The Uncomputable Father Prandtl

· fluid mechanics
Authors

Ludwig Prandtl sentencing fluid mechanics to 100 years of uncomputability.

The Paralyzing Dictate by the Father

Ludwig Prandtl (1875 – 1953) took on the role of Father of Modern Fluid Mechanics by breaking away from the paradigm of classical fluid mechanics formed by Euler and d’Alembert in the mid 18th century based on describing slightly viscous flow by the inviscid Euler equations.

Prandtl did this by insisting that in mathematical modeling  of slightly viscous flow, a no-slip boundary condition with a boundary layer must be used which excludes the Euler equations, since only the normal velocity can be prescribed in a slip boundary condition.  By this dictate Prandtl circumvented d’Alembert’s Paradox of the existence of non-physical solutions of the Euler equations (so-called potential solutions) allowing a bluff body to move through a fluid without drag, which had troubled classical fluid mechanics from start.

Prandtl thus dictated that drag (and lift) originate from a very thin boundary layer caused by a no-slip boundary condition with fluid particles sticking to a solid boundary. The dictate saved fluid mechanics from collapse in the beginning of the 20th century when lift and drag of a wing required an explanation, but at the same time made slightly viscous flow uncomputable, since resolution of thin boundary layers requires quadrillions of mesh points.

Prandtl’s no-slip dictate as a Father has served as a main obstacle in Computational Fluid Dynamics CFD for 100 years, but is now being challenged by our work showing that slightly viscous flow can be accurately simulated computationally using millions of mesh points by using a slip boundary condition without boundary layers. This opens to take CFD out of its paralyzing grip from Prandtl’s uncomputable boundary layers.

As a major step forward we uncover the Secret of Flight by solving the Navier-Stokes equations with a slip boundary condition.

Prandtl and Einstein

We may compare Prandtl’s negative impact on science by that of Einstein with his relativity theory. Einstein broke away from Newtonian mechanics with its apparent Michelson-Morley paradox by dictating that space and time are to be connected by the Lorentz transformation. This dictate like threw physics into a 20th century nightmare of “curved space-time” which is still holding its grip.

Prandtl and Einstein were not trained mathematicians and fell in the trap of giving trivial mathematics a deep physical significance.

Prandtl: I am an Engineer

The Navier-Stokes equations can be combined with different boundary conditions, velocity (Dirichlet) or force (Neumann) conditions, and slip expressing zero skin friction is a rational choice, when it is known that the skin friction is small as is the case in slightly viscous flow. This was not properly understood by Prandtl. The limitations of his mathematical training is evidenced by his former student von Karman:

  • Prandtl, an engineer by training, was endowed with rare vision for the understanding of physical phenomena and unusual ability in putting them into relatively simple mathematical form. His control of mathematical method and tricks was limited; many of his collaborators and followers surpassed him in solving difficult mathematical problems. But his ability to establish systems of simplified equations which expressed the essential physical relations and dropped the nonessentials was unique, I believe, even when compared with his great predecessors in the field of mechanics – men like Leonhard Euler and d’Alembert. 

And also by himself:

  • I am an engineer. If you want that way, a theoretician in engineering subjects, and I have used mathematics in the different cases of problems that I have addressed. But I have never furthered the science of mathematics through any contribution. 

Einstein expressed that he was well aware of the fact that he did something he should not have done, by his famous: Forgive me, Newton! Similarly, Prandtl should have said: Forgive me, Euler!

See also Dr Faustus of Modern Physics.

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