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 deformation tensor governs/encodes the local flow topology
Deformation: kinematics Deformation
Differential forms
Deformation/Strain rate/Rotation *These are inverse time scales=internal bootstrap frequencies* Strain rate: shear dissipation Compression: bulk dissipation Rotation: No Dissipation
Whiteboard Example: Compute S,D,Omega, div for Couette, Poiseuille, Rigid rotation, Irrotational vortex, Elongational (torture) flow
Radial Flow
Vortex Flow
Elongational Flow
Velocity-Vorticity Degree of local rotation Eliminates pressure Useful for nearly-inviscis flows
Rotational/Irrotational
Rotational/Potential Flow Potential ~ Inviscid Potential & Incompressible Analytic function: very useful for 2D low-viscous hydrodynamics
Kelvin theorem
Turbo-jungle vorticity
What’s vorticity good for? Pressure-free! Vortex stretching: Take curl of both sides and use identity To obtain:
Enstrophy Vorticity Stretch: Finite-time blow-up? 2d Beltrami flows:
What’s vorticity good for? Pressure-free! Vortex Collection: Long-range (electrostatic) interactions
Helicity Swirl motion, Dynamo 1d 2d Beltrami flows:
2d: Vorticity-Streamfunction Two-dimensional Potential-->Irrotational: Built-in incompressibility:
Potential Flow: 2D Conformal mapping:
Body-fitted coordinates
2d: Enstrophy conserved Vortex stretching identically zero: Enstrophy is conserved:
2d turbulence: Vorticity-Stream Two-dimensional: spectral methods Nonlinear depletion: coherent structures (vortices)
Coherent structures Non-linear depletion Cascade blocking; Long-lived metastable states Enstrophy cascade: REGULAR!
Ideal 2d: Hamiltonian Symplectic dynamics: Borrow a lot from particle dynamics! Hamiltonian streaming + vortex mergers/breakup
Enstrophy: inverse cascade Cascade blocking; Long-lived metastable states Enstrophy cascade: REGULAR! Energy cascade: SINGULAR
End of Lecture