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
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
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
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
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
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
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
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
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
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.