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L3: The Navier-Stokes equations II: Topology

L3: The Navier-Stokes equations II: Topology. Prof. Sauro Succi. Topological Fluid Dynamics. Deformation / Strain rate/ Rotation. * These are inverse time scales = internal bootstrap frequencies * The d eformation tensor governs / encodes the local flow topology.

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L3: The Navier-Stokes equations II: Topology

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  1. L3: The Navier-Stokesequations II:Topology Prof. Sauro Succi

  2. TopologicalFluid Dynamics

  3. Deformation/Strain rate/Rotation *These are inverse time scales=internal bootstrap frequencies* The deformationtensorgoverns/encodes the local flow topology

  4. Deformation: kinematics Deformation

  5. Differentialforms

  6. Differentialforms

  7. Deformation/Strain rate/Rotation Strain rate: sheardissipation Compression: bulk dissipation Rotation: No Dissipation *These are inverse time scales=internal bootstrap frequencies*

  8. WhiteboardExample: Compute S,D,Omega, div for Couette, Poiseuille, Rigidrotation, Irrotationalvortex, Elongational (torture) flow

  9. Radial Flow

  10. Vortex Flow

  11. Elongational Flow

  12. Velocity-Vorticity Degree of localrotation Eliminates pressure Useful for nearly-inviscisflows

  13. Rotational/Irrotational

  14. Rotational/Potential Flow Potential ~ Inviscid Potential & Incompressible Analyticfunction: veryuseful for 2D low-viscoushydrodynamics

  15. Kelvin theorem

  16. Turbo-junglevorticity

  17. What’svorticitygood for? Take curl of bothsides and use identity To obtain: Pressure-free! Vortex stretching:

  18. Enstrophy Vorticity Stretch: Finite-time blow-up? 2d Beltrami flows:

  19. What’svorticitygood for? Pressure-free! Vortex Collection: Long-range (electrostatic) interactions

  20. Helicity 1d Swirlmotion, Dynamo 2d Beltrami flows:

  21. 2D is different!

  22. 2d: Vorticity-Streamfunction Two-dimensional Built-in incompressibility: Potential-->Irrotational:

  23. Potential Flow: 2D Conformalmapping:

  24. Body-fitted coordinates

  25. 2d: Enstrophyconserved Vortex stretching identically zero: Enstrophyisconserved:

  26. 2d turbulence: Vorticity-Stream Two-dimensional: spectralmethods Nonlineardepletion: coherentstructures (vortices)

  27. Coherent structures Non-linear depletion Cascadeblocking; Long-livedmetastablestates Enstrophycascade: REGULAR!

  28. Ideal 2d: Hamiltonian Symplecticdynamics: Borrow a lot from particledynamics! Hamiltonian streaming + vortexmergers/breakup

  29. Enstrophy: inverse cascade Cascadeblocking; Long-livedmetastablestates Enstrophycascade: Energy cascade: SINGULAR REGULAR!

  30. End of Lecture

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