onsdag 5 december 2012

Liberation from the Prandtl Spell 1

The New Theory of Flight explaining for the first time the mathematics and physics of the flight of birds and airplanes is based on solving the Navier-Stokes equations with slip boundary conditions as a model for slightly viscous flow.

The Navier-Stokes equations with slip boundary conditions and vanishingly small viscosity are scale invariant in the sense that a change of the scale in space leaves the equations invariant (assuming a vanishingly small viscosity has no scale). We see this effect as large-scale features of bluff body flow reflecting the geomety of the body, which remain the same under mesh refinement (modulo gradient sharpening effects) such as the 3d rotational separation pattern of the flow around a circular cylinder shown above, which is also seen at the trailing edge of a wing and what lies behind the secret of flight.

It is the scale invariance of solutions to the Navier-Stokes equations with slip boundary condition and vanishing viscosity, which make solutions computable on meshes resolving only the large scale features of the flow in what can be referred to as Large Eddy Simulation (LES), without the need of user-defined turbulence models.

This makes computational solution of slightly viscous flow possible by removing the Prandtl spell of having to resolve thin boundary layers from no-slip boundary conditions, which has parlalyzed fluid mechanics for so long time. Follow the story in the upcoming book The Secret of Flight.

In fact, a no-slip boundary condition is mathematically incompatible with vanishing viscosity and Prandtl's insistence on using no-slip can be seen as a consequence of Prandtl's limitation as mathematician witnessed by e.g. von Karman.

2 kommentarer:

  1. Isn't LES an abbreviation for Large Eddie Simulation?

  2. Yes, Large Eddy Simulation. Corrected.