Assignment Information
This assignment is split into two sections. Your assignment is to be presented as a concise report in a Word document or PDF. 10% of the assignment marks are allocated for overall clarity and presentation of the report. Marks will be deducted for handwritten answers and screenshots of equations.
In your report, be sure to clearly identify the start of each section and question, and to provide a sub-heading (e.g. Section 1 Question 1“1.1 First Derivative of f(x)”). Each question will be marked separately, so make sure all relevant working and final answers are in that section.
1 Section 1: Numerical Approximations: 40%
Consider the function;
In this section you are required to compare the exact solution of the first derivative of this function, f′ (x), to the four-point forward, backward and central finite difference stencils given respectively as;
where ∆x is the grid spacing, fi = f(xi). The analytic expression for the first derivative of f(x) is;
For the written report:
1. Use the Taylor Series expansion to derive the leading order error for the stencils provided. Leave your final solution in terms of ∆x. You will receive marks for showing appropriate working. (Tip: You only need to expand the Taylor series up to the fifth derivative) (15%)
2. Using MATLAB, write a code that computes the first derivative of f(x) using the three stencils over the domain x = [-3, 3] for a variable number of points, n. Create a plot showing the forward, backward and central finite difference stencils compared to the analytic solution for n = 8, 16, 32. Make sure for the analytic solution you use at least n = 1000 points for the plot so that the complete curvature is shown. Comment on any differences you notice between the 3 stencils, and how they change with increasing grid resolution. (Tip: To avoid putting too many curves into one plot for comparison, it might be clearer to present
3 separate plots, one each for n = 8, 16, 32. On each of these plots you would then have 4 curves overlaid.) (10%)
3. Examine the L2-norm of the error ϵ for the numerical approximations by undertaking calcu- lations with n points where n = 2p + 1 for p = 2, 3, 4, .., 8, 9, 10. Create a log-log plot of the L2-norm verses ∆x and discuss the behaviour in light of your result from Question 1 and 2. Which stencil should you choose and why? (10%)
4. Include your Matlab script as an appendix to the report. Marks will be given for clarity of programming and commenting. (5%)
2 Section 2: Wind Turbine Blade Aerodynamics: 50%
2.1 Introduction
Figure 1: The Lillgrund Wind Farm (left) consisting of 48 Siemens SWT-2.3-93 2.3MW (right)
Donard Wind Turbines Ltd are developing new wind turbine blades. They are considering hiring you for the analysis of the aerodynamic performance of their new turbine blades, but first of all they want to have confidence in your ability to establish a workflow to compute the lift and drag of a turbine blade and understand your expected accuracy against experimental data. For this, they have provided you with a text file containing a 2D airfoil section for the Siemens SWT-2.3-93 2.3MW wind turbine blade section FFA-W3-211. This wind turbine is shown in Figure 1 and is employed in multiple offshore wind farms including the Lillgrund wind farm pictured.
Your key tasks are (i) undertake a calculation of the provided airfoil to validate your computational approach, (ii) apply your validated approach to generate performance metrics for the airfoil which will help Donard Wind Turbines Ltd understand your capabilities.
2.2 Assignment
Your tasks are:
1. Using your knowledge of CFD, undertake a validation of ANSYS Fluent for the provided airfoil section with Reynolds 1.8 × 106 and Mach 0.15 with freestream T = 288.15K and P = 101, 325Pa. You each have a different angle of attack as specified in the accompanying document‘ClassListAngles.pdf’. In the report [30 marks total]:
(a) present the computational grid employed, and concisely articulate your reasons for choice of grid type and spacing. Hint: you may have to reduce the defeaturing size to 1 × 10-4m to prevent the mesher from defeaturing the thin trailing edge. [5 marks]
(b) present and discuss your choice of governing models and boundary conditions. [5 marks]
(c) detail what steps you have taken to ensure that you are confident that your CFD solution is converged employing quantitative measures of grid convergence. Support your conclusions with appropriate quantitative figures and/or tables of key quantities [10 marks].
(d) compare your converged solution with the available experimental data for CL and CD detailed on page 76 onwards of the provided technical report and plotted in Figures 102 and 103. Comment on the quality of the agreement and discuss possible reasons for any discrepancies. [5 marks]
(e) present a contour flood showing the variation of static pressure around the airfoil and variation of velocity. Briefly discuss the results. [5 marks]
2. Based on the now validated approach, extend your analysis to examine the flow around the airfoil the same Reynolds number but at a Mach number and angle of incidence of your choice, with freestream T = 288.15K and P = 101, 325Pa. The Mach number should be greater than 0.4 but I suggest less than 5 (although if you’d like to try higher please do so just make sure you can address the items below). The angle of incidence should be ≤ 20。 if you are running at low Mach to enable some comparison with experimental data. [20 marks total]:
(a) calculate the new inflow conditions, and describe clearly the calculation resulting in the new viscosity you must use to ensure that the Reynolds number does not change from the previous case. [5 marks]
(b) detail any differences in the computational approach (grids, models, solution methods) compared with your previous calculation and demonstrate grid convergence [5 marks]
(c) calculate CL and CD for this new Mach number and/or angle and compare them to the original case, discussing any differences and the possible reason for those. [5 marks]
(d) present and discuss the static pressure and velocity fields highlighting differences with the original case. [5 marks]
3 Presentation: 10%
10 marks are allocated to the structure of the report, correct spelling and grammar, fluency of sentences, presentation of equations, clarity of figures and captions, axes and labels, quality of contour floods and any legends included.
3.1 Report and Hand-in
Write a report summarising your findings in a PDF. Structure your report such that each sub- question is a separate chapter. The report should not exceed 2500 words, any additional words will not be marked. The code in the appendix is not included in the 2500 words. It can contain any number of figures, but figures and tables in the report must be numbered and discussed. An electronic copy of the report should be submitted to Canvas by midnight on the 8th April 2025.
This assignment should take the average student 20 hours to complete.