SCUOLA DI INGEGNERIA
Dipartimento di Ingegneria dell’Energia, ei Sistemi, del
Territorio e delle Costruzioni
CORSO DI LAUREA MAGISTRALE IN INGEGNERIA ENERGETICA
TESI DI LAUREA MAGISTRALE
Improvement of an Actuator Line Model for Vertical Axis
Tidal Turbine and sensitivity to variations of blade shape,
solidity and operating conditions
Relatori Candidato
Stefania Zanforlin Marika Francesconi
Benedetto Rocchio
Anno accademico 2018/2019
Appello di Laurea 06 Giugno 2019
This thesis deals with the calibration of an Actuator Line Model, applied to the geometry of a three-straight-bladed Darrieus Vertical Axis Tidal Turbine developed by the Italian start-up company WindCity.
This model, introduced in the solver ANSYS Fluent thanks to a User Defined Function developed by researchers of University of Pisa, contains two main sub-routines for Vertical Axis Turbines: virtual camber and dynamic stall. In this thesis an improvement has been made on virtual camber model by implementing the conformal mapping technique in the code, which should be more accurate and should not need ad hoc calibration even changing blade profile, turbine geometry or operating conditions. 2-D transient CFD simulations have been performed at different operating conditions and with two blade profiles (NACA0018 and NACA4418) in order to use them as reference for the calibration of the ALM, since at the moment no experimental data are available for this turbine.
ALM parameters are calibrated for a reference case for both symmetrical and cambered blades and then the response of the model to changes in operating conditions has been investigated. Modelled blades and resolved blades results are compared in terms of mean power coefficient, single-blade power coefficient and single-blade relative angle of attack as functions of the blade azimuthal position θ and through visualization of some variable fields (X-Velocity, Static Pressure, Vorticity, Turbulent Kinetic Energy), in order to verify the sensitivity of the model and in particular the goodness of the sub-models introduced.
Table of Contents
1 Introduction 1
2 State of the art and general principles 3
2.1 Hydrokinetic energy 3
2.2 Hydrokinetic turbines: HATTs and VATTs. 5
3 Models 8
3.1 DMST 9
3.2 ACM and ALM 11
3.2.1 Virtual Camber 15
3.2.2 Dynamic Stall 17
3.2.3 Performance curves of the profiles 19
4 Geometry and Computational Domain 21
4.1 Sensitivity to grid refinement. 25
5 Computational settings 30
6 Results 34
6.1 NACA0018 38
Reference case: U∞= 1.75 m/s , TSR= 1.85 38
Virtual Camber Improvement 43
U∞= 1.75 m/s , TSR= 1.4 46 U∞= 1.75 m/s , TSR= 2.2 47 U∞= 0.5 m/s , TSR= 1.85 49 U∞= 0.8 m/s , TSR= 1.85 50 U∞,rated= 2.547 m/s , TSR= 1.85 52 U∞= 3.25 m/s , TSR= 1.45 53 U∞= 4 m/s , TSR= 1.18 54 Maximum solidity: R= 0.40 m 56
U∞= 1.75 m/s , TSR= 1.4 64 U∞= 1.75 m/s , TSR= 2.2 65 U∞= 0.5 m/s , TSR= 1.85 67 U∞= 0.8 m/s , TSR= 1.85 68 U∞= 3.25 m/s , TSR= 1.45 69 U∞= 4 m/s , TSR= 1.178 71
6.3 Resume of global performance 72
7 Conclusions and future research 74
1 Introduction
1
1 Introduction
In the recent decades worldwide focus has moved to environmental issues in addition to resources’ availability. The fast growth of energy demand, caused by a growth of population and a general improvement of welfare and of industrialisation , has also led to an increase in the awareness of the consequences that energy production chain has on the environment. Some international organisations, such as IEA (International Energy Agency), IPCC (Intergovernmental Panel on Climate Change), monitor global consumption of primary energy and resources exploited in order to produce this energy; through these data such organisations predict trends for the future in terms of produced and consumed energy, availability of resources and environmental impact. Such reports (for example [1][2][3][4]) have highlighted in the past years that the previous trend was leading to irreparable global warming and pollution levels.
This major awareness makes governments all over the world take measures and promote different economic and technological policies [5], aimed at containing global warming and environmental pollution and exploitation.
The consequence is a deep change in the mix of resources employed [4], in the logic and priorities of projects and on research subjects; in particular renewable sources should be preferred in order to limit carbon dioxide emissions in atmosphere.
Among renewable energies, this thesis work deals with the employment of hydrokinetic turbines in order to use and transform tidal energy. This technology has only recently developed and is therefore subject for research, but is very promising because it can already benefit from the good technical know-how available from wind energy and at the same time it has one important advantage, i.e. the predictability of tides both in magnitude and in direction with respect to wind ([6][7][8][9][10]) .
Anyway, there is always an aleatory component when dealing with renewable energies and with a system that cannot be controlled: the nature. In such a system the different devices hardly never work at their nominal working conditions, so the prediction of the response of technology to these changes is extremely important.
2 For this aim, help comes from numerical simulation models, which have to be as accurate and fast as possible. Many models exist, belonging to different levels of complexity, and among them is the class of so-called “hybrid” models, the union of Blade Element Momentum (BEM) method and of Computational Fluid Dynamics (CFD). The model described and used in this thesis belongs to the class of BEM-CFD methods and it is here compared with “full CFD” as regards computational time requirement and accuracy of the results.
These numerical models are applied in this thesis on the geometry of an innovative vertical axis tidal turbine developed by the Italian start-up company WindCity (Rovereto, Trento, Italy) [11]. It is a three-straight-bladed vertical-axis Darrieus-type turbine and while the hydrokinetic version of this technology is at stage-1, the wind version has reached a greater Technology Readiness Level (TRL). The full scale floating hydrokinetic version is under construction and will be tested in September 2019 at the Ifremer laboratory (Boulogne-sur-Mer, France) in the frame of the European project Marinet2. The upcoming experimental campaign will allow the TRL to be updated from 3 to 4 (stage-2).
For this reason there are not any experimental data on the studied turbine yet, to be used as reference; therefore CFD simulations have been carried out in order to have some reference for calibration of the BEM-CFD model. It is important to underline that CFD results are affected by numerical errors that depend on the adopted spatial discretization grid, on turbulence model adopted and on numerical parameters set, so they need to be checked in future with experimental data, when they will be available. Once the hybrid model and its subroutines are calibrated to this reference, the response of the model to possible changes in operating conditions and in geometry are investigated. In particular, tests with the modelled blades are conducted with different airfoil shape, different radius (thus solidity), different inflow velocity and rotational speed.
2 State of the art and general principles
3
2 State of the art and general principles
This chapter summarises the general features of hydrokinetic source under an energetic point of view and the present possible ways of extracting power from it. The state of the art of the devices, with their advantages and disadvantages are here also presented.
2.1 Hydrokinetic energy
Hydrokinetic turbines use the kinetic energy contained in river and marine currents. Since only kinetic energy is used, this technology share the same principles of wind energy extraction and therefore the developed devices are very similar to the existent wind turbines.
There are of course some differences, connected to the presence of water instead of air. They are here briefly summarised:
- Mean velocity of river currents is about 2-4 m/s and marine currents have even lower velocities, while mean velocities of wind are about 11-13 m/s;
- Density of water is 835 times the density of air; - Viscosity of water is higher;
- Aggressive and difficult environment leads faster to damage of the components; - Cavitation;
- More turbulent environment, due to irregular seabed and eventually presence of obstacles;
- Variation of velocity with depth and influence of the presence and shape of the seabed/riverbed.
As a consequence, the available energy per square meter of frontal area of the turbine (A=1 m2)
𝑃 =1 2𝜌𝐴𝑈
4 is about 10 times higher than the energy contained in wind. This is of course a positive aspect but has influence on the size of the devices: these devices are subjected to higher loads and fatigue, so their cost is affordable only for small scales.
Because of small size and modest velocity, the use of diffusers around these turbines is widely recommended and studied [12].
In addition to that, underwater installation makes maintenance very difficult and expensive, because the device is hard to reach and could eventually need maintenance campaigns more frequently because of the aggressivity of marine environment.
On the other hand, such devices have of course much lower visual impact than offshore wind towers and lower environmental impact than hydroelectric power stations. Another fundamental aspect of the resource is the variability. This is shared by all renewable energies and nowadays the predictability of power production by these resources is one of the central issues in the global energy scenario.
Speaking about rivers, although their turbulence levels are quite high and influence the turbine’s wake, they have the advantage of unidirectionality, which deeply simplifies the process and the project of the device itself, as it will be better explained in the next paragraph. Despite that, the flow velocity is influenced by rains and seasonal cycle, so it has a stochastic variation component.
Tidal energy instead has a precise cyclical behaviour, which can be well predicted: tides are generated by gravitational interactions between Earth, Moon and, to lower degree, Sun. Their relative positions affect both magnitude and duration of a cycle; the amplitude of the tide has a double sinusoid trend, one with a period of about 12 hours and the other with a period of about 15 days.
2 State of the art and general principles
5
Figure 1 Sinusoidal behaviour of velocity due to tides
Water velocity inverts about every 6 hours but it may also change its direction because of the shape of the shoreline. The devices that use this kind of energy should be able to follow this changes of direction in order to maximize power extraction and production.
2.2 Hydrokinetic turbines: HATTs and VATTs.
Hydrokinetic turbines descend from wind technology because they work following the same principle: kinetic energy of the flow is converted into torque on the shaft of an electric generator by the blades.
There are two main families: Horizontal Axis (HATT) and Vertical Axis (VATT) Tidal Turbines.
The Horizontal Axis Wind Turbine technology has been developed for decades, so it is more mature than that of Vertical Axis Turbines, which was almost abandoned for a long time because of problems of efficiency and self-starting.
The shaft of HATTs is parallel to the incoming flow; they have higher efficiency than VATTs and are self-starting, which are of course fundamental features in such devices, but their optimal working condition is in flow aligned with the axis. As a consequence they are usually preferred whenever the direction of the flow is constant, otherwise they need sophisticated active yaw devices, which are expensive and decrease the net power production. For this reason, HATTs are a good option when they are installed in rivers, for example.
6 As negative aspects, VATTs have low efficiency and fluctuating torque because each blade during its down-stream path is invested cyclically by the wake of the other blades and this produces counter-torque. As a consequence, they are affected by low starting-torque and self-starting problems.
Much research at the moment is focused on self-starting issue of Vertical Axis Turbines and in this scenario the Italian start-up company WindCity conceived and developed an innovative vertical axis turbine for wind energy harvesting with passively variable blade pitch and turbine diameter.
Anyway, VATTs are particularly suitable for marine installations because they can operate with every direction of the incoming current, so they don’t need any yaw device during a tidal cycle, in which water velocity inverts and changes direction. This implicates also simplicity of construction and cost-effectiveness.
One of the most important issues that have to be considered when working in water is the phenomenon of cavitation. It is promoted by high tip velocities, or equivalently high tip-speed-ratios (TSR) and by little depth of installation of the device under the free surface. Unfortunately, this is also where the velocity of the flow and consequently the achievable power are the greatest because the no-slip condition of the seabed has almost extinguished. Since HATTs usually work with higher TSRs, VATTs can possibly be installed nearer to the free surface (always making sure that cavitation is avoided), with the just mentioned advantages.
Another difference between Horizontal Axis and Vertical Axis Turbines is that the latter show good response also to vertical component of incoming velocity and in marine environment, affected by waves, this feature is far from negligible.
The use of VATT is also useful because it allows to install the electrical motor and the gearbox on a floating platform, reducing costs related to additional electrical security components for underwater operation and facilitating maintenance operations.
Vertical Axis Tidal Turbines can be “lift driven” or Darrieus, or “drag driven” or Savonius. “Lift driven” or “drag driven” means that in the vector composition of both drag and lift forces, respectively lift or drag give the greater tangential contribution.
Darrieus turbines have aerofoil-shaped blades, symmetric or cambered. The blades can be furthermore straight or curved as in the “troposkien” turbine.
2 State of the art and general principles
7
Figure 2 Types of Vertical Axis Tidal Turbines: (a) troposkien Darrieus, (b) straight-bladed Darrieus, (c) Savonius
Straight-bladed turbines have a simpler geometry and this allows bidimensional analysis, except for tip losses. In facts, a 2-D simulation represents ideally what happens on the midplane of a straight-bladed turbine with infinitely long blades.
Savonius turbines don’t have proper blades, but scoops or curved metal sheets. They are of course more robust than Darrieus turbines, but have lower efficiency because each scoop harvests power only for half cycle and is slowed down by the negative contribution of the opposite scoop.
8
3 Models
There are many possible ways of predicting Vertical Axis Tidal or Wind Turbines’ performance, both regarding energy productivity and mechanical fatigue.
Computational Fluid Dynamics (CFD) in 2 or 3 dimensions is mostly used; it is able to solve pressure and velocity field up to the length scale of the boundary layer when fed with a fine enough discretized domain. From the solution of the Reynolds Averaged Navier-Stokes (RANS) equations, it is possible to extrapolate the value of the forces acting on the walls, i.e. the blades, and consequently the torque and power produced. This kind of solver needs a very fine grid, especially near to solid walls, where gradients of variables at stake are considerable, and so a large number of cells and high computational cost are needed.
On the contrary, there are approaches with very low computational cost, which are based on the BEM (Blade Element Momentum) method: they are mathematical-analytical models that relate the forces acting on the blades to the variation of velocity of the flow and to the pressure drop through the rotor. Such models need as input accurate curves of performance CL-α and CD-α of the employed foil for many values of
Reynolds number, but can take advantage and of lower computational effort. An example of BEM model is the Double Multiple Stream Tubes Model (DMST), which will be described in the following paragraph.
The two approaches just described can be combined in the so-called BEM-CFD models: in this case the blades are not present in the grid but are replaced by negative source terms in the momentum equations, which should represent the fluid-body interaction. This new term is only applied in the region where the blades are found, while in the rest of the flow-field it is neglected.
A group of researchers of University of Pisa has developed a BEM-based User Defined Function (UDF) to be involved in CFD code ANSYS Fluent for vertical axis turbines [13]. The UDF contains the information required about the blade (chord, distance from leading edge to mounting point, position at each time step and movement law, performance curves of the foil at different Reynolds number) and about the zones of the domain, i.e. in which zone the source terms have to be applied and how this zone is
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9 discretised. Furthermore, the UDF contains the equations for the calculation of the actual relative velocity and angle of attack and consequently of the source terms to apply in the domain, starting from the elaborated information.
It contains too some additional sub-models concerning typical VATT unsteady and 3D phenomena. Finally, the particular spreading method and parameters of the source terms are also contained in the UDF.
The positive aspect of BEM-CFD models is undoubtedly the lower computational effort with respect to the fully-resolved blades simulations, brought by the use of BEM method in the blades’ region, where the main differences in the grid refinement are observable. This feature makes this approach suitable also for performance prediction of wide wind farms or for variable geometry, i.e. for cases in which only CFD would lead to prohibitive computational cost. Anyway it must be underlined that BEM-CFD solver gives considerably less accurate results in the wake representation than CFD only.
Research work all around the world led to the development of two main BEM-CFD approaches for Vertical-Axis Turbines, with different level of accuracy:
- Actuator Cylinder Model (ACM) [14][15];
- Actuator Line Model (ALM) [16][17][18][19][20].
3.1 DMST
In this model, the cylindric area internal to the blades’ trajectory is divided in tubes, parallel to the direction of the incoming potential flow; each tube is independent from the adjacent ones and is divided in two halves: the upstream tube and the downstream tube [21].
10
Figure 3 Schematic of Double-Multiple Streamtube Method [22]
The model is based on the following hypothesis [21]: - Incompressible flow, hence there is no drag force;
- The wake created from the upstream blade is fully developed when it reaches the downstream blade;
- The velocity is uniform in the section of the tube and is perpendicular to the cross-section itself;
- Two tuning coefficients (one for upstream path and one for downstream path) are evaluated to calculate the actual incident velocity on the blade.
Despite its simplicity and the assumptions done, with a right definition of the tuning coefficients, the model gives accurate results at low TSRs and for low solidity turbines: in these two conditions in facts the real behaviour tends to the above hypothesis. Far from the optimal TSR, DMST model tends to over-predict power, torque and forces, so it gives significantly less accurate results with respect to CFD [23]. Despite that, this model can be used to evaluate the general performance of a turbine under a range of operating conditions. Anyway it is unable to predict local effects of the flow around the
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11 blades, since it is based only on analytical evaluations, and for this is reason it cannot be used when dealing with geometry optimization issues.
3.2 ACM and ALM
The ACM (Actuator Cylinder Model) and the ALM (Actuator Line Model) are BEM-CFD methods, in which the blades are absent in the simulations and they are mimicked through the introduction of negative source terms of momentum in the domain. They are based on the same principles and share the same solution scheme, which will be hereafter described, but they apply momentum source terms to different geometric portions of the domain.
The ACM is the vertical-axis version of the pre-existent Actuator Disc Model for horizontal-axis turbines. In this model, BEM method is applied in the cylindrical region swept during the blades’ rotation. In 2-D simulations this zone appears as a ring with its centrum coincident with the axis position of the turbine and its radius equal to the distance from the axis and the mounting point of the blade; in the ring’s thickness, covered by a single layer of cells, is contained the thickness of the blade.
ACM strongly reduces resolution requirements of the mesh, but on the other hand, it misses important information about the blades’ position and their aerodynamic properties.
12
Figure 4 Example of a grid used with ACM
The ALM is a good compromise between full blade resolution and ACM, both in terms of wake representation and of computational effort.
In the ALM the source terms are applied only in the cylindrical zone around a single blade, which hence appears as a circle around the pretended blade section in a 2-D domain. The diameter of this circle, i.e. the base of the cylinder, is the fundamental parameter that needs to be accurately chosen.
Despite the absence of their walls in the domain as proper boundaries, the position of the blades is known at each time step, since their movement is imposed in the UDF. For every blade at each time step the solver calculates the resultant inflow velocity in magnitude and direction, at a specified distance ahead of the blade itself. From these quantities and from the current position of the blade, it obtains the angle of attack and the chord-based Reynolds number:
𝑅𝑒𝑐 = 𝜌 ∙ 𝑊 ∙ 𝑐ℎ𝑜𝑟𝑑 𝜇
Where ρ and µ in this case are respectively density and dynamic viscosity of liquid water at normal conditions.
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13 Differently from horizontal-axis turbine, in facts, the angle of attack varies during the turbine’s revolution because the blades’ inclination varies with respect to the flow. Reynolds number and local angle of attack are used to find instantaneous lift and drag coefficient via interpolation of the data sets of the foil section. From these coefficients, the values of the forces are then calculated:
𝐿 = 𝐶𝐿∙ 1 2𝜌𝑈∞ 2𝐴 𝐷 = 𝐶𝐷∙ 1 2𝜌𝑈∞ 2𝐴
Being 𝐴 = 𝑐 ∙ 𝑠𝑝𝑎𝑛, where c is the chord length. For 2-D simulations, a blade length span = 1 m is considered.
These forces are then used to calculate the momentum source terms:
𝑆𝑥= 𝐿 ∙ (sin 𝛼 cos 𝜃 − cos 𝛼 sin 𝜃) − 𝐷 ∙ (cos 𝛼 sin 𝜃 − sin 𝛼 cos 𝜃)
𝑆𝑦 = 𝐿 ∙ (sin 𝛼 sin 𝜃 + cos 𝛼 cos 𝜃) − 𝐷 ∙ (cos 𝛼 cos 𝜃 + sin 𝛼 sin 𝜃)
Where 𝛼 is the local angle of attack and 𝜃 is the azimuthal position of the blade. This latter angle 𝜃 is zero at the beginning of the upstream path and grows counter-clockwise, as shown in Figure 5.
14
Figure 5 Reference system
In the ALM, the real forces are replaced by an imposed body force that is typically smoothed over several grid points by using a Gaussian kernel whose standard deviation is ε, the so-called spreading parameter. The choice of ε depends basically on the grid refinement but, although in literature it has been found that an optimum value is on the order of 14%-25% of the chord length [24], a sensitivity analysis has to be carried on about this value, since a different concentration of the force in the field gives a different contribution to the turbulence introduced in the flow and to the induced velocity distributions.
The formulation of the Gaussian kernel is the following:
𝜂𝜀 = 1 𝜀2𝜋𝑒−𝑟
2⁄𝜀2
It can be demonstrated that this formula comes from the definition of the double normal distribution in the variables x and y, where 𝑟 = √𝑥2+ 𝑦2 is the local distance from the
application point of the forces and 𝜀2 = 2𝜎2, being σ the variance of the normal
distribution of each variable.
For the application in the ALM, ε is the width of the kernel in which each source term is spread. Since the width of the zone of application of the model is known, a relation can
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15 be found between the variance σ of the normal distribution and the parameter ε to give as input to the UDF.
The grid zone in which the source terms are applied is a ring with a thickness of 2 chords; this area has to contain almost all the normal distribution in order to avoid discontinuity at the boundaries between the ring and the adjacent zones. We impose that at these boundaries the source term applied is only the 0.03% of the calculated value: this means that the width of the ring contains 6σ of the normal distribution. From this follows this simple assumption:
6 ∙ 𝜎 = 2 ∙ 𝑐ℎ𝑜𝑟𝑑
The relation between the width of the Gaussian kernel and the variance is:
𝜀 = √2𝜎 Consequently:
𝜀 = √2𝜎 = √2 ∙1
3∙ 𝑐ℎ𝑜𝑟𝑑 = 0.118 𝑚
The User Defined Function (UDF) contains also some sub-routines that are fundamental to obtain a correct representation of the real behaviour of the turbine and that are the outcome of years of research about BEM-CFD models; in particular for vertical-axis turbines in a 2-D analysis, the “Virtual camber” model and the “Dynamic stall” model play a fundamental role.
3.2.1 Virtual Camber
The performance of Darrieus turbines is characterised by the so-called “flow curvature effects”, caused by the circular trajectories of the blades. The curvature of the flow has influence on the aerodynamic properties of the blades, which have different behaviours when immersed in a curvilinear or in a rectilinear flow. Since the performance curves CL
-α and CD-α are obtained in wind gallery, i.e. with rectilinear flow, a correction of these
16 chord to turbine radius ratio, c/R, increases, as demonstrated by Migliore et al. [25] [26]. Their study also demonstrates, thanks to kinematic analysis, that curvilinear flow introduces an effective angle of incidence in addition to the traditionally defined angle of attack and that it alters the camber of the blade. This means, for example, that a symmetrical airfoil with zero incidence in a curvilinear flow behaves as a “virtual” cambered airfoil with non-zero incidence in a rectilinear flow. This comes from the fact that the angle and the relative inflow velocity not only vary with azimuthal position, but they also have a chordwise distribution, because the radial distance from the rotation axis changes along the chord, and so also the tangential velocity due to rotation does. In their study they propose the use of conformal mapping technique for transforming the geometric foil in the curvilinear flow into a “virtual” foil in a rectilinear flow, calculating virtual incidence and virtual camber of the mean line. Figure 6 shows an example of how conformal mapping works.
Figure 6 Qualitative example of conformal transformation [25]
While in their study Migliore and Wolfe only consider the mean values of these parameters over a rotation and assume that this simplification leads to negligible errors, this assumption cannot be applied to the present case because c/R ratio is larger than the value recommended by Migliore and Wolfe in [25] already at the minimum turbine’s solidity.
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17 The sub-routine here developed applies conformal mapping technique to every blade at every time step, using the velocity components calculated at the previous time step for defining the local TSR.
The first step is to apply the conformal transformation to the curved streamlines, in order to make them rectilinear. Later the reference system and the mean line of the employed profile are transformed, in order to obtain the virtual chord of the profile and the function of the modified mean line yML = f (x, c, TSRlocal, θ, R) in the new coordinates.
This function is later used in the following definition of the “zero-lift angle of attack”, i.e. the angle of which the CL-α curve of a symmetrical profile has to be translated leftwards
to obtain the curve of a profile with a precise camber:
𝛼𝑧𝑒𝑟𝑜−𝑙𝑖𝑓𝑡 = 1 𝜋∫ 𝑑𝑦𝑀𝐿 𝑑𝑥 (1 − cos 𝜃) 𝑑𝜃 𝜋 0
Where the subscript ML stands for “mean line” and x is the dimensionless position along the virtual chord obtained with the conformal transformation of the real chord.
Finally, this angle is added to the flow angle of attack measured by the ALM, resulting in a higher input angle and consequently in different lift and drag coefficients. These corrected coefficients are used in the calculation of the source terms at every time step as described in the previous paragraph.
This method is more complicated from an analytical point of view than Migliore’s one, but this technique is general, so, once implemented in the software, it can be applied to turbines with every solidity, with different profiles, both symmetrical and cambered, with different mounting pitch angles and different mounting points, without the need for checking its validity at every try.
3.2.2 Dynamic Stall
Another consequence of the cyclic variation of the angle of attack due to turbine’s rotation is the dynamic stall [27] [28]. This phenomenon occurs when the angle of attack varies with a certain frequency, so it is typical for vertical axis turbines but also for
18 variable pitching airfoils. Dynamic stall strongly affects the turbine’s performance, especially at low TSRs, because it causes hysteresis in the lift coefficient and reduction of the power coefficient. This deterioration is particularly evident at low TSRs, when the maximum value of angle of attack is very high. At the optimum TSR and at higher TSRs this effect is negligible, since the flow is almost fully attached.
The hysteresis curve becomes more pronounced as the frequency of oscillation of the angle increases. Dynamic stall is thus classified as “light stall” at low frequencies and as “deep stall” at high frequencies. This latter case is the worst as concerns mean lift coefficient and global performance.
Figure 7 Lift coefficient in dynamic stall: (a) on set of stall, (b) light stall, (c) deep stall [27] - .- static stall - - dynamic stall
As the angle of attack increases because of the turbine’s rotation, the flow begins to separate and the first vortexes create. At the static stall angle though, the presence of the vortex induces additional suction, which makes the lift coefficient grow further, even if the flow is separated. Up to a certain angle, called “dynamic stall angle”, 𝛼𝑑𝑠, the
profile experiences higher lift force, which can be convenient in some practical applications. At a certain point, the vortexes move towards the trailing edge and finally leave the profile: this effect coincides with a drop of the lift coefficient, which later remains low because the vortexes in the near wake induce shape drag force. This effect persist while the angle of attack again decreases and it causes hysteresis of the CL, which
assumes again the linear behaviour when the angle decreases about under the static stall angle, i.e. when the flow reattaches to the profile. As a consequence of the
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19 hysteresis, the average lift coefficient will be lower than the static one and also the produced power will suffer from this decrease.
In the ALM this complex phenomenon needs to be simplified in separate stages, each one independent from the others, and implemented in the UDF.
In this work, the method proposed by Rocchio, Chicchiero, Salvetti and Zanforlin [29] was followed, which in turn was derived by models by Leishman and Beddoes [30], by Sheng et al. [31] and by Larsen, Nielsen and Krenk [32].
The stages are: creation of the tip vortex, detachment of the vortex from the blade’s trailing edge, stall, reattachment of the boundary layer.
Dynamic stall depends strongly on the static lift coefficient of the profile used [33] (in this case, NACA0018 and NACA4418) and on the TSR, in particular on the rotational speed of the turbine.
The calibration of the model has been conducted at the optimal TSR=1.85 for NACA0018 and for NACA4418 with their respective static lift coefficient curves as reference. The results obtained have been later compared with the related CFD-2D resolved blades simulations’ results. After the optimal values of the sensible parameters are found at the optimal TSR, the parameters are kept constant and the response of this method to different TSRs, inflow velocities and solidities is investigated, again comparing the results with the resolved blades simulations’ ones.
3.2.3 Performance curves of the profiles
One of the negative aspects of BEM-CFD methods is that they all need accurate information about the blade profile employed, in particular the performance curves CL
-α and CD-α as functions of chord Reynolds number are needed. These curves have to be
given as input to the model, which will use them to make interpolations over Reynolds number and angle of attack in order to find the local and instantaneous values of lift and drag coefficients.
The above mentioned curves have been found in literature ([34] per NACA0018 and [35] for NACA4418) and are reported hereafter. Such curves are generally obtained with wind tunnel experiments.
20
Figure 8 Lift and drag coefficients at different Reynolds number, NACA0018
4 Geometry and Computational Domain
21
4 Geometry and Computational Domain
In this thesis work a vertical-axis, Darrieus type turbine with three straight blades is examined. Maintaining the same geometry, two different profiles are tested: symmetrical NACA0018 and cambered NACA4418 with its curvature pointed outwards, which is the final project of the company. The peculiarity of this turbine is that its radius varies with potential flow velocity in order to maximize the turbine’s performance. It means that the turbine always works at its optimal Tip Speed Ratio (TSR), i.e. the ratio between the blade’s tangential velocity and the potential flow velocity:
𝑇𝑆𝑅 = 𝑅 𝛺 𝑈∞
The different geometries tested have radius of 0.75 m and 0.40 m; at these two radiuses the turbine has a solidity respectively of 15.9 % and 29.8 %, being the solidity σ defined as:
𝜎 = 𝑁 ∙ 𝑐 2 𝜋 𝑅
Where N is the number of blades, c is the chord and R is the radius of the turbine.
Despite the different geometry, the same grid for the external domain has been used for every set of simulations and it has been scaled up following the variation of the rotating disk’s diameter from case to case; in facts, different grids for the rotating zones were necessary for the two methods employed and for both the radiuses of the turbine (R=0.75 m and R=0.40 m). As a consequence, the effective dimensions of the external domain are not always the same, but they are fixed multiples of the rotating zone’s diameter.
The domain is 39.3 times the turbine’s zone diameter long in the direction of the incoming velocity, and 42 times wide in the perpendicular direction. The centre of the turbine is located symmetrically with respect to the width and at 64% of the longitudinal length from the inlet, which is at the left side (Figure 10).
22
Figure 10 Scheme and dimensions of the external domain.
With this constant structure, the domain is always greater enough than the turbine in order to avoid every blockage effect, so there is no need to create a different external grid for every case, because it wouldn’t affect the results anyway.
To increase accuracy of the results, the grid is refined near the turbine and in the wake region.
The grid is unstructured and it is made up of 82 786 quadrilateral cells. For the simulations with modelled blades instead, the external domain grid has been slightly modified: in order to avoid discontinuity problems at the interface between the ring containing the blades and the external domain, the number of nodes on the interface has been set equal to that of the ring and some layers have been added near the interface. The total number of cells with this modify is 71 285.
As regards the turbine’s zone, many grids have been tested both for the resolved blades and for the modelled blades approaches, both for maximum and minimum radius, both for NACA0018 and for NACA4418 profiles.
4 Geometry and Computational Domain
23 For the resolved blades method, a first unstructured grid is implemented: it is refined near the blades by placing 220 nodes on each sidewall of the blade plus 18 nodes near the trailing edge. The first layer of cells is 1 μm thick and the growth ratio of the layers’ thickness is 1.2. Near the walls there are 40 layers in total. The total number of cells for both turbine’s diameters and for both profiles are summarised in the following table. From now on, this type of grid will be called “mesh-A”, regardless of the turbine’s radius and the blade profile.
Profile Turbine Radius Number of cells
NACA 0018 R = 0.75 m 116 793 R = 0.40 m 104 007 NACA 4418 R = 0.75 m 115 773 R = 0.40 m 94 308
Table 1 Number of cells of mesh-A
As highlighted in the study by Maître, Amet and Pellone [7], strong influence on the results is held by the dimensionless wall distance parameter y+, which is defined as:
𝑦+ = 𝜌 𝑢𝜏 𝑦
𝜇
Where y is the normal distance from wall at cell centre, 𝑢𝜏 is the friction velocity and 𝜇
is the dynamic viscosity of the fluid. The parameter y+ is useful to distinguish zones of
the boundary layer with different law-of-the-wall and for this reason it influences the solution in presence of physical walls. As demonstrated by Maître, Amet and Pellone [7] through comparison with experimental data, consistent results are obtained, when using k-ω SST turbulence model, with average y+ on the profile smaller than 1 and with
maximum y+ around 1.5. In their work they calculate y+ on a fixed blade in a steady flow,
24 instead, y+ is calculated for every cell adjacent to the blade’s wall directly from the solver
ANSYS Fluent, from the values of the solved velocity field. The tests on the first grid implemented have given average y+ always smaller than 0.1 and maximum values
around 0.11 for all TSRs. Despite the different method of calculating y+, i.e. the different
incoming velocity involved, the values obtained are conservatively included in the range advised by Maître et al., so no changes of the first layer of cells are needed. A new grid has been created maintaining the same thickness for the first layer’s cells but increasing the refinement: the number of nodes on the blades is brought to 340 per side plus 23 nodes in the trailing edge for each blade; the number of layers has been incremented from 40 to 73 and the new layers’ thickness growth ratio is 1.1 instead of 1.2. The total number of cells of the new grid for the rotating disk is reported in the following table for the four cases of different geometry and profiles. From now on, this type of grid will be called “mesh-B”, regardless of the turbine’s radius and the blade profile.
Profile Turbine Radius Number of cells
NACA 0018
R = 0.75 m 203 046
R = 0.40 m 243 789
NACA 4418 R = 0.75 m 252 984
Table 2 Number of cells of mesh-B
As an example, the following picture shows the refinement of the grid near one blade for the case R = 0.75 m and NACA 0018, respectively for mesh-A and mesh-B, while Table 3 summarises the sensible parameters in the mesh construction.
4 Geometry and Computational Domain
25
Figure 11 Blade refinement; mesh-A (left), mesh-B (right)
Mesh-A Mesh-B Number of nodes on
one blade 440 + 18 680 + 23 Thickness of the first
layer of cells (µm) 1 1
Number of layers 40 73
Growth ratio of the
layers’ thickness 1.2 1.1
Table 3 Parameters of mesh-A and mesh-B grid refinement
4.1 Sensitivity to grid refinement.
Numerical simulations with these two grids have been conducted for different inflow velocities and TSRs. Here follow the results of one of these cases as an example of the general difference between the used grids that can be found in every simulation. With inflow velocity of 4 m/s, maximum turbine’s radius and maximum rotational speed (TSR = 1.178), the following figure shows the trend of a symmetrical single blade’s power coefficient within a rotation. Azimuthal position between 0 and 180° is the upstream path, while between 180 and 360° is the downstream path.
It can be observed from the figure that with the more coarse grid the maximum value of power coefficient is lower and is reached earlier in the rotation. The difference in the
26 single blade’s behaviour leads also to a different average value of total power coefficient: with mesh-A, Cpav,tot is 0.145, while with mesh-B it is 0.276.
It has been calculated that the refinement of the grid leads to an increase in power coefficient that depends on the working conditions: at TSRs near to the optimal one, the increase is very low (8% at TSR= 1.85 and U∞= 1.75 m/s), while with very low and very
high inflow velocity the power coefficient can even double itself.
Figure 12 Single blade’s power coefficient obtained with two different grids
In order to understand this difference in the single blade’s power coefficient, especially intense between θ = 80° and θ = 180°, a deeper investigation of the flow is presented in the following pictures, which show the streamlines coloured by vorticity magnitude at positions θ = 80°, θ = 90°, θ = 120°, θ = 200° and θ = 240° for mesh-A (left) and mesh-B (right) for the same working conditions (U∞= 4 m/s TSR= 1.178).
It can be observed that the separation bubble that is created on the blade at about θ = 80° and the vortex related to it, are larger and more persistent with mesh-A than with mesh-B. The consequence is a greater drag force on the blade, which causes the decrease in performance at those blade positions and consequently the decrease of the global value of power coefficient. This effects were also observed by Maître, Amet and Pellone and, since they had the possibility to compare numerical and experimental
4 Geometry and Computational Domain
27 results and they found out good consistency with their finest grid and their smallest y+,
from now on mesh-B will be assumed as reference mesh for validating the ALM. It is important to underline that these numerical results will have to be compared with experimental data when they will be available in the future; eventually these data will be useful to correct again the calibration of the BEM-CFD model.
θ = 80°
θ = 90°
28 θ = 200°
θ = 240°
Figure 13 Streamlines coloured by vorticity magnitude for mesh-A (left) and mesh-B (right) at different blade position
Also for the ALM simulations some grids have been created and compared. For this cases though, the grid is much simpler and more coarse: the mesh is structured and it is made up of an external ring, where the momentum source terms will be applied in the model, and an internal O-grid.
This is one of the most important pros of BEM-CFD approaches: thanks to the absence of the blades, the rotating disk has no discontinuities and can be easily reproduced by a structured grid. In addition to that, the absence of physical walls makes it useless to have a very fine grid, which will increase the calculation time.
The parameters of the adopted grid are summarised in Table 4, while Figure 14shows the mesh for the case with minimum solidity: the blue part of the mesh is the ring in
4 Geometry and Computational Domain
29 which the source terms are applied. The ring is 2 chords thick and its midline coincides with the trajectory of the mounting point of the blades.
Thickness of the ring 2 chords’ length
N° azimuthal cells 192
N° radial cells in the ring 30
Total n° cells of the disk 13 248
Table 4 Features of the turbine zone’s mesh for modelled blades simulations
30
5 Computational settings
All the simulations have been conducted with the software ANSYS Fluent and are based on the transient solution of pressure-based Reynolds Averaged Navier-Stokes (RANS) equations of transport of mass, momentum and energy. The considered equations are the following: 𝜕 𝜕𝑡𝜌 + ∇(𝜌𝑢⃗ ) = 0 𝜕 𝜕𝑡(𝜌𝑢⃗ ) + ∇ ∙ (𝜌𝑢⃗ 𝑢⃗ ) = ∇ ∙ 𝜏̿ − ∇𝑝 + 𝜌𝑔 𝜕 𝜕𝑡(𝜌𝑒) + ∇ ∙ (𝜌𝑒𝑢⃗ ) = ∇ ∙ (𝜏̿ − 𝑝𝐼̿)𝑢⃗ − ∇𝑞 + 𝜌𝑔 𝑢⃗ + 𝑆𝑒
Since the RANS solve only the mean flow, a loss of information about turbulent fluctuations of the main quantities involved in the flow follows from this assumption. In order to take account of these fluctuating quantities, this kind of solvers usually involve a turbulence model when applying RANS equations with the aim of reintroducing the effects of the unknown Reynolds stress tensor. In this case the k-ω SST (Shear Stress Transport) turbulence model has been chosen because it fits quite well both the resolved blades and the modelled blades applications. In facts, this model solves the partial differential equations for turbulent kinetic energy (k) and specific dissipation rate (ω) in proximity of physical walls and it shifts to a k-ε model while moving away from them. In the modelled blades simulations, no physical walls are present so the k-ω SST model is reduced to a k-ε model for this case.
The equations introduced by this model are the following:
𝜕 𝜕𝑡(𝜌𝑘) + 𝜕 𝜕𝑥𝑖(𝜌𝑘𝑢𝑖) = 𝜕 𝜕𝑥𝑗(𝛤𝑘 𝜕 𝑘 𝜕𝑥𝑗) + 𝐺𝑘− 𝑌𝑘+ 𝑆𝑘
5 Computational settings 31 𝜕 𝜕𝑡(𝜌𝜔) + 𝜕 𝜕𝑥𝑖(𝜌𝜔𝑢𝑖) = 𝜕 𝜕𝑥𝑗(𝛤𝜔 𝜕 𝜔 𝜕𝑥𝑗) + 𝐺𝜔 − 𝑌𝜔 + 𝑆𝜔 𝜕 𝜕𝑡(𝜌𝜀) + 𝜕 𝜕𝑥𝑖(𝜌𝜀𝑢𝑖) = 𝜕 𝜕𝑥𝑗((𝜇 + 𝜇𝑡 𝜎𝜀) 𝜕 𝜀 𝜕𝑥𝑗) + 1.44 𝜀 𝑘(𝐺𝑘) − 1.92𝜌 𝜀2 𝑘 + 𝑆𝜀
Where G is a generic generation term, 𝛤 is the generic diffusivity, 𝑌 is a turbulence-driven dissipation term and 𝑆 is a source term.
While in the BEM-CFD case all the information about the movement of the turbine’s blades is included in the User Defined Function (UDF), which will be described later, in the CFD case it is necessary to specify the rotational velocity as a Cell Zone Condition for the circular region of the mesh that contains the turbine. In the modelled blades simulations however, the Cell Zone condition of X-Momentum and Y-Momentum Source terms are turned on in the ring that virtually contains the blades.
The boundary conditions of the domain are:
- velocity inlet: at the left side boundary, from which the flow enters the domain with specified uniform velocity magnitude and direction normal to the boundary itself; the turbulent intensity entering the domain is set to 4% and the viscosity ratio is set to 100, which is a typical value when simulating in water;
- symmetry: at the upper and lower sides of the domain, parallel to the undisturbed flow;
- pressure outlet: at the right side boundary, with backflow direction normal to the boundary itself, backflow turbulent intensity of 4% and backflow turbulent viscosity ratio of 100;
- no-slip wall: at the walls of the blades. This condition is only present in the resolved blades simulations.
For the calculation of some specific coefficients of performance, such as moment coefficient or force coefficients on the blades, the solver needs some reference values, which are related to the employed fluid (density, dynamic viscosity), to the state
32 (pressure, temperature) and to the examined object. In particular for a vertical axis turbine in a 2-D analysis the input values given are the diameter of the turbine for the Area reference value (considering 1 m blade length) and the radius of the turbine for the Length.
The coupling of pressure and velocity is solved through the SIMPLEC (SIMPLE – Consistent) algorithm, which may accelerate convergence in some cases of turbulent flows, in particular when the pressure-velocity coupling is the main deterrent to obtaining the solution [36].
Spatial and time discretization methods, which are essential for convergence time and solution accuracy, are here summarised:
- Gradient – Least Squares cell based - Pressure – 2nd order
- Momentum – 2nd order upwind
- Turbulent kinetic energy – 2nd order upwind
- Specific dissipation rate – 2nd order upwind
- Transient formulation – 2nd order implicit
During each time step the solver calculates the solution and stores the local absolute residuals that it obtains for every variable’s balance. Convergence is reached when all the residuals are lower than the set value, which is 5e-05 for this set of simulations. Anyway, in order to contain the time of calculation in the first time steps, when the initialized field of velocity and pressure is very far from the regime solution, the maximum number of iterations per time step is set to 200.
The time step for every simulation is calculated starting from the rotational speed. For the simulations with resolved blades we impose that the turbine rotates of 0.5° every time step. As a consequence, the time step (s) is calculated with the following:
𝑇𝑆 = 2 𝜋
𝛺 ∙ 0.5 360
5 Computational settings
33 and a complete revolution of the turbine is reached after 720 time steps.
For the modelled blades simulations, every time step must correspond with a rotation angle equal to the angle between two consecutive nodes in the structured grid.
Hence:
𝑇𝑆 = 2 𝜋
34
6 Results
For both profile NACA 0018 and NACA 4418 profiles, several URANS of the resolved blades have been carried out in order to find out the optimal working conditions with an imposed inlet velocity of 1.75 m/s. In particular, a range of TSR from 1.4 to 2.2 has been tested for minimum solidity turbine (R= 0.75 m/s). The optimal TSR turns out to be 1.85 for both symmetrical and cambered blades. With this optimal TSR, many simulations with different inlet velocities are performed, to analyse how the incoming velocity influences the turbine performance. In the following the simulations with the resolved blades are used as reference solutions.
For each simulation, the same analysis is conducted; it is organised in order to compare the general performance and to investigate the goodness of the sub-routines implemented in the UDF and of the chosen parameters.
In particular, much information can be extracted by the single-blade power coefficient and by the single-blade relative angle of attack as functions of the blade azimuthal position θ.
The power coefficient can be easily obtained with the solver ANSYS Fluent, considering the following equation:
𝐶𝑃 = 𝐶𝑀 ∙ 𝑇𝑆𝑅
Where CM is the moment coefficient and is directly obtained as a report of the solver
([36]):
𝐶𝑀 = 𝑀
1
2 𝜌 ∙ 𝑈𝑟𝑒𝑓2∙ 𝐴 ∙ 𝐿
Where ρ, Uref, A, L are density, velocity, area and length expressed in the “Reference
Values” section in the solver.
While the power coefficient can be directly obtained with the solver ANSYS Fluent, the angle of attack for each azimuthal position has been evaluated with a particular procedure, which has been derived from dynamic and kinematic considerations.
6 Results
35 The approach hereafter described follows the idea proposed by Benini et al. [37]: blade relative velocity is determined as the vector sum of the local absolute wind velocity U investing the blade and the opposite of the tangential velocity of the blade at its mounting point, -ΩR. A circle, corresponding to the trajectory of the mounting points of the blades, is used to save and export each 6 time steps the values of absolute velocity magnitude and absolute velocity angle with respect to the x-axis. These values, calculated on the nodes or edges of the cells crossed by the circle, along with the tracking points’ coordinates, are used to calculate the mean velocity magnitude and direction over a revolution for each azimuthal position by means of the software Matlab. In the calculation of the average values, however, the points close to the blades’ walls could give misleading information about the incoming velocity that invests the blade, because there the flow is strongly deflected and accelerated. For this reason the points immediately before each leading edge and after each trailing edge were excluded. The spatial arch that contains these points is 16 degrees large in case of minimum solidity turbine (R= 0.75 m): this value is the one recommended in Benini’s work [37]. For maximum solidity turbine instead (R= 0.40 m), acceptable results are obtained eliminating the points contained in 34 degrees’ arch before and after each blade. Finally the mean values of magnitude and angle of absolute velocity, respectively U and δ, are used to calculate the angle of attack α as a function of θ with the following construction.
We define
𝛾 = 𝜃 − 𝛿
the angle between the local absolute velocity U and the tangential direction. Since there is no mounting pitch angle, the tangential direction coincides with the chord.
The angle of attack α, i.e. the angle between relative velocity W and the chord, is given by:
𝛼 = 𝑈 sin 𝛾 𝑈 cos 𝛾 + 𝛺𝑅
36 The geometric construction of relative velocity vector and of angle of attack is reported in Figure 15.
Figure 15 Blade velocity vectors and angle of attack
This procedure has been applied to each simulation with the resolved blades and the curves CP-θ and α-θ obtained are used as reference for the parameters’ calibration and
the analysis of the sensitivity of the model to changes in operating conditions.
The comparison on the trend of angle of attack has however just a qualitative meaning because for the modelled blades simulations the angle of attack in the graphs is calculated with the same kinematic construction above presented, but starting from values of velocity that are obtained differently. The velocity of the field in facts is taken at a certain distance ahead of the blade itself at every time step for every blade and is the value that is then used for all the necessary calculations in the routine. Despite being the result of the optimization of the parameters, the two procedures cannot be compared from a quantitative point of view, but they are anyway useful to understand how the model works and how it represents reality.
Along with CP-θ and α-θ curves, also average power coefficient over a revolution and
contours of velocity, static pressure, turbulent kinetic energy (TKE) and vorticity are considered in the calibration procedure.
For a deeper analysis and for investigating the effect of a single sub-routine, also the trend of lift and drag force on a single blade with azimuthal position and the resulting CL-α curve have been compared when needed.
6 Results
37 For both blade profiles, the ALM parameters have been optimized at TSR=1.85: in particular, the sensible variables are:
- The position at which the incoming velocity for each blade is stored, in order to be used in source terms’ calculation;
- The width of the zone over which the incoming velocity is averaged, around the above-mentioned position;
- The width of the kernel ε; - The angle of dynamic stall, 𝛼𝑑𝑠;
- The frequency of the dynamic law of increase of CL,d, 𝜔3, before the leading-edge
vortex leaves the blade.
The best results with ALM were found with different sets of parameters in case of NACA0018 and NACA4418 profiles at the same inlet velocity and at optimal TSR. These sets of parameters are then kept constant, while tests with different TSRs and inlet velocities are performed.
Keeping the same inlet velocity, TSR=1.4 and TSR=2.2 are tested, which are two extreme conditions of the CP – TSR curve.
The other purpose is to obtain the Power Curve as function of the incoming velocity. In particular, it is supposed that the turbine works at its optimal TSR until it reaches its maximum allowed rotation speed. The flow velocity at which we have this condition is called “rated velocity”, 𝑈∞,𝑟𝑎𝑡𝑒𝑑 and it can be calculated as follows:
𝑈∞,𝑟𝑎𝑡𝑒𝑑 =
𝑅 ∙ 𝛺𝑚𝑎𝑥 𝑇𝑆𝑅𝑜𝑝𝑡
For higher inflow velocities, rotational speed is kept constant, therefore TSR will be lower than the optimal one.
For CFD only and for CFD-ALM, this is how the tests can be summarised: - U∞= 0.5 m/s , TSR= 1.85
- U∞= 0.8 m/s , TSR= 1.85
- U∞= 1.75 m/s , TSR= 1.85
38 - U∞= 3.25 m/s , TSR= 1.45
- U∞= 4 m/s , TSR= 1.18
For symmetrical blades’ case only, the model has been tested also with maximum solidity turbine (R= 0.40 m) at its optimal working condition (U∞= 1.75 m/s, TSR= 1.04).
6.1 NACA0018
Reference case: U
∞= 1.75 m/s , TSR= 1.85
Figure 16 and Figure 17show respectively the angle of attack and the power coefficient for a blade during a revolution (the reference system of azimuthal angle θ is the same as in Figure 5). The solution here reported is the result of the calibration over many parameters and it has been chosen as a compromise between correspondence of CP-θ
curve and mean value of power coefficient.
Figure 16 NACA0018, U∞=1.75 m/s, TSR= 1.85: comparison of angle of attack during a revolution
6 Results
39
Figure 17 NACA0018, U∞=1.75 m/s, TSR= 1.85: comparison of single blade power coefficient during a
revolution between resolved blades and modelled blades simulation
It can be observed from these figures that modelled blades simulation overpredicts the maximum value of power coefficient and that it is reached at a lower value of θ. This behaviour is coherent with the trend of the angle of attack: in facts, it is always a bit overestimated during the whole upstream path, i.e. for azimuthal positions between 0° and 180°.
In addition to that, a second peak of the angle of attack is visible in the downstream path, i.e. for azimuthal position between 180° and 360°. Since blades’ solid walls are absent in ALM, in the turbine’s region it is difficult to represent turbulent vortexes and to obtain the same level of perturbation of the flow. For this reason the blades in their downstream path are not really invested by the wake of the upstream blades, as it happens in reality. This is why, while using ALM, the angle of attack’s trend in the downstream path is similar to the upstream path, but with a lower peak value, due to the slowdown of the flow. This difference is found in every modelled blades simulation, but it does not affect the power coefficient because at those azimuthal angles drag force is anyway predominant on lift and negative torque is produced.
40 With this set of parameters, the mean CP, ALM value of the turbine is 0.311, against the
result of the resolved blade simulation, that gives a mean CP, ref of 0.337. The related
percentual error, calculated as
∆𝐶𝑃,%=𝐶𝑃,𝐴𝐿𝑀− 𝐶𝑃,𝑟𝑒𝑓
𝐶𝑃,𝑟𝑒𝑓 ∙ 100 Is -7.71 %.
The following pictures show the contours of x-velocity, pressure, vorticity and turbulent kinetic energy (TKE) for the two cases. The reference simulations results are recognisable because the blades are visible as “holes” in the domain; in the pictures representing the CFD-ALM cases, the grid of the zone in which the source terms are applied is superimposed on the contours.
Figure 18 Contours of Static Pressure (-3000 ÷ 1500 kPa): CFD (left), BEM-CFD (right)
6 Results
41
Figure 20 Contours of X-Velocity (-1 ÷ 3 m/s): CFD (left), BEM-CFD (right)
42
6 Results
43 It can be seen that the ALM reproduces quite accurately the various variable fields, except of course very close to the “virtual blades” and in the regions that would be occupied by the blades themselves.
Despite the ALM is not able to reproduce the so-called near wake, the vorticity field agrees well with the reference field in the region where the blades are found. In particular, the almost same vortical structures caused by the blades’ movement are present and recognisable. Moreover, at about ϑ= 120° and ϑ= 200° the separation of the blades’ boundary layer, i.e. vorticity which sheds from the body boundaries, is well reproduced although the “separation” is a consequence of the velocity gradients introduced by the distributed source term.
In addition to that, also two vorticity zones at the sides of the turbine are present and are very alike the ones obtained in the reference simulations, except for the fact that they diffuse earlier.
With ALM there is more turbulence in the wake immediately behind the turbine; as a consequence, there is greater dissipation and the wake recoveries earlier. These two effects, observable in the TKE contours, are caused by the introduction of concentrated forces in the domain, while in resolved blades simulations instead, turbulence is created along the entire “solid wall” boundary. In the turbine’s region of course no coherent information can be obtained by the contours of turbulent kinetic energy, because of the absence of the blades’ solid walls.
Virtual Camber Improvement
Only for this case the comparison between different virtual camber approaches and the reference simulation is presented. During this thesis the conformal mapping technique proposed by Migliore was implemented in the code. This paragraph shows how this improvement has affected the results on one single case. The result obtained by resolved blades simulation is compared with the results obtained without virtual camber subroutine, with the previous virtual camber model and with the present one, which led to the results already shown in the previous paragraph.
44 The following pictures show the curve of single blade power coefficient over a revolution for the four cases and the four contours of static pressure in the turbine’s zone. The brown circle visible in the contours of modelled blades simulations’ results is the trajectory of the mounting points of the blades.
(a) (b)
(c) (d)
Figure 23 Contours of Static Pressure (-3000 ÷ 1500 kPa) in the turbine zone: (a) reference CFD, (b) without virtual camber, (c) previous virtual camber model, (d) improved virtual camber model
6 Results
45
Figure 24 Comparison of single blade power coefficient over a revolution with different virtual camber approaches
From the pressure fields it can be observed that without virtual camber the blade that at the moment occupies the position ϑ=0° has opposite pressure distribution with respect to the reference (case (b)): the absence of virtual incidence and virtual camber induced by flow curvature has given an opposite force on the blade and a global positive tangential force instead of a negative one and this is in facts proved by a positive instantaneous power coefficient in the curve.
The virtual camber model that was already present in the code has been later modified because it gives an overprediction in the angle of attack. The consequence is an overestimation of lift coefficient and a consequent higher power coefficient over almost the entire revolution. In the pressure field (case (c)) it can be observed that the zones of pressure and depression are more marked than in the other cases. In addition to that, the blade that occupies the position ϑ=240° has a wrong pressure distribution in picture (c), that corresponds to the power coefficient peak in the curve at that azimuthal position and that is absent in the other cases and in the reference.
46
U
∞= 1.75 m/s , TSR= 1.4
In the first part of this work it is explained that the optimal TSR has been found out thanks to CFD studies at different TSRs. The results obtained during those tests are now compared with the results obtained with ALM at the respective operating conditions, but with the set of optimal parameters found for the reference case (optimal TSR). One of the aims of this work is in facts to investigate the response of the model to changes in operating conditions and that means inflow velocity but also TSR.
As shown in Figure 26,the use of the parameters optimized for TSR= 1.85 leads to great error in case of low TSR (red curve). Therefore an attempt of improvement has been made by varying only the parameters of dynamic stall. In particular these parameters are changed in order to induce dynamic stall earlier, because with low TSRs the angles of attack are generally higher, so stall happens earlier and is predominant.
The green curve in Figure 26is the result of this attempt. There is still much difference in the mean value of power coefficient and the peak is reached at a different azimuthal position.
This suggests that also other parameters, not only related to dynamic stall, should be changed.
Figure 25 NACA0018, U∞=1.75 m/s, TSR= 1.4: comparison of angle of attack during a revolution between
6 Results
47
Figure 26 NACA0018, U∞=1.75 m/s, TSR= 1.4: comparison of single blade power coefficient during a
revolution between resolved blades and modelled blades simulation with reference and modified set of parameters
The following table summarises the power coefficient mean values ad percentual errors obtained.
C
P,average[-]
Δ
CP,%[%]
CFD only 0.219 - CFD-ALM 0.538 145.2 CFD-ALM modified 0.380 73.2 Table 5 NACA0018, U∞=1.75 m/s, TSR= 1.4U
∞= 1.75 m/s , TSR= 2.2
At TSRs higher than the optimal, dynamic stall is less important, so the performance does not change very much with a different choice of parameters. The following pictures show the results obtained with ALM with the set of parameters optimized for the reference case. The percentual error in the mean global power coefficient is -59.7%,
48 being CP, ref= 0.293 and CP, ALM= 0.118. In this conditions, instead, virtual camber effect is
predominant, because the higher TSR is reached with the same inlet velocity, so rotational speed has grown.
Figure 27 NACA0018, U∞=1.75 m/s, TSR= 2.2: comparison of angle of attack during a revolution between
resolved blades and modelled blades simulation
Figure 28 NACA0018, U∞=1.75 m/s, TSR= 2.2: comparison of single blade power coefficient during a
