Universit`a di Pisa

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Universit`

a di Pisa

Dottorato di Ricerca in Ingegneria Industriale

Curriculum in Ingegneria Aerospaziale Ciclo XXXII

Modeling and simulation of massively separated wakes

Supervisors Author

Prof. Maria Vittoria Salvetti Benedetto Rocchio

Dr. Stefania Zanforlin

Coordinator of the PhD Program Prof. Giovanni Mengali

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Alice laughed: “There’s not use trying” - she said - “one can’t believe impossible things.” “I daresay you haven’t had much practice.”

-said the Queen - “When I was your age, I always did it for half-an hour a day. Why, sometimes I’ve believed as many as six im-possible things before breakfast.”

Lewis Carroll Through the looking-glass, and what Alice found there (1871)

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Abstract

This thesis is aimed at contributing to the numerical modeling and simulation of large turbulent wakes originating from flow separation. Two classes of problems have been investigated, namely the flow around wind/hydro-kinetic turbines and the flow around an elongated cylinder.

As for the first problem, on difficulty in the numerical simulation of wind/hydro-kinetic turbines is related to the presence of the rotating blades. Therefore, ac-tuator models are widely used to avoid the solution of boundary layers around the blades, in which the blades are replaced by the aerodynamic forces they exert on the flow. Among the models available, the most accurate one is the actuator line model (ALM), whose accuracy and reliability is further investigated for the predictions of turbulent separated wakes. To this aim, well resolved Large-Eddy Simulations (LES) of the flow around a NACA0009 airfoil have been performed mimicking the geometry with the immersed boundary method; these simulations are used as a reference for the appraisal of the ALM results. Since the ALM accuracy depends on some free parameters, a systematic investigation of the sen-sitivity to them has been performed, by employing a stochastic approach. A calibration of the model set-up is also carried out for high angle of attack flow conditions. When the blade works in design condition, the angle of attack is low and the aerodynamic moment does not impact too much on the prediction of the wake. However, it may happen that the incoming velocity has some fluctu-ations bringing the blade to increase the angle of attack or to stall. With the calibrated set-up, the ALM gives good predictions of the body wake in terms of velocity and turbulence. However, the vertical position of the wake is shifted compared to the real one. This is due to the lack of the aerodynamic moment in the ALM approach, which at high angles of attack can be significant also for symmetric bodies. Thus, an improvement of the ALM is obtained by embedding the aerodynamic moment through a couple of opposite forces symmetric respect to the aerodynamic center of the airfoil. The modified ALM parameters are, once again, calibrated against the results obtained for the NACA0009 simulations. The proposed ALM modifications enhance the agreement with the reference case.

Vertical axis turbines (VATs) are characterized by additional unsteady phe-nomena, i.e. virtual camber effects and dynamic stall, which should be taken into account to have a good prediction of the their performances. An extensive study has been done to model the so-called dynamic stall under pitching conditions. Thus, it is obvious the importance of a model to predict this phenomenon. In particular, the “deep” stall regime, for a pitching airfoil, is considered. A model is proposed, which has a low implementation and computational complexity, but yet is able to deal with different types of dynamic stall conditions, including those

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characterized by multiple vortex shedding at the airfoil leading edge. The pro-posed model is appraised against an extensive data set of experimental curves for different airfoils and against other models, available in the literature.

Different actuator models, together with the new dynamic stall model other existing models to account for the virtual camber effects and tip-losses, have been successfully implemented in a commercial CFD software. Numerical simulations of fixed and floating vertical turbines have been carried out for validation, showing that the developed tool is promising for numerical simulations of vertical turbines at affordable computational costs and that it is able to capture the effect of a pitching motion on a vertical axis wind turbine power production, which is inter-esting for off-shore oscillating platform applications. Clearly, more applications are needed in the future to complete the assessment of the developed tool.

The second problem, investigated in this thesis, is the flow around a rectangu-lar cylinder, having a chord-to-depth ratio equal to 5, which is the object of the benchmark BARC. From previous works, a large dispersion was observed in the numerical predictions of the flow features on the cylinder side and of some related quantities of interest. Some contributors to the benchmark highlighted a sort of paradox: high fidelity numerical simulations, having limited subgrid-scale dissi-pation and fine grid resolution, mismatch with the experiments, which conversely are in good agreement with less accurate simulations. In particular, by increasing the spanwise grid resolution, the length of the mean bubble on the cylinder side becomes shorter than the one observed in the experiments. We first check that a further refinement of the grid in the horizontal planes has only a minor effect on the characteristics of the flow on the cylinder side and it does not improve the agreement with the experiments. The main contribution to the benchmark by the present thesis is that this paradox can be explained by the fact that the upstream edges in the numerical simulations are perfectly sharp while they have a certain degree of roundness in experiments. This is shown through a sensitivity analysis of the LES results to the curvature radius. It is observed that even a very small rounding edges leads to a significant increase of the length of the mean recirculation region and it enhances the agreement with the experimental data.

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Acknowledgements

Finalmente eccoci qua, a tirare le somme di un percorso che `e stato a tratti tortuoso. Le persone da ringraziare sono tante e probabilmente dimenticher`o qualcuno, ma proviamo ad andare con ordine!

Prima di tutti, vorrei ringraziare la Professoressa Maria Vittoria Salvetti. Beh, cosa dire. Grazie per la fiducia che hai dimostrato nei miei confronti in questi anni e per tutto il sostegno didattico ed emotivo che mi hai dato. Grazie per tutti i “che ca**o fai” o per i “testone” che mi sono molte volte meritato, ma grazie anche per avermi ascoltato e per aver preso in considerazione le mie idee e le mie proposte. Ogni tanto un “bravo” sono riuscito a strappartelo. Mi hai aiutato a crescere nel mio approccio alla ricerca e come persona. Ti meriti, indubbiamente, il primo posto tra i ringraziamenti! Spero che tutto questo continui.

Restando all’interno del nostro laboratorio, vorrei ringraziare Alessandro Mar-iotti. In realt`a dovresti essere tu a ringraziarmi per ogni volta che, in missione, sono diventato il tuo fotografo personale! Scherzi a parte, grazie per tutto l’aiuto che mi hai dato in questi ultimi anni!

Non posso non ringraziare Alessandro Anderlini, che purtroppo non fa più parte di questo laboratorio. In questi tre anni e mezzo, sei stato più un amico che un collega. Sei stato un sostegno nei momenti difficili del mio percorso, mi hai sempre ascoltato e ti sei sempre interessato a me. Ammetto che a volte io sia stato un po’ duro nei tuoi confronti, ma sai quanto bene ti voglio e quanto io apprezzi il tuo modo di essere una persona fantastica. Grazie. Vabbè lo sai che non sono bravo in queste cose, basta!

Rullo di tamburi! Tocca a Claudio, quello “de Derni”. Ormai sono quasi 10 anni che ci sopportiamo a vicenda. Tra poco dobbiamo andare a cena per fes-teggiare, ricordalo! Tra un “io proprio un de capisco” e un “Claudio, ma cavolo!”, abbiamo sempre imparato molto stando insieme e confrontandoci. So molto bene che non ami questo genere di cose, pertanto mi limiter`o a un semplice: grazie per essere ancora qui.

Un grazie al gruppo di meccanica del volo spaziale: Marco, Lorenzo e il dis-perso Andrea (anche detto Signor Lupara). Grazie per i caff`e, i pettegolezzi, le risate e le tutte le nostre chiacchierate che hanno contribuito a rendere questo percorso indimenticabile.

Facciamo un volo oltreoceano e atterriamo a Dallas. Vorrei ringraziare il Professor Stefano Leonardi per avermi dato l’opportunità di continuare a lavorare insieme e per aver sempre creduto nelle mie idee e nelle mie capacità. Sei stato il primo ad avermi mostrato questo mondo e ti ringrazio. Se oggi mi trovo qui, a scrivere questo pezzo di storia personale, è soprattutto per merito tuo.

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ogni momento, nonostante a volte potrei non dimostrarlo come volete. E ancora una volta non lo far`o! Mwahahaha. Davvero, non ci sono parole per dirvi quanto io vi voglia bene e quanto vi devo per tutto quello che avete fatto e continuate a fare per me, ogni momento. Non sono stati anni semplici, ma li abbiamo superati insieme, come sempre. Tutto questo `e anche per voi, spero di avervi reso fieri di me e spero di continuare a farlo.

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Chapter

1

Introduction

Unsteady and turbulent flows and wakes originating from boundary-layer sepa-ration are encountered in many engineering and natural applications. The aim of this thesis is to give some contributions to the numerical modeling and simulation of two classes of problems, both involving large turbulent wakes.

The first considered problem is the flow around wind/hydro-kinetic turbines having horizontal or vertical axes. Harvesting power from the wind is one of the oldest activities humans have carried out over the history (sails, windmill etc.). First wind turbines were constructed by the end of the 19th century in Scotland and United States. They were characterized by rotor diameters of 15 − 25m with an extractable power of 10 − 20kW. Recently, after the loss of interest during the 20th century, the attention on renewable energy resources increased exponentially, with a particular focus on wind energy. We can divide the turbines in two cat-egories: horizontal- and the vertical-axis turbines, see Figure 1.1. Nowadays the horizontal axis turbines (HAT) are the most used in practical applications thanks to their reachable power production (up to 7.5MW) [1]. However, the interest in vertical-axis turbines (VATs) is growing, since they have some advantages over HATs. First, due to their limited number of rotating and static parts [2], the costs of design, manufacturing and maintenance are reduced. The VATs are able to remain operational for an extended range of incoming flow conditions, with a low sensitivity to wind direction. Moreover, it has been shown in [3] that VATs placed in a close proximity may lead to an increase of the performance of the two turbines (packing factor ).

Indeed, to fulfill the huge demand of energy, the turbines are usually clustered in farms (see Figure 1.2), sometimes with a cumulative power production com-parable to the one of a nuclear powerplant [4]. However, the overall production is always lower than the one expected, with losses around 10 − 20% of the nominal total power. The reason behind this setback lies in the interaction between the

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(a) (b)

Figure 1.1: Examples of: (a) horizontal-axis and (b) vertical-axis turbines.

turbines in farms. When the flow passes trough a turbine, a highly turbulent wake is generated. The averaged distance between two in-line turbines is gener-ally lower than the distance required to obtain a perfect recovery of the turbulent wake. As a consequence, the rear (or waked ) turbines are impinged by an inho-mogeneous and less energetic flow, which reduces their performances. Moreover, due to the fluctuations of the incoming turbulent flow, the waked turbines are subjected to fatigue loads, which reduce their operational life.

Figure 1.2: Picture of a wind farm.

It is clear how a good prediction of the dynamics and features of the turbine wakes plays a key-role in the power production of the entire farm. Simplified

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3

tools, which are still attracting for industries thanks to their low computational cost, are the so-called wake models [5–7]. These are semi-empirical models based on several physical simplifications and they do not directly solve the flow around turbines. Due to these simplifications, semi-empirical corrections are needed.

In order to achieve a better knowledge of the wake interactions, a more ac-curate flow description is needed. Computational Fluid Dynamics (CFD) has become a popular research tool in wind energy. CFD simulations are able to capture the three-dimensional, unsteady character of the flow past wind tur-bines. Due to the high Reynolds numbers and the wide range of turbulent scales which characterize these kind of flows, Direct Numerical Simulations (DNS) are still unaffordable. Thus, Large-Eddy Simulations (LES) or Reynolds Averaged Navier-Stokes (RANS) simulations appear to be the only tools that can be used in practice. Independently of the turbulence modeling, one of the main difficulty of the numerical simulation of the flow past wind/hydro-kinetic turbines is given by the presence of the rotating blades. The resolution of the flow around the rotating turbine blades with body fitted codes or with other techniques is still practically difficult ore even impossible due to the extremely high computational costs (particularly for farm configurations). Therefore, simplified models such as the actuator line or actuator disk [8–10] models are usually employed to mimic the presence of the blades, by replacing them with the forces they apply to the flow. To avoid numerical instabilities, the forces are spread by means of Gaus-sian kernels, depending on the adopted model, and they are added to the fluid governing equations.

In the actuator line model (ALM) the aerodynamic forces are distributed in a small region centered into the line representing the blade position. The kernel is typically a Gaussian function, whose standard deviation determines the spreading of the forces. Despite its high computational cost, the ALM is well known to provide the most accurate description of the turbine wake compared to other actuator models. However, the ALM contains some free parameters, whose values change its predicting capability. There are no definitive guidelines to set the free parameters. For numerical stability, it is suggested that the standard deviation of the Gaussian kernel be at least twice the grid spacing [11]. This was confirmed by Mart´ınez et al. [10], who investigated the sensitivity to the spreading to grid-spacing ratio as well as to the grid resolution for a fixed spreading value. Shives and Crawford suggested [12] that the spreading parameter should be related to the local chord length and thus vary along the radial direction for a common wind turbine blade geometry. Recently, Mart`ınez et al. [13] found an optimal ALM set-up for conditions in which the angle of attack of each section of the blade is such that the boundary layer is attached. They also considered a correction to the use

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of the velocity sampled at the actuator line point based as a reference velocity to evaluate the aerodynamic loads of each blade section. In this context, an extension of the set-up proposed by Mart´ınez et al. [13] is presented for high angles of attack, when separation of the boundary layer occurs. The proposed ALM set-up is calibrated against a well resolved LES simulation of the flow around a NACA0009 airfoil at high angle of attack. The calibrated set-up is also appraised in the simulation of the turbulent separated wake of a flat plate. Moreover, an improvement of the ALM is obtained by embedding the aerodynamic moment. The parameters are once again calibrated against the results obtained for the NACA0009 simulation.

The studies on the VATs show how the aerodynamic of these machines is more challenging than the initial expectations and more complicated compared to their direct competitors (HATs). These complications are associated with the orbital motion of the blades. In this conditions, the blades behave differently than if they were in a rectilinear flow. Therefore, even if the cross-section of the turbine blade is a symmetric airfoil, it behaves as a cambered one because the rotation changes the “virtual” angle of attack seen by the airfoil [14]. It follows that the aerodynamic loads acting on the blades during the motion can be very different from the ones predicted by the classical aerodynamic curves. Moreover, under unsteady conditions the lift and drag curves of the airfoil differ from the static ones, due to the so-called “dynamic stall” phenomenon. The dynamic stall is characterized by larger loads on the structures then the static one, by larger recirculation areas and by the shedding of multiples vortices, which can lead to the failure of the structures [15, 16]. On the other hand, dynamic stall occurs at larger values of the lift coefficient than in static conditions, and this may be convenient to obtain larger lift values in practical applications [17]. Most of the current knowledge on the dynamic stall relies on experiments, since numerical simulations are difficult due to the intrinsic unsteadiness coupled with turbulence at high Reynolds numbers (∼ 106). Thus, simplified models giving a prediction of the aerodynamic loads having low computational costs are clearly attractive for practical applications. Currently, several empirical and semi-empirical models are available in literature. One of the most popular is the one proposed by Leishman and Beddoes [15]. Their model evaluates the aerodynamic coefficients by dividing them into two contributions. The first is the one that follows the attached flow behavior, while the second contribution takes into account the separation of the boundary layer, namely the formation and evolution of the leading-edge vortex (LEV) and the trailing-edge separation, which gives losses in the circulation and non-linear forces, evaluated by means of the attachment degree of the boundary layer [18, 19]. The model well predicts a lot of flow

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5

conditions, but the amount of a-priori unknown input parameters makes the model difficult to be used. Sheng et al. [20] proposed modifications to make the model accurate also for low Mach applications. Hansen et al. [21] modified the Leishman and Beddoes model as well, by neglecting compressibility effects and the leading edge separation of the flow. The model proposed by Tran and Petot [22], the so-called ONERA model, evaluates the aerodynamic coefficients by summing two contributions: one that follows a first order differential equation, which describes the behavior for oscillations below the static stall angle, and the other that follows a second-order differential equation, which describes the force for oscillations over the stall angle. One of the simplest empirical model is the one proposed by Gormont [23]. The idea is to evaluate the forces at an angle of attack (αref) which differs from the actual one. Berg [24] proposes a modification of the Gormont model, by averaging the lift and drag coefficients, computed through the Gormont correction and those at steady conditions. Larsen et al. [25] proposed a semi-empirical simple and computationally cheap model, without considering compressibility effects. This model takes into account the presence of a leading-edge vortex, or LEV, whose growth is governed by a simple first order differential equation. Moreover, it is based on the concept of degree of separation of the boundary layer, which modifies the behavior of the lift coefficient in most part of the dynamic cycle. More recently Modarres et al. [26] proposed a second order empirical model, which is able to well capture the second peak of the dynamic lift coefficient due to the shed of a second vortex from the airfoil leading-edge. This thesis proposes a model having a low implementation and computational complexity yet being able to give accurate predictions for different dynamic stall conditions, including dynamic stall with multiple vortex shedding. The model is validated against experimental curves for NACA0012 and for other airfoil geometries, to prove its robustness. The model is also compared to the one proposed by Larsen et al. [25], which has a comparable complexity, and to the more complex model by Modarres et al. [26], which is designed to capture the the secondary dynamic stall peak.

The models to mimic the presence of the rotating blades, together with the proposed dynamic stall model and other existing models to account for additional phenomena characterizing the VATs, have been successfully implemented in a commercial CFD software. Several preliminary numerical simulations of fixed and floating vertical turbines have been carried out for validation.

The second considered problem is a configuration which can be considered as representative of bluff bodies of interest in civil engineering, such as long-span bridge decks or high-rise buildings (see Figure 1.3). More precisely, the flow around an elongated rectangular cylinder, having a chord-to-depth ratio

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equal to 5, is considered. It is the object of the benchmark BARC, launched in 2008 [27], collecting several experimental and numerical results. As for wind turbines, the BARC problem involve massively separated and turbulent flow. Indeed, despite its simple geometry, the BARC configuration is characterized by a quite complex flow. First, the boundary layer shows separation from the upstream edges becoming a separated shear-layer. Then, the shear-layer starts to roll-up forming vortical structures of different size and the flow reattaches along the lateral surface of the cylinder. Furthermore, downstream in the wake the classical vortex shedding occurs. The BARC realizations obtained by the benchmark contributors over the years, up to 2014, were reviewed in Bruno et al. [28]. A large dispersion was observed in the numerical predictions of the flow features and quantities on the cylinder lateral sides. Sensitivity studies carried out by the BARC contributors were not conclusive and in some cases controversial. In particular, in LES perfomed by different groups [29] [30] it was found that increasing grid resolution or decreasing subgrid-scale dissipation leads to a deterioration of the agreement with the experiments with a too short mean recirculation region on the cylinder side. A similar behavior was also observed by Mannini et al. [31] in Detached-Eddy Simulations when decreasing the numerical viscosity.

As for wind-turbine flow, the aim of the present thesis is to give a contribution to assess and improve the reliability of numerical simulations. Indeed, the main contribution is to show that this paradox may be explained by the fact that the upstream edges in the numerical simulations are perfectly sharp while they have a certain degree of roundness in experiments. LES simulations have been carried out with rounded upstream edges for small values of the curvature radius and the impact on the flow features has been investigated both deterministically and stochastically.

The thesis is organized as follows.

In Chapter 2 the modeling and the numerical methodologies used in this thesis for the simulation of the problems under consideration, i.e. wind turbines and the BARC configuration, are recalled. The approach used to model the blades in the wind turbine simulations is also presented together with the stochastic methodology adopted for sensitivity analysis and parameter calibration.

In Chapter 3 the optimal set-up of the ALM proposed by Mart´ınez et al. [13] is extended to high angles of attack, for which the flow may separate along the airfoil. The proposed ALM set-up is calibrated against a well resolved LES sim-ulation of the flow around a NACA0009 airfoil at high angle of attack. The calibrated set-up is also appraised in the simulation of the turbulent separated wake of a flat plate.

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7

Figure 1.3

In Chapter 4 the actuator line model is modified in order to take into account the aerodynamic moment. The aerodynamic moment is embedded in the ALM by using a couple of opposite forces symmetric respect to the aerodynamic center of the airfoil. The same NACA0009 flow, as in the previous Chapter, at high angle of attack has been considered and used to calibrated the new ALM parameters.

In Chapter 5 a model to mimic the presence of the rotating blades for VATs is presented. Then, different phenomena characterizing the aerodynamics of vertical axis turbines are presented and discussed.

In Chapter 6 the attention is focused on modeling the dynamic stall behavior of a pitching airfoil. The deep stall regime is in particular considered. A model is proposed, which has a low implementation and computational complexity, but yet is able to deal with different types of dynamic stall conditions, including those characterized by multiple vortex shedding at the airfoil leading edge.

In Chapter 7 the implementation of the models, described in Chapters 5 and 6, in the commercial CFD software ANSYS Fluent is validated and appraised in the numerical simulations of Vertical Axis Wind Turbines. The function is then used to preliminary study a floating vertical axis turbine.

In Chapter 8 the Benchmark on the Aerodynamics of a Rectangular 5:1 Cylin-der (BARC) is presented. LES having a further grid refinement in the cross-sectional planes of the cylinder is presented and the impact of this refinement on the flow features on the cylinder side is discussed.

In Chapter 9 the effect of rounding of the upstream edges on the flow around the elongated rectangular cylinder is studied. This analysis is aimed to check if

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the manufacturing tolerances of the real geometry of the upstream edges could explain the discrepancy between the high fidelity numerical simulations and the experiments.

Finally Concluding remarks are presented by summing up the contributions given by the present work to the numerical modeling and simulation of two classes of problems.

List of publications

- Rocchio B., Ciri U., Salvetti M. V., Leonardi S.: Appraisal and calibration of the actuator line model for the prediction of turbulent separated wakes. Wind Energy, 1-18 (2020). https://doi.org/10.1002/we.2483;

- Rocchio B., Chicchiero C., Salvetti M. V., Zanforlin S.: A simple model for deep dynamic stall conditions. Wind Energy, 1-24 (2020). https://doi. org/10.1002/we.2463;

- Mariotti A., Rocchio B., Pasqualetto E., Mannini C., Salvetti M. V.: Flow around a 5 : 1 rectangular cylinder: effects of the rounding of the upstream corners. Direct and Large-Eddy Simulation XII ERCOFTAC Series (in press);

- Rocchio B., Ciri U., Salvetti M. V., Leonardi S.: Large Eddy Simulation of a wind farm experiment. Direct and Large-Eddy Simulation XI ERCOFTAC Series, Springer International Publishing Switzerland, (2019);

- Deluca S., Zanforlin S., Rocchio B., Haley P. J., Fourcat C., Mirabito C., Lermusiaux F.J.: Scalable Coupled Ocean and Water Turbine Mod-eling for Assessing Ocean Energy Extraction. OCEANS 2018 MTS/IEEE Charleston, OCEAN, (2019);

- Rocchio B., Deluca S., Salvetti M. V., Zanforlin S.: Development of a BEM-CFD tool for Vertical Axis Wind Turbines based on the Actuator Disk model. Energy Procedia, (2018).

Submitted

- Rocchio B., Mariotti A., Salvetti M. V.: Flow around a 5:1 rectangular cylinder: effects of upstream-edge rounding, Journal of Wind Engineering & Industrial Aerodynamics.

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Chapter

2

Methodology

In this Chapter we briefly recall the modeling and the numerical methodologies used in this thesis for the simulation of the problems under consideration, i.e. wind turbines and the BARC configuration. The approach used to model the blades in the wind turbine simulations is also presented in Section 2.3. Finally, the stochastic methodology adopted for sensitivity analysis and model calibration is described in Section 2.4.

2.1 Large-Eddy Simulations

2.1.1 Governing equations

The simulations of horizontal-axis wind turbines and of the BARC problem have been carried out by using the large-eddy simulation (LES) approach. LES is an approach in which the governing equations are filtered in order to separate the turbulent scales. In these simulations, the larger scales of the motion are directly resolved, while the effect of the smaller scales is represented by closure models. The scales are separated by applying a filter to the flow variables. In the physical space the filter is obtained by the convolution, which is a product in the Fourier space. Thus, any flow variable φ(x, t) can be written as a sum of two contributions:

φ(x, t) = ¯φ(x, t) + φ0(x, t) (2.1) where ¯φ(x, t) is the filtered/resolved part and φ0(x, t) is the residual/modeled part. A generic filter of kernel F (x, t) is characterized by a cut-off scale in space

¯

∆ in the physical space, while in the Fourier space the filter is characterized by a cut-off wave number ¯κ = π/ ¯∆.

By applying the filter to the incompressible Navier-Stokes equations, the LES governing equations can be written, with the Einstein notation, as follows:

∂ ¯ui ∂xi

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∂ ¯ui ∂t + ∂ ¯uiu¯j ∂xj = −∂ ¯p ∂xi + ν ∂ ∂xj ∂ ¯ui ∂xj +∂ ¯uj ∂xi ! +∂τ r ij ∂xj (2.2b) and the problem is unclosed because of τ_ijr, which appears from the filtering of the convective term. It is the Subgrid-Scale (SGS) stress tensor, containing the effects of the unresolved turbulent fluctuations (u0): the closure is achieved by modeling τ_ijr with the use of the so-called SGS models.

The LES simulations performed in this thesis have been carried out by using two different codes, which are described herein.

2.1.2 Subgrid-scale model and numerical discretization in UTD-WS

The UTD-WS is a proprietary code developed at the University of Texas at Dallas. The space discretization method consists of a central second-order finite-difference approximation on a Cartesian orthogonal grid for the spatial derivatives.

The computational grid used with this code is Cartesian and staggered, which means that the velocity is placed on the cell faces, while the pressure is in the center of the cell (Fig. 2.1). A staggered grid is employed because this allows to prevent spurious pressure modes in the solution and while keeping the scheme compact to increase numerical accuracy.

p

u

w

v

x

y

z

Figure 2.1: Staggered grid cell

Time advancement is obtained by a fractional step method which employs the Crank-Nicholson scheme for the linear terms, which are treated implicitly, and a low-storage third order Runge-Kutta scheme for the nonlinear convective term, treated explicitly. It is low-storage because during the evaluation of the solution just two positions of memory are occupied. The large sparse matrix resulting from the implicit treatment of the linear terms is then inverted by an

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2.1 Large-Eddy Simulations 11

approximate factorization technique. A projection method is employed for the pressure correction, in order to satisfy the continuity equation.

Some constraints have to be satisfied to obtain stable simulations. The first constraint is given by the CFL (the Courant Friedrichs and Lewy number [32]). Due to the explicit treatment of the convective term together with the use of the Runge-Kutta scheme, the following condition holds for the UTD-WS code:

CFL = ui∆t ∆x max ≤√3 (2.3)

A lager value of the CFL allows to have larger ∆t, with the same grid size, and to reduce the computational time. If an implicit scheme was used for the convective terms, this limit could be overcomed. However, this is not possible without a linearization because the convective term is non-linear, which usually leads to loss in the accuracy.

Conversely, the linear viscous terms are treated implicitly, to avoid the fol-lowing additional stability condition:

∆t ∆x2_Re ≤

1

2n (2.4)

where n is the number of the problem dimension (2D or 3D). This condition becomes more restrictive than the first one for low Reynolds numbers and 3D flows.

The presence of a generic body is taken into account, in this code, by using the Immersed Boundary Method (IBM) [33]. This technique allows to simulate the flow around complex geometries by employing a simple Cartesian grid, without the need of a body-fitted grid or a curvilinear grid, more complicated to be implemented in a numerical code. The present implementation consists of finding the grid-points that fall inside the body boundaries and in imposing that the velocity in these points is equally identical to the body velocity, Vb.

ui= Vb (2.5)

Since the the body surface does not necessarily coincide with the computational grid, attention must be paid to avoid the body to be represented in a stepwise way. The real body contour is reproduced by using for the discretization of the first and second derivatives of velocity the distance between the velocity points and the boundary of the body (∆x0) rather than using the mesh size (∆x), see Figure 2.2. The IBM is used because it allows reproducing with good accuracy the flow around bodies of arbitrary shape in computationally efficient orthogonal grids, as opposed to body-fitted grids, which lead to large sparse matrices for the numerical solution, which are numerically intensive. An assessment of the

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accuracy of the present IBM method against experimental data can be found in previous works on rough walls ( [33–36]) and also on wind turbines ( [37–39]). For the whole numerical method we refer to [40].

v

Δx'

u

Figure 2.2: Immersed Boundary Method for a 2D body: (•) points inside the body, (×) points outside the body

As for the SGS closure the Smagorinsky model has been employed. Joseph Smagorinsky in 1963 [41] developed the most widely used and simple eddy-viscosity model in the LES approach. The SGS stress tensor is modeled by an eddy-viscosity assumption as follows:

τ_ijr = −2νSGSS¯ij (2.6)

in which νSGS is the SGS viscosity and ¯Sij is the resolved strain rate tensor, which can be written:

¯ Sij = 1 2 s ∂ ¯uj ∂xi + ∂ ¯ui ∂xj (2.7) The subgrid viscosity is assumed to be:

νSGS = (Cs∆)2 q

2 ¯SijS¯ij (2.8)

where ∆ =√3_{∆x∆y∆z. ∆x, ∆y and∆z are the dimensions of the grid cell along}

the directions x,y and z. Cs is an arbitrary constant that changes case by case. Values between 0.1 and 0.12 are commonly used for shear flows, as reported in [43].

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2.1 Large-Eddy Simulations 13

One of the drawbacks of the Smagorinsky model is its behavior near the wall. The subgrid viscosity νSGS has to vanish as a suitable function of the distance from the wall. This is not the case for the Smagorinsky model and, as a consequence, the SGS dissipation is overestimated in that region. A solution to this problem is to correct the behavior of this model in the near-wall region using a damping function, i.e. the Smagorinsky constant Cs∆ is multiplied by f (y) to assure a correct trend in the vicinity of the solid boundary. In the UTD-WS code, the Van Driest damping [42] is employed and the damped constant is given by: (Cs∆)damp= (Cs∆) 1 − exp −uτy Aν ! (2.9) where A=25 and uτ =

√

τw/ρ is the friction velocity near the wall and τw is the wall shear stress.

Another possibility is to use the so-called implicit LES approach, in which no subgrid scale model is employed. In this case, the numerical dissipation is assumed to to act as a SGS model. The UTD-WS code with the above described methodology and with both the approaches to turbulence has been used in the present thesis for the study on the actuator line model.

2.1.3 Subgrid-scale model and numerical discretization in Nek5000

For the simulations of the BARC flow, Nek5000 is adopted. Nek5000 is a mas-sively parallel open-source code for the solution of the incompressible Navier-Stokes equations, and it is based on high-order spectral element method, devel-oped by Fischer et al. [44]. The solution is projected over a polynomial basis of N th-order defined inside each grid element, which is a rectangular or a suitable coordinate mapping of a rectangular. The basis functions inside each element consist of Legendre polynomials of order N for velocity, and generally N − 2 for pressure, where typically N ≥ 6; the different order of the polynomials is employed to avoid spurious modes for the pressure field. When dealing with this method, two different refinements are possible. The so-called p-refinement means that the order of the polynomial base is increased by keeping unchanged the number of spectral elements, this method has a spectral convergence. The other is the h-refinement in which the number of spectral elements is increased and the order of the polynomial base is fixed and the method has a convergence of a high-order finite-element method.

The temporal discretization in Nek5000 is a k-th order splitting method, de-veloped in [45]. Let us consider the momentum equation in which, for the sake of simplicity, the nonlinear convective term is N , the diffusive one is L:

ρ∂u

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The k-th order backward differentiation formula which is used to approximate the time derivative can be written as follows:

k X j=0 bj ∆tu n+1−j_{= −∇p}n+1_{+ L(u}n+1_{) +} k X j=1 ajN (un+1−j) (2.11)

where bj and aj are coefficients, which depend on the discretization order. Here, the third order for the time derivative discretization is used (k = 3). By using the continuity equation ∇ · u = 0 and taking the divergence of Equation (2.11), the pressure field can be obtained by solving the following Poisson problem:

∆pn+1= ∇ · F(un) (2.12)

which depends on the previous time-step. By using the computed pressure and by taking in mind that L(un+1) = µ∆un+1, Equation (2.11) can be rearranged as follows: b0 ∆tu n+1_{− µ∆u}n+1 _{= −∇p}n+1_{+ F(u}n₎ _(2.13) where F(un) = k X j=1 bj ∆tu n+1−j₊ k X j=1 ajN (un+1−j) (2.14)

contains all the explicit terms. Equation (2.13) is the Helmhotz system to solve. As usual, for the stability and efficiency of the numerical simulation, the nonlinear term is solved explicitly (F(un)), i.e. they depend only on the velocity at previous time step, while the linear ones are treated implicitly (at n + 1).

As for the LES formulation, a low-pass explicit filter in the modal space is applied at the end of each step of the Navier-Stokes time integration [46]. The adopted filter is a sharp cut-off one for the modes up the unfiltered ones (kc) and it has a quadratic transfer function for the modes inside kc ≤ k ≤ N . The transfer function of the filter can be written as follows:

   σk= 1 k < kc σk= 1 − w k−kc N −kc 2 kc≤ k ≤ N (2.15)

where w is a tunable weighting parameter. This modal filter provides a dissipation in the highest resolved modes, which is usually interpreted as a “SGS dissipation” [47] [48].

2.2 Reynolds Averaged Navier-Stokes equations

The simulations of the vertical-axis wind turbines were carried out by using the Reynolds-Averaged Navier-Stokes equations and the commercial code ANSYS Fluent.

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2.2 Reynolds Averaged Navier-Stokes equations 15

2.2.1 Governing equations

The Reynolds Averaged Navier-Stokes (RANS) equations govern the evolution of the mean flow quantities. To obtain these equations, the Reynolds decomposition is considered:

u(x, t) = U(x, t) + u0(x, t) (2.16)

where U(x, t) is the mean value of the considered quantity, usually obtained by averaging in time and in the homogeneous directions, and u0(x, t) is its fluctuating part. The difference with the filtered quantities in the LES approach is that when considering the Reynolds decomposition all the turbulent scales are contained in u0(x, t). By using this decomposition in the averaged incompressible Navier-Stokes equations it is possible to write:

∂hUii ∂xi = 0 (2.17a) ∂hUji ∂t + hUii ∂hUji ∂xi = −∂hpi ∂xj + ν∇2hUji + 1 ρ ∂τij ∂xj (2.17b) where τij = −ρhu0iu0ji is called “Reynolds stresses” arising from the nonlinear convective term. Since all the component of the Reynolds stress cannot be directly evaluated, the problem is unclosed. As for the LES approach, a model is needed to determine τij and to obtain a closure.

2.2.2 Turbulence model and numerical discretization in ANSYS Flu-ent

The numerical solution of these governing equations is obtained through the soft-ware ANSYS Fluent, which is a widely used finite-volume commercial softsoft-ware, by means of a second-order accurate space discretization and time advancing. This method divides the computational domain in a finite number of cells or control volumes, over which the equations are integrated. Let us consider a con-vective term of a generic quantity φ, integrated in a generic control volume, which can be written applying the divergence theorem as:

Z Vi ∇ · (ρuφ)dVi= Z Si dSi· (ρuφ) (2.18)

where Si is the contour of Vi. By discretizing this equation: Z Si dSi· (ρuφ) = X fi Si· (ρuf)φf = X fi Fiφf (2.19)

where Fi = Si · (ρuf) the mass flow rate through each interface between two neighbor cells and subscript f indicates that the quantities are evaluated at the cell faces.

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n (U) f f PB PA Cell A Cell B

Figure 2.3: Two-dimensional sketch of two neighbor cells in the finite-volumes method.

As for the spatial discretization, Fi is evaluated at the interface through a centered interpolation, while the diffusion term in the momentum equation (see Equation (2.17b)) is central-differenced and it is second-order accurate. A dif-ferent treatment is needed for the nonlinear convective term, which involves the evaluation of the quantities at the cell faces φf (in Equation (2.19)), interpo-lated from the cell center values. This is reached using an upwind scheme. In particular, among the possible upwind schemes available in ANSYS Fluent, the second-order is chosen, in which all the quantities at cell faces are computed using a multidimensional linear reconstruction approach [49]. In this approach, higher-order accuracy is achieved at cell faces through a Taylor series expansion of the cell-centered solution about the cell centroid. Thus, the generic quantity at the cell face, referred to the upwind one, can be written as follows:

φf,SOU = φ + ∇φ · r (2.20)

where SOU indicates the secondo-order upwind scheme, r is the distance between the upstream cell-centroid and its face centroid, φ and ∇φ are the cell-centered value and its gradient. Equation (2.20) requires the evaluation of ∇φ in each cell, which is computed by means of Green-Gauss node-based gradient evaluation.

The temporal discretization belongs to the Backward-Differentiation Formula family, which is a group of constant time-integration step schemes involving a

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2.3 Rotor modeling 17

certain number of time-steps. The second-order scheme can be written as follows: ∂u

∂t =

3un+1− 4un_{+ ui}n−1

2∆t (2.21)

In this case, the discretization of the transient term involves the time-levels n+1, n and n−1. In order to avoid CFL restrictions typical of the explicit approaches, an implicit formulation is employed herein, i.e the spatial discretization is computed as a function of the variables at level n + 1. By using this approach, the scheme is unconditionally stable with respect the size of the time step. The solution at each time level is obtained solving sequentially the governing equations adopting the SIMPLEC algorithm [50] to deal with the pressure-velocity coupling.

To close the RANS equations a model for the Reynolds stresses is needed. The largely used shear-stress transport SST k − ω model, developed by Menter [51], is employed in this thesis. The SST k − ω model is a blending between the k − ω model [52], in the near-wall region, and the k − model [53], in the free-stream region. The blending function is designed to be one in the region near the wall, by activating the standard k − ω model, and zero away from the surface, which activates the transformed k − model. Moreover, the eddy viscosity is modified to have congruent coefficients in the two models.

2.3 Rotor modeling

Computational Fluid Dynamics (CFD) has become a popular research tool in wind energy. CFD simulations are able to capture the three-dimensional, un-steady character of the flow past wind turbines and predict their performances. However, the solution of the flow around the rotating turbine blades with body fitted codes or immersed boundary methods is still practically unaffordable due to the extremely high computational cost (particularly for multiple machines in wind farms). Therefore, the actuator line or actuator disk models [8–10] are typically used to represent the effect of the blades.

In these models, the aerodynamic interaction between the flow and the blades is modeled by means of body forces added to the fluid governing equations. Using a blade element approach, the blade is abstracted into a line and divided into sections. At each section, the force per unit length depends on the blade geometry (e.g. chord and airfoil shape) and on the local flow conditions (e.g. relative velocity and angle of attack). The angle of attack is determined sampling the flow velocity from the numerical simulation and computing the relative velocity. The airfoil shape and the local angle of attack are used to determine the aerodynamic coefficients (lift and drag) from airfoil look-up tables at each section. The force is then distributed in the computational domain using a spreading kernel.

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Depending on the kernel size and how the aerodynamic loads are distributed, two main possible models are possible: the Actuator Disk model (ADM) [54] and the actuator line one (ALM) [55]. The ADM has been the first model to be employed in LES of wind turbine array and it is widely used in the literature especially for the simulation of large arrays of wind turbines [56,57]. In this model the aerodynamic forces are spread by means of two Gaussian functions: one that distributes the forces on the whole rotor area, i.e. the area swept by the blades, the other which allows to spread the loads along the streamwise direction, giving a sort of rotor thickness. Recently, the Rotating Actuator Disk model (RADM) has been developed [37, 58], which, contrarily to the classic ADM formulation, takes explicitly into accounts the rotation of the blades and it is thus expected to provide a more realistic description of the wake characteristics.

In the ALM the aerodynamic forces are distributed in a small region centered into the line representing the blade position. The spreading kernel is typically a Gaussian function of the distance from the actuator line position. The standard deviation of the Gaussian kernel, ε, determines the width of the spreading. The major drawback of the ALM approach is that, since the forces are spread in a small area, the required time-steps are smaller than the ones needed by the RADM (and, of coarse, by the ADM) [10, 37, 58, 59]. However, the increase of the simulation computational cost is associated with a gain in accuracy in depicting wake details such as, for instance, the tip vortices.

All previous models are not accurate in the representation of the flow in the near-body region, due to the fact that the body is replaced through local forces, although spread over some space regions. Recently, a technique called actuator surface model (ASM) was developed by Zhang [60] and then improved by Shen et al. [61, 62], in which the body forces are first distributed along the chord of the turbine blade. The distribution is obtained through a pre-determined function which depends on the angle of attack and on the geometry of the blade. Compared to the ALM, the new model allows to achieve a more accurate description of the flow structures near the blades and in the tip vortex region, but it requires an a-priori knowledge of the aerodynamic force distribution along the chord. Actuator Line Model

In the ALM each blade is represented as a rotating line, which goes from the hub of the blade up to the tip. The line is divided in a finite number of segments and for each one the geometrical characteristic of the blade, and its cross-section, are known (Figure 2.4). The angle of attack of each segment is given by:

α(r) = tan−1 Ux Uθ− ωr − φ(r) (2.22)

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Figure 2.4: Blade discretization.

where φ(r) is the local twist of the blade, r is the local radius of the blade seg-ment, ω is the rotational speed and Ux and Uθ are the streamwise and azimuthal components of flow velocity impinging on the blade, Uref (see Figure 2.5).

otor Plane T L D Vrel _U (1-a r(1-a') U -r U_x

Figure 2.5: Cross-section of the airfoil.

Once the local angle of attack (α(r)) is known, the aerodynamic coefficients of the airfoil are obtained from available (α, CL) and (α, CD) look-up tables. For the present work we used a NACA0009 because experimental measurements were available in literature [63]. Hence, the aerodynamic forces for each blade segment are evaluated through a two-dimensional approach as follows:

Fi,r= 1 2ρcrV

2

relCFi,r(αr) (2.23)

where i = 1, 2 denotes the lift and drag forces respectively, CFi,r is the related

dimensionless coefficient, ρ is the fluid density, cr is the local chord of the blade, depending on the radial position r. By using the velocities triangle shown in Figure 2.5, it is evaluated as follows:

Vrel= p

U2

x + (Uθ− ωr)2 (2.24)

Since the forces are 2D and all the 3D effects are neglected in this formula-tion, a modified Prandtl correction factor [64] is employed to reproduce the real

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behavior of the blade at the tip.

Figure 2.6: Aerodynamic forces distributions: actuator line model.

The total aerodynamic force F is spread among the grid-points near the ac-tuator line position in order to avoid numerical instabilities, which may occur if a concentrated force is applied to the flow-field. The spreading of the force is given by a Gaussian kernel (η) as follows:

f = R RFη(ε) Aη(ε)dA

(2.25) where ε is the standard deviation of the kernel and it is well-known as the spread-ing parameter and A is the area of the Gaussian kernel (see Figure 2.6). The Gaussian kernel in the ALM is written as follows:

η(ε) = e−(¯rε) 2

(2.26) where ¯r is the distance between a generic grid-point and the actuator line mea-sured in a plane perpendicular to the blade axis. The point at which the velocity is sampled changes the aerodynamic loads of the blades, while the spreading pa-rameter changes the force distribution and strongly influences the evolution of the turbine wake in terms of velocity and turbulence. Therefore, for studying the performances of a single turbine, or of the entire wind farm, a good choice of these parameters is required. Although the spreading parameter is strictly dependent from the grid resolution, there is uncertainty in its value [13] [11]. A complete study of the sensibility of the actuator line model to its parameters is presented in Chapter 3.

Actuator Disk Model

The Actuator Disk Model is another technique used to spread the aerodynamic forces. The total force F, evaluated from Equation (2.23), is now spread on the

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whole rotor area by using the following Gaussian regularization kernel: η(εx, θ) = e x−xc εx 2 e θ− ¯θ 2π/N 2 (2.27) where N is the number of blades, x − xcrepresents the distance between a generic grid-point in streamwise direction and the position of the rotor plane (xc), εx is the spreading parameter determining the thickness of the actuator disk and θ − ¯θ is the relative angular distance from the instantaneous angular position of the blade. In the conventional configuration of a wind turbine there are three blades, thus the forces of each blade are spread in circular sector of angle 2π/3 and over the thickness of the rotor (see Figure 2.7). The rotating actuator disk model (RADM) takes into account the rotation of the blades; thus the position of each circular sector changes in time.

Figure 2.7: Aerodynamic forces distributions: actuator disk model.

Actuator Surface Model

In this model the body forces, at each cross-section of the blade, are evaluated by a two-dimensional approach as in Equation (2.23), but they are now distributed along the airfoil chord as follows:

ˆ

Fi,r= Fi,rFdistr(ˆx) (2.28)

where ˆx is a chordwise coordinate, which goes from 0 to 1 (from the leading edge and the trailing edge), and Fdistr(ˆx) is a semi-empirical formula describing the pressure distribution along the airfoil boundary. In order to obtain an expression for Fdistr(ˆx), a parametric study of the pressure distribution on the airfoil under various conditions is needed. This expression needs to take into account all the relevant parameters, such as the Reynolds number, the angle of attack, the airfoil thickness and camber. Depending on the angle of attack, it is written as follows:

Fdistr(ˆx) =

f (ˆx) R1

0 f (ˆx)dˆx

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in which:                          f (ˆx) = A + B exp −ln 2 ln2(1+(ˆx−c)(E2−1)/(DE)) ln2E |α| fT |α| ≤ 10◦ f (ˆx) = A + B exp −ln 2 ln2(1+(ˆx−c)(E2−1)/(DE)) ln2_E fT 10◦< |α| < 25◦ f (ˆx) =hA+C ˆ_{1+B ˆ}_xx0.50.5_{+D ˆ}+E ˆ_{x+F ˆ}x+Gˆ_xx1.51.5 i |α| ≥ 25◦ (2.30) where the coefficients A, B, C, D, E, F and G are given in [60], and fT is a function of the airfoil thickness and camber. These are curve-fitted using the TableCurve2D [65] software and they are based on the pressure distribution ob-tained from XFOIL code (see [60–62]).

Despite the aerodynamic forces are distributed along the chord of the blade cross-section, numerical instability and wiggles may arise in the simulation [62]. As done for the other models, to overcome these difficulties, the total aerodynamic force ˆF is spread as follows:

f = Z c

0 ˆ

F(l) · η(ε)(|x − l|)dl (2.31)

where the regularization kernel is the following normalized Gaussian distribution:

η(ε) = 1 ε2_π3/2e −₍r ε) 2 (2.32)

2.4 Stochastic approach

To carry out systematic sensitivity analysis and model calibration, a stochas-tic approach has been used in the present thesis based on generalized Polyno-mial Chaos. The generalized Polynomial Chaos expansion is a non-intrusive interpolant method based on the spectral projection of a given random pro-cess over a known orthogonal polynomial base [66]. More in detail, let us con-sider a generic quantity of interest R(ξ(γ)), with γ being a random event and ξ(γ) = [ξ1, ξ2, ..., ξM] the M −dimensional vector consisting of the independent random variables in the event space, Γ. The gPC expansion of R can be written

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2.4 Stochastic approach 23 as follow: R(γ) = a0B0+ + M X i1=1 ai1B1(ξi1) + + M X i1=1 i1 X i2=1 ai1i2B2(ξi1, ξi2) + + M X i1=1 i1 X i2=1 i2 X i3=1 ai1i2i3B3(ξi1, ξi2, ξi3) + ... (2.33)

where Bk is the polynomial of maximum order k containing the interaction of a set of k parameters among M . This expression is written using a order-based notation and it can be simplified by using a term-based indexing, as follow:

R(γ) = +∞ X

j=0

αjΦj(ξ(γ)) (2.34)

where a bijective relation exists between ai1,i2,...,in and αj. Each of the Φj(ξ(γ))

is a multivariate polynomial of index j, which involves products of the one-dimensional polynomials, and αj is the corresponding Galerkin projection co-efficient. In addition to its simple notation, the term-based indexing allows to easily compute the Galerkin coefficient by taking advantage of the orthogonality property of the polynomials basis (Φj). The j coefficient can be written as follow:

αj =

hR, Φ_ji hΦj, Φji

(2.35) where h·, ·i denotes the usual L2 scalar product involving a weight function ρ(ξ) depending on the chosen polynomial family.

hf, gi = Z

Γ

f (ξ)g(ξ)ρ(ξ)dξ (2.36)

The weighting function is the product of the weight functions associated with the chosen polynomial family for each uncertain parameter:

ρ(ξ) = M Y

i=1

ρi(ξi) (2.37)

As we can see from Equation (2.34), to obtain the exact expansion for the quantity of interest R(γ) an infinite number of computation is required. From a practical point of view, it is necessary to truncate the expansion up to a certain order T such that the spectral decomposition can be truncated to a finite number

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of terms. Two truncation methods are commonly used: the total-order and the tensor-product expansion. If the total-oder expansion is employed, the truncated gPC expansion includes a complete base of M −dimensional polynomials up to a specified fixed total-order P and the truncation order is given by:

T = (M + P )!

M !P ! − 1 (2.38)

where P is the maximum polynomial degree for each parameter and, again, M is the polynomial dimension. Otherwise, in the tensor-product expansion the polynomial order bound is applied to each parameter polynomial base. The tensor-product expansion contains all the multi-dimensional polynomials up to order P and some of the polynomials of higher order up to order M P and the truncation order is:

T = M Y

i=1

(P + 1) − 1 (2.39)

No matter what truncation rule is employed, the truncated gPC expansion can be written as: RgP C(γ) = T X j=0 αjΦj(ξ(γ)) (2.40)

in which the evaluation of T + 1 terms is required. In the present work, each stochastic analysis was carried out with a gPC truncated with a tensor-product approach.

The choice of the polynomial base, over which the random process is pro-jected, plays a key-role in this analysis and it depends on the Probability Density Functions (PDFs) associated with the random variables. If the PDF can be approximated with one of the classical continuous distribution, reported in Ta-ble 2.1, the “optimal” polynomial base can be derived from the family of hyper-geometric orthogonal polynomials, known as the Askey scheme [67]. By following this scheme, the optimal polynomial family is orthogonal respect the weighting function ρ(ξ), which is equal, or at most different for a multiplicative factor, to the PDF of the random variable. The optimal polynomial family is that giving the highest rate of convergence of the Galerkin projection, i.e. the best accuracy for a given truncation order.

The integrals in the scalar products are computed numerically, in this the-sis, by using Gaussian quadrature with (P + 1)M quadrature points, which are defined according to the chosen polynomial family. For each quadrature point a deterministic computation of the quantity of interest R(ω) must be carried out.

Once each projection coefficient αj of Equation (2.40) has been calculated, it is interesting to quantify the statistical moments of the quantity of interest RgP C. When the orthonormal polynomial bases are used, the first and second

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2.4 Stochastic approach 25

Table 2.1: Optimal polynomial base for some classical continuous probability dis-tribution [67]

Distribution PDF Polynomial Weight Support

Normal √1 2e −x2 2 Hermite e− x2 2 [−∞; +∞] Uniform 1₂ Legendre 1 [−1; +1] Beta ₂α+β+1(1−x)_B(α+1,β+1)α(1+x)β Jacobi (1 − x) α_{(1 + x)}β _{[−1; +1]} Exponential e−x Laguerre e−x [0; +∞] Gamma β−αxα−1e − x β Γ(α) Generalized Laguerre xαe−x [0; +∞]

order statistic, i.e. the mean value RgP C and the variance σgP C2 , can be evaluated through simple relations:

RgP C = α0 (2.41) σ2_{gP C} = T X j=1 α2_j (2.42)

2.4.1 Partial sensitivity analysis

When a generic quantity of interest R depends on a certain number or random parameters, not all of them have the same impact on the result variability. Thus, it could be interesting to have a way to estimate the effect of each input variables or groups of variables on the output.

A methodology to study this effect was proposed by Sobol in [68]. Let us consider a generic integrable function f (x) defined in Ω which can be written as:

f (x) = f0+ M X s=1 M X i1<...<is fi1...is(xi1, ..., xis) (2.43)

where 1 < xi1 < ... < xis ≤ M and (xi1, ..., xis) is a unique subset of the input

variables. This expression is called ANOVA-representation if: Z 1

0

fi1,..is(xi1, ..., xis)dxk = 0 k = i1, ..., is (2.44)

which indicates that all the terms in Equation (2.43) are orthogonal. The ANOVA-representation can be re-written in a simple way as follows:

f (x) = f0+ M X i=1 fi(xi) + M X i<j fij(xi, xj) + ... + f12...n(x1, x2, ...xn) (2.45)

Assuming that the function f (x) is also square integrable, then all the terms

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term of the ANOVA-representation, the variance σ2 f of f (x) can be written as follows: σ2_f = Z f (x)2− f₀2= M X s=1 M X i1<...<is Z f_i2₁_...i_sdxi1...dxis (2.46) where eachR f2

i1...isdxi1...dxis is called “partial variance” and the following relation

holds: σ_f2 = M X s=1 M X i1<...<is σ_i2₁_...i_s (2.47)

Thus, a single partial variance σ_i2

1...is is the contribution of a single term of the

ANOVA-representation to the total variance σ2

f; the higher a partial variance is, the stronger is the contribution of the variable subset to which the ANOVA term is related.

The so-called “Sensitivity or Sobol indices” are defined as: Si1...is = σ_i2 1...is σ2 f (2.48) which are nonnegative and their sum is:

M X s=1 M X i1<...<is Si1...is = 1 (2.49)

As shown in Equation (2.48), each Sobol index shows how much a single input variable, or a subset of variables, contributes to the total variance σ_f2 of the considered function f (x). Moreover, Si1...is = 0 indicates that the function is not

sensitive at all to the considered group of variable, while Si1...is = 1 means that

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Part I

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Chapter

3

Appraisal and calibration of the Actuator Line

Model for the prediction of turbulent separated

wakes

The aim of this Chapter is to further investigate the accuracy and the reli-ability of the actuator line model (ALM) predictions for turbulent separated wakes. Large-Eddy Simulations (LES) of the flow around a NACA0009 airfoil are performed mimicking the geometry with the immersed boundary method. Results are validated against experiments and used to assess the accuracy of the ALM predictions for the same airfoil, with different values of the spreading parameter and of the reference velocity and for two values of the angle of at-tack. It is found that the ALM set-up recently derived from linearized inviscid analysis leads to accurate results for the lower angle of attack, while at the higher one, for which a significant separation of the boundary layer occurs, the ALM requires a different set of model parameters. This calls for a systematic investigation of the sensitivity to the ALM parameters for separated flows, which is carried out herein through a stochastic approach allowing continuous response surfaces to be obtained in the parameter space. The ALM parame-ters are calibrated against the results obtained with the immersed boundaries. With the calibrated model parameters, the ALM gives good predictions of the velocity and turbulent kinetic energy in the far wake. Finally, the proposed model parameters are used to predict the flow past a different geometry, a flat plate, at high angle of attack. The accuracy of the prediction of the far wake is again good, showing the robustness of the identified set-up.

3.1 Introduction

As shown in Section 2.3, the spreading parameter and on the reference velocity used to determine the angle of attack and the aerodynamic forces are two key

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parameters of the ALM. However, there are no definitive guidelines to set these input parameters. For numerical stability, it is suggested that the spreading pa-rameter be at least twice the grid spacing, in order to avoid singularities due to a pointwise force [11]. This was confirmed by Mart´ınez et al. [10], who investi-gated the sensitivity to the spreading to grid-spacing ratio as well as to the grid resolution for a fixed spreading value. Shives and Crawford suggested [12] that the spreading parameter should be related to the local chord length (ε/c ' 1/8– 1/4) and thus vary along the radial direction for a common wind turbine blade geometry. An analogous approach was also pursued by Jha et al. [69] and Jha and Schmitz [70] defining an equivalent elliptic blade planform.

Similarly, there is not a general consensus for the reference velocity. The reference velocity is usually sampled at the center of the spreading region (that is at the actuator line point), although the airfoil aerodynamic coefficients are classically defined based on the freestream velocity. This may introduce the need for a correction of the sampled velocity. Other approaches have also been proposed consisting in averaging the velocity sampled at multiple points [71] or averaging the velocity in space using the spreading kernel as a weight [72, 73]. Other possible methods are presented in [74–76]. Some of the existing approaches have been compared by Merabet and Laurendeau in [77], focusing on how they are affected by some parameters, such as the grid resolution or the employed solver.

Recently, Mart`ınez et al. [13] carried out analytical computations using the linearized Euler equations. They found the optimal set-up which minimizes the error between the potential flow around an airfoil and that generated by a body force as in an ALM. They also considered a correction to the reference velocity sampled at the actuator line point based on the drag coefficient. Two differ-ent spreading parameters were obtained related to the airfoil chord and do not strongly depend on the angle of attack in the attached-flow range.

The aim of this Chapter is to extend the results of Mart´ınez et al. [13] to higher angle of attack, where the flow may separate along the airfoil and hence the use of a potential flow as a target solution is no more appropriate. These conditions may occur on wind turbine blades when operating in off-design condi-tions or when the incoming flow is highly turbulent, for example due to impinging wakes from upstream turbines in wind farms. We use a similar set-up as the one of Mart´ınez et al. [13], i.e. we consider a stationary two-dimensional wing, and perform large-eddy simulations to compare the flow around the actual body with the flow generated by the ALM approach. Since the computational cost of each single LES is very large, a systematic deterministic analysis of the sensitivity to the ALM parameters, namely spreading and reference velocity, is difficult.

Universit`a di Pisa

Universit`

a di Pisa

Modeling and simulation of massively separated wakes

Abstract

Acknowledgements

Contents

Chapter

1

Introduction

Chapter

2

Methodology

2.1

Large-Eddy Simulations

p

u

w

v

x

y

z

v

Δx'

u

2.2

Reynolds Averaged Navier-Stokes equations

2.3

Rotor modeling

2.4

Stochastic approach

Part I

Chapter

3

Appraisal and calibration of the Actuator Line

Model for the prediction of turbulent separated

wakes

3.1

Introduction