# Coupling Numerical Methods and Analytical Models for Ducted Turbines to Evaluate Designs

^{1}

^{2}

^{2}Wind Inc., Boston, MA 02116, USA

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{−6}m

^{2}/s, and a density of $\rho $ = 999.1 kg/m

^{3}[19]. The 0.8 m diameter turbine is a three-bladed horizontal axis turbine, with a twenty degree root pitch, a five degree tip pitch, and NACA 63-8xx sections [15]. The effects of the hub and the upright support are assumed to be small and are ignored. The turbulent intensity for the freestream turbine is 2.9%. The ducted model uses the same turbine geometry and water parameters with the addition of an efficient duct design provided by V

^{2}Wind Inc. This duct is designed for a ducted wind turbine operating in near ground conditions. Therefore, ambient turbulent intensity is assumed 10% for the inlet boundary condition of the ducted CFD cases. The specifics of the duct geometry are discussed in Section 2.3.

#### 2.1. Derivation of the Analytical Model for Ducted Turbines

#### 2.2. Freestream CFD Model

#### 2.3. Ducted CFD Model

#### 2.4. Numerical Methods and Turbulence Modeling

^{5}at the lowest tip speed ratio (TSR) to 2.5 × 10

^{5}at the highest TSR. This is in the transitional regime. The equations for TSR, blade Reynolds number, $R{e}_{\mathrm{blade}}$, and duct Reynolds number, $R{e}_{\mathrm{duct}}$, are respectively shown in Equations (9)–(11). TSR is calculated as a function of the turbine radius, r, the rotational speed, $\omega $, and freestream velocity, ${v}_{\infty}$. $R{e}_{\mathrm{blade}}$ is calculated as a function of the chord at 70% of the radius, ${c}_{{r}_{o}}$ = 0.03 m, ${v}_{\infty}$, $\nu $, the radius at 70%, ${r}_{0}$ = 0.28 m, and $\omega $. $R{e}_{\mathrm{duct}}$, calculated with duct maximum diameter, ${D}_{\mathrm{duct}}$ = 1.536 m, at freestream velocity, is 1.9 × 10

^{6}. The power is calculated as the product of the torque and rotational speed.

## 3. Results

#### 3.1. Freestream CFD Predictions versus Experiments

_{T}, and the coefficient of power, C

_{P}, respectively.

#### 3.2. CFD Results of the Ducted Turbine

#### 3.3. Comparison of the Analytical Model to Experimental and CFD Results of Ducted Turbine Cases

#### 3.3.1. Analytical Model Prediction of Screen Tests for a Ducted Turbine

#### 3.3.2. Analytical Model Prediction of the RANS CFD Results for a Turbine in a Uniform Tube

#### 3.3.3. The Analytical Model’s Ability to Predict RANS CFD Results for a Ducted Turbine

## 4. Discussion

#### Design Implications

## 5. Conclusions

## 6. Patents

^{2}Wind Inc.’s patent application: US20160305247A1.

## Acknowledgments

^{2}Wind Inc.’s contributions was made possible by the United States Army Small Business Innovation Research Contract W911QY-13-C-0054.

## Author Contributions

## Conflicts of Interest

^{2}Wind Inc., which is a wind turbine company. This research was motivated by a desire to better understand how power is extracted from ducts and to develop a better analytical method to analyze ducted turbines.

## Abbreviations

AMI | arbitrary mesh interface |

CFD | computational fluid dynamics |

LES | large eddy simulation |

MRF | moving reference frame |

RANS | Reynolds averaged Navier–Stokes |

URANS | unsteady Reynolds averaged Navier–Stokes |

## Appendix A. Effect of Modifying the Pitch of the Turbine in a Tube

**Figure A1.**Effect of pitch on the power ratio and ${C}_{P,{A}_{\mathrm{max}}}$. (

**a**) ${C}_{P,{A}_{\mathrm{max}}}/{C}_{P,\mathrm{eff}}$ vs. TSR; (

**b**) ${C}_{P,{A}_{\mathrm{max}}}/{C}_{P,\mathrm{eff}}$ vs. ${C}_{T{v}_{1}}$; (

**c**) ${C}_{P,{A}_{\mathrm{max}}}$ vs. ${C}_{T{v}_{1}}$.

**Figure A2.**Streamlines at 70% span for the 35° root pitch turbine in the tube. (

**a**) Peak TSR; (

**b**) Over spin.

## Appendix B. Design Twist and Pitched Blade Details for the Ducted Turbine

**Figure A4.**Pitch distribution for the original blade at different root pitch angles and for the design twist blade at different root pitch angles. (

**a**) Design twist; (

**b**) all blades.

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**Figure 2.**Freestream mesh. (

**a**) The whole domain. (

**b**) The mesh near the freestream turbine with the wake refinement and the cylindrical AMI.

**Figure 3.**Ducted turbine mesh. (

**a**) The whole meshed domain sliced in the middle. (

**b**) The mesh near the ducted turbine. (

**c**) Cross section of the duct.

**Figure 5.**Steady state and transient results for freestream ${C}_{P}$ and ${C}_{T}$ as a function of TSR. (

**a**) ${C}_{P}$ vs. TSR; (

**b**) ${C}_{T}$ vs. TSR.

**Figure 6.**Ducted results. (

**a**) ${C}_{P,{A}_{\mathrm{max}}}$ vs. TSR; (

**b**) ${C}_{P,{A}_{\mathrm{max}}}$ vs. ${C}_{T{v}_{1}}$; (

**c**) ${v}_{1}/{v}_{\infty}$ vs. ${C}_{T{v}_{1}}$; (

**d**) ${C}_{P}$ vs. TSR.

**Figure 7.**Flow visualization of ducted turbine. (

**a**) Velocity field of ducted turbine at different TSRs. (

**b**) Coarse mesh flow and base mesh flow at TSR = 5.714.

**Figure 8.**Analytical predictions along with the screen tests of Gilbert et al. [2]. (

**a**) ${C}_{P,{A}_{\mathrm{max}}}$ vs. ${C}_{T{v}_{1}}$; (

**b**) ${v}_{1}/{v}_{\infty}$ vs. ${C}_{T{v}_{1}}$.

**Figure 9.**Analytical models along with the CFD of Bahaj et al. turbine [15] in a tube with diameter equal to duct throat diameter. (

**a**) ${C}_{P,{A}_{\mathrm{max}}}$ vs. ${C}_{T{v}_{1}}$; (

**b**) ${C}_{P,\mathrm{eff}}$ vs. ${C}_{T{v}_{1}}$; (

**c**) ${v}_{1}/{v}_{\infty}$ vs. ${C}_{T{v}_{1}}$.

**Figure 10.**Analytical model and ducted CFD predictions. (

**a**) ${v}_{1}/{v}_{\infty}$ vs. ${C}_{T{v}_{1}}$; (

**b**) ${C}_{P,{A}_{\mathrm{max}}}$ vs. ${C}_{T{v}_{1}}$.

**Figure 11.**Analytical model and ducted CFD predictions with calibrated velocity. (

**a**) ${v}_{1}/{v}_{\infty}$ vs. ${C}_{T{v}_{1}}$; (

**b**) ${C}_{P}$ vs. ${C}_{T{v}_{1}}$.

**Figure 13.**Effect of pitch on ${C}_{P,{A}_{\mathrm{max}}}$ and ${C}_{T,{v}_{1}}$ for a ducted turbine.

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**MDPI and ACS Style**

Knight, B.; Freda, R.; Young, Y.L.; Maki, K.
Coupling Numerical Methods and Analytical Models for Ducted Turbines to Evaluate Designs. *J. Mar. Sci. Eng.* **2018**, *6*, 43.
https://doi.org/10.3390/jmse6020043

**AMA Style**

Knight B, Freda R, Young YL, Maki K.
Coupling Numerical Methods and Analytical Models for Ducted Turbines to Evaluate Designs. *Journal of Marine Science and Engineering*. 2018; 6(2):43.
https://doi.org/10.3390/jmse6020043

**Chicago/Turabian Style**

Knight, Bradford, Robert Freda, Yin Lu Young, and Kevin Maki.
2018. "Coupling Numerical Methods and Analytical Models for Ducted Turbines to Evaluate Designs" *Journal of Marine Science and Engineering* 6, no. 2: 43.
https://doi.org/10.3390/jmse6020043