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Governing equation LES (Large Eddy Simulation) is adopted to solve the

computational fluid dynamics:

𝜕𝜌𝑢 𝑖𝜕𝑥𝑖

= 0 1

𝜕

𝜕𝑡𝜌𝑢 𝑖 +

𝜕

𝜕𝑥𝑗𝜌𝑢 𝑖𝑢 𝑗 =

𝜕

𝜕𝑥𝑗𝜇𝜕𝑢 𝑖𝜕𝑥𝑗

−𝜕𝑝

𝜕𝑥𝑖−𝜕𝜏𝑖𝑗

𝜕𝑥𝑗+ 𝑓𝑖 (2)

where 𝑢 𝑖 and 𝑝 are respectively filtered velocities and pressure, 𝜌 is the air density, 𝜇 is the dynamic viscosity, 𝜏𝑖𝑗 is the SGS (subgrid-scale Reynolds

stress) and calculated by the Smagorinsky-Lill model and 𝑓𝑖 is the source term to present the wind turbine induced forces.

System configuration

The ADM-R (Actuator Disk Model with Rotation) is adopted to parameterize rotor-induced forces under low (WT-L) and high Reynolds number (WT-H).

Wake properties downwind the turbine

A Numerical Study of Wind Turbine Wake by Large Eddy Simulation and Proposal for a New Analytical Wake Model

Guowei QIAN, Takeishi ISHIHARA Department of Civil Engineering, School of Engineering, The University of Tokyo

1. Ishihara, T., Yamaguchi, A. & Fujino, Y., 2004. Development of a new wake model based on a wind tunnel experiment. Global Wind Power, p.6.

2. Wu, Y.T. & Porté-Agel, F., 2011. Large-Eddy Simulation of Wind-Turbine Wakes: Evaluation of Turbine Parametrisations. Boundary-Layer Meteorology, 138(3), pp.345–366.

3. Katic, I., Højstrup, J. & Jensen, N.O., 1986. A simple model for cluster efficiency. EWEA 1986, pp.407–410.

4. Crespo, A. & Hernández, J., 1996. Turbulence characteristics in wind-turbine wakes. Journal of Wind Engineering and Industrial Aerodynamics, 61(1), pp.71–85.

5. Bastankhah, M. & Porté-Agel, F., 2014. A new analytical model for wind-turbine wakes. Renewable Energy, 70, pp.116–123.

Wind turbines operating in the downwind wake flow are subjected to two main problems: decreased energy production due to the velocity deficit and increased fatigue loading due to the added turbulence intensity, therefore an accurate and efficient evaluation of the wake effect is essential. In previous studies, the effects of the ambient turbulence intensity and thrust coefficient have not been analyzed systematically both for velocity deficit and added turbulence intensity[1,2]．In addition, existing wake models still have problems of accuracy and universality[3,4].

In this study, a series of numerical simulations to investigate the wake characteristics are performed and an analytical model to predict the wake effect for a single wind turbine under various operation and ambient turbulence conditions is subsequently proposed.

Based on a series of numerical simulations of wind turbine wake a new analytical wake model is proposed. Following conclusions are obtained:

(1) The numerical wind tunnel and the wind turbine model built in this study show high accuracy, which reveals that the thrust coefficient and ambient turbulence intensity can be the two main dominant parameters for wake effects.

(2) The proposed wake model well predicts the mean velocity and turbulence intensity in both near and far wake region, which can be an effective tool to predict the wake effects for the wind turbine with varying thrust coefficients under offshore and onshore conditions.

PO.022

Introduction Verification

Numerical Simulation and Results

Conclusions

References

Center line velocity deficit & Top tip height added turbulence intensity

The proposed model shows good agreement with the experiment and LES data. The Katic.’s model[3] generally underestimates the velocity deficit. The Crespo’s model[4] predicts well the maximum value in the near wake region, however it underestimates the decay rate of ∆𝐼1 in the far wake region for large 𝐶𝑇 cases.

Horizontal profiles in the wake region

Compared with existing models based on the top-hat assumption, the proposed model gives more reasonable distribution for both velocity and turbulence intensity.

Case WT-Type Ia λ Pitch(°) CT

1 - 0.035 - - -

2 - 0.137 - - -

3 WT-L 0.035 5.52 0 0.37

4 WT-L 0.035 9.69 0 0.81

5 WT-L 0.137 5.52 0 0.37

6 WT-L 0.137 9.69 0 0.81

7 WT-H 0.035 5.66 7.4 0.36

8 WT-H 0.035 8.89 0 0.84

9 WT-H 0.137 5.66 7.4 0.36

10 WT-H 0.137 8.89 0 0.84

Analytical Model

Velocity deficit The velocity deficit ∆𝑈 𝑥, 𝑟 induced by the rotor is

assumed to be axial symmetric and self-similar following the distribution function 𝜙 𝑟 𝜎 with a Gaussian shape in each wake cross section , which are expressed by the following equations:

𝑈 𝑥, 𝑦, z = 𝑈0 y, z − ∆𝑈 𝑥, 𝑟 3

𝛥𝑈 𝑥, 𝑟 𝑈ℎ = 𝑓 𝐶𝑇, 𝐼𝑎, 𝑥 𝐷 𝜙 𝑟 𝜎 (4)

𝜙 𝑟 𝜎 = exp −𝑟2

2𝜎2 (5)

where, 𝑈 𝑥, y, z is the mean velocity in the wake region, 𝑈0 y, z is the ambient mean velocity D is the rotor diameter and r is the radial distance form the center of the wake, 𝜎 is the standard deviation of the velocity deficit distribution at each cross section.

𝑓 𝐶𝑇, 𝐼𝑎, 𝑥 𝐷 representing the maximum normalized velocity deficit occurring at the wake center is modeled as the function of 𝐶𝑇 and I𝑎 with the expression as :

𝑓 𝐶𝑇, 𝐼𝑎, 𝑥 𝐷 =1

𝑎 + 𝑏 ⋅ 𝑥 𝐷 + 𝑝 2 6

𝑎 = 0.93𝐶𝑇−0.75𝐼𝑎

0.17, 𝑏 = 0.42𝐶𝑇0.6𝐼𝑎

0.2, 𝑝 =0.15𝐶𝑇

−0.25𝐼𝑎−0.7

1 + 𝑥 𝐷 2 (7)

where a and b are the model parameters and p is a correction term proposed to consider the velocity deficit in the new wake region, the expressions of which are obtained by parameter identification based on the numerical simulation results.

The wake region is assumed to expand linearly and 𝜎 is expressed by the same formula in Bastankhah & Porte-Agle’s[5] model as follows:

𝜎

𝐷= 𝑘∗

𝑥

𝐷+ 𝜀 8

𝑘∗ = 0.11𝐶𝑇1.07𝐼𝑎

0.20, 𝜀 = 0.23𝐶𝑇−0.25𝐼𝑎

0.17 (9)

where, 𝑘∗ and 𝜀 can be determined by comparing the parameter 𝑎 and b in

the Equation (7) with the parameters in the model proposed by Bastankhah

& Porte-Agle[5].

Added turbulence intensity Regardless of the incoming condition, the added

turbulence intensity ∆𝐼1 𝑥, 𝑟 can be also assumed axial symmetric and self-similar following the supposed distribution function 𝜙 𝑟 𝜎 with a dual-Gaussian shape, which are described by the following equations:

𝐼1 𝑥, 𝑦, 𝑧 = 𝐼02(𝑦, 𝑧) + 𝛥𝐼1

2 𝑥, 𝑟 10

∆𝐼1 𝑥, 𝑟 = 𝑔 𝐶𝑇, 𝐼𝑎, 𝑥 𝐷 𝜑 𝑟 𝜎 (11)

𝜑 𝑟 𝜎 = 𝑘1exp −𝑟−𝐷 2 2

2𝜎2 + 𝑘2exp −𝑟+𝐷 2 2

2𝜎2 (12)

𝑘1 = cos2 𝜋 2 ⋅ 𝑟 𝐷 − 0.5 𝑟 𝐷 ≤ 0.5

1 𝑟 𝐷 > 0.5 (13)

𝑘2 = cos2 𝜋 2 ⋅ 𝑟 𝐷 + 0.5 𝑟 𝐷 ≤ 0.5

0 𝑟 𝐷 > 0.5 (14)

where, 𝐼1 𝑥, y, z is the turbulence intensity in the wake region, 𝐼0 y, z is the ambient turbulence intensitywhere, 𝜎 is the standard deviation of the added turbulence intensity at each cross section which can be determined by the same expression as Equation (8), 𝑘1 and 𝑘2 are the parameters to combine the added turbulence intensity induced by each tip side.

𝑔 𝐶𝑇, 𝐼𝑎, 𝑥 𝐷 denotes the maximum added turbulence intensity occurring at the tip side position, which is also modeled as the function of 𝐶𝑇 and I𝑎 with the expression as:

𝑔 𝐶𝑇, 𝐼𝑎, 𝑥 𝐷 =1

𝑑 + 𝑒 ⋅ 𝑥 𝐷 + 𝑞 15

𝑑 = 2.3𝐶𝑇−1.2, 𝑒 = 1.0𝐼𝑎

0.1, 𝑞 =0.7𝐶𝑇

−3.2𝐼𝑎−0.45

1 + 𝑥 𝐷 2 (16)

where 𝑑 and 𝑒 are the model parameters and 𝑞 is a correction term proposed to consider the velocity deficit in the new wake region, the expressions of which are also obtained by parameter identification based on the numerical simulation results.

Numerical simulation domain: the setting generally follows Ishihara’s[1] wind tunnel experiment.

Wind turbine model in wind tunnel experiment

(WT-L)

Choshi offshore 2.4MW wind turbine (WT-H)

Parameters for cases: 𝐼𝑎 - ambient turbulence intensity, λ - tip speed ratio, 𝐶𝑇 - thrust coefficient.

0.2

0.4

0.6

0.8

1Proposed

Katic

RSM

LES

EXP

U

/U

Case 3&7

0.2

0.4

0.6

0.8

1

E

I

Q

B

R

EXP

T

LES

RSM

0 2 4 6 8 10

U

/U

Case 5&9

0.2

0.4

0.6

0.8

1

E

I

Q

B

R

EXP

T

LES

RSM

0 2 4 6 8 10

x/D

U

/U

Case 6&10

0.2

0.4

0.6

0.8

1C

G

O

B

B

EXP

D

LES

RSM

0 2 4 6 8 10

U

/U

Case 4&8

0.1

0.2

0.3

Proposed

CH

LES

EXP

D

0 2 4 6 8 10

I 1

Case 3&7

0.1

0.2

0.3

EXP

CH

LES

WT-Real

IEC

New_IEC

Proposed

0 2 4 6 8 10

I 1

Case 4&8

0.1

0.2

0.3CH

EXP

LES

WT-Real

IEC

New_IEC

Proposed

0 2 4 6 8 10x/D

I 1

Case 6&10

0.1

0.2

0.3

LES

WT-Real

CH

EXP

IEC

New_IEC

Proposed

0 2 4 6 8 10

I 1

Case 5&9

Exp LES(WT-L) LES(WT-H)

ProposedCrespo & Hernandez

0.5 1 1.5 2 2.5 3 3.5 4 4.50.98

11.021.041.061.08

1.11.12

Exp LES(WT-L) LES(WT-H)

ProposedKatic

0.5 1 1.5 2 2.5 3 3.5 4 4.50.98

11.021.041.061.08

1.11.12

-1

-0.5

0

0.5

1

Katic(2)

Katic(4)

Katic(6)

Katic(8)

p2

p4

p6

p8

L

M

N

O

LES(2D)

LES(4D)

LES(6D)

LES(8D)

RSM(2D)

RSM(4D)

RSM(6D)

RSM(8D)

-2 0 2 4 6 8 10

y/D

Case 3&7

-1

-0.5

0

0.5

1

Katic(2)Katic(4)

Katic(6)Katic(8)

p2p4

p6p8

LM

NO

LES(2D)LES(4D)

LES(6D)LES(8D)

RSM(2D)RSM(4D)

RSM(6D)RSM(8D)

-2 0 2 4 6 8 10

y/D

Case 4&8

-1

-0.5

0

0.5

1

Katic(2)Katic(4)Katic(6)

Katic(8)LM

N

Op2

p4p6

p8LES(2D)

LES(4D)LES(6D)

LES(8D)RSM(2D)

RSM(4D)RSM(6D)RSM(6D)

-2 0 2 4 6 8 10

y/D

Case 5&9

-1

-0.5

0

0.5

1

Katic(2)

Katic(4)Katic(6)

Katic(8)L

MN

Op2

p4p6

p8LES(2D)

LES(4D)LES(6D)

LES(8D)RSM(2D)

RSM(4D)RSM(6D)

RSM(8D)

-2 0 2 4 6 8 10

y/D

0 1

U/Uh x/D

Case 6&10

-1

-0.5

0

0.5

1

Katic(2)

Katic(4)

Katic(6)

Katic(8)

p2

p4

p6

p8

IEC2

IEC4

IEC6

IEC8

LES(2D)

LES(4D)

LES(6D)

LES(8D)

RSM(2D)

RSM(4D)

RSM(6D)

RSM(8D)

-2 0 2 4 6 8 10

y/D

Case 3&7

-1

-0.5

0

0.5

1

Katic(2)Katic(4)Katic(6)Katic(8)LMNOp2p4p6p8LES(2D)LES(4D)LES(6D)LES(8D)RSM(2D)RSM(4D)RSM(6D)RSM(8D)

-2 0 2 4 6 8 10

y/D

Case 4&8

-1

-0.5

0

0.5

1

Katic(2)

Katic(4)

Katic(6)

Katic(8)

p2

p4

p6

p8

L

M

N

O

LES(2D)

LES(4D)

LES(6D)

LES(8D)

RSM(2D)

RSM(4D)

RSM(6D)

RSM(8D)

-2 0 2 4 6 8 10

y/D

Case 5&9

-1

-0.5

0

0.5

1Katic(2)

Katic(4)Katic(6)

Katic(8)LMNOp2

p4p6p8LES(2D)LES(4D)LES(6D)

LES(8D)RSM(2D)

RSM(4D)RSM(6D)RSM(8D)

-2 0 2 4 6 8 10

y/D

0 0.3x/DI

1

Case 6&10

LES(WT-L)

LES(WT-H)

Proposed

Katic

0.5 1 1.5 2 2.5 3 3.5 4 4.50.98

1

1.02

1.04

1.06

1.08

1.1

1.12

LES(WT-L)

LES(WT-H)

Proposed

Crespo & Hernandez

0.5 1 1.5 2 2.5 3 3.5 4 4.50.98

1

1.02

1.04

1.06

1.08

1.1

1.12

Fence

Wind turbine

Fence

Spires

Outlet

Inlet

7.5m

18.5m

5.5m

1.8m

1.5m

-2 0 2 4 6 8 100

0.5

1

1.5

2

2.5Exp(2D)

Exp(8D)

Katic(2D)

Katic(4D)

Katic(6D)

Katic(8D)

Proposed(2D)

Proposed(4D)

Proposed(6D)

Proposed(8D)

LES(2D)

LES(4D)

LES(6D)

LES(8D)

RSM(2D)

RSM(4D)

RSM(6D)

RSM(8D)

z/H

Case 3&7

-2 0 2 4 6 8 100

0.5

1

1.5

2

2.5Exp(2D)

Exp(8D)

Katic(2D)

Katic(4D)

Katic(6D)

Katic(8D)

Proposed(2D)

Proposed(4D)

Proposed(6D)

Proposed(8D)

LES(2d)

LES(4d)

LES(6d)

LES(8d)

RSM(2D)

RSM(4D)

RSM(6D)

RSM(8D)

A

A

z/H

Case 4&8

-2 0 2 4 6 8 100

0.5

1

1.5

2

2.5

Exp(2D)

Exp(8D)

Katic(2D)

Katic(4D)

Katic(6D)

Katic(8D)

Proposed(2D)

Proposed(4D)

Proposed(6D)

Proposed(8D)

new nacell(2D)

new nacell(4D)

new nacell(6D)

new nacell(8D)

new_n(2)

new_n(4)

new_n(6)

new_n(8)

z/H

Case 5&9

-2 0 2 4 6 8 10

0

0.5

1

1.5

2

2.5Exp(2D)

Exp(8D)

Katic(2D)

Katic(4D)

Katic(6D)

Katic(8D)

Proposed(2D)

Proposed(4D)

Proposed(6D)

Proposed(8D)

LES(2D)

LES(4D)

LES(6D)

LES(8D)

RSM(2D)

RSM(4D)

RSM(6D)

RSM(8D)

z/H

0 1U/U

hx/D

Case 6&10

-2 0 2 4 6 8 100

0.5

1

1.5

2

2.5Exp(2D)

Exp(8D)

Katic(2D)

Katic(4D)

Katic(6D)

Katic(8D)

Proposed(2D)

Proposed(4D)

Proposed(6D)

Proposed(8D)

LES(2D)

LES(4D)

LES(6D)

LES(8D)

RSM(2D)

RSM(4D)

RSM(6D)

RSM(8D)

z/H

Case 3&7

-2 0 2 4 6 8 100

0.5

1

1.5

2

2.5Exp(2D)

Exp(8D)

Katic(2D)

Katic(4D)

Katic(6D)

Katic(8D)

Proposed(2D)

Proposed(4D)

Proposed(6D)

Proposed(8D)

LES(2D)

LES(4D)

LES(6D)

LES(8D)

RSM(2D)

RSM(4D)

RSM(6D)

RSM(8D)

z/H

Case 4&8

-2 0 2 4 6 8 100

0.5

1

1.5

2

2.5

Exp(2D)

Exp(8D)

CH(2D)

CH(4D)

CH(6D)

CH(8D)

Proposed(2D)

Proposed(4D)

Proposed(6D)

Proposed(8D)

LES(2D)

LES(4D)

LES(6D)

LES(8D)

RSM(2D)

RSM(4D)

RSM(6D)

RSM(8D)

z/H

Case 5&9

-2 0 2 4 6 8 100

0.5

1

1.5

2

2.5

Exp(2D)

Exp(8D)

CH(2D)

CH(4D)

CH(6D)

CH(8D)

Proposed(2D)

Proposed(4D)

Proposed(6D)

Proposed(8D)

LES(2D)

LES(4D)

LES(6D)

LES(8D)

RSM(2D)

RSM(6D)

RSM(4D)

RSM(6D)

z/H

0 0.3I1

x/D

Case 6&10

Exp

LES(WT-L)

LES(WT-H)

1.5 2 2.5 3 3.5 4 4.50.98

1

1.02

1.04

1.06

1.08

1.1

1.12

The above figure presents the resolved normalized mean velocity and turbulence intensity downwind the turbine, and it can be seen that LES simulation results generally show good agreement with the experiment data. Wind turbine model under low and high Reynolds number within the same ambient turbulence flow show almost same wake properties when the thrust coefficient is close.