1. Incompressible Flows Sauro Succi

2. Incompressible flows

4. Incompressible constraint Kinematic Constraint: elliptic (time-consuming) “Slow” flows: sound speed to infinity (fluid<<sound)

5. Matrix Formulation Cruelly non-local: no way!

7. Many options… Colocated/Staggered Explicit/Implicit, Exactly/Quasi Incompressible, ……

8. Colocated; Control Volume Hourglass in simple geos No hourglass in complex ones

9. Staggered: stronger VP coupling

10. Staggered Laborious, good for surfint > simple geos No hourglass, VP coupled

11. Isotropic Laplacians

12. Colocated: Hourglass instability

13. Complex geos Spherical cows! Staggered: complicated Colocated: no hourglass

14. Modern FV: Implicit diffusion with structured colocated FV leads to 9-diag regular matrices, Can be solved efficiently with ADI. Poisson solver has no hourglass, but still very Expensive because the coeff’s are inhomogeneous

15. Handling non-locality Rapid Poisson Solvers Artificial compressibility Predictor-Corrector methods Explicit/Implicit time marching

16. Rapid Poisson: Spectral Fourier transform: f(x) to f(k) And back : f(k) to f(x)

17. Differential to algebraic problem 1. FT 2. Solve 3. IFT

18. 2d homog. Inc. turbulence

19. Spectral: plus and minus Problems: N^2 complexity Periodic Geometries Remedies FFT: N^2 to N*logN Periodic constraint basically remains

20. Two basic families Exactly Incompressible (EI) Artificial Compressibility (AC)

21. Exactly incompressible Strictly incompressible: elliptic Two hyperbolic+one elliptic, stiff matrix

22. EI: Explicit Divfree is enforced in time, but Poisson very CPU intensive -> Rapid Elliptic Solvers (RES) Solve Poisson for p^0, then advance U^0 to U^1

23. Artificial Compressibility Fictitious (pseudo)-time Exact at steady state Hard to soft constraint

24. Full Time-dependent Exact at steady-state (only)

25. Divergence dynamics Small-amplitude oscillations around epsilon=O(Mach^2) “Hydrodynamic Charge” Similar to gravity: curvature of u

26. AC: Chorin Pseudodyn is stable: small flucts around p0 divu>0 p goes down and viceversa Divfree remains O(epsilon) all along, No Poisson, but dt very small

27. AC: another version ? Pseudodyn is stable: small flucts around p0 divu>0 p goes down and viceversa Divfree remains O(epsilon) all along, No Poisson, but dt very small

28. AC: Explicit: WRONG! Divfree is not conserved in time, No Poisson, but p1 not ok: iteration needed: WRONG: if p0 obeys poisson divu frozen = 0!!! Wrong: divfree frozen to 0

29. Hard vs Soft Constraints Electronic structure: Born-Oppenheimer, Car-Parrinello: soft Orbital Orthogonality : hard Biomolecular dynamics: hard Fluid Compressibility: soft With f hard to invert Hard: Soft:No need to invert f

30. CFL stability conditions Diffusion is very-constraining Advection: Explicit Diffusion: Implicit

31. EI: Linearly-Implicit Poisson less of a drag: implicit anyway

32. Predict-Correct Predict u*(p=0): Correct u*: Require: (Projection) u^{n+1} is now div-free

33. AC: Implicit Diffusion (Linear)

34. Summary Exactly Incompressible: Explicit: Divfree is forced via Poisson, but Poisson solver is a drag Remedies: RPS: Rapid Poisson Solver (simple geo’s) Implicit: large dt, Poisson less of a drag, implicit anyway Artificial Compressibility: Exact only at steady-state. Divfree is only quasi-conserved to O(eps) Can leave with it if steady-state is the only target Less so for dynamics Implicit: PS no longer a drag, implicit anyway

35. Nonlinearity

36. Nonlinearity-Picard iteration

37. The face of the discrete operators: Finite Differences

38. MAC staggered grid(FD)

39. Pressure equation

40. Staggered grid: X component

41. Y-component

42. Explicit/Implicit

43. Boundary conditions Spherical cows! ? ?

44. Boundary Conditions: Dirichlet

45. Boundary Conditions: Neumann

46. One-sided derivatives

47. End of Lecture

48. Colocated: Hourglass instability

49. Colocated Simple, economic > complex geos Hourglass, VP uncoupled

50. Incompressible/Compressible Viscous/Inviscid Steady/Unsteady Navier-Stokes equations

51. Special features of NSE Vector 3d Non-Linear Non-local (incompressible) Complex geos

52. Mathematical structure 3 explicit: soft and matrix-free. But … Incompressibility holds only at steady-state, OK if steady-state is the only target

53. Fully Explicit (AC)

54. nw ne d sw n e w s P E N W S NE SE SWSW Vertex-centered Colocated

55. Nonlinearly-Implicit Nonlinear iterations, k=0,1,…