turbine planes, is defined by the parameter 𝑛𝑧 , therefore, adopting a uniform distribution:

Δ𝑧 = 𝐻 𝑛𝑧

(33) It is sufficient to apply the same procedure as the one previously described on each turbine plane in order to obtain the total turbine power harvested by the rotor. Supposing a turbine with 𝑧 variable from −𝐻/2 to 𝐻/2:

𝑃 =𝑁𝑏 2𝜋∫ ∫ 𝑃(𝜃, 𝑧)𝑑𝜃𝑑𝑧 2𝜋 0 𝐻/2 −𝐻/2 ~𝑁𝑏Δ𝑧 𝑛𝑟𝑖𝑛𝑔 ∑ ∑ 𝑃(𝜃𝑖, 𝑧𝑘) 𝑛𝑟𝑖𝑛𝑔 𝑖=1 𝑛_𝑧 𝑘=1 (34) 𝐶𝑃= 𝑃 1 2 𝜌𝐴𝑈̅̅̅̅∞ 3= 𝑃 1 2 𝜌(2𝑅𝐻)𝑈̅̅̅̅∞ 3 (35)

Where 𝑈̅̅̅̅ is the depth-averaged undisturbed velocity, which needs to be evaluated ∞

should 𝑈∞ vary along the 𝑧-direction.

𝑈∞ ̅̅̅̅ = 1 𝐻∫ 𝑈∞(𝑧)𝑑𝑧 𝐻/2 −𝐻/2 ~1 𝐻∑ 𝑈∞(𝑧𝑘)Δ𝑧 𝑛𝑧 𝑘=1 = 1 𝑛𝑧 ∑ 𝑈∞(𝑧𝑘) 𝑛𝑧 𝑘=1 (36)

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2.4. Additional aerodynamic phenomena

Despite the DMST model has all the elements necessary to achieve the full description of the aerodynamic load on a VATT, additional phenomena must be accounted for during computation in order to obtain results comparable with experimental and CFD data.

The ones that most heavily influence the rotor performance prediction have been taken into account in our computations via a series of submodels.

2.4.1. Dynamic stall

Compared to horizontal axis turbines, VATTs cannot be considered to work under steady conditions since, during operation, the blades encounter ample and generally fast variation in angle of attack. This may lead to a noticeable variation of the airfoil aerodynamic behavior, i.e. 𝐶𝐿 and 𝐶𝐷 coefficients.

This phenomenon, generally referred to as dynamic stall, is currently under heavy investigation by academic research and, while not entirely understood, is of great importance since it determines a noticeable increase in the aerodynamic load, formation of recirculation zones and vortices on the airfoil that heavily alter the flow dynamics on the blade profile and possibly dangerous unforeseen stresses on the blade materials [44]. For 𝛼 increasing in time, i.e. the upstroke phase when 𝛼̇ > 0, the airfoil may reach angles of attack higher than the angle at which stall occurs in static conditions 𝛼𝑠

without actually stalling. Moreover, a vortex starts to form near the leading edge of the profile, named herein leading edge vortex (LEV), whose intensity grows as 𝛼 continues to increase. This leads to the complete separation of the boundary layer together with an increase in 𝐶𝐿. When a threshold angle of attack is reached, the LEV is convected

towards the trailing ledge determining an additional rise and then sudden drop in 𝐶𝐿,

further increased by the formation and following detachment of a vortex near the trailing edge of the airfoil (trailing edge vortex, TEV). The blade is now completely stalled (deep stall).

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During the downstroke phase, in which 𝛼̇ < 0, the profile persists in its stalled state until reaching a value of 𝛼 generally lower than 𝛼𝑠. Afterwards, the boundary layer

becomes progressively reattached.

The overall effect of these dynamics is a hysteresis of the 𝐶𝐿(𝛼) curve which can deviate

noticeably from the same curve under static conditions. The entity of this variation is related to many factors among which the most important are the frequency of 𝛼 variation and the incoming flow Mach number [46].

A series of empirical and semiempirical models have been developed to mathematically describe dynamic stall such as the one by Leishman et al. [44], later simplified and tailored for vertical axis wind turbine applications by Larsen et al. [47].

The model adopted in this work has been recently developed by Chicchiero [46] to describe the dynamic stall phenomenon of a pitching airfoil, i.e. rotating around a fixed point, described by the following law of motion.

𝛼(𝑡) = 𝐴0+ 𝐴1sin(𝜔𝑡) ( )

Specifically, the model has been developed as an improvement over the one by Larsen et al. to describe an airfoil undergoing deep stall. Its aim is to obtain a good description at a reasonable computational cost and without the need for an extensive calibration. The 𝐶𝐿(𝛼) curve is divided into 4 parts, which represent the various dynamic stall

phases, and are presented in Figure 2.7. • Phase 1 (𝟏 → 𝟐)

The 𝐶𝐿(𝛼) curve follows a linear behavior until the dynamic stall angle 𝛼𝑑𝑠 is

reached.

• Phase 2 (𝟐 → 𝟑)

The LEV cause a further increase of 𝐶𝐿 followed by a drop towards static conditions.

• Phase 3 (𝟑 → 𝟒)

The minimum value of 𝐶𝐿 is reached.

• Phase 4 (𝟒 → 𝟏)

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Figure 2.7: Dynamic stall model phases division [46].

Only the main mathematical details will be dealt with in this work. For further information, the reader is directed to [46]. The generic form of the 𝐶𝐿 term in the various

phases is the following.

𝐶𝐿(𝑡) = 𝐶𝐿,𝑑(𝑡) + 𝐶𝐿,𝑣𝑠(𝑡) (37)

Where the first term represents the base lift coefficient while the second the oscillatory variations due to vortex shedding.

𝐶𝐿,𝑑 is described by different equations for different phases. Except for the first phase,

a first order dynamic is used.

𝐶𝐿,𝑑 = 𝐶𝐿0,𝑠(𝛼) 𝑝ℎ𝑎𝑠𝑒 1

𝐶𝐿,𝑑 = 𝑓1(𝑡, 𝜔1, 𝐶𝐿,𝑠, 𝛼) 𝑝ℎ𝑎𝑠𝑒 2

𝐶𝐿,𝑑 = 𝑓2(𝑡, 𝜔2, 𝐶𝐿,𝑚𝑖𝑛, 𝛼𝑚𝑖𝑛, 𝛼) 𝑝ℎ𝑎𝑠𝑒 3

𝐶𝐿,𝑑 = 𝑓3(𝑡, 𝜔3, 𝐶𝐿,𝑠, 𝛼) 𝑝ℎ𝑎𝑠𝑒 4

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Where 𝐶𝐿0,𝑠 is the linear lift coefficient, 𝐶𝐿,𝑠 is the static lift coefficient, 𝐶𝐿,𝑚𝑖𝑛 is the

minimum lift coefficient reached for 𝛼𝑚𝑖𝑛 and 𝜔𝑖 are free parameters.

On the other hand, 𝐶𝐿,𝑣𝑠 is evaluated as follows.

𝐶𝐿,𝑣𝑠= 0 𝛼̇ > 0 ∧ α < αds

𝐶𝐿,𝑣𝑠= 𝑓4(𝑡, 𝜔4, 𝑆𝑡, 𝑈𝐿𝐸𝑉) 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (39)

Where 𝑆𝑡 is the Strouhal number and 𝑈𝐿𝐸𝑉 is the convection velocity of the LEV.

Overall, this model showed a better agreement with experimental data on deep stall conditions compared to the model by Larsen et al. and therefore has been adopted in this work.

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2.4.2. Tip losses

An airfoil of finite length experiences a loss in lift and drag forces as the blade tip is approached. This is due to the fact that, near the end of the blade, the flow starts crossing over laterally from the pressure side to the suction side of the airfoil due to the pressure gradient that normally induces the lift and drag forces. In the process, the aerodynamic forces drop and the flow acquires a velocity component that is parallel to the blade spanwise direction and perpendicular to the chord line which causes distinctive helical

tip vortices.

Figure 2.8: Schematic representation of tip vortex.

The mathematical aspects of tip losses were first analyzed by Prandtl [48] during his study of propellers with a finite number of blades. He demonstrated that the circulation of a rotor tends to zero exponentially when approaching the blade tip and this result was later embedded by Glauert in his BEM theory for horizontal axis wind turbines by imposing a suitable radial variation of the axial induction factor.

Namely, he defined 𝐹 as the ratio of average induced velocity over the induced velocity at the considered blade position and imposed its variation between 0 and 1 with the following law, represented in Figure 2.9.

𝐹 =2 𝜋cos

−1_{[exp (−}𝑁𝑏(𝑅 − 𝑟)

2𝑟𝑠𝑖𝑛(𝛼))] (40)

Where 𝑅 is the rotor radius, 𝑟 is the radial position, 𝑁𝑏 is the number of blades and 𝛼 is

the angle of attack. A series of more refined approaches have been developed and are illustrated in [49].

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Vertical axis wind and tidal turbines suffer from the same performance degradation near the blade tips, which needs to be accounted for. A literature review showed that no specific relations for tip losses modeling on vertical rotors have been defined, despite their influence on turbine performance has been investigated by Zanforlin et al. [50] and Balduzzi et al. [22].

Figure 2.10: Flow streamlines in the tip region, skin friction lines and z velocity component on the blade suction surface [22].

Most of the DMST codes available in literature, which mainly focus on vertical axis wind turbine analysis, are either 2D [35–37,51], therefore completely neglect this aspect, or try to replicate the Glauert formula on the VATT geometry [33,52].

Two approaches to describe tip losses on a VATT have been adopted in this work. Both are based on the assumption reported by Shen et al. [49] according to which tip losses can be implemented via a modification on the aerodynamic coefficients 𝐶𝐿 and 𝐶𝐷,

namely:

𝐶𝐿𝑇𝐿 = 𝑓𝑇𝐿𝐶𝐿 (41)

𝐶𝐷𝑇𝐿 = 𝑓𝑇𝐿𝐶𝐷 (42)

Where 𝑓𝑇𝐿 is a function that varies from 1 near the center of the turbine to 0 approaching

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The first approach consists of applying the following Glauert-inspired formula, after introducing modifications in order to account for the peculiarities of a vertical axis rotor geometry, analogously to [33,52]. 𝑓𝑇𝐿= 2 𝜋cos −1_{[exp (−}𝑁𝑏(1 − |𝜇|) 2|𝜇||𝑠𝑖𝑛(𝛼)|)] (43)

Where 𝜇 = 𝑧/(𝐻/2) is the non-dimensional vertical position along the blade that equals 0 at the turbine equatorial plane, 1 and -1 at the top and bottom blade tip respectively. A geometrical representation of this formula is presented in Figure 2.11.

Figure 2.11: Tip loss factor Glauert formula applied to VATT.

The second approach is CFD based in that data from 3D CFD simulations of VATT have been used in order to empirically determine 𝑓𝑇𝐿 as the following quantity.

𝑓𝑇𝐿= 1 2𝜋 ∫ 𝐶𝑃𝐶𝐹𝐷(𝜃, 𝜇)𝑑𝜃 2𝜋 0 1 2𝜋 ∫ 𝐶𝑃𝐶𝐹𝐷(𝜃, 0)𝑑𝜃 2𝜋 0 (44) 𝐶_𝑃𝐶𝐹𝐷_{(𝜃, 𝜇) =} 1 Δ𝑡∫ 𝑇(𝜃, 𝜇, 𝑡)𝜔𝑅 1 2 𝜌𝐴𝑈∞3 𝑑𝑡 Δ𝑡 0 (45)

Where Δ𝑡 is the simulation time and 𝑇(𝜃, 𝜇, 𝑡) is the torque force acting a blade during the simulation.

This approach allows the DMST model to accurately mimic the CFD results through

data injection which is going to be faithfully replicated since the DMST model is 2D.

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simulations employed in this work will be illustrated in Section 3.2 while it will now be discussed the general methodology adopted to obtain 𝑓𝑇𝐿.

The CFD solver employed is ANSYS Fluent. It has an integrated torque evaluation feature that allows to log the torque force acting on impermeable domain surfaces, i.e.

wall zones, through time. Therefore, it was sufficient to define a set of separated wall

zones, similar to ribbons, at different 𝜇 along the blade span, as illustrated in Figure 2.12, and to record the torque acting on each ribbon during the turbine rotation during the CFD simulation.

Figure 2.12: Details of ribbons along which the torque has been logged during 3D CFD simulations. A number of simulations for a defined turbine design operating at different 𝑇𝑆𝑅 and 𝐴𝑅 has been carried out and the torque force data through time have been produced for all cases. Knowledge of 𝜔 and the number of time-steps required to cover a revolution allows converting time data to azimuthal positions 𝜃 data so that previous equations can be employed. The result is a database of 𝑓𝑇𝐿 evaluated at discrete values of 𝜇, 𝑇𝑆𝑅 and

𝐴𝑅, slices of which are presented in Figure 2.13.

Figure 2.13: Examples of fTL data for two investigated rotor geometries

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It is now possible to use 3D linear interpolation to obtain 𝑓𝑇𝐿 values at a desired

(𝜇, 𝑇𝑆𝑅, 𝐴𝑅) coordinate, provided that it falls into the analyzed working conditions, i.e. extrapolation is not allowed.

The predictive value of each approach, i.e. ability to reproduce the turbine performance evaluated with CFD methods, will be investigated in Section 3.4.2.

2.4.3. Streamlines curvature

Let us consider the VATT scheme of Figure 2.14. As the flow approaches the rotor, the DMST model predicts a reduction in flow velocity from 𝑈∞ to 𝑈1 along the 𝑥-axis only.

However, the hypothesis of a strictly one-dimensional flow is unrealistic, especially in the upstream part of the turbine where a strong 𝑦-velocity component 𝑉1 may arise,

generally positive for 0° ≤ 𝜃 ≤ 90° and negative for 90° < 𝜃 < 180°. The magnitude of the streamline curvature strongly depends on the rotor geometry and 𝑇𝑆𝑅. This is clearly visible by analyzing any flow field produced via CFD simulations, for example [53] or Figure 2.15 (Refer to Section 3.2 for CFD simulations details and settings).

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Figure 2.15: 2D CFD streamlines plot colorized according to velocity angle from -20° (Blue) to 20° (Red); (Top image) TSR = 2.1 (Bottom image) TSR = 3.6.

Therefore, the presence of 𝑉1 should not be neglected as, for a fixed 𝑈1, it entails a

variation of 𝛼 → 𝛼𝑠𝑐. To the author’s knowledge, this effect has not been accounted for

by any DMST-based code in literature.

To increase the accuracy of the results obtained via DMST modeling, an original submodel to account for streamlines curvature in the upstream part of the rotor has been developed and validated. However, the flow in the downstream path is still considered monodimensional.

The curvature of the streamlines is imposed by assuming that, in each front streamtube, the flow is slowed down while acquiring an arbitrary inclination given by 𝑓𝑠𝑐. In other

words, for a generic streamtube at position 0° ≤ 𝜃 ≤ 180°,the following conditions are imposed.

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30 { √𝑈₁2_{+ 𝑉} 12= 𝑎1𝑈∞ 𝑉1 𝑈1 = 𝑓𝑠𝑐(𝜃) → {𝑈1= 𝑎1𝑈∞ √1 + 𝑓𝑠𝑐2(𝜃) = 𝑏1𝑈∞ 𝑉1= 𝑓𝑠𝑐(𝜃)𝑈1 (46)

Moreover, Equations 14,15 are modified as follows in order to account for the presence of 𝑉1.

𝑈𝑛(𝜃) = 𝑈1𝑠𝑖𝑛(𝜃) − 𝑉1cos (𝜃) (47)

𝑈𝑐(𝜃) = 𝜔𝑅 + 𝑈1cos(𝜃) + 𝑉1sin (𝜃) (48)

The rest of the Blade Element theory formulas are applied normally so that the virtual 𝑦-velocity component 𝑉1 effectively leads to a modification of the calculated angle of

attack 𝛼.

In order for the Actuator disc theory to remain applicable, the limits and physical considerations previously reserved to 𝑎1 must be now applied to 𝑏1. Namely, Equations

7, 8, 13 become the following.

0.5 ≤ 𝑏1≤ 1 → 0.5√1 + 𝑓𝑠𝑐2(𝜃) ≤ 𝑎1≤ √1 + 𝑓𝑠𝑐2(𝜃) (49) 𝐶𝑋,1= 𝑓𝐴𝐷(𝑏1) (50) 𝑈𝑒= (2𝑏1− 1)𝑈∞→ 𝑈𝑒= ( 2𝑎1 √1 + 𝑓𝑠𝑐2(𝜃) − 1) 𝑈∞ (51)

In this work, 𝑓𝑠𝑐(𝜃) has been derived from data obtained via 2D CFD simulations,

whose details are reported in Section 3.2, through the following procedure.

For a chosen rotor, a series of 2D simulations have been run with different 𝑇𝑆𝑅. Time-averaged velocity angle data 𝛽(𝜃, 𝑇𝑆𝑅) = atan (𝑈𝑦/𝑈𝑥) have been exported along the

following parametric curve.

{(𝑥 − 𝑥0)2+ (𝑦 − 𝑦0)2= 𝑅2

0° ≤ 𝜃 ≤ 180°

Where (𝑥0, 𝑦0) is the turbine center, 𝑅 its radius and 𝜃 is defined as usual according to

Figure 2.14.

Afterwards, the data has been processed with MATLAB via its Curve Fitting Toolbox in order to find a 2D function of (𝜃, 𝑇𝑆𝑅) that could best fit the available data through least-square method. MATLAB also provides indicators such as Adjusted R-Square and RMSE to analyze the goodness of the fit, for the description of which the reader is referred to [54].

The definition of 𝑓𝑠𝑐(𝜃) and overall validation of this model against 2D CFD data will

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2.4.4. Streamtube expansion

As previously noted, the DMST model imposes the absence of variation between the areas of an upstream and its respective downstream streamtube which leads to a violation of the mass balance.

In vertical axis wind turbine computations, this effect is generally regarded as negligible, despite models have been developed to account for this phenomenon [41] and have been adopted in literature [52], but no study has been conducted on VATTs to prove the same. As a result, an original approach has been adopted to consider streamtube expansion in VATT DMST computations by evaluating a new equilibrium velocity profile that accounts for mass balance corrections without the need for actually altering the DMST scheme, i.e, number, positions of streamtubes, etc.

Let us consider the scheme of Figure 2.1 and write the mass balance between the state of undisturbed flow and the equilibrium condition, after the first actuator disc, on a single generic streamtube at position 𝜃.

𝜌𝐴∞(𝜃)𝑈∞= 𝜌𝐴𝑒(𝜃)𝑈𝑒(𝜃) (52)

From which we can obtain. 𝐴𝑒(𝜃) = 𝑈∞ 𝑈𝑒(𝜃) 𝐴∞(𝜃) = 𝑅Δ𝜃|sin(𝜃)| 2𝑎1(𝜃) − 1 (53) The overall actuator disc surfaces for the undisturbed and equilibrium flow are:

𝐴∞ = ∫ 𝐴(𝜃)𝑑𝜃 𝜋 0 = ∫ 𝑅|sin (𝜃)|𝑑𝜃 𝜋 0 = 2𝑅 (54) 𝐴𝑒= ∫ 𝐴𝑒(𝜃)𝑑𝜃 𝜋 0 = ∫ 𝑅|sin(𝜃)| 2𝑎1(𝜃) − 1 𝑑𝜃 𝜋 0 (55) Each streamtube can be also identified via its 𝑦-coordinate 𝑦𝑠𝑡(𝜃), which is placed in the middle of the streamtube.

𝑦∞𝑠𝑡(𝜃) = 𝑅 𝑐𝑜𝑠(𝜃) (56)

Therefore, the equilibrium velocity profile can be described by the curve 𝑓1= (𝑦∞𝑠𝑡, 𝑈𝑒).

This model supposes a uniform displacement of the single streamtubes, i.e. a modification of 𝑦∞𝑠𝑡(𝜃) to 𝑦𝑒𝑠𝑡(𝜃), by imposing the following relation.

𝑦𝑒𝑠𝑡(𝜃) 𝑦∞𝑠𝑡(𝜃) = 𝐴𝑒 𝐴∞ → 𝑦𝑒𝑠𝑡(𝜃) = ( 𝐴𝑒 𝐴∞ ) 𝑦∞𝑠𝑡(𝜃) (57)

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The new velocity profile after the streamtube expansion is supposed to be 𝑓2= (𝑦𝑒𝑠𝑡, 𝑈𝑒)

and is used to evaluate the flow velocities at the undisturbed flow positions 𝑦∞𝑠𝑡. In other

words, the corrected equilibrium velocity profile due to streamtube expansion 𝑈𝑒𝑠𝑒(𝜃)

is evaluated as follows.

𝑈𝑒𝑠𝑒(𝜃) = 𝑓2(𝑦∞𝑠𝑡(𝜃)) (58)

It is now possible to use 𝑈𝑒𝑠𝑒(𝜃) to solve the downstream part of the rotor without further

modifications to the computational algorithm.

A graphical example of the velocity profile alteration is presented in Figure 2.16. The blue curve indicates a generic 𝑓1= (𝑦∞𝑠𝑡, 𝑈𝑒) whereas the orange curve represents a

generic 𝑓2= (𝑦𝑒𝑠𝑡, 𝑈𝑒). The white dots on each curve represent 𝑦∞𝑠𝑐(𝜃) and 𝑦𝑒𝑞𝑠𝑐(𝜃)

respectively.

Figure 2.16: Example of streamtube expansion correction.

This streamtube expansion model allows to account for the mass conservation in the downstream part of the VATT, albeit in a simplified way, without the need to change the downstream streamtubes scheme in the DMST theory and in the computational code, unlike other approaches [41].

This model has been further modified by adding a tuning factor 𝑘 in Equation 57. 𝑦𝑒𝑠𝑡(𝜃) = [(1 − 𝑘) + 𝑘 (

𝐴𝑒

𝐴∞

)] 𝑦∞𝑠𝑡(𝜃) (59)

The parameter 𝑘 can vary between 0, when the submodel is deactivated, and 1, when no tuning is introduced.

Tuning procedure and overall model validation are reported in Section 3.3.3.

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2.5. Computational implementation

The DMST performance evaluation algorithm, together with the previously mentioned submodels, has been coded into a MATLAB program.

The code is capable of analyzing a straight-bladed vertical axis rotor of any geometry (radius 𝑅, height 𝐻) at generic 𝑇𝑆𝑅 equipped with any airfoil shape under generic flow conditions, i.e. any flow nature (density 𝜌, viscosity 𝜇) and vertical velocity profile 𝑈∞(𝑧) can be implemented.

Two parameters define the geometrical discretization of the rotor. 𝑛𝑟𝑖𝑛𝑔 represents the

total, i.e. upstream plus downstream, number of streamtubes while 𝑛𝑧 represents the

number of turbine planes in which the rotor is divided. The turbine planes are identified by the index 𝑘 = [1 ÷ 𝑛𝑧] while the streamtubes of the 𝑘-th plane are identified by the

index 𝑖 = [1 ÷ 𝑛𝑟𝑖𝑛𝑔].

Figure 2.17: Flow diagram of MATLAB implementation of VATT performance evaluation

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Let us examine the code flow diagram which is presented in Figure 2.17. Once the input data has been defined, the VATT is divided into planes of uniform height Δ𝑧 = 𝐻/𝑛𝑧

where streamtubes of angular width Δ𝜃 = 2𝜋/𝑛𝑟𝑖𝑛𝑔 are analyzed.

For each 𝑘-th plane, the front streamtubes are solved first, i.e. for 𝑖 going from 0 to 𝑛𝑟𝑖𝑛𝑔/2 the streamtube solver function is called, after the local

𝑇𝑆𝑅𝑘 = 𝜔𝑅/𝑈∞(𝑧𝑘) is evaluated (Figure 2.18). This function is based upon the

MATLAB function fzero [55] which finds the roots of a non-linear function through a combination of bisection, secant, and inverse quadratic interpolation methods.

Figure 2.18: Flow diagram of MATLAB implementation of streamtube solver function “DMST Solve”. In this case, the non-linear function to find the roots of is given by the difference of the thrust force evaluated via Blade Element theory 𝐹_𝑥,𝑖𝐵𝐸𝑇 and Actuator Disc theory 𝐹_𝑥,𝑖𝐴𝐷 for the 𝑖-th streamtube.

Development and validation of a tool for vertical axis tidal turbine performance evaluation in marine environments