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ME3341: Numerical Methods Project Background Overall production of energy in wind turbines operating in a wind farm is inhibited primarily through energy decreases from the wake velocity deficit and fatigue increases due to turbulence intensity of upwind turbines

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ME3341: Numerical Methods Project

1. Background

Overall production of energy in wind turbines operating in a wind farm is inhibited primarily through energy decreases from the wake velocity deficit and fatigue increases due to turbulence intensity of upwind turbines. Prediction of these is crucial to planning, design, and optimization of power production of wind farms. Wake models are used to approximate the velocity and turbulence intensity in the wake in order to provide valuable information about the average power production [1]. Wake models are created via curve fitting of measurements or simulation data from the wakes of turbines. In this project, we will use data obtained via high-fidelity simulation of a model wind turbine with diameter D = 1.1m. in a wind tunnel operating in the regime where power production is optimal [2].

2. Data

Data of the average streamwise velocity deficit, (u_in(z/D) − u(z/D))/u_in(z/D = 0), and streamwise velocity variance, ⟨u_′u_′⟩(z/D)/u²_in(z/D = 0), where u_in(z/D) is the incoming velocity, are stored in files ‘turbine_velocity_deficit.txt’ and ‘turbine_velocity_variance.txt’, respectively. Each contain 12 columns of data. The first column is the transverse direction coordinates, (z/D). The next 11 columns the profiles at downwind locations. The coordinates of all eleven locations (x/D) are stored in ‘turbine_downwind_locations.txt’.

3. Project

Prepare a well-organized report that has an introduction, numerical methods, results and conclusion section. For more description of what to include in each section see the class syllabus. There should be a subsection for each question below in the results section. Answer all questions and provide code used for curve-fitting as needed in an appendix.

1. Using a non-linear curve-fitting, fit all eleven velocity deficit profiles to the following Gaussian equation:               (1)

where you will find coefficients for amplitude A₁, standard deviation σ₁, and c.

1. • Provide a plot for each profile comparing the data to the curve-fit with z/D on the abscissa and velocity deficit on the ordinate. How well does the curve-fitting work?
   • Provide a plot of the residual norm with respect to the downwind location x/D. What trends are there in the variance of the parameters?
   • Provide a plot of the standard deviation σ₁ with respect to the downwind location x/D. Are there any trends? Given the variance in the previous plot, how well is σ₁ approximated?
2. Using non-linear curve-fitting all eleven velocity variance profiles to the following equation:

 (2)

where parameters are σ₂ is the standard deviation, A₂ is the amplitude, and d. Model coefficients b₁ and b₂ are subject to the equations [3]:

 (3)

and

 (4)

2. • Provide a plot for each profile comparing the data to the curve-fit with z/D on the abscissa and velocity variance on the ordinate. How well does the curve-fitting work?
   • Provide a plot of the residual norm with respect to the downwind location x/D. What trends are there in the variance of the parameters?
   • Provide a plot of the standard deviation σ₂ with respect to the downwind location x/D. Are there any trends? Given the variance in the previous plot, how well is σ₂ approximated?
3. The standard deviations σ₁ and σ₂ provide a metric to quantify the width of the wake at a given downwind location. The wake expands (grows spanwise) and the velocity deficit reduces (or recovers) with downwind distance from the turbine. This is important because wake expansion and recovery play a role in where downwind turbines can be placed. Using a linear regression and the equation y = a₁(x/D)+a₂, fit the standard deviations σ₁ and σ₂ as function of x/D in order to investigate the expansion of the wake.
   • Provide a plot of the linear fit with data.
   • How do the slopes differ from one another? Do you have an explanation?
   • What are the coefficients of determination? Is this a good fit? Why or why not?
   • If we could exclude some of the data (e.g. all data where x/D < 5), would the fit be any better?
4. Using the fit y = a₁(x/D)^β1, linearize the problem, use linear regression for the standard deviations σ₁ and σ₂. How does β₁ compare to the linear regression where it was assumed that σ ∼ x¹? Does a linear assumption of wake recover hold up?

References

1. Majid Bastankhah and Fernando Porté-Agel. A new analytical model for wind-turbine wakes. Renewable Energy, 70:116–123, 2014.
2. Daniel Foti, Xiaolei Yang, Filippo Campagnolo, David Maniaci, and Fotis Sotiropoulos. Wake meandering of a model wind turbine operating in two different regimes. Phys. Rev. Fluids, 3(5):054607, 2018.
3. Takeshi Ishihara and Guo-Wei Qian. A new Gaussian-based analytical wake model for wind turbines considering ambient turbulence intensities and thrust coefficient effects. J. Wind Eng. Ind. Aerod., 177:275–292, 2018.