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

2. Topological Fluid Dynamics

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

4. Deformation: kinematics Deformation

5. Differential forms

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

8. Whiteboard Example: Compute S,D,Omega, div for Couette, Poiseuille, Rigid rotation, Irrotational vortex, Elongational (torture) flow

9. Radial Flow

10. Vortex Flow

11. Elongational Flow

12. Velocity-Vorticity Degree of local rotation Eliminates pressure Useful for nearly-inviscis flows

13. Rotational/Irrotational

14. Rotational/Potential Flow Potential ~ Inviscid Potential & Incompressible Analytic function: very useful for 2D low-viscous hydrodynamics

16. Kelvin theorem

17. Turbo-jungle vorticity

18. What’s vorticity good for? Pressure-free! Vortex stretching: Take curl of both sides and use identity To obtain:

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

20. What’s vorticity good for? Pressure-free! Vortex Collection: Long-range (electrostatic) interactions

21. Helicity Swirl motion, Dynamo 1d 2d Beltrami flows:

24. 2d: Vorticity-Streamfunction Two-dimensional Potential-->Irrotational: Built-in incompressibility:

25. Potential Flow: 2D Conformal mapping:

26. Body-fitted coordinates

27. 2d: Enstrophy conserved Vortex stretching identically zero: Enstrophy is conserved:

28. 2d turbulence: Vorticity-Stream Two-dimensional: spectral methods Nonlinear depletion: coherent structures (vortices)

29. Coherent structures Non-linear depletion Cascade blocking; Long-lived metastable states Enstrophy cascade: REGULAR!

30. Ideal 2d: Hamiltonian Symplectic dynamics: Borrow a lot from particle dynamics! Hamiltonian streaming + vortex mergers/breakup

31. Enstrophy: inverse cascade Cascade blocking; Long-lived metastable states Enstrophy cascade: REGULAR! Energy cascade: SINGULAR

32. End of Lecture