Journal cover
Journal topic
**Wind Energy Science**
The interactive open-access journal of the European Academy of Wind Energy

Journal topic

- About
- Editorial board
- Articles
- Special issues
- Manuscript tracking
- Subscribe to alerts
- Peer review
- For authors
- For reviewers
- Imprint
- Data protection

- About
- Editorial board
- Articles
- Special issues
- Manuscript tracking
- Subscribe to alerts
- Peer review
- For authors
- For reviewers
- Imprint
- Data protection

WES | Articles | Volume 3, issue 1

Wind Energ. Sci., 3, 221-229, 2018

https://doi.org/10.5194/wes-3-221-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/wes-3-221-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Wind Energy Science Conference 2017

**Research articles**
25 Apr 2018

**Research articles** | 25 Apr 2018

Ducted wind turbine optimization

- Mechanical and Aeronautical Engineering Department, Clarkson University, Potsdam, NY 13699-5725, USA

Abstract

Back to toptop
The design of a ducted wind turbine modeled using an actuator disc was studied using Reynolds-averaged Navier–Stokes (RANS) computational fluid dynamics (CFD) simulations. The design variables included the rotor thrust coefficient, the angle of attack of the duct cross section, the radial gap between the rotor and the duct, and the axial location of the rotor in the duct. Two different power coefficients, the rotor power coefficient (based on the rotor swept area) and the total power coefficient (based on the exit area of the duct), were used as optimization objectives. The optimal value of thrust coefficients for all designs was nearly constant, having a value between 0.9 and 1. The rotor power coefficient was sensitive to rotor gap but was insensitive to the rotor's axial location for positions ranging from upstream of the throat to nearly half the distance down the duct. Compared to the design that maximized rotor power coefficient, the design for maximal total power coefficient was characterized by a smaller angle of attack, a smaller rotor gap, and a downstream placement of the rotor. The insensitivity of power output to the rotor position implies that a rotor placed further downstream in the duct could produce the same power with a considerably smaller duct exit area and thus a greater total power coefficient. The design for that maximized total power coefficient exceeded Betz's limit with a total power coefficient of 0.67.

How to cite

Back to top
top
How to cite.

Bagheri-Sadeghi, N., Helenbrook, B. T., and Visser, K. D.: Ducted wind turbine optimization and sensitivity to rotor position, Wind Energ. Sci., 3, 221-229, https://doi.org/10.5194/wes-3-221-2018, 2018.

1 Introduction

Back to toptop
A properly designed duct placed around a wind turbine can increase power output by increasing the mass flow rate through the rotor. Ducted wind turbines (DWTs) are also called diffuser-augmented wind turbines (DAWT) or shrouded wind turbines. Lilley and Rainbird (1956) performed a one-dimensional momentum analysis of DWTs and concluded that higher expansion ratios of the duct and more subatmospheric pressures at the exit plane of the duct result in higher power outputs. They also suggested wind tunnel tests with screens of different porosities to model the pressure drop across the rotor. Such experimental tests were performed by Igra (1976, 1977, 1981), Foreman et al. (1978), Gilbert et al. (1978), and Gilbert and Foreman (1979). The negative effect of flow separation on the power output of DWTs was observed, and various methods of preventing separation were investigated. Also, experimental tests with real turbines were performed, and the power augmentation of DWTs was demonstrated (Gilbert and Foreman, 1979, 1983; Igra, 1981). As the duct can be considered an annular wing (de Vries, 1979) with higher lift, meaning more suction and circulation, high-lift airfoils were used from early experimental studies.

Using lifting line theory for the rotor and modeling the duct as a superposition of vortex and source rings, Koras and Georgalas (1988) and Georgalas et al. (1991) studied the power output of DWTs with airfoil cross sections and large rotor gaps (the clearance between the tip of the rotor and the duct) as a function of several design variables including the angle of attack of the duct cross section, the chord length of the duct, the maximum camber of the duct cross section, and the relative position of the rotor with respect to the maximum camber point of the duct cross section. They found a linear increase of power with duct chord length and angle of attack of the duct cross section. They also concluded that the effect of rotor position on the power output was weak. Politis and Koras (1995) extended the previous work to DWTs with any rotor gap.

Axisymmetric computational fluid dynamics (CFD) models were used (Phillips et al., 1999, 2002 and
Phillips, 2003) to improve the design of the first full-scale DWT
built (the Vortec 7). Hansen et al. (2000) performed a CFD study of DWTs and
used the *k*−*ω* shear stress transport (SST) turbulence model for the axisymmetric model as it is more
sensitive to adverse pressure gradients (Menter, 1994) and can be
more accurate in predicting flow separation. Another similar CFD study was
performed by Abe and Ohya (2004), where effects of rotor loading and the
incidence angle of the duct on power output of a flanged DWT were examined
and compared with experimental data. Ohya et al. (2012) and Kardous et al. (2013)
did further CFD simulations of the flanged DWT with the rotor modeled as an
actuator disc and found good agreement with wind tunnel data.

Van Bussel (1999, 2007) analyzed DWTs using 1-D momentum theory
and concluded that optimal coefficient of thrust in a DWT is similar to an
open rotor equal to 8∕9. He also concluded that experimental power
coefficients based on the exit area of the duct (the total power coefficient)
above 0.5 have not been achieved yet and that very significant back pressure
reductions are needed to achieve values of total power coefficients
significantly above Betz's limit. Jamieson (2009) also used a similar
momentum analysis and derived the same value of 8∕9 for optimal loading on
the rotor and noted that it should be independent of duct design.
Werle and Presz (2008) in another study based on 1-D momentum analysis found that
the maximum attainable power from a DWT is determined by shroud force
coefficient, ${C}_{\mathrm{s}}={F}_{\mathrm{s}}/T$, where *F*_{s} is the axial force on the duct
(shroud) and *T* is the thrust of the rotor.

Hjort and Larsen (2014) used an axisymmetric CFD model with an actuator disc modeling the wind turbine for a multi-element DWT. They characterized the performance of the DWT using power coefficients based on the exit area of the duct with values well above Betz's limit. Aranake and Duraisamy (2017) also utilized an axisymmetric Reynolds-averaged Navier–Stokes (RANS) solver with an actuator disc model for the turbine to optimize the airfoils used for the duct cross section and blades and verified the result with 3-D simulations. Venters et al. (2017) investigated the optimal design of a DWT using the same approach (i.e., using a RANS solver and actuator disc model). The design variables investigated were the rotor loading, the angle of attack of the duct cross section, the rotor gap, and the axial position of the rotor. They used a response surface fitted to a number of design point calculations, and then the surface was searched using the NLPQL (Nonlinear Programming by Quadratic Lagrangian) algorithm (Schittkowski, 1986). They concluded that rotor loading is the main factor defining the performance of the DWT with the coefficient of thrust almost constant (close to 1) for different duct sizes. The power output of DWT was sensitive to the angle of attack of the duct cross section. However, the results for effect of the rotor gap and axial position of rotor were not conclusive. This paper improves on the work of Venters et al. (2017) with a more accurate CFD model, a direct optimization technique, and a wider range of design variables. One of the goals of this study is to continue the investigation of Venters et al. (2017) into how the objective of the optimization changes the optimal design. Specifically, Venters et al. (2017) examined two objective functions, the rotor power coefficient and the total power coefficient. Their results indicated that the optimal design changes significantly depending on the objective function, but the results for optimizing the total power coefficient did not converge to an optimal solution. The goal of this work is to identify an optimal configuration for this objective function.

The paper is organized as follows. The details of the CFD model along with an evaluation of two different pattern search optimization methods are given in Sect. 2. Optimization results with the objective of maximizing the rotor power coefficient are given in Sect. 3, and the variation of the rotor power coefficient and flow field with different design variables is presented. In Sect. 4, optimization results are presented for the objective of maximizing the total power coefficient. These results are compared with the goal of understanding how the optimal axial position of the rotor depends on the optimization objective.

2 Method

Back to toptop
A two-dimensional axisymmetric numerical model was
developed in ANSYS Fluent 17.1 to simulate the flow field of a
DWT. The wind turbine rotor was modeled as an actuator disc with a
pressure drop, Δ*p*, given by

$$\begin{array}{}\text{(1)}& \mathrm{\Delta}p={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathit{\rho}{V}_{z}^{\mathrm{2}}{C}_{\mathrm{T},\phantom{\rule{0.125em}{0ex}}\mathrm{rotor}},\end{array}$$

where *ρ* is the air density and *C*_{T, rotor} is the thrust coefficient
based on the axial velocity, *V*_{z}, at the rotor. The thrust force, *T*, is
given by

$$\begin{array}{}\text{(2)}& T=\mathrm{2}\mathit{\pi}\underset{\mathrm{0}}{\overset{D/\mathrm{2}}{\int}}\mathrm{\Delta}pr\mathrm{d}r,\end{array}$$

where *D* is the rotor diameter. The extracted power, *P*, is given by

$$\begin{array}{}\text{(3)}& P=\mathrm{2}\mathit{\pi}\underset{\mathrm{0}}{\overset{D/\mathrm{2}}{\int}}{V}_{z}\mathrm{\Delta}pr\mathrm{d}r.\end{array}$$

Clearly, with an actuator disc model, rotor blade efficiency losses are not
considered. The design variables, shown in Fig. 1,
were the thrust coefficient of the rotor ${C}_{\mathrm{T},\phantom{\rule{0.125em}{0ex}}\mathrm{rotor}}=\frac{T}{\frac{\mathrm{1}}{\mathrm{2}}\mathit{\rho}{V}_{z}^{\mathrm{2}}{A}_{\mathrm{rotor}}}$, the angle of attack of the duct
cross section *α*, the radial gap of the rotor Δ*r*∕*D*, and the
axial location of the rotor *z*∕*c*. Because the thrust coefficient based on
the freestream velocity, *V*_{∞}, is easier to interpret, most results are
presented in terms of ${C}_{\mathrm{T}}=\frac{T}{\frac{\mathrm{1}}{\mathrm{2}}\mathit{\rho}{V}_{\mathrm{\infty}}^{\mathrm{2}}{A}_{\mathrm{rotor}}}$. All results are made nondimensional by the rotor diameter, the
freestream velocity, and the fluid density. The conditions studied correspond
to air with a free-stream velocity of 11 m s^{−1}, a rotor diameter of 2.5 m,
and a duct chord length *c* such that $c/D=$ 27.6 %. This corresponds to
*Re*${}_{\mathrm{D}}=\mathrm{1.88}\times {\mathrm{10}}^{\mathrm{6}}$ and *Re*${}_{\mathrm{c}}=\mathrm{5.20}\times {\mathrm{10}}^{\mathrm{5}}$, where *Re*_{D} and *Re*_{c}
are the Reynolds numbers based on the rotor diameter and duct chord length,
respectively. An Eppler E423 airfoil was chosen as the cross section of the
duct. This airfoil is designed to create high lift and operate at low
Reynolds numbers. The operating range of Eppler E423 is *Re*${}_{\mathrm{c}}>\mathrm{2}\times {\mathrm{10}}^{\mathrm{5}}$
(Selig et al., 1996). Two power coefficients, ${C}_{\mathrm{P}}=\frac{P}{\frac{\mathrm{1}}{\mathrm{2}}\mathit{\rho}{V}_{\mathrm{\infty}}^{\mathrm{3}}{A}_{\mathrm{rotor}}}$ and ${C}_{\mathrm{P},\phantom{\rule{0.125em}{0ex}}\mathrm{total}}=\frac{P}{\frac{\mathrm{1}}{\mathrm{2}}\mathit{\rho}{V}_{\mathrm{\infty}}^{\mathrm{3}}{A}_{\mathrm{total}}}$, were used as objective functions for the optimizations and to
compare the performance of different DWT designs.

The domain and mesh used for the simulations are shown in Fig. 2. These were defined to ensure mesh independence for power
coefficients as the design variables were varied. The domain extended 15 duct
chord lengths upstream of the rotor and 25 chord lengths downstream.
Numerical tests showed that this domain size gave power coefficient values
that were independent of the domain size to two significant digits. As all
optimizations were done with the same domain, this was deemed large enough to
accurately calculate changes in the solutions with the design variables. The
shape of the domain at its top made a distinct transition between inflow and
outflow boundaries, which eliminated convergence issues due to reverse flow
through outlet boundaries. The reverse-flow issue occurred when the inlet and
outlet were smoothly connected. The mesh of Fig. 2 consisted of
about 500 000 elements. The duct boundary layer mesh had a growth rate of
1.1, and the first mesh point was set at ${y}^{+}\approx \mathrm{1}$. The boundary layer
thickness was calculated as a function of *Re*_{c} for each case, and enough
inflation layers were used to span the entire boundary layer. The
quality-based smoothing option in Fluent was used to improve the mesh
quality.

ANSYS Fluent's *k*−*ω* SST turbulence model was used to solve the
incompressible Navier–Stokes equations. The pressure-based solver was chosen
with the coupled scheme used for the pressure–velocity coupling. Gradients
were calculated using the Green–Gauss node-based method, and second-order
discretization schemes were used for pressure, momentum, turbulent kinetic
energy, and specific dissipation rate. The output power, thrust, and drag
coefficient of the duct were calculated and monitored at each iteration to
ensure convergence.

For most of the optimization results, a pattern search
method (Powell, 1964) was used to find the optimal design of the DWT.
Optimizations were first performed with *C*_{P} as the objective function and
then with *C*_{P, total} as the objective function. The optimization for both
objective functions started from the same set of design variables
(${C}_{\mathrm{T},\phantom{\rule{0.125em}{0ex}}\mathrm{rotor}}=\mathrm{0.816}$; *α*=25^{∘}; $\mathrm{\Delta}r/D=\mathrm{0.03}$; and $z/c=\mathrm{0.14}$). In our implementation of Powell's method a quadratic interpolation of
the function values is used to identify the optimal step length to move the
design point in the coordinate or pattern directions. The optimization was
stopped when the improvements obtained from the optimization methods were
within a specified tolerance. The termination criterion was
$\frac{{C}_{\mathrm{P},\phantom{\rule{0.125em}{0ex}}\mathrm{optimal}}-{C}_{\mathrm{P},\mathrm{0}}}{{C}_{\mathrm{P},\mathrm{0}}+{C}_{\mathrm{P},\phantom{\rule{0.125em}{0ex}}\mathrm{optimal}}}<\mathrm{0.005}$, where
*C*_{P,0} is the initial value of *C*_{P} at the beginning of a search cycle.
The design variables were then varied to determine the sensitivity of the
objective function to the design parameters in the vicinity of the optimal
design point.

Powell's method is known to be slow converging for objective functions that
are discontinuous. As shown in Sect. 3 and Sect. 4, it was observed that the optimal design points were on
the verge of flow separation along the duct airfoil and that separation was
accompanied with a large drop in power output. Therefore, the objective
functions were nearly discontinuous at the optimal design point. With such an
objective function, the optimizer worked inefficiently in finding the optimal
step length. Also, when the optimizer moved the design point close to a
discontinuity, it moved away from that point very slowly.
Figure 3a shows the history of an optimization using Powell's
method with $z/c=\mathrm{0.05}$ fixed and the design variables being
*C*_{T, rotor}, *α* and Δ*r*∕*D*. The optimization method
was stopped at about 100 iterations without meeting its termination
tolerance. At that point the search algorithm was jumping around
significantly. This is shown in Fig. 3b, which shows the
search history of *C*_{T},*α* points. The optimal point is shown
as a red triangle which occurs at *C*_{T}≈1 and *α*≈27^{∘}. The points close to each other in Powell's method that
are not near the optimum point are design points close to separation where
the optimizer had trouble finding the optimal step length and was stuck close
to the function discontinuity. The maximum *C*_{p} obtained by the
search method was 1.031.

The same problem was subsequently approached with the Hooke and Jeeves method
(Hooke and Jeeves, 1961) with the same termination criterion and starting
point. The optimization history is shown in Fig. 4a. This
time the optimization algorithm reached the optimal design in only 16
function evaluations and reached the termination criterion in 32 function
evaluations. In addition, a better design with *C*_{P}=1.053 was
found. The more efficient performance of the Hooke and Jeeves method can also
be observed in Fig. 4b, which shows the
(*C*_{T}, *α*) search points. The optimum point is again shown as a red triangle and
occurred at *C*_{T}=0.97 and *α*=30^{∘}. Because the
Hooke and Jeeves method does not fit an analytic function to the function
values, it did not face the same difficulty when it got close to sharp
variations in the objective function.

Although the Hooke and Jeeves method was more efficient, unless stated otherwise, most of the results obtained below were found using Powell's method as this was the first method implemented. As this method did not always satisfy the optimization stopping criterion, we call the optimized designs “near-optimum” points. The search history shown in Fig. 3b is fairly typical for the cases shown below. For each search direction, function evaluations at design points in the search directions of roughly ±1 % were evaluated with no increases in the optimal value. This, however, does not preclude the possibility of a slow variation along a ridge in the optimization function, which is essentially the reason we see 5 % differences in the optimized design variables between the two different optimization methods for this example. In many of the figures below, individual design variables are varied around the optimal point to give a further sense of the sensitivity to the design variables.

3 Design for optimal *C*_{P}

Back to toptop
The middle column of Table 1 shows the
near-optimal design found with Powell's method when optimizing for maximum
*C*_{P}. The design values are close to what was observed by
Venters et al. (2017). Venters et al. (2017) used a smaller chord length for the
duct ($c/D=\mathrm{22.5}$ %) and a different turbulence model (*k*−*ϵ*
realizable) and obtained a maximal value for *C*_{P}=1.00 at *C*_{T}=1.08 and
*α*=37.5^{∘}. Our results predicted a lower value of optimal *C*_{T}
and *α*, which could be because of the more accurate turbulence model as
the *k*−*ω* SST turbulence model is known to be more accurate in
prediction of flows with significant adverse pressure gradient and flow
separation (Menter, 1994).

The results for the variation of *C*_{P} with *C*_{T} are shown in
Fig. 5. The highest *C*_{P} in this plot is at *C*_{T}≈0.93. When
using the Hooke and Jeeves optimization, optimal *C*_{T} values very close to 1
were observed, which is closer to that observed by Venters. One-dimensional momentum
analysis done by van Bussel (1999) and Jamieson (2009) predicted
that the optimal *C*_{T} for a ducted turbine would be independent of duct design and
have a value of 8∕9, which is the same as that of an open rotor. The plot of
Fig. 5 also shows the curve for an open rotor as predicted by
actuator disc theory. Similar to an open rotor, increasing the loading on the
rotor beyond the near-optimal design point of the DWT reduced the mass flow
rate through the rotor and thus its output power. Also similar to an open
rotor, at loadings less than the near-optimal design point, the flow rate
through the rotor was larger, but the pressure drop was too low to obtain
optimal power. In the ducted case, however, the reduction in *C*_{T} had an
additional effect, which was to cause flow separation in the duct. As shown
next, there is a strong coupling between the coefficient of thrust, the angle
of attack of the duct, and separation. Increasing the angle of attack or
decreasing the coefficient of thrust can lead to separation.

The effect of changing *α* is shown in Fig. 6. When
*α* was increased beyond the near-optimal design point, a large flow
separation resulted, which was accompanied by a sharp decrease in the output
power. The flow field of the near-optimal design is shown in Fig. 7. The effect of increasing *α* on the flow
field is shown in Fig. 8. Comparing the two flow
fields, it is apparent that the small increase in angle of attack leads to a
large separated region at the trailing edge of the airfoil. The separated
region effectively reduces the exit area area of the duct, resulting in the
capture of a smaller upstream flow area and a smaller power extraction.
Similarly, reducing *α* from the near-optimal design also resulted in a
decrease of *C*_{P} because of the decreased exit area.

Likewise, as shown in Fig. 9, if rotor gap, Δ*r*∕*D*, was
increased beyond the near-optimal design point (while keeping other design
variables constant), a large power drop was observed due to flow separation
and the streamlines appeared similar to Fig. 8.
Decreasing Δ*r*∕*D* also reduced the power output of the rotor. Reduction
of the rotor gap results in a decrease in the exit area of the duct, which
could be the reason for the reduction in power.

The dependence of *C*_{P} on the axial position of the rotor, *z*∕*c*, can be seen
from Fig. 10. As *z*∕*c* was varied from the near-optimal design,
the power output did not change significantly. To better understand the
effect of axial location on the power output of the rotor, the design was
optimized using the Hooke and Jeeves pattern search method at a number of fixed
*z*∕*c* values from 0.05 to 0.35. The results shown in Fig. 10
confirm that *C*_{P} within the range of *z*∕*c* values shown is not very
sensitive to the axial position of the rotor. The higher values of *C*_{P}
shown are due to better performance of the Hooke and Jeeves search algorithm
as discussed in Sect. 2.1. This result shows that one can
place the rotor anywhere from upstream of the throat to halfway down the duct
and obtain similar performance.

4 Design for optimal *C*_{P, total}

Back to toptop
The last column of Table 1 shows the near-optimal design
parameters when *C*_{P, total} was the objective function, and
Fig. 11 shows the geometry and flow field of the near-optimal
design. Compared to the design for optimal *C*_{P}, when a DWT was designed for
optimal *C*_{P, total}, the values of *α* and Δ*r*∕*D* were
decreased, whereas *z*∕*c* was increased. All of these changes have a similar effect: to
decrease the exit area of the duct, which is in the denominator of the
objective function.

The value of *C*_{T} of the near-optimal design (0.87) was close to the optimal
*C*_{T} when *C*_{P} was optimized using Powell's method (0.93). It is also close
to the optimal value for an open rotor which van Bussel (1999) and
Jamieson (2009) predicted. However, there is some ambiguity in the
preciseness of this value because both Venters and the Hooke and Jeeves
optimization showed values near 1.00 when optimizing *C*_{P}.

The variation of *C*_{P, total} with *z*∕*c* is presented in Fig. 12. All other design variables were fixed at the near-optimal
design point for *C*_{P, total} as given in Table 1 as
*z*∕*c* was varied. Since *C*_{P, total} depends on both power output and the
exit area of the duct, the values of *C*_{P} at each design point are also
shown so that variations due to changes in exit area or power can be better
understood. Similar to Fig. 10 the power of the rotor, *C*_{p}, is
not very sensitive to *z*∕*c* when $z/c<\mathrm{0.5}$. The exit area decreases as
*z*∕*c* is increased, which makes *C*_{P, total} increase as the rotor is moved
towards the exit of the duct. When *z*∕*c* is increased past 0.5, the power
extracted decreases, but *C*_{P, total} continues to increase because of the
decreasing exit area. Just past the optimal value of *z*∕*c* the flow
separates, leading to a sharp decrease in both *C*_{P} and *C*_{P, total}.

The best value obtained here for ${C}_{\mathrm{P},\phantom{\rule{0.125em}{0ex}}\mathrm{total}}=\mathrm{0.67}$ was above Betz's limit and
was also higher than the previous result by Venters et al. (2017) of 0.621.
Thus, it is possible to extract more power per unit device area using a ducted
turbine than when using an open rotor. This is in agreement with theoretical
predictions by van Bussel (2007) at high back pressure reductions.
To obtain this value of *C*_{P, total}, the rotor must be at the rear of the duct.

5 Conclusions

Back to toptop
The optimal design of a ducted wind turbine characterized by the thrust
coefficient of the rotor *C*_{T, rotor}, the angle of attack of the duct
cross section *α*, the rotor gap Δ*r*∕*D*, and the axial location
of the rotor *z*∕*c* was investigated. The optimal design was significantly
different when different power coefficients *C*_{P} (based on rotor area) and
*C*_{P, total} (based on the exit area of the duct) were used as design
objectives. Compared to the design for optimal *C*_{P}, the design for optimal
*C*_{P, total} resulted in a duct with smaller *α* and Δ*r*∕*D* and a
rotor placed at the rear of the duct rather than towards the front. This type
of design has been experimentally investigated in Kanya and Visser (2018).

The design for optimal *C*_{P, total} attained ${C}_{\mathrm{P},\phantom{\rule{0.125em}{0ex}}\mathrm{total}}=\mathrm{0.67}$, which was
above Betz's limit. This optimal design was on the brink of flow separation;
increases in *α*, decreases in *C*_{T}, or increases in Δ*r*∕*D* all
resulted in flow separation and a sharp decrease in power output. The Hooke
and Jeeves optimization method was found to be more efficient in finding the
optimal designs than Powell's method, which was attributed to this
sharp variation in *C*_{P} around the design point.

Although optimal design data obtained from Powell's method should represent
characteristics of a good design based on *C*_{P} or *C*_{P, total}, due to
convergence issues they should be considered approximate. Also at higher
Reynolds numbers (i.e., with larger DWTs) the flow field and separation
characteristics of the DWT may change, which in turn can change the
characteristics of the optimal design. Additionally, including effect like
swirl and center body in the CFD model should give more realistic results and
can change the optimal design.

Data availability

Back to toptop
Data availability.

A set of ANSYS Fluent case and data files is publicly available online (Bagheri-Sadeghi, 2018).

Competing interests

Back to toptop
Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

Back to toptop
Special issue statement.

This article is part of the special issue “Wind Energy Science Conference 2017”. It is a result of the Wind Energy Science Conference 2017, Lyngby, Copenhagen, Denmark, 26–29 June 2017.

Acknowledgements

Back to toptop
Acknowledgements.

We are grateful for the funding support of this project from the New York
State Energy and Research Development Authority (NYSERDA) through NEXUS-NY.
We would also like to thank Ken Willmert of Clarkson University for
providing the numerical implementation of Powell's method.

Edited by: Jens Nørkær Sørensen

Reviewed by: two anonymous referees

References

Back to toptop
Abe, K.-i. and Ohya, Y.: An Investigation of Flow Fields around Flanged Diffusers Using CFD, J. Wind Eng. Ind. Aerod., 92, 315–330, https://doi.org/10.1016/j.jweia.2003.12.003, 2004. a

Aranake, A. and Duraisamy, K.: Aerodynamic Optimization of Shrouded Wind Turbines, Wind Energy, 20, 877–889, https://doi.org/10.1002/we.2068, we.2068, 2017. a

Bagheri-Sadeghi, N., Helenbrook, B. T., and Visser, K. D.: Data associated with publication “Ducted wind turbine optimization and sensitivity to rotor position”, https://doi.org/10.17605/OSF.IO/XFSZM, 2018.

de Vries, O.: Fluid Dynamic Aspects of Wind Energy Conversion, Advisory Group for Aerospace Research and Development (AGARD), Neuilly-sur-Seine, France, Technical Report no. 243, 1979. a

Foreman, K. M., Gilbert, B., and Oman, R. A.: Diffuser Augmentation of Wind Turbines, Sol. Energy, 20, 305–311, https://doi.org/10.1016/0038-092X(78)90122-6, 1978. a

Georgalas, C. G., Koras, A. D., and Raptis, S. N.: Parametrization of the Power Enhancement Calculated for Ducted Rotors with Large Tip Clearance, Wind Engineering, 15, 128–136, http://www.jstor.org/stable/43749450, 1991. a

Gilbert, B. L. and Foreman, K. M.: Experimental Demonstration of the Diffuser-Augmented Wind Turbine Concept, J. Energy, 3, 235–240, https://doi.org/10.2514/3.48002, 1979. a, b

Gilbert, B. L. and Foreman, K. M.: Experiments With a Diffuser-Augmented Model Wind Turbine, J. Energ. Resour.-ASME, 105, 46–53, https://doi.org/10.1115/1.3230875, 1983. a

Gilbert, B. L., Oman, R. A., and Foreman, K. M.: Fluid Dynamics of Diffuser-Augmented Wind Turbines, J. Energy, 2, 368–374, https://doi.org/10.2514/3.47988, 1978. a

Hansen, M. O. L., Sørensen, N. N., and Flay, R. G. J.: Effect of Placing a Diffuser around a Wind Turbine, Wind Energy, 3, 207–213, https://doi.org/10.1002/we.37, 2000. a

Hjort, S. and Larsen, H.: A Multi-Element Diffuser Augmented Wind Turbine, Energies, 7, 3256–3281, https://doi.org/10.3390/en7053256, 2014. a

Hooke, R. and Jeeves, T. A.: “Direct Search” Solution of Numerical and Statistical Problems, J. ACM, 8, 212–229, https://doi.org/10.1145/321062.321069, 1961. a

Igra, O.: Shrouds for Aerogenerators, AIAA Journal, 14, 1481–1483, https://doi.org/10.2514/3.61486, 1976. a

Igra, O.: Compact Shrouds for Wind Turbines, Energ. Convers., 16, 149–157, https://doi.org/10.1016/0013-7480(77)90022-5, 1977. a

Igra, O.: Research and Development for Shrouded Wind Turbines, Energ. Convers. Manage., 21, 13–48, https://doi.org/10.1016/0196-8904(81)90005-4, 1981. a, b

Jamieson, P. M.: Beating Betz: Energy Extraction Limits in a Constrained Flow Field, J. Sol. Energ.-T. ASME, 131, 031008-1–031008-6, https://doi.org/10.1115/1.3139143, 2009. a, b, c

Kanya, B. and Visser, K. D.: Experimental Validation of a Ducted Wind Turbine Design Strategy, Wind Energy Science Journal, in review, 2018. a

Kardous, M., Chaker, R., Aloui, F., and Nasrallah, S. B.: On the Dependence of an Empty Flanged Diffuser Performance on Flange Height: Numerical Simulations and PIV Visualizations, Renew. Energ., 56, 123–128, https://doi.org/10.1016/j.renene.2012.09.061, 2013. a

Koras, A. D. and Georgalas, C. G.: Calculation of the Influence of Annular Augmentors on the Performance of a Wind Rotor, Wind Engineering, 12, 257–267, http://www.jstor.org/stable/43750035, 1988. a

Lilley, G. and Rainbird, W.: A Preliminary Report on the Design and Performance of Ducted Windmills, Tech. rep., College of Aeronautics, Cranfield, UK, 1956. a

Menter, F. R.: Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, 32, 1598–1605, https://doi.org/10.2514/3.12149, 1994. a, b

Ohya, Y., Uchida, T., Karasudani, T., Hasegawa, M., and Kume, H.: Numerical Studies of Flow around a Wind Turbine Equipped with a Flanged-Diffuser Shroud Using an Actuator-Disk Model, Wind Engineering, 36, 455–472, https://doi.org/10.1260/0309-524X.36.4.455, 2012. a

Phillips, D., Flay, R., and Nash, T.: Aerodynamic Analysis and Monitoring of the Vortec 7 Diffuser-Augmented Wind Turbine, Transactions of the Institution of Professional Engineers New Zealand: Electrical/Mechanical/Chemical Engineering Section, 26, 13–19, 1999. a

Phillips, D., Richards, P., and Flay, R.: CFD modelling and the development of the diffuser augmented wind turbine, Wind Struct., 5, 267–276, 2002. a

Phillips, D. G.: An Investigation on Diffuser Augmented Wind Turbine Design, PhD thesis, The University of Auckland, Auckland, New Zealand, 2003. a

Politis, G. K. and Koras, A. D.: A Performance Prediction Method for Ducted Medium Loaded Horizontal Axis Windturbines, Wind Engineering, 19, 273–288, http://www.jstor.org/stable/43749587, 1995. a

Powell, M. J. D.: An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives, Computer J., 7, 155–162, https://doi.org/10.1093/comjnl/7.2.155, 1964. a

Schittkowski, K.: NLPQL: A fortran subroutine solving constrained nonlinear programming problems, Ann. Oper. Res., 5, 485–500, https://doi.org/10.1007/BF02022087, 1986. a

Selig, M. S., Guglielmo, J. J., Broeren, A. P., and Giguere, P.: Summary of Low-Speed Airfoil Data – Vol. 2, SoarTech Publications, Virginia Beach, VA, USA, 1996. a

van Bussel, D. G. J. W.: The Science of Making More Torque from Wind: Diffuser Experiments and Theory Revisited, J. Phys. Conf. Ser., 75, 012010, http://stacks.iop.org/1742-6596/75/i=1/a=012010, 2007. a, b

van Bussel, G.: An Assessment of the Performance of Diffuser Augmented Wind Turbines (DAWT's), in: Proceedings of the Third ASME/JSME Joint Fluids Engineering Conference, 18–23 July 1999, San Francisco, CA, USA, 1999. a, b, c

Venters, R., Helenbrook, B. T., and Visser, K. D.: Ducted Wind Turbine Optimization, J. Sol. Energ.-T. ASME, 140, 011005–011005–8, https://doi.org/10.1115/1.4037741, 2017. a, b, c, d, e, f, g

Werle, M. J. and Presz, W. M.: Ducted Wind/Water Turbines and Propellers Revisited, J. Propul. Power, 24, 1146–1150, https://doi.org/10.2514/1.37134, 2008. a