Abstract
This work analyses the link between the aspect ratio of a verticalaxis straightbladed (HRotor) wind turbine and its performance (power coefficient). The aspect ratio of this particular wind turbine is defined as the ratio between blade length and rotor radius. Since the aspect ratio variations of a verticalaxis wind turbine cause Reynolds number variations, any changes in the power coefficient can also be studied to derive how aspect ratio variations affect turbine performance. Using a calculation code based on the Multiple Stream Tube Model, symmetrical straightbladed wind turbine performance was evaluated as aspect ratio varied. This numerical analysis highlighted how turbine performance is strongly influenced by the Reynolds number of the rotor blade. From a geometrical point of view, as aspect ratio falls, the Reynolds number rises which improves wind turbine performance.
Highlights

This paper evaluates VAWT performance;

How Reynolds number influences rotor performance has been studied;

How Reynolds number is linked to rotor aspect ratio has been investigated;

A new design procedure governing the rotor’s aspect ratio has been presented;

The new design procedure maximizes wind turbine efficiency.
Introduction
There are two types of wind turbine which produce electrical energy from the wind: they are horizontalaxis wind turbines (HAWTs) and verticalaxis wind turbines (VAWTs). The second, and in particular straightbladed VAWTs, have a simplified geometry with no yaw mechanism or pitch regulation, and have neither twisted nor tapered blades [1]. VAWTs may be utilized to generate electricity and pump water, as well as in many other applications [1]. Furthermore, they can handle the wind from any direction regardless of orientation and are inexpensive and quiet [2]. Wind turbines have aroused the interest of both industry and the academic community [3–15, 29–31], which have developed different numerical codes for designing and evaluating wind rotor performance. Recent studies [16–20] have highlighted that VAWTs can achieve significant improvements in efficiency.
VAWT can work even when the wind is very unstable making them suitable for urban and smallscale applications [21]. Their particular axial symmetry means they can obtain energy where there is high turbulence.
Their optimum operating conditions (maximum power coefficient) depend on rotor solidity and tip speed ratio [22]. For a VAWT rotor solidity depends on the number of blades, airfoil chord and rotor radius. Tip speed ratio is a function of angular velocity, undisturbed wind speed and rotor radius.
In the design process of a verticalaxis wind turbine it is crucial to maximize the aerodynamic performance [22, 26]. The aim is to maximize the annual energy production by optimizing the curve of the power coefficient varying with the tip speed ratio [25]. For a fixed crosssectional area of the turbine, to optimize the curve of the power coefficient it is possible to use different airfoil sections and/or rotors with different solidity [26].
To maximize energy extraction, other authors introduced guide vanes [27] and/or blade with a variable pitch angle [28] in verticalaxis wind turbines.
In the design process of a verticalaxis wind turbine, a wrong choice of the aspect ratio of the wind turbine may cause a low value of the power coefficient (wind turbine efficiency). This parameter (the aspect ratio) is often chosen empirically on the basis of the experience of the designer, and not on scientific considerations.
In this work, the link between the aspect ratio of a wind turbine and its performance has been studied, and a correlation between the aspect ratio and the turbine’s performance has been found.
Designing an HRotor
Designing a verticalaxis wind turbine with straight blades requires plotting power coefficient c_{ p } against tip speed ratio λ, as a function of rotor solidity σ (Fig. 1).
Figure 1 shows the behaviour of the power coefficient for a wind turbine with straight blades and a NACA 0018 airfoil.
Figure 1 curves were obtained using a calculation code based on MSTM theory [23].
From the graph in Fig. 1, the solidity which maximizes power coefficient σ = 0.3 can be identified, which has a c_{ pmax } = 0.51 corresponding to λ = 3.0.
Since solidity σ equals:
chord c can be expressed as a function of solidity, rotor radius and blade number N_{ b }, as per Eq. 2:
The power of a wind turbine with a vertical axis can be expressed as per Eq. 3:
Having defined the turbine’s aspect ratio (AR) as the ratio between blade height and rotor radius (AR = h/R), rotor radius can be derived from Eq. 3:
(in Eq. 4 power P and wind velocity V_{ 0 } are design data and ρ is air volume mass).
This design approach is iterative and from time to time it will be necessary to reevaluate the blade’s Reynolds number and if necessary repeat the procedure with new power coefficient curves.
The local Reynolds number is:
where c is the chord from Eq. 2, ν is the kinematic air viscosity, and w is the air speed relative to the airfoil as Fig. 2 shows.
Adopting a mathematical approximation, to evaluate the Reynolds number, w can be substituted by ωR with the advantage of having a mean Reynolds number independent of the angle $\mathit{\vartheta}$ of rotation (see Fig. 2).
To conclude the design cycle, simply calculate ωR directly from TSR relative to c_{ pmax } identifiable in Fig. 1,
If the Reynolds number thus calculated is different to the one for the power coefficient curve adopted initially (Fig. 1), a new power coefficient curve should be plotted for a different Reynolds number (second attempt). Usually, the iterative design process needs only 2 or 3 iterations.
Figure 3 shows the power coefficient curves for the wind turbine with the NACA 0018 airfoil, at high Reynolds numbers.
In conclusion, the Reynolds number strongly influences the power coefficient of a verticalaxis wind turbine. Furthermore, it changes as the main dimensions of the turbine rotor change. Increasing rotor diameter rises the Reynolds number of the blade.
The importance of aspect ratio
In Eq. 4 note how radius R increases as ratio AR decreases. In Eq. 2 if R increases, chord c increases too, and in Eq. 7 note how increasing the chord rises the Reynolds number. Finally, Fig. 3 shows how the power coefficient increases as the Reynolds number rises.
Moreover, the rotational velocity ω can be derived from Eq. 6:
Equation (8) shows how ω is inversely proportional to R. From the Fig. 3 graph, note how λ_{cpmax} decreases as Reynolds number rises.
So, to maximize the power coefficient, the rotor’s aspect ratio should be as small as possible. As aspect ratio diminishes there are two advantages: the local Reynolds number rises and simultaneously the rotational velocity diminishes. The effect of the rotor’s aspect ratio on Reynolds number and rotational velocity are shown in Fig. 4 for a twinbladed 1 kW turbine in a wind velocity of 10 m/s.
Figure 5 shows two verticalaxis turbines with identical design power, blade number and aerodynamic profile (NACA 0018) but with two different aspect ratios (AR_{ 1 } = 2; AR_{ 2 } = 0.4). As stated above, the turbine with the lowest AR will have the highest power coefficient and the lowest rotational velocity. This turbine will display two further advantages: firstly, a structural one (thicker blades are more stress resistant) and secondly, inservice stability (greater rotor inertia).
Figure 6 shows the same graphs as Fig. 4 but parameterized for design power variation. Note how the turbines with the highest power have higher Reynolds numbers and lower rotational velocity.
To conclude, VAWT performance is directly related to the blade’s Reynolds number. The higher this is, the better the turbine’s performance. This is because as the Reynolds number increases, the blade’s lift coefficient [24] rises and the drag coefficients decrease (Fig. 7a, b) thus providing greater torque.
Application for a case study
To better explain the design procedure investigated in this work, a case study is presented. From Fig. 4, it is possible to notice that the lower the AR, the higher the Reynolds number; this can improve wind turbine performance. In the design procedure, a choice of a low value for the AR is therefore suitable. In this case study, a comparison between two wind rotors, designed with two different AR values (AR = 2 and AR = 0.4), will be presented.
Suppose, for simplicity, we want to design a VAWT with a rated power of 1 kW, with only two straight blades with symmetrical aerodynamic profile (NACA 0018), and that the installation site is characterized from a mean wind speed of 10 m/s. The design procedure is showed in the Fig. 8. In this figure a flowchart for designing the wind turbine is adopted. In this design procedure, first of all, a Reynolds number must be supposed. This Reynolds number will be called: “the first attempt Reynolds number”. Supposed a first attempt Reynolds number equal to 5 × 10^{6}, from the graph in Fig. 3a it is possible to read the highest performance (c_{ pmax } = 0.51) in correspondence of a particular rotor solidity and tip speed ratio: σ_{ cpmax } = 0.3 e λ_{ cpmax } = 3.0. For an aspect ratio of 2 (h/R = 2) (square VAWT cross section), Eq. 4 will provide the rotor radius R = 0.904 m (ρ = 1.2 kg/m^{3}). From Eq. 2, we obtain a chord c = 0.136 m, and from Eq. 8 we obtain a rotational velocity of 317 rpm. Now a new Reynolds number (second attempt) must be evaluated from Eq. 7: Re = 2.8 × 10^{5}.
Now the design procedure should be repeated with the new value of the Reynolds number. This Reynolds number will be called: “the second attempt Reynolds number”. Taking into account Fig. 3c, we will obtain the new geometric values for the wind turbine. These values are reported in Table 1. The design procedure will end when the Reynolds number does not change in a meaningful manner. The design procedure goes to numerical convergence with only two iterations.
From the data reported in Table 1 for h/R = 2, the design procedure gives rise to a power coefficient of 0.464. Repeating the procedure described above for h/R = 0.4, we will obtain an higher efficiency: c_{ pmax } = 0.475 (see Table 2)
Conclusion
This work looks at designing a verticalaxis wind turbine to maximize its power coefficient. It has been seen that the power coefficient of a wind turbine increases as the blade’s Reynolds number rises. Using a calculation code based on the Multiple Stream Tube Model, it was highlighted that the power coefficient is influenced by both rotor solidity and Reynolds number.
By analysing the factors which influence the Reynolds number, it was found that the ratio between blade height and rotor radius (aspect ratio) influences the Reynolds number and as a consequence the power coefficient.
It has been highlighted that a turbine with a lower aspect ratio has several advantages over one with a higher value.
The advantages of a turbine with a lower aspect ratio are: higher power coefficients, a structural advantage by having a thicker blade (less height and greater chord), greater inservice stability from the greater inertia moment of the turbine rotor.
Abbreviations
 a :

Interference factor (–)
 V _{ 0 } :

Free stream wind speed (m/s)
 R :

Rotor radius (m)
 ω :

Rotor angular velocity (s^{−1})
 h :

Blade length (m)
 w :

Relative airfoil wind speed (m/s)
 α :

Angle of attack (°)
 L :

Lift (N)
 D :

Drag (N)
 R* :

Resultant force (N)
 ϑ :

Blade angular position (°)
 P :

Power (W)
 N _{ b } :

Number of blades (–)
 n :

Rotor rotational velocity (rpm)
 Re :

Reynolds number (–)
 ρ :

Air density (kg/m^{3})
 ν :

Kinematic air viscosity (m^{2}/s)
 c _{ p } :

Power coefficient (–)
 λ :

Tip speed ratio (–)
 σ :

Rotor solidity (–)
 σ _{ cpmax } :

σ that maximizes c_{ p } (–)
 λ _{ cpmax } :

λ that maximizes c_{ p } (–)
 c _{ pmax } :

Maximum c_{ p } (–)
 c :

Airfoil chord (m)
 C _{ L } :

Lift coefficient (–)
 C _{ D } :

Drag coefficient (–)
 NACA:

National Advisory Committee for Aeronautics
 VAWT:

Verticalaxis wind turbine
 HRotor:

VAWT with straight blades
 TSR:

Tip speed ratio
 MSTM:

Multiple Stream Tube Model
 AR :

Aspect ratio
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Brusca, S., Lanzafame, R. & Messina, M. Design of a verticalaxis wind turbine: how the aspect ratio affects the turbine’s performance. Int J Energy Environ Eng 5, 333–340 (2014). https://doi.org/10.1007/s400950140129x
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Keywords
 VAWT
 MSTM
 HRotor
 Reynolds number
 Aspect ratio