Airfoil Lab

Picture of Flettner’s boat, which used rotating cylinders instead of sails.

·         Important: Due the the heavy use of the "Symbol" font (for Greek letters), this lab will likely not display correctly in any web browser other than Internet Explorer.  When viewing or printing this lab, please use IE.  If the Greek letters q, p, w, r, y, j and W do not appear correctly as theta, pi, omega, rho, psi, phi, and Omega, please use a different web browser!

1. Instructions

This lab is due Monday, May 1 at the beginning of lecture.  TA's will have lab hours from Monday April 24  until Sunday the 30th.  The TAs in the lab are there to help you with Mathematica syntax; they are not there to debug your files.  Please keep this in mind.

Late labs are not accepted. You are encouraged to work in groups of no more than 3 people, though you can work alone.  You need hand in only one paper copy of your lab per group.  Labs must also be submitted electronically through AMESS; Failure to do so will result in a grade of 0 for the lab.

On your title page, clearly mark

  • Names
  • Student ID numbers
  • Professor
  • TA for recitation and recitation number
  • If you do not include this information, as much as 10% will be deducted from your final score

Format is worth 20% of your grade.  Please refer to the following writing guidelines for the expository sections of this report, You are required to know and follow the Writing Guidelines for all labs.

Writing Guidelines for Calculus Labs
 Sample Lab Reports

DO NOT LEAVE ANY OF YOUR FILES SAVED ON THE COMPUTERS IN THE LAB!
IF YOUR FILES ARE FOUND SAVED ON A LAB MACHINE,
IT MAY RESULT IN YOUR GROUP GETTING A 0 (ZERO) FOR THE LAB.
You can either e-mail them to yourself, use webfiles, or save on a USB 'memory stick'

2. Lab Goals

The purpose of this lab is to apply several Calculus III techniques to a physical problem.  The problem, described below, concerns airfoils and lift-producing rotating cylinders. The primary mathematical technique in this lab is line integrals.  Mathematica can be used to compute these integrals.  Make sure you fully understand the material from sections 14.1 and 14.2 in the text, as you will need this knowledge to complete this lab.

3. Mathematica Commands

Below is a partial list of Mathematica commands that you may (...will...) find useful for this lab.  You should read Mathematica's help file for each of these, and make sure you understand what they do, before starting the lab.

·         Which[]

·         Table[]

·         NIntegrate[]
   Note: This will work quickly when Integrate[] takes a very long time.

·         ImplicitPlot[]
    Note: You must load this command first, <<Graphics`ImplicitPlot`

·         PlotVectorField[]
    Note: You must load this command first, <<Graphics`PlotField`

4. Fluid Dynamics Background

The following four subsections briefly introduce you to some of the basic ideas related to airfoils in fluid dynamics. Airfoil theory is a vast field, and the basic theory usually needs a one-year course to be covered. The information below is the minimum background needed for you to solve this lab. Make sure you understand all the concepts below, otherwise you don’t stand a chance in completing this lab.

I. Airfoil Nomenclature

An airfoil can be represented by the functions f and g, with the latter describing its lower surface, and the former, its upper surface. 

The leading edge and the trailing edge are the points on the airfoil where the upper and lower surfaces meet, with the former encountering some fluid particle before the latter. 

The chord line is the straight line connecting the leading edge to the trailing edge. 

The mean camber line c is the curve that contains the points halfway between the upper and lower surfaces as measured perpendicular to the mean camber line itself. If we are given f and g, we can approximate c as follows:

The angle of attack a is the angle between the chord line and the freestream velocity V¥ , which represents the velocity of the fluid far from the airfoil. 

The lift L is the component of the resultant aerodynamic force on the airfoil perpendicular to V¥. One usually defines L’ to be the lift per unit span of the airfoil (e.g. if an airplane wing has L’= 255 N/m, and if such a wing has 10m from one side to another, the total lift it generates is 2550N). The lift per unit span can be calculated by the following formula: 

 

Where r¥ is the density of the fluid far from the airfoil, and G is the circulation (to be defined later) in m2/s.

 

II. 2-Dimensional Velocity Field, Stream Function and Velocity Potential 

The velocity field V represents the velocity of a fluid around an airfoil. In the case of a two-dimensional flow, we may write V = u i + v j. 

The stream function y represents the paths of a fluid (streamlines) around an airfoil. In order to get the equation for one these paths, one simply needs to set y = constant to get an implicit function of x and y (or r and q ). Then, by using different constant values, different implicit functions will be created, with each one representing a different streamline. Then, one can visualize the streamlines by implicit plotting the equations obtained. Every point along a streamline is parallel to the fluid velocity.

One can also show the following relation between the velocity field and the stream function: 

 

 

The velocity potential f is another way of representing a fluid flow around an airfoil. The equipotentials f = constant are curves orthogonal to the streamlines. It can be shown that the velocity potential relates to the velocity field by the following equations: 

 Therefore, given some velocity potential function f, one can readily compute the velocity field V and the stream function y associated with that velocity potential.

 

III. Circulation 

The circulation G is defined as: 

 

Where V is the velocity field around an airfoil and W is a closed path that encloses the airfoil. In aerodynamics it is convenient to consider positive circulation to be clockwise. Thus, the minus sign in the equation above, since the integral is computed in the counter-clockwise direction.

 

So, the circulation is simply the negative of the line integral of the fluid velocity around a closed path. 

Sometimes, the velocity field around some airfoil may not be known, but the pressure on the airfoil may be measured. Then, one can compute the strength of the vortex sheet g (which involves mathematical tools and computing skills beyond Calculus 3), and compute the circulation of the airfoil using the equation below: 

 

Where C is the curve described by the mean camber line. 

 

IV. Flow about a Rotating Cylinder 

The flow about a cylinder of radius R, centered at the origin and rotating at an angular velocity w is given by the following velocity potential in polar coordinates: 

The stream function representing this flow in polar coordinates is:

In both equations, w is taken to be positive counter-clockwise, and the freestream velocity is taken to be V¥=V¥ i. 

According to the equations above, if we spin a cylinder clockwise and insert it in a uniform fluid flow going from left to right, the cylinder generates lift. This phenomenon is called the Magnus effect, named after the German engineer who first observed it and explained it in 1852. This is why baseballs and soccer balls travel in curved paths when given a spin. 

In 1920, the German engineer Anton Flettner replaced the sail on a boat with a spinning cylinder, which, in combination with the wind, provided propulsion for the boat (see picture on first page).

5. Problem Statement

Suppose you are an Aerospace Engineer. Your job will be to analyze the performance of a given airfoil and compare it to the performance of a rotating cylinder of same surface area. 

The following tasks will guide you in applying your Calculus knowledge to Fluid Dynamics, so that you can accomplish your job. 

I. Airfoil

a.

The airfoil to be analyzed is described by the following two functions: 

Where f(x) represents the upper surface, and g(x) the lower surface; note that lengths are in meters
The function cosh(x) is the hyperbolic cosine function, in Mathematica, it is Cosh[x].
Compute the mean camber line c(x). You may find the Mathematica command Which[] useful in defining a piecewise function.

b.

Generate a plot of the airfoil with its mean camber line. Make sure that the y-axis is in the same scale as the x-axis, so that you can have a good idea of what the airfoil looks like. You might also want to turn the axes off.  In Mathematica, type  ?Axes  for more information.

c.

For the entire lab, use the values: Density of air at sea level: r¥=1.23kg/m3,  Freestream velocity: V¥=132 m/s (in the positive x direction).
Let the strength of vortex sheet for the airfoil at zero angle of attack (a=0) under the above conditions be given by: 

 

in units of m/s. Compute the lift per unit span L’ (in N/m) of the airfoil at zero angle of attack (don’t forget to specify the parameterizations being used i.e, how you set up the integral).

d.

From experimental data, you know that when a = -2 degrees, L’ = 0. Assuming that, in the range –2 < a < 15 degrees, L’ is a linear function of a, compute L’ as a function of a.

e.

Make a plot of L’ vs. a (using the function and domain above). Calculate the maximum value of L’.

II. Rotating Cylinder

a.

Now, you need to determine the radius of the cylinder to be used in the comparison with the airfoil. The specification is that the surface area of the cylinder needs to be the same as the surface area of the airfoil. Since we are considering a 2-D cross section, this means the perimeter of the airfoil must be the same length as the circumference of the cylinder. Use arc-length integrals to determine the perimeter of the airfoil and then find the radius R of the corresponding cylinder.

b.

Compute the velocity field V around the cylinder (which is centered at the origin.) Hint: you may want to change f from polar to Cartesian coordinates, otherwise you will have to endeavor yourself in doing the computations in polar coordinates.

c.

Plot the velocity field V (it is just a vector field) over the domain {(x,y): -0.6 < x < 0.5 , -0.5 < y < 0.5} with 0=w (non-rotating cylinder.) On this plot, overlay a cylinder of radius R.  Now, do the same for a rotating cylinder with angular velocity =w -300 rad/s.  Comment on any differences.

d.

Convert the stream function y from polar to Cartesian. Then, plot some (about 20) streamlines ( y(x,y) = c ) for several values of c, -100 < c < 100 over the domain {(x,y): -1 < x < 1 , -1 < y < 1}.  Overlay these streamlines on the same graph.  As in part c., do this both for w=0 and w=-300 rad/s.  Describe what these plots tell you about the airflow around the cylinder.
Hint: One easy way to do this in Mathematica is to start your command with,  Show[Table[ImplicitPlot[ ........  or   ImplicitPlot[Table[ ........
Hint: If that way isn't easy enough, consider the contours of y.

e.

Compute the circulation G of the cylinder as a function of its angular velocity w.  Make sure you fully explain the curve you are integrating over, and provide details of the integral used. (See section 14.2).

f.

Could one compute the circulation G of the cylinder using Green’s Theorem?  If so, set up the integral, and compute it; if not, fully explain why.

g.

Compute L’ of the cylinder as a function of its angular velocity w.

h.

Suppose the equation computed in g. holds for –2V¥ /R < w < 0. Make a plot of L’ vs. w. Compute the maximum value of L’.

III. Comparison

a.

Compare your results for both the airfoil and the cylinder, and decide which one has the best performance according to your calculations.

b.

You don’t usually see airplanes with rotating cylinders in place of regular wings flying in the sky. From your calculations and your knowledge of Physics, can you explain why? Should anything else be taken into account in the previous analysis to help you make a better performance comparison? Note: it may help to convert angular velocities from radians/s to rpm. Also note that there are several reasons why airfoils are used instead of rotating cylinders -- try to think of as many as you can (and report several of the most significant).

6. Lab Report

You need to describe your computations and your results in a report. Your report must be written as a scientific report, with introduction, body and conclusion. Here are a few reminders:

  • Explain the purpose of the lab. State the problem.
  • You need not teach fluid dynamics to the reader. You may assume the reader is familiar with the notation and concepts.
  • When doing calculations, you must state which equations you are using (e.g. if you are computing circulation, you must state ).
  • Solve the tasks in a logical concise manner. Write your report based on these tasks, but you must link them as if you were telling a story to the reader (which you are).  Simply spitting out answers to these items is not an acceptable report.
  • Always explain what you are doing and why.  
  • When performing numerical calculations, figures less than 10-9 should be considered to be zero (they are due to round-off errors).

Remember the Following:

  • Always include units in your answers;
  • Always label plots and refer to them in the text;
  • Labs must be typed. Including the equations in the main body (part of your learning experience is to learn how to use an equation editor). An exception can be made for lengthy calculations in the appendix, which can be hand written (as long as they are neat and clean), and minor labels on plots, arrows in the text and a few subscripts. 
  • Your report doesn't have to be long. You need quality, not quantity of work. Of course you cannot omit any important piece of information, but you need not add any extras.
  • DO NOT include print outs of computer software screens. This will be considered as garbage. You simply need to state which software you used in each step, and what it did for you.
  • You must include any plot that supports your conclusions or gives you insight in your investigations.
  • Write your report in an organized and logical fashion. Section headers such as Introduction, Background, Problem Statement, Calculations, Results, Conclusion, Appendix, etc... are not mandatory, but are highly recommended. They not only help you write your report, but help the reader navigate through your paper, besides giving it a cleaner look. 
Lab Created by Marcio Carvalho, Mar 2001. Modified by David Beltran-del-Rio.
Last modified by Jason Sherman and Elizabeth Green, April 2006.
Special thanks to Prof. Brian Argrow, for going over the main ideas of the lab.