Airfoil Lab

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

Make sure to check occasionally for any changes that might be made. This version is correct as of 04/6/2010

1. Instructions

This lab is due Monday, April 19 in lecture. Late labs are not accepted. You are encouraged to work in groups of no more than 3 people, though you may work alone.

On your title page, clearly mark

• Names
• Student ID numbers
• Recitation number and professor
• Time and TA for recitation

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.

2. Lab Goals

The purpose of this lab is to apply several Calculus III techniques to a physical problem. The physical problem, described below, concerns airfoils and lift producing rotating cylinders. The primary mathematical technique in this lab is the computation of line integrals. Mathematica can be used to compute these integrals.

3. Fluid Dynamics Background

The following four subsections briefly introduce some of the ideas related to airfoils in fluid dynamics. Airfoil theory is a vast field, and the basics are usually covered in a one-year course. The field of fluid dynamics covers a lot of ground as well, and airfoil theory is only a part of it. In the case of airfoil theory, the "fluid" in question is air, although the results obtained apply not only to Earth's atmosphere, but to any "air" in general. The information below is the minimum background in both fluid dynamics and airfoil theory needed to solve this lab.

I. Airfoil Nomenclature

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

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 α is the angle between the chord line and the freestream velocity V_∞, which represents the velocity field of the fluid through which the airfoil travels.

The lift L is the component of the aerodynamic force on the airfoil perpendicular to V_∞. One usually defines L’ (this is standard notation in airfoil literature; It is NOT the derivative of 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 is 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 ρ_∞ is the density of the fluid far from the airfoil, and Γ is the circulation (to be defined later).

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 ψ represents the paths of a fluid (streamlines ) around an airfoil. In order to obtain the equation for one of these paths, one simply need set ψ = constant to get an implicit function of x and y (or r and θ). Then, by using different constant values, different implicit functions will be created with each one representing a different streamline. One can visualize the streamlines by implicitly plotting the equations obtained.

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

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

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

III. Circulation

The circulation Γ is defined as:

Where V is the velocity field around an airfoil and Ω 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. In this case, one can find the "strength of the vortex sheet" - γ and compute the circulation of the airfoil using the equation below:

Where C is the curve described by the mean camber line. The derivation of this property involves material beyond the scope of Calculus III, but we can use it to answer the questions below.

Note that we have provided two different ways to compute the circulation (Γ), in the lab you will need to use them both for the two different cases you are presented with. Make sure it is clear which computation to use in each case!

IV. Flow about a Rotating Cylinder

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

Where ω is taken to be positive counter-clockwise, and the freestream velocity is taken to be V_∞ = V_∞ i.

According to the equation above, if we spin a cylinder clockwise and insert it in a uniform fluid flow moving from left to right, the cylinder generates lift. This phenomenon is called the Magnus effect, named after the German engineer who first observed and explained it in 1852. It 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.

4. 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 steps will guide you through the process:

(a) Airfoil

1.

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.

Compute the mean camber line c(x).

2.

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 have a good idea of what the airfoil looks like. You may find the Mathematica commands Which[] or Piecewise[] to be useful in defining a piecewise function, as in f(x) and c(x). However, be careful with Which[], it will not work well when you are integrating over c(x) below.

3.

The density of air at sea level is ρ_∞=1.23kg/m³. Let the freestream velocity be V_∞=130 m/s (in the positive x direction). Let the strength of the vortex sheet for the airfoil at zero angle of attack (α=0) under the above conditions be given by:

Compute the lift per unit span L’ (in N/m) of the airfoil at zero angle of attack. Don’t forget to specify the parameterization being used.

4.

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

5.

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

(b) Rotating Cylinder

6.

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 that the circumference of the cylinder must equal the perimeter of the airfoil. Use arc-length integrals to determine the perimeter of the airfoil and then find the radius R of the corresponding cylinder.

7.

Compute the velocity field V around the cylinder (which is centered at the origin). Hint: You may want to change φ from polar to Cartesian coordinates.

8.

Compute the circulation Γ of the cylinder as a function of its angular velocity ω. You must use a choose a closed path Ω that encloses the cylinder, and is centered at the origin (A square box is perhaps the easiest to use). Take special care in parametrizing this path and in setting up the corresponding path integral. Provide details of the path integral used. (See section 14.2). Instead of the methods used in section 14.2 for computing line integrals, give an alternative way of computing the circulation Γ as some double integral over the region enclosed by Ω. You must set up the integral, but you do not need to compute it. Keep in mind that in order for this alternative representation of circulation to hold true, we must have that Ω be a simple and closed curve.

9.

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

10.

Suppose the equation for L’ computed in exercise 9. holds for -2V _∞/R < ω < 0. Make a plot of L’ vs. ω. Compute the maximum value of L’.

(c) Comparison Analysis

11.

Compare your results for both the airfoil and the cylinder, and decide which one has the best performance according to your calculations. Does this make sense physically? Justify your answer.

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^-7 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.