Characterization of aerodynamic performance of wind-lens turbine using high-fidelity CFD simulations

Abstract

Wind-lens turbines (WLTs) exhibit the prospect of a higher output power and more suitability for urban areas in comparison to bare wind turbines. The wind-lens typically comprises a diffuser shroud coupled with a flange appended to the exit periphery of the shroud. Wind-lenses can boost the velocity of the incoming wind through the turbine rotor owing to the creation of a low-pressure zone downstream the flanged diffuser. In this paper, the aerodynamic performance of the wind-lens is computationally assessed using high-fidelity transient CFD simulations for shrouds with different profiles, aiming to assess the effect of change of some design parameters such as length, area ratio and flange height of the diffuser shroud on the power augmentation. The power coefficient (C_p) is calculated by solving the URANS equations with the aid of the SST k–ω model. Furthermore, comparisons with experimental data for validation are accomplished to prove that the proposed methodology could be able to precisely predict the aerodynamic behavior of the wind-lens turbine. The results affirm that wind-lens with cycloidal profile yield an augmentation of about 58% increase in power coefficient compared to bare wind turbine of the same rotor swept-area. It is also emphasized that diffusers (cycloid type) of small length could achieve a twice increase in power coefficient while maintaining large flange heights.

Graphical abstract

Keywords

shroud / diffuser-augmented wind turbine (DAWT) / Betz limit / aerodynamics / computational fluid dynamics (CFD)

1 Introduction

As wind energy tends to grow rapidly, the wind poses some challenges to power generation in some sites. Limited sites suitable for wind farm construction, fluctuating low-wind speed, and turbulent nature of the local-wind and extremely gusty-wind are common examples. The new generations of turbines make sites, which are slightly less windy to be feasible economically, but doing so requires even more precise investigation of the resource characteristics and energy potential. Therefore, a new technique to enhance the possibility of wind energy use in sites characterized with lower wind speeds and complex terrains is highly needed. The concept of the diffuser augmented wind turbine (DAWT) has been proposed to augment the generated power from wind. The principle of DAWTs is introduced to increase the power of wind turbine by enclosing the turbine by a duct. It is known that the power produced is proportional to the approaching-wind speed cubed. Therefore, a large increase in power output can be obtained, when there is a possibility of creating even a minor increase in the oncoming wind velocity. If the wind speed increased by means of the dynamic nature of fluid around a structure or topography is called ‘shroud’, the power output could be enhanced locally. In other words, if the wind flow is concentrated locally, the power generated from the wind turbine will be increased substantially, exceeding the Betz limit.

In regards to the development of an efficient wind energy conversion system, this paper aims at finding out an effective method for collecting more wind flow and a suitable type of wind turbines which can generate reasonable amount of energy from the wind. One of the DAWT concepts used for wind acceleration with the name of ‘wind-lens’ has been developed by Ohya et al. [1–10]. Subsequently, a diffuser-shaped configuration able to collect and accelerate the oncoming wind has been developed. Besides, they have proposed a diffuser shroud with a broad-ring flange that has the capability of increasing the wind speed from oncoming wind substantially by the creation of a low-pressure zone as a result of vortex formation. Although it takes the shape of a diffuser structure enveloping a wind turbine rotor like one developed in Refs. [11–17], the feature that makes it quite different is the presence of a large flange on the exit periphery of the diffuser shroud. The shrouded wind turbine equipped with a flanged diffuser (long-style) increases the power output by a factor of about 4–5 times as compared to the un-shrouded wind turbine for the same wind speed and rotor swept-area [6]. Moreover, for practical applications to a micro- and small-scale wind turbine, a compact-style flanged diffuser has been developed. The configuration that consists of a diffuser shroud and a broad-ring flange is significantly modified from the one adopting a long diffuser with a large flange. Shrouded wind turbines with short flanged diffusers (compact-style) have introduced power enhancement by a factor of about 2–3 times in comparison to bare wind turbines [2].

Many computational studies concerning the DAWT are reported by several authors within the historical review. A computational study was developed by Fletcher [18] including both wake rotation and blade Reynolds number effects. Loeffler [19,20] showed that the radial exit flange presented more economic option than extension of the conical diffuser structure to an equivalent exit-area-ratio. In Ref. [21], Georgalas and Koras demonstrated that the performance of a ducted turbine depended mainly on duct chord to duct diameter ratio and the shape of the duct. Hansen et al. [22] conducted CFD simulations to prove that the Betz limit could be beaten. Philips [23] optimized the design of the DAWT by means of CFD computations alongside with some experiments on small scale wind-turbines. Abe and Ohya [8] carried out numerical investigations to predict the flow pattern around diffuser shrouds to promote small-scale wind turbines.

Watson et al. [24] used one-dimensional theory and experimental work to prove that very little improvement could be made by using a concentrator only at the inlet of the turbine. A modification of diffuser structure was done in Ref. [25] by adopting an optimized airfoil to be the sectional profile of the shroud. Gomis [26] used CFD techniques to investigate that different factors had an influence on the aerodynamic performance of the DAWTs. Ohya et al. [10] carried out CFD simulations in order to model the flow around two types of wind-lens turbines (i.e., long and compact styles). Takahashi et al. [9] performed a three-dimensional numerical simulation using the LES approach to predict the blade-tip vortices of the wind-lens turbine.

A new class of DAWTs was presented in Ref. [27]. The diffuser in the new DAWT was designed based on the idea of utilizing high-lift airfoils taken from the aircraft applications. Oka et al. [28] introduced an optimization procedure via the genetic algorithm (GA) coupled with quasi-three-dimensional CFD simulations for designing the wind-lens turbine aerodynamically. A self-adaptive flange was proposed in Ref. [29] to reduce the aerodynamic loads on the flanged diffuser of the wind-lens particularly at high approaching wind velocity. Liu et al. [30] introduced an optimization methodology by combining the CFD and GA methods to develop the shape design of the wind-lens. A multi-objective genetic algorithm (MOGA) with the aid of blade element momentum (BEM) model was employed in Ref. [31] to maximize the wind-lens efficiency through optimizing turbine blade, shroud shape, and flange height.

In summary, the historical review since 1980s showed that the concept of wind turbine equipped with a flanged diffuser shroud or the so-called ‘wind-lens’ is more efficient than any concepts discussed before. In the last decade, many works related to accelerating the approaching were released. The wind-lens turbine was considered to reap remarkably more power output than a bare wind turbine, even if the power coefficient C_p is normalized with respect to the exit diameter of the flanged diffuser. Thus, the authors of the present paper decided to focus only on the concept of ‘wind-lens’ for horizontal-axis wind turbines (HAWTs). To the best of the authors’ knowledge, no reliable computational study used to solve the full set of Navier-Stokes equations in transient mood was dedicated to assess the performance of this type of wind turbines. The novelty of this paper lies in the fact that the adopted approach here is depending on the full unsteady Reynolds-averaged Navier-Stokes (URANS) or high-fidelity CFD simulations to demonstrate the aerodynamic characteristics of the wind-lens turbine developed by Ohya et al. [1–10]. Since most of the research in the existing literature about wind-lens concept have been conducted based on wind-tunnel measurements [2,4–7], even the computational ones have adopted low-fidelity CFD simulation including the use of actuator disk (AD) model or RANS-BEM [10,31] to avoid higher computational cost and to reduce the computational time for their studies. The AD model or RANS-BEM usually model the action of the rotor by introducing body ‘circular disk’ and the body forces are regarded to be originating from 2D airfoil data in combination with the BEM technique. Therefore, high-fidelity CFD simulations using complete URANS are highly needed to precisely predict the aerodynamic performance in more detail. The main targets of the present paper are to develop an accurate 3D model to assess the performance of such wind turbines, and to obtain the optimal design parameters for a wind-lens turbine from the aerodynamic performance point of view. Therefore, the present paper can be considered as a guide for other researchers interested in studying the behavior of such turbines computational to help them verify their numerical models.

2 Theoretical background

The fundamental equations used to govern the fluid motion can be solved using ANSYS-Fluent® in all flow moods, steady or transient, compressible or incompressible and adiabatic or flow including heat transfer. The conservation of mass ‘the continuity of flow’ and conservation of momentum ‘the Newton’s Second Law of Motion’ are two of the most common basic laws in fluid dynamics. The incompressible version (r = constant) of the conservation of mass and momentum equations of the fluid are respectively written as

(1){\displaystyle \frac{\partial u_i}{\partial x_i}}=\mathrm{0},

(2){\displaystyle \frac{\partial u_i}{\partial t}}+u_j{\displaystyle \frac{\partial u_i}{\partial x_j}}=-{\displaystyle \frac{\mathrm{1}}{\rho }}{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\mu }{\rho }}{\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_i}{\partial x_j\partial x_j}}-\mathrm{2}\Omega u_i+F_i,

where u is velocity vector, t is time, x is Cartesian coordinate in the x-direction, i is the vector notation in x, j is the vector notation in y, p is static pressure, μ is absolute viscosity, Ω is angular velocity, and F is external body forces.

The shear-stress transport (SST) k–ω is classified as a hybrid RANS model, combining the strength of the k–ω model in the farfield region and the k–ω model in the near-wall region to improve the flow property calculations [32]. The SST k–ω model was developed by Menter [33] to heal the problem of insufficiency of the k–ω model in modeling the near-wall flow accurately and predicting the boundary layer flow properly. Several authors claimed that the SST k–ω model is superior to the k–ω model especially when adverse pressure gradients are present. A standard k–ω model is utilized to calculate flow variables in regions far away from the wall, while a modified k–ω model is used near the wall by considering the turbulence frequency (ω) as a second variable instead of variable ε which acts for the turbulent kinetic energy dissipation (ε). The SST k–ω model computes the Reynolds stresses in the same manner as in the k–ω model. The transport equations of the SST k–ω model including low-Re correction can be written as [34,35]

(3){\displaystyle \frac{\partial k}{\partial t}}+u_i{\displaystyle \frac{\partial k}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_i}}\left[{\displaystyle \frac{\mathrm{1}}{\rho }}(\mu +{\sigma }_k{\mu }_t){\displaystyle \frac{\partial k}{\partial x_i}}\right]+{\displaystyle \frac{\mathrm{1}}{\rho }}{\tilde{P}}_k-{\beta }^{*}k\omega ,

(4){\displaystyle \frac{\partial \omega }{\partial t}}+u_i{\displaystyle \frac{\partial \omega }{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_i}}\left[{\displaystyle \frac{\mathrm{1}}{\rho }}(\mu +{\sigma }_{\omega }{\mu }_t){\displaystyle \frac{\partial \omega }{\partial x_i}}\right]+{\displaystyle \frac{\mathrm{1}}{\rho }}{\displaystyle \frac{\alpha }{v_t}}{\tilde{P}}_k-\beta {\omega }^{\mathrm{2}}+\mathrm{2}(\mathrm{1}-F_{\mathrm{1}}){\displaystyle \frac{{\sigma }_{\omega ,\mathrm{2}}\partial k\partial \omega }{\omega \partial x_i\partial x_i}},

where k is turbulence kinetic energy,ω is specific dissipation rate, μ_{t is eddy viscosity, {\tilde{P}}_k is turbulence kinetic energy due to mean velocity gradients, and F1 is blending function. Model constants are σk = 2.0, σω= 2.0,β*= 0.09, β= 0.072, α= 1.0, and σω,2= 0.856.}

The advantages of the SST k–ω model over other RANS turbulence models have been reported in the field of numerical simulation and wake modeling of wind turbines [36]. Furthermore, it was the first RANS model to entirely predict the performance of a wind turbine [37,38]. Therefore, the SST k–ω model has been selected to model the turbulence with high accuracy in the present paper. Table 1 presents the different turbulence models employed by several authors to investigate the performance of DAWTs.

3 Methodology

3.1 Geometry

A rotor blade is created depending on the one-dimensional theory of wind-lens proposed by Inoue et al. [1]. The flow structure around the two common styles of wind-lens turbines are illustrated in Fig. 1. The data of the real physical dimensions represented by thickness (t), chord length (c) and the blade length (r) of the blade along surface coordinates, simply x, y, and z, respectively are shown in Fig. 2. The upwind configuration of a wind-lens turbine, which comprises a shortened diffuser coupled with a broad-ring flange, is presented in Fig. 3. The rotor blade consists of airfoil series arranged in the manner of MEL20M01, MEL18M31, and MEL12M84 from root to tip of the blade [10]. The airfoil shapes that forms the rotor and the empirical values of lift coefficient C_l and drag coefficient C_d of the MEL airfoils can be found in Refs. [54,55]. Both the rotor tip and hub diameters are designed to be 1.0 m and 0.13 m, respectively. The rotor blade needs to be pitched by 4° toward the direction of wind from the rotor plane in order to match the experimental setup.

For wind-lens, some geometrical parameters like flange height (h), diffuser length (L_N+ L_D) and area ratio (A_exit/A_throat) are investigated. A schematic diagram of the wind-lens turbine and its corresponding parameters are clarified in Fig. 4. The computational analysis is initiated with four designs of WLTs introduced in Ref. [2]. The four shapes are assigned to Aii, Bii, Cii, and Sii as considered in the experimental tests, each of which differs from the other in terms of length and area of the diffuser shroud as shown in Fig. 5. All wind-lenses have roughly exact (L_N+ L_D)/D and dissimilar A_exit/A_throat. In particular, the Sii-type wind-lens is designed to be of straight linear profile. For Aii and Bii types, the curved sectional shape is selected while a cycloid sectional shape is adopted in the case of Cii wind-lens. The length to diameter ratio (L_N+ L_D)/D and the area ratio A_exit/A_throat corresponding to each type are reported in Table 2 [2].

From the experimental results obtained in Ref. [2], it can be realized that the C-type (cycloid type) wind-lens is the most efficient compact collection-acceleration structure. Hence, the length effect of the C-type is aerodynamically examined to find the most efficient configuration. For this purpose, another four models with cycloidal cross-section ranging from C0 to Ciii are constructed as depicted in Fig. 6. Table 3 gives the values of length to diameter ratios (L_N+ L_D)/D and area ratios A_exit/A_throat for the C-type models as stated in Ref. [2].

Table 4 presents the features and required data for the wind-lens turbines included in the current paper 2. Both the Reynolds number (Re) and the wind velocity (U₀) are maintained to be similar to those in the wind-tunnel measurements performed in Ref. [2].

3.2 Flow domain

The flow domain consists of two cylindrical regions: an inner region containing the rotor blade, which is rotating with varied angular velocity to evaluate the turbine performance and a stationary outer region. The diameter of the flow domain is equal to 20 times the rotor diameter to eliminate the far field effects by keeping the brimmed diffuser far enough from boundaries. The flow domain is extended to 7.5 rotor diameters upwind of rotor where the inlet is located and 12.5 diameters downwind where the outlet is located based on Ref. [10]. For simplicity and computational resources saving, the original flow domain was reduced to be one-third of the flow domain of the whole geometry of a wind-lens turbine. This technique has been previously applied in Refs. [50,56]. By taking the advantage of the 120 degrees periodicity of a three-bladed rotor inside the wind-lens, only one blade is explicitly considered as indicated in Fig. 7.

3.3 Unstructured mesh

The computational meshes considered are created using ANSYS-Meshing, the automated tool supplied by ANSYS Inc. Workbench. The unstructured mesh is employed for the whole flow domain with 1327255 tetrahedral cells in the rotating region and less tetrahedral cells of 897395 contained in the stationary one (see Fig. 8(a)). An intensive mesh density is maintained around the wall boundaries to get y⁺ less than unity through the implementation of inflation/prismatic layers. Fifteen inflation/prismatic layers are created on the blade while 10 layers are assigned to the surface of the diffuser with a growth rate of 1.2. The first cell near the walls is located at a normal distance of about 1 × 10⁻⁵ m to achieve y⁺–1. Figure 8(b) demonstrates the detailed view of the mesh around blade tip and near the hub.

Unfortunately, the implementation of inflation layers adversely affects the mesh quality as it produces very thin prisms of high cell aspect ratio. A group of cells with severely non-orthogonal faces appeared around the sharp trailing edge of the turbine blade. This might cause instability and inaccuracy in computational calculations. To weed out the highly non-orthogonal cells, the blade is slightly modified by employing the blunt trailing edge instead of the sharp trailing edge [57]. This modulation is simply carried out via reducing the chord length of each blade section by roughly 2% and, therefore, the cell skewness is kept below 0.85.

Four levels of mesh are tested to ensure that the obtained results are mesh-independent. The Cii model as an example of wind-lens is tested at l = 4.3 to guarantee the mesh independency. The maximum power coefficient (C_p,max) corresponding to each mesh level is recorded in Table 5. As seen, the relative difference between the predicted values of C_p,max obtained from the M-4 and M-3 meshes is 0.44% which is considered to be in the range of the experimental error so that the medium mesh M-3 with 2.22 × 10⁶ cells can be chosen for wind-lens CFD simulations and sounds to introduce an adequate mesh independency.

3.4 Boundary conditions

Boundary conditions (BCs) define the appropriate flow properties on faces of boundaries for the chosen model. Therefore, they are the corner stone of any simulation, and it is important to specify the BCs properly. Many kinds of underlying BCs are employed in the present simulations i.e., velocity-inlet, pressure-outlet, symmetry, non-conformal interface, and no-slip condition for both stationary and moving walls.

3.5 Solver settings

The CFD solver is set to operate in transient mood for the high-fidelity CFD simulations. The well-known software ANSYS-Fluent® is used to solve the URANS equations based on the finite-volume approach. The fluid could be treated as incompressible since the inlet flow velocity does not exceed 8 m/s during the simulations. The pressure-based solver is considered to calculate such an incompressible flow field.

3.6 Pressure-velocity coupling

Several approaches for pressure-velocity coupling are possible. The coupling between velocity and pressure is done by using the PISO algorithm (pressure implicit with splitting of operator). It should be noted that no any skewness correction is needed as the quality of the implemented mesh is good enough.

3.7 Spatial and temporal discretization

The least square cell-based scheme is used for the spatial discretization of the gradients. All variables are solved by the second-order upwind scheme, except for the pressure, which is interpolated by the standard scheme. The standard pressure interpolation scheme is previously used to characterize the performance of HAWTs as reported in Ref. [58]. The 2nd-order implicit scheme is employed to account for transient formulation. Table 6 introduces an overview of the spatial as well as temporal discretization schemes implemented in the present paper.

3.8 Time marching step

The unsteady performance of the turbine rotor is simulated with a constant inlet velocity, while changing the rotor rotational speed to obtain different tip speed ratios. The time step (∆t) is set for each rotational speed corresponding to an azimuthal angular step (Dq) based on Ref. [59]. Each physical time step is set to perform about 30 sub-iterations in order to obtain a converged solution in the simulations. Similar to the mesh density, the effect of the time step (∆t) on the CFD solution should be investigated. For this purpose, four different values of time step (∆t) equivalent to azimuthal angular steps (Dq) of 0.25°, 0.5°, 1° and 3° are included. As shown in Table 7, the difference in C_p is less than 1% for the 0.5° and 1° time steps, while it exceeds this threshold for the 3° time step. Therefore, a time step of 1° has been systematically retained for the rest of the CFD simulations.

3.9 Convergence criteria

A double check is performed to ensure solution convergence. The first criterion of convergence relies on the torque coefficient C_Q based on the rotor swept-area (A = pR²):

(5)C_Q={\displaystyle \frac{Q}{{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho AR{U}_{\mathrm{0}}^{\mathrm{2}}}}.

Globally, each simulation executes when the relative variation between the value of torque coefficient at the (N)-cycle and its value at the (N–1)-cycle follows the relation of

(6)

The torque coefficient C_Q is obtained by averaging its values over the last two cycles. It is indicated that each CFD-simulation needs from 4 up to 9 cycles depending on the value of tip-speed ratio in order to obtain a quasi-steady-state solution. Figure 9 displays an example of the process by which the torque coefficient C_Q follows to attain the quasi-steady-state at l = 4.3.

A further test is conducted on the scaled residuals, which ought to be less than 10⁻⁶ for equations of momentum, turbulence kinetic energy (k), and turbulence specific dissipation rate (w) while 10⁻⁴ is used for continuity equation.

4 Results and discussion

Two main indicators, i.e., power and torque alongside their variations with wind speed usually describe the aerodynamics of any wind turbine. However, it is more appropriate to express the aerodynamic performance in terms of non-dimensional parameters like power and torque coefficients, which are functions of tip speed ratio. CFD methods as a widely accepted analytical tool applied for wind turbines are used to assess the performance characteristics of wind turbines and to reveal the flow-field around the wind-turbine blades. Therefore, the experimental data reported in Ref. [2] are selected to validate the high-fidelity CFD results obtained from the present model. As mentioned before, the wind-lens employed is a perfect replica of the one adopted during the experiments conducted in Ref. [2]. The computational results are calculated at both the same wind speed (U₀ = 8 m/s) and swept-area of the wind-lens turbine (A = 0.785 m²) as those in the experiment.

4.1 Effect of wind-lens shape

Figure 10 shows a comparison of the computational results against the experimental results recorded in Ref. [2] for the Aii, Bii, Cii and Sii wind-lens turbines. The power coefficient in the present paper is calculated based on the rotor swept-area (πR²) as C_{\mathrm{p}}=P/{\displaystyle (\frac{\mathrm{1}}{\mathrm{2}}}\rho A{U}_{\mathrm{0}}^{\mathrm{3}}). The current high-fidelity CFD results are consistent with the experimental measurements, in particular, the power coefficient of Aii, Bii, and Cii types until incipient stall condition, at \lambda= 4.3. The results indicate that when a wind-lens technology is adopted, an emphatic increase in the power coefficient is successfully achieved to be approximately 2.5 times as large as a wind turbine without diffuser shroud. The maximum power coefficient (C_p,max) obtained from the high-fidelity CFD simulation is 0.39 for a bare wind turbine compared to C_p,max of wind-lens turbines which is in the range of 0.70–0.92. The comparison of Aii (circular type) against Sii (linear type) which have almost the same area ratio A_exit/A_throat, indicates that Aii type is more efficient than Sii type. In addition, it can be noted that both Bii (circular type) and Cii (cycloid type) give higher values of C_p compared to Aii diffuser (circular type). The present CFD results are consistent with the experimental measurements except that CFD results suggested that Bii gives the highest C_p whereas experimental measurements indicated that the C_p of Cii is relatively higher than that of Bii. The flow-field analysis reveals that the boundary-layer flow on the inner surface of the curved diffuser Cii does not show a remarkable flow separation. Consequently, Cii wind-lens is preferred over Aii, which has less area ratio A_exit/A_throat compared to that of Cii.

According to experimental measurements, C-type with a cycloidal shape is found to be the most efficient collection-acceleration device compared to the others. Thus, Ohya et al. [2] studied further the C-type to reach the optimum proportions of the diffuser shroud. In the same way, the cycloid type is numerically investigated with the new dimensions in the current paper. The effect of the change of shroud length to diameter ratio (L_N + L_D)/D on the performance of C-type wind-lens is examined. Figure 11 compares the output of the CFD simulations against the experimental data reported in Ref. [2]. For all cycloid types (i.e., C0, Ci, Cii, and Ciii), the flange height (h = 0.1D) is maintained in this paper. It can be noticed in Fig. 11 that the Ciii wind-lens shows higher values of C_p,max = 0.960 compared to other wind-lens turbines of cycloid sectional shape.

It is clearly seen from Figs. 10 and 11 that in some cases where CFD fails to predict the C_p of wind-lens turbines at high tip-speed ratios, the simulation results in significantly higher values. It can be seen that the power coefficient exhibits a jump in the experimental results through the tip-speed ratios from l = 3 to l = 3.25 for all tested WLTs. However, the results obtained from the current CFD simulations does not exhibit this jump. The maximum power coefficient predicted by the high-fidelity CFD approach occurs at l = 4.6, whereas for the experimental results it occurs at l = 4.3.

The difference between experimental and CFD data could be globally explained by the fact that experimental data represents the real behavior of the object under test with specific measuring errors while CFD data represents the behavior of the same object based on its theoretical model. In the current paper, the deviation between experimental and simulation results could be caused by insufficient mesh refinements within the wake region behind the wind-lens turbine. It is known that the wake has a great influence on the blade aerodynamics especially the aerodynamic forces on the rotor blade like drag and lift forces. Therefore, the wake must be accurately resolved to provide a good prediction of the output power, which mainly depends on the blade forces. In the present paper, a fair mesh refinement is adopted because of the lack of the computational resources. Another reason could be related to the blunt trailing edge of the rotor blade used to simplify the meshing process, while during the experiments the trailing edge of the rotor blade is not modified. This might have affected the nature of the flow around the rotor blade. It can be concluded that, in spite of some deviation, the employed CFD model could generally predict the power coefficient (C_p) with a high accuracy.

Table 8 introduces the comparison of the CFD predicted and experimental measured values of the maximum power coefficient C_p,max for different wind-lens shapes. The Bii type shows the highest power coefficient of C_p,max = 0.920 at λ= 4.6 as compared to the bare turbine of C_p,max = 0.392 at λ= 3.6.

Table 9 illustrates the comparison of the CFD predicted and experimental measured values of the maximum power coefficient C_p,max for C-type wind-lens shapes. The Ciii type shows the highest power coefficient of C_p,max = 0.960 at λ= 4.6 as compared to the bare turbine of C_p,max = 0.392 at λ= 3.6.

The complex flow in the blade-tip region of both bare and wind-lens turbines is reasonably predicted. To better observe the vortex development, a 3D volume rendering (Iso-surface) of the circumferential vorticity (ζ_x) at the optimum tip-speed ratio is illustrated in Fig. 12. The circumferential vortices are identified using the {\lambda }_{\mathrm{2}} criterion. The vortex colored in red is defined as blade-tip vortex since it flows from the front side of the blade toward the rear side. On the other hand, a strong counter-rotating vortex colored in blue is induced between the blade-tip and the inside-wall of the diffuser shroud. This counter-rotating vortex is possibly generated owing to the interaction between the blade-tip vortex and the inside-wall of the diffuser shroud.

It can be observed that WLTs generate stronger blade-tip vortices than those of bare wind turbines. This is attributed to the augmented influx near the inner-surface of the diffuser, which is collected and accelerated by the wind-lens device. It is worth mentioning that the induced counter-rotating vortices colored in blue totally disappear in the case of bare wind turbines unlike the wind-lens turbines.

Initially, the blade-tip vortex is found to be in alignment with the induced counter-rotating vortex. Thereafter, the induced counter-rotating is partly rolled up by the blade-tip vortex. More downstream, the blade-tip vortices, in turn, roll up and form a ring-structure with the induced counter-rotating vortices. As the wake progresses, the blade-tip vortices and the counter-rotating vortices are weakened together and do not break down before they reach the last portion of the diffuser shroud. As shown in Fig. 12, Cii (cycloid type) wind-lens (WL) generate much stronger blade-tip vortices which break down faster than Aii and Sii types. Similarly, Ciii-type shows the strongest blade-tip vortices among all the cycloid types. On the other side, the blade-tip vortices of the C0 wind-lens are characterized by weaker, more stable vortices.

Contour plots for acceleration factor K=U_z/U₀ around bare and various wind-lens turbines on the central plane are computed in Fig. 13. As shown, the highest velocities which are higher than the approaching wind velocity (U₀) can be observed on the throat section of all wind lenses because of the diffuser shroud effect. In other words, the maximum acceleration factor (K) on the throat appears within the region located between the blade-tip and the internal face of the diffuser shroud.

A recirculation region with a lower speed (area in blue) is clearly visible between blade-tip vortices and the wake core in the case of the WLT. However, in the case of bare wind turbine, this decelerated region almost disappears. For all types of wind-lens turbines, a low-speed region is formed behind the flange attached to the diffuser shroud. Bare wind turbine has no flanged diffuser so that no separated flow can be detected except for the flow located downstream the hub of the rotor. It is worth highlighting that the effect of the separated flow on the turbine performance might be ignored because of its orientation on the rotor axis with minimal torque arm.

Remarkably, the Sii type has a lower wake effect compared to the other wind-lens turbines as the size of the circulation zone is smaller. It can be noticed that the smallest zone of circulation is assigned to the C0 wind-lens. On the other hand, it is noticed that diffusers with a high length to diameter ratio (L_N + L_D)/D like Cii and Ciii-types have the highest wake effect and coupled with a wide circulation zone.

Figure 14 compares the static pressure distribution for bare turbine and different configurations of wind-lens turbines. It is indicated that the flange of the wind-lens promotes the pressure drop at the rotor plane due to the appearance of the negative pressure region by the action of the vortices formed downstream the wind-lens. This negative pressure allows to draw more mass flow to the wind turbine inside the diffuser shroud and consequently the power output can be increased. As seen in Fig. 14, the lowest negative pressure is noticed in the case of Sii wind-lens while Ciii-type possesses the highest negative pressure among all C-shape wind-lenses. Many previous researches have proved that the size of the negative pressure region behind the wind-lens usually plays a big role in the mechanism of wind acceleration. The higher the value of negative pressure, the greater effect of wind acceleration can be obtained. Therefore, the power augmentation using this acceleration mechanism should be interpreted through depiction of the vortices formed downstream the flange of the diffuser which directly affects the negative pressure.

The streamlines in the rear area of the flange for different wind-lens turbines are illustrated in Fig. 15. Generally, the flange of the wind-lens is considered to be an obstacle for the airflow to enter inside the diffuser smoothly. A pair of two opposite direction vortices is generated behind the wind-lens, producing a negative pressure region, which increases the pressure in front of the flange of the diffuser and consequently increases the power coefficient. This draws more airflow into the inlet part of the shroud where the maximum wind speed is noted. Aside from these two opposite-direction vortices, small eddies emerge inside the diffuser due to the boundary layer flow separation which reduces power coefficient. Consequently, there are two main effects that judge the optimum wind-lens design for a better aerodynamic performance. The higher or lower power coefficient depends on which of these two effects is the dominant. It can be observed that the two opposite direction vortices behind the Cii wind-lens (cycloid type) is much stronger than those of Aii, Bii, and Sii. Considering the C-shape wind-lenses, Fig. 18 shows that the two opposite-direction vortices downstream the flanged diffuser is not strong enough. This is the main reason of the lower maximum power coefficient C_p in the case of short diffusers (i.e., C0 and Ci) compared to the diffusers with a high length to diameter ratio (L_N + L_D/D) such as Cii and Ciii for a prescribed flange height (h). The current streamlines pattern around the wind-lens visualized in Fig. 15 is confirmed in many previous numerical and experimental researches [2,6,8,10,31,33,34].

4.2 Effect of flange height to diameter ratio (h/D)

Another important parameter namely ‘the flange height’ is taken into consideration in the present paper. The flange height represented in terms of non-dimensional flange height to diameter ratio (h/D) is changed in the range of 0 to 0.2 with a step of 0.05 to demonstrate its effect on both the power augmentation and the flow pattern. Figure 16 is a comparison of the C_p,_max obtained from the experimental measurements [2] and the C_p,_max predicted in the current paper at different diffuser length to diameter ratios (L_N + L_D)/D. C_p,_max is defined as the maximum value of C_p corresponding to the optimal tip-speed ratio λ for each type. It is found that the C_p,_max value decreases as the diffuser length to diameter ratio (L_N + L_D)/D decreases. The results indicate that a wind-lens turbine of large flanges is more efficient than one equipped with small flanges.

Figure 16 shows that wind-lens turbines with a flange height to diameter ratio of h/D = 0.2 perform better and achieve the maximum power augmentation. Furthermore, when the flange height is larger than 10% (h>0.1D), the power coefficient C_p of C0-type wind-lens has approximately a doubled increase when compared to a bare wind turbine, while the Ciii-type wind-lens has approximately a 2.6-times increase. Therefore, a 2–3 times increase in output power coefficient C_p can be expected, even if a very compact brimmed diffuser is used as the default shroud for a wind-lens turbine. Both experimental [2] and present computational results reach the same conclusion of achieving a twice increase in power coefficient for a flanged diffuser with a small length to diameter ratio (L_N+ L_D)/D while maintaining a large brim height to diameter ratio h/D. The Cii-type is the chosen diffuser type to explain in more detail the flow field around wind-lens turbine when the flange height (h) becomes variable.

As illustrated before, the blade-tip vortices of the bare turbine are collected near the rotor, while the blade-tip vortices in the case of the wind-lens are shifted further downstream as the accelerated influx through the rotor pulls these vortices away from the blade region. The interaction between the blade-tip vortices and the turbine rotor weakens behind the wind-lens. Consequently, the aerodynamic loads of the wind-lens are higher in comparison to the bare wind turbine.

The effect of flange height (h) on the vortical pattern of the wind-lens turbine in the near wake region is illustrated in Fig. 17. The 3D visualization of this vortical pattern is explained in terms of the circumferential vorticity (ζ_x) using the l₂-criterion. For h = 0.2D, the wind-lens generates strong blade-tip vortices, however the vortices break down rapidly compared to others. Conversely, the vortices of the wind-lens equipped with a short flange (i.e., h = 0.05D) are weaker but relatively stable with ring structure and can propagate a longer downstream distance.

Figure 18 shows the contour plots of the acceleration factor (K) around the C-shape wind-lens turbines at h/D = 0.05, 0.1, 0.15 and 0.2. By increasing the flange height (h), the flow through the throat is strongly accelerated, leading to a vortices formation that causes the negative pressure located behind the flange. Consequently, this lowered pressure draws much more airflows into the turbine rotor. It is clearly seen that the size of the low-speed recirculation zone for wide flanges is remarkably larger than the size of this zone in the wake region of wind-lens configurations with small flange heights. Therefore, the power coefficient (C_p) for h = 0.2D shows higher values in comparison to those for the wind-lens configurations with a small flange height like h = 0.05D.

The flange height (h) also affects the static pressure distribution inside the wind-lens turbine. Figure 19 shows the contours of static pressure for wind-lenses of different flange height to diameter ratios (h/D). It is observed from Fig. 19 that increasing the flange height (h) from 0.1D to 0.2D causes an increase in the flange frontal pressure of the diffuser and a substantial reduction in the pressure behind the diffuser. It can be affirmed that wind-lens with a flange height (h) equal to 0.2D have the maximum possible flange frontal pressure with the minimum possible back pressure. This explains why the C-type wind-lens shows the best performance in terms of the output power and operating range at h/D = 0.2 as seen before in Fig. 16.

Figure 20 shows that as the flange height increases the flow inside the wind-lens tends to separate from the inner wall of the diffuser especially in the case of diffusers of large lengths. Wind-lens turbines with a high flange height of 0.2D has the widest high-pressure region upstream characterized with the strongest vorticity shedding behind the diffuser. The configuration of Ciii-type with a flange height of 0.2D performs better and shows the highest power coefficient compared to other configurations. By increasing the flange height (h) more than 0.2D, the flow begins to separate strongly from the inner surface of the diffuser. Additionally, a severe increase in the flange frontal pressure is noticed when the the flange height (h) becomes greater than 0.2D. This high pressure extends to the main streamflow and impedes the fluid flowing inside the wind-lens turbine. As a result, the output power coefficient (C_p) is dramatically reduced in spite of the lower back pressure created when keeping the wider flange of the diffuser.

5 Conclusions

This paper studies the aerodynamic performance of the so-called wind-lens turbine using high-fidelity CFD simulations. The use of low-fidelity models such as actuator disk and RANS-BEM models would significantly affect the results because of the approximations associated with the application of such models. Therefore, CFD simulations relying on full URANS could be a more efficient tool to characterize the aerodynamic performance of wind-lens turbines in a more accurate way. The present CFD results match well with the available experimental data in the open literature. At the same time, the effect of the wind-lens shape represented by the length to diameter ratio ((L_N + L_D)/D) and area ratio (A_exit/A_throat) on the flow-field is computationally analyzed. Based on the findings, the following conclusions can be reached:

The Cii (cycloid type) has higher values of C_p compared to Aii (circular type), Bii (circular type), and Sii (linear type). Therefore, Cii diffuser is considered to be the suitable wind-lens shape with maximum power coefficient of 0.92.

The effect of changing the length of the C-type shroud on the performance of wind-lens turbines is computationally assessed, and the Ciii type shows higher values of power coefficient than other C-type diffusers with C_p,_max= 0.96.

The flange height (h) for C-type wind-lenses is varied from 0.05D to 0.2D, and it is found that the maximum power coefficient (C_p,_max) increases as the flange widens.

About a twice increase is achieved in C_p,_max for wind-lens (cycloid type) with a small length to diameter ratio (L_D + L_N)/D while maintaining a large flange height to diameter ratio h/D. In other words, C0 and Ci types can be a suitable choice if the flange height (h) is kept greater than 0.1D.

As a future study, it is recommended more advanced turbulence models like hybrid RANS/LES or pure large eddy simulation (LES) be employed to predict the performance characteristics of the wind-lens and compare it to those from URANS simulations.

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