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Research Papers: Flows in Complex Systems

Analytical Solution for the Optimal Spacing of Wind Turbines

[+] Author and Article Information
Ajay K. Prasad

Department of Mechanical Engineering
and Center for Carbon-free Power Integration,
University of Delaware,
Newark, DE 19716
e-mail: prasad@udel.edu

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received April 9, 2013; final manuscript received September 11, 2013; published online November 12, 2013. Assoc. Editor: Mark F. Tachie.

J. Fluids Eng 136(1), 011107 (Nov 12, 2013) (5 pages) Paper No: FE-13-1230; doi: 10.1115/1.4025648 History: Received April 09, 2013; Revised September 11, 2013

The lateral and longitudinal spacing between individual turbines in a wind turbine array must be large enough to minimize the wake effects caused by an upstream turbine on those that lie downstream from it. Here, the flow downstream of a single wind turbine is examined by modeling its far-field development as a turbulent axisymmetric wake which is well described in the turbulence literature. In particular, the velocity defect profile in the wake is approximated by a Gaussian function in the radial coordinate. Scaling laws are used to derive closed-form solutions for the wake diameter, wake velocity defect, and wake power recovery as functions of downstream distance from the rotor. Our results show that at a downstream distance of 10 rotor diameters, the wake centerline velocity will recover to 77% of the free-stream value. It is also seen that the power within the wake recovers quickly for small downstream distances, but beyond about 10 rotor diameters, the rate of power recovery slows down. Implications for the optimal spacing of wind turbines are discussed based on these findings.

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Figures

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Fig. 1

Axisymmetric wake development behind a bluff body in a uniform airflow showing increase of wake radius b, and decrease of velocity deficit Vc with increasing downstream distance x. The region of velocity deficit is shaded gray.

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Fig. 2

Wake development behind a single rotor

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Fig. 3

Variation of turbine power coefficient with velocity ratio v

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Fig. 4

Variation of wake diameter with x/Do for various v

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Fig. 5

Variation of wake centerline velocity with x/Do for various v

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Fig. 6

The recovery of power in the wake with x/Do for various v′. The power is integrated from the wake centerline to the nominal wake width (r = b).

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Fig. 7

The recovery of power in the wake with x/Do for various v′. The power is integrated from the wake centerline to the rotor tip (r = Do/2).

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Fig. 8

A laterally staggered array might alleviate the wake effect on downstream turbines

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