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ObjectiveThe objective of the present study is to model the turbulent air flow around a horizontal axis wind turbine using a modified form of the second-order Reynolds Stress Model (RSM), known as the Algebraic Stress Model (ASM). ANSYS Fluent and MATLAB software packages are utilized in this computational study. Results of different versions of ASM are compared against those found from a fully-converged RSM simulation.

Introduction Wind energy is one of the cleanest forms of the renewable energy, and is used for

electricity generation. Computational Fluid Dynamics (CFD) has proven to be a simple, cheap and relatively

accurate tool for prediction of wind turbine performance, where suitability of different designs can be evaluated at a low cost.

Turbulent flow around a wind turbine blade with NACA4412 airfoil is modeled numerically. This airfoil is well known for a very high lift to drag ratio.

Algebraic Stress Model is employed along with k- turbulence model, to obtain Reynolds Stresses updated for the near-blade region.

Analysis of turbulent flow around horizontal axis wind turbines using Algebraic Stress Model

Ryoichi S. Amano and Saman Beyhaghi

Department of Mechanical EngineeringUniversity of Wisconsin-Milwaukee

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Algebraic Stress Model (Implicit):

2

3

i jk ij ij ij

u uP P

k

There are no longer partial derivatives of <uiuj> terms in the equations. The system of equations for all the Reynolds stresses need to be solved simultaneously.

Final form:

Different ASM versions: Complete model : No simplification

Compact ij,2 :

No Wall effect : ij = ij,1 + ij,2

22

3

ij , ij ij kP P

1) Solve the flow field and turbulence everywhere around the blade using the RANS based k- turbulence model.

2) Store all the mean velocity gradients, as well as k and distribution in the areas of interest (near-blade region).

3) Solve the flow field only for a few iterations to find somewhat reasonable estimate for the Reynolds stresses, to be used as initial guesses for ASM. Alternately some arbitrary constant values can be used, perhaps average values from a previous simulation.

4) Solve the system of algebraic equations for six unknown Reynolds stresses.5) Finally, in order to assess the accuracy of the ASM results, solve the flow field

everywhere around the blade using the RSM turbulence model, and compare the results with those obtained via ASM in the near-blade region.

Numerical Algorithm:

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Axial velocity contours in the vertical mid plane:

Distribution of the normalized Reynolds stresses around the blade at the height z/L=0.5, and distance from the blade h/L= 2.5e-6 (full ij,2 form)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

X coordinates of data points [m]

Y c

oord

inat

es o

f da

ta p

oint

s [m

]

10 20

30

40 50

60 70

80

90

100

110120

130140

150

160

170

180

190

200210

220 230 240 250 260 270 280290

Indices of grid points of interest around the blade at z/L=0.5, and h/L=5e-5

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

<u

u>

/U2

For Reynolds stress component <uu>

RSM Fluent

ASM

0 50 100 150 200 250

-0.1

-0.05

0

0.05

0.1

<u

v>/U

2

For Reynolds stress component <uv>

RSM Fluent

ASM

0 50 100 150 200 250

-0.1

-0.05

0

0.05

0.1

cell number (along curve length)

<u

w>

/U2

For Reynolds stress component <uw>

RSM Fluent

ASM

0 50 100 150 200 2500

0.05

0.1

<vv

>/U

2

For Reynolds stress component <vv>

RSM Fluent

ASM

0 50 100 150 200 250

-0.1

-0.05

0

<vw

>/U

2

For Reynolds stress component <vw>

RSM Fluent

ASM

0 50 100 150 200 2500

0.1

0.2

0.3

cell number (along curve length)

<w

w>

/U2

For Reynolds stress component <ww>

RSM Fluent

ASM

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Conclusion

Based on the current results, it appears that the full ij,2 with wall effects shows slightly better agreements with RSM data.

The compact ij,2 model turns out to be the least accurate model. The main point of this study is that without going through the time consuming RSM

simulation, one can use computationally cheaper methods (such as k-), and then solve a set of algebraic equations in order to get reasonable values for Reynolds stresses in the near wall region.

It is recommended to repeat the simulations using other forms of wall correction terms, and explicit ASM.

To have a better assessment of the various ASM models, this study needs to be performed for all of the nodes within a cylinder-like volume encompassing the entire blade.

The next step is to employ a hybrid LES/RANS model, by combining ASM with large eddy simulation (LES) for analysis of flow around a wind turbine blade, to see if there is any improvement in torque and power generation predictions.

Future Work