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Mathematics of flying

I am having to brush up on Principles of Flight and finding that as I dig deeper there are various competing theories – and if I understand correctly we do not have complete equations that explain flight.

I have ordered a Calculus refresher (nearly forty years since I had to look at a differential equation, and they are still quite strange organisms), with the title Calculus without limits, by J Sparks – hopefully gets me beyond the rote learning of integration rules of my youth.

Whether this will arm me with a little knowledge to understand Navier Stokes or Kutta Zhukovsky, only time will tell.

Having had to carry out a few projects in Romania, I have always had a soft spot for Coanda, so hopefully will come across his equations as well. If I understand correctly the weight of the boundary layer control technology on the Buccaneer, neatly cancelled out the benefit!

Any way purpose of the thread starter is to get a debate going on which theory of flight do people subscribe to: Newton’s second law, Bernoulli, free stream tube, Coanda, Kutta, the weight of the paperwork, pilot skill/prayer, etc

Oxford (EGTK), United Kingdom

Navier Stokes equations are hardcore. I think they’re the hardest differential equation types that are out there. I had tried to program a computer simulation using the equations once but quit after months of reading and trying to understand them. See my background was computer science and while we did loads of Math, the type of Math required for CFD is a different animal.

which theory of flight do people subscribe to: Newton’s second law, Bernoulli, free stream tube, Coanda, Kutta

I think all of these are correct.

All the fluid-dynamics equations can be derived from Newtonian mechanics. Actually, to be pedantic, everything in the universe that isn’t small enough for quantum effects or fast enough for relativistic effects does behave fully in accordance with Newtonian mechanics.

NM is just impractical for practical uses because there are far too many molecules banging about

People like Bernoulli worked out simplified methods (mostly empirically I think in his case; I am sure he didn’t know what a molecule was) which could be used by anybody with a piece of pipe and some stuff flowing through it.

I couldn’t differentiate or integrate if you paid me for it, but some stuff I was reading recently said that the Navier Stokes equations do actually work for all aircraft work at all speeds – if you know how to use them.

Last Edited by Peter at 20 Feb 16:35
Administrator
Shoreham EGKA, United Kingdom

Any way purpose of the thread starter is to get a debate going on which theory of flight do people subscribe to: Newton’s second law, Bernoulli, free stream tube, Coanda, Kutta, the weight of the paperwork, pilot skill/prayer, etc

The only proper theory of lift is Mary Shafer’s lift demons.

Bookworm – if I read it carefully this theory reduces to Newton’s 2nd law – so Peter is right!

I like the point about the fairer demons being kept in the dark on the cost impact of aviation…

Oxford (EGTK), United Kingdom

Hey Robert,

if you are interested in aerodynamics beyond the knowledge a pilot needs to fly, I may recommend Fundamentals of Aerodynamics by John Anderson.

In general, with Navier-Stokes you can describe every flow. But Navier-Stokes isn’t that handy outside the numerical world. (Even with numeric flow calculations, Navier-Stokes equations are not trivial to hanlde). You have to make some assumptions about your flow, to get an equation, that you can actually handle (i.e. inviscid flow, incompressible flows, etc.).

And this is with all other models, too. You have to make some assumptions on your flow to begin with (and you have to keep it in mind) . For instance for Kutta-Jhoukowski, the flow has to be in a steady state (d/dt = 0), inviscid, incompressible and non-rotating to begin with. Or Bernoulli, which is true for incompressible, inviscid flow, only. So you can’t use all of these for calculations in any flow.

Then, you have to know what you want to describe. Kutta-Jhoukowski and Blasius can’t calculate any drag, for instance. Biot-Savart or Bernoulli can not calculate lift or drag, and so on.

For a pilot, Newtons laws of motion are perfect suitable to describe what happens, as described by Wolfgang Langewiesche in his book ‘stick and rudder’. (You probably know this one :-)

Cheers,

Last Edited by mh at 21 Feb 06:25
mh
Aufwind GmbH
EKPB, Germany

mh many thanks – stick and rudder gets looked at quite often, the book is very good at explaining what the aircraft is doing – so good both on theory and practice.

I thought an edit with a quote from Langewiesche would be on point (no pun intended)

“Trying to understand the piloting of airplanes by concentrating on Bernoulli and Prandtl is like trying to catch on to tennis by studying just exactly how the rubber molecules behave in a tennis ball when the ball hits the court and just exactly how the catgut behaves in the racket when the ball strikes: instead of simply observing that it bounces!”

Last Edited by RobertL18C at 21 Feb 08:28
Oxford (EGTK), United Kingdom

This non-pilot article on Bernoulli vs Newton is quite good.

https://www.scientificamerican.com/article/no-one-can-explain-why-planes-stay-in-the-air/

There is the question of how well one needs to understand something to use it. I’m not ashamed to admit I don’t really understand how, say, gravity works: the effects are obvious, but how does it really work? “Yes, but why” used to be my favourite expression (I must have been an annoying child )

EGHO-LFQF-KCLW, United Kingdom

The substantial money prize to complete the Navier-Stokes partial differential equations on fluid dynamics is still there for the taking for over a century just round the corner at Oxford.

The Kutta-Zhukovski condition ignores the physics negation of circulation theory to come up with some mathematical work arounds.

Oxford (EGTK), United Kingdom

I see this is a holy thread resurrection Batman and I was the OP! Wheels of time.

Oxford (EGTK), United Kingdom
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