Bernoulli’s Principle

Fluid mechanics got started with the Euler equations formulated by Euler in 1755 building on Daniel Bernoulli’s Hydrodynamik from 1738 and Johann Bernoulli’s (father of Daniel) Hydraulica. As a first major result Euler derived Bernoulli’s Principle for potential flow as stationary incompressible inviscid irrotational flow as certain solutions of Euler equations: 

  • \frac{1}{2}U^2 + P = C,

where U is the speed (modulus of the velocity), P the pressure and C a constant. In short Bernoulli’s Principle states that the pressure is high where the fluid speed is low and vice versa.

This was a wonderful result predicting in particular the following potential flow around a circular cylinder showing velocity (arrows) and pressure (color: high pressure red and low pressure blue) in a section of the cylinder:

We see that the potential flow satisfies Bernoulli’s Priniciple with high pressure where the speed is small (in particular at stagnation in front and back with zero speed) and low pressure where the speed is high on top and bottom. In particular, the pressure is equal at the stagnations points in front and back for any shape, as a most remarkable statement.

This was a masterpiece of mathematics by one of history’s greatest mathematicians, but the happiness did not last long. D’Alembert pointed out that the drag of the potential solution was zero which did not at all correspond to massive undeniable observations of substantial drag.

The mathematical euphoria was then turned into deepest pessimism, because nobody could figure out what was wrong with the mathematics and its potential solution visibly satisfying the Euler equations visible describing fluid mechanics visibly based on Newton’s 2nd Law. If mathematics could not trusted, what could then be trusted in the era of Enlightenment?

It took more than 250 years to resolve the mystery by understanding that what is unphysical and mathematically unsound about the potential solution, is that it is unstable at separation and thus cannot be observed in reality. The potential solution satisfies the Euler equations but it will cease to be a solution under vanishingly small perturbations and turn into a solution with a turbulent pattern of 3d rotational separation which makes the flow different in the back than in the front thereby causing substantial drag. This is all explained carefully in Resolution of d’Alembert’s Paradox and Separation.

But the potential solution is not equally unstable everywhere, for example not at attachment  in the front, and as long as the potential solution does not turn turbulent it offers excellent information and aids understanding. The potential solution is a great gift to humanity but it it has to be handled with care and enjoyed without excess.

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