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Analytic analysis of team race performance


Book Review - by Joe Supercool

Low Speed Aerodynamics 
by Katz and Plotkin

Buy now from the bookshop

This review may not be quite what you would expect from a book reviewer. Indeed, you may decide that this review is really a load of rubbish. On the other hand, I may be able to stir the sympathies of messer's Katz and Plotkin to give me my own copy. But that is getting ahead of things...

Many years ago, in a place far away.... Sydney Australia, 1984; I was walking across the turf farm at Windsor that we used for free flight trimming, looking for my wayward F1C power model. Instead of finding it, I found instead Jim Christie, who was at the time unbeatable in F1B Wakefield. Andy Kerr had shown me how to make fibreglass propellers, so I began to take more and more interest in the subject of propeller design. Since at that point in time I knew absolutely nothing about propeller design, my ignorance was quite patent. Noticing that the prop on Jimís model had rather elegant curves, I was moved to comment:

  "Nice looking prop Jim"

  To which Jim replied:

  "Well, its not an accident you know"

This rather shocked me, as this was 6 words more than he usually spoke to me. You might say I found Jim to be rather reserved! Since I was on a roll, I carelessly threw back the rejoinder:

  "What do you mean, Jim?"

  "Its been designed that way"

Revelation. Here was a man who knew how to design propellers!! How the rest of the conversation progressed I do not recall; however, it turned out that Jim had the Larrabee presentation from the 1979 National Free Flight Symposium. This presentation included everything one needed to know to design a propeller. Furthermore, it was all laid out as a set of algorithms ready to be plugged into a computer program.

He was kind enough to post this information to me. This proved to be a turning point in my life. I was to change from being a well-paid Nuclear Standards Scientist to a poorly paid, self-employed propeller designer and manufacturer. So poorly paid that this year (2002) I received from the Federal Government a low-income benefit of $65. Let no one say our politicians are less than generous!

Moving right along, in the Larrabee dissertation were the algorithms for analysing a propeller by blade element analysis. To the best of my knowledge, the first presentation of this theory was given in 1894 by Drzeweicki (pronounced Jay-vee-yet-ski), and it is still the theory one finds in any number of fluid dynamics textbooks. I took Larabee's version and basically hammered it to death in my computer. It proved to be very instructive indeed, and I learned a great deal from that about propeller behaviour.

But there was a devil in the detail. The theory divided the propeller into a series of short airfoils, each of which was treated as being independent of the adjacent airfoil. The overall performance of the propeller was obtained by adding up the performance of all the little short airfoils. Now you can do that in wing theory as well, but there is just one little problem. The wing has to be of infinite span, or, at least, so long that one could lose interest in building it. A bit like an Irish runway, short but very wide. I could not accept the accuracy of this method as applied to propellers. 

This was one devil I wanted to put back in Hell. But how to do it numerically, I had no idea. There is an excellent volume by Houghton and Stock called "Aerodynamics for Engineering Students" (Edward Arnold (Publishers) Ltd., London, 1960) which describes the vortex theory of wings. No doubt, the maths in this book could be used to test the validity of the independent blade element idea, but it was not in a form that I found accessible. Very frustrating.

My good friend Stuart Maxwell in 1997 showed me Katz and Plotkin's book, which is the subject of this review. I was to find the answers therein.

But first, before I delve further into this subject, I must make the necessary disclaimers. I do not own a copy of "Low Speed Aerodynamics" by Katz and Plotkin. I only have read a borrowed copy, and that was 5 years ago. Now you will understand how vague this review is going to be!

To continue: Stu asked me if I could understand the maths in this book. Here was a challenge. I did three years of Pure Mathematics at University back in the '60's, maybe all that study was going to pay off now! Yes, it did!

The book is written as a highly rigorous and formalised study of vortex theory, which is applied to the so-called "panel method". The natural language of this theory is vector integral calculus. It is totally without meaning to the layman. If you don't have the maths, you can't read this book.

However, if you do, it is a delight. One chapter in particular stands out, I wish I could remember which one!! The feature of this chapter is that it breaks down the vector cross-products into computational form which permit the calculation of the induced axial, radial and tangential velocities due to a finite line vortex of nominated strength.

What a mouthful, but how brilliantly useful! I'll skip the "panel method" component of the book, as being of no interest to me. Although there is no reference to propellers, the maths is right there to do a neat analysis of the independent blade element theory that was so bugging me.

It works this way. Each propeller blade element is treated as being a short little wing. It has its own set of tip vortices, and a vortex around its length that generates lift. This configuration of vortices is called a "horseshoe vortex", as the three components I have mentioned join together to form just such a horseshoe shape, which trails off into the distance. In fact, the shape trails all the way back to where the propeller first started moving, where may be found the "starting vortex, which closes the pattern of the vortex. Let us look at this for a moment.

 This diversion may be a bit esoteric, but vortices are rather beautiful and interesting. They even have their own set of laws. For example, vortices cannot be open ended: or expressed in another way, they may only end on a hard surface. Take for instance a smoke ring: this is a called a ring vortex. If you pass your finger thru it, it immediately is destroyed. It is closed upon itself, and that is the property that allows it to exist.

  Consider also a tornado. This is a vortex that ends attached to a hard surface: the surface prevents high pressue air from entering the vortex and likewise destroying it.  A waterspout ends on a fluid surface, the water being drawn into the low pressure in the vortex core. It can continue to exist only because extra vorticity is supplied by the weather system that first generated the vortex.

Which introduces another law, the conservation of vorticity. This is rather like the conservation of angular momentum: it is of interest because it can have some beautiful and quite extraordinary consequences.

Ask yourself: what happens when 2 ring vortices collide? Think hard, I have given you a clue already ! Give up?  I did not know. 

Back in '92, Lim and Nickels wanted to know real bad. They invented a machine to create ring vortices in a fluid! Not only that, they could colour them red or blue, and fire them at each other. How utterly, utterly cool!

They have received the all-time greatest award from Joe Supercool, for the most Supercool experiment ever done in the known Universe!

So what does happen when a red ring vortex collides with a blue ring vortex? Remember, vorticity is conserved in the collision, so they just cannot go away!

 Well, they very briefly writhe together and separate as 8 new ring vortices. Most wonderful of all, each of the new vortices has one half red and one half blue!

Don't believe me? If I remember correctly, the photographs of this mating were published in Nature, Vol 357, pages 225-227 in 1992. The photos can also be found in 'Fluid Vortices' edited by Sheldon I. Green, 1995 (ISBN 0-7923-3376-4).

Now back to propeller vortex theory. We were fitting horse-shoe vortices to each blade element of the propeller. This is a little tricky, as the propeller has twist, so that the side vortices are not aligned. Also they trail back in a circular path, so pity the poor horse that had to wear these shoes! As it happens, the strength of the vortex at the airfoil falls off fairly quickly as it trails back, so we do not have to consider the whole of space when adding up the vortex forces.

This is not true of the vortices on adjacent blade elements. They interact strongly. Indeed, the vortex at the tip element interacts all the way along the blade: all the elements affect each other. The blade elements are definitely not independent. The effect of a blade element on the other blade elements falls off inversely with its distance from those elements. This is a result used by Katz and Plotkin, as part of the vortex description.

Words are inadequate to describe this process any further, so I will desist from further description and move to results.

The qualities of most interest in propeller design are the interference factors.  Simply put, that is the air movements caused by the passage of the propeller and their accompanying forces. These cause the thrust and torque forces as familiar parameters. I computed the interference factors using the method of "Low Speed Aerodynamics" and obtained values 4 times higher than I expected. This was disappointing, as I had hoped to do better. Just what went wrong I do not know.

This disappointment was offset by the clear functional dependence obtained from the  "mathematical model". I could see the tip and root vortices, and the distribution of the downwash along the blade. This downwash is associated with the "slip" of the propeller, which is in essence the axial induced velocity produced by the propeller action. Ideally, this velocity would be uniform along the blade, a condition that yields the highest possible propeller efficiency.

The model showed that this uniformity could not be obtained, a result also stated by Larrabee. Propeller design by the independent blade element method does permit the attainment of a uniform "slip", so that the method must be in error.

Now to conclude the theory side. One glorious feature of the Katz and Plotkin maths is that one can model the prop in three dimensions. That is, if one is to model the propeller as a "lifting lineĒ of blade elements, then the propeller need not be straight: it can be coned and lagged and the effect of this curvature evaluated.

I have one criticism of the book. It may be to my own lack of perspicacity, but I did not find the treatment of the "co-location point" to be satisfactory. The concept was used heavily, but I had to go back and struggle thru the text to find the derivation of the concept. Even then, I found it to be superficial and I had little confidence in applying it to situations more complex than that in which it was derived. Possibly if I was to read it again, I may do better, but that is my recollection and I cannot resile from it.

Have you noticed all the sentences ending with "it"? Lucky my grammar teacher is dead, or I would be writing this standing up. 

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