Solutions stationnaires des équations de Navier-Stokes en domaines extérieurs dans le régime des faibles nombres de Reynolds Peter Wittwer Département.

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Presentation transcript:

Solutions stationnaires des équations de Navier-Stokes en domaines extérieurs dans le régime des faibles nombres de Reynolds Peter Wittwer Département de Physique Théorique Université de Genève

images: R. P. Feynman, Vol. II G. K. Batchelor, An Introduction to Fluid Mechanics M. Van Dyke, An Album of Fluid Motion collaborations: Guillaume Van Baalen Frédéric Haldi Vincent Heuveline Sebastian Bönisch

─ Introduction to the problem ─ Asymptotic analysis ─ Applications

Exterior Flows

Navier-Stokes

Re=0.16

Re=1.54

Re=56.5

Re=118

Re=7000

Case of finite volume

Case of infinite volume, I

Case of infinite volume, II

Asymptotic analysis

Results (d=2)

Interpretation:

Results (d=3)

Two steps: ─ construct downstream asymptotics dynamical system invariant manifold theory renormalization group universality ─ determines asymptotics everywhere

Vorticity:

Vorticity equation

Fourier transform

Diagonalize

Stable and unstable modes

use contraction mapping principle

Large time asymptotics:

Two steps: ─ construct downstream asymptotics dynamical system invariant manifold theory renormalization group universality ─ determines asymptotics everywhere

Determines asymptotics everywhere:

Applications in collaboration with: Sebastian Bönisch Rolf Rannacher Vincent Heuveline Heidelberg & Karlsruhe

Adaptive boundary conditions

To second order:

Comparison with Experiment:

Cloud Microphysics and Climate M. B. Baker, SCIENCE, Vol. 276, 1997

Work in progress: d=2 case with lift (numerical) d=2 second order asymptotics (theory) d=3 (numerical) d=2, 3: free fall problem (numerical) d=3 case with rotation at infinity (theory; see P. Galdi (2005) for recent results) Other research groups: d=2 time periodic (theory)

Large time asymptotics: