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.
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
- On the Uniqueness of Weak Solutions of the Navier-Stokes Equations
- Is the Clay Navier-Stokes Problem Wellposed?
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.