Aerodynamics Computational Assignment #2: Flow Over Thin Airfoils

Matlab Code Policy:

Computational codes must be written individually and are expected to be written in MAT-LAB. If you have collaborated with others while writing your code be sure to acknowledge them in the header of your code, otherwise you may receive a zero for plagiarism. All code les required to successfully run the computational assignment driver script should be sub-mitted via the course website by 11:59pm on the due date. Code les will not be accepted after the given due date.

Re ection Questions:

In this assignment, there are multiple re ection questions. These re ection questions are provided to help you review the functionality of your code, help you analyze and understand your results, and to test your understanding of the concepts being studied.

Learning Outcomes:


    1. Understand how thin airfoil theory is used to approximate aerodynamic forces.

    2. Practice using the superposition of elementary  ows to complete analysis of an airfoil.

    3. Understand the e ect of ow parameters on streamlines, equipotential lines, and pres-sure contours.











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ASEN 3111    Aero Computational Assignment # 2    Spring 2021






The    theory, as

along

the

z



UPPER SURFACE

OF THIN AIRFOIL

x
α    0    CHORD LINE    c
V∞

Then, the    ow is represented by a vortex sheet whose strength    (x) is given by:



s
1



x

(x) = 2 V
1





c














x












c



where is the angle of attack of the incoming ow relative to the x-axis and V1 is the free-



separated

by a depicted in the

z


Vortex of Strength

Γi

x
0    x1    x2    x3    x4             xi    xN  c

Δx


Write a MATLAB function which plots the stream lines, equipotential lines, and pressure contours for ow about a thin symmetric airfoil using the approximations detailed above.


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ASEN 3111    Aero Computational Assignment # 2    Spring 2021



Your function should take the form:

function Plot_Airfoil_Flow(c,alpha,V_inf,p_inf,rho_inf,N)

where c is the chord length c (in meters), alpha is the angle of attack (in degrees), V_inf is the free-stream ow speed V1 (in meters per second), p_inf is the free-stream pressure p1 (in Pascals), rho_inf is the free-stream density 1 (in kilograms per meter cubed), and N is the number of discrete vortices N employed to approximate the vortex sheet.

Using your MATLAB function,

Visualize or generate plots of the stream lines, equipotential lines, and pressure con-tours for ow about a thin symmetric airfoil with c = 5 m, = 10 , V1 = 50 m/s, p1 = 101:3 103 Pa, and 1 = 1:225 kg/m3.

Conduct a study of the e ect of the number of discrete vortices N on the resulting ow and pressure eld accuracy for the aforementioned values. The assessment of accuracy should not only be qualitative in nature but also quantitative. Namely, a study of error in velocity and pressure as a function of N should be conducted. It is up to the student to de ne a measure (or measures) of error for this problem. Generate a plot of the convergence of the chosen error assessment versus the the number of discrete vortices N.

Conduct a study on how stream lines and equipotential lines are a ected by changes in: 1) chord length, 2) angle of attack, and 3) free-stream  ow speed. For this study, it is recommended that you graphically compare these variables using a  gure with sub-plots so that the changes in each variable can be compared side-by-side. To best facilitate this comparison ensure that your plots are consistently scaled and rendered.


Re ection: Evaluate the pressure contours, and identify the locations of minimum and maximum pressure. How do these change as the above conditions are altered? Consider the streamlines and equipotential lines; are the elds continuous? What does this imply?

Hint: You will need to use the principle of superposition within your MATLAB function. This principle is illustrated in the MATLAB le Lifting_Cylinder.m located on the course website, wherein the stream lines for the ow around a cylinder with circulation are plotted by superposing ow from a uniform ow, a dipole, and a vortex. It is recommended that you follow the logic of Lifting_Cylinder.m when building your function Plot_Airfoil_Flow.











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