The aerodynamic design of a ducted wind turbine for maximum total power coefficient was studied numerically using the axisymmetric Reynolds-averaged Navier–Stokes equations and an actuator disc model. The total power coefficient characterizes the rotor power per total device area rather than the rotor area. This is a useful metric to compare the performance of a ducted wind turbine with an open rotor and can be an important design objective in certain applications. The design variables included the duct length, the rotor thrust coefficient, the angle of attack of the duct cross section, the rotor gap, and the axial location of the rotor. The results indicated that there exists an upper limit for the total power coefficient of ducted wind turbines. Using an Eppler E423 airfoil as the duct cross section, an optimal total power coefficient of 0.70 was achieved at a duct length of about 15 % of the rotor diameter. The optimal thrust coefficient was approximately 0.9, independent of the duct length and in agreement with the axial momentum analysis. Similarly independent of duct length, the optimal normal rotor gap was found to be approximately the duct boundary layer thickness at the rotor. The optimal axial position of the rotor was near the rear of the duct but moved upstream with increasing duct length, while the optimal angle of attack of the duct cross section decreased.

The power output of a wind turbine can be augmented by surrounding it with a duct, typically referred to as a ducted wind turbine (DWT), a diffuser augmented wind turbine, or a shrouded wind turbine. The effect of the duct is to increase the mass flow rate through the rotor. For a given rotor area, significantly more power can be obtained for a DWT compared to an open wind turbine. However, by adding a duct, the total area of the device facing the wind direction is increased. If the power produced per total projected frontal area of the device is calculated for DWTs, often values closer to that of open wind turbines are found

Since the experimental demonstration of the power augmentation provided by shrouding wind turbines

The design variables of a ducted wind turbine.

Many of the numerical studies use axisymmetric computational fluid dynamics (CFD)
models

As most studies focus on maximizing

The effect of the angle of attack of the duct cross section has been included in most studies

There are a few studies on the optimization of the shape of the duct cross section for optimal

The effect of the rotor gap as a design variable is considered in a few studies

This study investigates the effect of the duct length on the optimal design for maximizing the total power coefficient (

The computational domain.

The paper is organized as follows: Sect. 2 discusses the CFD model used for RANS simulations including a validation study of the actuator disc model used followed by the details of the DWT design parameterization and the optimization method used. The optimization results and how the optimal design changes with the duct length are discussed in the third section. This section also involves a sensitivity analysis of

Ansys Fluent 17.1 was used to solve the incompressible RANS equations with the

The grid near the duct, which uses both structured and unstructured elements, is depicted in Fig.

Grid near the duct and actuator disc.

Two metrics were used to characterize the performance of a DWT: first, the power coefficient based on the swept area of the rotor,

In order to validate the actuator disc model, the axisymmetric actuator disc without a duct was simulated in the domain shown in Fig.

Comparison of axisymmetric RANS actuator disc model with 1-D momentum actuator disc theory.

The performance of the DWT was considered to be a function of a number of design variables, mentioned in the introduction, including the chord length of the duct cross section (

When

The Hooke and Jeeves direct search optimization technique

The optimization technique starts by modifying the initial design, one design variable at a time. These exploratory moves in the design space are called steps in coordinate directions. All the design variables were scaled by their initial values, and the initial step size in each coordinate direction was set to 5 %. Based on the success or failure of these steps in the coordinate directions of the five-dimensional design space, the algorithm creates pattern directions, moves the base point, and increases or decreases the step sizes. The stopping criterion was

Convergence of the optimization technique to the optimal design.

The design variables for optimal

The power coefficients of designs for optimal

The optimization was performed at a constant

To verify that there is indeed a maximum in

Both here and in

The streamlines and contours of nondimensional velocity magnitude

The geometry and streamlines of the first three configurations in Table

The optimal normal rotor gap (

Similar to

The sensitivity of the total power coefficient of the optimal design to different design variables

The sensitivity of the total power coefficient of the optimal design to different design variables.

Figure

Comparison of

To examine Reynolds number sensitivity, the simulation of the optimal design was repeated at

The result of optimization at this higher Reynolds number is shown in the second row of Table

The Reynolds number sensitivity of optimal design for

The contour plot of eddy viscosity ratio

Eddy viscosity ratio contours near the leading edge of the duct for the optimal design (

The optimal design of a ducted wind turbine with the Eppler E423 airfoil as its cross section was investigated using CFD simulations of axisymmetric RANS equations with the

The results demonstrate the existence of a maximum

Additionally, the results of optimization at fixed

The optimal angle of attack of the duct cross section decreased significantly with increasing the duct length. Additionally, the optimal design was on the verge of flow separation with respect to the angle of attack of the duct cross section and the axial position of the rotor.

The optimal normal rotor gap was close to the boundary layer thickness at the rotor. Therefore, the optimal normal rotor gap scaled proportional to the chord length of the duct cross section as the turbulent boundary layer thickness almost linearly increases with the chord length of the duct cross section. This gap is needed to create the high-velocity annular jet, which helps keep the boundary layer attached.

The optimal rotor position was at the rear of the duct, but at greater values of duct length it moved further upstream in the duct. This further upstream position was more effective at eliminating flow separation and hence allowed greater values of

Data are available upon request from the corresponding author.

NBS contributed to the methodology, ran the simulations, post-processed the data, and wrote the first draft of the paper. BTH supervised the study and contributed to the conceptualization, methodology, writing, and revision of the paper. KDV contributed to the conceptualization and revision of the paper.

The authors declare that there is no conflict of interest.

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This paper was edited by Jens Nørkær Sørensen and reviewed by Peter Jamieson, David Wood, and one anonymous referee.