How to calculate the drag on an airfoil from momentum equation (Fundamental of Aerodynamics 5th edition, J.D.Anderson)?

Question

If you got the book, please read 2.6 section (page 135). Here is my sumary of this section:

An airfoil in a wind tunnel (wingtips were butted against both sidewalls of wind tunnel). The control volume is 'abcdefghia', it has unit depth in the z direction (perpendicular to the page). Through some steps that you can read from pages, we got: $$ Drag\space on \space the \space airfoil = -\unicode{x222F}_{ai+bh}(\rho \vec V.\vec {ds})u $$ In experinment, how can we know where the line bh is, how long it is..?

In the picture above, can we move line bh backward (to the right) ?

How big is the wind tunnel or how far from top to bottom walls of the wind tunnels (with given airfoil)is in oder to have the lines ab and hi ?

Do you feel this method is very inaccurate with a lot of negligibly and pitot rake thing (page 141) ?

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@Peter: Thank you, nice to hear that !! . Actually, I thought I understood this section until I face the problem 2.2 at page 198 (by the end of chapter 2). I reviewed this section and a lot of questions arised. First, sorry because English is not my mother language.

1st question: I really have no idea where the line bh is (both in vertically and horizontally) and how is the length of bh ?.

2nd question: Suppose we already have the line bh located properly. My physical feel tell me that the line bh can move backward and the velocity gradinent $ u_2 $ = f(y) stay the same. But imagine in real life, if we move the line bh far enough, the velocity gradient $u_2$ will be uniform and equal the freestream velocity $u_1$ !!

About the third question, I think: we must have top and bottom walls of the wind tunnel that the pressure over them is equal to the free stream pressure, I mean: move them far from the airfoil in vertical direction so that the pressure over them is uniform and equal to freestream pressure.

4th question: I still had no idea because I never use my hands to do these kind of experiments, but my instinct tell me it will have a lot of errors. My word maybe too dumb but I'd like to learn what people do in real experiment.

Answer 1

You are basically asking two questions here:

1. How do we choose an appropriate control volume?
2. What effects does the physical geometry of a wind tunnel have on our ability to make accurate measurements?

Contol-volume selection

Young, et al., have some nice words:

Any volume in space can be considered as a control volume. The ease of solving a given fluid mechanics problem is often very dependent upon the choice of the control volume used. Only by practice can we develop skill at selecting the "best" control volume. None are "wrong," but some are "much better" than others.

The researchers who came up with the control volume in your image had plenty of practice and knew how to cleverly decrease their computational load through control-volume selection. They could have selected a control volume outside the wind tunnel—or even 1000 miles from it—but this approach would have required a lot more work to obtain their results. I suggest looking through this set of notes for a mathematically inclined presentation on the control-volume selection in this particular case (it will also help you with problem 2.2 and your second question). Alv's answer discusses some of the more qualitative factors influencing the control-volume selection in this instance.

So to answer your first two questions specifically, you can put $bh$ wherever you want and make it as long as you want, but there is an advantage to setting up the control volume the way it is in your image.

Wind-tunnel experiments

To answer your third question, the walls of the wind tunnel would ideally be infinitely far away from the test article. Such an arrangement is obviously impossible, so various wall-interference corrections are developed. Here is a brief overview of those corrections; a simple Google search will yield much more information. Individual tunnels will develop their own set of corrections that include things such as the appropriate size for a model and the distance it should be from the walls (also search blockage ratio).

Wind-tunnel design can also help mitigate wall interference. Many higher-speed tunnels feature slotted walls in their test sections to strike a compromise between directing the flow of air and allowing pressure disturbances to dissapate.

Specifically to your fourth question, the accuracy of your results will be determined by how judiciously you determine and apply your corrections. Given that wind tunnels are still used in industry to develop new airplanes, we can be certain that there is a way to get reasonable results. Even further, the data produced by the very studies Anderson describes is still used by engineers and scientists; the pressure-integration method has proven quite robust over the years.

Answer 2

those are some great questions. I will try to be very concise.

Firstly, regarding to the line ‘bh’, there is no fixed length you can calculate. Ideally you want to make sure you get measurements of the entire region where there is a momentum loss. Therefore, you seek to take measurements until you reach a point far enough so $u_1=u_2$.

Note that this momentum loss is directly related to the size of the wake, thus the length of ‘bh’ would be determined by the Reynolds number, separation point, incidence, etc. One common problem, especially at high AoA, is that the rake is so large that the wake rake does not cover this ‘bh’ line. This would mean that the momentum loss measured, and therefore the drag, is smaller than it is physically.

With regards to the distance from the trailing edge: You must avoid being too close since phenomena like recirculation bubbles or the pressure difference between the upper and lower part of the aerofoil would basically ruin your measurements (the velocity is not normal to the probes , etc). On the other hand, the flow in the rake induces an additional momentum loss, thus if you take the measurements farther aft, you would be also measuring this contribution, apart from the drag of the aerofoil. In addition, the size of the wake increases, leading to the problems mentioned in the first question. Typically, a distance of approximately one chord is used for the experiments.

The next question is related to a different topic, wind tunnel corrections. From the picture, due to the momentum loss, using the equation of continuity, you will always have $ai<bh$. What is defined as ‘”streamlines far away from the body”, lines ab and hi, are assuming constant pressure and velocity. If you add the wind tunnel boundaries, imposing the streamlines to be parallel to the surface of the tunnel, there would be an acceleration of the flow, $u_1<u_2$, thus this p=p_infinity will no longer be achieved.

This change in the velocity profile will affect the measurements at the rake, therefore some corrections are needed. The bigger the wind tunnel is, the smaller the corrections are. In the book you will find a nice chapter about this.

Regarding to the last question, unfortunately I don’t have the book anymore… But this method has indeed some drawbacks (in addition to the ones aforementioned). Firstly, the pressure drag is almost neglected. Moreover, measuring the turbulent flow at the wake is difficult and leads to errors if the velocity is not normal to the probe (this happens especially at high AoA).

Hope this has helped!