I'm doing a project (dissertation) on the mathematics of Aerofoil Theory. I wonder if I could get some advice on a possible structure. I'm new to fluid dynamics, so it's quite hard to know where to start.

So far I've looked at different types of flows; streamlines and velocity potentials. I've also covered a fair amount of complex variable theory - and can (loosely) see how conformal mappings are used to map for instance a cylinder to an aerofoil shape using Joukowski's transform. I would guess from here, I would want to look at Bernoulli's equation for lift and start looking at modelling an example.

It would be useful to know if this is the right kind of structure to follow or if I'm approaching this the wrong way! So any help would be appreciated. Thanks!


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  • $\begingroup$ This might be helpful. $\endgroup$ – Mike Dunlavey Sep 9 '13 at 13:25
  • $\begingroup$ Not really sure what you mean by structure. Could you clarify? $\endgroup$ – Pranav Hosangadi Sep 9 '13 at 17:51
  • $\begingroup$ @PranavHosangadi By structure, I mean the order in which I should study the subject so I can understand it. For instance, I don't currently have the knowledge to apply a Joukowski transform to a cylinder and work out the lift etc. I hope that makes sense. $\endgroup$ – Brit Miller Sep 9 '13 at 17:58
  • $\begingroup$ @BritMiller I see. I took that to mean structural types. $\endgroup$ – user6972 Sep 9 '13 at 18:51
  • $\begingroup$ @user6972 Thanks anyway, that could be useful later in my project. :-) $\endgroup$ – Brit Miller Sep 9 '13 at 19:19

To use the Joukowski concept of an airfoil I have learnt from the professors Claes Johnson and his student professor Johan Hoffman is the "old" and wrong theory of flight because it is totally based on 2D calculation. It is a beautiful theory in complex variables but completely unphysical it is by no means based on physical laws (complete mumbo-jumbo, so to say) according to these two experts on Numerical analysis from the calculation center of (NADA/KTH). Their road to the truth is stony but correct. Remember marquis Pierre Simon de Laplace's word of wisdom: La verite seul est belle (only the truth is beautiful).

So instead of trying to develop the wrong 2D (or even worse) 3D ideas of Prandtl I advise You to study the "New theory of flight" of Johnson-Hoffman. This theory is based on FEM-analysis which has it's origin in the famous Russian mathematician and engineer Galerkin. This is the only way to solve this problem of fluid dynamics to start with Navier-Stokes equation with a slip-boundary-condition which is totally different to the no-slip condition of Dirichlet type. With this You can reach a more profound understanding of the fluid dynamics of airfoils. The Görtler vortices then goes from secondary to primary phenomenons and important in the understanding of stall and flow separation for instance.

  • $\begingroup$ A bit late for me as I finished this project nearly a year ago - but thank you for the insight. From the work I did, yes the 2D model was unphysical but the model could be a physical model when considering boundary layer theory - but I didn't get that far. $\endgroup$ – Brit Miller Jan 20 '15 at 18:08
  • $\begingroup$ I feel compelled to note that the so-called "new theory of flight" by Claes Johnson and friends is complete and utter nonsense, and is not taken serious by anyone with an understanding of the pertinent fluid mechanics or aerodynamics. I will also note that Johnson has tried to peddle this stuff to a number of reputable journals in the field, and almost all of them have rejected his manuscript, with the sole exception of the somewhat obscure Journal of Mathematical Fluid Mechanics (I really don't know how that could happen; I'll talk to Rannacher about this when I get a chance...) $\endgroup$ – Pirx Jan 4 '17 at 22:34

Most airfoils are built using the NACA formulas/tables where each parameter is specified in the type of foil.


Each of the 4 parameters specific the foil type NACA XYZZ where

X--First digit describing maximum camber as percentage of the chord.
Y--Second digit describing the distance of maximum camber from the airfoil leading edge in tens of percents of the chord.
ZZ--Last two digits describing maximum thickness of the airfoil as percent of the chord.  

A symmetrical foil would be NACA 0010 would mean 0% camber, 0% percent camber distance and 10% thickness to chord length.

There is also a 5 digit series of NACA foils for more complex shapes. And there are 16-series, 6-series, 7-series and 8-series foils as well. The benefits of using a NACA shape is they have been well characterized. Here's a summary of the types and the pros/cons depending on your application.

Edit: From an application perspective (non-academic) Foil design and analysis is usually done by defining what type of lift parameters you want: range of lift coefficients, Reynolds numbers, where the airfoil should perform best, stall characteristics, moment coefficient, thickness, low drag, high lift... This is where NACA foils are handy because you don't have to start from scratch.

On a custom foil the next step is usually to analyze it numerically with one of two popular methods:

  • PROFIL by Professor Richard Eppler, University of Stuttgart, Germany.

  • XFOIL by Professor Mark Drela, Massachusetts Institute of Technology, USA.

Take a look at how each tool works and you'll get an idea of methods that might be useful for you to learn. Other references the vortex lattice method, basic thin foil aerodynamics, thin foil aerodynamics derivations text book

  • $\begingroup$ The sort of thing I'm looking to do is go right from the beginning. Initially I'm looking at ideal fluids and fluid flows. I was thinking of starting with a fluid flow around a cylinder, and using Joukowski's transformation to construct different aerofoils. From there I want to go on to looking at lift and different shapes. If that makes sense, I'm looking at how to construct an aerofoil via Joukowski's transform first and then looking at the effects of different shapes etc. (Or maybe this is the wrong approach?) $\endgroup$ – Brit Miller Sep 9 '13 at 19:43
  • 1
    $\begingroup$ It's not my specialty, but you might want to take a look at the last link to see some of the methods used. It starts using the Biot-Savart Law. Actually here is the link to the first page of the book desktop.aero/appliedaero/preface/welcome.html $\endgroup$ – user6972 Sep 11 '13 at 1:36

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