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Research Papers

Ducted Wind Turbine Optimization OPEN ACCESS

[+] Author and Article Information
Ravon Venters

Mechanical and Aeronautical
Engineering Department,
Clarkson University,
Potsdam, NY 13699-5725
e-mail: venterrm@clarkson.edu

Brian T. Helenbrook

Professor
Mechanical and Aeronautical
Engineering Department,
Clarkson University,
Potsdam, NY 13699-5725
e-mail: helenbrk@clarkson.edu

Kenneth D. Visser

Associate Professor
Mechanical and Aeronautical
Engineering Department,
Clarkson University,
Potsdam, NY 13699-5725
e-mail: visser@clarkson.edu

1Corresponding author.

Contributed by the Solar Energy Division of ASME for publication in the JOURNAL OF SOLAR ENERGY ENGINEERING: INCLUDING WIND ENERGY AND BUILDING ENERGY CONSERVATION. Manuscript received March 10, 2016; final manuscript received August 1, 2017; published online November 29, 2017. Assoc. Editor: Douglas Cairns.

J. Sol. Energy Eng 140(1), 011005 (Nov 29, 2017) (8 pages) Paper No: SOL-16-1123; doi: 10.1115/1.4037741 History: Received March 10, 2016; Revised August 01, 2017

This study presents a numerical optimization of a ducted wind turbine (DWT) to maximize power output. The cross section of the duct was an Eppler 423 airfoil, which is a cambered airfoil with a high lift coefficient (CL). The rotor was modeled as an actuator disk, and the Reynolds-averaged Navier–Stokes (RANS) k–ε model was used to simulate the flow. The optimization determined the optimal placement and angle for the duct relative to the rotor disk, as well as the optimal coefficient of thrust for the rotor. It was determined that the optimal coefficient of thrust is similar to an open rotor in spite of the fact that the local flow velocity is modified by the duct. The optimal angle of attack of the duct was much larger than the separation angle of attack of the airfoil in a freestream. Large angles of attack did not induce separation on the duct because the expansion caused by the rotor disk helped keep the flow attached. For the same rotor area, the power output of the largest DWT was 66% greater than an open rotor. For the same total cross-sectional area of the entire device, the DWT also outperformed an open rotor, exceeding Betz's limit by a small margin.

Ducted wind turbines (DWTs), in comparison to open-rotor turbines, increase the power obtained by a fixed sized rotor by increasing the momentum through the rotor. Figure 1 shows an isometric view of a conceptual DWT design. The DWT is not a new idea as it was first introduced by Lilley and Rainbird [1], with the earliest experimental research performed by Foreman et al. [2,3]. Foreman et al. designed a DWT with a theoretical power output, based on one-dimensional (1D) analysis, of two times that of a bare turbine with the same rotor size. Igra [4] also conducted 1D analysis and provided supporting insight into the advantages of a DWT, introducing the use of an airfoil as the cross section of the duct. It took a number of years before the ideas of Lilley and Rainbird were used in a full scaled DWT. The development of the full-scaled turbine was conducted by Vortec Energy, supported by research conducted by Phillips et al. [5]. They determined that the efficiency of the turbine, based on the rotor size, exceeded that of Betz's limit. However, no viable commercial product was ever realized.

Recently, there has been renewed interest in DWT systems [627]. Hansen et al. [6] verified the research conducted by Foreman and Gilbert [3], extending the 1D analysis with a computational fluid dynamics (CFD) study. The results showed an increase in mass flow at the plane of the rotor with the inclusion of a duct. Using one-dimensional actuator disk theory, Jamieson [15] developed an ideal limiting theory for any ducted rotor. The study established that the maximum amount of energy which can be extracted is independent of the ducted system as long as the duct itself does not extract energy. Research by Ohya et al. [16,19] was directed toward the use of a flanged duct. The studies involved experimental and computational analysis of the duct, concluding that the addition of the duct to the turbine could increase the power output by a factor of five. Mansour and Meskinkhoda also investigated the use of a flanged diffuser reporting similar results as Ohya et al. The studies mentioned thus far used a duct with the cross-sectional geometry of a flat plate. There have been a number of studies which used an airfoil for the cross-sectional geometry of the duct. Franković and Vrsalović [7] observed that the airfoil provides a lift force, which allows greater expansion of the rotor wake, amplifying the power output. Shives and Crawford [28] also used an airfoil shaped cross section. The study focused on the effects of viscous loss, flow separation, and the base pressure for a DWT. An analytical model was developed based on CFD results to assess the effects. The results suggested that the inlet efficiency is unchanged with the design parameters, flow separation negatively affects the performance of the duct, and the base pressure effect can increase the performance. A more recent study by Aranake et al. [24] performed three-dimensional simulations adopting the airfoil cross-sectional geometry. The results substantiated prior studies that used two-dimensional and axisymmetric geometries. Through the use of analytical and numerical methods Bontempo and Manna [2527] extended nonlinear actuator disk theory, which they applied to open and ducted wind turbines. They verified results from axial momentum theory and showed how a cambered airfoil duct can significantly increase the power output of an open-rotor turbine.

Based on the studies of DWTs utilizing an airfoil geometry as the cross section of the duct, the ideal airfoil geometry is a cambered airfoil, because it provides high lift coefficients. In most designs, the airfoil is typically positioned at an angle of attack such that the lift coefficient is maximal. Thus, the flow over the airfoil is close to separating. Beyond this angle of attack, performance of the airfoil will decrease. Predicting the maximal lift coefficient is critical for creating an optimally designed duct.

The goal of this work was to determine the optimum orientation of a duct to increase the overall efficiency of a DWT. None of the studies discussed above performed a full CFD optimization study addressing the ducts orientation. To perform the optimization, Reynolds-averaged Navier–Stokes (RANS) CFD simulations were used with the k–ε realizable turbulence model. An Eppler 432 airfoil was chosen for the cross section of the duct. This airfoil was selected from the University of Illinois at Urbana-Champaign (UIUC) airfoil database for its high maximum lift coefficient, CL. Based on the experimental results, it has a maximum CL of 2. A validation study for the airfoil was conducted to verify the accuracy of the RANS model near stall conditions. To model the rotor, an actuator disk approximation was used. The ANSYS Workbench optimization package was used to determine the optimal position and angle of attack for the duct, as well as the optimal coefficient of thrust for the rotor.

The Eppler airfoil was chosen for its high lift coefficient based on the experimental results from the UIUC wind tunnel experiments [29]. The E423 airfoil is designed using the Eppler design method [30], which determines the airfoil shape that will produce a specified pressure distribution and exhibit desired boundary-layer characteristics. The E423 is designed to produce a high lift coefficient at lower Reynolds numbers. Table 1 gives the airfoil's geometric parameters and the corresponding maximum CL attained at a Reynolds (Re) number of 3.0 × 105. The geometry of the airfoil is shown in Fig. 2.

Numerical Validation.

All numerical simulations were performed using the Reynolds-averaged Navier–Stokes equations with the k–ε closure model, and the flow was assumed incompressible. The commercial code Fluent [31] was used with the SIMPLE algorithm to update the solution and Fluent's second-order accurate unstructured scheme for quadrilaterals to discretize the equations. To verify the accuracy of this formulation, numerical results for the E423 airfoil were compared to the experimental data taken in the UIUC open return subsonic wind tunnel [29]. The overall experimental uncertainty was estimated to be 1.5% for the lift and 4.5% for the drag. Further discussion of the experimental data can be found in Selig and Guglielmo [29].

To perform the airfoil simulations, a domain was constructed which encompassed the airfoil as shown in Fig. 3(a). The domain extended 20 chord lengths upstream and 30 chord lengths downstream of the airfoil. The shape of the domain was chosen such that there was a distinct transition between the inflow and outflow boundaries. This eliminated convergence difficulties associated with having reverse flow through outlet boundaries. An unstructured quadrilateral mesh was generated for the domain. The mesh shown in Fig. 3(a) was composed of about 100,000 elements.

On the airfoil, a wall boundary condition was applied with enhanced wall treatment, which required y+ ≤ 5 to accurately describe the flow. To satisfy this condition, a boundary layer mesh with 15 layers and a growth rate of 1.2 were added on the airfoil surface. The initial wall spacing was 1.2 × 10–4 in chord units, yielding a maximum y+ of 4.43. The mesh near the airfoil is shown in Fig. 3(b).

At the inflow, the turbulence intensity was set to 1.5% of the inlet velocity with a dissipation rate of 6.0 × 10–4 (nondimensionalized using the flow velocity and the chord length). As the flow is uniform, the turbulence intensity decays as the flow approaches the airfoil.

Before comparing to the experiment, a grid refinement study was conducted for an angle of attack of 12 degrees. The mesh was refined by subdividing the initial mesh of 20,000 elements. Table 2 shows mesh sizes accompanied by their corresponding CL and CD values, and Fig. 4 shows a plot of the lift coefficient versus mesh size. With a mesh resolution of 400,000 elements, the changes in CL with further grid refinement were less than 1%. To reduce the computational cost, the simulations were conducted using a mesh composed of 100,000 elements, which gave an error on the order of 2% in the lift values. The error in the drag values was slightly higher at 4%. Since determining the lift is the priority, this discrepancy was accepted.

Experimental Validation.

The experimental results for the E423 were measured at a Reynolds number of 300,000. Figure 5 shows a comparison between the experimentally measured CL values and the RANS predictions. The figure also shows RANS predictions at Re = 1 × 106 to give an idea of the sensitivity of the RANS results to the Reynolds number. The experimental results indicate that the angle of attack at maximum lift is between 13 deg and 14 deg, while the RANS model predicts that it is around 16 deg. The RANS model uniformly over-predicts the experimental results by about 15% at Re = 3 × 105. There is no experimental data at higher Reynolds numbers, so it is difficult to validate the sensitivity to the Reynolds number predicted by the RANS model. The results indicate that the magnitude of CL is sensitive to Re but the angle of maximum CL is fairly insensitive to Re. Overall the RANS model predicts the angle of maximum lift to within a few degrees. This will cause errors in the DWT optimization results of a similar magnitude.

To verify that the peak in the lift coefficient is due to separation effects, Fig. 6 shows the coefficient of pressure, Cp, and the coefficient of friction, Cf on the airfoil surface for an angle of attack of 16 deg and Re = 1.0 × 106. On both the top and bottom surface of the airfoil, Cf is defined as positive if the shear stress on the airfoil is toward the trailing edge. Cf on the bottom surface is negative at the leading edge of the bottom surface, because the stagnation point is at x/c = 0.025 on the bottom surface, and thus, the shear stress between x/c = 0 and 0.025 on the bottom is toward the leading edge. Cf on the top surface peaks at a value near 0.02 and then decreases until becoming negative at around x/c = 0.6. This indicates that there is a recirculation zone on the upper airfoil surface from x/c = 0.6 to x/c = 1.0. Consistent with this, Cp remains nearly constant at around 0 in this region also indicating separation. There are no experimental data available for the location of the separation point so it is difficult to know the accuracy of the prediction. Previous studies on other airfoils have shown that the k–ε model can not accurately predict the point of separation [32]. However, given that the angle of peak lift coefficient is reasonable, it was concluded that the k–ε model could be used to predict the optimal angle of attack for a DWT to within a few degrees. Ideally, one would use an shear stress transport turbulence model to predict a flow which involves separation [28,33]. However, as we only need to predict the onset of separation, the less computationally expensive k–ε model was used.

The optimization study was conducted to find the ideal configuration of the duct to maximize the power output of a DWT. The rotor was approximated as an actuator disk, which made the entire geometry axisymmetric. Four design parameters were chosen: the Δr and Δz displacements of the duct with respect to the rotor tip, the angle (θ) of the duct in relation to the free stream velocity, and the coefficient of thrust of the rotor disk, CT. Figure 7 shows the definition of the geometric design parameters. The Δr and Δz displacements were measured to the leading edge of the airfoil. The fourth design variable, CT, determines the pressure drop across the actuator disk as

Display Formula

(1)Δp=CT12ρu2

This parameter captures the effect different rotor blade designs would have on the power output. Table 3 gives the allowed range for the design variables in the optimization process. For all cases, the optimal values were found within these ranges.

The diameter of the rotor, D, was fixed during the optimization, and all results were nondimensionalized by D, the free-stream velocity u, and the free-stream density, ρ. As mentioned above, the cross section of the duct was an Eppler423. Individual optimizations were performed for three different chord lengths of the duct, c/D = 12.5%, 17.5%, and 22.5%. The chord length itself was not optimized because, as shown in the following section, the power attainable with a DWT monotonically increases with the length of the duct. For all cases, the Reynolds number based on the rotor diameter was 1 × 106. The Reynolds number of the airfoils based on their chord length for the three different chord lengths is 1.25 × 105, 1.75 × 105, and 2.20 × 105, which falls into the range of the conditions examined in the airfoil validation study.

The power of the rotor was the output used for the goal-driven optimization (GDO). The power produced is given by Display Formula

(2)P=2π0D/2Δpuzrdr

where uz is the flow velocity at the rotor normal to the rotor plane, r is the radial coordinate, and dr represents the differential radius. Results are presented in terms of the coefficient of power, CP, which is defined as Display Formula

(3)CP=P12ρu3Arotor

where Arotor is the area of the rotor not including the duct. In some cases, the total cross-sectional area of the duct, Atotal was used to nondimensionalize the power. This is denoted by CP,total—the power nondimensionalized by the total cross-sectional area of the device.

The simulations were performed with the same methodology as used for the airfoil calculations except in this case the problem was assumed axisymmetric. Figure 8 shows a schematic of the domain and mesh used for the calculations. The turbulence parameters at the inlet were the same as those used for the airfoil simulations. The turbine (actuator disk) was located 30 diameters from the inlet and 50 diameters from the zero gage pressure outlet boundary condition. An axisymmetric boundary condition was applied to the bottom of the geometry. Similar to the airfoil simulations, a boundary layer mesh was added to the duct with 15 layers, a growth rate of 1.2, and an initial spacing in units of chord length of 1.2 × 10–4. The average y+ value of the first point in the boundary layer was 3.65 which is within the required range of y+ ≤ 5 for the enhanced wall treatment. The mesh was refined around the actuator disk composed of quadrilaterals with sides ≈1.0% of the disk diameter, spanning in the normal and tangential directions.

Validation.

Because this was an optimization study, the mesh size for each simulation varied. However, the average size was 100,000 elements. The boundary layer mesh surrounding the duct was the same as in the airfoil simulations. To verify grid convergence, a mesh sensitivity test was conducted for a design point, with a Δr/D and Δz/D at 0.006, θ equal to 30 deg, CT = 1, and c/D = 22.5%. Figure 9 shows the convergence of CP with respect to the number of elements in the mesh. At 100,000 elements, the mesh size is sufficient, having a 5% difference in solution between 100,000 and 400,000 elements.

Optimization Results.

The optimization was conducted using ANSYS GDO, which is a component of ANSYS Workbench [31]. The GDO required first the construction of the geometry and mesh, followed by the generation of design points. The design points were generated using the central composite design (described in Ref. [31]). Additional design points were input manually to refine the response surface. The design points all fell within the parameter ranges given in Table 3. From the design points, a response surface was constructed using a second-order polynomial, which predicted the behavior of the DWT device. The response surface was then searched with the NLPQL algorithm [31] to obtain an optimal solution.

Table 4 shows the results for each chord length. As expected, the longest chord length generated the highest CP. The optimum value of CT is basically independent of chord length. Of all the design variables, CT had the strongest effect on output power over the ranges of design variables specified in Table 3, thus, the design of the rotor blades is the most critical aspect for obtaining optimal power. Figure 10 shows a cross section of the response surface of CP versus θ and CT for the largest chord length duct. As CT varies between 0.4 and 1.2, the power varies by more than a factor of 2. θ had the next strongest influence on the power, but only caused about a 10% change over the 15 deg range.

The independence of the optimal CT with respect to chord length was somewhat surprising. The average normal flow velocity over the rotor plane varies significantly with the chord length of the duct. This is shown in the table as u¯z/u. For an open rotor, the value of u¯z/u for optimum power is 2/3 based on the Betz's analysis. The higher values obtained here are a direct result of the greater flow induced through the rotor by the duct. Defining a coefficient of thrust based on the local conditions at the rotor as Display Formula

(4)CT,r=Δp1/2ρu¯z2

we see that the local thrust coefficients (shown in the table) vary significantly as the duct size increases. This indicates that although CT is nearly constant, the rotor blades needed to produce this value would be different for different duct sizes. The blades for a larger duct should produce the same Δp as the blades used for a smaller duct, but at a higher local flow velocity.

The duct angle was the next most influential design variable. For all duct lengths, the optimal angle was around 37.5 deg. This is much larger than the angle one would expect simply based on the angle of attack for peak CL in a uniform stream. To understand why this is the case, a velocity contour plot with streamlines from the optimized solution is displayed in Fig. 11 for c/D = 22.5%. The angle of attack relative to the incoming streamlines is large relative to the separation angle of attack; the flow approaches the duct at an angle of approximately 5.4 deg, and as a result, the effective angle of attack is roughly 32 deg (37.5–5.4). This is because the actuator disk causes the streamlines to expand and more easily stay attached to the duct. Figure 12 shows the coefficient of pressure and the coefficient of friction on the duct surface. At this extreme angle of attack, the flow is able to stay attached for virtually the entire length of the duct. This is verified by the fact that Cf stays positive on the top surface all the way to x*/c = 0.97 (x* is the coordinate along the chord of the airfoil) where the airfoil ends at x*/c = 1. The Cp profile also continuously decreases toward the trailing edge consistent with the lack of a significant recirculation zone.

The output power was least sensitive to the position of the duct relative to the rotor tip. Figure 13 shows the response surface as a function of the radial displacement and the axial displacement. Over the ranges studied, there was at most a 3% variation in CP which is basically negligible compared to the numerical errors in the computations. Of the two variables, the power was more sensitive to radial position than axial position. Table 4 shows that the optimal value of Δr is roughly constant when scaled with the diameter of the rotor. This indicates that the gap between the rotor and the duct is primarily determined by the size of the rotor not the length of the duct. This is consistent with the fact that the rotor wake has a strong influence on keeping the flow attached to the duct. Given the weak sensitivity of the power to the axial position of the duct, it is difficult to make conclusions about how this position should scale with the size of the duct.

Comparison to an Open Rotor.

The CP values obtained with a duct are clearly larger than the maximum one can obtain using an open rotor (0.593). This is an appropriate comparison to make if the blades are the most costly component of the device, and the cost of the duct is less significant; however, several other metrics may provide additional insight for other situations. Another interesting metric to consider is Display Formula

(5)CP,total=CPArotorAtotal

which is the power one can obtain for a given cross-sectional area of the DWT including the exit area (Atotal) of the duct. This metric is appropriate when the cost is proportional to the total area of the device. This is a slightly different objective than optimizing for power, because increasing the angle of the duct and the radial displacement both increase Atotal and thus decrease the objective, CP,total. For this reason, an additional optimization was run for the duct with length c/D = 22.5% and the goal of maximizing CP,total.

Figure 14 shows a cross section of the response surface as function of CT and θ. The optimization was performed with the same limits given in Table 3. As shown in the figure, because of the dependence of Atotal on θ, the optimum value of θ has shifted to lower values whereas CT optimum is still near 1. The actual optimum value of θ seems to be slightly outside of the range, but was judged to be close enough to 30 deg to use this value as the optimum value.

Figure 15 shows a cross section of the response surface as function of Δr/D and Δz/D. The optimal value of Δr in this case is much lower than in the case where power was the objective function due to the dependence of Atotal on Δr. The optimal value of Δr is beyond the lower limit of the allowable optimization range for Δr. It was difficult to allow lower values of Δr because slight adjustments in θ then resulted in intersections between the airfoil and the rotor. For the purposes of comparing results between the two types of optimization objectives, Δr was set to the lower limit of the range for this case. The optimal value of Δz/D was around 7% which is the same as what was found when optimizing for power.

Table 5 shows a comparison of the results between an open rotor, the case where the optimization objective was CP, and the case when the optimization objective was CP,total. When one compares the three cases based on CP, the ducted turbine provides the largest value. This advantage increases as one increases the length of the duct. However, if one compares based on CP,total, the ducted turbine that was optimized for CP is less effective than an open rotor. When one optimizes based on CP,total, the ducted turbine outperforms an open rotor. Thus, a ducted rotor is able to extract more power per unit cross-sectional area than an open rotor and exceeds Betz's limit of power per area. This is in agreement with the theory proposed by Jamieson [15].

Another metric of importance in wind turbine design is the total thrust exerted on the tower. This is relevant when the cost of the tower is a significant fraction of the cost of the turbine. Unlike an open rotor, where the force acting on the rotor is equal to the thrust force, the force acting on a DWT must include the drag on the duct. A reasonable metric that can be used to compare different designs is the power per unit tower force. This has units of velocity and thus should be nondimensionalized by the free-stream velocity as Display Formula

(6)Cu=PFu

where F is the total axial force on the rotor and the duct. We have not tried to optimize this objective, but simply compared the previous designs based on this metric. The results are shown in the second last column of Table 5. For an open rotor, Cu = CP/CT = 2/3. For the ducted cases around 45% of the axial force comes from the duct, this reduces Cu significantly. Thus, at least for these designs which have not been optimized to maximize Cu, a stronger tower would be needed to extract the same power.

A numerical optimization was conducted of a ducted wind turbine. Several conclusions and recommendations can be made as a result of this study. Of the four parameters optimized, namely, the rotor coefficient of thrust, the duct angle of attack, and the horizontal and vertical placement of the duct relative to the rotor tip, the power output was most sensitive to the coefficient of thrust. The optimal coefficient of thrust was not significantly different from that of an open rotor in spite of the fact that the local flow conditions were different.

The second most sensitive parameter was the duct angle of attack. It was found that the optimal angle of attack was much larger than one would expect based on the angle of attack of maximum lift for an airfoil in a free-stream flow. Large angles of attack did not induce separation on the duct, because the expansion caused by the rotor disk helped keep the flow attached to the duct.

The least sensitive parameters were the position of the duct relative to the tip of the airfoil. Of these the vertical gap between the rotor and the duct was more important and optimal values were between 5% and 10% of the chord length of the duct. However, it was difficult to determine how this distance scaled with the size of the rotor disk or the length of the duct. Similarly, the optimal streamwise position was to have the leading edge of the duct between 10% and 25% of the chord length upstream of the rotor, but there was only a weak sensitivity of the power to this parameter.

For the same rotor area, the power output of the DWT was greater than an open rotor. The increase in power relative to an open rotor depends on the size of the duct with larger ducts giving progressively larger power increases. If one considers a DWT with the same total cross-sectional area as an open rotor, the DWT still produces more power. For the duct with a chord length of 22.5% of the rotor diameter, a Cp value of 0.658 was obtained which exceeds the Betz value of 16/27.

A final observation was that a ducted turbine optimized to produce maximal power per unit cross-sectional area requires a stronger tower to produce the same power as an open rotor.

  • Arotor =

    projected rotor area

  • Atotal =

    projected exit area

  • c =

    chord length of airfoil

  • CD =

    coefficient of drag

  • Cf =

    coefficient of friction

  • CL =

    coefficient of lift

  • Cp =

    coefficient of pressure

  • CP =

    coefficient of power

  • CP,total =

    coefficient of power considering duct area

  • CT =

    coefficient of thrust

  • CT,r =

    coefficient of thrust at the rotor plane

  • Cu =

    power per unit tower force

  • D =

    diameter of turbine rotor

  • P =

    power

  • uz =

    velocity normal to rotor

  • u =

    free-stream velocity

  • x* =

    airfoil coordinate parallel to chord

  • y+ =

    dimensionless wall distance

  • Δp =

    pressure drop across rotor

  • Δr =

    radial displacement

  • Δz =

    axial displacement

  • ρ =

    density of fluid

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Jafari, S. A. , and Kosasih, B. , 2014, “ Flow Analysis of Shrouded Small Wind Turbine With a Simple Frustum Diffuser With Computational Fluid Dynamics Simulations,” J. Wind Eng. Ind. Aerodyn., 125, pp. 102–110. [CrossRef]
Aranake, A. C. , Lakshminarayan, V. K. , and Duraisamy, K. , 2015, “ Computational Analysis of Shrouded Wind Turbine Configurations Using a 3-Dimensional RANS Solver,” Renewable Energy, 75, pp. 818–832. [CrossRef]
Bontempo, R. , and Manna, M. , 2013, “ Solution of the Flow Over a Non-Uniform Heavily Loaded Ducted Actuator Disk,” J. Fluid Mech., 728, pp. 163–195. [CrossRef]
Bontempo, R. , and Manna, M. , 2014, “ Performance Analysis of Open and Ducted Wind Turbines,” Appl. Energy, 136, pp. 405–416. [CrossRef]
Bontempo, R. , and Manna, M. , 2016, “ Effects of the Duct Thrust on the Performance of Ducted Wind Turbines,” Energy, 99, pp. 274–287. [CrossRef]
Shives, M. , and Crawford, C. , 2011, “ Developing an Empirical Model for Ducted Tidal Turbine Performance Using Numerical Simulation Results,” Proc. Inst. Mech. Eng., Part A, 226(1), pp. 112–125.
Selig, M. S. , and Guglielmo, J. J. , 1997, “ High-Lift Low Reynolds Number Airfoil Design,” J. Aircraft, 34(1), pp. 72–79. [CrossRef]
Eppler, R. , and Somers, D. M. , 1980, “ Supplement to: A Computer Program for the Design and Analysis of Low-Speed Airfoils,” NASA Langley Research Center, Hampton, VA, Report No. NASA-TM-81862.
ANSYS, 2011, “ 14.0 Theory Guide,” ANSYS, Canonsburg, PA, accessed Nov. 15, 2017, https://www.scribd.com/doc/140163383/Ansys-Fluent-14-0-Users-Guide
Rezaei, F. , Roohi, E. , and Pasandideh-Fard, M. , 2013, “ Stall Simulation of Flow Around and Airfoil Using a LES Model and Comparison of RANS Models at Low Angles of Attack,” 15th Conference on Fluid Dynamics, Bandar Abbas, Iran, Dec. 18–20, pp. 1–10.
Menter, F. R. , 2009, “ Review of the Shear-Stress Transport Turbulence Model Experience From an Industrial Perspective,” Int. J. Comput. Fluid Dyn., 23(4), pp. 305–316. [CrossRef]
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References

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Foreman, K. , and Gilbert, B. , 1979, “ Technical Development of the Diffuser Augmented Wind Turbine (DAWT) Concept,” Wind Eng., 3(3), pp. 153–166.
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Phillips, D. , 2003, “ An Investigation on Diffuser Augmented Wind Turbine Design,” Ph.D. thesis, University of Auckland, Auckland, New Zealand.
Anzai, A. , Nemoto, Y. , and Ushiyama, I. , 2004, “ Wind Tunnel Analysis of Concentrators for Augmented Wind Turbines,” Wind Eng., 28(5), pp. 605–613. [CrossRef]
Abe, K. , Nishida, M. , Sakurai, A. , Ohya, Y. , Kihara, H. , Wada, E. , and Sato, K. , 2005, “ Experimental and Numerical Investigations of Flow Fields Behind a Small Wind Turbine With a Flanged Diffuser,” J. Wind Eng. Ind. Aerodyn., 93(12), pp. 951–970. [CrossRef]
Watson, S. , Infield, D. , Barton, J. , and Wylie, S. , 2007, “ Modelling of the Performance of a Building-Mounted Ducted Wind Turbine,” J. Phys.: Conf. Ser., 75, p. 012001. [CrossRef]
Jamieson, P. , 2008, “ Beating Betz-Energy Extraction Limits in a Uniform Flow Field,” European Wind Energy Conference (EWEC), Brussels, Belgium, Mar. 31–Apr. 3, pp. 1–10.
Ohya, Y. , Karasudani, T. , Sakurai, A. , Abe, K. , and Inoue, M. , 2008, “ Development of a Shrouded Wind Turbine With a Flanged Diffuser,” J. Wind Eng. Ind. Aerodyn., 96(5), pp. 524–539. [CrossRef]
Kosasih, B. , and Tondelli, A. , 2012, “ Experimental Study of Shrouded Micro-Wind Turbine,” Procedia Eng., 49, pp. 92–98. [CrossRef]
Gaden, D. L. , and Bibeau, E. L. , 2010, “ A Numerical Investigation Into the Effect of Diffusers on the Performance of Hydro Kinetic Turbines Using a Validated Momentum Source Turbine Model,” Renewable Energy, 35(6), pp. 1152–1158. [CrossRef]
Ohya, Y. , and Karasudani, T. , 2010, “ A Shrouded Wind Turbine Generating High Output Power With Wind-Lens Technology,” Energies, 3(4), pp. 634–649. [CrossRef]
Takahashi, S. , Hata, Y. , Ohya, Y. , Karasudani, T. , and Uchida, T. , 2012, “ Behavior of the Blade Tip Vortices of a Wind Turbine Equipped With a Brimmed-Diffuser Shroud,” Energies, 5(12), pp. 5229–5242. [CrossRef]
Kannan, T. S. , Muthasher, S. A. , and Lau, Y. K. , 2013, “ Design and Flow Velocity Simulation of Diffuser Augmented Wind Turbine Using CFD,” J. Eng. Sci. Technol., 8(4), pp. 372–384.
Mansour, K. , and Meskinkhoda, P. , 2014, “ Computational Analysis of Flow Fields Around Flanged Diffusers,” J. Wind Eng. Ind. Aerodyn., 124, pp. 109–120. [CrossRef]
Jafari, S. A. , and Kosasih, B. , 2014, “ Flow Analysis of Shrouded Small Wind Turbine With a Simple Frustum Diffuser With Computational Fluid Dynamics Simulations,” J. Wind Eng. Ind. Aerodyn., 125, pp. 102–110. [CrossRef]
Aranake, A. C. , Lakshminarayan, V. K. , and Duraisamy, K. , 2015, “ Computational Analysis of Shrouded Wind Turbine Configurations Using a 3-Dimensional RANS Solver,” Renewable Energy, 75, pp. 818–832. [CrossRef]
Bontempo, R. , and Manna, M. , 2013, “ Solution of the Flow Over a Non-Uniform Heavily Loaded Ducted Actuator Disk,” J. Fluid Mech., 728, pp. 163–195. [CrossRef]
Bontempo, R. , and Manna, M. , 2014, “ Performance Analysis of Open and Ducted Wind Turbines,” Appl. Energy, 136, pp. 405–416. [CrossRef]
Bontempo, R. , and Manna, M. , 2016, “ Effects of the Duct Thrust on the Performance of Ducted Wind Turbines,” Energy, 99, pp. 274–287. [CrossRef]
Shives, M. , and Crawford, C. , 2011, “ Developing an Empirical Model for Ducted Tidal Turbine Performance Using Numerical Simulation Results,” Proc. Inst. Mech. Eng., Part A, 226(1), pp. 112–125.
Selig, M. S. , and Guglielmo, J. J. , 1997, “ High-Lift Low Reynolds Number Airfoil Design,” J. Aircraft, 34(1), pp. 72–79. [CrossRef]
Eppler, R. , and Somers, D. M. , 1980, “ Supplement to: A Computer Program for the Design and Analysis of Low-Speed Airfoils,” NASA Langley Research Center, Hampton, VA, Report No. NASA-TM-81862.
ANSYS, 2011, “ 14.0 Theory Guide,” ANSYS, Canonsburg, PA, accessed Nov. 15, 2017, https://www.scribd.com/doc/140163383/Ansys-Fluent-14-0-Users-Guide
Rezaei, F. , Roohi, E. , and Pasandideh-Fard, M. , 2013, “ Stall Simulation of Flow Around and Airfoil Using a LES Model and Comparison of RANS Models at Low Angles of Attack,” 15th Conference on Fluid Dynamics, Bandar Abbas, Iran, Dec. 18–20, pp. 1–10.
Menter, F. R. , 2009, “ Review of the Shear-Stress Transport Turbulence Model Experience From an Industrial Perspective,” Int. J. Comput. Fluid Dyn., 23(4), pp. 305–316. [CrossRef]

Figures

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Fig. 1

Conceptual design of a DWT. The geometry of the duct's cross section is that of a highly cambered airfoil.

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Fig. 2

Eppler 423 airfoil geometry

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Fig. 3

Domain and mesh used for the airfoil validation study: (a) domain and mesh and (b) boundary layer mesh

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Fig. 4

Convergence of k–ε model

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Fig. 5

E423 at Re = 3.0 × 105 and 1.0 × 106 compared to experimental data

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Fig. 6

E423 at Re = 1.0 × 106: coefficient of pressure and shear stress versus chord length

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Fig. 7

Detail of parameters for duct optimization

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Fig. 8

Domain and mesh for the optimization studies: (a) domain and mesh and (b) magnified region around actuator disk

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Fig. 9

Convergence plot: Cp versus mesh resolution (number of elements)

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Fig. 10

Response surface: CT versus AOA for DWT for chord length of 45% at the optimal CP

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Fig. 11

Velocity contours overlaid by streamlines for the optimal duct configuration

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Fig. 12

Coefficient of pressure and friction for the optimal duct design with c/D = 22.5%

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Fig. 13

Response surface: Δr/D versus Δz/D or chord length of 45% at the optimal CP

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Fig. 14

Response surface: CT versus AOA for DWT for chord length of 45% at the optimal CP,total

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Fig. 15

Response surface: Δr/D versus Δz/D or chord length of 45% at the optimal CP,total

Tables

Table Grahic Jump Location
Table 1 Airfoil data
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Table 2 Grid refinement for RANS model
Table Grahic Jump Location
Table 3 Parameter value range
Table Grahic Jump Location
Table 4 Optimal design of the DWT at varying chord lengths
Table Grahic Jump Location
Table 5 Optimal design of the DWT at varying chord lengths

Errata

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