# Navier-Stokes Equations

## Slightly Viscous Incompressible Flow

The flow of air around a wing is lightly viscous with a Reynolds number ranging from hundreds of thousands for birds to a billion for a jumbo jet at cruising speed. The flow attaches as laminar flow at the leading edge of the wing and separates as turbulent flow at the trailing edge. The skin friction is small since the viscosity of air is small.

Aerodynamics is accurately modeled by the incompressible Navier-Stokes equations for flow speeds well below sonic speeds (up to about 300 km/h).  A slip boundary condition with zero skin friction can be used as an approximation of the actual small skin friction.

The essential features of the aerodynamics of subsonic flight are thus:

• high Reynolds number
• incompressible flow
• turbulent flow
• small skin friction: slip boundary condition.

## Navier-Stokes/Euler with Slip

The new theory of flight is evidenced by the fact that the incompressible Navier-Stokes equations with slip boundary conditions are computable using less than a million mesh points without resolving thin boundary layers in DFS as Direct Finite Element Simulation, and that the computations agree with experiments. DFS computes best possible solutions to Euler’s equations as Navier-Stokes equations with vanishing viscosity.

The lift and drag of a wing, or a car, is computable to within the experimental accuracy of a few percent.

The story is told in the book Computational Turbulent Incompressible Flow.

## Scale Invariance

Navier-Stokes equations with slip boundary conditions and vanishingly small viscosity are scale invariant in the sense that smooth solutions remain smooth solutions under a change of scale in space and time. Bluff body flow thus depends on the form of the body but not on size.

In particular, the separation pattern and at the trailing edge of a wing is remains the same and thus also lift and drag as the radius of the trailing edge tends to zero.

## Clay Institute \$1 Million Millennium Problem

One of the 7 Clay Mathematics Institute Millennium Problems concerns existence and regularity of solutions to the Navier-Stokes equations. The problem formulation is discussed on

We here suggest a reformulation of the Clay Problem into 1 million €1 problems, one of which we have solved on our way to reveal the Secret of Flight, see Clay Navier-Stokes Millennium Problem Solved.

## 1 thought on “Navier-Stokes Equations”

1. I would be happy if the group on KTH under professors Claes Johnson and Johan Hoffman would have more supporters in the University world. Empirical scientists in Göttingen seem to have arrived to the same conclusion as them. The main problem is maybe that the Swedish group are world leading experts in FEM and Numerical analysis and work aposteriori when others seem to believe the question How aeroplanes can fly still is possible to solve with potentials in 2D (complex variables) and apriori expressions. I admire Claes Johnson who had the guts to refused to get bribed with the Prandtl medal. Anyhow I would like to see at least another University in the world to support the idea of elegant separation, which in my eyes is not only elegant but brilliant an completely relevant and logical!
Gunnar W Bergman
technical journalist
Norsborg – Sweden